Overview

Statistics and probability are important tools for the analysis of data.

 

Basic Statistics

Sample Mean

$$ \bar{x} = \dfrac{1}{n} \sum\limits_{i=1} x_i$$

data <- c(1,2,3,4,5)
mean(data)
sum(data)/length(data)

Sample Variance

$$ s^2 = \dfrac{1}{n-1} \sum\limits_{i=1} (x_i - \bar{x}^2) $$

var(data)
sum( (data - mean(data))^2 ) / (length(data) - 1)

Sample Standard Deviation

$$ s = \sqrt{s^2}$$

sd(data)
sqrt( var(data) )

 

Linear Regression

Linear regression models assume some relationship exists between a response variable ( \(y\) ) and its predictor (or independent) variables ( \(\mathbf{x} \) ). $$ \begin{align} y &= \mathbf{x} \beta + \epsilon \\[1em] &= \beta_0 + \beta_1 x_1 + \beta_2 x_2 + ... + \beta_p x_p + \epsilon \\[1em] \end{align} $$

To estimate the coefficients, we can use the least-squares approach:

$$ \begin{align} \hat{\beta} &= (\mathbf{x}^T \mathbf{x})^{-1} \, \mathbf{x}^T \mathbf{y} \\[1em] &= \beta_0 + \beta_1 x_1 + \beta_2 x_2 + ... + \beta_p x_p + \epsilon \\[1em] \end{align} $$

We can use the following code in R to estimate the coefficients manually.

	betahat <- solve( t(x) %*% x ) %*% t(x) %*% y

Linear Regression Model Predictions

Once we obtain the \( \beta \) weights for our model (either by deriving them or through Gradient Descent), we can have the model estimate/predict values for given input data:

$$\hat{y} = \mathbf{x} \beta$$ $$h_\mathbf{\beta}(\mathbf{x}) = \hat{y} = \mathbf{x}\beta$$

The hypothesis function \( h_\beta(\mathbf{x}) \) (outputs are often denoted as \(\hat{y} \)) accepts some record \(\mathbf{x} = \begin{bmatrix} 1& x_1 & x_2 & ... & x_p \end{bmatrix}\) and multiplies it by the corresponding weights \( \beta = \begin{bmatrix} \beta_0 \\ \beta_1 \\ \beta_2 \\ \vdots \\ \beta_p \end{bmatrix}\).