Appendix A. Penalty matricies. A pdf version is also available.
Spline Definition (following Wood 2001)
A cubic spline, , is a smoothed curve through a set of n points , where . In our model, these points are not the data, but rather the knot values for the rates that best fit the modified objective function. We use the standard first derivative basis (see Lancaster and Salkaukas 1986 for a good introduction) to represent cubic splines:
where pi is the value corresponding to position xi and is the first derivative of at (i.e., ). The functions are:
, where .
and are set equal to zero to yield a so called “natural” spline. Fitting a spline through a set of simply requires that the be determined via the linear equation:
where and the matrix and the matrix have zeroes everywhere except as follows:
, , and
Therefore f(x) can be rewritten
as , where ,
This yields a general description of spline function representation for .
In the stage-structured model (Eq. 1) the mortality rates are represented by cubic splines. To create the cohort criteria to control sensitivity, a value of mortality (or the gradient of mortality) in one stage needs to be compared to a value of mortality at a different time in the previous or following stage. The realization of this leads to the introduction of a new index , representing the stage in a stage-structured framework. will now stand for the per capita mortality of the kth stage at time x.
, where , , and and .
In the description of the deviance and gradient criterion below, we use and to represent the per capita mortality and the derivative of the per capita mortality respectively. The corresponding spline functions are , , , and , , , .
From Eqs. 4 and 5 in the paper we have,
To explain the derivation of these two terms we will first show the calculation of the deviance (the gradient criterion works by analogy) in mortality for one cohort at time t and then generalize the case for an arbitrary number of cohorts. The deviance criterion for one cohort at time t is:
where , and n represents the number of stages. This has the potential to be computationally expensive. However, if the parameter vector is removed, then the majority of the calculations only need to be done once.
Then can be written as .
For an arbitrary number of cohorts, say m, the equation is
The penalty matrix only needs to be calculated once during a fitting routine.
Wood, S. N. 2001. Partially
specified Ecological Models. Ecological Monographs 71:125.
Lancaster, P., and K. Salkauskas. 1986. Curve and Surface Fitting: an Introduction. Academic Press, San Diego, California, USA.