Principal Component Analysis Using Singular Value Decomposition (SVD) Method

When performing Principal Component Analysis (PCA) on loaded data using the Singular Value Decomposition (SVD) method, we can complete the process through the following key steps:

1. First, we need to standardize the data to ensure different features have equal weighting. This preprocessing step typically involves mean-centering and scaling to unit variance using functions like StandardScaler from sklearn.preprocessing or manual implementation with numpy operations.

2. Next, we apply SVD to decompose the standardized data matrix and extract principal components. In Python, this can be implemented using numpy.linalg.svd() function, which returns three matrices: U (left singular vectors), S (diagonal singular values matrix), and Vt (right singular vectors transposed). The columns of V represent the principal component directions.

3. We then reduce data dimensionality by retaining the largest principal components. This involves selecting the top-k singular vectors based on explained variance ratio, typically calculated from the squared singular values divided by their total sum. The cumulative variance can guide the choice of optimal dimensions.

4. Finally, we implement the complete workflow through source code that combines these steps using libraries like numpy and scikit-learn. The code should include data standardization, SVD computation, variance calculation, and transformation to principal component space using matrix multiplication of original data with selected components.

Thus, when using SVD for PCA, these systematic steps help us effectively understand and analyze data patterns while maintaining computational efficiency through proper dimensionality reduction techniques.