*Ecological Archives*
E085-119-A1

**Lorenzo Ciannelli, Kung-Sik
Chan, Kevin M. Bailey, and Nils Chr. Stenseth. 2004. Nonadditive
effects of the environment on the survival of a large marine fish population.
***Ecology* 85:3418–3427.

Appendix A. A description
of the method of penalized least squares..

The method of penalized least squares
balances the goodness of model fit with the smoothness of the function estimators.
For simplicity, assume that *f*_{2} vanishes and there is only
one *g* so that there are two unknown functions denoted as *f* and
*g*. Model identification requires that all functions be centered, that
is, the sample mean of each function is zero over the data points. Formally,
the model is estimated by minimizing where
*RSS *is the residual sum of squares and the roughness of a function is
measured by the integral of its squared second derivative; the smoothness coefficients
are positive
numbers that quantify the trade-off of badness of fit with the roughness of
the functions. For fixed smoothness coefficients, the minimization admits a
closed-form solution with the estimators of *f *and *g* being natural
cubic splines, i.e., piecewise cubic polynomials that are linear at the two
extremes and twice continuously differentiable (Wahba 1990,
Green and Silverman 1994). In particular, for zero smoothness
coefficients, the function estimators are generally extremely wiggly as the
functions try to follow every details of the data pattern including their random
aspects. On the other hand, with infinite smoothness coefficients, the functions
become straight lines. Thus, the determination of the smoothness coefficients
is pivotal. Two popular approaches to determining the are
the methods of cross validation (CV) and generalized cross validation (GCV)
(Wahba 1990, Green and Silverman 1994,
Wood 2000). The method of CV determines the by
minimizing the predictive residual sum of squares; the predictive residual of
a specific data case is defined as the observed response minus the predicted
value based on the model fitted to all data except the data case under prediction.
The method of CV then, in principle, requires repeating the model estimation
as many times as the number of data cases, and hence is generally computationally
intensive. For "linear"estimators, the predictive residuals can be
related to the residuals from the overall model fitted to all data, in which
case the method of CV can be done efficiently. (This is the case for the TGAM
when the threshold line is known). In fact, in such cases, the predictive residuals
can be further approximated using an appealing scheme, resulting in the method
of generalized cross validation (GCV; see Wahba 1990, Green
and Silverman 1994). Notice that within a parametric framework and under
some regularity conditions the method of CV is asymptotically equivalent to
the Akaike Information Criterion (AIC; Stone 1977, Tong
1990). However, the methods of CV and GCV are applicable in both parametric
and nonparametric settings. Indeed, the method of CV can be shown to be consistent
under very general conditions (Wahba 1990, Chan
and Tong 2001).

LITERATURE
CITED

Chan, K. S., and H. Tong. 2001.
Chaos: a statistical perspective. Springer Verlag, New York, New York, USA.

Green, P. J., and B. W. Silverman.
1994. Nonparametric regression and generalized linear models: a roughness penalty
approach. Chapman and Hall, New York New York, USA.

Stone, M. 1977. An asymptotic equivalence
of choice of model by cross-validation and Akaike's criterion. Journal of the
Royal Statistical Society, Series B, Methodological **39**:44–47.

Tong, H. 1990. Non-linear time series:
a dynamical system approach. Oxford University Press, Oxford, UK.

Wahba, G. 1990. Spline models for
observational data (SIAM).

Wood, S. J. R. 2000. Modelling and
smoothing parameter estimation with multiple quadratic penalties. Journal of
the Royal Statistical Society B **62**(2):413–428.

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