Multiple Linear Regression

Each of independent measurements produces the response value

along with set of values $x_{i1},\ldots,x_{ik}$ for each $i=1,\ldots,n$ .

Predictor	Response
$x_{11},\ldots,x_{1k}$
$\vdots$	$\vdots$
$x_{n1},\ldots,x_{nk}$

We call the variables for $x_{ij}$ 's explanatory variables or "predictor" and the variable for 's dependent variable or "response." When the relationship between the predictor $x_{ij}$ and the response can be modeled by

$\displaystyle Y_i = \beta_0 + \beta_1 x_{i1} + \cdots + \beta_k x_{ik} + \epsilon_i,$

it is called a multiple linear regression model. Here $\epsilon_i$ is introduced as “random error.”