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    <title>DEV Community: vindianadoan</title>
    <description>The latest articles on DEV Community by vindianadoan (@vindianadoan).</description>
    <link>https://dev.to/vindianadoan</link>
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      <title>DEV Community: vindianadoan</title>
      <link>https://dev.to/vindianadoan</link>
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    <item>
      <title>Fixing AWS CLI Version Conflicts on Windows</title>
      <dc:creator>vindianadoan</dc:creator>
      <pubDate>Wed, 12 Mar 2025 02:05:06 +0000</pubDate>
      <link>https://dev.to/vindianadoan/fixing-aws-cli-version-conflicts-on-windows-5fl8</link>
      <guid>https://dev.to/vindianadoan/fixing-aws-cli-version-conflicts-on-windows-5fl8</guid>
      <description>&lt;p&gt;Symptom: You install AWS CLI v2 but aws --version still shows v1.x.x.&lt;/p&gt;

&lt;p&gt;Cause: An older aws.exe appears earlier in your PATH than the new CLI. On Windows, the system uses the first match it finds.&lt;/p&gt;

&lt;h2&gt;
  
  
  Quick Check
&lt;/h2&gt;

&lt;p&gt;Run:&lt;br&gt;
&lt;/p&gt;

&lt;div class="highlight js-code-highlight"&gt;
&lt;pre class="highlight shell"&gt;&lt;code&gt;where aws
&lt;/code&gt;&lt;/pre&gt;

&lt;/div&gt;



&lt;p&gt;You might see:&lt;br&gt;
&lt;/p&gt;

&lt;div class="highlight js-code-highlight"&gt;
&lt;pre class="highlight plaintext"&gt;&lt;code&gt;C:\Users\YourName\anaconda3\Scripts\aws
C:\Users\YourName\anaconda3\Scripts\aws.cmd
C:\Program Files\Amazon\AWSCLIV2\aws.exe
&lt;/code&gt;&lt;/pre&gt;

&lt;/div&gt;



&lt;p&gt;The first path is always used (in this example, Anaconda’s AWS CLI v1).&lt;/p&gt;

&lt;h2&gt;
  
  
  Two Fixes
&lt;/h2&gt;

&lt;h3&gt;
  
  
  1. Remove the Old CLI
&lt;/h3&gt;

&lt;p&gt;If installed via conda:&lt;br&gt;
&lt;/p&gt;

&lt;div class="highlight js-code-highlight"&gt;
&lt;pre class="highlight shell"&gt;&lt;code&gt;conda remove awscli
&lt;/code&gt;&lt;/pre&gt;

&lt;/div&gt;



&lt;p&gt;If installed via pip:&lt;br&gt;
&lt;/p&gt;

&lt;div class="highlight js-code-highlight"&gt;
&lt;pre class="highlight shell"&gt;&lt;code&gt;pip uninstall awscli
&lt;/code&gt;&lt;/pre&gt;

&lt;/div&gt;



&lt;p&gt;Then, confirm by running:&lt;br&gt;
&lt;/p&gt;

&lt;div class="highlight js-code-highlight"&gt;
&lt;pre class="highlight shell"&gt;&lt;code&gt;where aws
&lt;/code&gt;&lt;/pre&gt;

&lt;/div&gt;



&lt;p&gt;You should now see only:&lt;br&gt;
&lt;/p&gt;

&lt;div class="highlight js-code-highlight"&gt;
&lt;pre class="highlight plaintext"&gt;&lt;code&gt;C:\Program Files\Amazon\AWSCLIV2\aws.exe
&lt;/code&gt;&lt;/pre&gt;

&lt;/div&gt;



&lt;p&gt;And &lt;code&gt;aws --version&lt;/code&gt; should show 2.x.x.&lt;/p&gt;

&lt;h3&gt;
  
  
  2. Reorder PATH
&lt;/h3&gt;

&lt;p&gt;If you can’t remove the old CLI:&lt;/p&gt;

&lt;ul&gt;
&lt;li&gt;Open System Properties → Environment Variables → Path.&lt;/li&gt;
&lt;li&gt;Move: &lt;code&gt;C:\Program Files\Amazon\AWSCLIV2\&lt;/code&gt; above any Python or Anaconda paths.&lt;/li&gt;
&lt;li&gt;Restart your terminal and run aws --version again.&lt;/li&gt;
&lt;/ul&gt;

</description>
    </item>
    <item>
      <title>Setting anaconda Python as the default</title>
      <dc:creator>vindianadoan</dc:creator>
      <pubDate>Wed, 12 Mar 2025 01:42:52 +0000</pubDate>
      <link>https://dev.to/vindianadoan/setting-anaconda-python-as-the-default-51ja</link>
      <guid>https://dev.to/vindianadoan/setting-anaconda-python-as-the-default-51ja</guid>
      <description>&lt;p&gt;Just documenting a quick note for me to refer back later. &lt;/p&gt;

&lt;h1&gt;
  
  
  Problem
&lt;/h1&gt;

&lt;p&gt;When I was trying to check the AWS CLI version on my terminal, I get this warning in return: &lt;/p&gt;

&lt;p&gt;&lt;a href="https://media2.dev.to/dynamic/image/width=800%2Cheight=%2Cfit=scale-down%2Cgravity=auto%2Cformat=auto/https%3A%2F%2Fdev-to-uploads.s3.amazonaws.com%2Fuploads%2Farticles%2Fa06ekow6t8okn6w90rgr.png" class="article-body-image-wrapper"&gt;&lt;img src="https://media2.dev.to/dynamic/image/width=800%2Cheight=%2Cfit=scale-down%2Cgravity=auto%2Cformat=auto/https%3A%2F%2Fdev-to-uploads.s3.amazonaws.com%2Fuploads%2Farticles%2Fa06ekow6t8okn6w90rgr.png" alt="Image description" width="800" height="82"&gt;&lt;/a&gt;&lt;br&gt;
which made me go: "Hang on, pretty sure I have anaconda installed". &lt;/p&gt;

&lt;p&gt;It must be because when I was installing anaconda, I didn't pick the setting so that Python is added to the environment. &lt;/p&gt;
&lt;h1&gt;
  
  
  How to Fix It
&lt;/h1&gt;
&lt;h2&gt;
  
  
  Method 1: Run Command Prompt as Administrator:
&lt;/h2&gt;

&lt;p&gt;Type the following commands, substituting the path to your Anaconda Python:&lt;br&gt;
&lt;/p&gt;

&lt;div class="highlight js-code-highlight"&gt;
&lt;pre class="highlight plaintext"&gt;&lt;code&gt;assoc .py=Python.File
ftype Python.File="C:\Users\YourName\anaconda3\python.exe" "%1" %*
&lt;/code&gt;&lt;/pre&gt;

&lt;/div&gt;



&lt;p&gt;These commands tell Windows to associate .py with a “Python.File” type, and then map “Python.File” to your Anaconda Python executable.&lt;/p&gt;

&lt;h2&gt;
  
  
  Method 2: Use the Windows Default Apps Settings (No Admin Required):
&lt;/h2&gt;

&lt;ul&gt;
&lt;li&gt;Open Settings → Apps → Default apps.&lt;/li&gt;
&lt;li&gt;Click Choose default apps by file type and scroll to .py.&lt;/li&gt;
&lt;li&gt;Select the current default app (or “No default”) and browse to your Python interpreter (e.g. C:\Users\YourName\anaconda3\python.exe).&lt;/li&gt;
&lt;/ul&gt;

&lt;p&gt;Now it's gone.&lt;/p&gt;

&lt;p&gt;&lt;a href="https://media2.dev.to/dynamic/image/width=800%2Cheight=%2Cfit=scale-down%2Cgravity=auto%2Cformat=auto/https%3A%2F%2Fdev-to-uploads.s3.amazonaws.com%2Fuploads%2Farticles%2Fjjy78ri4ozfv2mtc4ume.png" class="article-body-image-wrapper"&gt;&lt;img src="https://media2.dev.to/dynamic/image/width=800%2Cheight=%2Cfit=scale-down%2Cgravity=auto%2Cformat=auto/https%3A%2F%2Fdev-to-uploads.s3.amazonaws.com%2Fuploads%2Farticles%2Fjjy78ri4ozfv2mtc4ume.png" alt="Image description" width="800" height="58"&gt;&lt;/a&gt;&lt;/p&gt;

</description>
    </item>
    <item>
      <title>[Boost]</title>
      <dc:creator>vindianadoan</dc:creator>
      <pubDate>Mon, 10 Mar 2025 06:53:54 +0000</pubDate>
      <link>https://dev.to/vindianadoan/-4mgj</link>
      <guid>https://dev.to/vindianadoan/-4mgj</guid>
      <description>&lt;div class="ltag__link"&gt;
  &lt;a href="/vindianadoan" class="ltag__link__link"&gt;
    &lt;div class="ltag__link__pic"&gt;
      &lt;img src="https://media2.dev.to/dynamic/image/width=800%2Cheight=%2Cfit=scale-down%2Cgravity=auto%2Cformat=auto/https%3A%2F%2Fdev-to-uploads.s3.amazonaws.com%2Fuploads%2Fuser%2Fprofile_image%2F1033749%2F07501b91-4e55-4142-b23b-7dfd1f95a084.png" alt="vindianadoan"&gt;
    &lt;/div&gt;
  &lt;/a&gt;
  &lt;a href="https://dev.to/vindianadoan/deploying-an-r-model-as-a-containerised-api-with-docker-plumber-and-renv-5eo4" class="ltag__link__link"&gt;
    &lt;div class="ltag__link__content"&gt;
      &lt;h2&gt;Part 1: Deploying an R Model as a Containerised API with Docker, Plumber, and renv&lt;/h2&gt;
      &lt;h3&gt;vindianadoan ・ Mar 10&lt;/h3&gt;
      &lt;div class="ltag__link__taglist"&gt;
      &lt;/div&gt;
    &lt;/div&gt;
  &lt;/a&gt;
&lt;/div&gt;


</description>
      <category>docker</category>
      <category>api</category>
      <category>tutorial</category>
      <category>devops</category>
    </item>
    <item>
      <title>Part 1: Deploying an R Model as a Containerised API with Docker, Plumber, and renv</title>
      <dc:creator>vindianadoan</dc:creator>
      <pubDate>Mon, 10 Mar 2025 02:54:51 +0000</pubDate>
      <link>https://dev.to/vindianadoan/deploying-an-r-model-as-a-containerised-api-with-docker-plumber-and-renv-5eo4</link>
      <guid>https://dev.to/vindianadoan/deploying-an-r-model-as-a-containerised-api-with-docker-plumber-and-renv-5eo4</guid>
      <description>&lt;p&gt;In this post, I’ll walk you through the entire process of building an R model, containerising it as a RESTful API using the Plumber package, and adding detailed logging. We’ll cover every step, including troubleshooting common issues (like missing system libraries) and ensuring reproducibility with renv. By the end of this tutorial, you’ll have a Docker image that serves your model via an API endpoint—ready for local testing or deployment to cloud services like AWS. In this part, we will build the components required for local testing, deployment to AWS will come in part 2. &lt;/p&gt;

&lt;p&gt;You can find the entire code and Dockerfiles in the &lt;a href="https://github.com/vindianadoan/deploy-R-model-to-AWS-ECR/tree/master" rel="noopener noreferrer"&gt;GitHub repo&lt;/a&gt;.&lt;/p&gt;

&lt;h2&gt;
  
  
  Table of Contents
&lt;/h2&gt;

&lt;ol&gt;
&lt;li&gt;Overview &amp;amp; Prerequisites&lt;/li&gt;
&lt;li&gt;Building &amp;amp; Saving an R Model&lt;/li&gt;
&lt;li&gt;Building the Base Image with System Dependencies and renv&lt;/li&gt;
&lt;li&gt;Iterative Model Image for Rapid Development&lt;/li&gt;
&lt;li&gt;Automating Docker Builds with a Shell Script&lt;/li&gt;
&lt;li&gt;Managing R Dependencies with renv&lt;/li&gt;
&lt;li&gt;Creating a Plumber API Endpoint&lt;/li&gt;
&lt;li&gt;
Testing the API

&lt;ul&gt;
&lt;li&gt;Running a Container of the Built Image&lt;/li&gt;
&lt;li&gt;Using Curl&lt;/li&gt;
&lt;li&gt;Using Postman&lt;/li&gt;
&lt;li&gt;Using R’s httr Package&lt;/li&gt;
&lt;/ul&gt;
&lt;/li&gt;
&lt;li&gt;
Troubleshooting

&lt;ul&gt;
&lt;li&gt;Package Installation Issues&lt;/li&gt;
&lt;li&gt;Docker Volume Mount&lt;/li&gt;
&lt;li&gt;Interact with Running Container&lt;/li&gt;
&lt;/ul&gt;
&lt;/li&gt;
&lt;/ol&gt;

&lt;h2&gt;
  
  
  1. Overview &amp;amp; Prerequisites &lt;a&gt;&lt;/a&gt;
&lt;/h2&gt;

&lt;p&gt;In this guide, we build a predictive model using the classic iris dataset and train a random forest classifier. We then containerise our R code with Docker and expose a RESTful API via Plumber. We’ll add logging using the logger package and manage dependencies with renv for reproducibility.&lt;br&gt;
Key components include:&lt;/p&gt;

&lt;ul&gt;
&lt;li&gt;R Model Training: Using randomForest to train on the iris dataset.&lt;/li&gt;
&lt;li&gt;Containerisation: Creating a reproducible R environment using Docker.&lt;/li&gt;
&lt;li&gt;Dependency Management: Using renv to lock package versions.&lt;/li&gt;
&lt;li&gt;Plumber API: Exposing a /predict endpoint with Swagger documentation.&lt;/li&gt;
&lt;li&gt;Logging: Detailed logging of API requests and responses, with logs mounted on the host.&lt;/li&gt;
&lt;/ul&gt;

&lt;p&gt;To effectively monitor our deployment workflow, I have established a dual logging system. The &lt;code&gt;outside_container_logs&lt;/code&gt; directory captures logs generated during local development—this includes running and testing individual R scripts, building the Docker container, and future deployments. Meanwhile, logs from processes running inside the container are directed to &lt;code&gt;inside_container_logs&lt;/code&gt;, which is mounted to /app/logs within the container. This separation enables us to clearly differentiate between pre-deployment activities and runtime behavior inside the container, allowing us to efficiently troubleshoot and ensure that both our development and production environments are operating as expected.&lt;/p&gt;

&lt;p&gt;&lt;a href="https://media2.dev.to/dynamic/image/width=800%2Cheight=%2Cfit=scale-down%2Cgravity=auto%2Cformat=auto/https%3A%2F%2Fdev-to-uploads.s3.amazonaws.com%2Fuploads%2Farticles%2Fksi51mwnccgu5w46zvu4.png" class="article-body-image-wrapper"&gt;&lt;img src="https://media2.dev.to/dynamic/image/width=800%2Cheight=%2Cfit=scale-down%2Cgravity=auto%2Cformat=auto/https%3A%2F%2Fdev-to-uploads.s3.amazonaws.com%2Fuploads%2Farticles%2Fksi51mwnccgu5w46zvu4.png" alt="Image description" width="800" height="352"&gt;&lt;/a&gt;&lt;/p&gt;

&lt;p&gt;Before you begin, ensure you have the following installed:&lt;/p&gt;

&lt;ul&gt;
&lt;li&gt;Docker: For container building.&lt;/li&gt;
&lt;li&gt;R (4.4.1 or compatible): For model training and package management.&lt;/li&gt;
&lt;li&gt;renv: For dependency management.&lt;/li&gt;
&lt;li&gt;Plumber and logger packages: To expose the API and manage logs.&lt;/li&gt;
&lt;/ul&gt;
&lt;h2&gt;
  
  
  2. Building &amp;amp; Saving an R Model &lt;a&gt;&lt;/a&gt;
&lt;/h2&gt;

&lt;p&gt;First, we create a training script (train.R) that trains a random forest model on the iris dataset and logs each step. This script automatically detects whether it’s running inside a container (by checking for /app/logs) and sets up the appropriate logging directory.&lt;br&gt;
&lt;code&gt;train.R&lt;/code&gt;&lt;br&gt;
&lt;/p&gt;

&lt;div class="highlight js-code-highlight"&gt;
&lt;pre class="highlight r"&gt;&lt;code&gt;&lt;span class="c1"&gt;# train.R&lt;/span&gt;&lt;span class="w"&gt;
&lt;/span&gt;&lt;span class="n"&gt;library&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="n"&gt;randomForest&lt;/span&gt;&lt;span class="p"&gt;)&lt;/span&gt;&lt;span class="w"&gt;
&lt;/span&gt;&lt;span class="n"&gt;library&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="n"&gt;logger&lt;/span&gt;&lt;span class="p"&gt;)&lt;/span&gt;&lt;span class="w"&gt;

&lt;/span&gt;&lt;span class="c1"&gt;# Choose the log directory based on whether /app/logs exists (inside container) or not (outside)&lt;/span&gt;&lt;span class="w"&gt;
&lt;/span&gt;&lt;span class="n"&gt;log_dir&lt;/span&gt;&lt;span class="w"&gt; &lt;/span&gt;&lt;span class="o"&gt;&amp;lt;-&lt;/span&gt;&lt;span class="w"&gt; &lt;/span&gt;&lt;span class="k"&gt;if&lt;/span&gt;&lt;span class="w"&gt; &lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="n"&gt;dir.exists&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="s2"&gt;"/app/logs"&lt;/span&gt;&lt;span class="p"&gt;))&lt;/span&gt;&lt;span class="w"&gt; &lt;/span&gt;&lt;span class="p"&gt;{&lt;/span&gt;&lt;span class="w"&gt;
  &lt;/span&gt;&lt;span class="s2"&gt;"/app/logs/R_logs"&lt;/span&gt;&lt;span class="w"&gt;
&lt;/span&gt;&lt;span class="p"&gt;}&lt;/span&gt;&lt;span class="w"&gt; &lt;/span&gt;&lt;span class="k"&gt;else&lt;/span&gt;&lt;span class="w"&gt; &lt;/span&gt;&lt;span class="p"&gt;{&lt;/span&gt;&lt;span class="w"&gt;
  &lt;/span&gt;&lt;span class="s2"&gt;"logs/outside_container_logs/R_logs"&lt;/span&gt;&lt;span class="w"&gt;
&lt;/span&gt;&lt;span class="p"&gt;}&lt;/span&gt;&lt;span class="w"&gt;

&lt;/span&gt;&lt;span class="c1"&gt;# Ensure the directory exists; create it if necessary&lt;/span&gt;&lt;span class="w"&gt;
&lt;/span&gt;&lt;span class="k"&gt;if&lt;/span&gt;&lt;span class="w"&gt; &lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="o"&gt;!&lt;/span&gt;&lt;span class="n"&gt;dir.exists&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="n"&gt;log_dir&lt;/span&gt;&lt;span class="p"&gt;))&lt;/span&gt;&lt;span class="w"&gt; &lt;/span&gt;&lt;span class="p"&gt;{&lt;/span&gt;&lt;span class="w"&gt;
  &lt;/span&gt;&lt;span class="n"&gt;dir.create&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="n"&gt;log_dir&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt;&lt;span class="w"&gt; &lt;/span&gt;&lt;span class="n"&gt;recursive&lt;/span&gt;&lt;span class="w"&gt; &lt;/span&gt;&lt;span class="o"&gt;=&lt;/span&gt;&lt;span class="w"&gt; &lt;/span&gt;&lt;span class="kc"&gt;TRUE&lt;/span&gt;&lt;span class="p"&gt;)&lt;/span&gt;&lt;span class="w"&gt;
&lt;/span&gt;&lt;span class="p"&gt;}&lt;/span&gt;&lt;span class="w"&gt;

&lt;/span&gt;&lt;span class="c1"&gt;# Create a timestamp for the log file name&lt;/span&gt;&lt;span class="w"&gt;
&lt;/span&gt;&lt;span class="n"&gt;timestamp&lt;/span&gt;&lt;span class="w"&gt; &lt;/span&gt;&lt;span class="o"&gt;&amp;lt;-&lt;/span&gt;&lt;span class="w"&gt; &lt;/span&gt;&lt;span class="n"&gt;format&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="n"&gt;Sys.time&lt;/span&gt;&lt;span class="p"&gt;(),&lt;/span&gt;&lt;span class="w"&gt; &lt;/span&gt;&lt;span class="s2"&gt;"%Y%m%d_%H%M%S"&lt;/span&gt;&lt;span class="p"&gt;)&lt;/span&gt;&lt;span class="w"&gt;
&lt;/span&gt;&lt;span class="n"&gt;log_file&lt;/span&gt;&lt;span class="w"&gt; &lt;/span&gt;&lt;span class="o"&gt;&amp;lt;-&lt;/span&gt;&lt;span class="w"&gt; &lt;/span&gt;&lt;span class="n"&gt;file.path&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="n"&gt;log_dir&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt;&lt;span class="w"&gt; &lt;/span&gt;&lt;span class="n"&gt;paste0&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="n"&gt;timestamp&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt;&lt;span class="w"&gt; &lt;/span&gt;&lt;span class="s2"&gt;"_training.log"&lt;/span&gt;&lt;span class="p"&gt;))&lt;/span&gt;&lt;span class="w"&gt;

&lt;/span&gt;&lt;span class="c1"&gt;# Configure logger with a unique file name for each run&lt;/span&gt;&lt;span class="w"&gt;
&lt;/span&gt;&lt;span class="n"&gt;log_appender&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="n"&gt;appender_file&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="n"&gt;log_file&lt;/span&gt;&lt;span class="p"&gt;))&lt;/span&gt;&lt;span class="w"&gt;

&lt;/span&gt;&lt;span class="n"&gt;log_info&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="s2"&gt;"Starting model training..."&lt;/span&gt;&lt;span class="p"&gt;)&lt;/span&gt;&lt;span class="w"&gt;

&lt;/span&gt;&lt;span class="c1"&gt;# Train a random forest model to classify iris species&lt;/span&gt;&lt;span class="w"&gt;
&lt;/span&gt;&lt;span class="n"&gt;data&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="n"&gt;iris&lt;/span&gt;&lt;span class="p"&gt;)&lt;/span&gt;&lt;span class="w"&gt;
&lt;/span&gt;&lt;span class="n"&gt;model&lt;/span&gt;&lt;span class="w"&gt; &lt;/span&gt;&lt;span class="o"&gt;&amp;lt;-&lt;/span&gt;&lt;span class="w"&gt; &lt;/span&gt;&lt;span class="n"&gt;randomForest&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="n"&gt;Species&lt;/span&gt;&lt;span class="w"&gt; &lt;/span&gt;&lt;span class="o"&gt;~&lt;/span&gt;&lt;span class="w"&gt; &lt;/span&gt;&lt;span class="n"&gt;.&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt;&lt;span class="w"&gt; &lt;/span&gt;&lt;span class="n"&gt;data&lt;/span&gt;&lt;span class="w"&gt; &lt;/span&gt;&lt;span class="o"&gt;=&lt;/span&gt;&lt;span class="w"&gt; &lt;/span&gt;&lt;span class="n"&gt;iris&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt;&lt;span class="w"&gt; &lt;/span&gt;&lt;span class="n"&gt;ntree&lt;/span&gt;&lt;span class="w"&gt; &lt;/span&gt;&lt;span class="o"&gt;=&lt;/span&gt;&lt;span class="w"&gt; &lt;/span&gt;&lt;span class="m"&gt;100&lt;/span&gt;&lt;span class="p"&gt;)&lt;/span&gt;&lt;span class="w"&gt;

&lt;/span&gt;&lt;span class="n"&gt;log_info&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="s2"&gt;"Model training complete."&lt;/span&gt;&lt;span class="p"&gt;)&lt;/span&gt;&lt;span class="w"&gt;

&lt;/span&gt;&lt;span class="c1"&gt;# Capture the output of print(model)&lt;/span&gt;&lt;span class="w"&gt;
&lt;/span&gt;&lt;span class="n"&gt;model_output&lt;/span&gt;&lt;span class="w"&gt; &lt;/span&gt;&lt;span class="o"&gt;&amp;lt;-&lt;/span&gt;&lt;span class="w"&gt; &lt;/span&gt;&lt;span class="n"&gt;capture.output&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="n"&gt;print&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="n"&gt;model&lt;/span&gt;&lt;span class="p"&gt;))&lt;/span&gt;&lt;span class="w"&gt;
&lt;/span&gt;&lt;span class="c1"&gt;# Log the captured output&lt;/span&gt;&lt;span class="w"&gt;
&lt;/span&gt;&lt;span class="n"&gt;log_info&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="s2"&gt;"Model details:\n{paste(model_output, collapse = '\n')}"&lt;/span&gt;&lt;span class="p"&gt;)&lt;/span&gt;&lt;span class="w"&gt;

&lt;/span&gt;&lt;span class="c1"&gt;# Save the trained model to disk&lt;/span&gt;&lt;span class="w"&gt;
&lt;/span&gt;&lt;span class="n"&gt;saveRDS&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="n"&gt;model&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt;&lt;span class="w"&gt; &lt;/span&gt;&lt;span class="n"&gt;file&lt;/span&gt;&lt;span class="w"&gt; &lt;/span&gt;&lt;span class="o"&gt;=&lt;/span&gt;&lt;span class="w"&gt; &lt;/span&gt;&lt;span class="s2"&gt;"data/model.rds"&lt;/span&gt;&lt;span class="p"&gt;)&lt;/span&gt;&lt;span class="w"&gt;
&lt;/span&gt;&lt;span class="n"&gt;log_info&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="s2"&gt;"Model saved to model.rds."&lt;/span&gt;&lt;span class="p"&gt;)&lt;/span&gt;&lt;span class="w"&gt;
&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;

&lt;/div&gt;



&lt;p&gt;Run this script (manually or as part of your Docker build) to generate your model and record the training process.&lt;/p&gt;

&lt;h2&gt;
  
  
  3. Creating a Plumber API Endpoint &lt;a&gt;&lt;/a&gt;
&lt;/h2&gt;

&lt;p&gt;Next, create an API using Plumber. We’ll define an endpoint &lt;code&gt;/predict&lt;/code&gt; that:&lt;/p&gt;

&lt;ul&gt;
&lt;li&gt;Loads the pre-trained model.&lt;/li&gt;
&lt;li&gt;Parses the incoming JSON request.&lt;/li&gt;
&lt;li&gt;Makes a prediction.&lt;/li&gt;
&lt;li&gt;Logs request and response details.&lt;/li&gt;
&lt;/ul&gt;

&lt;p&gt;&lt;code&gt;serve.R&lt;/code&gt;&lt;br&gt;
&lt;/p&gt;

&lt;div class="highlight js-code-highlight"&gt;
&lt;pre class="highlight r"&gt;&lt;code&gt;&lt;span class="c1"&gt;# serve.R&lt;/span&gt;&lt;span class="w"&gt;
&lt;/span&gt;&lt;span class="n"&gt;library&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="n"&gt;plumber&lt;/span&gt;&lt;span class="p"&gt;)&lt;/span&gt;&lt;span class="w"&gt;
&lt;/span&gt;&lt;span class="n"&gt;library&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="n"&gt;randomForest&lt;/span&gt;&lt;span class="p"&gt;)&lt;/span&gt;&lt;span class="w"&gt;
&lt;/span&gt;&lt;span class="n"&gt;library&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="n"&gt;logger&lt;/span&gt;&lt;span class="p"&gt;)&lt;/span&gt;&lt;span class="w"&gt;
&lt;/span&gt;&lt;span class="n"&gt;library&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="n"&gt;jsonlite&lt;/span&gt;&lt;span class="p"&gt;)&lt;/span&gt;&lt;span class="w"&gt;

&lt;/span&gt;&lt;span class="c1"&gt;# Set up logging&lt;/span&gt;&lt;span class="w"&gt;
&lt;/span&gt;&lt;span class="n"&gt;log_dir&lt;/span&gt;&lt;span class="w"&gt; &lt;/span&gt;&lt;span class="o"&gt;&amp;lt;-&lt;/span&gt;&lt;span class="w"&gt; &lt;/span&gt;&lt;span class="k"&gt;if&lt;/span&gt;&lt;span class="w"&gt; &lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="n"&gt;dir.exists&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="s2"&gt;"/app/logs"&lt;/span&gt;&lt;span class="p"&gt;))&lt;/span&gt;&lt;span class="w"&gt; &lt;/span&gt;&lt;span class="p"&gt;{&lt;/span&gt;&lt;span class="w"&gt;
  &lt;/span&gt;&lt;span class="s2"&gt;"/app/logs/R_logs"&lt;/span&gt;&lt;span class="w"&gt;
&lt;/span&gt;&lt;span class="p"&gt;}&lt;/span&gt;&lt;span class="w"&gt; &lt;/span&gt;&lt;span class="k"&gt;else&lt;/span&gt;&lt;span class="w"&gt; &lt;/span&gt;&lt;span class="p"&gt;{&lt;/span&gt;&lt;span class="w"&gt;
  &lt;/span&gt;&lt;span class="s2"&gt;"logs/outside_container_logs/R_logs"&lt;/span&gt;&lt;span class="w"&gt;
&lt;/span&gt;&lt;span class="p"&gt;}&lt;/span&gt;&lt;span class="w"&gt;
&lt;/span&gt;&lt;span class="k"&gt;if&lt;/span&gt;&lt;span class="w"&gt; &lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="o"&gt;!&lt;/span&gt;&lt;span class="n"&gt;dir.exists&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="n"&gt;log_dir&lt;/span&gt;&lt;span class="p"&gt;))&lt;/span&gt;&lt;span class="w"&gt; &lt;/span&gt;&lt;span class="p"&gt;{&lt;/span&gt;&lt;span class="w"&gt;
  &lt;/span&gt;&lt;span class="n"&gt;dir.create&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="n"&gt;log_dir&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt;&lt;span class="w"&gt; &lt;/span&gt;&lt;span class="n"&gt;recursive&lt;/span&gt;&lt;span class="w"&gt; &lt;/span&gt;&lt;span class="o"&gt;=&lt;/span&gt;&lt;span class="w"&gt; &lt;/span&gt;&lt;span class="kc"&gt;TRUE&lt;/span&gt;&lt;span class="p"&gt;)&lt;/span&gt;&lt;span class="w"&gt;
&lt;/span&gt;&lt;span class="p"&gt;}&lt;/span&gt;&lt;span class="w"&gt;
&lt;/span&gt;&lt;span class="c1"&gt;# Create a timestamp for the log file name&lt;/span&gt;&lt;span class="w"&gt;
&lt;/span&gt;&lt;span class="n"&gt;timestamp&lt;/span&gt;&lt;span class="w"&gt; &lt;/span&gt;&lt;span class="o"&gt;&amp;lt;-&lt;/span&gt;&lt;span class="w"&gt; &lt;/span&gt;&lt;span class="n"&gt;format&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="n"&gt;Sys.time&lt;/span&gt;&lt;span class="p"&gt;(),&lt;/span&gt;&lt;span class="w"&gt; &lt;/span&gt;&lt;span class="s2"&gt;"%Y%m%d_%H%M%S"&lt;/span&gt;&lt;span class="p"&gt;)&lt;/span&gt;&lt;span class="w"&gt;
&lt;/span&gt;&lt;span class="n"&gt;log_file&lt;/span&gt;&lt;span class="w"&gt; &lt;/span&gt;&lt;span class="o"&gt;&amp;lt;-&lt;/span&gt;&lt;span class="w"&gt; &lt;/span&gt;&lt;span class="n"&gt;file.path&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="n"&gt;log_dir&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt;&lt;span class="w"&gt; &lt;/span&gt;&lt;span class="n"&gt;paste0&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="n"&gt;timestamp&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt;&lt;span class="w"&gt; &lt;/span&gt;&lt;span class="s2"&gt;"_request.log"&lt;/span&gt;&lt;span class="p"&gt;))&lt;/span&gt;&lt;span class="w"&gt;
&lt;/span&gt;&lt;span class="n"&gt;log_appender&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="n"&gt;appender_file&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="n"&gt;log_file&lt;/span&gt;&lt;span class="p"&gt;))&lt;/span&gt;&lt;span class="w"&gt;

&lt;/span&gt;&lt;span class="c1"&gt;# Load the trained model&lt;/span&gt;&lt;span class="w"&gt;
&lt;/span&gt;&lt;span class="n"&gt;model&lt;/span&gt;&lt;span class="w"&gt; &lt;/span&gt;&lt;span class="o"&gt;&amp;lt;-&lt;/span&gt;&lt;span class="w"&gt; &lt;/span&gt;&lt;span class="n"&gt;readRDS&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="s2"&gt;"data/model.rds"&lt;/span&gt;&lt;span class="p"&gt;)&lt;/span&gt;&lt;span class="w"&gt;

&lt;/span&gt;&lt;span class="c1"&gt;#* @filter log_requests&lt;/span&gt;&lt;span class="w"&gt;
&lt;/span&gt;&lt;span class="k"&gt;function&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="n"&gt;req&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt;&lt;span class="w"&gt; &lt;/span&gt;&lt;span class="n"&gt;res&lt;/span&gt;&lt;span class="p"&gt;)&lt;/span&gt;&lt;span class="w"&gt; &lt;/span&gt;&lt;span class="p"&gt;{&lt;/span&gt;&lt;span class="w"&gt;
  &lt;/span&gt;&lt;span class="n"&gt;log_info&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="s2"&gt;"Incoming request: {req$REQUEST_METHOD} {req$PATH_INFO}"&lt;/span&gt;&lt;span class="p"&gt;)&lt;/span&gt;&lt;span class="w"&gt;
  &lt;/span&gt;&lt;span class="n"&gt;log_info&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="s2"&gt;"Request body: {req$postBody}"&lt;/span&gt;&lt;span class="p"&gt;)&lt;/span&gt;&lt;span class="w"&gt;
  &lt;/span&gt;&lt;span class="n"&gt;plumber&lt;/span&gt;&lt;span class="o"&gt;::&lt;/span&gt;&lt;span class="n"&gt;forward&lt;/span&gt;&lt;span class="p"&gt;()&lt;/span&gt;&lt;span class="w"&gt;
&lt;/span&gt;&lt;span class="p"&gt;}&lt;/span&gt;&lt;span class="w"&gt;

&lt;/span&gt;&lt;span class="c1"&gt;#* @post /predict&lt;/span&gt;&lt;span class="w"&gt;
&lt;/span&gt;&lt;span class="c1"&gt;#* @serializer json&lt;/span&gt;&lt;span class="w"&gt;
&lt;/span&gt;&lt;span class="k"&gt;function&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="n"&gt;req&lt;/span&gt;&lt;span class="p"&gt;)&lt;/span&gt;&lt;span class="w"&gt; &lt;/span&gt;&lt;span class="p"&gt;{&lt;/span&gt;&lt;span class="w"&gt;
  &lt;/span&gt;&lt;span class="n"&gt;data&lt;/span&gt;&lt;span class="w"&gt; &lt;/span&gt;&lt;span class="o"&gt;&amp;lt;-&lt;/span&gt;&lt;span class="w"&gt; &lt;/span&gt;&lt;span class="n"&gt;fromJSON&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="n"&gt;req&lt;/span&gt;&lt;span class="o"&gt;$&lt;/span&gt;&lt;span class="n"&gt;postBody&lt;/span&gt;&lt;span class="p"&gt;)&lt;/span&gt;&lt;span class="w"&gt;
  &lt;/span&gt;&lt;span class="n"&gt;input_data&lt;/span&gt;&lt;span class="w"&gt; &lt;/span&gt;&lt;span class="o"&gt;&amp;lt;-&lt;/span&gt;&lt;span class="w"&gt; &lt;/span&gt;&lt;span class="n"&gt;as.data.frame&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="n"&gt;data&lt;/span&gt;&lt;span class="p"&gt;)&lt;/span&gt;&lt;span class="w"&gt;
  &lt;/span&gt;&lt;span class="n"&gt;pred&lt;/span&gt;&lt;span class="w"&gt; &lt;/span&gt;&lt;span class="o"&gt;&amp;lt;-&lt;/span&gt;&lt;span class="w"&gt; &lt;/span&gt;&lt;span class="n"&gt;predict&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="n"&gt;model&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt;&lt;span class="w"&gt; &lt;/span&gt;&lt;span class="n"&gt;input_data&lt;/span&gt;&lt;span class="p"&gt;)&lt;/span&gt;&lt;span class="w"&gt;
  &lt;/span&gt;&lt;span class="n"&gt;log_info&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="s2"&gt;"Prediction: {paste(pred, collapse=', ')}"&lt;/span&gt;&lt;span class="p"&gt;)&lt;/span&gt;&lt;span class="w"&gt;
  &lt;/span&gt;&lt;span class="nf"&gt;list&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="n"&gt;prediction&lt;/span&gt;&lt;span class="w"&gt; &lt;/span&gt;&lt;span class="o"&gt;=&lt;/span&gt;&lt;span class="w"&gt; &lt;/span&gt;&lt;span class="nf"&gt;as.character&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="n"&gt;pred&lt;/span&gt;&lt;span class="p"&gt;))&lt;/span&gt;&lt;span class="w"&gt;
&lt;/span&gt;&lt;span class="p"&gt;}&lt;/span&gt;&lt;span class="w"&gt;
&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;

&lt;/div&gt;



&lt;p&gt;Plumber automatically generates Swagger documentation (accessible at &lt;a href="http://localhost:8080/__docs__/" rel="noopener noreferrer"&gt;http://localhost:8080/__docs__/&lt;/a&gt;) from these annotations.&lt;/p&gt;

&lt;h2&gt;
  
  
  4. Building the Base Image with System Dependencies and renv &lt;a&gt;&lt;/a&gt;
&lt;/h2&gt;

&lt;p&gt;To ensure that the heavy lifting of installing system dependencies and R packages is done only once, we created a base image. This image includes R 4.4.1, necessary system libraries (for packages such as textshaping and ragg), and restores R packages using an &lt;code&gt;renv.lock&lt;/code&gt; file.&lt;br&gt;
&lt;code&gt;Dockerfile.base&lt;/code&gt;&lt;br&gt;
&lt;/p&gt;

&lt;div class="highlight js-code-highlight"&gt;
&lt;pre class="highlight shell"&gt;&lt;code&gt;&lt;span class="c"&gt;# Use a specific R base image&lt;/span&gt;
FROM r-base:4.4.1

&lt;span class="c"&gt;# Install system dependencies required by R packages&lt;/span&gt;
RUN apt-get update &lt;span class="o"&gt;&amp;amp;&amp;amp;&lt;/span&gt; apt-get &lt;span class="nb"&gt;install&lt;/span&gt; &lt;span class="nt"&gt;-y&lt;/span&gt; &lt;span class="se"&gt;\&lt;/span&gt;
    libcurl4-openssl-dev &lt;span class="se"&gt;\&lt;/span&gt;
    libssl-dev &lt;span class="se"&gt;\&lt;/span&gt;
    libxml2 &lt;span class="se"&gt;\&lt;/span&gt;
    libxml2-dev &lt;span class="se"&gt;\&lt;/span&gt;
    libz-dev &lt;span class="se"&gt;\&lt;/span&gt;
    libsodium-dev &lt;span class="se"&gt;\&lt;/span&gt;
    libfontconfig1-dev &lt;span class="se"&gt;\&lt;/span&gt;
    libfreetype6-dev &lt;span class="se"&gt;\&lt;/span&gt;
    libharfbuzz-dev &lt;span class="se"&gt;\&lt;/span&gt;
    libfribidi-dev &lt;span class="se"&gt;\&lt;/span&gt;
    libpng-dev &lt;span class="se"&gt;\&lt;/span&gt;
    libtiff5-dev &lt;span class="se"&gt;\&lt;/span&gt;
    libjpeg-dev &lt;span class="se"&gt;\&lt;/span&gt;
    build-essential &lt;span class="se"&gt;\&lt;/span&gt;
    pkg-config &lt;span class="se"&gt;\&lt;/span&gt;
    wget &lt;span class="se"&gt;\&lt;/span&gt;
    apt-transport-https &lt;span class="se"&gt;\&lt;/span&gt;
    ca-certificates &lt;span class="se"&gt;\&lt;/span&gt;
 &lt;span class="o"&gt;&amp;amp;&amp;amp;&lt;/span&gt; &lt;span class="nb"&gt;rm&lt;/span&gt; &lt;span class="nt"&gt;-rf&lt;/span&gt; /var/lib/apt/lists/&lt;span class="k"&gt;*&lt;/span&gt;

&lt;span class="c"&gt;# Create a directory for your application&lt;/span&gt;
WORKDIR /app

&lt;span class="c"&gt;# Copy your renv lock file to restore R packages&lt;/span&gt;
COPY renv.lock /app/renv.lock

&lt;span class="c"&gt;# Install renv and restore packages using renv.lock&lt;/span&gt;
RUN R &lt;span class="nt"&gt;-e&lt;/span&gt; &lt;span class="s2"&gt;"install.packages('renv', repos='http://cran.rstudio.com/'); &lt;/span&gt;&lt;span class="se"&gt;\&lt;/span&gt;&lt;span class="s2"&gt;
          renv::restore(repos = c(CRAN = 'http://cran.rstudio.com/'))"&lt;/span&gt;
&lt;/code&gt;&lt;/pre&gt;

&lt;/div&gt;



&lt;p&gt;This image (tagged as, for example, &lt;code&gt;r-base-mlops:4.4.1&lt;/code&gt;) serves as a stable foundation.&lt;/p&gt;

&lt;h2&gt;
  
  
  5. Iterative Model Image for Rapid Development &lt;a&gt;&lt;/a&gt;
&lt;/h2&gt;

&lt;p&gt;Building on the base image, we created a smaller model iteration image that contains only our model code. This image allows us to rapidly iterate on our model (or API) without reinstalling all the system dependencies and R packages every time.&lt;br&gt;
We’ll create a Docker image that:&lt;/p&gt;

&lt;ul&gt;
&lt;li&gt;Uses the R base image (here, version 4.4.1).&lt;/li&gt;
&lt;li&gt;Copies your scripts and model into the container.&lt;/li&gt;
&lt;li&gt;Exposes the Plumber API on port 8080.&lt;/li&gt;
&lt;/ul&gt;

&lt;p&gt;&lt;code&gt;Dockerfile.model&lt;/code&gt;&lt;br&gt;
&lt;/p&gt;

&lt;div class="highlight js-code-highlight"&gt;
&lt;pre class="highlight shell"&gt;&lt;code&gt;&lt;span class="c"&gt;# Use the prebuilt base image that contains R 4.4.1 and all system dependencies &amp;amp; renv packages&lt;/span&gt;
FROM r-base-mlops:4.4.1

&lt;span class="c"&gt;# Set the working directory&lt;/span&gt;
WORKDIR /app

&lt;span class="c"&gt;# Create directories for logs and model data&lt;/span&gt;
RUN &lt;span class="nb"&gt;mkdir&lt;/span&gt; &lt;span class="nt"&gt;-p&lt;/span&gt; /app/logs/docker_logs /app/logs/R_logs /app/logs/aws_logs
RUN &lt;span class="nb"&gt;mkdir&lt;/span&gt; &lt;span class="nt"&gt;-p&lt;/span&gt; data

&lt;span class="c"&gt;# Copy your model scripts into the container&lt;/span&gt;
COPY ./R/train.R /app/train.R
COPY ./R/serve.R /app/serve.R
&lt;span class="c"&gt;# Copy the runner script to start the API&lt;/span&gt;
COPY ./R/run_api.R /app/run_api.R

&lt;span class="c"&gt;# Run the training script to generate or update your model (e.g. data/model.rds)&lt;/span&gt;
RUN Rscript train.R

&lt;span class="c"&gt;# Expose the port on which your Plumber API will run&lt;/span&gt;
EXPOSE 8080

&lt;span class="c"&gt;# Start the Plumber API using the runner script&lt;/span&gt;
CMD &lt;span class="o"&gt;[&lt;/span&gt;&lt;span class="s2"&gt;"Rscript"&lt;/span&gt;, &lt;span class="s2"&gt;"run_api.R"&lt;/span&gt;&lt;span class="o"&gt;]&lt;/span&gt;
&lt;/code&gt;&lt;/pre&gt;

&lt;/div&gt;



&lt;p&gt;The run_api.R script simply loads serve.R and starts the API:&lt;br&gt;
&lt;/p&gt;

&lt;div class="highlight js-code-highlight"&gt;
&lt;pre class="highlight r"&gt;&lt;code&gt;&lt;span class="c1"&gt;# run_api.R&lt;/span&gt;&lt;span class="w"&gt;
&lt;/span&gt;&lt;span class="n"&gt;pr&lt;/span&gt;&lt;span class="w"&gt; &lt;/span&gt;&lt;span class="o"&gt;&amp;lt;-&lt;/span&gt;&lt;span class="w"&gt; &lt;/span&gt;&lt;span class="n"&gt;plumber&lt;/span&gt;&lt;span class="o"&gt;::&lt;/span&gt;&lt;span class="n"&gt;plumb&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="s2"&gt;"serve.R"&lt;/span&gt;&lt;span class="p"&gt;)&lt;/span&gt;&lt;span class="w"&gt;
&lt;/span&gt;&lt;span class="n"&gt;pr&lt;/span&gt;&lt;span class="o"&gt;$&lt;/span&gt;&lt;span class="n"&gt;run&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="n"&gt;host&lt;/span&gt;&lt;span class="w"&gt; &lt;/span&gt;&lt;span class="o"&gt;=&lt;/span&gt;&lt;span class="w"&gt; &lt;/span&gt;&lt;span class="s2"&gt;"0.0.0.0"&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt;&lt;span class="w"&gt; &lt;/span&gt;&lt;span class="n"&gt;port&lt;/span&gt;&lt;span class="w"&gt; &lt;/span&gt;&lt;span class="o"&gt;=&lt;/span&gt;&lt;span class="w"&gt; &lt;/span&gt;&lt;span class="m"&gt;8080&lt;/span&gt;&lt;span class="p"&gt;)&lt;/span&gt;&lt;span class="w"&gt;
&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;

&lt;/div&gt;



&lt;p&gt;This setup ensures that when you change your model code (in &lt;code&gt;train.R&lt;/code&gt; or &lt;code&gt;serve.R&lt;/code&gt;), you only need to rebuild the model image—not the entire base image.&lt;/p&gt;

&lt;h2&gt;
  
  
  6. Automating Docker Builds with a Shell Script &lt;a&gt;&lt;/a&gt;
&lt;/h2&gt;

&lt;p&gt;To build the Docker images, you can run these command lines:&lt;br&gt;
&lt;/p&gt;

&lt;div class="highlight js-code-highlight"&gt;
&lt;pre class="highlight shell"&gt;&lt;code&gt;&lt;span class="c"&gt;# 1. Build the Docker base image &lt;/span&gt;
docker build &lt;span class="nt"&gt;--no-cache&lt;/span&gt; &lt;span class="nt"&gt;-f&lt;/span&gt; docker/Dockerfile.base &lt;span class="nt"&gt;-t&lt;/span&gt; r-base-mlops:4.4.1 &lt;span class="nb"&gt;.&lt;/span&gt;
&lt;span class="c"&gt;# 2. Build the Docker model image &lt;/span&gt;
docker build &lt;span class="nt"&gt;--no-cache&lt;/span&gt; &lt;span class="nt"&gt;-f&lt;/span&gt; docker/Dockerfile.model &lt;span class="nt"&gt;-t&lt;/span&gt; my-r-sagemaker-model &lt;span class="nb"&gt;.&lt;/span&gt;
&lt;/code&gt;&lt;/pre&gt;

&lt;/div&gt;



&lt;p&gt;However, to simplify the process of building your Docker images above and ensure thorough logging, I’ve created an automation script. This script lets you choose between building the model image or the base image with a simple command-line argument. It also saves detailed build logs for later review.&lt;br&gt;
&lt;code&gt;docker_build.sh&lt;/code&gt;&lt;br&gt;
&lt;/p&gt;

&lt;div class="highlight js-code-highlight"&gt;
&lt;pre class="highlight shell"&gt;&lt;code&gt;&lt;span class="c"&gt;#!/bin/bash&lt;/span&gt;

&lt;span class="c"&gt;# Change working directory to the repository root.&lt;/span&gt;
&lt;span class="c"&gt;# This works if your script is in ./bash/ and your repo root is one level up.&lt;/span&gt;
&lt;span class="nv"&gt;REPO_ROOT&lt;/span&gt;&lt;span class="o"&gt;=&lt;/span&gt;&lt;span class="s2"&gt;"&lt;/span&gt;&lt;span class="si"&gt;$(&lt;/span&gt;&lt;span class="nb"&gt;cd&lt;/span&gt; &lt;span class="s2"&gt;"&lt;/span&gt;&lt;span class="si"&gt;$(&lt;/span&gt;&lt;span class="nb"&gt;dirname&lt;/span&gt; &lt;span class="s2"&gt;"&lt;/span&gt;&lt;span class="nv"&gt;$0&lt;/span&gt;&lt;span class="s2"&gt;"&lt;/span&gt;&lt;span class="si"&gt;)&lt;/span&gt;&lt;span class="s2"&gt;/.."&lt;/span&gt; &lt;span class="o"&gt;&amp;amp;&amp;amp;&lt;/span&gt; &lt;span class="nb"&gt;pwd&lt;/span&gt;&lt;span class="si"&gt;)&lt;/span&gt;&lt;span class="s2"&gt;"&lt;/span&gt;
&lt;span class="nb"&gt;cd&lt;/span&gt; &lt;span class="s2"&gt;"&lt;/span&gt;&lt;span class="nv"&gt;$REPO_ROOT&lt;/span&gt;&lt;span class="s2"&gt;"&lt;/span&gt; &lt;span class="o"&gt;||&lt;/span&gt; &lt;span class="nb"&gt;exit &lt;/span&gt;1

&lt;span class="c"&gt;# Check if an argument is provided&lt;/span&gt;
&lt;span class="k"&gt;if&lt;/span&gt; &lt;span class="o"&gt;[&lt;/span&gt; &lt;span class="nv"&gt;$# &lt;/span&gt;&lt;span class="nt"&gt;-eq&lt;/span&gt; 0 &lt;span class="o"&gt;]&lt;/span&gt;&lt;span class="p"&gt;;&lt;/span&gt; &lt;span class="k"&gt;then
  &lt;/span&gt;&lt;span class="nb"&gt;echo&lt;/span&gt; &lt;span class="s2"&gt;"Usage: &lt;/span&gt;&lt;span class="nv"&gt;$0&lt;/span&gt;&lt;span class="s2"&gt; &amp;lt;model|base&amp;gt;"&lt;/span&gt;
  &lt;span class="nb"&gt;exit &lt;/span&gt;1
&lt;span class="k"&gt;fi

&lt;/span&gt;&lt;span class="nv"&gt;container_type&lt;/span&gt;&lt;span class="o"&gt;=&lt;/span&gt;&lt;span class="nv"&gt;$1&lt;/span&gt;

&lt;span class="c"&gt;# Ensure the logs directory exists&lt;/span&gt;
&lt;span class="nb"&gt;mkdir&lt;/span&gt; &lt;span class="nt"&gt;-p&lt;/span&gt; logs/outside_container_logs/docker_logs/

&lt;span class="c"&gt;# Create a unique log filename using a timestamp and the container type&lt;/span&gt;
&lt;span class="nv"&gt;timestamp&lt;/span&gt;&lt;span class="o"&gt;=&lt;/span&gt;&lt;span class="si"&gt;$(&lt;/span&gt;&lt;span class="nb"&gt;date&lt;/span&gt; +&lt;span class="s1"&gt;'%Y%m%d_%H%M%S'&lt;/span&gt;&lt;span class="si"&gt;)&lt;/span&gt;
&lt;span class="nv"&gt;log_file&lt;/span&gt;&lt;span class="o"&gt;=&lt;/span&gt;&lt;span class="s2"&gt;"logs/outside_container_logs/docker_logs/&lt;/span&gt;&lt;span class="k"&gt;${&lt;/span&gt;&lt;span class="nv"&gt;timestamp&lt;/span&gt;&lt;span class="k"&gt;}&lt;/span&gt;&lt;span class="s2"&gt;_&lt;/span&gt;&lt;span class="k"&gt;${&lt;/span&gt;&lt;span class="nv"&gt;container_type&lt;/span&gt;&lt;span class="k"&gt;}&lt;/span&gt;&lt;span class="s2"&gt;_build.log"&lt;/span&gt;

&lt;span class="nb"&gt;echo&lt;/span&gt; &lt;span class="s2"&gt;"Building starting. Build log will be saved to &lt;/span&gt;&lt;span class="nv"&gt;$log_file&lt;/span&gt;&lt;span class="s2"&gt;"&lt;/span&gt;

&lt;span class="c"&gt;# Check the argument to determine which Docker build command to use&lt;/span&gt;
&lt;span class="k"&gt;if&lt;/span&gt; &lt;span class="o"&gt;[&lt;/span&gt; &lt;span class="s2"&gt;"&lt;/span&gt;&lt;span class="nv"&gt;$container_type&lt;/span&gt;&lt;span class="s2"&gt;"&lt;/span&gt; &lt;span class="o"&gt;=&lt;/span&gt; &lt;span class="s2"&gt;"model"&lt;/span&gt; &lt;span class="o"&gt;]&lt;/span&gt;&lt;span class="p"&gt;;&lt;/span&gt; &lt;span class="k"&gt;then
    &lt;/span&gt;docker build &lt;span class="nt"&gt;--no-cache&lt;/span&gt; &lt;span class="nt"&gt;-f&lt;/span&gt; docker/Dockerfile.model &lt;span class="nt"&gt;-t&lt;/span&gt; my-r-sagemaker-model &lt;span class="nb"&gt;.&lt;/span&gt; &lt;span class="o"&gt;&amp;gt;&lt;/span&gt; &lt;span class="s2"&gt;"&lt;/span&gt;&lt;span class="nv"&gt;$log_file&lt;/span&gt;&lt;span class="s2"&gt;"&lt;/span&gt; 2&amp;gt;&amp;amp;1
&lt;span class="k"&gt;elif&lt;/span&gt; &lt;span class="o"&gt;[&lt;/span&gt; &lt;span class="s2"&gt;"&lt;/span&gt;&lt;span class="nv"&gt;$container_type&lt;/span&gt;&lt;span class="s2"&gt;"&lt;/span&gt; &lt;span class="o"&gt;=&lt;/span&gt; &lt;span class="s2"&gt;"base"&lt;/span&gt; &lt;span class="o"&gt;]&lt;/span&gt;&lt;span class="p"&gt;;&lt;/span&gt; &lt;span class="k"&gt;then
    &lt;/span&gt;docker build &lt;span class="nt"&gt;--no-cache&lt;/span&gt; &lt;span class="nt"&gt;-f&lt;/span&gt; docker/Dockerfile.base &lt;span class="nt"&gt;-t&lt;/span&gt; r-base-mlops:4.4.1 &lt;span class="nb"&gt;.&lt;/span&gt; &lt;span class="o"&gt;&amp;gt;&lt;/span&gt; &lt;span class="s2"&gt;"&lt;/span&gt;&lt;span class="nv"&gt;$log_file&lt;/span&gt;&lt;span class="s2"&gt;"&lt;/span&gt; 2&amp;gt;&amp;amp;1
&lt;span class="k"&gt;else
    &lt;/span&gt;&lt;span class="nb"&gt;echo&lt;/span&gt; &lt;span class="s2"&gt;"Unknown container type: &lt;/span&gt;&lt;span class="nv"&gt;$container_type&lt;/span&gt;&lt;span class="s2"&gt;. Use 'model' or 'base'."&lt;/span&gt;
    &lt;span class="nb"&gt;exit &lt;/span&gt;1
&lt;span class="k"&gt;fi

&lt;/span&gt;&lt;span class="nb"&gt;echo&lt;/span&gt; &lt;span class="s2"&gt;"Build log saved to &lt;/span&gt;&lt;span class="nv"&gt;$log_file&lt;/span&gt;&lt;span class="s2"&gt;"&lt;/span&gt;

&lt;/code&gt;&lt;/pre&gt;

&lt;/div&gt;



&lt;p&gt;To run this script:&lt;br&gt;
&lt;/p&gt;

&lt;div class="highlight js-code-highlight"&gt;
&lt;pre class="highlight shell"&gt;&lt;code&gt;&lt;span class="c"&gt;# To give it permission&lt;/span&gt;
&lt;span class="nb"&gt;chmod&lt;/span&gt; +x ./bash/docker_build.sh 
&lt;span class="c"&gt;# To build the base image &lt;/span&gt;
./bash/docker_build.sh base
&lt;span class="c"&gt;# To build the model image &lt;/span&gt;
./bash/docker_build.sh model
&lt;/code&gt;&lt;/pre&gt;

&lt;/div&gt;



&lt;h2&gt;
  
  
  7. Managing R Dependencies with renv &lt;a&gt;&lt;/a&gt;
&lt;/h2&gt;

&lt;p&gt;Using renv ensures that your container installs the exact versions of R packages required by your project. To set this up locally:&lt;br&gt;
&lt;/p&gt;

&lt;div class="highlight js-code-highlight"&gt;
&lt;pre class="highlight plaintext"&gt;&lt;code&gt;# 1. Initialise renv:
renv::init()
# 2. Install any packages you use (eg. plumber, randomForest, logger, tidyverse). 
# 3. Snapshot the environment:
renv::snapshot()
&lt;/code&gt;&lt;/pre&gt;

&lt;/div&gt;



&lt;h2&gt;
  
  
  8. Testing the API &lt;a&gt;&lt;/a&gt;
&lt;/h2&gt;

&lt;h3&gt;
  
  
  Running a Container of the Built Image
&lt;/h3&gt;

&lt;p&gt;Normally, to run a container, this is all you need to run:&lt;br&gt;
&lt;/p&gt;

&lt;div class="highlight js-code-highlight"&gt;
&lt;pre class="highlight shell"&gt;&lt;code&gt;docker run &amp;lt;your_container_image&amp;gt;
&lt;/code&gt;&lt;/pre&gt;

&lt;/div&gt;



&lt;p&gt;Let's say our container image name is called &lt;code&gt;my-r-sagemaker-model&lt;/code&gt;.&lt;/p&gt;

&lt;p&gt;But since we want to expose the container's API specifically to port 8080, we need to explicitly map the container's port 8080 to port 8080 on our host machine.&lt;br&gt;
&lt;/p&gt;

&lt;div class="highlight js-code-highlight"&gt;
&lt;pre class="highlight plaintext"&gt;&lt;code&gt;docker run -p 8080:8080 my-r-sagemaker-model
&lt;/code&gt;&lt;/pre&gt;

&lt;/div&gt;



&lt;p&gt;Without the &lt;code&gt;-p 8080:8080 flag&lt;/code&gt;, the container will still run, but Docker does not automatically expose any ports to your host. This means that even though your API is running inside the container on port 8080, you wouldn’t be able to connect to it from your host machine.&lt;/p&gt;

&lt;p&gt;Finally, since we also want to mount the &lt;code&gt;app/logs/&lt;/code&gt; folder within the container with our external log folder (&lt;code&gt;/logs/inside_container_logs&lt;/code&gt;):&lt;br&gt;
&lt;/p&gt;

&lt;div class="highlight js-code-highlight"&gt;
&lt;pre class="highlight plaintext"&gt;&lt;code&gt;docker run -p 8080:8080 -v "$(pwd)/logs/inside_container_logs:/app/logs" my-r-sagemaker-model
&lt;/code&gt;&lt;/pre&gt;

&lt;/div&gt;



&lt;h3&gt;
  
  
  Testing the Container Using Curl
&lt;/h3&gt;



&lt;div class="highlight js-code-highlight"&gt;
&lt;pre class="highlight plaintext"&gt;&lt;code&gt;curl -X POST "http://localhost:8080/predict" \
     -H "Content-Type: application/json" \
     -d '{"Sepal.Length": 5.1, "Sepal.Width": 3.5, "Petal.Length": 1.4, "Petal.Width": 0.2}'
&lt;/code&gt;&lt;/pre&gt;

&lt;/div&gt;



&lt;p&gt;You should be getting this result:&lt;br&gt;
&lt;/p&gt;

&lt;div class="highlight js-code-highlight"&gt;
&lt;pre class="highlight plaintext"&gt;&lt;code&gt;Prediction: setosa
&lt;/code&gt;&lt;/pre&gt;

&lt;/div&gt;



&lt;h3&gt;
  
  
  Testing the Container Using Postman
&lt;/h3&gt;

&lt;ol&gt;
&lt;li&gt;Set the HTTP method to POST.&lt;/li&gt;
&lt;li&gt;Enter the URL: &lt;a href="http://localhost:8080/predict" rel="noopener noreferrer"&gt;http://localhost:8080/predict&lt;/a&gt;.&lt;/li&gt;
&lt;li&gt;Under the Headers tab, add &lt;code&gt;Content-Type&lt;/code&gt; for &lt;strong&gt;Key&lt;/strong&gt; with value &lt;code&gt;application/json&lt;/code&gt;.&lt;/li&gt;
&lt;li&gt;In the Body tab, select raw and choose JSON from the dropdown.&lt;/li&gt;
&lt;li&gt;Paste the following JSON:
&lt;/li&gt;
&lt;/ol&gt;

&lt;div class="highlight js-code-highlight"&gt;
&lt;pre class="highlight json"&gt;&lt;code&gt;&lt;span class="w"&gt;   &lt;/span&gt;&lt;span class="p"&gt;{&lt;/span&gt;&lt;span class="w"&gt;
     &lt;/span&gt;&lt;span class="nl"&gt;"Sepal.Length"&lt;/span&gt;&lt;span class="p"&gt;:&lt;/span&gt;&lt;span class="w"&gt; &lt;/span&gt;&lt;span class="mf"&gt;5.1&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt;&lt;span class="w"&gt;
     &lt;/span&gt;&lt;span class="nl"&gt;"Sepal.Width"&lt;/span&gt;&lt;span class="p"&gt;:&lt;/span&gt;&lt;span class="w"&gt; &lt;/span&gt;&lt;span class="mf"&gt;3.5&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt;&lt;span class="w"&gt;
     &lt;/span&gt;&lt;span class="nl"&gt;"Petal.Length"&lt;/span&gt;&lt;span class="p"&gt;:&lt;/span&gt;&lt;span class="w"&gt; &lt;/span&gt;&lt;span class="mf"&gt;1.4&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt;&lt;span class="w"&gt;
     &lt;/span&gt;&lt;span class="nl"&gt;"Petal.Width"&lt;/span&gt;&lt;span class="p"&gt;:&lt;/span&gt;&lt;span class="w"&gt; &lt;/span&gt;&lt;span class="mf"&gt;0.2&lt;/span&gt;&lt;span class="w"&gt;
   &lt;/span&gt;&lt;span class="p"&gt;}&lt;/span&gt;&lt;span class="w"&gt;
&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;

&lt;/div&gt;



&lt;ol&gt;
&lt;li&gt;Click Send to view the response.&lt;/li&gt;
&lt;/ol&gt;

&lt;h3&gt;
  
  
  Testing the Container Using R’s httr Package
&lt;/h3&gt;



&lt;div class="highlight js-code-highlight"&gt;
&lt;pre class="highlight r"&gt;&lt;code&gt;&lt;span class="n"&gt;library&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="n"&gt;httr&lt;/span&gt;&lt;span class="p"&gt;)&lt;/span&gt;&lt;span class="w"&gt;
&lt;/span&gt;&lt;span class="n"&gt;library&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="n"&gt;jsonlite&lt;/span&gt;&lt;span class="p"&gt;)&lt;/span&gt;&lt;span class="w"&gt;

&lt;/span&gt;&lt;span class="n"&gt;url&lt;/span&gt;&lt;span class="w"&gt; &lt;/span&gt;&lt;span class="o"&gt;&amp;lt;-&lt;/span&gt;&lt;span class="w"&gt; &lt;/span&gt;&lt;span class="s2"&gt;"http://localhost:8080/predict"&lt;/span&gt;&lt;span class="w"&gt;
&lt;/span&gt;&lt;span class="n"&gt;input&lt;/span&gt;&lt;span class="w"&gt; &lt;/span&gt;&lt;span class="o"&gt;&amp;lt;-&lt;/span&gt;&lt;span class="w"&gt; &lt;/span&gt;&lt;span class="nf"&gt;list&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="w"&gt;
  &lt;/span&gt;&lt;span class="n"&gt;Sepal.Length&lt;/span&gt;&lt;span class="w"&gt; &lt;/span&gt;&lt;span class="o"&gt;=&lt;/span&gt;&lt;span class="w"&gt; &lt;/span&gt;&lt;span class="m"&gt;5.1&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt;&lt;span class="w"&gt;
  &lt;/span&gt;&lt;span class="n"&gt;Sepal.Width&lt;/span&gt;&lt;span class="w"&gt;  &lt;/span&gt;&lt;span class="o"&gt;=&lt;/span&gt;&lt;span class="w"&gt; &lt;/span&gt;&lt;span class="m"&gt;3.5&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt;&lt;span class="w"&gt;
  &lt;/span&gt;&lt;span class="n"&gt;Petal.Length&lt;/span&gt;&lt;span class="w"&gt; &lt;/span&gt;&lt;span class="o"&gt;=&lt;/span&gt;&lt;span class="w"&gt; &lt;/span&gt;&lt;span class="m"&gt;1.4&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt;&lt;span class="w"&gt;
  &lt;/span&gt;&lt;span class="n"&gt;Petal.Width&lt;/span&gt;&lt;span class="w"&gt;  &lt;/span&gt;&lt;span class="o"&gt;=&lt;/span&gt;&lt;span class="w"&gt; &lt;/span&gt;&lt;span class="m"&gt;0.2&lt;/span&gt;&lt;span class="w"&gt;
&lt;/span&gt;&lt;span class="p"&gt;)&lt;/span&gt;&lt;span class="w"&gt;
&lt;/span&gt;&lt;span class="n"&gt;response&lt;/span&gt;&lt;span class="w"&gt; &lt;/span&gt;&lt;span class="o"&gt;&amp;lt;-&lt;/span&gt;&lt;span class="w"&gt; &lt;/span&gt;&lt;span class="n"&gt;POST&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="w"&gt;
  &lt;/span&gt;&lt;span class="n"&gt;url&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt;&lt;span class="w"&gt;
  &lt;/span&gt;&lt;span class="n"&gt;body&lt;/span&gt;&lt;span class="w"&gt; &lt;/span&gt;&lt;span class="o"&gt;=&lt;/span&gt;&lt;span class="w"&gt; &lt;/span&gt;&lt;span class="n"&gt;toJSON&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="n"&gt;input&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt;&lt;span class="w"&gt; &lt;/span&gt;&lt;span class="n"&gt;auto_unbox&lt;/span&gt;&lt;span class="w"&gt; &lt;/span&gt;&lt;span class="o"&gt;=&lt;/span&gt;&lt;span class="w"&gt; &lt;/span&gt;&lt;span class="kc"&gt;TRUE&lt;/span&gt;&lt;span class="p"&gt;),&lt;/span&gt;&lt;span class="w"&gt;
  &lt;/span&gt;&lt;span class="n"&gt;encode&lt;/span&gt;&lt;span class="w"&gt; &lt;/span&gt;&lt;span class="o"&gt;=&lt;/span&gt;&lt;span class="w"&gt; &lt;/span&gt;&lt;span class="s2"&gt;"raw"&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt;&lt;span class="w"&gt;
  &lt;/span&gt;&lt;span class="n"&gt;add_headers&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="s2"&gt;"Content-Type"&lt;/span&gt;&lt;span class="w"&gt; &lt;/span&gt;&lt;span class="o"&gt;=&lt;/span&gt;&lt;span class="w"&gt; &lt;/span&gt;&lt;span class="s2"&gt;"application/json"&lt;/span&gt;&lt;span class="p"&gt;)&lt;/span&gt;&lt;span class="w"&gt;
&lt;/span&gt;&lt;span class="p"&gt;)&lt;/span&gt;&lt;span class="w"&gt;
&lt;/span&gt;&lt;span class="n"&gt;result&lt;/span&gt;&lt;span class="w"&gt; &lt;/span&gt;&lt;span class="o"&gt;&amp;lt;-&lt;/span&gt;&lt;span class="w"&gt; &lt;/span&gt;&lt;span class="n"&gt;content&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="n"&gt;response&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt;&lt;span class="w"&gt; &lt;/span&gt;&lt;span class="n"&gt;as&lt;/span&gt;&lt;span class="w"&gt; &lt;/span&gt;&lt;span class="o"&gt;=&lt;/span&gt;&lt;span class="w"&gt; &lt;/span&gt;&lt;span class="s2"&gt;"parsed"&lt;/span&gt;&lt;span class="p"&gt;)&lt;/span&gt;&lt;span class="w"&gt;
&lt;/span&gt;&lt;span class="n"&gt;print&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="n"&gt;result&lt;/span&gt;&lt;span class="p"&gt;)&lt;/span&gt;&lt;span class="w"&gt;
&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;

&lt;/div&gt;



&lt;h2&gt;
  
  
  9. Troubleshooting &lt;a&gt;&lt;/a&gt;
&lt;/h2&gt;

&lt;h3&gt;
  
  
  Container fails to install a package
&lt;/h3&gt;

&lt;p&gt;If a package fails to install (e.g., textshaping or ragg), check the error message for missing system libraries. Update your Dockerfile.base to install the required development packages (such as libharfbuzz-dev, libfribidi-dev, libpng-dev, libtiff5-dev, and libjpeg-dev).&lt;br&gt;
If you don't have the logs setup, you can check by running the container, then call the library to see if it's installed, and try to reinstall it inside the container to see if it's having any issues:&lt;/p&gt;

&lt;p&gt;Run the container interactively:&lt;br&gt;
&lt;/p&gt;

&lt;div class="highlight js-code-highlight"&gt;
&lt;pre class="highlight plaintext"&gt;&lt;code&gt;docker run -it my-r-sagemaker-model R
&lt;/code&gt;&lt;/pre&gt;

&lt;/div&gt;



&lt;p&gt;Then you will be in the R console. Here you can check if something is installed (Eg. plumber):&lt;br&gt;
&lt;/p&gt;

&lt;div class="highlight js-code-highlight"&gt;
&lt;pre class="highlight plaintext"&gt;&lt;code&gt;library(plumber)
&lt;/code&gt;&lt;/pre&gt;

&lt;/div&gt;



&lt;p&gt;If the package wasn't installed correctly, it will say there's no package. Here you can try to reinstall it to see what error comes out, if it's an issue with missing system requirements, it should tell you what is missing, then you just need to that package's installation to the &lt;code&gt;Dockerfile.base&lt;/code&gt; script. If the package loads without error, the issue is resolved. To exit, type:&lt;br&gt;
&lt;/p&gt;

&lt;div class="highlight js-code-highlight"&gt;
&lt;pre class="highlight plaintext"&gt;&lt;code&gt;q()
&lt;/code&gt;&lt;/pre&gt;

&lt;/div&gt;



&lt;p&gt;and then press &lt;code&gt;n&lt;/code&gt; when prompted to save the workspace.&lt;/p&gt;

&lt;h3&gt;
  
  
  Docker Volume Mount
&lt;/h3&gt;

&lt;p&gt;When running your container, use the Docker -v (or --volume) flag to map the host directory to the container directory. For example, if your host directory is in the current working directory, you could run:&lt;br&gt;
&lt;/p&gt;

&lt;div class="highlight js-code-highlight"&gt;
&lt;pre class="highlight plaintext"&gt;&lt;code&gt;docker run -v "$(pwd)/logs/inside_container_logs:/app/logs" your_container_image
&lt;/code&gt;&lt;/pre&gt;

&lt;/div&gt;



&lt;p&gt;This command tells Docker to mount your host’s &lt;code&gt;logs/inside_container_logs&lt;/code&gt; directory into the container at &lt;code&gt;/app/logs&lt;/code&gt;.&lt;br&gt;
After running the container, use &lt;code&gt;docker container inspect &amp;lt;container-id&amp;gt;&lt;/code&gt; to verify that the mount is correctly set, and check the local folder to ensure that logs are being written there. &lt;br&gt;
Working on Windows brought its own challenges. When mounting a host directory to capture logs, my Docker run command was appending extra characters like ;C to the path. For example, instead of a clean path, I saw:&lt;br&gt;
&lt;/p&gt;

&lt;div class="highlight js-code-highlight"&gt;
&lt;pre class="highlight plaintext"&gt;&lt;code&gt;"Mounts": [
            {
                "Type": "bind",
                "Source": "C:\\Users\\Vince\\repos\\AWS-projects\\deploy-R-model-to-AWS-ECR\\logs\\inside_container_logs;C",
                "Destination": "\\Program Files\\Git\\app\\logs",
                "Mode": "",
                "RW": true,
                "Propagation": "rprivate"
            }
&lt;/code&gt;&lt;/pre&gt;

&lt;/div&gt;



&lt;p&gt;This was due to Git Bash’s interpretation of Windows paths, so it's recommended to use another terminal/powershell for this. &lt;/p&gt;

&lt;p&gt;Any logs written to &lt;code&gt;/app/logs&lt;/code&gt; by your R scripts (e.g., &lt;code&gt;logs/R_logs/training.log&lt;/code&gt; and &lt;code&gt;logs/R_logs/requests.log&lt;/code&gt;) will now be saved in your local logs folder (&lt;code&gt;logs/inside_container_logs/R_logs&lt;/code&gt;), allowing you to access them even after the container stops.&lt;/p&gt;

&lt;h3&gt;
  
  
  Interact with Running Container
&lt;/h3&gt;

&lt;p&gt;If you want to interact directly with a running container, for instance, you need to inspect logs directly within a running container without mounting a volume, you can use the docker exec command:&lt;br&gt;
&lt;/p&gt;

&lt;div class="highlight js-code-highlight"&gt;
&lt;pre class="highlight shell"&gt;&lt;code&gt;docker &lt;span class="nb"&gt;exec&lt;/span&gt; &lt;span class="nt"&gt;-it&lt;/span&gt; &amp;lt;container-id&amp;gt; bash
&lt;span class="nb"&gt;ls&lt;/span&gt; /app/logs
&lt;/code&gt;&lt;/pre&gt;

&lt;/div&gt;



&lt;p&gt;For example, I first call &lt;code&gt;docker ps&lt;/code&gt; to get the &lt;code&gt;container-id&lt;/code&gt;, then run the above &lt;code&gt;docker exec&lt;/code&gt; command to enter the container, where you can do various things with it, such as inspecting the logs:&lt;/p&gt;

&lt;p&gt;For example:&lt;br&gt;
&lt;/p&gt;

&lt;div class="highlight js-code-highlight"&gt;
&lt;pre class="highlight shell"&gt;&lt;code&gt;Vince@tanjiro &lt;span class="o"&gt;[&lt;/span&gt;~/.../AWS-projects/deploy-R-model-to-AWS-ECR] ± master ✓           &lt;span class="o"&gt;[&lt;/span&gt;12:35:08]&amp;gt; docker ps
CONTAINER ID   IMAGE                         COMMAND                  CREATED       STATUS
     PORTS                    NAMES
04221d808cc1   my-r-sagemaker-model:latest   &lt;span class="s2"&gt;"R -e 'pr &amp;lt;- plumber…"&lt;/span&gt;   4 hours ago   Up 4 hours   0.0.0.0:8080-&amp;gt;8080/tcp   distracted_nobel
Vince@tanjiro &lt;span class="o"&gt;[&lt;/span&gt;~/.../AWS-projects/deploy-R-model-to-AWS-ECR] ± master ?:1 ✗       &lt;span class="o"&gt;[&lt;/span&gt;16:31:35]1 &lt;span class="o"&gt;&amp;gt;&lt;/span&gt; docker &lt;span class="nb"&gt;exec&lt;/span&gt; &lt;span class="nt"&gt;-it&lt;/span&gt; 04221d808cc1 bash
root@04221d808cc1:/app# &lt;span class="nb"&gt;ls
&lt;/span&gt;data  logs  renv.lock  serve.R  train.R
root@04221d808cc1:/app# &lt;span class="nb"&gt;cd &lt;/span&gt;logs/
root@04221d808cc1:/app/logs# &lt;span class="nb"&gt;ls
&lt;/span&gt;R_logs
root@04221d808cc1:/app/logs# &lt;span class="nb"&gt;cd &lt;/span&gt;R_logs/
root@04221d808cc1:/app/logs/R_logs# &lt;span class="nb"&gt;ls
&lt;/span&gt;20250310_012557_request.log
root@04221d808cc1:/app/logs/R_logs#
&lt;span class="nb"&gt;exit

&lt;/span&gt;What&lt;span class="s1"&gt;'s next:
    Try Docker Debug for seamless, persistent debugging tools in any container or image → docker debug 04221d808cc1
    Learn more at https://docs.docker.com/go/debug-cli/
&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;

&lt;/div&gt;



</description>
    </item>
    <item>
      <title>A cheat sheet on Pandas DataFrame Attributes</title>
      <dc:creator>vindianadoan</dc:creator>
      <pubDate>Sat, 18 Jan 2025 16:04:54 +0000</pubDate>
      <link>https://dev.to/vindianadoan/a-cheat-sheet-on-pandas-dataframe-attributes-297h</link>
      <guid>https://dev.to/vindianadoan/a-cheat-sheet-on-pandas-dataframe-attributes-297h</guid>
      <description>&lt;p&gt;This post summarises a range of commonly used attributes of a pandas DataFrame object. &lt;br&gt;
Using this DataFrame as an example:&lt;br&gt;
&lt;/p&gt;

&lt;div class="highlight js-code-highlight"&gt;
&lt;pre class="highlight plaintext"&gt;&lt;code&gt;import pandas as pd 
df = pd.DataFrame({
    'name': ["A", "B"],
    'value': [1,2]
})
&lt;/code&gt;&lt;/pre&gt;

&lt;/div&gt;



&lt;h1&gt;
  
  
  1. Core Attributes
&lt;/h1&gt;

&lt;div class="table-wrapper-paragraph"&gt;&lt;table&gt;
&lt;thead&gt;
&lt;tr&gt;
&lt;th&gt;Attribute&lt;/th&gt;
&lt;th&gt;Description&lt;/th&gt;
&lt;th&gt;Output&lt;/th&gt;
&lt;/tr&gt;
&lt;/thead&gt;
&lt;tbody&gt;
&lt;tr&gt;
&lt;td&gt;&lt;code&gt;.shape&lt;/code&gt;&lt;/td&gt;
&lt;td&gt;Returns a tuple representing the dimensionality of the DataFrame in the form &lt;code&gt;(number_of_rows, number_of_columns)&lt;/code&gt;.&lt;/td&gt;
&lt;td&gt;
&lt;code&gt;print(df.shape)&lt;/code&gt;&lt;br&gt;&lt;code&gt;# Output: (2,2)&lt;/code&gt;
&lt;/td&gt;
&lt;/tr&gt;
&lt;tr&gt;
&lt;td&gt;&lt;code&gt;.columns&lt;/code&gt;&lt;/td&gt;
&lt;td&gt;Returns an &lt;code&gt;Index&lt;/code&gt; object containing the column labels of the DataFrame&lt;/td&gt;
&lt;td&gt;
&lt;code&gt;print(df.columns)&lt;/code&gt;&lt;br&gt;&lt;code&gt;# Output: Index(['name', 'value'], dtype='object')&lt;/code&gt;
&lt;/td&gt;
&lt;/tr&gt;
&lt;tr&gt;
&lt;td&gt;&lt;code&gt;.index&lt;/code&gt;&lt;/td&gt;
&lt;td&gt;Represents the row labels of the DataFrame, encapsulated in an &lt;code&gt;Index&lt;/code&gt; object.&lt;/td&gt;
&lt;td&gt;
&lt;code&gt;print(df.index)&lt;/code&gt;&lt;br&gt;&lt;code&gt;# Output: RangeIndex(start=0, stop=2, step=1)&lt;/code&gt;
&lt;/td&gt;
&lt;/tr&gt;
&lt;tr&gt;
&lt;td&gt;&lt;code&gt;.dtypes&lt;/code&gt;&lt;/td&gt;
&lt;td&gt;Provides the data type of each column in the DataFrame.&lt;/td&gt;
&lt;td&gt;
&lt;code&gt;print(df.dtypes)&lt;/code&gt; &lt;br&gt;&lt;code&gt;# Output&lt;/code&gt;&lt;br&gt;&lt;code&gt;name     object&lt;/code&gt;&lt;br&gt;&lt;code&gt;value     int64&lt;/code&gt;&lt;br&gt;&lt;code&gt;dtype: object&lt;/code&gt;
&lt;/td&gt;
&lt;/tr&gt;
&lt;/tbody&gt;
&lt;/table&gt;&lt;/div&gt;

&lt;h1&gt;
  
  
  2. Data Inspection Attributes
&lt;/h1&gt;

&lt;div class="table-wrapper-paragraph"&gt;&lt;table&gt;
&lt;thead&gt;
&lt;tr&gt;
&lt;th&gt;Attribute&lt;/th&gt;
&lt;th&gt;Description&lt;/th&gt;
&lt;th&gt;Output&lt;/th&gt;
&lt;/tr&gt;
&lt;/thead&gt;
&lt;tbody&gt;
&lt;tr&gt;
&lt;td&gt;&lt;code&gt;.values&lt;/code&gt;&lt;/td&gt;
&lt;td&gt;Returns a NumPy array representation of the DataFrame's data.&lt;/td&gt;
&lt;td&gt;
&lt;code&gt;print(df.values)&lt;/code&gt;&lt;br&gt;&lt;code&gt;# Output:&lt;/code&gt;&lt;br&gt;&lt;code&gt;[['A' 1]&lt;/code&gt;&lt;br&gt;&lt;code&gt;['B' 2]]&lt;/code&gt;
&lt;/td&gt;
&lt;/tr&gt;
&lt;tr&gt;
&lt;td&gt;&lt;code&gt;.size&lt;/code&gt;&lt;/td&gt;
&lt;td&gt;Returns the total number of elements in the DataFrame (calculated as rows x columns).&lt;/td&gt;
&lt;td&gt;
&lt;code&gt;print(df.size)&lt;/code&gt;&lt;br&gt;&lt;code&gt;# Output: 4&lt;/code&gt;
&lt;/td&gt;
&lt;/tr&gt;
&lt;tr&gt;
&lt;td&gt;&lt;code&gt;.ndim&lt;/code&gt;&lt;/td&gt;
&lt;td&gt;Returns the number of dimensions of the DataFrame. For DataFrames, this is always 2.&lt;/td&gt;
&lt;td&gt;
&lt;code&gt;print(df.ndim)&lt;/code&gt;&lt;br&gt;&lt;code&gt;# Output: 2&lt;/code&gt;
&lt;/td&gt;
&lt;/tr&gt;
&lt;tr&gt;
&lt;td&gt;&lt;code&gt;.empty&lt;/code&gt;&lt;/td&gt;
&lt;td&gt;Returns &lt;code&gt;True&lt;/code&gt; if the DataFrame is empty (ie. has no elements), otherwise &lt;code&gt;False&lt;/code&gt;.&lt;/td&gt;
&lt;td&gt;
&lt;code&gt;print(pd.DataFrame().empty)&lt;/code&gt;&lt;br&gt;&lt;code&gt;# Output: True&lt;/code&gt;
&lt;/td&gt;
&lt;/tr&gt;
&lt;/tbody&gt;
&lt;/table&gt;&lt;/div&gt;

&lt;h1&gt;
  
  
  3. Others
&lt;/h1&gt;

&lt;div class="table-wrapper-paragraph"&gt;&lt;table&gt;
&lt;thead&gt;
&lt;tr&gt;
&lt;th&gt;Attribute&lt;/th&gt;
&lt;th&gt;Description&lt;/th&gt;
&lt;th&gt;Output&lt;/th&gt;
&lt;/tr&gt;
&lt;/thead&gt;
&lt;tbody&gt;
&lt;tr&gt;
&lt;td&gt;
&lt;code&gt;.T&lt;/code&gt; (Transpose)&lt;/td&gt;
&lt;td&gt;Returns the transpose of the DataFrame, swapping rows with columns.&lt;/td&gt;
&lt;td&gt;
&lt;code&gt;print(df.values)&lt;/code&gt;&lt;br&gt;&lt;code&gt;name   A  B&lt;/code&gt;&lt;br&gt;&lt;code&gt;value  1  2&lt;/code&gt;
&lt;/td&gt;
&lt;/tr&gt;
&lt;/tbody&gt;
&lt;/table&gt;&lt;/div&gt;

</description>
    </item>
    <item>
      <title>ARIMA Modelling: Theory</title>
      <dc:creator>vindianadoan</dc:creator>
      <pubDate>Wed, 24 May 2023 19:13:12 +0000</pubDate>
      <link>https://dev.to/vindianadoan/arima-modelling-a-comprehensive-guide-1gm6</link>
      <guid>https://dev.to/vindianadoan/arima-modelling-a-comprehensive-guide-1gm6</guid>
      <description>&lt;h2&gt;
  
  
  Introduction
&lt;/h2&gt;

&lt;p&gt;ARIMA models, standing for AutoRegressive Integrated Moving Average, is a great model for predicting temporal time series data with a complex nature. Unlike basic regression models, which may struggle to capture complexity with just a few variables (usually due to autocorrelation, trend and seasonality), ARIMA excels in treating autocorrelation, trend and seasonality through a built-in differencing function (the "Integrated" part of ARIMA); thus, it removes the need for specifying explanatory variables and extra transformation. Examples of application include finance, weather predictions, and even anticipating website traffic. &lt;br&gt;
&lt;br&gt;&lt;/p&gt;
&lt;h2&gt;
  
  
  Define the model
&lt;/h2&gt;

&lt;p&gt;Let's build the ARIMA model from scratch. As discussed, the ARIMA model is made of three parts: an autoregressive part, an integrated part and a moving average part. &lt;/p&gt;

&lt;p&gt;For example, given some time series &lt;br&gt;


&lt;/p&gt;
&lt;div class="katex-element"&gt;
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&lt;/div&gt;
&lt;br&gt;
The autoRegressive (AR) component says that an observation 
&lt;span class="katex-element"&gt;
  &lt;span class="katex"&gt;&lt;span class="katex-mathml"&gt;(Xt)(X_t)&lt;/span&gt;&lt;span class="katex-html"&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mopen"&gt;(&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;X&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mathnormal mtight"&gt;t&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mclose"&gt;)&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/span&gt;
 at certain point in time 
&lt;span class="katex-element"&gt;
  &lt;span class="katex"&gt;&lt;span class="katex-mathml"&gt;tt&lt;/span&gt;&lt;span class="katex-html"&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord mathnormal"&gt;t&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/span&gt;
, can be described as a linear combination of its lagged observations (ie. prior time points): &lt;br&gt;

&lt;div class="katex-element"&gt;
  &lt;span class="katex-display"&gt;&lt;span class="katex"&gt;&lt;span class="katex-mathml"&gt;Xt=α1Xt−1+α2Xt−2...X_t = \alpha_1 X_{t-1} + \alpha_2 X_{t-2} ...&lt;/span&gt;&lt;span class="katex-html"&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;X&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mathnormal mtight"&gt;t&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mrel"&gt;=&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;α&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;1&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;X&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mathnormal mtight"&gt;t&lt;/span&gt;&lt;span class="mbin mtight"&gt;−&lt;/span&gt;&lt;span class="mord mtight"&gt;1&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mbin"&gt;+&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;α&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;2&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;X&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mathnormal mtight"&gt;t&lt;/span&gt;&lt;span class="mbin mtight"&gt;−&lt;/span&gt;&lt;span class="mord mtight"&gt;2&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mord"&gt;...&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/div&gt;
&lt;br&gt;
where 
&lt;span class="katex-element"&gt;
  &lt;span class="katex"&gt;&lt;span class="katex-mathml"&gt;αi\alpha_i&lt;/span&gt;&lt;span class="katex-html"&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;α&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mathnormal mtight"&gt;i&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/span&gt;
 are the parameters or coefficients of such regression. &lt;br&gt;
(Note: if you find the terms linear combination and linear regression confusing, a linear regression is model that outputs predictions based on a linear combination of input features). &lt;br&gt;
Using the lag operator (
&lt;span class="katex-element"&gt;
  &lt;span class="katex"&gt;&lt;span class="katex-mathml"&gt;LpXt=Xt−qL^pX_t = X_{t-q}&lt;/span&gt;&lt;span class="katex-html"&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;L&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mathnormal mtight"&gt;p&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;X&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mathnormal mtight"&gt;t&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mrel"&gt;=&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;X&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mathnormal mtight"&gt;t&lt;/span&gt;&lt;span class="mbin mtight"&gt;−&lt;/span&gt;&lt;span class="mord mathnormal mtight"&gt;q&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/span&gt;
), you can also express the above as: &lt;br&gt;

&lt;div class="katex-element"&gt;
  &lt;span class="katex-display"&gt;&lt;span class="katex"&gt;&lt;span class="katex-mathml"&gt;Xt=α1(Xt−1)+α2(Xt−2)...+αp(Xt−p)
X_t = \alpha_1 (X_{t-1}) + \alpha_2 (X_{t-2}) ... + \alpha_p (X_{t-p})
&lt;/span&gt;&lt;span class="katex-html"&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;X&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mathnormal mtight"&gt;t&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mrel"&gt;=&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;α&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;1&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mopen"&gt;(&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;X&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mathnormal mtight"&gt;t&lt;/span&gt;&lt;span class="mbin mtight"&gt;−&lt;/span&gt;&lt;span class="mord mtight"&gt;1&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mclose"&gt;)&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mbin"&gt;+&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;α&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;2&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mopen"&gt;(&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;X&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mathnormal mtight"&gt;t&lt;/span&gt;&lt;span class="mbin mtight"&gt;−&lt;/span&gt;&lt;span class="mord mtight"&gt;2&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mclose"&gt;)&lt;/span&gt;&lt;span class="mord"&gt;...&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mbin"&gt;+&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;α&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mathnormal mtight"&gt;p&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mopen"&gt;(&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;X&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mathnormal mtight"&gt;t&lt;/span&gt;&lt;span class="mbin mtight"&gt;−&lt;/span&gt;&lt;span class="mord mathnormal mtight"&gt;p&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mclose"&gt;)&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/div&gt;
&lt;br&gt;

&lt;div class="katex-element"&gt;
  &lt;span class="katex-display"&gt;&lt;span class="katex"&gt;&lt;span class="katex-mathml"&gt;Xt=α1LXt+α2L2Xt...+αpLpXt
X_t = \alpha_1 LX_{t} + \alpha_2 L^2X_{t} ... + \alpha_p L^pX_{t}
&lt;/span&gt;&lt;span class="katex-html"&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;X&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mathnormal mtight"&gt;t&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mrel"&gt;=&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;α&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;1&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mord mathnormal"&gt;L&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;X&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mathnormal mtight"&gt;t&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mbin"&gt;+&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;α&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;2&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;L&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;2&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;X&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mathnormal mtight"&gt;t&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mord"&gt;...&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mbin"&gt;+&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;α&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mathnormal mtight"&gt;p&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;L&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mathnormal mtight"&gt;p&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;X&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mathnormal mtight"&gt;t&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/div&gt;
&lt;br&gt;

&lt;div class="katex-element"&gt;
  &lt;span class="katex-display"&gt;&lt;span class="katex"&gt;&lt;span class="katex-mathml"&gt;Xt=(∑i=1pαiLi)Xt(1)
X_t = (\sum_{i=1}^{p} \alpha_i L^i)X_{t} \hspace{1cm} (1)
&lt;/span&gt;&lt;span class="katex-html"&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;X&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mathnormal mtight"&gt;t&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mrel"&gt;=&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mopen"&gt;(&lt;/span&gt;&lt;span class="mop op-limits"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mathnormal mtight"&gt;i&lt;/span&gt;&lt;span class="mrel mtight"&gt;=&lt;/span&gt;&lt;span class="mord mtight"&gt;1&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="mop op-symbol large-op"&gt;∑&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mathnormal mtight"&gt;p&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;α&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mathnormal mtight"&gt;i&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;L&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mathnormal mtight"&gt;i&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mclose"&gt;)&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;X&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mathnormal mtight"&gt;t&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mopen"&gt;(&lt;/span&gt;&lt;span class="mord"&gt;1&lt;/span&gt;&lt;span class="mclose"&gt;)&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/div&gt;
&lt;br&gt;
An optimisation of the AR part for the ARIMA model is an optimisation of the number of previous terms - or the number of time lags to be included in the linear combination. We call 
&lt;span class="katex-element"&gt;
  &lt;span class="katex"&gt;&lt;span class="katex-mathml"&gt;pp&lt;/span&gt;&lt;span class="katex-html"&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord mathnormal"&gt;p&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/span&gt;
 the order of the autoregressive model. &lt;br&gt;
For example, when we say the AR part has an order of two, we symbolise it as 
&lt;span class="katex-element"&gt;
  &lt;span class="katex"&gt;&lt;span class="katex-mathml"&gt;AR(2)AR(2)&lt;/span&gt;&lt;span class="katex-html"&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord mathnormal"&gt;A&lt;/span&gt;&lt;span class="mord mathnormal"&gt;R&lt;/span&gt;&lt;span class="mopen"&gt;(&lt;/span&gt;&lt;span class="mord"&gt;2&lt;/span&gt;&lt;span class="mclose"&gt;)&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/span&gt;
, and express it as: &lt;br&gt;

&lt;div class="katex-element"&gt;
  &lt;span class="katex-display"&gt;&lt;span class="katex"&gt;&lt;span class="katex-mathml"&gt;Xt=α1LXt+α2L2XtX_t = \alpha_1 LX_{t} + \alpha_2 L^2X_{t}&lt;/span&gt;&lt;span class="katex-html"&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;X&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mathnormal mtight"&gt;t&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mrel"&gt;=&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;α&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;1&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mord mathnormal"&gt;L&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;X&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mathnormal mtight"&gt;t&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mbin"&gt;+&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;α&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;2&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;L&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;2&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;X&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mathnormal mtight"&gt;t&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/div&gt;


&lt;p&gt;Now, equivalently, 
&lt;span class="katex-element"&gt;
  &lt;span class="katex"&gt;&lt;span class="katex-mathml"&gt;(Xt)(X_t)&lt;/span&gt;&lt;span class="katex-html"&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mopen"&gt;(&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;X&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mathnormal mtight"&gt;t&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mclose"&gt;)&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/span&gt;
 can also be expressed as a combination of previous error terms. This is described in the Moving Average part of the ARIMA model:&lt;br&gt;

&lt;/p&gt;
&lt;div class="katex-element"&gt;
  &lt;span class="katex-display"&gt;&lt;span class="katex"&gt;&lt;span class="katex-mathml"&gt;Xt=ϵt+θ1ϵt−1+θ2ϵt−2+...+θqϵt−q
X_t = \epsilon_t + \theta_1\epsilon_{t-1} + \theta_2\epsilon_{t-2} + ... + \theta_q \epsilon_{t-q}
&lt;/span&gt;&lt;span class="katex-html"&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;X&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mathnormal mtight"&gt;t&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mrel"&gt;=&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;ϵ&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mathnormal mtight"&gt;t&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mbin"&gt;+&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;θ&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;1&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;ϵ&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mathnormal mtight"&gt;t&lt;/span&gt;&lt;span class="mbin mtight"&gt;−&lt;/span&gt;&lt;span class="mord mtight"&gt;1&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mbin"&gt;+&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;θ&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;2&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;ϵ&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mathnormal mtight"&gt;t&lt;/span&gt;&lt;span class="mbin mtight"&gt;−&lt;/span&gt;&lt;span class="mord mtight"&gt;2&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mbin"&gt;+&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;...&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mbin"&gt;+&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;θ&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mathnormal mtight"&gt;q&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;ϵ&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mathnormal mtight"&gt;t&lt;/span&gt;&lt;span class="mbin mtight"&gt;−&lt;/span&gt;&lt;span class="mord mathnormal mtight"&gt;q&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/div&gt;



&lt;div class="katex-element"&gt;
  &lt;span class="katex-display"&gt;&lt;span class="katex"&gt;&lt;span class="katex-mathml"&gt;Xt=ϵt+θ1Lϵt+θ2L2ϵt+...+θqLqϵt
X_t = \epsilon_t + \theta_1 L \epsilon_{t} + \theta_2 L^2\epsilon_{t} + ... + \theta_q L^q \epsilon_{t}
&lt;/span&gt;&lt;span class="katex-html"&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;X&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mathnormal mtight"&gt;t&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mrel"&gt;=&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;ϵ&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mathnormal mtight"&gt;t&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mbin"&gt;+&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;θ&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;1&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mord mathnormal"&gt;L&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;ϵ&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mathnormal mtight"&gt;t&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mbin"&gt;+&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;θ&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;2&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;L&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;2&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;ϵ&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mathnormal mtight"&gt;t&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mbin"&gt;+&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;...&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mbin"&gt;+&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;θ&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mathnormal mtight"&gt;q&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;L&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mathnormal mtight"&gt;q&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;ϵ&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mathnormal mtight"&gt;t&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/div&gt;



&lt;div class="katex-element"&gt;
  &lt;span class="katex-display"&gt;&lt;span class="katex"&gt;&lt;span class="katex-mathml"&gt;Xt=ϵt+(∑i=1qθqLq)ϵt(2)
X_t = \epsilon_t + (\sum_{i=1}^{q} \theta_q L^q)\epsilon_{t} \hspace{1cm} (2)
&lt;/span&gt;&lt;span class="katex-html"&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;X&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mathnormal mtight"&gt;t&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mrel"&gt;=&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;ϵ&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mathnormal mtight"&gt;t&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mbin"&gt;+&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mopen"&gt;(&lt;/span&gt;&lt;span class="mop op-limits"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mathnormal mtight"&gt;i&lt;/span&gt;&lt;span class="mrel mtight"&gt;=&lt;/span&gt;&lt;span class="mord mtight"&gt;1&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="mop op-symbol large-op"&gt;∑&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mathnormal mtight"&gt;q&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;θ&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mathnormal mtight"&gt;q&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;L&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mathnormal mtight"&gt;q&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mclose"&gt;)&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;ϵ&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mathnormal mtight"&gt;t&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mopen"&gt;(&lt;/span&gt;&lt;span class="mord"&gt;2&lt;/span&gt;&lt;span class="mclose"&gt;)&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/div&gt;
&lt;br&gt;
where:&lt;br&gt;

&lt;span class="katex-element"&gt;
  &lt;span class="katex"&gt;&lt;span class="katex-mathml"&gt;θi\theta_i&lt;/span&gt;&lt;span class="katex-html"&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;θ&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mathnormal mtight"&gt;i&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/span&gt;
 are the parameters/ coefficients of the linear combination of the error terms&lt;br&gt;

&lt;span class="katex-element"&gt;
  &lt;span class="katex"&gt;&lt;span class="katex-mathml"&gt;ϵt\epsilon_t&lt;/span&gt;&lt;span class="katex-html"&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;ϵ&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mathnormal mtight"&gt;t&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/span&gt;
 are the error terms at time 
&lt;span class="katex-element"&gt;
  &lt;span class="katex"&gt;&lt;span class="katex-mathml"&gt;tt&lt;/span&gt;&lt;span class="katex-html"&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord mathnormal"&gt;t&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/span&gt;
&lt;br&gt;

&lt;span class="katex-element"&gt;
  &lt;span class="katex"&gt;&lt;span class="katex-mathml"&gt;qq&lt;/span&gt;&lt;span class="katex-html"&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord mathnormal"&gt;q&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/span&gt;
 is the order of this moving average. &lt;br&gt;
Note that the concept of moving average here can be misleading, as moving average in this context simply refers to the moving window of the previous errors, it does not take an average of anything. 

&lt;p&gt;Since both (1) and (2) describe 
&lt;span class="katex-element"&gt;
  &lt;span class="katex"&gt;&lt;span class="katex-mathml"&gt;XtX_t&lt;/span&gt;&lt;span class="katex-html"&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;X&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mathnormal mtight"&gt;t&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/span&gt;
, we can add them together: &lt;br&gt;

&lt;/p&gt;
&lt;div class="katex-element"&gt;
  &lt;span class="katex-display"&gt;&lt;span class="katex"&gt;&lt;span class="katex-mathml"&gt;Xt=(∑i=1pαiLi)Xt+ϵt+(∑i=1qθqLq)ϵt
X_t = (\sum_{i=1}^{p} \alpha_i L^i)X_{t} + \epsilon_t + (\sum_{i=1}^{q} \theta_q L^q)\epsilon_{t}
&lt;/span&gt;&lt;span class="katex-html"&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;X&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mathnormal mtight"&gt;t&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mrel"&gt;=&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mopen"&gt;(&lt;/span&gt;&lt;span class="mop op-limits"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mathnormal mtight"&gt;i&lt;/span&gt;&lt;span class="mrel mtight"&gt;=&lt;/span&gt;&lt;span class="mord mtight"&gt;1&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="mop op-symbol large-op"&gt;∑&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mathnormal mtight"&gt;p&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;α&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mathnormal mtight"&gt;i&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;L&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mathnormal mtight"&gt;i&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mclose"&gt;)&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;X&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mathnormal mtight"&gt;t&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mbin"&gt;+&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;ϵ&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mathnormal mtight"&gt;t&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mbin"&gt;+&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mopen"&gt;(&lt;/span&gt;&lt;span class="mop op-limits"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mathnormal mtight"&gt;i&lt;/span&gt;&lt;span class="mrel mtight"&gt;=&lt;/span&gt;&lt;span class="mord mtight"&gt;1&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="mop op-symbol large-op"&gt;∑&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mathnormal mtight"&gt;q&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;θ&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mathnormal mtight"&gt;q&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;L&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mathnormal mtight"&gt;q&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mclose"&gt;)&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;ϵ&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mathnormal mtight"&gt;t&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/div&gt;


&lt;p&gt;This is often expressed as: &lt;br&gt;

&lt;/p&gt;
&lt;div class="katex-element"&gt;
  &lt;span class="katex-display"&gt;&lt;span class="katex"&gt;&lt;span class="katex-mathml"&gt;Xt−(∑i=1pαiLi)Xt=ϵt+(∑i=1qθqLq)ϵt
X_t - (\sum_{i=1}^{p} \alpha_i L^i)X_{t} = \epsilon_t + (\sum_{i=1}^{q} \theta_q L^q)\epsilon_{t}
&lt;/span&gt;&lt;span class="katex-html"&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;X&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mathnormal mtight"&gt;t&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mbin"&gt;−&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mopen"&gt;(&lt;/span&gt;&lt;span class="mop op-limits"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mathnormal mtight"&gt;i&lt;/span&gt;&lt;span class="mrel mtight"&gt;=&lt;/span&gt;&lt;span class="mord mtight"&gt;1&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="mop op-symbol large-op"&gt;∑&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mathnormal mtight"&gt;p&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;α&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mathnormal mtight"&gt;i&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;L&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mathnormal mtight"&gt;i&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mclose"&gt;)&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;X&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mathnormal mtight"&gt;t&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mrel"&gt;=&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;ϵ&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mathnormal mtight"&gt;t&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mbin"&gt;+&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mopen"&gt;(&lt;/span&gt;&lt;span class="mop op-limits"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mathnormal mtight"&gt;i&lt;/span&gt;&lt;span class="mrel mtight"&gt;=&lt;/span&gt;&lt;span class="mord mtight"&gt;1&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="mop op-symbol large-op"&gt;∑&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mathnormal mtight"&gt;q&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;θ&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mathnormal mtight"&gt;q&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;L&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mathnormal mtight"&gt;q&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mclose"&gt;)&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;ϵ&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mathnormal mtight"&gt;t&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/div&gt;



&lt;div class="katex-element"&gt;
  &lt;span class="katex-display"&gt;&lt;span class="katex"&gt;&lt;span class="katex-mathml"&gt;(1−∑i=1pαiLi)Xt=(1+∑i=1qθqLq)ϵt
(1 - \sum_{i=1}^{p} \alpha_i L^i)X_{t} = (1 + \sum_{i=1}^{q} \theta_q L^q)\epsilon_{t}
&lt;/span&gt;&lt;span class="katex-html"&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mopen"&gt;(&lt;/span&gt;&lt;span class="mord"&gt;1&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mbin"&gt;−&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mop op-limits"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mathnormal mtight"&gt;i&lt;/span&gt;&lt;span class="mrel mtight"&gt;=&lt;/span&gt;&lt;span class="mord mtight"&gt;1&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="mop op-symbol large-op"&gt;∑&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mathnormal mtight"&gt;p&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;α&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mathnormal mtight"&gt;i&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;L&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mathnormal mtight"&gt;i&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mclose"&gt;)&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;X&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mathnormal mtight"&gt;t&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mrel"&gt;=&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mopen"&gt;(&lt;/span&gt;&lt;span class="mord"&gt;1&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mbin"&gt;+&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mop op-limits"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mathnormal mtight"&gt;i&lt;/span&gt;&lt;span class="mrel mtight"&gt;=&lt;/span&gt;&lt;span class="mord mtight"&gt;1&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="mop op-symbol large-op"&gt;∑&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mathnormal mtight"&gt;q&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;θ&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mathnormal mtight"&gt;q&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;L&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mathnormal mtight"&gt;q&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mclose"&gt;)&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;ϵ&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mathnormal mtight"&gt;t&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/div&gt;


&lt;p&gt;We say that, given time series data 
&lt;span class="katex-element"&gt;
  &lt;span class="katex"&gt;&lt;span class="katex-mathml"&gt;XtX_t&lt;/span&gt;&lt;span class="katex-html"&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;X&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mathnormal mtight"&gt;t&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/span&gt;
 where 
&lt;span class="katex-element"&gt;
  &lt;span class="katex"&gt;&lt;span class="katex-mathml"&gt;tt&lt;/span&gt;&lt;span class="katex-html"&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord mathnormal"&gt;t&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/span&gt;
 is an integer index and the 
&lt;span class="katex-element"&gt;
  &lt;span class="katex"&gt;&lt;span class="katex-mathml"&gt;XtX_t&lt;/span&gt;&lt;span class="katex-html"&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;X&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mathnormal mtight"&gt;t&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/span&gt;
 are real numbers, an 
&lt;span class="katex-element"&gt;
  &lt;span class="katex"&gt;&lt;span class="katex-mathml"&gt;ARIMA(p,q)ARIMA(p,q)&lt;/span&gt;&lt;span class="katex-html"&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord mathnormal"&gt;A&lt;/span&gt;&lt;span class="mord mathnormal"&gt;R&lt;/span&gt;&lt;span class="mord mathnormal"&gt;I&lt;/span&gt;&lt;span class="mord mathnormal"&gt;M&lt;/span&gt;&lt;span class="mord mathnormal"&gt;A&lt;/span&gt;&lt;span class="mopen"&gt;(&lt;/span&gt;&lt;span class="mord mathnormal"&gt;p&lt;/span&gt;&lt;span class="mpunct"&gt;,&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mord mathnormal"&gt;q&lt;/span&gt;&lt;span class="mclose"&gt;)&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/span&gt;
 model is given by: &lt;br&gt;

&lt;/p&gt;
&lt;div class="katex-element"&gt;
  &lt;span class="katex-display"&gt;&lt;span class="katex"&gt;&lt;span class="katex-mathml"&gt;(1−∑i=1pαiLi)Xt=(1+∑i=1qθqLq)ϵt
(1 - \sum_{i=1}^{p} \alpha_i L^i)X_{t} = (1 + \sum_{i=1}^{q} \theta_q L^q)\epsilon_{t}
&lt;/span&gt;&lt;span class="katex-html"&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mopen"&gt;(&lt;/span&gt;&lt;span class="mord"&gt;1&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mbin"&gt;−&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mop op-limits"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mathnormal mtight"&gt;i&lt;/span&gt;&lt;span class="mrel mtight"&gt;=&lt;/span&gt;&lt;span class="mord mtight"&gt;1&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="mop op-symbol large-op"&gt;∑&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mathnormal mtight"&gt;p&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;α&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mathnormal mtight"&gt;i&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;L&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mathnormal mtight"&gt;i&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mclose"&gt;)&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;X&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mathnormal mtight"&gt;t&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mrel"&gt;=&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mopen"&gt;(&lt;/span&gt;&lt;span class="mord"&gt;1&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mbin"&gt;+&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mop op-limits"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mathnormal mtight"&gt;i&lt;/span&gt;&lt;span class="mrel mtight"&gt;=&lt;/span&gt;&lt;span class="mord mtight"&gt;1&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="mop op-symbol large-op"&gt;∑&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mathnormal mtight"&gt;q&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;θ&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mathnormal mtight"&gt;q&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;L&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mathnormal mtight"&gt;q&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mclose"&gt;)&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;ϵ&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mathnormal mtight"&gt;t&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/div&gt;


&lt;h3&gt;
  
  
  Why do we need both an AR component and an MA component?
&lt;/h3&gt;

&lt;p&gt;Since the AR(p) component is a combination of the past 'p' values, and the values of the series become more and more dependent on its past values when moving along the time series. Hence, it captures the trend and autocorrelation in the series. &lt;br&gt;
Meanwhile, the MA(q) part is a combination of the past 'q' forecast errors, so it captures the "shock" (unexpected changes) to the model instead. The impact of a shock in the series decreases over time (called "shock decay"). Thus, the effect of an error made at a particular point in time diminishes as we move further away from that point. &lt;br&gt;
In short, the AR(p) part describes the overall and long term changes, whereas the MA(q) describes the short-term changes. Together, they allow the ARIMA model to capture a wide range of time series patterns. &lt;/p&gt;

</description>
    </item>
    <item>
      <title>Filtering data in R</title>
      <dc:creator>vindianadoan</dc:creator>
      <pubDate>Mon, 27 Feb 2023 10:28:49 +0000</pubDate>
      <link>https://dev.to/vindianadoan/filtering-data-in-r-5ah0</link>
      <guid>https://dev.to/vindianadoan/filtering-data-in-r-5ah0</guid>
      <description>&lt;p&gt;&lt;strong&gt;Base R&lt;/strong&gt;&lt;/p&gt;

&lt;ul&gt;
&lt;li&gt;subset() function:
&lt;/li&gt;
&lt;/ul&gt;

&lt;div class="highlight js-code-highlight"&gt;
&lt;pre class="highlight plaintext"&gt;&lt;code&gt;subset(df, column_name &amp;gt; value)
&lt;/code&gt;&lt;/pre&gt;

&lt;/div&gt;



&lt;ul&gt;
&lt;li&gt;subset operator:
&lt;/li&gt;
&lt;/ul&gt;

&lt;div class="highlight js-code-highlight"&gt;
&lt;pre class="highlight plaintext"&gt;&lt;code&gt;filtered_df &amp;lt;- df[df$column_name &amp;gt; 5, ]
&lt;/code&gt;&lt;/pre&gt;

&lt;/div&gt;



&lt;ul&gt;
&lt;li&gt;indices in subset operator:
&lt;/li&gt;
&lt;/ul&gt;

&lt;div class="highlight js-code-highlight"&gt;
&lt;pre class="highlight plaintext"&gt;&lt;code&gt;filtered_df &amp;lt;- df[indices, ]
&lt;/code&gt;&lt;/pre&gt;

&lt;/div&gt;



&lt;ul&gt;
&lt;li&gt;which() function to extract indices:
&lt;/li&gt;
&lt;/ul&gt;

&lt;div class="highlight js-code-highlight"&gt;
&lt;pre class="highlight plaintext"&gt;&lt;code&gt;# create a data frame
df &amp;lt;- data.frame(name = c("Alice", "Bob", "Charlie", "David"),
                 age = c(25, 35, 40, 30))

# use which() to extract indices of rows where age &amp;gt; 30
indices &amp;lt;- which(df$age &amp;gt; 30)

# use the indices to extract the subset of rows
subset_df &amp;lt;- df[indices, ]

# print the resulting data frame
subset_df
&lt;/code&gt;&lt;/pre&gt;

&lt;/div&gt;



&lt;p&gt;&lt;strong&gt;Tidyverse/dplyr&lt;/strong&gt;&lt;/p&gt;

&lt;ul&gt;
&lt;li&gt;filter() function:
filter() function filters rows based on certain conditions.
&lt;/li&gt;
&lt;/ul&gt;

&lt;div class="highlight js-code-highlight"&gt;
&lt;pre class="highlight plaintext"&gt;&lt;code&gt;subset_df &amp;lt;- filter(df, column_name &amp;gt; 30)
subset_df &amp;lt;- filter(df, column1_name &amp;gt; 30 
                  &amp;amp; column2_name == TRUE)
&lt;/code&gt;&lt;/pre&gt;

&lt;/div&gt;



&lt;ul&gt;
&lt;li&gt;slice() function:
slice() function selects specific rows.
&lt;/li&gt;
&lt;/ul&gt;

&lt;div class="highlight js-code-highlight"&gt;
&lt;pre class="highlight plaintext"&gt;&lt;code&gt;subset_df &amp;lt;- slice(df, indices)
subset_df &amp;lt;- slice(df, c(1,3))
&lt;/code&gt;&lt;/pre&gt;

&lt;/div&gt;



&lt;ul&gt;
&lt;li&gt;select() function: 
select() function selects specific columns.
&lt;/li&gt;
&lt;/ul&gt;

&lt;div class="highlight js-code-highlight"&gt;
&lt;pre class="highlight plaintext"&gt;&lt;code&gt;subset_df &amp;lt;- select(df, column_name)
&lt;/code&gt;&lt;/pre&gt;

&lt;/div&gt;



</description>
    </item>
    <item>
      <title>Why is SSE commonly used in linear regression?</title>
      <dc:creator>vindianadoan</dc:creator>
      <pubDate>Sun, 26 Feb 2023 00:37:42 +0000</pubDate>
      <link>https://dev.to/vindianadoan/why-is-sse-commonly-used-in-linear-regression-1oh3</link>
      <guid>https://dev.to/vindianadoan/why-is-sse-commonly-used-in-linear-regression-1oh3</guid>
      <description>&lt;p&gt;SSE (sum of squared errors) is commonly used in linear regression because it has some desirable mathematical properties and is relatively easy to optimize using techniques like gradient descent.&lt;/p&gt;

&lt;p&gt;One reason SSE is used is because it is a squared error metric. Squaring the errors has a few benefits. First, it makes the errors positive, which means that errors in both directions (i.e., overpredictions and underpredictions) are treated equally. Second, it penalizes large errors more heavily than small errors. This is desirable because we typically care more about reducing large errors than small errors.&lt;/p&gt;

&lt;p&gt;Another reason SSE is used is because it leads to a convex optimisation problem. In other words, the cost function has a single global minimum that can be found using gradient descent or other optimization techniques. This is important because it guarantees that the optimisation algorithm will converge to the optimal solution.&lt;/p&gt;

&lt;p&gt;Finally, SSE is easy to work with mathematically. The cost function is differentiable, which means that we can calculate its gradient (i.e., the partial derivatives of the cost function with respect to the weights and bias) analytically. This makes it possible to use optimization techniques like gradient descent to find the optimal values of the model parameters.&lt;/p&gt;

&lt;p&gt;In addition to SSE, other types of errors used in machine learning include:&lt;/p&gt;

&lt;p&gt;Mean Absolute Error (MAE): This measures the absolute differences between the predicted values and the actual values. MAE is less sensitive to outliers compared to SSE.&lt;/p&gt;

&lt;p&gt;Root Mean Squared Error (RMSE): This measures the square root of the average of the squared differences between the predicted values and the actual values. RMSE is used when the errors follow a normal distribution.&lt;/p&gt;

&lt;p&gt;Mean Absolute Percentage Error (MAPE): This measures the absolute percentage difference between the predicted values and the actual values. MAPE is commonly used in forecasting and time series analysis.&lt;/p&gt;

&lt;p&gt;R-squared (R2): This measures the proportion of the variance in the dependent variable that is explained by the independent variables. R2 ranges from 0 to 1, where 1 indicates a perfect fit.&lt;/p&gt;

&lt;p&gt;The choice of error metric depends on the type of problem and the nature of the data. Some errors are more appropriate for certain types of data, while others may be more robust to outliers or more intuitive to interpret. It is important to choose the appropriate error metric that aligns with the problem and evaluate the model's performance using that metric.&lt;/p&gt;

</description>
      <category>learning</category>
      <category>inspiration</category>
    </item>
    <item>
      <title>Supervised learning - Linear regression model</title>
      <dc:creator>vindianadoan</dc:creator>
      <pubDate>Sat, 25 Feb 2023 15:29:14 +0000</pubDate>
      <link>https://dev.to/vindianadoan/simple-explanation-of-machine-learning-mathematics-46i</link>
      <guid>https://dev.to/vindianadoan/simple-explanation-of-machine-learning-mathematics-46i</guid>
      <description>&lt;h3&gt;
  
  
  Linear Regression Model
&lt;/h3&gt;

&lt;p&gt;Concept map: &lt;br&gt;
Linear regression model 

&lt;span class="katex-element"&gt;
  &lt;span class="katex"&gt;&lt;span class="katex-mathml"&gt;→\rightarrow&lt;/span&gt;&lt;span class="katex-html"&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mrel"&gt;→&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/span&gt;
 SSE 
&lt;span class="katex-element"&gt;
  &lt;span class="katex"&gt;&lt;span class="katex-mathml"&gt;→\rightarrow&lt;/span&gt;&lt;span class="katex-html"&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mrel"&gt;→&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/span&gt;
 Gradient descent 
&lt;span class="katex-element"&gt;
  &lt;span class="katex"&gt;&lt;span class="katex-mathml"&gt;→\rightarrow&lt;/span&gt;&lt;span class="katex-html"&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mrel"&gt;→&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/span&gt;
 Training the model&lt;/p&gt;

&lt;p&gt;&lt;strong&gt;1. Linear Regression Model:&lt;/strong&gt;&lt;br&gt;
Linear regression is a type of supervised learning algorithm used to predict a continuous target variable based on one or more input features. Other common supervised learning algorithms include logistic regression, decision trees, and neural networks. &lt;br&gt;
The goal of linear regression is to find the best linear relationship between the input features and the target variable. The model takes the form of a linear equation:&lt;br&gt;

&lt;/p&gt;
&lt;div class="katex-element"&gt;
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class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mbin"&gt;+&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;b&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;1&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;x&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;1&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mbin"&gt;+&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;b&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;2&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;x&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;2&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mbin"&gt;+&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;...&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mbin"&gt;+&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;b&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mathnormal mtight"&gt;n&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;x&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mathnormal mtight"&gt;n&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/div&gt;
&lt;br&gt;
where:&lt;br&gt;

&lt;span class="katex-element"&gt;
  &lt;span class="katex"&gt;&lt;span class="katex-mathml"&gt;yy&lt;/span&gt;&lt;span class="katex-html"&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord mathnormal"&gt;y&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/span&gt;
 is the target variable 
&lt;span class="katex-element"&gt;
  &lt;span class="katex"&gt;&lt;span class="katex-mathml"&gt;x1,x2,...,xnx_1, x_2, ..., x_n&lt;/span&gt;&lt;span class="katex-html"&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;x&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;1&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mpunct"&gt;,&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;x&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;2&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mpunct"&gt;,&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mord"&gt;...&lt;/span&gt;&lt;span class="mpunct"&gt;,&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;x&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mathnormal mtight"&gt;n&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/span&gt;
 are the input features &lt;br&gt;

&lt;span class="katex-element"&gt;
  &lt;span class="katex"&gt;&lt;span class="katex-mathml"&gt;b0b_0&lt;/span&gt;&lt;span class="katex-html"&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;b&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;0&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/span&gt;
 is the y-intercept (also known as the bias) &lt;br&gt;

&lt;span class="katex-element"&gt;
  &lt;span class="katex"&gt;&lt;span class="katex-mathml"&gt;b1,b2,...,bnb_1, b_2, ..., b_n&lt;/span&gt;&lt;span class="katex-html"&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;b&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;1&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mpunct"&gt;,&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;b&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;2&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mpunct"&gt;,&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mord"&gt;...&lt;/span&gt;&lt;span class="mpunct"&gt;,&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;b&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mathnormal mtight"&gt;n&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/span&gt;
 are the coefficients (also known as weights) of the input features

&lt;p&gt;The objective of linear regression is to find the values of 
&lt;span class="katex-element"&gt;
  &lt;span class="katex"&gt;&lt;span class="katex-mathml"&gt;b0,b1,b2,...,bnb_0, b_1, b_2, ..., b_n&lt;/span&gt;&lt;span class="katex-html"&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;b&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;0&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mpunct"&gt;,&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;b&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;1&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mpunct"&gt;,&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;b&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;2&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mpunct"&gt;,&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mord"&gt;...&lt;/span&gt;&lt;span class="mpunct"&gt;,&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;b&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mathnormal mtight"&gt;n&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/span&gt;
 that minimize the difference between the predicted and actual values of 
&lt;span class="katex-element"&gt;
  &lt;span class="katex"&gt;&lt;span class="katex-mathml"&gt;yy&lt;/span&gt;&lt;span class="katex-html"&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord mathnormal"&gt;y&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/span&gt;
. This is typically done by minimizing the sum of squared errors (SSE) between the predicted and actual values.&lt;br&gt;&lt;/p&gt;

&lt;p&gt;&lt;strong&gt;2. SSE&lt;/strong&gt;&lt;br&gt;
By definition:&lt;br&gt;

&lt;/p&gt;
&lt;div class="katex-element"&gt;
  &lt;span class="katex-display"&gt;&lt;span class="katex"&gt;&lt;span class="katex-mathml"&gt;SSE=∑i=1n(yi−yi^)2SSE = \sum_{i=1}^n (y_i - \hat{y_i})^2&lt;/span&gt;&lt;span class="katex-html"&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord mathnormal"&gt;SSE&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mrel"&gt;=&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mop op-limits"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mathnormal mtight"&gt;i&lt;/span&gt;&lt;span class="mrel mtight"&gt;=&lt;/span&gt;&lt;span class="mord mtight"&gt;1&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="mop op-symbol large-op"&gt;∑&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mathnormal mtight"&gt;n&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mopen"&gt;(&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;y&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mathnormal mtight"&gt;i&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mbin"&gt;−&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord accent"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;y&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mathnormal mtight"&gt;i&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="accent-body"&gt;&lt;span class="mord"&gt;^&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mclose"&gt;&lt;span class="mclose"&gt;)&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;2&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/div&gt;
&lt;br&gt;
where:&lt;br&gt;

&lt;span class="katex-element"&gt;
  &lt;span class="katex"&gt;&lt;span class="katex-mathml"&gt;yiy_i&lt;/span&gt;&lt;span class="katex-html"&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;y&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mathnormal mtight"&gt;i&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/span&gt;
 is the actual value of the target variable for the 
&lt;span class="katex-element"&gt;
  &lt;span class="katex"&gt;&lt;span class="katex-mathml"&gt;ithi^{th}&lt;/span&gt;&lt;span class="katex-html"&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;i&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mathnormal mtight"&gt;t&lt;/span&gt;&lt;span class="mord mathnormal mtight"&gt;h&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/span&gt;
 data point &lt;br&gt;

&lt;span class="katex-element"&gt;
  &lt;span class="katex"&gt;&lt;span class="katex-mathml"&gt;yi^\hat{y_i}&lt;/span&gt;&lt;span class="katex-html"&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord accent"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;y&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mathnormal mtight"&gt;i&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="accent-body"&gt;&lt;span class="mord"&gt;^&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/span&gt;
 is the predicted value of the target variable for the 
&lt;span class="katex-element"&gt;
  &lt;span class="katex"&gt;&lt;span class="katex-mathml"&gt;ithi^{th}&lt;/span&gt;&lt;span class="katex-html"&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;i&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mathnormal mtight"&gt;t&lt;/span&gt;&lt;span class="mord mathnormal mtight"&gt;h&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/span&gt;
 data point&lt;br&gt;
The minimization of SSE is typically achieved using gradient descent, which is an optimization algorithm that iteratively adjusts the values of 
&lt;span class="katex-element"&gt;
  &lt;span class="katex"&gt;&lt;span class="katex-mathml"&gt;b0,b1,b2,...,bnb_0, b_1, b_2, ..., b_n&lt;/span&gt;&lt;span class="katex-html"&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;b&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;0&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mpunct"&gt;,&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;b&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;1&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mpunct"&gt;,&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;b&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;2&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mpunct"&gt;,&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mord"&gt;...&lt;/span&gt;&lt;span class="mpunct"&gt;,&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;b&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mathnormal mtight"&gt;n&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/span&gt;
 to find the optimal values that minimize SSE.&lt;br&gt;
&lt;br&gt;&lt;br&gt;
&lt;strong&gt;3. Gradient Descent:&lt;/strong&gt;&lt;br&gt;
Gradient descent is an iterative optimization algorithm that is used to find the optimal values of the model parameters (weights and biases) that minimize the cost function (SSE in this case). It works by updating the parameters in the opposite direction of the gradient of the cost function with respect to the parameters.

&lt;p&gt;The update rule for the weights in gradient descent is:&lt;br&gt;

&lt;/p&gt;
&lt;div class="katex-element"&gt;
  &lt;span class="katex-display"&gt;&lt;span class="katex"&gt;&lt;span class="katex-mathml"&gt;wi=wi−α∂Cost∂wiw_i = w_i - \alpha \frac{\partial Cost}{\partial w_i}&lt;/span&gt;&lt;span class="katex-html"&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;w&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mathnormal mtight"&gt;i&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mrel"&gt;=&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;w&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mathnormal mtight"&gt;i&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mbin"&gt;−&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord mathnormal"&gt;α&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mopen nulldelimiter"&gt;&lt;/span&gt;&lt;span class="mfrac"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord"&gt;∂&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;w&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mathnormal mtight"&gt;i&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="frac-line"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord"&gt;∂&lt;/span&gt;&lt;span class="mord mathnormal"&gt;C&lt;/span&gt;&lt;span class="mord mathnormal"&gt;os&lt;/span&gt;&lt;span class="mord mathnormal"&gt;t&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mclose nulldelimiter"&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/div&gt;
&lt;br&gt;
where:&lt;br&gt;

&lt;span class="katex-element"&gt;
  &lt;span class="katex"&gt;&lt;span class="katex-mathml"&gt;wiw_i&lt;/span&gt;&lt;span class="katex-html"&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;w&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mathnormal mtight"&gt;i&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/span&gt;
 is the 
&lt;span class="katex-element"&gt;
  &lt;span class="katex"&gt;&lt;span class="katex-mathml"&gt;ithi^{th}&lt;/span&gt;&lt;span class="katex-html"&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;i&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mathnormal mtight"&gt;t&lt;/span&gt;&lt;span class="mord mathnormal mtight"&gt;h&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/span&gt;
 weight &lt;br&gt;

&lt;span class="katex-element"&gt;
  &lt;span class="katex"&gt;&lt;span class="katex-mathml"&gt;α\alpha&lt;/span&gt;&lt;span class="katex-html"&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord mathnormal"&gt;α&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/span&gt;
 is the learning rate or step size for each iteration of gradient descent &lt;br&gt;

&lt;span class="katex-element"&gt;
  &lt;span class="katex"&gt;&lt;span class="katex-mathml"&gt;∂Cost∂wi\frac{\partial Cost}{\partial w_i}&lt;/span&gt;&lt;span class="katex-html"&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mopen nulldelimiter"&gt;&lt;/span&gt;&lt;span class="mfrac"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mtight"&gt;∂&lt;/span&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mathnormal mtight"&gt;w&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size3 size1 mtight"&gt;&lt;span class="mord mathnormal mtight"&gt;i&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="frac-line"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mtight"&gt;∂&lt;/span&gt;&lt;span class="mord mathnormal mtight"&gt;C&lt;/span&gt;&lt;span class="mord mathnormal mtight"&gt;os&lt;/span&gt;&lt;span class="mord mathnormal mtight"&gt;t&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mclose nulldelimiter"&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/span&gt;
 is the partial derivative of the cost function with respect to the 
&lt;span class="katex-element"&gt;
  &lt;span class="katex"&gt;&lt;span class="katex-mathml"&gt;ithi^{th}&lt;/span&gt;&lt;span class="katex-html"&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;i&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mathnormal mtight"&gt;t&lt;/span&gt;&lt;span class="mord mathnormal mtight"&gt;h&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/span&gt;
 weight&lt;br&gt;
The update rule for the bias term is similar:&lt;br&gt;

&lt;div class="katex-element"&gt;
  &lt;span class="katex-display"&gt;&lt;span class="katex"&gt;&lt;span class="katex-mathml"&gt;b0=b0−α∂Cost∂b0b_0 = b_0 - \alpha \frac{\partial Cost}{\partial b_0}&lt;/span&gt;&lt;span class="katex-html"&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;b&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;0&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mrel"&gt;=&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;b&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;0&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mbin"&gt;−&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord mathnormal"&gt;α&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mopen nulldelimiter"&gt;&lt;/span&gt;&lt;span class="mfrac"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord"&gt;∂&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;b&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;0&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="frac-line"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord"&gt;∂&lt;/span&gt;&lt;span class="mord mathnormal"&gt;C&lt;/span&gt;&lt;span class="mord mathnormal"&gt;os&lt;/span&gt;&lt;span class="mord mathnormal"&gt;t&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mclose nulldelimiter"&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/div&gt;
&lt;br&gt;
where:&lt;br&gt;

&lt;span class="katex-element"&gt;
  &lt;span class="katex"&gt;&lt;span class="katex-mathml"&gt;b0b_0&lt;/span&gt;&lt;span class="katex-html"&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;b&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;0&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/span&gt;
 is the bias term &lt;br&gt;

&lt;span class="katex-element"&gt;
  &lt;span class="katex"&gt;&lt;span class="katex-mathml"&gt;∂Cost∂b0\frac{\partial Cost}{\partial b_0}&lt;/span&gt;&lt;span class="katex-html"&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mopen nulldelimiter"&gt;&lt;/span&gt;&lt;span class="mfrac"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mtight"&gt;∂&lt;/span&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mathnormal mtight"&gt;b&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size3 size1 mtight"&gt;&lt;span class="mord mtight"&gt;0&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="frac-line"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mtight"&gt;∂&lt;/span&gt;&lt;span class="mord mathnormal mtight"&gt;C&lt;/span&gt;&lt;span class="mord mathnormal mtight"&gt;os&lt;/span&gt;&lt;span class="mord mathnormal mtight"&gt;t&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mclose nulldelimiter"&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/span&gt;
 is the partial derivative of the cost function with respect to the bias term

&lt;p&gt;The partial derivatives of the cost function with respect to the weights and bias can be calculated using calculus.&lt;br&gt;&lt;/p&gt;

&lt;p&gt;&lt;strong&gt;4. Training the Model:&lt;/strong&gt;&lt;br&gt;
To train the model, we first initialize the weights and bias to some random values. We then iteratively update the weights and bias using the gradient descent algorithm until the cost function reaches a minimum or a predefined stopping criterion is met (e.g., maximum number of iterations reached).&lt;/p&gt;

&lt;p&gt;Once the model is trained, we can use it to make predictions on new data by plugging in the values of the input features into the linear equation and calculating the predicted value of the target variable. &lt;/p&gt;

</description>
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