Can you describe the Least Squares method?

The Least Squares method, also known as Ordinary Least Squares (OLS), is a statistical technique used for estimating the parameters of a linear regression model. The goal is to find the best-fitting linear relationship between the dependent variable (usually denoted as y) and one or more independent variables (denoted as x). "Best-fitting" means that the linear relationship minimizes the sum of the squared differences (residuals) between the actual value of the dependent variable (y) and the predicted value (ŷ).

Given a dataset with n observations ((x₁, y₁), …, (x_n, y_n)) and a linear regression model defined as:

$$\hat{y_i} = \beta_0 + \beta_1 x_{i1} + \beta_2 x_{i2} + \cdots + \beta_k x_{ik}$$

where $\hat{y_i}$ is the predicted value, β₀ is the intercept, β₁, …, β_k are the coefficients of the k independent variables x_i1, …x_ik, and i = 1, …, n.

The least squares method aims to minimize the sum of the squared residuals:

$$RSS = \sum_{i=1}^{n} (y_i - \hat{y_i})^2$$

To solve for the coefficients β₀, β₁, …, β_k, we need to find the partial derivatives of the RSS with respect to each β, and set them equal to zero:

$$\frac{\partial RSS}{\partial \beta_0} = 0, \frac{\partial RSS}{\partial \beta_1} = 0, \ldots, \frac{\partial RSS}{\partial \beta_k} = 0$$

This will result in a system of k + 1 normal equations. Solving this system provides the values of the regression coefficients that minimize the residual sum of squares.

In matrix notation, the linear regression model can be written as:

y = Xβ + ϵ

where y is a column vector of dependent variable values, X is a matrix with columns containing the independent variable values, β is a column vector containing the regression coefficients, and ϵ is a column vector containing the residuals.

Using least squares, the coefficients β can be estimated as:

$$\hat{\boldsymbol{\beta}} = (\mathbf{X}^\top \mathbf{X})^{-1} \mathbf{X}^\top \mathbf{y}$$

Here, X^⊤ represents the transpose of matrix X, and (X^⊤X)^− 1 is the inverse of matrix X^⊤X.

In summary, the Least Squares method is used to estimate the parameters of a linear regression model by minimizing the sum of the squared differences between the actual and predicted values of the dependent variable. The estimated parameters can be computed using normal equations or matrix algebra.