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 
,
, 
and 
and 
.
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 
.
Penalty matrices
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,
| Deviance criterion: |  |
| Gradient criterion: |  . |
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.
Define 
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.
Literature cited
Wood, S. N. 2001. Partially
specified Ecological Models. Ecological Monographs 71:1–25.
Lancaster, P., and K. Salkauskas. 1986. Curve
and Surface Fitting: an Introduction. Academic Press, San Diego, California,
USA.