Linear and Quadratic Discriminant Analysis

Linear Discriminant Analysis (LDA) stands as one of the most elegant and foundational algorithms in statistical pattern recognition. Developed by Ronald Fisher in 1936 for classifying iris species, LDA exemplifies how strong modeling assumptions can lead to simple, interpretable, and computationally efficient classifiers.

Unlike discriminative models such as logistic regression that directly model the decision boundary, LDA takes a generative approach: it models the underlying probability distribution of each class and uses Bayes' theorem to make predictions. This fundamental distinction carries profound implications for how LDA learns, generalizes, and behaves under different data conditions.

However, the power of LDA comes at a price—a set of strong assumptions that must hold (at least approximately) for the method to work well. Understanding these assumptions deeply is not merely academic; it is essential for knowing when LDA is appropriate, how to diagnose problems, and when to consider alternatives like QDA or more flexible methods.

Before diving into LDA's specific assumptions, we must understand the generative classification paradigm. In generative classification, we model:

1. Class priors $P(Y = k)$: The probability of each class occurring in the population
2. Class-conditional densities $P(X | Y = k)$: The probability distribution of features given each class

Once these components are estimated, Bayes' theorem gives us the posterior probability for classification:

$$P(Y = k | X = x) = \frac{P(X = x | Y = k) \cdot P(Y = k)}{\sum_{j=1}^{K} P(X = x | Y = j) \cdot P(Y = j)}$$

The classification rule assigns each observation to the class with the highest posterior:

$$\hat{y} = \arg\max_k P(Y = k | X = x)$$

This framework is powerful because once we have good estimates of the class-conditional densities, we can:

• Generate synthetic data from each class
• Understand the data-generating process
• Compute posterior probabilities, not just class labels
• Handle missing data through marginalization

The modeling challenge:

The core difficulty is estimating the class-conditional densities $P(X | Y = k)$. In principle, we could use any density estimation technique—kernel density estimation, Gaussian mixtures, normalizing flows, etc. However:

• Nonparametric methods suffer from the curse of dimensionality
• Flexible parametric methods require many parameters
• Simple parametric methods impose strong restrictions

LDA takes the third approach: it assumes a very specific, constrained parametric form for the class-conditional densities. This constraint is simultaneously LDA's greatest strength (few parameters, easy estimation) and its greatest limitation (potential model misspecification).

The first and most fundamental assumption of LDA is that the features within each class follow a multivariate Gaussian (normal) distribution. Mathematically, for each class $k$:

$$X | Y = k \sim \mathcal{N}(\mu_k, \Sigma_k)$$

where:

• $\mu_k \in \mathbb{R}^p$ is the mean vector for class $k$
• $\Sigma_k \in \mathbb{R}^{p \times p}$ is the covariance matrix for class $k$
• $p$ is the number of features

The multivariate Gaussian probability density function is:

$$P(X = x | Y = k) = \frac{1}{(2\pi)^{p/2}|\Sigma_k|^{1/2}} \exp\left(-\frac{1}{2}(x - \mu_k)^T\Sigma_k^{-1}(x - \mu_k)\right)$$

This density defines elliptical contours of equal probability in feature space, centered at $\mu_k$ and shaped by $\Sigma_k$.

Why Gaussian?

The Gaussian distribution is the most commonly assumed form for several compelling reasons:

Theoretical justification (Central Limit Theorem): When features are aggregates of many small, independent effects, the CLT suggests Gaussian behavior. Heights, weights, test scores, and many natural measurements exhibit approximately Gaussian distributions.

Maximum entropy property: Among all distributions with a given mean and variance, the Gaussian has maximum entropy (disorder). If we only know the first and second moments, the Gaussian is the 'least committed' distribution we can assume.

Analytical tractability: Gaussians are closed under linear operations, marginalization, and conditioning. The math works out beautifully—log-likelihoods become quadratic, posteriors remain Gaussian, and decision boundaries take simple forms.

Parameter estimation: With $n$ samples from class $k$, we can efficiently estimate:

$$\hat{\mu}k = \frac{1}{n_k}\sum{i: y_i = k} x_i \quad \text{(sample mean)}$$

$$\hat{\Sigma}k = \frac{1}{n_k - 1}\sum{i: y_i = k} (x_i - \hat{\mu}_k)(x_i - \hat{\mu}_k)^T \quad \text{(sample covariance)}$$

Diagnosing Gaussianity:

Before applying LDA, it's prudent to assess whether the Gaussian assumption is reasonable:

1. Univariate normality tests: Apply Shapiro-Wilk or Anderson-Darling tests to each feature within each class. However, note that these tests are sensitive to sample size and cannot assess multivariate normality.

2. Q-Q plots: For each feature and class, compare empirical quantiles against Gaussian quantiles. Systematic departures (S-curves, heavy tails) indicate violations.

3. Multivariate normality: Use tests like Mardia's test (measures multivariate skewness and kurtosis) or Henze-Zirkler test. Visual inspection via chi-squared Q-Q plots of Mahalanobis distances can also help.

4. Density visualization: For low-dimensional data, kernel density estimates can reveal multimodality or asymmetry that contradicts the unimodal, symmetric Gaussian.

Robustness: Despite being a parametric assumption, LDA often exhibits surprising robustness to mild Gaussian violations—especially if classes are well-separated. The shared covariance assumption (discussed next) matters more in practice.

The second core assumption of LDA—and the one that distinguishes it from QDA—is that all classes share the same covariance matrix:

$$\Sigma_1 = \Sigma_2 = \cdots = \Sigma_K = \Sigma$$

This is called the homoscedasticity assumption (from Greek 'homos' meaning same and 'skedasis' meaning dispersion). Each class-conditional Gaussian has the same shape and orientation; only the means differ.

Geometric interpretation:

The equal covariance assumption means that the elliptical contours of constant density have the same shape, size, and orientation across all classes—they are simply translated to different locations (the class means). Imagine identical ellipsoids placed at different centers in feature space.

Mathematical consequence: From QDA to LDA

Let's see how equal covariances lead to linear boundaries. For a two-class problem, the decision boundary is where the posterior probabilities are equal:

$$P(Y = 1 | X) = P(Y = 2 | X)$$

Using Bayes' theorem and taking logarithms:

$$\log P(X | Y = 1) + \log P(Y = 1) = \log P(X | Y = 2) + \log P(Y = 2)$$

Substituting the Gaussian densities:

$$-\frac{1}{2}(x - \mu_1)^T\Sigma_1^{-1}(x - \mu_1) - \frac{1}{2}\log|\Sigma_1| + \log\pi_1$$ $$= -\frac{1}{2}(x - \mu_2)^T\Sigma_2^{-1}(x - \mu_2) - \frac{1}{2}\log|\Sigma_2| + \log\pi_2$$

With different covariances ($\Sigma_1 eq \Sigma_2$), expanding the quadratic forms gives terms like $x^T\Sigma_1^{-1}x$ and $x^T\Sigma_2^{-1}x$ that don't cancel—leaving a quadratic function of $x$. This is QDA.

With equal covariances ($\Sigma_1 = \Sigma_2 = \Sigma$), these quadratic terms perfectly cancel:

$$-\frac{1}{2}x^T\Sigma^{-1}x + \mu_1^T\Sigma^{-1}x - \frac{1}{2}\mu_1^T\Sigma^{-1}\mu_1 + \log\pi_1$$ $$= -\frac{1}{2}x^T\Sigma^{-1}x + \mu_2^T\Sigma^{-1}x - \frac{1}{2}\mu_2^T\Sigma^{-1}\mu_2 + \log\pi_2$$

The $x^T\Sigma^{-1}x$ terms cancel, leaving:

$$(\mu_1 - \mu_2)^T\Sigma^{-1}x = \frac{1}{2}\mu_1^T\Sigma^{-1}\mu_1 - \frac{1}{2}\mu_2^T\Sigma^{-1}\mu_2 + \log\frac{\pi_2}{\pi_1}$$

This is a linear function of $x$—hence Linear Discriminant Analysis.

Estimating the pooled covariance matrix:

The maximum likelihood estimate of the shared covariance matrix pools data from all classes:

$$\hat{\Sigma} = \frac{1}{n - K}\sum_{k=1}^{K}\sum_{i: y_i = k}(x_i - \hat{\mu}_k)(x_i - \hat{\mu}_k)^T$$

Equivalently:

$$\hat{\Sigma} = \frac{\sum_{k=1}^{K}(n_k - 1)\hat{\Sigma}k}{\sum{k=1}^{K}(n_k - 1)}$$

This is a weighted average of the class-specific sample covariance matrices, where weights are $(n_k - 1)$ (degrees of freedom for each class). Larger classes contribute more to the estimate.

Testing the assumption:

Box's M-test formally tests the null hypothesis $H_0: \Sigma_1 = \Sigma_2 = \cdots = \Sigma_K$. However, in practice:

• The test is sensitive to normality violations (it tests both simultaneously)
• With large samples, even trivial differences become 'significant'
• With small samples, the test has low power

A practical approach is to compare the eigenvalues of class-specific covariance matrices. If the ratios of largest to smallest eigenvalues differ substantially across classes, the assumption is likely violated.

The third component of LDA is the specification of class prior probabilities $\pi_k = P(Y = k)$. While not an 'assumption' in the same sense as Gaussianity, how we estimate or specify priors significantly affects model behavior, especially for imbalanced classification problems.

Estimation approaches:

1. Sample proportions (most common): $$\hat{\pi}_k = \frac{n_k}{n}$$ where $n_k$ is the number of training samples in class $k$.

2. Equal priors: $$\pi_k = \frac{1}{K} \quad \forall k$$ This ignores training set class frequencies, useful when the test distribution differs.

3. Domain-specified priors: Set based on domain knowledge of population frequencies (e.g., disease prevalence).

Effect of priors on decision boundaries:

Priors shift the decision boundary. For a two-class problem, the log-odds ratio includes the term $\log(\pi_1/\pi_2)$. This shifts the boundary toward the class with lower prior—requiring more 'evidence' (in the form of likelihood) to classify into the less common class.

Implicit assumption:

Using sample proportions as priors implicitly assumes that:

1. The training data is a representative sample from the target population
2. The class distribution in training matches the class distribution at test time
3. The cost of misclassification is symmetric across classes

When any of these assumptions fail, adjusting priors becomes essential. In medical diagnosis, adjusting priors to reflect disease prevalence (rather than case-control study proportions) is critical for calibrated predictions.

Beyond the three core assumptions, LDA makes several additional implicit assumptions that practitioners should be aware of:

Feature preprocessing implications:

Certain preprocessing steps affect how well LDA assumptions hold:

• Standardization: Centering and scaling features doesn't change Gaussianity but can improve numerical stability.

• Log/power transformations: Can induce approximate normality for right-skewed features (common in count data, durations, monetary values).

• Box-Cox transformations: Optimal power transformation toward normality, but must be applied within each class separately for the class-conditional assumption.

• Principal component analysis: Decorrelates features, but doesn't guarantee normality of the components.

• Outlier removal/winsorization: Can substantially improve Gaussian fit but may lose information.

Understanding the consequences of assumption violations helps you decide when LDA is acceptable versus when alternatives are necessary.

Violation: Non-Gaussian distributions

Mild violations (slightly heavy tails, minor asymmetry):

• LDA often still performs reasonably well
• Mean and covariance capture most of the relevant structure
• Classification accuracy may be slightly suboptimal but not catastrophic

Severe violations (multimodality, very heavy tails, discrete data):

• Decision boundaries are fundamentally wrong
• Posterior probabilities are miscalibrated
• May perform much worse than discriminative alternatives
• Consider Gaussian mixture models, kernel methods, or nonparametric approaches

Violation: Unequal covariances

This is typically more consequential than non-Gaussianity:

When class covariances are substantially different:

• Linear boundaries are suboptimal; quadratic boundaries would fit better
• LDA may underperform on one class while overperforming on another
• Classes with smaller variance are often underrepresented in predictions
• The pooled covariance is a poor summary of either class's spread

Diagnosis:

• Compare determinants $|\hat{\Sigma}_k|$ across classes
• Visualize class-specific scatter ellipses
• Compute eigenvalue ratios for each class's covariance
• Use Box's M-test (with appropriate caveats)

Solutions:

• Use QDA if you have enough data (discussed next page)
• Use regularized discriminant analysis as a compromise
• Apply transformations that stabilize variance

Understanding how LDA relates to other classification methods illuminates its unique position:

LDA vs Logistic Regression:

Both produce linear decision boundaries, but from different directions:

• LDA: Generative—models $P(X|Y)$ and $P(Y)$, derives $P(Y|X)$
• Logistic Regression: Discriminative—directly models $P(Y|X)$

Key differences:

• LDA is more efficient when Gaussian assumptions hold (uses all data properties)
• Logistic regression is more robust to model misspecification
• LDA requires invertible covariance; logistic regression doesn't
• LDA naturally extends to multi-class; logistic regression requires extensions (softmax, OvR)
• LDA provides a useful dimensionality reduction (Fisher's discriminant)

LDA vs QDA:

Both are Gaussian generative classifiers:

• LDA: Shared covariance → linear boundary, $O(p^2)$ parameters
• QDA: Class-specific covariances → quadratic boundaries, $O(Kp^2)$ parameters

LDA is essentially a constrained QDA where the constraint is $\Sigma_1 = \Sigma_2 = \cdots = \Sigma_K$.

LDA vs Naive Bayes (Gaussian):

Both make Gaussian assumptions:

• Gaussian NB: Assumes features are conditionally independent given class (diagonal covariance)
• LDA: Models full covariance structure (including correlations)

Naive Bayes makes a stronger assumption (zero correlations) but requires even fewer parameters: $O(Kp)$ vs LDA's $O(p^2)$.

LDA as dimensionality reduction:

LDA can also be viewed as a technique that finds the projection maximizing class separation. In this view, it's an alternative to PCA for supervised dimensionality reduction—projecting onto the directions that best discriminate classes rather than maximize variance.

What's next:

Now that we understand the assumptions that justify linear decision boundaries, the next page explores the shared covariance structure in greater detail—how it's estimated, why it enables dimensionality reduction, and how to interpret the resulting discriminant functions geometrically.