10. Principal Component Analysis (PCA) | Dimensionality Reduction Technique | Machine Learning

Introduction to Dimensionality Reduction
Definition of Dimensionality Reduction:
Explain that dimensionality reduction is a technique used to reduce the number of input variables in a dataset while retaining as much information as possible.
Highlight that high-dimensional data can lead to the "curse of dimensionality," which makes models more complex and less interpretable.
Importance of Dimensionality Reduction:
Reduces computational costs, improves model performance, and helps in visualizing high-dimensional data.
Eliminates noisy or irrelevant features, improving the model’s accuracy and reducing overfitting.
Common Dimensionality Reduction Techniques:
Principal Component Analysis (PCA): Linear technique that transforms data to a new set of orthogonal axes (principal components).
t-SNE (t-Distributed Stochastic Neighbor Embedding): Non-linear method for visualizing high-dimensional data.
LDA (Linear Discriminant Analysis): Used when the data has labeled categories.
Autoencoders: Neural network-based method for non-linear dimensionality reduction.

Key Concepts Behind PCA
Variance and Covariance:
Variance: The measure of spread or dispersion of the data along a particular dimension.
Covariance: A measure of how two dimensions vary together. If the covariance is high, the features are correlated.
Principal Components:
Principal components (PCs) are linear combinations of the original features.
The first principal component is the direction with the highest variance, the second principal component is orthogonal to the first, and so on.
Eigenvalues and Eigenvectors:
Eigenvectors: The directions (principal components) that maximize variance.
Eigenvalues: The magnitude of variance along each eigenvector. Higher eigenvalues indicate that the principal component captures more variance.
PCA involves solving the eigenvalue decomposition of the covariance matrix of the data.
Dimensionality Reduction:
By selecting the top K principal components, the data is projected into a lower-dimensional space.
The amount of information retained is given by the sum of the selected eigenvalues divided by the sum of all eigenvalues.

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