Introduction
Discriminant Analysis is a powerful supervised classification technique used when the dependent (target) variable is categorical and the independent variables are continuous. Unlike cluster analysis, where group membership is unknown and discovered through patterns in data, discriminant analysis requires predefined class labels. The objective is to build a mathematical function that can accurately assign new observations to one of the known categories.
Among the different types of discriminant analysis, Linear Discriminant Analysis (LDA) is the most widely used due to its simplicity, interpretability, and effectiveness when its assumptions are satisfied. LDA works by forming a linear combination of predictor variables that maximizes the separation between classes. This article explores the origins of LDA, its theoretical foundation, assumptions, real-world applications, and practical case studies, concluding with insights into when and how it should be used effectively.
Origins of Linear Discriminant Analysis
Linear Discriminant Analysis was introduced by Sir Ronald A. Fisher in 1936 in his seminal paper “The Use of Multiple Measurements in Taxonomic Problems.” Fisher developed LDA as a method to distinguish between different species of flowers based on multiple physical measurements—work that famously involved the Iris dataset, which remains a benchmark dataset in machine learning and statistics today.
Fisher’s primary goal was dimensionality reduction with class separation. While techniques like Principal Component Analysis (PCA) reduce dimensions based on variance alone, Fisher’s LDA explicitly uses class information to find directions that best separate groups. This made LDA one of the earliest and most influential supervised learning algorithms, laying the foundation for modern classification techniques.
Conceptual Understanding of Discriminant Analysis
Discriminant analysis operates by computing a discriminant score, which is a weighted sum of the predictor variables. Based on this score and a predefined threshold, an observation is assigned to a specific class.
Mathematically, LDA assumes that:
Each class follows a multivariate normal distribution
All classes share a common covariance matrix
Classification boundaries are linear
For a two-class problem, LDA finds a linear decision boundary such that observations on one side are classified into class 1, while those on the other side belong to class 2. When more than two classes exist, multiple linear discriminant functions are computed.
Assumptions of Linear Discriminant Analysis
LDA makes several important assumptions that directly influence its performance:
Multivariate Normality Predictor variables should be normally distributed within each class. Violations of this assumption may reduce classification accuracy.
Homoscedasticity (Equal Covariance) The variance-covariance structure of predictors should be the same across all classes.
Independence of Observations Each data point must be independently sampled.
Absence of Multicollinearity Highly correlated predictors reduce the stability and interpretability of the discriminant function.
Balanced Group Sizes Extreme imbalance in class sizes can bias classification results.
Despite these assumptions, LDA is often robust enough to perform reasonably well even when some assumptions are mildly violated.
Fisher’s Linear Discriminant: The Core Idea
Fisher’s Linear Discriminant focuses on maximizing the ratio of between-class variance to within-class variance. In simple terms, it finds a projection where:
Class means are as far apart as possible
Data points within the same class are as close together as possible
This optimization leads to a discriminant function that provides maximum class separability. The result is a low-dimensional representation that preserves class discrimination power, making LDA useful both for classification and visualization.
Linear vs Quadratic Discriminant Analysis
While LDA assumes equal covariance matrices across classes, Quadratic Discriminant Analysis (QDA) relaxes this assumption. QDA allows each class to have its own covariance matrix, resulting in non-linear (quadratic) decision boundaries.
LDA performs better when data is linearly separable and sample size is limited.
QDA is more flexible but requires more data and is sensitive to overfitting.
Choosing between LDA and QDA depends on data structure, dimensionality, and sample size.
Real-Life Applications of Linear Discriminant Analysis
1. Healthcare and Medical Diagnosis
LDA is widely used in medical research to classify patients based on clinical measurements. For example:
Classifying tumors as benign or malignant
Diagnosing diseases based on blood test indicators
Differentiating between disease subtypes
2. Finance and Credit Risk Analysis
Financial institutions use LDA for:
Credit scoring and loan approval
Bankruptcy prediction
Fraud detection
By analyzing financial ratios and transaction patterns, LDA helps classify customers into risk categories.
3. Marketing and Customer Segmentation
Marketers apply LDA to:
Segment customers based on purchasing behavior
Predict customer churn
Identify high-value customer groups
The interpretability of LDA makes it especially valuable for business decision-making.
4. Image and Face Recognition
Before deep learning became dominant, LDA played a crucial role in face recognition systems. Combined with PCA, it produced Fisher Faces, which improved classification accuracy under varying lighting conditions.
5. Speech and Audio Recognition
LDA is used to reduce feature dimensions and improve class separation in speech recognition, speaker identification, and audio classification tasks.
Case Study 1: Iris Flower Classification
One of the most iconic applications of LDA is the classification of iris flower species based on four measurements:
Sepal length
Sepal width
Petal length
Petal width
Using LDA, researchers can reduce the four-dimensional feature space into two discriminant axes that clearly separate the three species. The first discriminant typically accounts for more than 99% of class separation, demonstrating the effectiveness of LDA when data is well-structured and linearly separable.
Case Study 2: Bankruptcy Prediction in Banking
A financial institution analysed historical data from firms that either went bankrupt or remained solvent. Predictor variables included:
Debt-to-equity ratio
Liquidity ratios
Profit margins
LDA was used to build a classification model that successfully separated high-risk firms from stable ones. The model enabled early intervention, reducing financial losses and improving credit risk management.
Case Study 3: Medical Diagnosis Using Biomarkers
In a biomedical study, LDA was applied to classify patients into disease and non-disease groups using biomarker measurements. The discriminant function identified a small subset of features that contributed most to classification accuracy, helping clinicians focus on the most relevant diagnostic indicators.
Strengths and Limitations of LDA
Strengths
Simple and computationally efficient
Highly interpretable
Effective for small to medium datasets
Works well when assumptions are met
Limitations
Sensitive to outliers
Performance degrades when assumptions are violated
Not suitable for highly non-linear data
Requires careful pre-processing
Conclusion
Linear Discriminant Analysis remains a cornerstone of statistical learning and supervised classification. Rooted in Fisher’s early work, LDA combines mathematical elegance with practical applicability across domains such as healthcare, finance, marketing, and pattern recognition. While modern machine learning techniques may outperform LDA in complex scenarios, its interpretability, efficiency, and theoretical foundation ensure its continued relevance.
For best results, practitioners should carefully evaluate data assumptions, perform exploratory analysis, and consider alternative methods like QDA when linear boundaries are insufficient. When applied correctly, LDA provides robust, explainable, and actionable insights from data.
This article was originally published on Perceptive Analytics.
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