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Penalized Least Squares Estimation

Penalized least squares estimation provides a way to balance fitting the data closely and avoiding excessive roughness or
rapid variation. A penalized least squares estimate is a surface that minimizes the penalized squared error over the class
of all surfaces that satisfy sufficient regularity conditions.

Define as a d-dimensional covariate vector from an matrix , as a p-dimensional covariate vector, and as the observation associated with . Assuming that the relation between and is linear but the relation between and is unknown, you can fit the data by using a semiparametric model as follows:

where f is an unknown function that is assumed to be reasonably smooth, , are independent, zero-mean random errors, and is a p-dimensional unknown parameter vector.

This model consists of two parts. The is the parametric part of the model, and the are the regression variables. The is the nonparametric part of the model, and the are the smoothing variables.
The ordinary least squares method estimates and by minimizing the quantity:

However, the functional space of is so large that you can always find a function f that interpolates the data points. In order to obtain an estimate that fits the data well and has some degree of smoothness,
you can use the penalized least squares method.

The penalized least squares function is defined as

where is the penalty on the roughness of f and is defined, in most cases, as the integral of the square of the second derivative of f.

The first term measures the goodness of fit and the second term measures the smoothness associated with f. The term is the smoothing parameter, which governs the tradeoff between smoothness and goodness of fit. When is large, it more heavily penalizes rougher fits. Conversely, a small value of puts more emphasis on the goodness of fit.

The estimate is selected from a reproducing kernel Hilbert space, and it can be represented as a linear combination of a sequence of basis
functions. Hence, the final estimates of f can be written as

where is the basis function, which depends on where the data are located, and and are the coefficients that need to be estimated.

For a fixed , the coefficients can be estimated by solving an system.

The smoothing parameter can be chosen by minimizing the generalized cross validation (GCV) function.

If you write

then is referred to as the *hat* or *smoothing* matrix, and the GCV function is defined as