# A Brief Look at Mixture Discriminant Analysis

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Lately, I have been working with finite mixture models for my postdoctoral work

on data-driven automated gating.

Given that I had barely scratched the surface with mixture models in the

classroom, I am becoming increasingly comfortable with them. With this in mind,

I wanted to explore their application to classification because there are times

when a single class is clearly made up of multiple subclasses that are not

necessarily adjacent.

As far as I am aware, there are two main approaches (there are lots and lots of

variants!) to applying finite mixture models to classfication:

Although the methods are similar, I opted for exploring the latter method. Here

is the general idea. There are classes, and each class is assumed to

be a Gaussian mixuture of subclasses. Hence, the model formulation is generative,

and the posterior probability of class membership is used to classify an

unlabeled observation. Each subclass is assumed to have its own mean vector, but

all subclasses share the same covariance matrix for model parsimony. The model

parameters are estimated via the EM algorithm.

Because the details of the likelihood in the paper are brief, I realized I was a

bit confused with how to write the likelihood in order to determine how much

each observation contributes to estimating the common covariance matrix in the

M-step of the EM algorithm. Had each subclass had its own covariance matrix, the

likelihood would simply be the product of the individual class likelihoods and

would have been straightforward. The source of my confusion was how to write

the complete data likelihood when the classes share parameters.

I decided to write up a document that explicitly defined the likelihood and

provided the details of the EM algorithm used to estimate the model parameters.

The document is available here

along with the LaTeX and R code.

If you are inclined to read the document, please let me know if any notation is

confusing or poorly defined. Note that I did not include the additional topics

on reduced-rank discrimination and shrinkage.

To see how well the mixture discriminant analysis (MDA) model worked, I

constructed a simple toy example consisting of 3 bivariate classes each having 3

subclasses. The subclasses were placed so that within a class, no subclass is

adjacent. The result is that no class is Gaussian. I was interested in seeing

if the MDA classifier could identify the subclasses and also comparing its

decision boundaries with those of linear discriminant analysis (LDA)

and quadratic discriminant analysis (QDA).

I used the implementation of the LDA and QDA classifiers in the MASS package.

From the scatterplots and decision boundaries given below,

the LDA and QDA classifiers yielded puzzling decision boundaries as expected.

Contrarily, we can see that the MDA classifier does a good job of identifying

the subclasses. It is important to note that all subclasses in this example have

the same covariance matrix, which caters to the assumption employed in the MDA

classifier. It would be interesting to see how sensitive the classifier is to

deviations from this assumption. Moreover, perhaps a more important investigation

would be to determine how well the MDA classifier performs as the feature

dimension increases relative to the sample size.

“` r Comparison of LDA, QDA, and MDA

library(MASS)

library(mvtnorm)

library(mda)

library(ggplot2)

set.seed(42)

n <- 500

# Randomly sample data

x11 <- rmvnorm(n = n, mean = c(-4, -4)) x12 <- rmvnorm(n = n, mean = c(0, 4)) x13 <- rmvnorm(n = n, mean = c(4, -4))

x21 <- rmvnorm(n = n, mean = c(-4, 4)) x22 <- rmvnorm(n = n, mean = c(4, 4)) x23 <- rmvnorm(n = n, mean = c(0, 0))

x31 <- rmvnorm(n = n, mean = c(-4, 0)) x32 <- rmvnorm(n = n, mean = c(0, -4)) x33 <- rmvnorm(n = n, mean = c(4, 0))

x <- rbind(x11, x12, x13, x21, x22, x23, x31, x32, x33) train_data <- data.frame(x, y = gl(3, 3 * n))

# Trains classifiers

lda_out <- lda(y ~ ., data = train_data) qda_out <- qda(y ~ ., data = train_data) mda_out <- mda(y ~ ., data = train_data)

# Generates test data that will be used to generate the decision boundaries via

# contours

contour_data <- expand.grid(X1 = seq(-8, 8, length = 300),
X2 = seq(-8, 8, length = 300))

# Classifies the test data

lda_predict <- data.frame(contour_data, y = as.numeric(predict(lda_out, contour_data)$class)) qda_predict <- data.frame(contour_data, y = as.numeric(predict(qda_out, contour_data)$class)) mda_predict <- data.frame(contour_data, y = as.numeric(predict(mda_out, contour_data)))

# Generates plots

p <- ggplot(train_data, aes(x = X1, y = X2, color = y)) + geom_point() p + stat_contour(aes(x = X1, y = X2, z = y), data = lda_predict) + ggtitle(“LDA Decision Boundaries”) p + stat_contour(aes(x = X1, y = X2, z = y), data = qda_predict) + ggtitle(“QDA Decision Boundaries”) p + stat_contour(aes(x = X1, y = X2, z = y), data = mda_predict) + ggtitle(“MDA Decision Boundaries”) ```

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