Quadratic Discriminant Analysis (original) (raw)

Last Updated : 7 Jan, 2022

Linear Discriminant Analysis

Now, Let's consider a classification problem represented by a Bayes Probability distribution P(Y=k | X=x), LDA does it differently by trying to model the distribution of X given the predictors class (I.e. the value of Y) P(X=x| Y=k):

P(Y=k | X=x) = \frac{P(X=x | Y=k) P(Y=k)}{P(X=x)}

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

In LDA, we assume that P(X | Y=k) can be estimated to the multivariate Normal distribution that is given by following equation:

f_k(x) = \frac{1}{(2\pi)^{p/2}|\mathbf\Sigma|^{1/2}} e^{-\frac{1}{2}(x-\mu_k)^T \mathbf{\Sigma}^{-1}(x-\mu_k)}

where, \mu_k = mean\, of\, the\, examples\, of \, category\, k \\ \mathbf{\sum} = covariance \, (we\, assume\, common\, covariance\, for\, all\, categories)

and P(Y=k) =\pi_k. Now, we try to write the above equation with the assumptions:

P(Y=k | X=x) = \frac{\pi_k \frac{1}{(2\pi)^{p/2}|\mathbf\Sigma|^{1/2}} e^{-\frac{1}{2}(x-\mu_k)^T \mathbf{\Sigma}^{-1}(x-\mu_k)}}{\sum_{j=1}^{K} \frac{1}{(2\pi)^{p/2}|\mathbf\Sigma|^{1/2}} e^{-\frac{1}{2}(x-\mu_j)^T \mathbf{\Sigma}^{-1}(x-\mu_j)}}

Now, we take log both sides and maximizing the equation, we get the decision boundary:

\delta_k(x) = \log \pi_k - \frac{1}{2}\mu_k^T \Sigma^{-1}\mu_k + x^T \Sigma^{-1}\mu_k

For two classes, the decision boundary is a linear function of x where both classes give equal value, this linear function is given as:

\left\{x: \delta_k(x) = \delta_{\ell}(x) \right\}, 1 \leq j,\ell \leq K

For multi-class (K>2), we need to estimate the pK means, pK variance, K prior proportions and \binom{p}{2}K = \left ( \frac{p(p-1)}{2} \right )K . Now, we discuss in more detail about Quadratic Discriminant Analysis.

Quadratic Discriminant Analysis

Quadratic discriminant analysis is quite similar to Linear discriminant analysis except we relaxed the assumption that the mean and covariance of all the classes were equal. Therefore, we required to calculate it separately.

Now, for each of the class y the covariance matrix is given by:

\Sigma_y = \frac{1}{N_y-1} \sum_{y_i = y} (x_i - \mu_y)(x_i -\mu_y)^T

By adding the following term and solving (taking log both side and ). The quadratic Discriminant function is given by:

\delta_k(x) = \log \pi_k - \frac{1}{2}\mu_k^T \mathbf{\Sigma}_k^{-1}\mu_k + x^T \mathbf{\Sigma}_k^{-1}\mu_k - \frac{1}{2}x^T \Sigma_k^{-1}x -\frac{1}{2}\log |\Sigma_k|

Implementation

import libraries

library(caret) library(MASS) library(tidyverse)

Code to plot decision plot

decision_boundary = function(model, data,vars, resolution = 200,...) { class='Species' labels_var = data[,class] k = length(unique(labels_var))

For sepals

if (vars == 'sepal'){ data = data %>% select(Sepal.Length, Sepal.Width) } else{ data = data %>% select(Petal.Length, Petal.Width) }

plot with color labels

int_labels = as.integer(labels_var) plot(data, col = int_labels+1L, pch = int_labels+1L, ...)

make grid

r = sapply(data, range, na.rm = TRUE) xs = seq(r[1,1], r[2,1], length.out = resolution) ys = seq(r[1,2], r[2,2], length.out = resolution) dfs = cbind(rep(xs, each=resolution), rep(ys, time = resolution))

colnames(dfs) = colnames(r) dfs = as.data.frame(dfs)

p = predict(model, dfs, type ='class' ) p = as.factor(p$class)

points(dfs, col = as.integer(p)+1L, pch = ".")

mats = matrix(as.integer(p), nrow = resolution, byrow = TRUE) contour(xs, ys, mats, add = TRUE, lwd = 2, levels = (1:(k-1))+.5)

invisible(mats) }

par(mfrow=c(2,2))

run the linear discriminant analysis and plot the decision boundary with Sepals variable

model = lda(Species ~ Sepal.Length + Sepal.Width, data=iris) lda_sepals = decision_boundary(model, iris, vars= 'sepal' , main = "LDA_Sepals")

run the quadratic discriminant analysis and plot the decision boundary with Sepals variable

model_qda = qda(Species ~ Sepal.Length + Sepal.Width, data=iris) qda_sepals = decision_boundary(model_qda, iris, vars= 'sepal', main = "QDA_Sepals")

run the linear discriminant analysis and plot the decision boundary with Petals variable

model = lda(Species ~ Petal.Length + Petal.Width, data=iris) lda_petal =decision_boundary(model, iris, vars='petal', main = "LDA_petals")

run the quadratic discriminant analysis and plot the decision boundary with Petals variable

model_qda = qda(Species ~ Petal.Length + Petal.Width, data=iris) qda_petal =decision_boundary(model_qda, iris, vars='petal', main = "QDA_petals")

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LDA and QDA visualization

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