Add Principal Component Analysis Algorithm in R

data_preprocessing/pca.r

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+# ==============================================
+# Principal Component Analysis (PCA)
+# ==============================================
+# Algorithm: Dimensionality reduction using orthogonal linear transformation
+# Framework: Base R
+#
+# Purpose:
+# - Reduce high-dimensional data into a smaller set of uncorrelated variables (principal components)
+# - Preserve as much variance as possible
+#
+# Steps:
+# 1. Standardize the dataset (zero mean, unit variance)
+# 2. Compute the covariance matrix of the standardized data
+# 3. Compute eigenvalues and eigenvectors of the covariance matrix
+# 4. Sort eigenvectors by decreasing eigenvalues (most variance first)
+# 5. Project original data onto top k eigenvectors to get reduced data
+#
+# Complexity:
+# - Time:  O(n * d^2 + d^3)  where n = samples, d = features
+# - Space: O(d^2 + n * d)
+#
+# Applications:
+# - Data visualization, noise reduction, feature extraction
+# - Preprocessing for machine learning models
+# ==============================================
+
+# PCA Algorithm Implementation (Algorithm only)
+pca_algorithm <- function(X, k) {
+  # Basic input validation (kept minimal to match repo style)
+  if (is.vector(X)) {
+    X <- matrix(X, ncol = 1)
+  }
+  if (!is.matrix(X) || !is.numeric(X)) {
+    stop("Input 'X' must be a numeric matrix or vector")
+  }
+  d <- ncol(X)
+  if (k <= 0 || k > d) {
+    stop("'k' must be between 1 and the number of columns of X")
+  }
+
+  # Step 1: Standardize the data (zero mean, unit variance per feature)
+  X_std <- scale(X)
+
+  # Step 2: Compute covariance matrix of standardized data
+  cov_matrix <- cov(X_std)
+
+  # Step 3: Eigen decomposition (covariance is symmetric)
+  eig <- eigen(cov_matrix)
+  eig_values <- eig$values
+  eig_vectors <- eig$vectors
+
+  # Step 4: Select top k principal components (eigenvectors)
+  top_vectors <- eig_vectors[, 1:k, drop = FALSE]
+
+  # Step 5: Project standardized data onto top k components
+  X_reduced <- X_std %*% top_vectors
+
+  return(list(
+    reduced_data = X_reduced,
+    components = top_vectors,
+    eigenvalues = eig_values
+  ))
+}
+
+# Example usage (algorithm only)
+# set.seed(42)
+# X <- matrix(rnorm(50 * 5), nrow = 50, ncol = 5)
+# pca_result <- pca_algorithm(X, k = 2)
+# head(pca_result$reduced_data)