Principal Component Analysis (PCA) is a statistical technique that simplifies complex datasets by reducing the number of variables while retaining as much original information as possible. Its core purpose is to transform data into a new set of uncorrelated variables, known as principal components, which capture the highest variance within the data. PCA is utilized across various fields, helping researchers and analysts make sense of large datasets.
Data Preparation
Data must undergo preparation before PCA calculations. The first step involves centering the data by subtracting the mean of each feature from all its values. This shifts the data’s origin to the center, focusing analysis on data spread.
Scaling the data is also often necessary, especially when features have different units or magnitudes. This involves dividing each centered value by its feature’s standard deviation. Standardization ensures all features contribute equally, preventing variables with larger values from disproportionately influencing the principal components. Without these preparation steps, features with larger scales could dominate the analysis, masking true underlying patterns.
Constructing the Covariance Matrix
After data preparation, the next step in PCA is constructing the covariance matrix. This square matrix summarizes the relationships between all pairs of features in the dataset. Each entry indicates how two variables change together.
The diagonal elements represent the variance of each individual feature, showing its deviation from the mean. Off-diagonal elements represent the covariance between different feature pairs, indicating if they tend to increase or decrease together. A positive covariance suggests variables increase together, while a negative covariance implies an inverse relationship. This matrix captures data spread and interdependencies, providing the foundational information PCA uses to identify directions of maximum variance.
Deriving Eigenvalues and Eigenvectors
The mathematical core of PCA involves deriving eigenvalues and eigenvectors from the covariance matrix. Eigenvectors are specific directions in the data space where data exhibits the most variance. These vectors represent the principal components, forming new axes for the dataset.
Each eigenvector has a corresponding eigenvalue, a numerical value quantifying the variance captured along that direction. A larger eigenvalue indicates its eigenvector captures more overall variability. Eigenvectors are ordered by their eigenvalues from largest to smallest. The first eigenvector (first principal component) points in the direction of greatest variance, the second captures the next most, and so on. This decomposition transforms original, possibly correlated features into a new set of uncorrelated principal components, revealing the underlying structure of the data.
Projecting Data onto Principal Components
After determining eigenvalues and eigenvectors, the next step is projecting the original data onto these principal components. This transforms the data from its original coordinate system to a new one, where the axes are the principal components. Each original data point is re-expressed as a combination of these components.
Projection is achieved by multiplying the centered original data by the matrix formed from the selected eigenvectors. This operation rotates the data, aligning it with the directions of maximum variance. The result is a new dataset where each row represents an observation, and each column corresponds to a principal component score. This transformation makes underlying patterns and relationships in the data more apparent.
Selecting Principal Components
The final stage of PCA involves selecting how many principal components to retain for dimensionality reduction. Eigenvalues are central to this decision, as they indicate the variance explained by each component. Components with larger eigenvalues capture more of the dataset’s information.
One common selection method is examining a scree plot, which graphically displays eigenvalues in descending order. Analysts look for an “elbow” where the decline in eigenvalue magnitude becomes less steep, suggesting subsequent components explain significantly less variance. Another approach is to choose components that collectively explain a predetermined percentage of total variance, such as 90% or 95%. Retaining only the most informative components simplifies the dataset while preserving its structure, making it easier to analyze and visualize.