On observation distributions for state space models of population survey data

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1 Journal of Animal Eology 2011, 80, doi: /j x On observation distributions for state spae models of population survey data Jonas Knape 1 *, Nilas Jonze n 1 and Martin Sköld 2 1 Department of Biology, Eology Building, Lund University, SE Lund, Sweden; and 2 Stokholms universitet, Matematisk statistik oh fo rsa kringsmatematik, SE Stokholm, Sweden Summary 1. State spae models are starting to replae more simple time series models in analyses of temporal dynamis of populations that are not perfetly ensused. By simultaneously modelling both the dynamis and the observations, onsistent estimates of population dynamial parameters may be obtained. For many data sets, the distribution of observation errors is unknown and error models typially hosen in an ad-ho manner. 2. To investigate the influene of the hoie of observation error on inferenes, we analyse the dynamis of a repliated time series of red kangaroo surveys using a state spae model with linear state dynamis. Surveys were performed through aerial ounts and, overdispersed, normal and log-normal distributions may all be adequate for modelling observation errors for the data. We fit eah of these to the data and ompare them using AIC. 3. The state spae models were fitted with maximum likelihood methods using a reent importane sampling tehnique that relies on the Kalman filter. The method relaxes the assumption of Gaussian observation errors required by the basi Kalman filter. Matlab ode for fitting linear state spae models with observations is provided. 4. The ability of AIC to identify the orret observation model was investigated in a small simulation study. For the parameter values used in the study, without repliated observations, the orret observation distribution ould sometimes be identified but model seletion was prone to mislassifiation. On the other hand, when observations were repliated, the orret distribution ould typially be identified. 5. Our results illustrate that inferenes may differ markedly depending on the observation distributions used, suggesting that hoosing an adequate observation model an be ritial. Model seletion and simulations show that for the models and parameter values in this study, a suitable observation model an typially be identified if observations are repliated. Model seletion and repliation of observations, therefore, provide a potential solution when the observation distribution is unknown. Key-words: maximum likelihood, model seletion, observation errors, population dynamis, proess errors, repliation, state spae models Introdution Monitoring programs often generate time series of population abundane that an be used for analysing, among other things, population trends, variability and extintion risks. A feature of almost all suh time series on natural populations is that population abundane is measured with errors; only in rare ases an the exat abundane of a *Correspondene author. jknape@berkeley.edu Present address: Department of Environmental Siene, Poliy and Management, University of California, 137 Mulford Hall #3114, Berkeley, CA 94720, USA. population be obtained. The dynamial properties of a time series, suh as autoorrelation, hange when observation errors are introdued. With inreasing magnitude of observation errors, a time series beomes inreasingly similar to white noise. The hange in dynamis aused by observation error has profound onsequenes for inferenes and an affet estimates of trends, extintion risks (Dennis, Poniano & Taper 2010) and environmental effets (Lindén & Knape 2009) as well as estimates of population abundane itself. These problems are well known (Kuno 1971; Walters & Ludwig 1981), but only reently have statistial methods to deal with them beome easily available and used by eologists. Observation errors in time series are handled Ó 2011 The Authors. Journal of Animal Eology Ó 2011 British Eologial Soiety

2 1270 J. Knape, N. Jonze n & M. Sko ld statistially by state spae models (de Valpine & Hastings 2002; Dennis et al. 2006; Bukland et al. 2007). The main feature, but also the main diffiulty, of state spae models is that they make a distintion between variane assoiated with the hanges in population abundane (proess error) and variane assoiated with estimating the population abundane (observation error). Under assumptions about model struture and the distributions of proess and observation error, the two soures of variane an often be statistially separated but usually with very low preision. Diret information about the sampling distribution inreases preision of parameter estimates (Knape 2008) and an, for instane, be obtained through repliating the sampling proess (Frekleton et al. 2006; Dennis, Poniano & Taper 2010). For ertain sampling designs, the distribution of the observation errors may be derived from first priniples. Suh is the ase with population size estimation from mark reapture data under assumptions of a losed population and independent sampling. For other types of data, there is no natural way of determining the distribution of the errors, and the hoie of distribution for the state spae model is instead typially made in an ad-ho fashion. Models of observation errors for time series that our in the literature inlude log-normal (Dennis et al. 2006), normal (Jamieson & Brooks 2004; Newman et al. 2009) and distributions (Stenseth et al. 2003). These distributions differ for instane in the saling of the variane with the mean, whih is likely to influene inferenes on population dynamis. The idea of state spae models is to separate observation errors from proess errors (Calder et al. 2003), but as the distribution of the observations is rarely known it is entral that an appropriate observation model an be identified from the data or, alternatively, that inferenes do not differ signifiantly among distributions. We analyse a time series with repliated observations on red kangaroos in New South Wales, Australia, and examine how different assumptions about the sampling proess affet estimates of population abundanes and population dynamial quantities. We further investigate how well these assumptions an be justified with and without repliated sampling using model seletion riteria. Simulations where the true sampling model is known are used to assess the performane of model seletion. Materials and methods DATA The data onsist of ounts of red kangaroos (Maropus rufus) inthe Kinhega National Park between 1973 and 1984 (Caughley, Shepherd & Short 1987). Kinhega is situated in western New South Wales, Australia, lose to the township of Menindee (32 33 S E). A fene and the natural boundary of a river prevented kangaroos from entering or leaving the park (Bayliss 1985). Approximately every third month, the red kangaroo population was surveyed by two observers ounting all kangaroos within 100 m on eah side of a plane flown along a transet line (Bayliss 1985). The survey was performed twie, on onseutive days, giving two measures of population abundane for eah of a total of 41 sampling oasions. Repliation is rare for eologial time series of population surveys (Dennis, Poniano & Taper 2010), and we utilize the double ounts to ompare analyses of models of both ounts to models involving only the first or only the seond ount at eah sampling oasion. We refer to the sequenes of first and seond day ounts as the first and seond time series, respetively. Only the month of sampling is given in Caughley, Shepherd & Short (1987), and times between sampling oasions varied over the time series, ranging from 2 to 5 months with 88% of intersampling times being 3 4 months. We ignore this ompliation in the models beause we are onerned with illustrating differenes among observation distributions rather than providing an in-depth analysis of kangaroo dynamis. MODELS We assume that the dynamis follow a simple Gompertz model with log-normal proess errors, N tþ1 ¼ N t expða þ b log N t þ e t Þ¼N t expða þ e t Þ; e t Nð0; s2 Þ; eqn 1 where =1 +b is the autoorrelation at the log sale (if )1 << 1), ands 2 is theproess errorvariane. The first population size was given a log-normal initial distribution, N 1 LN(5, 10). On the log sale, this is a linear Gaussian autoregressive model. Similar single lag models have been used previously for kangaroo population dynamis (Jonzén et al. 2005; Hauser, Pople & Possingham 2006). We denote the ount on the first day at time t, 1 t 41, by y 1t and the ount on the seond day at time t by y 2t. We fit six different models to the full data set, in whih pairs y Æt = (y 1t,y 2t ) are independent over time onditionally on the unobserved population abundanes, N Æ. Within pairs, we onsider two situations: either the two repliates y 1t and y 2t are independent onditionally on N t or they are independent onditionally on a salar state variable z t. The latter is introdued to model dependene between repliates onditionally on population abundane, beause of fators influening observations that may be onstant over the time of the survey (e.g. weather). We assume z t to be log-normally distributed and independent over time onditionally on N t, z t N t LN(log N t ) r 2 od 2, r2 od ). The variane parameter r 2 od ontrols the amount of overdispersion, and models inluding terms z t will, heneforth, be referred to as overdispersed models. The overdispersion of the observations for these models is relative to the population size, and the two repliates are not overdispersed relative to eah other. Note also that the distribution LN(l, r 2 ) has mean exp(l + r 2 2) and that z t is defined so that E(z t N t )=N t. The full models together with their implied observation error varianes are as follows: 1 No observation errors, here we set (y 1t +y 2t ) 2 =N t. 2 distribution, y it N t Po(N t ) and V(y it N t ) =N t. 3 Log-normal distribution, y it N t LN(log N t ) r 2 2, r 2 ) and V(y it N t ) =N 2 t (exp r 2 ) 1). 4 Overdispersed distribution, y it z t Po(z t ) and V(y it N t ) =N t + N 2 t (exp r2 od ) 1). 5 distribution, y it N t N(N t, r 2 )andv(y it N t ) = r 2. 6 Overdispersed log-normal distribution, y it z t LN (log z t ) r 2 2, r 2 )andv(y it N t ) =N 2 t (exp (r 2 od + r2 ) ) 1). To quantify the advantage of using repliates, models 1 5 were also fitted to the first and seond day series sets, y 1Æ and y 2Æ, separately. Model 6 was not fitted to these sets as it is not identifiable without repliates.

3 Observation models for population data 1271 Table 1. Model seletion Full data First time series Seond time series Observation model AIC Akaike weight Observation model AIC Akaike weight Observation model AIC Akaike weight Overdispersed log-normal 1079Æ0 0Æ57 Overdispersed 551Æ9 0Æ36 Overdispersed 556Æ4 0Æ43 Log-normal 1079Æ6 0Æ42 Log-normal 552Æ2 0Æ31 Log-normal 556Æ9 0Æ Æ0 0Æ00 553Æ0 0Æ21 558Æ3 0Æ16 Overdispersed 2083Æ5 0Æ00 555Æ3 0Æ07 561Æ1 0Æ Æ1 0Æ00 No observation error 555Æ4 0Æ06 No observation error 561Æ2 0Æ04 AIC values and Akaike weights for the models of the full data (left) and of the first (middle) and seond (right) time series modelled separately. Models are sorted aording to the AIC values with the top model preferred. Monte Carlo error standard deviation of the AIC is <0Æ08. MAXIMUM LIKELIHOOD ESTIMATES FOR NON-NORMAL STATE SPACE MODELS Gaussian state spae models an be relatively easily fitted numerially via the Kalman filter using either maximum likelihood (Dennis et al. 2006) or Bayesian (Calder et al. 2003) tehniques. Fitting more general models is nontrivial and usually requires Monte Carlo methods. Methods for general state spae models inlude the following: sequential Monte Carlo, Markov hain Monte Carlo (MCMC) and numerial integration tehniques (de Valpine & Hastings 2002). For model seletion in lassial settings, the exat likelihood value is needed to ompute model seletion riteria suh as the AIC or BIC. MCMC methods generally produe relative rather than exat likelihood values, and the latter an be diffiult to approximate (de Valpine 2008). We here use an importane sampling (a form of Monte Carlo integration) tehnique that allows maximum likelihood estimation of Gaussian state models with non-gaussian observations (Durbin & Koopman 1997; Shephard & Pitt 1997). The tehnique was developed by Durbin & Koopman (1997) and Shephard & Pitt (1997) and extended by Jungbaker & Koopman (2007). It requires Gaussian state models but plaes, in theory, no restritions on the observation distributions. The likelihood is omputed through Monte Carlo integration over the hidden population abundanes using importane sampling, i.e. via sampling from an approximating distribution. The Laplae approximation to the distribution of the hidden state vetor is used as the importane density. When the state model is Gaussian, the Kalman filter (Dennis et al. 2006) an be used to implement the Newton Raphson method to find the mode and the Hessian of the distribution for the Laplae τ Combined First series Seond series Combined First series Seond series N T Combined First series Seond series N T Combined First series Seond series Fig. 1. Parameter estimates (. 95% onfidene intervals) of density dependene (top left), proess error standard deviation (top right) and estimated population sizes at the time of the last sample (bottom left) and one time step after the last sample (bottom right). The left group of bars refers to the models of both repliates, the middle group to the first repliate series and the right group to the seond repliate series under no observation errors (4), observations (,), log-normal observations (s), overdispersed observations ()), normal observations (h) and overdispersed log-normal observations (+).

4 1272 J. Knape, N. Jonze n & M. Sko ld approximation. The Kalman filter an also be used to sample from the Laplae approximating distribution. Details about the method are given in Appendix S1(Supporting Information). The likelihood obtained by the importane sampling was maximized in Matlab using the funtion fminsearh, whih implements the Nelder Mead simplex algorithm. In finding the maximum likelihood value, we for eah evaluation of the likelihood used 20 iterations of the Newton Raphson method to obtain the Laplae approximation and 500 samples were drawn from the importane density for all but the normal models for whih 2000 samples were drawn. As importane sampling is a Monte Carlo method, the likelihood value for a fixed input differs slightly between different evaluations. This may ause onvergene problems for numerial optimization routines. To avoid this, the same random seed was used for all alls to the likelihood funtion, giving a smooth likelihood (Jungbaker & Koopman 2007). Likelihoods of state spae models may have multiple optima, and we therefore optimized the likelihood repeatedly using different starting values to fminsearh to derease the risk of finding estimates orresponding to a loal but nonglobal optimum. Approximately 95% onfidene intervals for the parameters were omputed from likelihood profiles by finding the values of the parameters that gave a likelihood profile value v Æ92 units from the maximum likelihood value, where v is the 95% quantile of the Chi-square distribution with one degree of freedom. This is based on minus two times the log of the likelihood ratio of the full model vs. a model with one parameter held fixed being asymptotially Chi-square distributed with one degree of freedom. In the event that more than two suh values were found, we hose the ones whose orresponding parameters were losest to the maximum likelihood estimates. When omputing onfidene intervals, we used 10 iterations of the Newton Raphson method and drew 100 samples from the importane density. Matlab ode, inluding an implementation of the Kalman filter, for fitting the Gompertz model with observations is available in Appendix S2 (Supporting Information). MODEL SELECTION Models were ompared using AIC, the Akaike information riterion defined as: AIC ¼ 2log likelihood þ 2k; where k is the number of parameters in the model. We further use model weights, defined by, X w i ¼ expð AIC i =2Þ expð AIC i =2Þ i where AIC i is the AIC of model i. Model weights an be informally interpreted as model probabilities (Burnham & Anderson 2002). Kangaroos Kangaroos Kangaroos Time Time Over-dispersed Fig. 2. Estimated relative population sizes for times 1 through 41 and one-step-ahead preditions of the population size at time 42 for the full data (top row) the first time series (seond row) and the seond time (bottom row) series with observations (left olumn) and overdispersed observations (right olumn). Cirles represent the observations from the first time series and rosses observations from the seond. The thik blak lines are the exponential of the onditional mean log population size, and the thin grey lines are the exponential of the same mean ± 1Æ96 of the onditional standard deviation of the population size at the log sale.

5 Observation models for population data 1273 od Fitted model od Simulation model Fig. 3. Bars represent the proportion of times the fitted model is the AIC best model for data simulated from the, log-normal, overdispersed and normal observation models. Points show the Akaike weights where best model weights are marked in orange. The left side of eah panel shows the results for unrepliated observation time series and the right side the results for time series with two repliate observations. To investigate the performane of model seletion, we for eah of the, log-normal, overdispersed and normal observation models (models 2 5) simulated 24 time series with unrepliated observations and 24 series with two repliate observations. The proess parameters were set to a=1æ6, =0Æ75 and s = 0Æ28 for all simulations, orresponding to the estimates from the best model of the repliated kangaroo data, and the initial population size was set to the mean of the two kangaroo ounts at time 1. All simulated series were of length 41. The observation error standard deviation was set to r = 0Æ25 for the log-normal model, r od = 0Æ27 for the overdispersed model and r = 138 for the normal model, whih orresponds to the respetive parameter estimates for the full data. Eah simulated data set was then fitted to all of the models 2 5, and parameter estimates and Akaike weights were reorded. When fitting the disrete observation distribution models to ontinuous simulated data, the data were rounded to the nearest integer. Results The model seletion riterion, AIC, gave the same ranking and similar Akaike weights for the unrepliated models of the first and seond time series (Table 1). The overdispersed model performed slightly better, for these two individual time series, than the log-normal and normal models. The and no observation error models had lower support but the maximal differene in AIC between any of the models is <5 meaning that there were only moderate to strong preferenes for any model over another. In ontrast, for the models of the full data, all the models with non- observations were highly preferred over the two models with observations (Table 1). The preferred models were the log-normal and overdispersed log-normal that had similar support. The standard deviation of the Monte Carlo error of the log likelihood is <0Æ04 for all models. Parameter estimates for the observation models were nearly idential to the parameter estimates from the model assuming no observation errors (Fig. 1, Table S1, Supporting Information). This might be expeted beause the highest numbers of ounted kangaroos in these data were around 1000 whih mean that the standard deviation of the observation errors is in pratie limited to being smaller

6 1274 J. Knape, N. Jonze n & M. Sko ld p than ffiffiffiffiffiffiffiffiffiffi In general, the models and the models with no observation errors gave larger estimates of the proess error standard deviation (s), smaller estimates of in (1) (stronger density dependene), higher preision in estimates of population sizes within the sampling period and lower preision in one-step-ahead preditions of population sizes than the other models. The differenes in density dependene are in line with well-known results regarding bias in estimates of density dependene in models without observation errors (e.g. Frekleton et al. 2006) but show that there may be biases even when observation errors are modelled. Interestingly, for the ombined data, the overdispersed log-normal model gives narrower onfidene intervals for the parameters and one-step-ahead preditions than the log-normal model. This is despite the fat that the onfidene interval for the overdispersion variane is wide and ontains zero (Table S1, Supporting Information). Plots of onditional population trajetories (Fig. 2, Figs S1, S2 and S3, Supporting Information) reveal the large differenes in unertainty about population abundane between models with observations and models allowing for more variations in the observation proess. Model seletion simulations showed that, for the parameter values used in the simulation, the true observation distribution an typially be readily identified when the true distribution is normal or log-normal and observations are repliated (Fig. 3). Model seletion also performed reasonably well when the true observations were distributed, for both repliated and unrepliated observations, and for unrepliated observations when they were normal. When the observations were simulated aording to the overdispersed, either the orret or the distribution was typially preferred by AIC. The simulations also showed that was negatively biased for most models (Fig. 4) but that s was not detetibly biased or only weakly biased when the true model is fitted (Fig. 5). Disussion Our results show that point estimates and preision of population dynamial parameters obtained from state spae models of time series an vary strongly depending on the model used for the observations. Hene, it an be essential that the hoie of model for the observations an be justified. od Fitted model Simulation model od Fig. 4. Estimates of obtained from the simulation study. Error bars denote the mean with 95% onfidene intervals of the estimator of. The horizontal line represents the true value of used in the simulations.

7 Observation models for population data 1275 od Fitted model τ τ τ τ od Simulation model Fig. 5. Estimates of s from the simulation study. Error bars denote the mean with 95% onfidene intervals of the estimator of s. The horizontal line represents the true value of s used in the simulations. Justifiation may arise from the details of sampling proedures but will often have to be based on the data. A priori, the sampling proess of aerial ounting of kangaroos ould suggest that the distribution may be a reasonable observation model. Considering eah of the two repliations of the series separately and omparing them with model seletion riteria against, for example, log-normal observation models would only give moderate support for the latter models over the models even though they give very different preditions about the dynamis. Considering both repliated series together, it is obvious that the distribution is not able to apture the variability in the observations. Repliated observations therefore are key to identifying the observations distribution. This onlusion is fortified by our model seletion simulations, whih however suggest that some distributions may be orretly identified even without repliated observations. Repliated observations also have been shown to improve preision and statistial properties of parameter estimates for the linear state spae model (Dennis, Poniano & Taper 2010) and, if the repliates are onditionally independent given population abundane, resolve the identifiability issue assoiated with the Gompertz state spae model for unrepliated observations (Knape 2008). In pratie, the benefits of repliation need to be weighed against osts, but Dennis, Poniano & Taper showed through simulations that better inferenes may, at least for ertain parameter ombinations, be obtained by repliation than by doubling the length of a time series. Albeit speulative, this indiates that in some instanes it might be more benefiial to trade sampling frequeny for repliation, e.g. by sampling twie every seond year instead of one every year (f. Hauser, Pople & Possingham 2006). A further pratial issue with repliation is the diffiulty in obtaining truly or nearly independent repliates (see Hurlbert 1984). Nonindependene of repliates may, for example, arise as a result of variation in sampling onditions between sampling times (Knape et al. 2009) or beause the proportion of individuals available for sampling varies between sampling times (Frekleton et al. 2006) and is modelled in our overdispersed and overdispersed log-normal models. The kangaroo data, indeed, seem to indiate nonindependene relative to a losed population. For instane, at the fourth

8 1276 J. Knape, N. Jonze n & M. Sko ld sampling oasion, 145 and 138 individuals were ounted but 3 months later 340 and 413 individuals were ounted. This suggests an inrease that is not possible attributed to only births, beause eah female gives rise to a maximum of 1Æ5 young per year. As the Kinhega population was losed to immigration and emigration, the nonindependene is presumably beause of variability in survey onditions suh as loud over or temperature affeting visibility (Caughley, Shepherd & Short 1987) or beause of kangaroos moving in groups. Large sale shifts in distribution have, for example, been desribed in response to spatio-temporal patterns in rainfall (Pople et al. 2007). From the model seletion simulations, it an be seen that, as ould be expeted, orrelations between repliates an hamper the ability to identify the observation distribution. In fat, the salar state variable z t in the overdispersed and log-normal models (4 and 6) is modelled as a univariate Gompertz state spae model (unrepliated model 3) and therefore will retain many of the estimation issues assoiated with that model. Repliation may however still provide some benefits beause parts of the observation error ould be filtered out from z t. The importane sampling method we use provides an alternative for fitting non-gaussian state spae models in lassial settings. While the method is restrited to linear state models, its main advantage is that it diretly provides the log-likelihood value. This is in ontrast to many other Monte Carlo tehniques where extra omputations are typially needed to estimate the exat likelihood (de Valpine 2008). The exat likelihood is neessary for model omparisons using likelihood ratios or model seletion riteria. Models with different observation distributions an therefore be ompared using the method. Apart from the models used here, we have found the method to work for gamma and negative binomial observation distributions. Beause the method is based on ML estimation, parameter estimates may be biased if sample size is small. Our simulations and simulations in Dennis, Poniano & Taper (2010) onfirm this. Model seletion for state spae models has been little explored (Bengtsson & Cavanaugh 2006). As the number of parameters in our observation models only ranges between zero and two, model seletion here is mainly a question of model fit and the omplexity penalty of AIC is therefore not put to ritial testing. The AIC is defined from the marginal likelihood and the number of fixed parameters, but for hierarhial models there are alternative ways of defining AIC like model seletion riteria. Instead of using the marginal likelihood, model seletion riteria an be foused at higher levels in the hierarhial model using the onditional likelihood given the values at that level and ounting the effetive number of parameters (Spiegelhalter et al. 2002). This is the idea behind the deviane information riterion (Spiegelhalter et al. 2002) for Bayesian models, where parameters are not easily ounted, and also underlies the onditional AIC for linear mixed models (Vaida & Blanhard 2005). Other viable approahes to model seletion for state spae models inlude Bayesian posterior model probabilities (Fru hwirth-shnatter 1995; Jamieson & Brooks 2004; Gimenez et al. 2009). Although our model seletion simulations are not omplete and apply to speifi models and parameter values, the results are enouraging for the prospet of using model seletion riteria to evaluate observation distributions if observations are repliated. 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