Although some of the EverForecast species models provide output on a daily time scale, water managers and scientists in the Everglades typically evaluate hydrologic conditions and make recommendations on a weekly to monthly time scale. To target managers’ needs, we summarized EverForecast outputs on a biweekly (14 day) time step.

This reflects a fundamental property of EDM in that forecast performance depends solely on the information content of the data rather than on how well assumed equations match reality.To clarify the concept of nonuniqueness, consider the canonical Lorenz attractor (SI Appendix, Fig. S1A). The behavior of this system is governed by three differential equations (SI Appendix, Eq. S1). However, the axes can be rotated to produce three new coordinates, x′, y′, and z′, and the equations rewritten in terms of these new coordinates, allowing the system to be described using either representation (x, y, and z or x′, y′, and z′) as well as mixed combinations (e.g., x, y, and z′). Thus, with an infinite number of ways to rotate the system, there are an unlimited number of “true variables” and “true models.” In the case of sockeye salmon, the similar performance of different models (SI Appendix, Table S4) does not mean that one or the other model is incorrect; instead, it reflects the fact that the environmental variables are indicators of the same general mechanism, and so different variable combinations can be equally informative for forecasting recruitment.Again, we emphasize that including a variable does not imply a direct causal link—variables in an EDM model improve forecasts because they are informative; it does not mean that the included variables are proximate causes. Importantly, the converse does not hold either: a variable could be causal and yet not appear in the multivariate EDM; this might occur when multiple stochastic drivers affect recruitment in an interdependent way, necessitating that a model include measurements of all of the drivers to account for their combined effect. For example, although none of the tested variables seem to improve forecasts for the Birkenhead stock (SI Appendix, Table S4), this does not mean that these sockeye salmon are insensitive to SST, river discharge, and the PDO. Rather, it suggests that the effect of these variables may be modulated by other factors not considered here.

Feedback mechanisms provide stability such that ecosystems appear stable during some time frames but can abruptly shift to express new structures in others (9)

One of the primary uses of a model like this one is to improve the conversation between stakeholders and managers. The model can be valuable in helping managers and citizens arrive at realistic goals and to realize that there will be inherent risks associated with meeting those goals. For example, our analysis shows that reducing the probability of transmission by one half in five years using vaccination is not likely when we include uncertainty in the ability of managers to treat a targeted number of seronegative females. Forecasts suggested that there was virtually no chance of meeting that goal (Table 12). Similarly there was a 7% chance of reducing adult female seroprevalence below 40% using vaccination. We can nonetheless use this work to articulate what level of brucellosis suppression is feasible given current technology. For example, managers and stakeholders might agree that it is enough to be moving in the right direction with efforts to reduce risk of infection from brucellosis. In this case, a reasonable goal might be “Reduce the probability of exposure by 10% relative to the current median value.” The odds of meeting that goal using vaccination increased to 26%. With this less ambitious goal, vaccination increases the probability that the goal would be met relative to no action by a factor of only 1.4. This illustrates a fundamental trade-off in making management choices in the face of uncertainty: less ambitious goals are more likely to be met, but they offer smaller improvements in the probability of obtaining the desired outcome relative to no action.

We show that these uncertainties combine to assure that long-term predictions, e.g., 20 years in Peterson et al. (1991); 35 years in Ebinger et al. (2011); 30 years in Treanor et al. (2010) will be unreliable because credible intervals on forecasts expand rapidly with increases in the forecast horizon (Table 9). Long-range forecasts will include an enormous range of probable outcomes. This finding urges caution in making long-term forecasts with ecological models.

Evaluation of alternatives proceeded in three steps. We first obtained the posterior process distribution of the state at some point in the future, given no action, and calculated the probability that the goal will be met (Fig. 3A). The no-action alternative can be considered a null model to which alternative actions can be compared. Next, we approximated the posterior process distribution at the same point in the future assuming that we have implemented an alternative for management and calculated the probability that the goal will be met (Fig. 3B). Finally, we calculated the ratio of the probability of meeting our goal by taking action over the probability if we take no action. This ratio quantifies the net effect of management (Fig. 3C) and permits statements such as “Taking the proposed action is five times more likely to reduce seroprevalence below 40% relative to taking no action.”This process for evaluating alternative actions explicitly incorporates uncertainties in the future state of the population in the presence and absence of management. A useful feature of this approach is that the weight of evidence for taking action diminishes as the uncertainty in forecasts increases. That is, increasing uncertainty in forecasts compresses the hatched area in Fig. 3C. This result encourages caution in taking action. Also useful is the inverse relationship between the absolute probability that a goal will be met by management and the probability that it will be met relative to taking no action. As the ambition of objectives increases (e.g., the dashed line in Fig. 3 moves to the left), the absolute probability that the management action will be achieved declines (the hatched area in Fig. 3B shrinks), but the probability of success relative to taking no action increases (the hatched area in Fig. 3C expands). This feature represents a fundamental trade-off in choosing goals and actions that are present in all management decisions: objectives that are not ambitious are easy to meet by applying management, but they might be met almost as easily by taking no action.

The model omits covariates describing weather conditions, e.g., drought severity, which have been included in other models of bison population dynamics in Yellowstone (Fuller et al. 2007a). We justify this omission because our central objective was to develop a forecasting model. We we use the term forecast to mean predictions of future states accompanied by coherent estimates of uncertainty arising from the failure of the model to represent all of the influences that shape the population's future trajectory.

The model is not spatially explicit. Although there is evidence that the population is made up of two different herds that spend their summers in the northern and central portions of Yellowstone National Park (Olexa and Gogan 2007), we justify our decision to treat the population without spatial structure as a first approximation of its behavior and because recent evidence suggests that substantial movement between herds occurs annually (Gates et al. 2005, Fuller et al. 2007, White and Wallen 2012).

Thus, predicting species responses to novel climates is problematic, because we often lack sufficient observational data to fully determine in which climates a species can or cannot grow (Figure 3). Fortunately, the no-analog problem only affects niche modeling when (1) the envelope of observed climates truncates a fundamental niche and (2) the direction of environmental change causes currently unobserved portions of a species' fundamental niche to open up (Figure 5). Species-level uncertainties accumulate at the community level owing to ecological interactions, so the composition and structure of communities in novel climate regimes will be difficult to predict. Increases in atmospheric CO2 should increase the temperature optimum for photosynthesis and reduce sensitivity to moisture stress (Sage and Coleman 2001), weakening the foundation for applying present empirical plant–climate relationships to predict species' responses to future climates. At worst, we may only be able to predict that many novel communities will emerge and surprises will occur. Mechanistic ecological models, such as dynamic global vegetation models (Cramer et al. 2001), are in principle better suited for predicting responses to novel climates. However, in practice, most such models include only a limited number of plant functional types (and so are not designed for modeling species-level responses), or they are partially parameterized using modern ecological observations (and thus may have limited predictive power in no-analog settings).

In eastern North America, the high pollen abundances of temperate tree taxa (Fraxinus, Ostrya/Carpinus, Ulmus) in these highly seasonal climates may be explained by their position at the edge of the current North American climate envelope (Williams et al. 2006; Figure 3). This pattern suggests that the fundamental niches for these taxa extend beyond the set of climates observed at present (Figure 3), so that these taxa may be able to sustain more seasonal regimes than exist anywhere today (eg Figure 1), as long as winter temperatures do not fall below the −40°C mean daily freezing limit for temperate trees (Sakai and Weiser 1973).

Comparison of ITCD algorithms is challenging when there are differences in study focus, study area, data applied, and accuracy assessment method used. Before 2005, the few studies that compared methods generally tested approaches on a common dataset.

Additionally, it is often challenging to apply an algorithm developed in one forest type to another area.

The most useful information that can be incorporated into ITCD studies is the expected crown size and stand density [25,67].

Only 23 studies actually integrated both active and passive data sources into the ITCD procedure since 2000 (Figure 2).

Another limitation is that few approaches take full advantage of the information contained within remotely sensed data, e.g., using only one band of multispectral imagery [29] or only the canopy height model derived from LiDAR data [30]. Significant amounts of information are dismissed or neglected during data preparation or processing. The integration of multispectral data and discrete LiDAR data is commonly used to improve tree species classification [10] and fusion of passive and active remotely sensed data may reduce commission and omission errors in ITCD results [31].

Comparison of methods continues to be complicated by both choice of reference data and assessment metric; it is imperative to establish a standardized two-level assessment framework to evaluate and compare ITCD algorithms in order to provide specific recommendations about suitable applications of particular algorithms.

If the rule is relaxed to embrace pattern alone, as explicitly advocated by Rensch and Mayr, exceptions can still be found both intra- and interspecifically. Within species, Rensch (1938) reported that 10–30% of the warm-blooded species examined by him were exceptions to Bergmann's rule. Ray (1960) reviewed the literature on body size variation in relation to climate for poikilotherms, and concluded that the rule was supported by 75% of species studied. Nevertheless, these percentages (see also James, 1970; Yom-Tov & Nix, 1986) support Mayr's (1956) contention that the rule would be proved if upheld by the majority of species, although his subsequent definition of a majority as more than 50%(Mayr, 1963) is rather generous in respect of a ‘rule’. Some studies, however, do find that the percentage of species in agreement with the intraspecific rule fails even this criterion (McNab, 1971).

It is the definition of Bergmann's rule, and specifically the taxonomic level at which the rule is considered to act, that has done most to cloud the degree of generality of the effect it describes. Bergmann himself (quoted in James, 1970) stated that ‘(i)f we could find two species of [homeothermic] animals which would only differ from each other with respect to size, . . . (t)he geographical distribution of the two species would have to be determined by their size . . . if there are genera in which the species differ only in size, the smaller species would demand a warmer climate, to the exact extent of the size difference.’ Also: ‘(a)lthough it is not as clear as we would like, it is obvious that on the whole the larger species live farther north and the smaller ones farther south.’Bergmann's formulation was later altered by Rensch (1938), whose revised definition was that ‘within a Rassenkreis [complex of races] of warm-blooded animals the races living in colder climates are generally larger than the races living in warmer regions.’ Rensch considered that the new definition better fitted the rule as then understood, but was quite clear that the revision was his own (‘I myself reduced it to the geographical races of a species’; Rensch, 1938). It was this revision that subsequently became the accepted formulation of Bergmann's rule; later definitions included ‘Races of warm blooded vertebrates from cooler climates tend to be larger than races of the same species from warmer climates’(Mayr, 1956), and ‘The smaller-sized geographic races of a species are found in the warmer parts of the range, the larger-sized races in the cooler districts’(Ray, 1960; see also definitions in Gittleman, 1985; Goudie & Ankney, 1986; Paterson, 1990; McDowall, 1994; Steudel, Porter & Sher, 1994; Smith, Betancourt & Brown, 1995; Atkinson & Sibly, 1997).The notion that the application of Bergmann's rule at the intraspecific level is a derived state was emphasized by James (1970), who noted that it was a considerable modification of Bergmann's original message, although one that fitted well with knowledge of intraspecific body size variation. Quotations from Bergmann (1847; given in translation by James, 1970) imply that he considered the effect to be interspecific, but between closely related species. Whether he intended the example he gave of ‘species within a genus’ to be literal is unclear. Whichever, it is clear that his formulation was not intraspecific, as he thought it ‘paradoxical that the effects of the same rule in races of animals are not very apparent’(Bergmann, 1847, quoted in James, 1970). In this context, it is interesting that Mayr (1956; see also Rensch, 1938) noted that many of the species considered by Bergmann were, when Mayr was writing, afforded only sub-specific status! Nevertheless, since James's paper, Bergmann's rule has been examined at a variety of taxonomic levels, for example within species (Barnett, 1977; Ralls & Harvey, 1985; Yom-Tov & Nix, 1986; Geist, 1987; Graves, 1991; Smith et al., 1995, 1998; Van Voorhies, 1996, 1997; Mousseau, 1997; Partridge & Coyne, 1997), between species within genera (Gittleman, 1985; Taylor & Gotelli, 1994), between functionally related species (Geist, 1987; Cotgreave & Stockley, 1994), and between species within a range of higher taxa (Zeveloff & Boyce, 1988; Cushman et al., 1993; Barlow, 1994; McDowall, 1994; Hawkins, 1995; Hawkins & Lawton, 1995; Poulin, 1995; Poulin & Hamilton, 1995; Blackburn & Gaston, 1996a).

We also did not allow different portions of our study area to respond to climate in different ways. Doing so would require spatially varying climate effects and a substantial increase in computational time. However, in future applications, it will be important to allow climate effects to vary over space to better capture reality. Conn et al. (2015) provide examples of how such spatiotemporal interactions can be included in abundance models. We might expect climate effects to interact with spatial covariates such as soil type, slope, and aspect.

To simulate equilibrium sagebrush cover under projected future climate, we applied average projected changes in precipitation and temperature to the observed climate time series. For each GCM and RCP scenario combination, we calculated average precipitation and temperature over the 1950–2000 time period and the 2050–2098 time period. We then calculated the absolute change in temperature between the two time periods (ΔT) and the proportional change in precipitation between the two time periods (ΔP) for each GCM and RCP scenario combination. Lastly, we applied ΔT and ΔP to the observed 28-year climate time series to generate a future climate time series for each GCM and RCP scenario combination. These generated climate time series were used to simulate equilibrium sagebrush cover.

Our process model (in Eq. (2)) includes a log transformation of the observations (log(yt − 1)). Thus, our model does not accommodate zeros. Fortunately, we had very few instances where pixels had 0% cover at time t − 1 (n = 47, which is 0.01% of the data set). Thus, we excluded those pixels from the model fitting process. However, when simulating the process, we needed to include possible transitions from zero to nonzero percent cover. We fit an intercept-only logistic model to estimate the probability of a pixel going from zero to nonzero cover: yi∼Bernoulli(μi)(8)logit(μi)=b0(9)where y is a vector of 0s and 1s corresponding to whether a pixel was colonized (>0% cover) or not (remains at 0% cover) and μi is the expected probability of colonization as a function of the mean probability of colonization (b0). We fit this simple model using the “glm” command in R (R Core Team 2014). For data sets in which zeros are more common and the colonization process more important, the same spatial statistical approach we used for our cover change model could be applied and covariates such as cover of neighboring cells could be included.

Our approach models interannual changes in plant cover as a function of seasonal climate variables. We used daily historic weather data for the center of our study site from the NASA Daymet data set (available online: http://daymet.ornl.gov/). The Daymet weather data are interpolated between coarse observation units and capture some spatial variation. We relied on weather data for the centroid of our study area.

Because SDMs typically rely on occurrence data, their projections of habitat suitability or probability of occurrence provide little information on the future states of populations in the core of their range—areas where a species exists now and is expected to persist in the future (Ehrlén and Morris 2015).

Our abilities to make observations are limited to a small range of space and time scales (8), limiting our capacity for understanding ecosystems and forecasting how they will respond to local and global change.

A range of information sources, which can include models, is used to develop alternative plausible trajectories of ecosystems; uncertainties about the future are represented by the range of conditions captured by the ensemble of scenarios. In contrast, forecasts narrowly limit uncertainties to those associated with a single potential outcome that is assumed to be predictable

Practices in the field of financial investing provide a good analogy to the stance we suggest for ecological predictions. A great deal of money and effort has been used to model the best ways to maximize investment returns (certainly more money and effort than has been used to refine ecological predictions). Although this work has resulted in greatly increased understanding of economic systems, the risks and limitations of using sophisticated economic models to make investments has led more and more investors to instead use simple, safe index funds. Essentially this is the recognition that the models and expert opinions are of exceptionally little value in making accurate, long-range predictions in this field and that precautionary strategies are a far better alternative.

First, major surprises are commonplace in the experience of field ecologists.

After explaining our project and providing several well-known examples of ecological surprises, we asked the recipients whether or not they had encountered any such events in the course of their field studies

Scenarios were initially developed by Herbert Kahn in response to the difficulty of creating accurate forecasts ( Kahn & Wiener 1967; May 1996 ). Kahn worked at the RAND Corporation, an independent research institute with close ties to the U.S. military. He produced forecasts based on several constructed scenarios of the future that differed in a few key assumptions ( Kahn & Wiener 1967 ). This approach to scenario planning was later elaborated upon at SRI International ( May 1996 ), a U.S. research institute, and at Shell Oil (  Wack 1985a, 1985b, Schwartz 1991; Van der Heijden 1996 ).

Prediction means different things to different technical disciplines and to different people ( Sarewitz et al. 2000 ). A reasonable definition of an ecological prediction is the probability distribution of specified ecological variables at a specified time in the future, conditional on current conditions, specified assumptions about drivers, measured probability distributions of model parameters, and the measured probability that the model itself is correct ( Clark et al. 2001 ). A prediction is understood to be the best possible estimate of future conditions. The less sensitive the prediction is to drivers the better ( MacCracken 2001 ). Whereas scientists understand that predictions are conditional probabilistic statements, nonscientists often understand them as things that will happen no matter what they do ( Sarewitz et al. 2000; MacCracken 2001 ).In contrast to a prediction, a forecast is the best estimate from a particular method, model, or individual. The public and decision-makers generally understand that a forecast may or may not turn out to be true ( MacCracken 2001 ). Environmental scientists further distinguish projections, which may be heavily dependent on assumptions about drivers and may have unknown, imprecise, or unspecified probabilities. Projections lead to “if this, then that” statements ( MacCracken 2001 ).

We hypothesize that precipitation levels during the preceding wet season and during the onset of the dry season in forests of Southern Hemisphere South America act as a key regulator of drought intensity during the subsequent dry season.

We defined our empirical predictive model as a linear combination of the two climate indices sampled during these months of maximum correlation:FSSpredicted(x,t)=a(x)×ONI[t,m(x)−τONI(x)]+b(x)×AMO[t,m(x)−τAMO(x)]+c(x)

MODIS provides consistent information on active fires, with omission and commission errors quantified in past work using ground observations and higher-resolution satellite imagery [e.g., (29–31)]

we used 2001–2009 fire counts detected by the Moderate Resolution Imaging Spectroradiometer (MODIS)

Fire season severity, here defined as the sum of satellite-based active fire counts in a 9-month period centered at the peak fire month, depends on multiple parameters that influence fuel moisture levels and fire activity in addition to precipitation, including vapor pressure deficits, wind speeds, ignition sources, land use decisions, and the duration of the dry season. As a result, the relationship between FSS and SSTs may be more complex than the relationships between precipitation and SSTs described above.

To predict FSS at or before the beginning of the fire season, we established a cutoff (minimum) lead time of 3 month

Whilst the consensus method we used provided the best predictions under AUC assessment – seemingly confirming its potential for reducing model-based uncertainty in SDM predictions [58], [59] – its accuracy to predict changes in occupancy was lower than most single models. As a result, we advocate great care when selecting the ensemble of models from which to derive consensus predictions; as previously discussed by Araújo et al. [21], models should be chosen based on aspects of their individual performance pertinent to the research question being addressed, and not on the assumption that more models are better.

Finally, by assuming the non-detection of a species to indicate absence from a given grid cell, we introduced an extra level of error into our models. This error depends on the probability of false absence given imperfect detection (i.e., the probability that a species was present but remained undetected in a given grid cell [73]): the higher this probability, the higher the risk of incorrectly quantifying species-climate relationships [73].

an average 87% of grid squares maintaining the same occupancy status; similarly, all climatic variables were also highly correlated between time periods (ρ>0.85, p<0.001 for all variables). As a result, models providing a good fit to early distribution records can be expected to return a reasonable fit to more recent records (and vice versa), regardless of whether relevant predictors of range shift have actually been captured. Previous studies have warned against taking strong model performance on calibration data to indicate high predictive accuracy to a different time period [20], [24]–[26]; our results indicate that strong model performance in a different time period, as measured by widespread metrics, may not indicate high predictive accuracy either.

Most variation in the prediction accuracy of SDMs – as measured by AUC, sensitivity, CCRstable, CCRchanged – was among species within a higher taxon, whilst the choice of modelling framework was as important a factor in explaining variation in specificity (Table 4 and Table S4). The effect of major taxonomic group on the accuracy of forecasts was relatively small.

The correct classification rate of grid squares that remained occupied or remained unoccupied (CCRstable) was fairly high (mean±s.d.  = 0.75±0.15), and did not covary with species’ observed proportional change in range size (Figure 3B). In contrast, the CCR of grid squares whose occupancy status changed between time periods (CCRchanged) was very low overall (0.51±0.14; guessing randomly would be expected to produce a mean of 0.5), with range expansions being slightly better predicted than range contractions (0.55±0.15 and 0.48±0.12, respectively; Figure 3C).

The consensus method Mn(PA) produced the highest validation AUC values (Figure 1), generating good to excellent forecasts (AUC ≥0.80) for 60% of the 1823 species modelled.

Quantifying the temporal transferability of SDMs by comparing the agreement between model predictions and observations for the predicted period using common metrics is not a sufficient test of whether models have actually captured relevant predictors of change. A single range-wide measure of prediction accuracy conflates accurately predicting species expansions and contractions to new areas with accurately predicting large parts of the distribution that have remained unchanged in time. Thus, to assess how well SDMs capture drivers of change in species distributions, we measured the agreement between observations and model predictions of each species’ (a) geographic range size in period t2, (b) overall change in geographic range size between time periods, and (c) grid square-level changes in occupancy status between time periods.

Although it is common knowledge that some of the modelling techniques we used (e.g., CTA, SRE) generally perform less well than others [32], [33], we believe that their transferability in time is not as well-established; therefore, we decided to include them in our analysis to test the hypothesis that simpler statistical models may have higher transferability in time than more complex ones.

We also considered including additional environmental predictors of ecological relevance to our models. First, although changes in land use have been identified as fundamental drivers of change for many British species [48]–[52], we were unable to account for them in our models – like most other published accounts of temporal transferability of SDMs [20], [21], [24], [25] – due to the lack of data documenting habitat use in the earlier t1 period; detailed digitised maps of land use for the whole of Britain are not available until the UK Land Cover Map in 1990 [53].

Great Britain is an island with its own separate history of environmental change; environmental drivers of distribution size and change in British populations are thus likely to differ somewhat from those of continental populations of the same species. For this reason, we only used records at the British extent to predict distribution change across Great Britain.

(1) Are climate-based SDMs transferable in time? (2) Can they capture drivers of expansion and contraction of species geographic ranges? (3) What is the relative effect of methodological and taxonomic variation on prediction accuracy?

Unfortunately, assessing whether they do is notoriously difficult since their main aim is to predict events that are yet to occur [20]; most studies thus measure the transferability of their models using a subset or re-sampled set of the distribution records used to build the models, a limited approach that can greatly inflate estimates of predictive accuracy [20]. For this reason, an emerging approach for estimating the true transferability of SDMs has been to validate model predictions against independent field records documenting shifts in species distributions to novel time periods [20]–[26] and regions [27]–[31]. However, published accounts of such independent model validation have generally lacked methodological or taxonomic breadth.

Despite the absence of mechanistic information about the underlying ecological processes, the relatively simple SSR method consistently outperformed the control models over near-term prediction horizons. This result was robust across all simulations and all life stages of the experimental data. Moreover, the SSR model achieved this feat using only a single time series, whereas the control model used all times series simultaneously (it is an ecologically unrealistic scenario to assume we know the model and have time series for all of the relevant variables). Other analyses have shown that multivariate SSR methods (35) improve with additional information (14, 36), suggesting that the performance of the SSR model tested here represents a lower bound on forecast accuracy attainable with this general approach.

In contrast, the linear forecasting methods were often no better than a prediction of the mean of the test set (represented by SRMSE = 1; Figs. 1 and 2). The performance of the control model varied depending on the system, but it too was often no better than a mean predictor.

To emulate ecologically realistic conditions, we added log-normal observation error with a CV of 0.2 to all of the training sets. The forecast accuracy was evaluated using the test-set time series without observation error.

All control models included log-normal process and observation error, and parameter values that resulted in chaotic or near-chaotic deterministic dynamics.

In real systems we often lack time series of important driving variables or species; thus, for this additional reason, this comparison represents a very optimistic scenario for the mechanistic modeling approach.

one can obtain good indications for theexistence of alternative attractors from field data, but theycan never be conclusive. There is always the possibilitythat discontinuities in time series or spatial patternsare due to discontinuities in some environmental factor.Alternatively, the system might simply have a thresholdresponse (Figure 2b).

Catastrophic regime shifts inecosystems: linking theory toobservation

A graphical model of a vegetation-water feedback

The obvious intuitive explanation for a sudden dra-matic change in nature is the occurrence of a sudden largeexternal impact. However, theoreticians have long stressedthat this need not be the case. Even a tiny incrementalchange in conditions can trigger a large shift in somesystems if a critical threshold known as ‘catastrophicbifurcation’ is passed[11].

In April 1950, Charney’s group made a series of successful 24-hour forecasts over North America, and by the mid-1950s, numerical forecasts were being made on a regular basis.

Charney determined that the impracticality of Richardson’s methods could be overcome by using the new computers and a revised set of equations, filtering out sound and gravity waves in order to simplify the calculations and focus on the phenomena of most importance to predicting the evolution of continent-scale weather systems.

Courageously, Richardson reported his results in his book Weather Prediction by Numerical Process, published in 1922.

Despite the advances made by Richardson, it took him, working alone, several months to produce a wildly inaccurate six-hour forecast for an area near Munich, Germany. In fact, some of the changes predicted in Richardson’s forecast could never occur under any known terrestrial conditions.

Imagine a rotating sphere that is 12,800 kilometers (8000 miles) in diameter, has a bumpy surface, is surrounded by a 40-kilometer-deep mixture of different gases whose concentrations vary both spatially and over time, and is heated, along with its surrounding gases, by a nuclear reactor 150 million kilometers (93 million miles) away. Imagine also that this sphere is revolving around the nuclear reactor and that some locations are heated more during one part of the revolution and other locations are heated during another part of the revolution. And imagine that this mixture of gases continually receives inputs from the surface below, generally calmly but sometimes through violent and highly localized injections. Then, imagine that after watching the gaseous mixture, you are expected to predict its state at one location on the sphere one, two, or more days into the future. This is essentially the task encountered day by day by a weather forecaster (Ryan, Bulletin of the American Meteorological Society,1982).

A more sophisticated version of training/test sets is cross-validation. For cross-sectional data, cross-validation works as follows. Select observation i for the test set, and use the remaining observations in the training set. Compute the error on the test observation. Repeat the above step for i=1,2,…,N where N is the total number of observations. Compute the forecast accuracy measures based on the errors obtained.