Linear Discriminant Analysis (LDA): Linear Discriminant Analysis(LDA) is a dimensionality reduction technique, that separates the best classes that are related to the dependent variable. Which makes it a supervised algorithm. In PCA, we do not consider the dependent variable. So this is the basic difference between the PCA and LDA algorithms.

If there are n number of independent variables, the LDA algorithm will extract p<=n new independent variables that separate most of the classes of the dependent variable.

LDA in Python: LDA is a very simple and popular algorithm in practice. In this tutorial, we will implement this algorithm alongside with Logistic Regression algorithm. For this task, we will use the famous "Wine.csv" dataset from the UCI machine learning repository. Our version of dataset contains thirteen independent variables that represent various aspects of wines and one dependent variable that represent the three types of buyers of the wine based on specific features. Now, we will implement LDA to reduce the number of independent variables to a predefined value (i.e. two).

You can download the whole dataset from here.

First of all, we will import some essential libraries.

# Importing the libraries
import numpy as np
import matplotlib.pyplot as plt
import pandas as pd

Then we will import our dataset and make the feature matrix and dependent variable vector.

# Importing the dataset
dataset = pd.read_csv('Wine.csv')
X = dataset.iloc[:, 0:13].values
y = dataset.iloc[:, 13].values

Now, we will split the dataset into training and test sets.

# Splitting the dataset into the Training set and Test set
from sklearn.model_selection import train_test_split
X_train, X_test, y_train, y_test = train_test_split(X, y, test_size = 0.2, random_state = 0)

Let's apply feature scaling onto the dataset.

# Feature Scaling
from sklearn.preprocessing import StandardScaler
sc = StandardScaler()
X_train = sc.fit_transform(X_train)
X_test = sc.transform(X_test)

We have come to the main part of our program. Now, we will implement the LDA algorithm to the dataset.

# Applying LDA
from sklearn.discriminant_analysis import LinearDiscriminantAnalysis as LDA
lda = LDA(n_components = 2)
X_train = lda.fit_transform(X_train, y_train)
X_test = lda.transform(X_test)

Note: n_component is the parameter that represents the number of independent variables we want in our model. Here we take it to 2, so our model will contain two independent variables.

Let's fit a linear regression algorithm to our model.

# Fitting Logistic Regression to the Training set
from sklearn.linear_model import LogisticRegression
classifier = LogisticRegression(random_state = 0)
classifier.fit(X_train, y_train)

# Predicting the Test set results
y_pred = classifier.predict(X_test)

Now, let's see how well our model is to make predictions using the confusion matrix.

# Making the Confusion Matrix
from sklearn.metrics import confusion_matrix
cm = confusion_matrix(y_test, y_pred)

                                                                       

Let's visualize the prediction of our model for both training and test sets.

# Visualising the Training set results
from matplotlib.colors import ListedColormap
X_set, y_set = X_train, y_train
X1, X2 = np.meshgrid(np.arange(start = X_set[:, 0].min() - 1, stop = X_set[:, 0].max() + 1, step = 0.01),
                     np.arange(start = X_set[:, 1].min() - 1, stop = X_set[:, 1].max() + 1, step = 0.01))
plt.contourf(X1, X2, classifier.predict(np.array([X1.ravel(), X2.ravel()]).T).reshape(X1.shape),
             alpha = 0.75, cmap = ListedColormap(('red', 'green', 'blue')))
plt.xlim(X1.min(), X1.max())
plt.ylim(X2.min(), X2.max())
for i, j in enumerate(np.unique(y_set)):
    plt.scatter(X_set[y_set == j, 0], X_set[y_set == j, 1],
                c = ListedColormap(('red', 'green', 'blue'))(i), label = j)
plt.title('Logistic Regression (Training set)')
plt.xlabel('LD1')
plt.ylabel('LD2')
plt.legend()
plt.show()

The graph will look like the following illustration

# Visualising the Test set results
from matplotlib.colors import ListedColormap
X_set, y_set = X_test, y_test
X1, X2 = np.meshgrid(np.arange(start = X_set[:, 0].min() - 1, stop = X_set[:, 0].max() + 1, step = 0.01),
                     np.arange(start = X_set[:, 1].min() - 1, stop = X_set[:, 1].max() + 1, step = 0.01))
plt.contourf(X1, X2, classifier.predict(np.array([X1.ravel(), X2.ravel()]).T).reshape(X1.shape),
             alpha = 0.75, cmap = ListedColormap(('red', 'green', 'blue')))
plt.xlim(X1.min(), X1.max())
plt.ylim(X2.min(), X2.max())
for i, j in enumerate(np.unique(y_set)):
    plt.scatter(X_set[y_set == j, 0], X_set[y_set == j, 1],
                c = ListedColormap(('red', 'green', 'blue'))(i), label = j)
plt.title('Logistic Regression (Test set)')
plt.xlabel('LD1')
plt.ylabel('LD2')
plt.legend()
plt.show()                                        

The output graph will be-