Multiple Linear Regression

Each of independent measurements produces the response value $ Y_i$ along with set of values $ x_{i1},\ldots,x_{ik}$ for each $ i=1,\ldots,n$.

Predictor Response
$ x_{11},\ldots,x_{1k}$ $ Y_1$
$ \vdots$ $ \vdots$
$ x_{n1},\ldots,x_{nk}$ $ Y_n$

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

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

it is called a linear regression model. Here $ \epsilon_i$ is introduced as ``random error.''

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