Linear Model

Linear Model#

Consider a dataset as a matrix of \(X \in \mathbb{R}^{m\times n}\) where we’ve extracted \(m\) numbers of data records/rows and \(n\) numbers of feature columns. We may map the dataset \(X_0\) to a feature function which outputs the design matrix \(X = f(X_0)\). Our linear model can easily be written in the form of a dot product,

\[ \hat y = X \beta \]

  • \(\hat y\) : Model output vector

  • \(\beta^T\) : Row vector of the transposed parameter vector

  • \(X\) : Feature or Design matrix

Solution to Square Design Matrix#

In rare cases, the design matrix is a square matrix. In this case the solution is possibly,

\[ \beta = X^{-1}\hat y \]

Normal Equation#

We may force the design matrix to be a square matrix by multiplying both sides by its transpose,

\[ X^T\hat y = X^T X \beta \]

We still have a linear equation and the solution for \(\beta\) is can be done in two ways:

  1. Assuming \(X^TX\) is an invertible matrix,

    \[ \beta = (X^TX)^{-1} X^T\hat y \]

  2. Gaussian Elimination (traditional way of solving linear algebra equations)

Contents