Principal Component Analysis (PCA)

Principal Component Analysis (PCA) is a popular unsupervised learning technique used for dimensionality reduction and feature extraction. PCA transforms a high-dimensional dataset into a lower-dimensional space while retaining the maximum amount of variance in the data.

Principal Component Analysis (PCA) works by finding a set of orthogonal vectors, called principal components. It captures the maximum amount of variance in the data. The first principal component is the direction in which the data varies the most. Each subsequent component is chosen to be orthogonal to the previous ones. It captures as much remaining variance as possible.

To perform Principal Component Analysis (PCA), the following steps are typically followed:

1. Standardize the data: Scale each feature so that it has a mean of 0 and a standard deviation of 1.
2. Calculate the covariance matrix: The covariance matrix is calculated to determine how the characteristics are related to each other.
3. Calculate the eigenvectors and eigenvalues of the covariance matrix: The eigenvectors are the principal components, and the eigenvalues indicate the proportion of the variance explained by each component.
4. Choose the number of principal components: The number of principal components to retain is selection based on the amount of variance explained and the desired dimensionality of the reduced data set.
5. Project the data onto the principal components: The original high-dimensional data set is projected onto the low-dimensional space spanning by the selected principal components.

PCA can be useful for reducing the computational complexity of a dataset, visualizing high-dimensional data, and identifying important features. However, it assumes that the data is linearly separable and may not perform well if the data has complex nonlinear relationships. Additionally, the interpretation of the principal components can be difficult, as they are linear combinations of the original features.