# Linear regression

**Linear regression** is the approximation of many data points using a single linear function of one or more variables.

## Mean squared error

Suppose we are given a multiset of dependent variable values and a multiset of corresponding independent variable values . We want to create a function that predicts with as little overall error as possible. We can quantify the error using **mean squared error**, defined by
Sometimes is notated , because is a prediction of .

Note the similarity of to the distance formula in Euclidean space; in fact, , so increases monotonically with (which is always nonnegative). Thus, minimizing corresponds to minimizing Euclidean distance between a point whose coordinates are the and a point whose coordinates are the .

If is a constant function equal to the arithmetic mean of , then the equals the variance of .

### Vector-valued functions

Sometimes multiple values are to be predicted in conjunction (for example, in a weather forecast, the wind components in both the north and east directions). In this case the are represented by vectors, so the predictor function should also be a vector-valued function. The formula is altered slightly to include magnitudes: The summands in the numerator, by the vector magnitude formula, are themselves the sum of squares of differences between components.

### Multiple regression

If there are multiple independent variables in conjunction (for example, if a student's past three AMC scores are used to predict the next score), then the regression becomes a **multiple regression**. Each must then be viewed as a sequence , where is the number of predictors. The terms of this sequence are passed one by one into as arguments; is therefore a function of variables.