Principal Component Analysis is a dimensionality reduction technique commonly used in machine learning and data analysis. Its primary goal is to transform high-dimensional data into a lower-dimensional representation, capturing the most important information. This reduction in dimensionality can lead to improved computational efficiency, visualization, and often better model performance.
steps to perform PCA:
- Standardize the Data: If the features in the dataset have different scales, it is important to standardize them by subtracting the mean and dividing by the standard deviation to give each feature equal importance.
- Compute the Covariance Matrix: Calculate the covariance matrix of the standardized data. The covariance matrix provides information about how variables change together.
- Compute Eigenvectors and Eigenvalues: Calculate the eigenvectors and eigenvalues of the covariance matrix. The eigenvectors represent the principal components, and the eigenvalues indicate the amount of variance captured by each principal component.
- Sort Eigenvectors by Eigenvalues: Sort the eigenvectors in descending order based on their corresponding eigenvalues. The higher the eigenvalue, the more variance is captured by the corresponding eigenvector.
- Select Principal Components: Choose the top k eigenvectors to form the new feature space. Typically, it would select the number of principal components that capture a sufficiently high percentage of the total variance.
Example:
Suppose the data on people’s heights and weights. We can find that most of the variation, along a diagonal line, representing a combination of height and weight. PCA helps us focus on this main trend and ignore less important details.