Discriminant analysis is a statistical method designed for predicting categorical outcomes based on several predictor variables. Unlike cluster analysis—where group memberships are unknown—discriminant analysis works with predefined categories. It essentially calculates a score derived from all predictor variables, and this score determines the category into which a data point falls. The technique, therefore, “discriminates” between data points and classifies them into distinct groups, which is why it carries the name.
This article focuses on linear discriminant analysis (LDA), its assumptions, and its practical uses in data science and beyond. We’ll also touch upon its extension, quadratic discriminant analysis (QDA), and discuss how these methods can be applied effectively.
Since discriminant analysis deals with multiple features, it operates under certain assumptions that are crucial for the accuracy of results:
Multivariate Normality – Each class should have features that are normally distributed. Because of this, the method is highly sensitive to outliers and imbalanced class sizes. Extremely small or large groups can distort classification performance.
Equal Variances (Homoscedasticity) – Variances of the features should remain consistent across all classes. This ensures fairness in how groups are compared.
Random Sampling – Features must be sampled randomly to maintain unbiased representation.
No Multicollinearity – Predictor variables should not be highly correlated with each other. Strong correlations reduce the discriminative power of the model.
If these assumptions are met, LDA can be a powerful tool for classification. Otherwise, violations can weaken its predictive strength.
At its core, LDA is based on the idea of conditional probability functions that are normally distributed for each class. If we consider two classes, the technique finds a linear combination of predictor variables that best separates them. A threshold is then applied: values below the threshold are assigned to one class, while those above are assigned to the other.
In simpler terms, LDA constructs a boundary line (or hyperplane in higher dimensions) that divides the classes. If the data points fall on one side of the line, they are classified into one category; if on the other side, they go into the alternate category.
Among the variations of LDA, Fisher’s Linear Discriminant is particularly well known. It identifies a linear combination of features that maximizes the separation between categories. This is done by maximizing the ratio of between-class variance to within-class variance.
Think of it as comparing the spread between groups versus the spread inside each group. The larger this ratio, the clearer the separation between categories.
Applications of Fisher’s LDA span across multiple fields:
Speech and Audio Recognition – Differentiating spoken words or sounds.
Face Recognition – Used in the creation of "Fisher’s Faces" and often combined with PCA for building "Eigenfaces."
Biomedical and Earth Sciences – For classifying biological or geological data.
Finance and Marketing – Applied in bankruptcy prediction, customer segmentation, and risk analysis.
This versatility explains why Fisher’s method continues to hold relevance decades after it was first introduced.
One of the biggest strengths of LDA is its ability to classify data with high accuracy when the classes are linearly separable. However, not all datasets follow this condition. Visual techniques, such as scatterplots, are often used to check whether the classes appear to be separable before applying LDA.
For linearly separable data, LDA usually achieves very high classification accuracy. In cases where data overlaps significantly, LDA can still perform reasonably well, though it may misclassify some observations.
While LDA assumes equal variance across classes and looks for linear boundaries, QDA relaxes this assumption. Instead, it allows for quadratic decision boundaries, which makes it more flexible in handling data that cannot be separated linearly.
QDA is especially useful when the underlying relationship between variables and categories is curved or more complex. However, because it requires estimating more parameters than LDA, QDA can sometimes be less stable, especially with smaller datasets.
LDA works best when the assumptions of normality and equal variances hold, and when the data is linearly separable.
QDA is more suitable when variances differ across classes or when data points cannot be separated with a straight line.
In practice, many analysts experiment with both techniques and choose the one that delivers higher accuracy and interpretability for their dataset.
Beyond classification, discriminant analysis offers a way to visualize how well data can be separated into categories. By projecting the data onto linear discriminants, we can see how distinct groups appear. Often, the first discriminant alone explains the majority of the separation, with subsequent discriminants adding minimal improvement.
Visualizations not only help in model interpretation but also in communicating results effectively to stakeholders who may not have a deep statistical background.
Discriminant analysis has been widely adopted in different domains due to its balance of simplicity and power:
Healthcare – For classifying patients into risk categories based on diagnostic indicators.
Marketing – To segment customers and predict buying behavior.
Finance – In credit scoring and identifying firms at risk of bankruptcy.
Security – For biometric systems such as facial recognition and fingerprint verification.
The technique’s adaptability ensures that it remains a go-to choice for both researchers and professionals dealing with classification problems.
Discriminant analysis—whether linear (LDA) or quadratic (QDA)—remains a cornerstone statistical technique for classification tasks. While LDA is simpler and effective for linearly separable datasets, QDA provides the flexibility needed for more complex cases.
Before applying either, it is critical to check whether the data meets the assumptions of normality, equal variance, and independence. If these assumptions are violated, results may lose accuracy. Data preprocessing, outlier detection, and exploratory analysis are therefore essential steps.
Ultimately, discriminant analysis not only helps in making predictions but also provides insights into the structure of the data itself. By balancing mathematical rigor with practical utility, it continues to be a reliable method for solving real-world classification problems.
This article was originally published on Perceptive Analytics.
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