Research has estimated associations between water temperature and the spatial distribution of marine fishes based upon correlations between temperature and the centroid of fish distribution (centre of gravity, *Gadus chalcogrammus* (Walleye pollock) in the Eastern Bering Sea, and use a vector‐autoregressive spatiotemporal model to attribute variation in

The Walleye pollock (*Gadus chalcogrammus*, Gadidae; hereafter referred to as pollock) stock in the Bering Sea supports an industry worth over $1 billion annually (in first sale wholesale prices; Fissel et al., ) and is the focus of extensive data collection and research. Previous authors (Kotwicki, Buckley, Honkalehto, & Walters, ; Wyllie‐Echeverria & Wooster, ) have shown that the spatial distribution of pollock varies between “warm” and “cool” years in the Eastern Bering Sea, where warm years commence with winter sea‐ice having a maximum extent that is skewed north relative to long‐term averages and cool years have a southward‐skewed distribution for winter sea‐ice. Maximum sea‐ice extent is strongly associated with the proportion of the Bering Sea with low bottom temperatures (the area of the “cold pool”; Stabeno, Bond, Kachel, Salo, and Schumacher, ), and the cold pool has been shown to affect the distribution of demersal fishes such as pollock (Kotwicki & Lauth, ). Previous analyses have generally studied the impact of temperature or cold pool area in isolation, without attempting to estimate what proportion of variance in pollock distribution is explained by temperature relative to other factors.

Ecological studies often seek to estimate a link between temperature and distribution, and then use this linkage to forecast shifts in distribution given changing global temperatures due to long‐term climate shifts. Examples include Perry, Low, Ellis, and Reynolds (), which regressed a sample‐based calculation for centre of gravity (COG) for North Sea fishes against a moving average of bottom temperature, and concluded that bottom temperature and northward COG were related. Similarly, Cheung et al. () fitted a climate envelope model using temperature as an explanatory variable to distribution for many marine species, and then projected likely changes in climate envelopes given different scenarios for future temperature. However, neither study quantifies the proportion of variation in distribution that is explained by temperature. By contrast, we argue that studies of spatial distribution and dynamics should include two features: (i) an analysis demonstrating a link between a causal variable (e.g., temperature) and a population variable (centre of distribution); and (ii) an attempt to quantify predictive uncertainty for the forecast. In particular, when a variable explains a small portion of total variance in distribution, forecasts will likely explain a correspondingly small portion of future variability.

In addition to temperature, the spatial distribution of fish populations may be affected by ontogenetic shifts as individuals age and grow (e.g., Gratwicke, Petrovic, & Speight, ; Hicks, Taylor, Grandin, Taylor, & Cox, ). Diet data from pollock have shown cannibalism which has led to hypotheses on possible evolutionary incentives for juveniles to minimize cannibalism by occupying different habitat than adults (Bailey, ). Given that juveniles and adults tend to have different spatial distributions, recruitment variation alone can affect the spatial distribution of pollock (e.g., where years following strong recruitment will be skewed towards the juvenile habitats). In the following, we term this mechanism a “size‐structured effect”. This affects distribution shifts whenever (i) juveniles and adults have strong spatial differences in distribution; (ii) recruitment variation is high; and (iii) generation time is low, due to high natural and/or fishing mortality rates. Conditions (ii) and (iii) both contribute to high variation in size‐structure over time.

The spatial distribution of fishes may change following many other density‐dependent and density‐independent processes in addition temperature and size‐structured effects. Previous studies report variability among years in timing of spawning and feeding migrations (e.g., Ernst, Orensanz, & Armstrong, ; Kotwicki et al., ; Nichol, ), in situ light conditions driving changes in distribution within feeding habitats (Kotwicki, De Robertis, von Szalay, & Towler, ), primary production patterns and fishing pressure (Garrison et al., ). Distribution may also change due to density‐dependent expansion of the population's range (Kotwicki & Lauth, ; Spencer, ), changes in food availability (e.g., Dorn, ; Nøttestad, Giske, Holst, & Huse, ) and interactions with other species including competition and predation (Ciannelli, Fauchald, Chan, Agostini, & Dingsør, ).

In this study, we develop a method to quantify the contribution of temperature, size‐structure and other factors that contribute to shifts in the spatial distribution of marine fishes. This method specifically decomposes variation in catch rates into components caused by changes in distribution (encounter/non‐encounter) and density (catch rates given that the species is encountered) to distinguish between these two mechanisms for changing distribution. We demonstrate this method using pollock as a case‐study example, and seek to answer: (i) Can changes in temperature or size‐structure generate substantial variation in COG over time for this species; and (ii) do the shifts caused by temperature or size‐structure explain observed patterns in COG for pollock?

We seek to model changes over time in the spatial distribution of different size‐categories for marine fishes. We chose to use a delta‐model, which separately estimates the probability *p* that a given sample will encounter the species, and the expected catch rate *r* given that the species is encountered (Maunder & Punt, ). Specifically, sampled biomass *b*_{i} (in kg) for each sample *i* is modelled as: *r*_{i} is the catch rate (in kg/km^{2}) for sample *i*, *B* given shape θ^{−2} and scale _{c} is the coefficient of variation for positive catches for size‐class *c*. We use this delta‐gamma function so that we can separately interpret predictors of spatial variation in encounter probability and positive catch rates. Given estimates of encounter probability and positive catch rate, we can then predict population density (in kg/km^{2}) at each sampled location as *d*_{i} = *p*_{i} × *r*_{i}.

We use a vector‐autoregressive spatiotemporal (VAST) model to decompose sampling variation into different biologically interpretable sources (Thorson & Barnett, ). We include three distinct sources of variation in both encounter probability and expected catch rates:

*Intercepts*—We incorporate annual variation in average encounter probability, *p*, and positive catch rates, *r*, across locations within the spatial domain of the model. This annual variation represents changes among years in the proportion of population abundance belonging to different size‐classes.*Covariates*—We incorporate measured habitat variables as predictors of *p* and *r*, which allows us to estimate the impact of measureable covariates on local density. Specifically, we incorporate responses to both local and regional variation in temperature.*Residual spatial and spatiotemporal variation—*We account for two sources of residual variation in *p* and *r* for each size‐class. The first source involves estimating spatial autocorrelation terms for each size‐class. These estimates are constant over time, and thus can be considered as representing unmeasured habitat variables that do not change over the modelled period (e.g., depth, sediment size). The second set of terms is to account for autocorrelation in *p* and *r* that changes among years and spatial locations. This latter spatiotemporal variation represents the influence of unmeasured habitat variables that change among years, and accounts for shifts in distribution for each size‐class beyond the shifts predicted by covariates (e.g., temperature).

We interpret this as a “semi‐parametric” model because it includes annually varying intercepts, measured covariates and residual spatial and spatiotemporal variation (Shelton, Thorson, Ward, & Feist, ). We use mixed‐effects methods for parameter estimation, and this approach identifies parameters that minimize residual spatial and spatiotemporal variation by maximizing the variation attributed to intercepts and covariates (items 1 and 2 above). We note that the model has some similarities to classical universal Kriging (Petitgas, ), but allows additional flexibility regarding the distribution of measurement errors and the covariance among size‐classes.

Encounter probability *p*_{i} and expected catch rates *r*_{i} for each sample *i* are affected by the predicted density of individuals or groups λ_{i} (in numbers/km^{2}) at the time *t*_{i}, location *s*_{i}, and length‐bin *c*_{i} associated with that sample: _{λ}(*t*_{i}, *c*_{i}) is an intercept that governs the expectation across space of numbers‐density for a given time and length‐bin, α_{k,c} is a coefficient that governs the effect of a covariate *x*_{k}(*s*_{i}, *c*_{i}, *t*_{i}) on numbers‐density associated with a given location, bin and time, ω_{λ}(*s*_{i}, *c*_{i}) represent spatial variation in numbers‐density λ_{i}, and _{i} is positive.

The probability of encounter *p*_{i} for sample *i* is derived from a Poisson process, given a random distribution of λ_{i} groups of individuals in the vicinity of a given location and time: *a*_{i} is the area sampled (in units of square kilometres) by the *i*‐th sample (i.e., a complementary log‐log‐link for *p*_{i}). The expected catch rate *r*_{i} given that a sample encounters a given species then depends upon expected numbers‐density and average weight *w*_{i} for each group of individuals: *w* to distinguish between parameters for the two components. Specifically, expected catch rate given an encounter is calculated as: *d* = λ × *w* = *p* × *r*. This model for numbers‐densities λ_{i} and average weight *w*_{i} has a similar number of parameters to a conventional delta‐generalized linear mixed model, although it offers the benefit of specifying covariates via a log‐link for both components (rather than a logit‐link for encounter probability in a conventional delta‐model) and covariate effects are therefore more easily interpretable.

Residual spatial variation is treated as a three‐dimensional Gaussian process (termed a “Gaussian random field”), with correlations over two spatial dimensions (e.g., eastings and northings) and among length‐bins: **Ω**_{λ} is a matrix composed of ω_{λ}(*s*_{i}, *c*_{i}) at every location *s* and length‐bin *c*,** R**_{λ} is the correlation among locations, and **V**_{ωλ} is the covariance among length‐bins for spatial variation in population density (where **Ω**_{w} is defined identically but with **R**_{w} and **V**_{ωw} in place of **R**_{w} and **V**_{ωw}). Spatial correlation **R**_{λ} for encounter probabilities follows a Matérn process: _{λ} governs the distance over which locations are uncorrelated (and where **R**_{r} is defined identically but with κ_{r} in place of κ_{λ}), *K*_{ν} is a Bessel function, and **H** is a two‐dimensional linear transformation representing geometric anisotropy (i.e., the tendency for correlations to decrease more slowly with distance along one axis than another, see Bez (: fig. ) or Thorson, Shelton, Ward, and Skaug ()). Covariance **V**_{ωλ} among length‐bins for **Ω**_{λ} (the random effect representing spatial variation in encounter probabilities) is approximated using a factor decomposition: **L**_{ωλ} is a *n*_{c} by *n*_{f} matrix where 0 ≤ *n*_{f} ≤ *n*_{c} is the rank of **V**_{ωλ} such that **V**_{ωλ} is a reduced‐rank approximation to covariance among bins (Thorson, Scheuerell, et al., ; Warton et al., ), and **V**_{ωr} is defined identically but with **L**_{ωr} in place of **L**_{ωλ}. Similarly, spatiotemporal variation in each year is treated as a three‐dimensional Gaussian process, following a Matérn process over space and a first‐order autoregressive process over length‐bins: **E**_{r}(*t*) is defined identically but with **R**_{r} and **R**_{λ} and **V**_{ωλ}) and spatiotemporal (

Previous analyses of distribution shifts have emphasized temperature as an important driver (Pinsky, Worm, Fogarty, Sarmiento, & Levin, ), so we include both the effect of temperature at the location and time of the survey (“local temperature”) and the effect of changes in average temperature throughout the Eastern Bering Sea in a given year (“regional temperature”). We model a local temperature effect by including a quadratic relationship between local temperature and both encounter probability and positive catch rates (Fig. S1). We also include the regional temperature effect by including the interaction of eastings (or northings) with the area of the cold pool (Figure ). Hypothetically, if a large cold pool is associated with a southward shift in distribution, then this would result in a significant estimate of the coefficient linking the northings‐cold pool variable to either λ or *w*. In summary, we include four covariates: *T*(*s*,* t*) is the bottom temperature associated with location *s* and year *t*,* T*^{2}(*s*,* t*) is bottom temperature‐squared (i.e., we include a quadratic effect of bottom temperature), *N*(*s*) and *E*(*s*) are northings and eastings for location *s*, and *A*(*t*) is an annual index of cold pool area (defined as the area in kilometres‐squared for bottom temperatures of 0°C or colder) for year *t*. Inspection of bottom temperatures shows that the cold pool was relatively large in 1999, and 2003–2004 and 2006, and essentially absent in 1987, 1996, 1998 and 2003 (Figure and Fig. S1).

We estimate parameters for this VAST model by maximizing the marginal likelihood of parameters given available data (see Table for a list of fixed and random effects). Fixed effects include intercepts for each year and size‐category for numbers‐density and average weight (_{w}(*c*,* t*)), the spatial scale for spatial correlations (κ_{λ} and κ_{w}) and geometric anisotropy (**H**), the covariance among length‐bins for spatial (**L**_{ωλ} and **L**_{ωw}) and spatiotemporal (_{c} for each size‐class *c*) and the effect of measured covariates (α_{k,c} and β_{k,c}). Mixed‐effect estimation will generally favour a solution that minimizes residual variation (i.e., a simultaneously low value for variance θ_{c}, **L**_{ωλ}, **L**_{ωw}, **Ω**_{λ} and **Ω**_{r}) and spatiotemporal variation (**E**_{λ} and **E**_{r}) as random effects, and approximate the marginal likelihood across these random effects using the Laplace approximation (Skaug & Fournier, ). Parameter estimation is conducted using Template Model Builder (Kristensen, Nielsen, Berg, Skaug, & Bell, ) in the R statistical environment (R Core Team, ). We use the stochastic partial differential equation approximation to the probability of random effects (Lindgren, Rue, & Lindström, ), and also use a “predictive process” approximation wherein spatial and spatiotemporal components are approximated as being piecewise constant at 100 locations (termed “knots”) that are selected using a k‐means algorithm applied to the location of sampling data. R package VAST for implementing this model for other data sets is publicly available on the first author's GitHub site (

As case‐study, we analyse catch rate data for pollock from the Eastern Bering Sea (EBS). Specifically, we use data collected during standardized EBS bottom trawl surveys conducted by the Alaska Fisheries Science Center since 1982 (Stauffer, ). Annual surveys were conducted in June and July over the fixed set of approximately 376 stations and used the same standard trawl (83–112 eastern otter trawl) during all years. Surveys started in the south‐eastern corner of the survey area and proceeded westward. Tow duration was approximately 30 min at 1.54 m/s (3 knots). A subsample of 150–200 individuals from each tow were measured to the nearest centimetre using fork length, and this subsample was then expanded to represent the entire tow catch (using the ratio of sampled weight and tow weight for that species). The catch rate in number per hectare was estimated using the area‐swept method (e.g., Alverson and Pereyra, ) by multiplying distance fished, as indicated by bottom contact sensor (Somerton & Weinberg, ), by the average distance between wing tips measured using acoustic spread sensors (see Weinberg and Kotwicki () for details). We analysed catch rate data for five size‐categories: 0–20 cm (mostly age‐1 individuals), 21–30 cm (mostly age‐2 individuals), 31–40 cm, 41–50 cm and 50+ cm individuals. Given the relatively small number of size‐categories, we used full rank for all covariance matrices (i.e., *n*_{f} = 5, such that **V**_{ωλ}, **V**_{ωr} and

Prior to analysis, we converted catch rate for each length category from numbers to weight (kilograms), using a year‐specific length–weight relationship obtained for years when length–weight samples were collected (i.e., 1991, and 1999–2016). For remaining years, the average length–weight relationship was obtained by pooling all available data. We also corrected catch rate estimates for density‐dependent sampling efficiency of the survey bottom trawl (Kotwicki, Ianelli, & Punt, ). We analyse data from 1982 to 2015, where this period includes a marked decrease and recovery of pollock abundance in the EBS between 2008 and 2015 (Ianelli, Honkalehto, Barbeaux, & Kotwicki, ).

We summarize shifts in distribution by calculating the centroid of the distribution for a given size‐category and year (termed centre of gravity, COG). To do so, we first calculate the biomass density *d*(*s*,* c*,* t*) for size‐class *c* associated with every location and year: *a*(*s*) is the area associated with modelled location *s*. We then calculate a model‐based estimate of the COG from predicted density throughout the population domain: *x*(*s*) is a suitable description of location for site *s*. This formula for COG standardizes by total abundance (in the denominator), so our analysis focuses on changes in distribution after controlling for changes in total abundance. Future research could alternatively explore temperature impacts on total abundance and/or density‐dependent impacts on COG. In the following, we use either eastings/northings in kilometres (when calculating summaries of distance, where eastings and northings are calculated using a projection that has minimal distortion of distance), or latitude/longitude in degrees (when plotting results on a map). Previous research suggests that model‐based estimates of COG are in some cases more statistically efficient than conventional sample‐based estimators (Thorson, Pinsky, & Ward, ).

We distinguish the following three hypotheses for explaining changes in distribution over time: (i) temperature; (ii) size‐structure; and (iii) residual variation. These hypotheses are not mutually exclusive, so we seek to estimate what proportion of variance in COG is explained by each of these factors. To do so, we take estimated values of fixed and random effects for the fitted model, and fix some subset to zero to exclude individual processes from the model. For each “counterfactual run,” we then recalculate the time series of northward and eastward COG that would result from that subset of parameters. Specifically:

*Temperature*—To isolate the effect of temperature, we eliminate any variation in size‐structure (i.e., fix intercepts for average numbers‐density and average weight for a given size‐class and year at its average over all years) and eliminate residual spatiotemporal variation, but leave temperature effects at their estimated value. We then use these values to predict biomass density *d* = λ × *r* for all sites, and calculate COG using these values.*Size‐structure*—To isolate the effect of variation in size‐structure, we eliminate temperature effects and residual spatiotemporal variation. We leave interannual variation in the proportion of total abundance belonging to different size‐classes and again calculate COG using these values.*Unexplained variation*—To isolate the portion of variation that is not explained by size‐structure or temperature, we eliminate variation in size‐structure and fix temperature effects at zero, and again calculate COG using these values.

Further details about how we implement these counterfactual models are provided in Table . In an exploratory run, we also confirm that eliminating variation in cohort strength, and fixing both temperature and unexplained variation at zero results in no variance in COG. This confirms that all estimated variation in COG is attributable to one of these three processes.

We first visualize northward and eastward COG for each of five size‐categories (Figure ). This shows that individuals with size 21–30 and 31–40 cm have a COG to the northwest, and individuals with size 50+ have a COG to the south‐east of the average COG for the entire population. The proportion of total biomass represented by individuals of size 0–40 cm has decreased over time (particularly during the early 1980s), while the proportion with size 50 cm or greater has increased (particularly prior to 1995). We therefore conclude that variation in size‐structure has caused a southward shift in distribution over time towards the COG for large individuals, with the greatest shift caused by size‐structure occurring from 1982 to 1990.

Next, we show density maps for total biomass, as well as the three counterfactual scenarios generated by excluding all mechanisms for variation in COG except temperature, recruitment variation or otherwise unexplained variation (Figure ). Relative to other years, density is high near the Alaskan peninsula in 1995, and is relatively high in the north‐western portion in 2008 and 2015 (Figure , 1st column). The counterfactual scenario restricting variation to temperature effects (Figure , 2nd column) shows that temperature in isolation explains relatively little variation in density among years, although it contributes to the elevated density in the southern boundary in 1995 and 2008. By contrast, changes in size‐structure in isolation (Figure , 3rd column) can generate the relatively high‐density inshore in 2008 relative to 1982 and 1989. Nevertheless, general patterns in spatiotemporal variation are mainly unexplained (Figure , 4th column); for example, the relative increase in densities in the northern portion of the populations range in 2008 and 2015 is attributed primarily to unexplained variation.

Next, we compute the standard deviation of COG for each counterfactual scenario (Table ). This shows the strong agreement between our model‐based estimates of COG and a simple abundance‐weighted average estimator, where both show a standard deviation of approximately 70–75 km east–west and 45–50 km north–south. Temperature in isolation accounts for more northward (21 km) than eastward (13 km) variation, while variation in size‐structure accounts for greater variation east–west (23 km) than north–south (7 km). The greatest standard deviation by far is attributed to “unexplained variation” in density.

Finally, inspection of COG estimates for each counterfactual scenario clearly confirms that observed variation in COG is primarily attributable to unexplained variation (Figure ). As expected, size‐structure causes a southward and eastward trend in COG from 1982 to nearly 2010, and causes greater variation east–west than north–south. By contrast, temperature generates strong interannual variation along the north–south axis. However, the estimated COG moved nearly 100 km north and west from 1982 to 2015, and neither temperature nor size‐structure captures this trend. Similarly, observed COG exhibits high interannual variation (e.g., a large shift northward from 1994 to 1997 and westward from 2003 to 2006), and neither temperature nor size‐structure explains this interannual variation.

In this study, we have developed a statistical approach for decomposing variation in species distribution into components attributable to local and regional temperature, variation in size‐structure and otherwise unexplained variation in density. We demonstrated this approach using the extensive and consistent data collections for pollock, one of the most valuable fisheries in the USA. Results showed that temperature and size‐structure by themselves generate a smaller variance in COG than is observed and that shifts in distribution predicted by temperature and size‐structure do not capture long‐term shifts to the north‐west for this species. This result is surprising, given the many studies that have previously attributed range shifts for this species to temperature affects (e.g., Kotwicki et al., ; Wyllie‐Echeverria & Wooster, ). However, this difference likely arises because previous studies have generally not attempted to quantify what portion of variation in distribution is attributable to temperature, and have therefore failed to recognize that previously identified effects of temperature explain a small portion of interannual or decadal variation. However, we note that Kotwicki and Lauth () had similarly concluded that the trend towards the north‐west in pollock is not directly related to changes in temperatures, and had hypothesized that fishing in the southern portion might be an alternative hypothesis. We therefore recommend future research to integrate spatial information regarding fishing effort into a VAST model (perhaps by using lagged fishing effort as a covariate) to explore whether the “unexplained variation” from this study can be attributed to human harvest.

More generally, we argue that any study regarding the likely impacts of future environmental changes (e.g., changes in sea‐ice cover or bottom temperatures) should first quantify the proportion of historical variation that can be explained by the variables that are available for forecasting. Given that the trend towards the north‐west for pollock is not explained by historical temperature effects, the likely impact of future temperature changes may also be swamped by otherwise unexplained factors affecting distribution for this species. Accurately quantifying uncertainty is increasingly recognized as vital to the use and credibility of any scientific model (Silver, ), and we similarly believe that it is vital for making credible predictions of climate impacts on marine fishes.

We acknowledge that it is unsettling that two dominant hypotheses (temperature and size‐structure) explain a relatively small portion of variation in COG for pollock, and we advocate for future research to decompose “unexplained variation” into different biological mechanisms. One avenue for research involves identifying patterns in distribution shifts for multiple species simultaneously, and cataloguing the physical or ecological variables that could simultaneously explain distributions shifts for multiple species. Spatiotemporal variation in density for pollock is strongly correlated with Pacific cod (*Gadus macrocephalus*, Gadidae; Thorson, Ianelli, et al. ()), and this suggests that there are shared environmental factors driving productivity for these phylogenetically related species. Productivity for pollock is likely impacted by species interactions (Spencer et al., ), so we recommend ongoing research regarding species interactions (competition and predation) in spatiotemporal statistical models. For example, interactions with the expanding spatial distribution of arrowtooth flounder could potential explain distribution shifts for juvenile pollock, either due to behavioural avoidance or direct predation (Spencer et al., ), and these competitive or predator/prey impacts could be integrated if species interactions were explicitly modelled (e.g., Thorson, Munch, & Swain, ).

Another potential mechanism that is missing in our model is the settlement of individual cohorts within particular spatial areas, which could cause COG to vary as it tracks spatial patterns in settlement of dominant cohorts over time. This process could presumably be modelled by explicitly modelling the settlement and growth of individual cohorts (Kristensen, Thygesen, Andersen, & Beyer, ; Thorson, Ianelli, Munch, Ono, & Spencer, ). However, integrating this hypothesis with habitat partitioning by different size‐classes would presumably require an explicit spatiotemporal model for individual movement rates by size‐class. Although spatiotemporal analysis of individual movement is feasible using advective‐diffusive modelling techniques (Sibert, Hampton, Fournier, & Bills, ; Thorson, Jannot, & Somers, ), its integration within a size‐structured spatiotemporal model remains a topic for future research. These three unmodelled mechanisms (additional environmental variables, species interactions and cohort‐specific distribution) will all be at least as difficult to forecast as global water temperature. Therefore, identifying mechanisms for historical variation in COG may not translate to improved forecasts of future variation. However, we still argue that disentangling the multiple drivers of historical COG is a necessary first step to forecasting future distribution shifts.

We also acknowledge several important caveats regarding our approach to identify temperature impacts on pollock distribution shifts in the EBS. Most importantly, we have not included lagged effects of local or regional temperatures (either within or among years), and recent research has highlighted the potentially important effect of lagged environmental conditions on explaining local density (Blonder et al., ). If bottom temperature varies among months within a given year, for example, then different months within a year may have weaker or stronger correlation with a given biological variable (e.g., Black, ). Best practices for including lagged effects is also an ongoing research topic when estimating environmental impacts on recruitment for marine fishes (Szuwalski, Vert‐Pre, Punt, Branch, & Hilborn, ), and future studies of pollock distribution shift could use exploratory methods to identify important lags for environmental variables. We also acknowledge that the impact of temperature on local density may be highly nonlinear. We have included a quadratic impact of local bottom temperature and a linear impact of regional cold pool size, and approximating temperature impacts in this way may have decreased the variance explained by temperature relative to using a highly nonlinear model for temperature impacts. Finally, we note that including additional missing variables may increase the explanatory power of temperature. For a hypothetical example, temperature impacts may have varied before and after the 1998 regime shift (Rodionov & Overland, ), so including the interactions of temperature and a dummy variable (representing before/after the regime shift) could increase the estimated importance of temperature. Future research could also use simulation testing to explore our ability to detect temperature signals of varying magnitude. We therefore interpret out results as a starting point for ongoing statistical and exploratory research seeking to attribute population‐wide distribution shifts for pollock to factors affecting local density.

Finally, we provide software to implement the vector‐autoregressive spatiotemporal model as an R package called *“*VAST”. This software can also be used to estimate size‐specific indices of biomass, for use as an input into size‐structured stock assessment models (Punt, Huang, & Maunder, ), or to jointly model the spatial distribution for multiple species simultaneously (Ovaskainen, Roy, Fox, & Anderson, ; Thorson, Scheuerell, et al., ). We encourage broader use of size‐structured spatiotemporal models to answer whether temperature generally explains a small or large portion of historical distribution shifts for a wide variety of marine fishes.

We thank Bob Lauth for data regarding the cold pool size in each year (shown in Figure ) and Kirstin Holsman for discussions regarding the distribution of pollock. We also thank Aaron Berger, Paul Spencer, Jim Hastie, and Michelle McClure and two anonymous reviewers for comments on an earlier draft.