Quadratic discriminant analysis is a method you can use when you have a set of predictor variables and you’d like to classify a response variable into two or more classes. It is considered to be the non-linear equivalent to linear discriminant analysis.
This tutorial provides a step-by-step example of how to perform quadratic discriminant analysis in R.
Step 1: Load Necessary Libraries
First, we’ll load the necessary libraries for this example:
library(MASS)
library(ggplot2)
Step 2: Load the Data
For this example, we’ll use the built-in iris dataset in R. The following code shows how to load and view this dataset:
#attach iris dataset to make it easy to work with attach(iris) #view structure of dataset str(iris) 'data.frame': 150 obs. of 5 variables: $ Sepal.Length: num 5.1 4.9 4.7 4.6 5 5.4 4.6 5 4.4 4.9 ... $ Sepal.Width : num 3.5 3 3.2 3.1 3.6 3.9 3.4 3.4 2.9 3.1 ... $ Petal.Length: num 1.4 1.4 1.3 1.5 1.4 1.7 1.4 1.5 1.4 1.5 ... $ Petal.Width : num 0.2 0.2 0.2 0.2 0.2 0.4 0.3 0.2 0.2 0.1 ... $ Species : Factor w/ 3 levels "setosa","versicolor",..: 1 1 1 1 1 1 1 ...
We can see that the dataset contains 5 variables and 150 total observations.
For this example we’ll build a quadratic discriminant analysis model to classify which species a given flower belongs to.
We’ll use the following predictor variables in the model:
- Sepal.length
- Sepal.Width
- Petal.Length
- Petal.Width
And we’ll use them to predict the response variable Species, which takes on the following three potential classes:
- setosa
- versicolor
- virginica
Step 3: Create Training and Test Samples
Next, we’ll split the dataset into a training set to train the model on and a testing set to test the model on:
#make this example reproducible set.seed(1) #Use 70% of dataset as training set and remaining 30% as testing set sample TRUE, FALSE), nrow(iris), replace=TRUE, prob=c(0.7,0.3)) train
Step 4: Fit the QDA Model
Next, we’ll use the qda() function from the MASS package to fit the QDA model to our data:
#fit QDA model
model #view model output
model
Call:
qda(Species ~ ., data = train)
Prior probabilities of groups:
setosa versicolor virginica
0.3207547 0.3207547 0.3584906
Group means:
Sepal.Length Sepal.Width Petal.Length Petal.Width
setosa 4.982353 3.411765 1.482353 0.2411765
versicolor 5.994118 2.794118 4.358824 1.3676471
virginica 6.636842 2.973684 5.592105 2.0552632
Here is how to interpret the output of the model:
Prior probabilities of group: These represent the proportions of each Species in the training set. For example, 35.8% of all observations in the training set were of species virginica.
Group means: These display the mean values for each predictor variable for each species.
Step 5: Use the Model to Make Predictions
Once we’ve fit the model using our training data, we can use it to make predictions on our test data:
#use QDA model to make predictions on test data
predicted predict(model, test)
names(predicted)
[1] "class" "posterior" "x"
This returns a list with two variables:
- class: The predicted class
- posterior: The posterior probability that an observation belongs to each class
We can quickly view each of these results for the first six observations in our test dataset:
#view predicted class for first six observations in test set head(predicted$class) [1] setosa setosa setosa setosa setosa setosa Levels: setosa versicolor virginica #view posterior probabilities for first six observations in test set head(predicted$posterior) setosa versicolor virginica 4 1 7.224770e-20 1.642236e-29 6 1 6.209196e-26 8.550911e-38 7 1 1.248337e-21 8.132700e-32 15 1 2.319705e-35 5.094803e-50 17 1 1.396840e-29 9.586504e-43 18 1 7.581165e-25 8.611321e-37
Step 6: Evaluate the Model
We can use the following code to see what percentage of observations the QDA model correctly predicted the Species for:
#find accuracy of model
mean(predicted$class==test$Species)
[1] 1
It turns out that the model correctly predicted the Species for 100% of the observations in our test dataset.
In the real-world an QDA model will rarely predict every class outcome correctly, but this iris dataset is simply built in a way that machine learning algorithms tend to perform very well on it.
You can find the complete R code used in this tutorial here.