Dimensionality Reduction

Large or highly correlated feature sets pose challenges in analysis. PCA reduces dimensionality by extracting principal components; PCR uses PCA in regression to mitigate multicollinearity; K-Means segmentation groups observations based on similarity. These techniques address dimension and clustering needs in exploratory and predictive contexts.

DATA REDUCTION METHODS

High-dimensional or correlated feature sets can obscure insights and hinder modeling. This section focuses on producing fewer, more robust features. First, Principal Component Analysis (PCA) identifies orthogonal components that capture variance. Then, Principal Component Regression (PCR) applies PCA within linear regression to manage correlated predictors effectively.

PRINCIPAL COMPONENT ANALYSIS (PCA)

Learning Objectives

• Perform PCA to reduce dimensions while preserving variance

• Interpret scree plots, loadings, variance explained

• Recognize PCA’s strengths/limitations (linear assumptions, interpretability)

Indicative Content

• Eigen Decomposition

  • Covariance/correlation matrix for principal components

• Scree Plot & Loadings

  • Determining component count, variable contributions

• Python Tools

  • sklearn.decomposition.PCA

PRINCIPAL COMPONENT REGRESSION (PCR)

Learning Objectives

• Apply PCA in linear regression to handle correlated predictors

• Decide how many PCs to retain (variance vs. parsimony)

• Compare PCR to standard OLS for improved stability under multicollinearity

Indicative Content

• PCR Steps

  • Standardize data → PCA → choose k PCs → regress Y on PCs

• Advantages/Disadvantages

  • Reduced correlation, less direct interpretability

• Implementation

  • PCA then regression (e.g., statsmodels or sklearn.linear_model)

TOOLS & METHODOLOGIES (DIMENSIONALITY REDUCTION)

• Python Libraries

  • sklearn.decomposition.PCA for component extraction

  • statsmodels or sklearn.linear_model for regression

• Workflow

  • Scale data → extract principal components → regress or interpret variance

• Evaluation

  • Scree plots to decide number of components

  • Explained variance vs. interpretability trade-offs