Discriminant Analysis: Statistics All The Way

Introduction
In the world of statistics and data science, one of the key challenges is to correctly classify data points into predefined categories. Discriminant Analysis, one of the classical techniques in multivariate statistics, plays a crucial role in this classification process. It is a method used when the dependent variable is categorical and the predictors are numerical.

Unlike cluster analysis, where the grouping criteria are unknown and must be discovered, discriminant analysis starts with known class labels and attempts to find mathematical functions that can best separate those classes.

At its core, Discriminant Analysis works by finding linear (or sometimes quadratic) combinations of features that best discriminate between categories. In this article, we’ll explore its origins, assumptions, mathematical foundations, and applications—with an example in R to tie theory to practice.

Historical Origins of Discriminant Analysis
The roots of discriminant analysis trace back to Sir Ronald A. Fisher in 1936, who introduced the concept while studying the famous Iris flower dataset. His objective was to find a mathematical rule that could distinguish between the three species of Iris flowers—Setosa, Versicolor, and Virginica—based on their physical measurements.

The result was the Fisher’s Linear Discriminant, which has since become a cornerstone in statistical pattern recognition and machine learning. Fisher’s approach focused on maximizing the separation between the means of classes relative to the variability within classes, a principle still used in modern classification algorithms.

Since then, discriminant analysis has evolved into several variants, the most popular being:

• Linear Discriminant Analysis (LDA) – assumes equal covariance across classes.
• Quadratic Discriminant Analysis (QDA) – relaxes the assumption of equal covariance, allowing for nonlinear boundaries.

These models paved the way for later developments in logistic regression, support vector machines, and deep learning classifiers, making Fisher’s work one of the foundations of predictive analytics.

Theoretical Foundations
Discriminant Analysis aims to classify observations into one of several known groups based on predictor variables. The basic idea is to compute discriminant functions—linear or quadratic equations—that produce a score for each observation. The observation is then assigned to the class with the highest probability or score.

Let’s consider two classes, C1C_1C1​ and C2C_2C2​, with feature vector xxx. The linear discriminant function can be expressed as:

w⋅x>cw cdot x > cw⋅x>c

where www represents the weight vector, and ccc is the threshold. If the condition holds true, the observation belongs to one class; otherwise, it belongs to the other.

When data can’t be separated linearly, Quadratic Discriminant Analysis (QDA) is used, which adds non-linear decision boundaries to handle complex relationships.

Assumptions of Discriminant Analysis
Before applying LDA or QDA, several assumptions need to be satisfied:

1. Multivariate Normality: Each class should follow a multivariate normal distribution.
2. Homoscedasticity: Variance-covariance matrices should be equal across groups (for LDA).
3. Random Sampling: Data points must be randomly and independently sampled.
4. No Multicollinearity: Predictor variables should not be highly correlated.

Violating these assumptions can lead to unreliable or biased classification results, so preprocessing steps such as normalization, outlier detection, and correlation checks are critical.

Fisher’s Linear Discriminant Analysis
Fisher’s LDA focuses on finding a projection that maximizes class separability. The optimal projection www is found by maximizing the ratio of between-class variance to the within-class variance:

S=(w⋅(μ2−μ1))2wT(Σ1+Σ2)wS = frac{(w cdot (mu_2 - mu_1))^2}{w^T (Sigma_1 + Sigma_2) w}S=wT(Σ1​+Σ2​)w(w⋅(μ2​−μ1​))2​

This ensures that the projected data points of different classes are as far apart as possible while maintaining compactness within each class.

Applications of Fisher’s LDA go far beyond academia—it has been used in face recognition, speech classification, and bioinformatics.

Real-World Applications of Discriminant Analysis
1. Finance and Credit Scoring
Banks and financial institutions use Discriminant Analysis to assess credit risk. By analyzing attributes such as income, debt ratio, and repayment history, LDA models classify borrowers as “low risk” or “high risk.”

Case Study: A study by Altman (1968) introduced the Z-score model, a form of discriminant analysis, to predict corporate bankruptcy. The model used five financial ratios to classify firms as solvent or insolvent, and it became a landmark in financial risk modeling.

2. Healthcare and Medical Diagnosis
In healthcare, LDA assists in diagnosing diseases by analyzing patient data. For example, it can distinguish between benign and malignant tumors based on measurements from medical imaging or lab tests.

Case Study: In a cancer detection study, discriminant analysis was applied to classify tissue samples using cell size and texture features. The model achieved high diagnostic accuracy and provided interpretable results for clinicians—highlighting which features contributed most to disease classification.

3. Marketing and Customer Segmentation
Marketers use discriminant analysis to understand consumer behavior and classify customers based on preferences or purchasing patterns.

Example: A retail company could apply LDA to categorize customers into “bargain seekers,” “brand loyalists,” and “premium buyers” using demographic and transactional data. This helps in tailoring marketing campaigns and pricing strategies effectively.

4. Environmental and Earth Sciences
In environmental studies, LDA has been applied to classify land use, soil types, or climate zones based on satellite imagery and sensor data.

Case Study: Researchers used discriminant analysis to classify regions in a watershed as forested or agricultural based on spectral data from satellites. The model achieved over 90% classification accuracy, helping policymakers monitor land degradation.

5. Forensic and Legal Sciences
Discriminant analysis is used in forensic anthropology to determine the sex or ancestry of skeletal remains. It analyzes bone measurements to classify individuals into demographic groups, aiding in criminal investigations and archaeological studies.

Discriminant Analysis in R: The Iris Dataset Example
To bring theory into practice, let’s use R’s MASS package to perform Linear Discriminant Analysis on the classic Iris dataset.

Load required library library(MASS)

Load dataset dataset <- iris

Perform LDA lda_model <- lda(Species ~ ., data = dataset)

Predict and check accuracy predictions <- predict(lda_model, dataset) table(predictions$class, dataset$Species)

The output shows that LDA correctly classifies all but a few points, achieving over 97% accuracy.

We can visualize how the discriminant functions separate species:

ldahist(data = predictions$x[,1], g = dataset$Species)

This histogram reveals clear separability among the three Iris species along the first discriminant axis.

If the data were not linearly separable, we could use Quadratic Discriminant Analysis (QDA) using the same MASS package:

qda_model <- qda(Species ~ ., data = dataset) predictions_qda <- predict(qda_model, dataset) table(predictions_qda$class, dataset$Species)

Since the Iris data are mostly linear, both LDA and QDA perform similarly well.

Visualizing Class Separation with the klaR Package
For exploratory visualization, R’s klaR package offers the partimat() function to illustrate how each feature pair partitions classes:

library(klaR) partimat(Species ~ ., data = dataset, method = "lda")

This produces a set of 2D scatterplots showing how combinations of features like petal length and width effectively separate species—mirroring Fisher’s original findings.

Conclusion
Discriminant Analysis remains one of the most powerful yet elegant classification techniques in statistics. From its inception in the 1930s by Ronald Fisher to its modern-day applications in finance, healthcare, marketing, and machine learning, it continues to provide interpretable and statistically sound models for categorical prediction.

While assumptions of normality and equal variances must be checked, tools like LDA and QDA remain essential for understanding data structure, reducing dimensionality, and enhancing classification accuracy.

For practitioners using R, the MASS and klaR packages make implementing and visualizing discriminant analysis straightforward. Whether you’re identifying credit risks, diagnosing diseases, or segmenting customers, discriminant analysis ensures that statistics truly go all the way in guiding decisions through data.

This article was originally published on Perceptive Analytics.

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