thusthe data lie approximately on -dimensional affine hyperplane bd xd { ( where we define [ bd ]the function ( bis linear in bbut not linear in the feature vector xdue to the constant howeveraugmenting the feature space with the constant the mapping [ ] ( :[ xb becomes linear in the feature space and so ( becomes linear model (see section most software packages for regression include as feature by default note that in ( we only specified the model for single pair (xythe model for the training set {( )(xn yn )is simply that each yi satisfies ( (with xi and that the {yi are independent setting [ yn ]we can write the multiple linear regression model for the training data compactly as xb emodel matrix ( where [ en ]is vector of iid copies of and is the model matrix given by > xn xn xnd > example (multiple linear regression modelfigure depicts realization of the multiple linear regression model yi xi xi ei where ~iid ( / the fixed feature vectors (vectors of explanatory variablesxi [xi xi ] lie in the unit square figure data from multiple linear regression model