Linear Regression

# Linear Regression #

## Infographic #

The linear regression infographic is shown below.

## Simple Linear Regression #

Ordinary Least Squares (OLS) can be used to fit a linear line to noisy data.

If $$y$$ is a vector of measured data, $$\beta_{0}$$ and $$\beta_{1}$$ are the actual linear model parameters (intercept and gradient) and $$\epsilon$$ represents the error vector, the model can be expressed as:

$y_{1} = \beta_{0} + \beta_{1}x_{1} + \epsilon_{1} \\y_{2} = \beta_{0} + \beta_{1}x_{2} + \epsilon_{2} \\\vdots \\y_{n} = \beta_{0} + \beta_{1}x_{n} + \epsilon_{n}$

Which can be expressed as:

$\begin{bmatrix} y_{1} \\ y_{2} \\ \vdots \\ y_{n} \end{bmatrix} = \begin{bmatrix} 1 & x_{1} \\ 1 & x_{2} \\ \vdots & \vdots \\ 1 & x_{n} \end{bmatrix} \begin{bmatrix} \beta_{0} \\ \beta_{1} \\ \end{bmatrix} + \begin{bmatrix} \epsilon_{1} \\ \epsilon_{2} \\ \vdots \\ \epsilon_{n} \end{bmatrix}$

Or in matrix form:

$y=X \beta + \epsilon$

For some estimate $$\hat{\beta}$$ of the model parameters, the error and error squared are defined by:

$\epsilon = y - X \hat \beta \\\epsilon^T\epsilon = (y - X \hat \beta)^T(y - X \hat \beta)$

Expanding the above equation gives:

$\\\epsilon^T\epsilon = y^Ty - y^T (X \hat \beta) - (X \hat \beta)^Ty + (X \hat \beta)^T(X \hat \beta) \\\epsilon^T\epsilon = y^Ty - (X \hat \beta)^T y - (X \hat \beta)^Ty + (X \hat \beta)^T(X \hat \beta) \\\epsilon^T\epsilon = y^Ty - 2(X \hat \beta)^T y + (X \hat \beta)^T(X \hat \beta) \\\epsilon^T\epsilon = y^Ty - 2\hat \beta ^T X^T y + \hat \beta^T X^T X \hat \beta$

To find $$\hat \beta$$ which minimises the square of the errors, the above equation can be differentiated and set equal to 0:

$\frac{\partial [\epsilon^T\epsilon]}{\partial \beta} = - 2 X^T y + 2 X^T X \hat \beta = 0$

Therefore,

$X^T X \hat \beta = 2 X^T$

Finally this can be rearranged to give the familiar OLS equation below which is the coefficient vector of model parameters:

$\\\hat \beta = (X^T X)^{-1} X^T y$

## Multiple Linear Regression #

The example above demonstrates fitting a linear model with intercept and gradient parameters to noisy data. We can also use the OLS equation to fit a higher order polynomial equation to noisy data where in this case the independent varaible is squared. Although this somewhat goes against intuition a linear relationship still remains between the independent variables ( $$x^0$$ , $$x^1$$ and $$x^2$$ ) and the dependent variable $$y$$ , so we can still use the OLS equation. We are using linear regression to fit a quadratic model, this is termed Miltiple Linear Regression since the number of independent variables is now > 1.

Along the same lines as the previous post, let’s consider a 2nd order polynomial model as follows, again $$y$$ is a vector of measured data, however this time $$\beta_{0}$$ , $$\beta_{1}$$ and $$\beta_{2}$$ are the quadratic model parameters, $$x$$ represents the independent variable and $$\epsilon$$ represents the error vector, this model can therefore be expressed as:

$y_{1} = \beta_{0} + \beta_{1}x_{1} + \beta_{2}x_{1}^2 + \epsilon_{1} \\y_{2} = \beta_{0} + \beta_{1}x_{2} + \beta_{2}x_{2}^2 + \epsilon_{2} \\\vdots \\y_{n} = \beta_{0} + \beta_{1}x_{n} + \beta_{2}x_{n}^2 + \epsilon_{n}$

As mentioned previously this non-linear model can effectively be considered a linear model if, instead of one independent variable $$x$$ , we consider the model to have 3 independent variables $$x^0$$ , $$x^2$$ and $$x^3$$ . Then we can write the model in the required linear form:

$\begin{bmatrix} y_{1} \\ y_{2} \\ \vdots \\ y_{n} \end{bmatrix} = \begin{bmatrix} 1 & x_{1} & x_{1}^2 \\ 1 & x_{2} & x_{2}^2 \\ \vdots & \vdots \\ 1 & x_{n} & x_{n}^2 \end{bmatrix} \begin{bmatrix} \beta_{0} \\ \beta_{1} \\ \beta_{2} \\ \end{bmatrix} + \begin{bmatrix} \epsilon_{1} \\ \epsilon_{2} \\ \vdots \\ \epsilon_{n} \end{bmatrix}$

Or in matrix form:

$y=X \beta + \epsilon$

And as per the simple linear regression example, we can calculate $$\hat \beta$$ from the OLS equation:

$\\\hat \beta = (X^T X)^{-1} X^T y$

## Python Implementation #

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