11.3 Principal component analysis (PCA)

Principal component analysis (PCA) is a powerful dimensionality reduction technique used in computational genomics. It transforms high-dimensional genomic data into a lower-dimensional space, preserving important information while enabling efficient analysis and visualization.

PCA identifies directions of maximum variance in data, using eigendecomposition of the covariance matrix. This process yields principal components that capture key patterns, allowing researchers to explore relationships between variables and samples in genomic datasets.

Dimensionality reduction techniques

• Dimensionality reduction aims to transform high-dimensional data into a lower-dimensional space while preserving important information
• Essential in computational genomics due to the high dimensionality of genomic data (gene expression, SNPs, epigenetic markers)
• Enables more efficient data storage, processing, and visualization, facilitating downstream analysis and interpretation

Feature selection vs feature extraction

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• Feature selection involves selecting a subset of the original features based on their relevance or importance
  • Filters irrelevant or redundant features, reducing data dimensionality
  • Examples include univariate statistical tests (t-tests, ANOVA), correlation-based methods, and regularization techniques (LASSO, elastic net)
• Feature extraction creates new features by combining or transforming the original features
  • Generates a lower-dimensional representation that captures the essential information
  • Examples include PCA, t-SNE, autoencoders, and matrix factorization methods

Unsupervised learning algorithms

• Unsupervised learning algorithms discover hidden patterns or structures in data without relying on predefined labels or outcomes
• Particularly useful in exploratory data analysis and hypothesis generation in genomics
• Examples include clustering algorithms (k-means, hierarchical clustering), self-organizing maps (SOMs), and topic modeling (latent Dirichlet allocation)

Mathematical foundations of PCA

• PCA is a linear dimensionality reduction technique that identifies the directions of maximum variance in the data
• Relies on the eigendecomposition of the data covariance matrix to find the principal components
• Projecting data onto the principal components allows for a lower-dimensional representation while preserving the most important information

Eigenvectors and eigenvalues

• Eigenvectors are vectors that, when a linear transformation is applied, only change in scale but not in direction
  • Mathematically, for a square matrix $A$, an eigenvector $v$ satisfies $Av = \lambda v$, where $\lambda$ is the corresponding eigenvalue
• Eigenvalues represent the scaling factor applied to the eigenvectors during the linear transformation
• In PCA, the eigenvectors of the covariance matrix represent the principal components, and the eigenvalues indicate the amount of variance explained by each component

Covariance matrices

• The covariance matrix captures the pairwise covariances between variables in a dataset
  • Covariance measures the joint variability of two variables, indicating how they change together
• For a dataset with $n$ observations and $p$ variables, the covariance matrix is a $p \times p$ symmetric matrix
• The diagonal elements of the covariance matrix represent the variances of individual variables, while the off-diagonal elements represent the covariances between variable pairs

Orthogonal transformations

• Orthogonal transformations are linear transformations that preserve the angles and distances between vectors
  • Examples include rotations, reflections, and permutations
• In PCA, the principal components form an orthogonal basis, meaning they are perpendicular to each other
• Orthogonality ensures that the principal components capture distinct and uncorrelated directions of variation in the data

PCA algorithm steps

• PCA involves a series of steps to transform the original data into a lower-dimensional representation
• The main steps include data normalization, covariance matrix computation, eigendecomposition, principal component selection, and data projection

Data normalization and centering

• Data normalization scales the variables to have similar ranges or magnitudes
  • Prevents variables with larger scales from dominating the analysis
  • Common normalization techniques include min-max scaling, z-score standardization, and log-transformation
• Data centering subtracts the mean of each variable from its corresponding values
  • Centers the data around the origin, which is necessary for PCA
  • Ensures that the principal components pass through the center of the data cloud

Computing covariance matrix

• The covariance matrix is computed from the normalized and centered data
  • For a dataset with $n$ observations and $p$ variables, the covariance matrix is calculated as $\frac{1}{n-1} X^T X$, where $X$ is the centered data matrix
• The covariance matrix captures the pairwise covariances between variables
• Diagonal elements represent the variances of individual variables, while off-diagonal elements represent the covariances between variable pairs

Eigendecomposition of covariance matrix

• Eigendecomposition is applied to the covariance matrix to obtain the eigenvectors and eigenvalues
  • Eigenvectors represent the principal components, which are the directions of maximum variance in the data
  • Eigenvalues indicate the amount of variance explained by each principal component
• The eigenvectors are sorted in descending order based on their corresponding eigenvalues
• The sorted eigenvectors form the columns of the principal component matrix

Selecting principal components

• The number of principal components to retain is determined based on the desired level of variance explanation or dimensionality reduction
• Scree plots, which display the eigenvalues in descending order, can help identify the "elbow" point where the variance explained by additional components diminishes
• Cumulative explained variance plots show the proportion of total variance explained by increasing numbers of components
• The selected principal components form a lower-dimensional subspace that captures the most important information in the data

Projecting data onto new subspace

• The original data is projected onto the selected principal components to obtain the lower-dimensional representation
  • The projection is performed by multiplying the centered data matrix by the principal component matrix
• The resulting projected data, also known as principal component scores, represent the original data in the new lower-dimensional subspace
• The projected data can be used for visualization, clustering, or as input for subsequent analyses

Interpreting PCA results

• Interpreting PCA results involves understanding the relationships between variables, observations, and principal components
• Several visualization and diagnostic tools can aid in the interpretation process

Scree plots and explained variance

• Scree plots display the eigenvalues of the principal components in descending order
  • The "elbow" point in the scree plot indicates the number of components that capture a significant portion of the total variance
• Explained variance plots show the proportion of total variance explained by each principal component
  • Cumulative explained variance plots illustrate the cumulative proportion of variance explained by increasing numbers of components
• These plots help determine the optimal number of components to retain for dimensionality reduction

Loadings and variable contributions

• Loadings represent the correlations between the original variables and the principal components
  • High absolute loadings indicate strong associations between variables and components
• Variable contributions measure the importance of each variable in defining the principal components
  • Variables with high contributions have a significant impact on the structure of the low-dimensional representation
• Examining loadings and variable contributions helps identify the key variables driving the patterns in the data

Biplots and data visualization

• Biplots simultaneously display the observations and variables in the principal component space
  • Observations are represented as points, while variables are represented as vectors
• The cosine of the angle between variable vectors approximates their correlation
  • Acute angles indicate positive correlations, obtuse angles indicate negative correlations, and right angles indicate no correlation
• The distance between observation points reflects their similarity in the low-dimensional space
• Biplots facilitate the identification of clusters, outliers, and relationships between observations and variables

Applications of PCA in genomics

• PCA has numerous applications in computational genomics, enabling the exploration and interpretation of high-dimensional genomic data

Gene expression data analysis

• PCA can be applied to gene expression data to identify patterns and sources of variation
  • Helps distinguish between biological conditions, cell types, or experimental treatments
• Principal components can capture batch effects, technical artifacts, or biological factors influencing gene expression
• Visualizing samples in the principal component space can reveal clusters or gradients related to biological processes or phenotypes

Population structure identification

• PCA is commonly used to identify population structure in genetic data
  • Detects genetic subgroups or ancestral populations based on patterns of genetic variation
• Principal components can capture geographic or ethnic differences among individuals
• Visualizing samples in the principal component space can reveal distinct clusters corresponding to different populations or admixture patterns

Genotype-phenotype associations

• PCA can be used as a preprocessing step in genome-wide association studies (GWAS) to control for population stratification
  • Population stratification can lead to spurious associations between genetic variants and phenotypes
• Principal components can be included as covariates in association tests to account for population structure and reduce false positives
• PCA can also be applied to identify genotype-phenotype associations by analyzing the relationship between genetic variants and phenotypic traits in the principal component space

Integration with other omics data

• PCA can be used to integrate multiple omics data types (transcriptomics, proteomics, metabolomics) for a comprehensive analysis
  • Helps identify common patterns or sources of variation across different molecular layers
• Principal components can capture the shared or complementary information between omics data types
• Integrative analysis using PCA can provide insights into the relationships between different biological processes and their impact on phenotypes or diseases

Limitations and considerations

• While PCA is a powerful tool for dimensionality reduction and data exploration, it has certain limitations and considerations that should be taken into account

Linearity assumption of PCA

• PCA assumes that the relationships between variables are linear
  • May not capture non-linear patterns or complex interactions in the data
• Non-linear dimensionality reduction techniques, such as t-SNE or kernel PCA, can be used when linear assumptions are violated
• It is important to assess the linearity of the data and consider alternative methods if necessary

Sensitivity to data scaling

• PCA is sensitive to the scaling of the variables
  • Variables with larger scales or variances can dominate the analysis and influence the principal components
• Proper data normalization and scaling (e.g., z-score standardization) should be applied before performing PCA
• The choice of scaling method can impact the interpretation of the results and should be carefully considered based on the nature of the data and research question

Dealing with missing data

• PCA requires complete data without missing values
  • Missing data can pose challenges and bias the analysis if not handled appropriately
• Common strategies for dealing with missing data include listwise deletion, pairwise deletion, and imputation methods (mean imputation, k-nearest neighbors, multiple imputation)
• The choice of missing data handling approach depends on the extent and pattern of missingness, as well as the assumptions about the missing data mechanism

Choosing optimal number of components

• Determining the optimal number of principal components to retain is a crucial decision in PCA
  • Retaining too few components may lead to loss of important information, while retaining too many components may introduce noise and hinder interpretation
• Scree plots, explained variance plots, and cumulative explained variance plots can guide the selection of the number of components
• Cross-validation techniques, such as leave-one-out cross-validation or k-fold cross-validation, can be used to assess the stability and predictive performance of different numbers of components

Extensions and variations of PCA

• Several extensions and variations of PCA have been developed to address specific challenges or incorporate additional information

Kernel PCA for non-linear data

• Kernel PCA is an extension of PCA that can capture non-linear patterns in the data
  • Maps the original data into a higher-dimensional feature space using a kernel function
  • Performs PCA in the transformed feature space, enabling the identification of non-linear relationships
• Common kernel functions include polynomial kernels, radial basis function (RBF) kernels, and sigmoid kernels
• Kernel PCA can be particularly useful when dealing with complex, non-linear data structures

Sparse PCA for feature selection

• Sparse PCA is a variation of PCA that incorporates feature selection by inducing sparsity in the principal components
  • Encourages some loadings to be exactly zero, effectively selecting a subset of the original variables
• Sparse PCA can be achieved through regularization techniques such as L1 (LASSO) or elastic net penalties
• Particularly useful when dealing with high-dimensional data with many irrelevant or noisy features
• Facilitates the identification of key variables contributing to the principal components and improves interpretability

Probabilistic PCA and latent variables

• Probabilistic PCA is a generative model that extends PCA to a probabilistic framework
  • Assumes that the observed data is generated from a set of latent variables with Gaussian noise
• Probabilistic PCA estimates the parameters of the generative model using maximum likelihood estimation
• Enables the handling of missing data, incorporation of prior knowledge, and estimation of uncertainty in the principal components
• Latent variables in probabilistic PCA can be interpreted as underlying factors or hidden causes of the observed data

Tensor decomposition methods

• Tensor decomposition methods extend PCA to higher-order tensors, which are multi-dimensional arrays
  • Examples include Tucker decomposition and CANDECOMP/PARAFAC (CP) decomposition
• Particularly useful when dealing with multi-way data, such as time-series gene expression data or multi-omics data integration
• Tensor decomposition methods can capture complex interactions and patterns across multiple dimensions or modalities
• Enable the identification of latent factors or components that jointly explain the variation in the higher-order data