#### Parameter estimation for the CSEG

Other parameter estimation approaches are also available. For example, the restricted maximum likelihood (REML) method corrects the finite-sample bias common to ML parameter estimates (Staples *et al.* 2004). Standard statistical packages can be used for REML estimation (Dennis *et al.* 2006). The ‘slope method’ (Holmes 2001) estimates simply by performing a linear regression on the variance with respect to *τ* . In our experience, the slope method is the most robust method when the data are generated from a strongly cyclic process, but this comes at the cost of an increased bias.

In the following cross-validations, we use the slope estimation method, because the Kalman filter and REML methods produced imprecise estimates for our strongly cyclic data. The Kalman filter was often unable to separate and even with 20–30 years of our simulated data. This problem was likely caused by autocorrelations in our simulated data; an extended Kalman filter might be able to correct this problem.

The principal challenge of CSEG parameterization lies in the fact that the actual data may differ from the CSEG model in very substantial ways. The CSEG is an approximation to model the quasi-extinction surface – it is not the underlying model of the data. Despite the existing work on estimation of CSEG approximations, cited above, there is still much research to be done on robust estimation methods. Research on incorporating age-structure information is helping in this regard (Hinrichsen 2002; Engen *et al.* 2007), but we also need to understand how to include non-quantitative life-history information, and how to adjust for autocorrelations generated by population dynamics.

#### Simulation studies of CSEG forecasts for noisy non-SEG processes

We used a Monte Carlo approach to study the performance of CSEGs estimated from various simulations of stochastic processes. Using a particular stochastic model (the ‘base model’), we randomly generated 1000 time series of 10, 20 and 30 years. Each simulation was allowed to run for 100 years (the ‘burn-in’ period) before any observations {*O*_{1},*O*_{2},…,*O*_{k}} were recorded. A CSEG was estimated from each series using the slope method (Holmes 2001, 2004), and used the estimated CSEG to give an estimate of the probability of an 80% population decline within the next 50 years. To determine the *actual* probability of 80% decline for a given series, we used the base model to simulate 1000 50-year trajectories forward from the population state found at the end of the period. In this way, we obtain a CSEG estimate for the expected probability of quasi-extinction, which we compare with the actual expected probability of quasi-extinction.

To calculate the quasi-extinction probabilities, we have to define the quasi-extinction threshold more explicitly. Specifically if our threshold is a percent decline, against what population size should the percent decline be measured? PVAs often use an average of the last three to five censuses, to reduce the effect of measurement errors and other noise. Another common threshold in practice is an absolute number, for example 250 individuals; in this case, the problem of defining the baseline is avoided. In this paper we define percent declines relative to , the maximum-likelihood estimate of the final count after the white noise process has been removed. is thus a fourth parameter that had to be estimated from the CSEG in our cross-validations.

As discussed in the introduction, the focus of this paper is on estimating the *propensity* of a given process to experience quasi-extinction. Thus, we cross-validate our CSEG estimates against the true *expected* probability of quasi-extinction:

- (12)

{*O*_{1},*O*_{2},…,*O*_{k}} denotes a specific simulated parameterization period, and Ω signifies the random sample of 1000 such periods. Pr(80% decline in next 50 years |{*O*_{1},*O*_{2},…,*O*_{k}}) is the actual probability of 80% decline after a given parameterization period, and Pr({*O*_{1},*O*_{2},…,*O*_{k}}) is the probability of observing the given parameterization period. Pr({*O*_{1},*O*_{2},…,*O*_{k}}) is simply 1/1000, as our parameterization periods are drawn randomly from the stationary distribution.

We simulated three very different cyclic population models: a stage-structured model, a density-dependent model with over-compensating dynamics and a model with four interacting species. For a stage-structured model, we used the Chinook salmon model (eqn 5) and estimated CSEGs from the spawner counts only. The parameters of the underlying model (given in Fig. 2) produce strongly fluctuating spawner counts (Fig. 2b). For an over-compensating, density-dependent process we used the stochastic Ricker model (eqn 11) using parameters (*K* = *N*_{0} = 1000, *r* = 0.02, ) taken from recently published PVA analyses of density-dependent processes (McCarthy *et al.* 2003; Sabo *et al.* 2004). This model experiences large density-dependent changes in population size, but is not tightly regulated (see Fig. 5a). To generate the multi-species time series, we used a first-order, multivariate, autoregressive (MAR-1) process. In this simulation, the dynamics of each species are described by a discrete stochastic Gompertz model (Dennis *et al.* 2006):

- (13)

where *g* is the growth rate, *b* is the strength of the density dependence and *ɛ*_{i,t} is a normally distributed variable with mean zero and process error variance of . Multi-species dynamics are then modelled by extending eqn 13 (Ives *et al.* 2003):

- (14)

Here **X**_{t} is a vector of the population sizes for each species at time *t*, **G** is a vector of their growth rates, **B** is the community matrix describing the strength of self-regulation (diagonal elements) and interspecies interactions (off-diagonals) and **E**_{t} is a process noise vector. We specified the MAR-1 process using estimates from a zooplankton community in Peter Lake, WI, USA (Ives *et al.* 2003). In estimating the CSEG and forecasting risk, we only used the time series for one of the four species (large phytoplankton). The time series of this process fluctuates tightly about an equilibrium value (similar to the process shown in Fig. 5b).

Figure 6 shows the estimated probability of decline, E[Pr(qe)], found using 10-, 20- and 30-year parameterization periods. The first thing to notice is that 10-year periods are insufficient for estimating the quasi-extinction risk. Although the estimates themselves are not terribly biased, their distributions are spread across 0–1 and are often bimodal. Thus, from now on we refer only to results for 20- or 30-year parameterization periods. It can be seen that the mean estimated CSEG risks (black dots) track the actual risks closely (solid black lines), but are not strictly accurate; each model generates its own flavour of bias. The slope method, as expected (Holmes & Fagan 2002), produces a biased estimate of . Initial tests suggest that this error could have been approximately halved by estimating the bias via parametric bootstrapping. Nonetheless, it is striking that the estimated and actual risks correspond so well given that all three models produce strongly oscillatory data and have high non-process variance.

The boxes and whiskers in Fig. 6 enclose 50% and 95% of the estimates respectively. The line in the middle of each box represents the median. The inner quartiles (covering 50% of all estimates) are quite small for the MAR and salmon simulations. In these two populations, the 95% ranges are also smaller than (0,1) for both short (10–20 year) and long (30+ year) forecast lengths. In the Ricker model, the long-term forecasts have very wide and bimodal 95% ranges while the short-term (10- and 20-year) forecasts have relatively low variability (Fig. 6e,f). This occurs because the *μ* parameter of the CSEG (which measures trend) should have been constrained to zero. Because this process reverts to the long-term mean population very slowly, the *μ* estimates took on highly uncertain positive and negative values.

Estimating and forecasting are known to be difficult for processes with *μ* = 0 (flat trend) and with slow fluctuations about the long-term mean. Small errors in the estimated value of *μ* lead to large errors in the estimated long-term population size and this leads to bimodal long-term extinction risk predictions. This problem has been noted frequently by others and motivates the common recommendation that policy-makers should rely on 10- to 30-year forecasts rather than 100-year. A recent paper introduced a discrete Gompertz state-space model to forecast extinction risks for density-dependent processes (Dennis *et al.* 2006). This may be a better approach for slowly reverting processes, for which the problem of bimodal qe estimates is especially apt to occur. We should note, however, that this is not necessarily a problem for the typical species being considered for an endangered, threatened or similar designation: such species are often declining.

#### Studies on CSEG forecasts for real-time series

Simulations are useful for studying population dynamics, but the models chosen are simplified and not necessarily representative of typical populations. In our second cross-validation study, we assembled a database of 63 time series at least 30 years long from species monitored for conservation or management reasons. These data were obtained from a literature search and through direct contact with governmental agencies across the world (Table S1). The majority are high-risk populations officially listed by one or more conservation agencies at the endangered or threatened level.

Our methods are similar to those of other cross-validations using real data (Brook *et al.* 2000; Holmes & Fagan 2002; Holmes *et al.* 2005), except that we examine the precision of our estimates as well as the bias. Each time series was divided into a 20-year parameterization period followed by a 10-, 20-, or 28-year forecast period. Very long time series were also segmented to increase the sample size; in this case the parameterization periods were allowed to overlap by 10 years. No more than five parameterization periods were taken from any one time series, so as not to over-represent any species or population. For each time series, a CSEG was estimated from the parameterization period using the slope method. The CSEG was then used to predict whether the quasi-extinction threshold would be reached during the forecast period. We validated the CSEG estimates in two different ways. First, the actual number of quasi-extinctions in the entire database was compared with the expected number derived from CSEG estimates at various thresholds and horizon lengths. This analysis assesses the systematic bias in the method and is analogous to the validation of E[Pr(qe)] estimates shown in Fig. 6. The goal of this analysis is to quantify whether we can properly estimate the risk on average. Unlike the simulations, we cannot assess the variability of the risk estimates directly. Instead, we use parametric bootstrapping (Dennis & Otten 2000; Engen *et al.* 2001) to estimate their precision. In this method, the estimated model is used to generate new data from which the parameters can be estimated again, and so on. This process creates a series of parameter estimates, which can be used to evaluate variability.

We calculated two measures of uncertainty. The first is the probability of 20%, 50%, or 80% decline within a 10-year forecast period, with 95% and 50% confidence intervals. These confidence intervals are notoriously difficult for decision makers to use – being essentially an uncertainty on an uncertainty. Population prediction intervals (PPI) have been proposed as an alternative (Engen *et al.* 2001; Sæther & Engen 2002). The PPI is the time interval that will include a decline to some quasi-extinction threshold 95% or 50% of the time. The PPI interval incorporates the uncertainty because of stochasticity as well as the uncertainty in the estimated parameters.

Figure 7 shows the estimated and actual proportion of quasi-extinctions over the entire database for 10-, 20- and 28-year forecasts following a 20-year parameterization period. The observed (line) and predicted (points) frequencies match very closely. These plots show that the CSEG estimates of E[Pr(qe)] are generally unbiased. Figure 8 shows the estimated precision of the estimates – via parametric bootstrapping. The left-hand column of Fig. 8 shows the estimated 95% (grey) and 50% (black) CIs of the CSEG-estimated quasi-extinction probabilities (80%, 50% and 20% declines, all over a 10-year projection period). The 50% CIs are quite narrow, and the 95% CIs are less than (0,1). Precision, as might be expected, is best when the estimated risk is high (> 75%) or low (< 25%). Precision is worst (wide CIs) when the estimated qe is near 50%. However, the PPIs in Fig. 8 indicate that the certainty of the risk estimation is often better than the CIs might suggest. The PPIs also highlight the fact that in most cases, a low estimated probability of quasi-extinction within 10 years corresponds to a long expected quasi-extinction time (< 50% of time series experience qe before 100 years) and conversely when the estimated qe risk is > 50%, half of time series will have experienced qe well before 50 years. The dataset used in this analysis is focused on populations of conservation concern. Such populations are atypical in that they tend to be at historically low levels or declining; however, PVA is generally done for precisely these types of populations.