We currently analyze and forecast rodent data at Portal using ten models:

## ESSS

ESSS (Exponential Smoothing State Space) is a flexible exponential smoothing state space model (Hyndman et al. 2008) 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 ets and forecast functions in the forecast package (Hyndman 2017) with the allow.multiplicative.trend argument set to TRUE and the ESSS function in our portalcasting package. 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.

## 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) and the AutoArima function in our portalcasting package.

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.

## 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) and the NaiveArima function in our portalcasting package.

## 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 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) and the nbGARCH function in our portalcasting package.

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 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) and the nbsGARCH function in our portalcasting package.

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 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) and the pevGARCH function in our portalcasting package.

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:

• max temp, mean temp, precipitation, NDVI
• max temp, min temp, precipitation, NDVI
• max temp, mean temp, min temp, precipitation
• precipitation, NDVI
• min temp, NDVI
• min temp
• max temp
• mean temp
• precipitation
• NDVI
• -none-

The final model is an intercept-only model. The single best model of the 11 is selected based on AIC.

## simplexEDM

simplexEDM (simplex projection using Empirical Dynamic Modeling) is a state-space reconstruction model adapted for forecasting and 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 method uses time-delay embedding to reconstruct a state-space for the dynamics underlying a time series (Packard et al. 1980, @Takens1981). A forecast from a point in the state space is then computed as a weighted average of the trajectories of nearest neighbors of that point, a minimal algorithm known as “simplex projection” (Sugihara and May 1990).

In applications to ecological time series, many of the parameters are set automatically, with the exception of the dimension of the time-delay embedding. Here, the embedding dimension ($$E$$) is selected as the value (between 1 and the max_E argument to simplexEDM()) that minimizes the mean absolute error over the in-sample portion of the data.

## GPEDM

GPEDM (Gaussian processes using Empirical Dynamic Modeling) is a state-space reconstruction model adapted for forecasting and 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 method uses time-delay embedding to reconstruct a state-space for the dynamics underlying a time series (Packard et al. 1980, @Takens1981). The forecast function is approximate using Gaussian processes.

As with simplexEDM(), many of the parameters are fit automatically, such as the length-scale, and variance parameters (see rEDM::block_gp() for details). One exception is the dimension of the time-delay embedding. Here, the embedding dimension ($$E$$) is selected as the value (between 1 and the max_E argument to GPEDM()) that minimizes the mean absolute error over the in-sample portion of the data.

## 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 and just control plots) spatial level and both the composite (community) and the articulated (species) ecological levels. 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. The jags_RW model is rather simple, but provides a starting template and underlying machinery for more articulated models using the JAGS infrastructure.

There are two process parameters: mu (the density of the species at the beginning of the time series) and tau (the precision (inverse variance) of the random walk, which is Gaussian on the log scale). The observation model has no additional parameters. The prior distributions for mu and tau are 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 variance that is twice as large as the observed variance. Due to the presence of 0s in the data and the modeling on the log scale, an offset of count + 0.1 is used prior to taking the log and then is removed after the reconversion (exponentiation) as density - 0.1 (where density is on the same scale as count, but can take non-integer values).

## Ensemble

In addition to the base models, we include a starting-point ensemble. In versions before November 2019, 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.

Packard, N. H., J. P. Crutchfield, J. D. Farmer, and R. S. Shaw. 1980. “Geometry from a Time Series.” Physical Review Letters 45 (9): 712–16.

Plummer, M. 2003. “A Program for Analysis of Bayesian Graphical Models Using Gibbs Sampling.” Proceedings of the 3rd International Workshop on Distributed Statistical Computing. https://bit.ly/33aQ37Y.

Sugihara, G., and R. M. May. 1990. “Nonlinear Forecasting as a Way of Distinguishing Chaos from Measurement Error in Time Series.” Nature 344: 734–41.

Takens, F. 1981. “Detecting Strange Attractors in Turbulence.” Dynamical Systems and Turbulence, Lecture Notes in Mathematics 898: 366–81.