## Model

Unlike the original noise vectors Ef, the residual vectors e( are correlated and their normality is approximate, with the quality of the approximation varying among different nonlinear time series models. Thus, autocorrelation tests and normality tests should be used only as rough guides to potential areas in which the model is not adequate. The residuals for each individual state variable should have small autocorrelations and approximate univariate normal distributions. In addition to standard normal probability plots, statistical tests for (univariate) normality such as the Lin-Mudholkar test are useful [117, 181]. Standard univariate autocorrelation tests are informative as well.

A useful scan for outliers from multivariate normality is to calculate the quadratic form st = e^E"1ei (2.22)

for each residual vector (where "t" denotes the transpose of a vector). If e( is indeed an observation from a multivariate normal (0, S) distribution, then st is an observation from a chi- square distribution with three degrees of freedom [30,163], We use the ML estimate of £ in (2.22) and therefore the chi-square distribution is only approximate.

Figure 2.3 shows times series plots of the data from replicate A (see Appendix A), together with the one-step LPA model predictions for each life-cycle stage. The predictions use the deterministic LPA model (2.4) with the parameter estimates given in Table 2.1. This plot, together with similar plots of the data from the other three replicates, shows, visually at least, that the one-step model predictions are reasonably accurate. Another visual way to inspect the residuals is to plot, for each stage, the differences between the logarithms of the observed and predicted numbers at time t + 1 against the numbers at time t of those stages on which they depend according to the LPA model (2.4). This is done in Fig. 2.4. We see from these plots that overall the residuals are not large in magnitude and, furthermore, that they do not seem to vary systematically with the sizes of the state variables.

Table 2.4 displays the results of a univariate normality analysis of the residuals. The residuals were calculated using data, and the LPA model (2.4) with ML parameter estimates in Table 2.1 (which we recall were obtained from these same data). Shown are first- and second-order autocorrelations and the Lin-Mudholkar normality statistic for each state variable in each of the four replicates. Only replicate B reveals some slight autocorrelation (and only second-order). Departure from normality is displayed only by the pupae of replicates A and C and the adults of 