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.

## Implementation

```
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[1]
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[0]
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)
```