## Models

We currently analyze and forecast rodent data at Portal using eleven models: AutoArima, ESSS, NaiveArima, nbGARCH, nbsGARCH, pevGARCH, jags_RW, jags_logistic, jags_logistic_covariates, jags_logistic_competition, jags_logistic_competition_covariates (WeecologyLab 2019). Each model has a function with its name (e.g., ESSS()) which is called from an associated R script in the models subdirectory with its name (e.g., ESSS.R)

### AutoArima

AutoArima (Automatic Auto-Regressive Integrated Moving Average) is a flexible Auto-Regressive Integrated Moving Average (ARIMA) model fit to the data at the composite (full site and just control plots) spatial level and both the composite (community) and the articulated (species) ecological levels. The model is selected and fitted using the auto.arima and forecast functions in the forecast package (Hyndman and Athanasopoulos 2013; Hyndman 2017) within the AutoArima() function. Generally, ARIMA models are defined according to three model structure parameters – the number of autoregressive terms (p), the degree of differencing (d), and the order of the moving average (q), and are represented as ARIMA(p, d, q) (Box and Jenkins 1970). While the auto.arima function allows for seasonal models, the seasonality is hard-coded to be on the same period as the sampling, which is not the case for the Portal rodent surveys. As a result, no seasonal models were evaluated. AutoArima is fit flexibly, such that the model parameters can vary from fit to fit.

### ESSS

ESSS (Exponential Smoothing State Space) is a flexible exponential smoothing state space model (Hyndman et al. 2008) fit to the interpolated data at the composite (full site and just control plots) spatial level and both the composite (community) and the articulated (species) ecological levels. The model is selected and fitted using the ets and forecast functions in the forecast package (Hyndman 2017) with the allow.multiplicative.trend argument set to TRUE within the ESSS() function. Models fit using ets implement what is known as the “innovations” approach to state space modeling, which assumes a single source of noise that is equivalent for the process and observation errors (Hyndman et al. 2008). In general, ESSS models are defined according to three model structure parameters – error type, trend type, and seasonality type (Hyndman et al. 2008). Each of the parameters can be an N (none), A (additive), or M (multiplicative) state (Hyndman et al. 2008). However, because of the difference in period between seasonality and sampling of the Portal rodents combined with the hard-coded single period of the ets function, we could not include the seasonal components to the ESSS model. ESSS is fit flexibly, such that the model parameters can vary from fit to fit.

### NaiveArima

NaiveArima (Naive Auto-Regressive Integrated Moving Average) is a fixed Auto-Regressive Integrated Moving Average (ARIMA) model of order (0,1,0) fit to the data at the composite (full site and just control plots) spatial level and both the composite (community) and the articulated (species) ecological levels. The model is selected and fitted using the Arima and forecast functions in the forecast package (Hyndman and Athanasopoulos 2013; Hyndman 2017) within the NaiveArima() function.

### nbGARCH

nbGARCH (Negative Binomial Auto-Regressive Conditional Heteroskedasticity) is a generalized autoregressive conditional heteroskedasticity (GARCH) model with overdispersion (i.e., a negative binomial response) fit to the interpolated data at the composite (full site and just control plots) spatial level and both the composite (community) and the articulated (species) ecological levels. The model for each species and the community total is selected and fitted using the tsglm function in the tscount package (Liboschik et al. 2017) within the nbGARCH() function. GARCH models are generalized ARMA models and are defined according to their link function, response distribution, and two model structure parameters – the number of autoregressive terms (p) and the order of the moving average (q), and are represented as GARCH(p, q) (Liboschik et al. 2017). The nbGARCH model is fit using the log link and a negative binomial response (modeled as an over-dispersed Poisson), as well as with p = 1 (first-order autoregression) and q = 12 (approximately yearly moving average). The tsglm function in the tscount package (Liboschik et al. 2017) uses a (conditional) quasi-likelihood based approach to inference and models the overdispersion as an additional parameter in a two-step approach. This two-stage approach has only been minimally evaluated, although preliminary simulation-based studies are promising (Liboschik, Fokianos, and Fried 2017).

### nbsGARCH

nbsGARCH (Negative Binomial Seasonal Auto-Regressive Conditional Heteroskedasticity) is a generalized autoregressive conditional heteroskedasticity (GARCH) model with overdispersion (i.e., a negative binomial response) with seasonal predictors modeled using two Fourier series terms (sin and cos of the fraction of the year) fit to the interpolated data at the composite (full site and just control plots) spatial level and both the composite (community) and the articulated (species) ecological levels. The model for each species and the community total is selected and fitted using the tsglm function in the tscount package (Liboschik et al. 2017) within the nbsGARCH() function. GARCH models are generalized ARMA models and are defined according to their link function, response distribution, and two model structure parameters – the number of autoregressive terms (p) and the order of the moving average (q), and are represented as GARCH(p, q) (Liboschik et al. 2017). The nbsGARCH model is fit using the log link and a negative binomial response (modeled as an over-dispersed Poisson), as well as with p = 1 (first-order autoregression) and q = 12 (approximately yearly moving average). The tsglm function in the tscount package (Liboschik et al. 2017) uses a (conditional) quasi-likelihood based approach to inference and models the overdispersion as an additional parameter in a two-step approach. This two-stage approach has only been minimally evaluated, although preliminary simulation-based studies are promising (Liboschik, Fokianos, and Fried 2017).

### pevGARCH

pevGARCH (Poisson Environmental Variable Auto-Regressive Conditional Heteroskedasticity) is a generalized autoregressive conditional heteroskedasticity (GARCH) model fit to the interpolated data at the composite (full site and just control plots) spatial level and both the composite (community) and the articulated (species) ecological levels. The response variable is Poisson, and a variety of environmental variables are considered as covariates. The model for each species is selected and fitted using the tsglm function in the tscount package (Liboschik et al. 2017) within the pevGARCH() function. GARCH models are generalized ARMA models and are defined according to their link function, response distribution, and two model structure parameters – the number of autoregressive terms (p) and the order of the moving average (q), and are represented as GARCH(p, q) (Liboschik et al. 2017). The pevGARCH model is fit using the log link and a Poisson response, as well as with p = 1 (first-order autoregression) and q = 12 (yearly moving average). The environmental variables potentially included in the model are min, mean, and max temperatures, precipitation, and NDVI. The tsglm function in the tscount package (Liboschik et al. 2017) uses a (conditional) quasi-likelihood based approach to inference. This approach has only been minimally evaluated for models with covariates, although preliminary simulation-based studies are promising (Liboschik, Fokianos, and Fried 2017). Each species is fit using the following (nonexhaustive) sets of the environmental covariates – [1] max temp, mean temp, precipitation, NDVI; [2] max temp, min temp, precipitation, NDVI; [3] max temp, mean temp, min temp, precipitation; [4] precipitation, NDVI; [5] min temp, NDVI; [6] min temp; [7] max temp; [8] mean temp; [9] precipitation; [10] NDVI; and [11] -none- (intercept-only). The single best model of the 11 is selected based on AIC.

### jags_RW

jags_RW fits a hierarchical log-scale density random walk model with a Poisson observation process using the JAGS (Just Another Gibbs Sampler) infrastructure (Plummer 2003) fit to the data at the composite (full site but just control plots) spatial level and the articulated (species) ecological level. Similar to the NaiveArima model, jags_RW has an ARIMA order of (0,1,0), but in jags_RW, it is the underlying density that takes a random walk on the log scale, whereas in NaiveArima, it is the raw counts that take a random walk on the observation scale. There are two process parameters – mu (the density of the species at the beginning of the time series) and sigma (the standard deviation of the random walk, which is Gaussian on the log scale). The observation model has no additional parameters. The prior distributions for mu is informed by the available data collected prior to the start of the data used in the time series. mu is normally distributed with a mean equal to the average log-scale density and a standard deviation of 0.04. Sigma was given a uniform distribution between 0 and 1.

### jags_logistic

jags_logistic fits a hierarchical log-scale density logistic growth model with a Poisson observation process using the JAGS (Just Another Gibbs Sampler) infrastructure (Plummer 2003) fit to the data at the composite (full site but just control plots) spatial level and the articulated (species) ecological level. Building upon the jags_RW model, jags_logistic expands upon the “process model” underlying the Poisson observations. There are four process parameters – mu (the density of the species at the beginning of the time series), sigma (the standard deviation of the random walk, which is Gaussian on the log scale), r (growth rate), and K (carrying capacity). The observation model has no additional parameters. The prior distributions for mu and K are informed by the available data collected prior to the start of the data used in the time series and sigma and r are set with vague priors. mu is normally distributed with a mean equal to the average log-scale density and a standard deviation of 0.04. K is modeled on the log-scale with a prior mean equal to the maximum of past counts and a standard deviation of 0.04. r is given a normal prior with mean 0 and standard deviation 0.04. Sigma was given a uniform distribution between 0 and 1.

### jags_logistic_covariates

jags_logistic_covariates fits a hierarchical log-scale density logistic growth model with a Poisson observation process using the JAGS (Just Another Gibbs Sampler) infrastructure (Plummer 2003) fit to the data at the composite (full site but just control plots) spatial level and the articulated (species) ecological level. Building upon the jags_logistic model, jags_logistic_covariates expands upon the “process model” underlying the Poisson observations. There are six process parameters – mu (the density of the species at the beginning of the time series), sigma (the standard deviation of the random walk, which is Gaussian on the log scale), and then intercept and slope parameters for r (growth rate) and K (carrying capacity) as a function of covariates (r being a function of the integrated warm rain over the past 3 months and K being a function of average NDVI over the past year). The observation model has no additional parameters. The prior distributions for mu and the K intercept are informed by the available data collected prior to the start of the data used in the time series and sigma and r are set with vague priors. mu is normally distributed with a mean equal to the average log-scale density and standard deviation 0.04. The K intercept is modeled on the log-scale with a prior mean equal to the maximum of past counts and standard deviation 0.04. The r intercept is given a normal prior with mean 0 and standard deviation 0.04. Sigma was given a uniform distribution between 0 and 1. The slopes for r and log-scale K were given priors with mean 0 and standard deviation 1.

### jags_logistic_competition

jags_logistic_competition fits a hierarchical log-scale density logistic growth model with a Poisson observation process using the JAGS (Just Another Gibbs Sampler) infrastructure (Plummer 2003) fit to the data at the composite (full site but just control plots) spatial level and the articulated (species) ecological level. Building upon the jags_logistic model, jags_logistic_competition expands upon the “process model” underlying the Poisson observations. There are six process parameters – mu (the density of the species at the beginning of the time series), sigma (the standard deviation of the random walk, which is Gaussian on the log scale), and then intercept and slope parameters for r (growth rate) and K (carrying capacity) as a function of competitior density (K being a function of current DO counts). The observation model has no additional parameters. The prior distributions for mu and the K intercept are informed by the available data collected prior to the start of the data used in the time series and sigma and r are set with vague priors. mu is normally distributed with a mean equal to the average log-scale density and standard deviation 0.04. The K intercept is modeled on the log-scale with a prior mean equal to the maximum of past counts and standard deviation 0.04. The r intercept is given a normal prior with mean 0 and standard deviation 0.04. Sigma was given a uniform distribution between 0 and 0.001. The slope for log-scale K was given a prior with mean 0 and standard deviation 1.

### jags_logistic_competition_covariates

jags_logistic_competition_covariates fits a hierarchical log-scale density logistic growth model with a Poisson observation process using the JAGS (Just Another Gibbs Sampler) infrastructure (Plummer 2003) fit to the data at the composite (full site but just control plots) spatial level and the articulated (species) ecological level. Building upon the jags_logistic model, jags_logistic_competition_covariates expands upon the “process model” underlying the Poisson observations. There are seven process parameters – mu (the density of the species at the beginning of the time series), sigma (the standard deviation of the random walk, which is Gaussian on the log scale), and then intercept and slope parameters for r (growth rate) and K (carrying capacity) as a function of competitior density (r being a function of the integrated warm rain over the past 3 months and K being a function of average NDVI over the past year as well as a function of current DO counts). The observation model has no additional parameters. The prior distributions for mu and the K intercept are informed by the available data collected prior to the start of the data used in the time series and sigma and r are set with vague priors. mu is normally distributed with a mean equal to the average log-scale density and standard deviation 0.04. The K intercept is modeled on the log-scale with a prior mean equal to the maximum of past counts and standard deviation 0.04. The r intercept is given a normal prior with mean 0 and standard deviation 0.04. Sigma was given a uniform distribution between 0 and 0.001. The slopes for r and log-scale K were given priors with mean 0 and standard deviation 1.

### Ensemble

In addition to the base models, we include a starting-point ensemble. Prior to v0.9.0, the ensemble was based on AIC weights, but in the shift to separating the interpolated from non-interpolated data in model fitting, we had to transfer to an unweighted average ensemble model. The ensemble mean is calculated as the mean of all model means and the ensemble variance is estimated as the sum of the mean of all model variances and the variance of the estimated mean, calculated using the unbiased estimate of sample variances.

## References

Box, G., and G. Jenkins. 1970. Time Series Analysis: Forecasting and Control. Holden-Day.

Hyndman, R. J. 2017. “forecast: Forecasting Functions for Time Series and Linear Models.” 2017. http://pkg.robjhyndman.com/forecast.

Hyndman, R. J., and G. Athanasopoulos. 2013. Forecasting: Principles and Practice. OTexts.

Hyndman, R. J., A. b. Koehler, J. K. Ord, and R. D. Snyder. 2008. Forecasting with Exponential Smoothing: The State Space Approach. Springer-Verlag.

Liboschik, T., K. Fokianos, and R. Fried. 2017. “tscount: An R Package for Analysis of Count Time Series Following Generalized Linear Models.” Journal of Statistical Software 82: 1–51. https://www.jstatsoft.org/article/view/v082i05.

Liboschik, T., R. Fried, K. Fokianos, and P. Probst. 2017. “tscount: Analysis of Count Time Series.” 2017. https://CRAN.R-project.org/package=tscount.

Plummer, M. 2003. “A Program for Analysis of Bayesian Graphical Models Using Gibbs Sampling.” Proceedings of the 3rd International Workshop on Distributed Statistical Computing. http://www.ci.tuwien.ac.at/Conferences/DSC-2003/Proceedings/Plummer.pdf.

WeecologyLab. 2019. “Portal Forecasting.” 2019. https://github.com/weecology/portalPredictions/.