13.1 Real-world case studies in data science using linear algebra

Linear algebra forms the backbone of data science, enabling powerful techniques for analysis and prediction. From representing data as vectors to complex matrix operations, it's essential for tasks like dimensionality reduction and recommendation systems.

This section dives into real-world applications, showing how linear algebra solves practical problems. We'll explore matrix factorization for recommendations, PCA for dimensionality reduction, and linear regression for predictive analytics, connecting theory to practice.

Linear Algebra for Data Science Problems

Data Representation and Preprocessing

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  Frontiers | A Comparison for Dimensionality Reduction Methods of Single-Cell RNA-seq Data View original

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  Frontiers | Linear and Non-linear Dimensionality-Reduction Techniques on Full Hand Kinematics View original

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  Frontiers | A Comparison for Dimensionality Reduction Methods of Single-Cell RNA-seq Data View original

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  Frontiers | Linear and Non-linear Dimensionality-Reduction Techniques on Full Hand Kinematics View original

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• Linear algebra represents data as vectors and matrices creating a powerful framework for solving complex data science problems
• Feature extraction and transformation techniques prepare data for linear algebra operations
  • One-hot encoding converts categorical variables into binary vectors
  • Normalization scales numerical features to a common range (0-1 or -1 to 1)
• Linear transformations and projections enable data visualization and dimensionality reduction in high-dimensional datasets
  • Example: Projecting 3D data onto a 2D plane for easier visualization
  • Example: Transforming RGB color space to grayscale using matrix multiplication

Matrix Operations and Decompositions

• Matrix operations form the foundation for implementing machine learning algorithms efficiently
  • Multiplication combines information from multiple sources (feature matrices and weight vectors)
  • Inversion solves systems of linear equations (least squares regression)
  • Transposition reorganizes data for specific computations (covariance matrix calculation)
• Eigenvalue decomposition and singular value decomposition (SVD) serve as fundamental matrix factorization methods
  • Eigenvalue decomposition: A = QΛQ^(-1), where Q contains eigenvectors and Λ contains eigenvalues
  • SVD: A = UΣV^T, where U and V are orthogonal matrices and Σ contains singular values
• Solving systems of linear equations underpins many optimization problems in data science
  • Least squares regression: minimize ||Ax - b||^2
  • Support vector machines: maximize margin between classes subject to linear constraints

Matrix Factorization for Recommendations

Collaborative Filtering Techniques

• Matrix factorization decomposes user-item interaction matrices into lower-dimensional latent factor matrices
  • Example: Netflix movie ratings matrix factored into user preferences and movie characteristics
• Singular Value Decomposition (SVD) identifies latent factors in user-item interactions
  • Decomposition: R ≈ U * Σ * V^T, where U represents user factors and V represents item factors
• Non-negative Matrix Factorization (NMF) handles non-negative data like user ratings or item features
  • Constraint: R ≈ W * H, where W and H contain non-negative elements
• Alternating Least Squares (ALS) solves matrix factorization problems in large-scale recommendation systems
  • Iteratively updates user and item factors while keeping the other fixed

Model Optimization and Evaluation

• Regularization techniques prevent overfitting in matrix factorization models
  • L1 regularization (Lasso) adds absolute value of coefficients to loss function
  • L2 regularization (Ridge) adds squared values of coefficients to loss function
• Evaluation metrics assess the performance of matrix factorization models
  • Mean Absolute Error (MAE): average absolute difference between predicted and actual ratings
  • Root Mean Square Error (RMSE): square root of average squared difference between predicted and actual ratings

Principal Component Analysis for Dimensionality Reduction

PCA Fundamentals and Computation

• Principal Component Analysis (PCA) identifies directions of maximum variance in high-dimensional data
  • Example: Reducing 1000-dimensional gene expression data to 10 principal components
• Covariance matrix and its eigendecomposition form the basis of PCA
  • Covariance matrix: C = (1/n) * X^T * X, where X is the centered data matrix
  • Eigendecomposition: C = V * Λ * V^T, where V contains eigenvectors (principal components) and Λ contains eigenvalues
• Singular Value Decomposition (SVD) provides an efficient method for computing principal components
  • SVD of centered data matrix: X = U * Σ * V^T, where V contains principal components

PCA Applications and Extensions

• Scree plots and cumulative explained variance ratios determine optimal number of principal components
  • Scree plot: eigenvalues vs. component number, look for "elbow" in the curve
  • Cumulative explained variance ratio: sum of explained variances up to k components divided by total variance
• PCA applications span various domains for feature extraction, noise reduction, and visualization
  • Image processing: compressing images by retaining top principal components
  • Bioinformatics: analyzing gene expression patterns across multiple experiments
• Kernel PCA extends PCA to nonlinear dimensionality reduction
  • Projects data into higher-dimensional spaces using kernel methods (polynomial, radial basis function)
  • Example: Separating concentric circles using RBF kernel PCA

Linear Regression for Predictive Analytics

Model Formulation and Estimation

• Linear regression models relationships between dependent and independent variables using linear equations
  • Single variable: y = β0 + β1x + ε
  • Multiple variables: y = β0 + β1x1 + β2x2 + ... + βnxn + ε
• Least squares method estimates coefficients by minimizing sum of squared residuals
  • Minimizes: Σ(yi - ŷi)^2, where yi are observed values and ŷi are predicted values
• Matrix formulation enables efficient computation of model parameters
  • y = Xβ + ε, where X is the design matrix and β is the coefficient vector
  • Closed-form solution: β = (X^T * X)^(-1) * X^T * y

Model Evaluation and Refinement

• Multicollinearity detection addresses issues with correlated predictor variables
  • Correlation analysis: compute pairwise correlations between predictors
  • Variance Inflation Factor (VIF): measures how much variance of a coefficient is inflated due to multicollinearity
• Regularization methods prevent overfitting and improve model generalization
  • Ridge regression (L2): adds penalty term λ * Σβj^2 to loss function
  • Lasso regression (L1): adds penalty term λ * Σ|βj| to loss function
• Model evaluation metrics assess predictive performance and model fit
  • R-squared: proportion of variance in dependent variable explained by the model
  • Adjusted R-squared: R-squared adjusted for number of predictors
  • Mean Squared Error (MSE): average squared difference between predicted and actual values
• Residual analysis validates assumptions of linear regression models
  • Residuals vs. fitted values plot: checks for homoscedasticity and linearity
  • Q-Q plot: assesses normality of residuals