In this Machine Learning from Scratch Tutorial, we are going to implement the LDA algorithm using only built-in Python modules and numpy. LDA (Linear Discriminant Analysis) is a feature reduction technique and a common preprocessing step in machine learning pipelines. We will learn about the concept and the math behind this popular ML algorithm, and how to implement it in Python.
All algorithms from this course can be found on GitHub together with example tests.
import numpy as np class LDA: def __init__(self, n_components): self.n_components = n_components self.linear_discriminants = None def fit(self, X, y): n_features = X.shape class_labels = np.unique(y) # Within class scatter matrix: # SW = sum((X_c - mean_X_c)^2 ) # Between class scatter: # SB = sum( n_c * (mean_X_c - mean_overall)^2 ) mean_overall = np.mean(X, axis=0) SW = np.zeros((n_features, n_features)) SB = np.zeros((n_features, n_features)) for c in class_labels: X_c = X[y == c] mean_c = np.mean(X_c, axis=0) # (4, n_c) * (n_c, 4) = (4,4) -> transpose SW += (X_c - mean_c).T.dot((X_c - mean_c)) # (4, 1) * (1, 4) = (4,4) -> reshape n_c = X_c.shape mean_diff = (mean_c - mean_overall).reshape(n_features, 1) SB += n_c * (mean_diff).dot(mean_diff.T) # Determine SW^-1 * SB A = np.linalg.inv(SW).dot(SB) # Get eigenvalues and eigenvectors of SW^-1 * SB eigenvalues, eigenvectors = np.linalg.eig(A) # -> eigenvector v = [:,i] column vector, transpose for easier calculations # sort eigenvalues high to low eigenvectors = eigenvectors.T idxs = np.argsort(abs(eigenvalues))[::-1] eigenvalues = eigenvalues[idxs] eigenvectors = eigenvectors[idxs] # store first n eigenvectors self.linear_discriminants = eigenvectors[0:self.n_components] def transform(self, X): # project data return np.dot(X, self.linear_discriminants.T)