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 *p _{i} *is the
value corresponding to position

, ,

, 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.