Page 186 - Applied Statistics with R

186 CHAPTER 10. MODEL BUILDING

10.1 Family, Form, and Fit

When modeling data, there are a number of choices that need to be made.

• What family of models will be considered?
• What form of the model will be used?
• How will the model be fit?

Let’s work backwards and discuss each of these.

10.1.1 Fit

Consider one of the simplest models we could fit to data, simple linear regression.

= ( , , , … , −1 ) + = + +
3
1
2
1 1
0
So here, despite having multiple predictors, we chose to use only one. How is
this model fit? We will almost exclusively use the method of least squares, but
recall, we had seen alternative methods of fitting this model.
argmin max | − ( + )|

1
0
0 , 1

argmin ∑ | − ( + )|

0
1
0 , 1 =1

argmin ∑( − ( + )) 2
0
1

0 , 1 =1
Any of these methods (we will always use the last, least squares) will obtain
estimates of the unknown parameters and . Since those are the only
0
1
unknowns of the specified model, we have then fit the model. The fitted model
is then

̂
̂
̂
̂ = ( , , , … , −1 ) = +
2
3
0
1 1
1
Note that, now we have dropped the term for the noise. We don’t make any
effort to model the noise, only the signal.