Singular Value Decomposition as a Test for Robustness of Models.

`Wisdom is the principle thing therefore get wisdom: and with all thy getting get understanding'

- Proverbs 4:7

(a) Introduction.

It is important in non-chaotic situations that the robustness of numerical models of ecological systems is determined over repeated runs the same results should be produced for dierent simulations of the same dynamical system. Although a spatial model with local stochasticity will vary in its details, the statistically-averaged spatial structures and population compositions should usually remain constant. A simple technique is presented for assessing the robustness

of models in this context. The method is based on singular value decomposition (SVD) of a matrix containing model output data and involves comparison withwhiteorGaussiannoise.

(b) Method.

This method relies on the generation of areplication matrixXfrom repeated runs of the model

that is to be tested. Suitable model output must be chosen. It is important that a sucient number of output variables is available, of the order of ten or more, so that a substantial matrix

X can be formed. If n denotes the number of variables, then n replicated simulations should

be carried out. Ifxi=fx (i) 1

x (i)

n gdenotes the output vector for the model, with x (i)

j being

the ith replicate of the jth variable, then the replication matrix is constructed as follows:

X = 0 B B B B @ x 1 ... xn 1 C C C C A :

The output vectors may correspond to the proportions of the state variables in a system. Alternatively, the vector may be a characterisation of the spatial structure of a system, such as a distribution of patch sizes. This allows the robustness of an emergent spatial pattern in a model to be assessed.

The standard technique of SVD is used to nd the singular values of the replication matrixX.

The matrix is decomposed into the form:

X =U:S:VT

whereU andV are orthogonal matrices andS is diagonal, with:

S = 0 B B B B @ e1 ... en 1 C C C C A

wherefeigare the singular values ofX. Standard computer algebra packages, such as Matlab

(The Maths Works Inc., 1992) produce the singular values in descending order of magnitude:

ke 1

kke 2

kkenk.

If the replicated runs are identical then rank(X) = 1 and hence e

2 = e3 =

= en 0. A

stochastic model will always have some error in all of the output variables and thus rank(X)

= n. However, if the replicates are accurate, then the vectorsfxigwill be similar and rank(X)

will be nearly1. This will be indicated by the presence of one dominant singular value, that is: ke 1 k ke 2 k > ke 3

k > > kenk. The following section quanties this singular value

dierence, characterised by ke2k ke

1

k, for dierent levels of noise.

(c) Application to Gaussian Noise.

Sample data can be generated to test the response of the singular value spectrum to varying levels of white (Gaussian) noise. A vector x is produced by assigning random values in the

interval !0,100] to the elementsfxig. Replicated `output' valuesfxigare produced by adding

Gaussian random variables of mean zero and specied standard deviation tox:

xi = x+ (i = 1n) where: = ( 1 n) and: j N(0 ):

The singular value spectrum may be found for a range of standard deviations . Here results are shown for = 12 100 . Figure 4 shows sample replication matrices for 151025

and n = 20. There is a clear dominant singular value in all cases, but the dominance decreases as noise increases. Figures 5a - b illustrate the relation between the Gaussian standard deviation and the ratio of the rst two singular values. This demonstrates a linear relationship, although with increasing uctuations at high . A standard linear regression (gure 5a) and a weighted regression (gure 5b) yield similar results. Weighting is achieved by using a factor of 1

2

x in the

regression sums, which causes the linear t to be biased towards the smaller noise levels. The ratio of the rst two singular values for the output from a model can be compared to the Gaussian results, specically the gradient g of the tted line, to obtain an estimate of the level of noise in the model. g represents the amount by which the ratio of the rst two singular values increases for each 1% of noise. Therefore, the percentage Gaussian noise pcorresponding to a

set of output data is given by:

p= ke 2 k gke 1 k :

The eect of matrix size n must now be taken into consideration, as g may depend on n. The variation of ke2k

ke 1

k with was found for a range of matrix sizes from 10

10 to 5555. Figure 5a

shows the plots of ke 2

k

ke1k against aggregated to form a surface. The surface roughly takes the

form of a at plane. The important feature is the variation in the slope of lines in the plane as the matrix size varies, that is, the form of the function g(n). The plane can be approximated by taking the slopes of the lines tted for all of the sizes of matrix and tting a line to these slopes (gure 5b). This demonstrates that g is linear in n. Thus the percentage of noise can be found by estimating g(n) from gure 5b and using:

p = ke 2 k g(n)ke 1 k :

2.2.1. Introduction.

`...a rich variety of behaviour, some of it very bizarre...'- Cartwright & Littlewood 1945

`In view of the inevitable inaccuracy and incompleteness of weather observations, precise very- long-range forecasting would seem to be non-existent'- Lorenz 1963

Anomalies in the behaviour of nonlinear systems were noted in the last century by Poincare, Kovalevskaya and Lyapunov (Percival, 1989) and later by Cartwright & Littlewood (1945). It was not until 1963 and the early days of digital computers, that Lorenz produced the rst small illustration of astrange attractor, the nely-structured geometric shape on which the dynamics of a nonlinear system move in phase space (Lorenz, 1963). Working in a restricted area of meteorology, there was little attention given to the Lorenz system, until it was rediscovered in the 1970s. Early mathematical treatment of chaotic nonlinearity was brought to the attention of the general scientic communityby May, using examples from population biology (May, 1974 1976 May & Oster, 1976). Since then the study of chaos has become a scientic discipline in its own right and is used throughout the physical and biological sciences.

The most inuential feature of chaotic dynamics is the sensitivity to initial conditions(SIC), popularly known as thebuttery eect (Markus, 1992). In a non-chaotic system, trajectories starting at nearby initial points will remain close together indenitely. This underpins classical science, which assumes that small starting errors will remain small. However, chaotic trajecto- ries that start close will rapidly diverge, so that perturbations are amplied (McGlade, 1994). The SIC of chaotic systems means that predictability is limited. Although chaos is determinis- tic, an initial condition can never be determined precisely, so the long term future of the system cannot be described.

This has produced two dierent reactions in ecologists: while some despair of predicting the future of any non-trivial ecological systems, others reject chaos as a natural phenomenon because they believe that prediction must be possible. There are, however, some clear compatibilities

between chaos and modern ecology. SIC shows that the past of a system fundamentally aects its future, which emphasises the importance of the history of an ecological system (chapter 7 Cohen & Stewart, 1991). Such ideas are also seen, for example, in community construction (Robinson & Edgemon, 1988 Facelli & Pickett, 1990) and succession (Clements, 1936 Horn, 1975 Clark, 1991). Chaos also has implications for the sampling of real systems: given a certain amount of data, the extra data needed to provide more information about the system increases exponentially.

It is, however, premature to abandon all hope of predicting behaviour and formulating general laws of a chaotic system (May, 1995). There needs to be a shift in the techniques used to analyse such systems (Stewart, 1995): rather than producing exact trajectories for given initial conditions, thetypeof behaviour should be investigated. Statistical and geometrical methods are suitable for producing information such as the range of possible dynamics, the structure of the underlying attractor and the probabilities of getting dierent outcomes. Judson (1994a) calls this thetextureof a system. Although chaotic dynamics are locally unstable, characteristically there is a globalstability. There are also dierent classes of chaos with dierent levels of stability, including fully-developed spatiotemporal chaos, which is robust to parametric perturbations (Kaneko, 1990).