How can discriminant analysis predict group membership?

1 What is discriminant analysis?

Discriminant analysis is a form of supervised learning, which means that you need to have a training dataset with known group labels. The goal of discriminant analysis is to find a linear or quadratic function that best separates the groups and maximizes the between-group variance relative to the within-group variance. This function is called the discriminant function, and it can be used to assign new observations to the most likely group. There are two main types of discriminant analysis: linear discriminant analysis (LDA) and quadratic discriminant analysis (QDA). LDA assumes that the groups have equal covariance matrices, while QDA allows for different covariance matrices.

2 What are the assumptions of discriminant analysis?

Before applying discriminant analysis, it is important to check some assumptions. These assumptions include that the predictor variables are continuous and follow a multivariate normal distribution within each group, that the groups are mutually exclusive and collectively exhaustive, and that the sample size is large enough to ensure reliable estimates of the group means and covariance matrices. Additionally, for LDA, the covariance matrices of the groups must be equal or similar in shape and size.

3 What are the limitations of discriminant analysis?

Discriminant analysis has some limitations to consider when interpreting the results. Firstly, it is sensitive to outliers and multicollinearity, which can distort the group separation and reduce accuracy. Additionally, it may not capture nonlinear relationships between the predictor variables and the group membership, thus reducing its predictive power. Lastly, it may not account for the unequal prior probabilities of the groups, potentially biasing the classification towards more frequent groups.

4 How to perform discriminant analysis in R?

To perform discriminant analysis in R, you can use the functions lda() and qda() from the MASS package. These functions take as arguments a formula that specifies the group variable and the predictor variables, and a data frame that contains the variables. The functions return an object that contains the coefficients of the discriminant function, the group means, the prior probabilities, and the classification results. You can also use the predict() function to apply the discriminant function to new data and obtain the predicted group labels and probabilities. Here is an example of how to perform LDA and QDA on the iris dataset, which contains four measurements of three species of iris flowers:

# Load the MASS package
library(MASS)
# Perform LDA on the iris dataset
lda.model <- lda(Species ~ Sepal.Length + Sepal.Width + Petal.Length + Petal.Width, data = iris)
# Print the summary of the LDA model
summary(lda.model)
# Perform QDA on the iris dataset
qda.model <- qda(Species ~ Sepal.Length + Sepal.Width + Petal.Length + Petal.Width, data = iris)
# Print the summary of the QDA model
summary(qda.model)
# Predict the group membership of a new observation using LDA
new.obs <- data.frame(Sepal.Length = 5.5, Sepal.Width = 3.2, Petal.Length = 1.5, Petal.Width = 0.4)
predict(lda.model, new.obs)
# Predict the group membership of a new observation using QDA
predict(qda.model, new.obs)        

6 Here’s what else to consider

This is a space to share examples, stories, or insights that don’t fit into any of the previous sections. What else would you like to add?