is: if are independent normal random variables, the statistic follows a chi-squared distribution with n − 1 degrees of freedom. Here, the degrees of freedom arises from the residual sum-of-squares in the numerator, and in turn the n − 1 degrees of freedom of the underlying residual vector . In the application of these distributions to linear models, the degrees of freedom parameters can take only integer values. The underlying families of distributions allow fractional values for the degrees-of-freedom parameters, which can arise in more sophisticated uses. One set of examples is problems where chi-squared approximations based on effective degrees of freedom are used. In other applications, such as modelling heavy-tailed data, a t or F-distribution may be used as an empirical model. In these cases, there is no particular degrees of freedom interpretation to the distribution parameters, even though the terminology may continue to be used. In non-standard regression Many non-standard regression methods, including regularized least squares (e.g., ridge regression), linear smoothers, smoothing splines, and semiparametric regression are not based on ordinary least squares projections, but rather on regularized (generalized and/or penalized) least-squares, and so degrees of freedom defined in terms of dimensionality is generally not useful for these procedures. However, these procedures are still linear in the observations, and the fitted values of the regression can be expressed in the form where is the vector of fitted values at each of the original covariate values from the fitted model, y is the original vector of responses, and H is the hat matrix or, more generally, smoother matrix. For statistical inference, sums-of-squares can still be formed: the model sum-of-squares is ; the residual sum-of-squares is . However, because H does not correspond to an ordinary least-squares fit (i.e. is not an orthogonal projection), these sums-of-squares no longer have (scaled, non-central) chi-squared distributions, and dimensionally defined degrees-of-freedom are not useful. The effective degrees of freedom of the fit can be defined in various ways to implement goodness-of-fit tests, cross-validation, and other statistical inference procedures. Here one can distinguish between regression effective degrees of freedom and residual effective degrees of freedom. Regression effective degrees of freedom For the regression effective degrees of freedom, appropriate definitions can include the trace of the hat matrix, tr(H), the trace of the quadratic form of the hat matrix, tr(H'H), the form tr(2H – H H'''), or the Satterthwaite approximation, . In the case of linear