Principal Component Analysis (PCA) is a popular dimensionality reduction technique, especially useful when you have a high number of correlated features.

By transforming the original features into a new set of non-correlated features, principal components (PCs) , PCA simplifies and speeds up machine learning algorithms such as clustering and regression.

The PCA Process

1. Standardization: Depending on the dataset, it might be necessary to standardize the features for better results.

2. Covariance Matrix Calculation: Determine the covariance among features.

3. Eigenvector & Eigenvalue Computation: From the covariance matrix, derive the eigenvectors and eigenvalues that signify the PCs:

   • Eigenvectors: These are the directions of the new feature space. They represent the PCs.
   • Eigenvalues: The magnitude of the eigenvectors, indicating the amount of variance explained by each PC.

4. Ranking of PCs: Sort the eigenvalues in descending order to identify the most important PCs (those responsible for the most variance).

5. Data Projection: Use the significant eigenvectors to transform the original features into the new feature space.

Variance and Information Loss

PCA aims to retain as much variance in the data as possible. The cumulative explained variance of the top $k$ (out of $n$) PCs gives a measure of the retained information:

$$\text{Explained Variance Ratio} = \frac{\text{Sum of } k \text{ highest eigenvalues}}{\text{Sum of all eigenvalues}}$$

Choosing the Right Number of Components

An important step before applying PCA is selecting the number of PCs to retain. Common methods include the “Elbow Method,” Visual Scree Test, and Kaiser-Guttman Criterion.

Code Example: PCA with scikit-learn

Here is the Python code:

from sklearn.decomposition import PCA
from sklearn.preprocessing import StandardScaler
import pandas as pd

# Load data
url = "https://archive.ics.uci.edu/ml/machine-learning-databases/iris/iris.data"
df = pd.read_csv(url, names=['sepal length', 'sepal width', 'petal length', 'petal width', 'target'])

# Standardize the Data
features = ['sepal length', 'sepal width', 'petal length', 'petal width']
x = df.loc[:, features].values
x = StandardScaler().fit_transform(x)

# PCA Projection to 2D
pca = PCA(n_components=2)
principalComponents = pca.fit_transform(x)

# Visualize in DataFrame
principalDf = pd.DataFrame(data=principalComponents, columns=['PC1', 'PC2'])
finalDf = pd.concat([principalDf, df[['target']]], axis=1)