Contributors
Ethan P. White, Glenda M. Yenni, Henry Senyondo, Shawn D. Taylor, Erica M. Christensen, Ellen K. Bledsoe, Juniper L. Simonis, and S. K. Morgan Ernest
Ecological Forecasting
Most forecasts for the future state of ecological systems are conducted once and never updated or assessed. As a result, many available ecological forecasts are not based on the most up-to-date data, and the scientific progress of ecological forecasting models is slowed by a lack of feedback on how well the forecasts perform. Iterative near-term ecological forecasting involves repeated daily to annual scale forecasts of an ecological system and regular assessment of the resulting predictions as new data become available. More frequent updating and assessment will advance ecological forecasting as a field by accelerating the identification of the best models for individual forecasts and improving our understanding of how to best design forecasting approaches for ecology in general.
The Portal Project
The Portal project, located in the Chihuahuan desert of southern Arizona, is a long-term experimental monitoring project in desert ecology. Established in 1977 by Jim Brown, we have over 40 years of data on rodents, plants, ants, and weather at the site. Rodent data are collected approximately monthly, an ideal scenario for short-term forecasts of rodent abundance.
Automated Predictions
Code and data underlying the predictions are housed in a GitHub repository, continuous integration and automated running of the code occurs via Travis, and completed forecasts are automatically archived on Zenodo. The translation of the raw Portal Data into model-ready formats is done via the portalr package and the portalcasting package is used to connect the data to the models, execute the models, synthesize the predictions, and produce the output figures.
For further detail, see our manuscript in Methods in Ecology and Evolution.
Acknowledgements
This research was supported in part by the National Science Foundation through grant DEB-1622425 to S.K.M. Ernest and by the Gordon and Betty Moore Foundation's Data-Driven Discovery Initiative through Grant GBMF4563 to E. P. White.
We thank Henry Senyondo for help with continuous integration, Hao Ye for feedback on documents and code, Heather Bradley for logistical support, John Abatzoglou for assistance with climate forecasts, and James Brown for establishing the Portal Project.
Current Models
We currently analyze and forecast rodent data at Portal using ten different models and an ensemble modeling combining all ten models into a single prediction.
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; Takens 1981). 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; Takens 1981). 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.” 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.” 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.
Rodents | Species | Common Name | Description |
---|---|---|---|
Baiomys taylori | northern pygmy mouse | Yes, apparently southeastern Arizona counts as 'northern' for this genus. One of two competitors for title of 'smallest Portal rodent,' B. taylori is currently an infrequent but welcome visitor to the site. As it is primarily a grassland species, individuals caught on the site rarely stay for long. Interestingly, the roughly 8-gram B. taylori is often found in association with cotton rats (Sigmodon), which can weight in excess of 100 grams. | |
Chaetodipus baileyi | Bailey's pocket mouse | Bailey's pocket mouse made its first appearance on the site in the mid-1990s. It is essentially a giant version of its congeneric relative, the desert pocket mouse (C. penicillatus). While close in size to Merriam's and Ord's kangaroo rats, it is an inferior competitor and, therefore, is found almost solely on kangaroo rat exclosure plots. | |
Chaetodipus hispidus | hispid pocket mouse | The hispid pocket mouse looks like a cactus mouse in pocket mouse form; it's the size of C. baileyi with a beautiful lateral orange stripe. C. hispidus has historically been the least frequently recorded species at the site. In the past few months, however, we've caught one or two each trapping period. Nevertheless, it remains a very transient speices; each individual caught this year has been caught once and then never seen again. | |
Chaetodipus intermedius | rock pocket mouse | C. intermedius is another pocket mouse, though very rarely found at the site as it prefers rockier habitats; it is very similar to C. penicillatus except in habitat preference. | |
Chaetodipus penicillatus | desert pocket mouse | Although C. penicillatus had been found at the site since the beginning of the study, it didn't become such an important player in the system until the 2000s. Now, this small pocket mouse is one of the main seasonal drivers of total rodent abundance at the site. The population explodes in the summers with known individuals, new adults, and brand new babies alike; they are ubiquitous across the site, regardless of plot type. In the winters, C. penicillatus enters a type of torpor (not quite hibernation), creating a large seasonal fluctuation in abundance. | |
Dipodomys merriami | Merriam's kangaroo rat | Kangaroo rats, with their giant hind feet and namesake bipedal movement, are some of the most beloved species at the site. Of the three species in the Dipodomys genus, D. merriami is currently the most abundant and, presumably, dominant in the system. | |
Dipodomys ordii | Ord's kangaroo rat | Ord's kangaroo rat is nearly indistinguishable from Merriam's kangaroo rat except for a tiny fifth toe on its hind feet that the Merriams lack. D. ordii can be thought of as D. merriami's kid brother who's always aspiring to be like his brother but can't quite keep up; a grass-loving species, D. ordii is often just trailing D. merriami in abundance. | |
Dipodomys spectabilis | banner-tailed kangaroo rat | The largest of the three kangaroo rats at the site, D. spectabilis (fondly referred to as 'spectabs' by those in the know) is known for its striking tail. It has by far the largest feet on the site at a whopping 48 mm average and often weighs in at over 100 grams, nearly twice the size of the other kangaroo rats. Spectabs were once running the show at Portal. As the site became shrubbier, however, the reign of D. spectabilis came to a slow end in the 1990's. Since then, a few individuals have popped up here and there but haven't stuck around, often heading for grassier pastures. | |
Neotoma albigula | white-throated woodrat | Along with being by far the largest rodent species at the site (my, what big teeth you have...), Neotoma albigula also has a big reputation---roughly 200 grams worth. Woodrats are quite different ecologically from the rest of the species at the site. Unlike the majority of our rodents which are granivorous, N. albigula is primarily a foliovore. (The resident woodrat in our ramada, however, seems to eat just about anything.) Woodrats---also commonly known as packrats---build middens, or large nests, out of sticks and any 'interesting' debris they can get their paws on. While their numbers at the site aren't prolific, research assistants know exactly which kangaroo rat exclosure plots this species tends to inhabit and secretly wish they'd brought leather gloves with them. | |
Onychomys leucogaster | northern grasshopper/scorpion mouse | Amongst smammal enthusiasts such as ourselves, the carnivorous grasshopper mice (Onychomys genus) are legendary. Their name(s) and this video basically say it all. In addition to eating insects and scorpions, they have been known to eat venomous centipedes and even other mice. Both species also have a distinctive high-pitched 'howl' when defending their territory. Mighty though they are, both species, and O. leucogaster in particular, have hilariously short tails. For unknown reasons, the northern grasshopper mouse is less common at the site than the southern. | |
Onychomys torridus | southern grasshopper/scorpion mouse | Aside from overall range, the southern species differs little from the northern (O. leucogaster) except for a slightly longer tail and somewhat more elongated appearance. This species is more abundant at the site and can be found in all plot types. | |
Perognathus flavus | silky pocket mouse | Perognathus flavus rivals only Baiomys taylorii for the title of smallest rodent at our site, weighing between 5 and 10 grams. Tiny, beautiful, and immensely soft, this aptly named silky pocket mouse is a favorite of previous and current research assistants alike. Its relatively low abundance and recent rarity makes it an especially exciting catch. | |
Peromyscus eremicus | cactus mouse | Often having a bold lateral orange stripe, the cactus mouse is one of the most colorful and striking mice at Portal. It is also the most commonly found Peromyscus species at the site, usually found in kangaroo rat exclosures and incidentally in the full removal plots. Unlike the white-footed mouse and deer mouse, however, it is mainly restricted to the deserts in the American Southwest and northern Mexico. | |
Peromyscus leucopus | white-footed mouse | This Peromyscus species, while very common throughout the majority of the United States, is not all that common at Portal.Peromyscusare good climbers and, therefore, frequently end up in the full rodent exclosure plots. | |
Peromyscus maniculatus | deer mouse | Like P. leucopus, P. maniculatus is nearly ubiquitous throughout the continental United States but is not a particularly common resident at the site. The desert variants of these two species are challenging to distinguish between, adding a fun challenge to trapping. | |
Reithrodontomys fulvescens | fulvous harvest mouse | The fulvous harvest mouse is an infrequent visitor to the site. It is the largest---and the rarest---harvest mouse species found at the site. Like many of the Reithrodontomys caught at the site, it is most often found in kangaroo rat exclosures and full rodent removals. | |
Reithrodontomys megalotis | western harvest mouse | R. megalotis is the most common harvest mouse found at Portal. Harvest mice abundance tends to remain fairly low throughout the site, though there is sometimes an increase in the winter when the desert pocket mice (C. penicillatus) are in torpor. | |
Reithrodontomys montanus | plains harvest mouse | While very similar to R. megalotis, R. montanus is notably smaller than the other harvest mice at the site. It is less commonly found at the site than R. megalotisbut plays a similar role ecologically. | |
Sigmodon fulviventer | tawny-bellied cotton rat | A favorite of many Portal research assistants, rodents in the Sigmodon genus are best described as 'squishy like cotton.' Okay, so maybe that's not the most descriptive identifier, but their size and coloration make them an easily identifiable genus. S. fulviventer is one of the two more frequently seen cotton rat species at the site. It is distinguishable by the yellow (rather than white) fur on its belly, as indicated by its descriptive common name. | |
Sigmodon hispidus | white-bellied cotton rat | S. hispidus, with a white rather than a tawny belly, is the other cotton rat species that somewhat frequently has been caught at the site. The Sigmodons seem to come through the site in waves; some years are boom years while others are bust years. What is driving these population cycles? How do Sigmodons move through the regional landscape? We don't know but sure would like to find out. | |
Sigmodon ochorognathus | yellow-nosed cotton rat | S. ochorognathus almost seems like an urban (erm...rural) legend nowadays. A few individuals were caught in the 1990s, but none have been seen since. |