Multiple Linear Regression

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

Predictor Response
$ x_{11},\ldots,x_{k1}$ $ Y_1$
$ \vdots$ $ \vdots$
$ x_{1n},\ldots,x_{kn}$ $ 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_j = \beta_0 + \beta_1 x_{1j} + \cdots + \beta_k x_{kj} + \epsilon_j,

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

© TTU Mathematics