Mathematical Foundations

Basic Matrix Operations:

• Matrix Transpose
• Matrix Multiplication
• Matrix Inverse
• Orthogonal Matrix
• Eigenvectors and Eigenvalues

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Dimension Analysis and Factor Analysis (PCA, SVD, LDA)

Features, variables, and columns are other terms for dimensions.

Dimensionality reduction is the process of extracting features from input variables that are composed of hundreds of variables.

Less dimensions provide rapid computations and clear interpretation, which reduces overfit situations and helps to avoid collinearity.

The ability to visualise multivariate data in a 2D space is another advantage of dimensionality reduction.

In this blog, among the various methods accessible

we will discuss the most popular methods:

• PCA - Principal Component Analysis
• SVD - Singular Value Decomposition
• LDA - Linear Discrimination Analysis
• Factor Analysis

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PCA - Principal Components Analysis

PCA is used to analyse dense data, which is data that is quantitative in nature and does not include many zeros.

Principal Component Analysis (PCA) is used to divide a large number of characteristics into an equal number of features known as Principal Components (PCs).

The initial group of PCs alone can gather the most information, but these PCs capture 100% of the data.

With minimal information loss, PCA enables us to considerably reduce the dataset size.

Applying PCA is ineffective if the initial dataset contains characteristics that are all associated.

Each PC will record all the data that is present in the original dataset's variables.

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Benefits of PCA

• Reduction of number of features & hence faster processing.
• Identify the relationship between multiple columns at one go by interpreting the weights of PCs.
• Visualizing multidimensional data using a 2D visualization technique.
• Inputs being correlated is called as collinearity and this is a problem, which is overcome by PCA because it makes the inputs uncorrelated.
• Helps in identifying similar columns.

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The i^th principal component is a weighted average of original measurements / columns:

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Weights (aij) are chosen such that:

• PCs are ordered by their variance (PC1 > PC2 > PC3, and so on)
• Pairs of PCs have correlation = 0
• For each PC, sum of squared weights = 1 (Unit Vector)

Data Normalization / Standardization should be performed before applying PCA.

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SVD - Singular Value Decomposition

Sparse data, or data with many items that are zeros, is reduced using a technique known as singular value decomposition, or SVD.

SVD is used to process photos and greatly aids in image processing by reducing the size of the images.

In the recommendation engine, SVD is frequently utilised.

It is a matrix decomposition method, represented as:

• Diagonal matrix (d) values are known as the singular values of the original matrix X.
• U matrix column values are called the left-singular vectors of X
• V matrix column values are called the right-singular vectors of X

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LDA - Linear Discriminant Analysis

For data with more features, dimensionality reduction is solved using linear discriminant analysis (LDA).

Considering the class label, linear discriminant analysis is a supervised algorithm.

Each class of datapoints has a centroid determined via LDA.

LDA determines a new dimension based on centroids in a way to satisfy two criteria:

• Maximize the distance between the centroid of each class.
• Minimize the variation (which LDA calls scatter and is represented by s2), within each category.