Gille Gauthier, Panagiotis Besbeas, Jean-Dominique Lebreton, and Byron J. T. Morgan. 2007. Population growth in Snow Geese: a modeling approach integrating demographic and survey information. Ecology 88:1420–1429.

Appendix A. Theory of the Kalman filter and smoother algorithms, and future perspectives.

We present here more details on the Kalman filter and smoother algorithms used in our paper. For additional details, the reader is referred to Harvey (1989). At the end, we also present some future perspectives on the development of integrated modeling using the Kalman filter.

Theory of the Kalman filter and smoother algorithms

In the same notation as the main text, we consider a state-space model comprising a state equation:

(A.1)

and an observation equation:

(A.2)

When used in “real time”, as is often done in engineering, filtering techniques attempt to estimate the state of the system at time t, i.e. , based on past information, i.e. on the survey values . A typical situation is to estimate the coordinates in space and the speed components of an object, such as a satellite, based on a model governing its movement (the state equation) and partial information (e.g., on speed only; the observation equation).

Over a time window from time 1 to , denoting as the vector of parameters, the overall survey likelihood is the probability density of the survey data (i.e. the observations of population size) viewed as a function of the parameters . This probability density can be written as a product of conditional probabilities:

(A.3)

where we also write f for the different conditional probability densities of the observations. The density of the initial population vector is given by g.

A major assumption in Kalman filtering involves assuming normal distributions for , and , over . Because of the linearity of the state-space equations, and of the invariance of the normal distribution under linear transformations, and , over , are also normally distributed, either conditionally or unconditionally. Expectations, variance and covariances are thus sufficient to fully determine the density probabilities in Eq. A.3. One first has to provide and to determine . This is exactly what is done in our case (see text) by the structured start approach, in a fashion compatible with . Then the density is easily obtained from Eq. A.2. The densities are obtained by recurrence in four steps:

1. First obtain the joint density of and ; At the first step, this is just ;

2. Then obtain the distribution of conditional on ; this step is alike a simultaneous multiple regression of the components of on ;

3. Apply Eq. A.1 to get the distribution of conditional on

4. Apply Eq. A.2 to get the distribution of conditional on , and indeed the joint distribution of and conditional on ; go back to step 1 after having incremented t.

The conditioning on is equivalent to conditioning on all the past data because of the Markovian nature of the state-space model. Again, in these four steps, it is sufficient to obtain the expectation and variance matrices, all distributions being normal. The Kalman filter can thus be viewed as a set of forward recurrence equations for these expectations and variance matrices. This process corresponds to the dotted box “Kalman filter” on Fig. A1 and thus leads to the survey likelihood (L_S).

Maximum likelihood estimates of are then obtained by numerically maximizing the combined likelihood, , which integrates the capture-recapture data (Fig. A1).

The Kalman smoother (second dotted box on Fig. A1) can then be used to obtain, by a recurrence running backwards, the expectation and variance matrices of and conditional on the whole set of observations , and not only on the past . The estimates being substituted for the parameters , one gets “smoothed estimates” of the whole series and .

Future perspectives

A logical next step would be to develop user-friendly software to implement the Kalman filter. Such software already exists to estimate demographic parameters from capture-recapture data (MARK, White and Burnham 1999; M-SURGE, Choquet et al. 2004) or to build matrix models (ULM, Legendre and Clobert 1995). Presently, one has to use these approaches independently, first to select the best model in a capture-recapture software, then use these parameter estimates in a matrix model software, and finally improve parameter values with the information contributed by the population survey. However, one can anticipate the merging of these approaches in a single software, where model selection would be based on an overall, combined likelihood function. In this context, the normal approximation to the likelihood is very useful (Besbeas et al. 2003) as it fully uses the idea that maximum likelihood estimates are asymptotically efficient and brings together all the information on the parameters existing in the data (Rao 1973). A preliminary version of such software, kalm, has been developed by P. Besbeas. Hopefully, in the long run, integrated monitoring and modeling will develop naturally, hand-in-hand.

FIG. A1. Flow chart summarizing the implementation of the Kalman filter to integrate demographic and population survey data.

LITERATURE CITED

Rao, C. R. 1973. Linear statistical inference and its applications. Wiley, New York, New York, USA.