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Appendix B. Selection of threshold values.

The threshold value for the threshold model formulations (Eq.3 in Methods) was chosen by minimizing the model Generalized Cross Validation – a measure of the mean squared predictive error of the model (Green and Silverman 1994). Namely, we run 20 different model formulations for as many threshold values contained within the upper 0.8 and lower 0.2 quantile range. Finally, we selected the model corresponding of the lowest GCV. Limiting the threshold covariate range to the upper 0.8 and lower 0.2 quantile ensured that at least 20% of the data were left within one of the two regimes. Figure B1 shows the GCV profile for each of the examined threshold formulations. In models were the effect of both current velocity and an additional environmental variable was assessed, we fixed the current threshold to its previously estimated value and, using the methodology just described, estimated the additional environmental threshold anew.

The so found lowest GCV however, cannot be compared with the GCV of a fully additive formulation, because it does not properly account for the presence of an additional parameter (i.e., the threshold) in the threshold formulation. Thus, to properly compare models with and without threshold we computed the genuine cross validation (CV) as follows. A random sample of 10% of the data was excluded from the observations and the remaining data were used to fit a new model (including a re-estimation of the threshold, when applicable). The so fitted model was then used to estimate the out-of-sample 10% data cases, and the mean squared predictive error was recorded. The same routine was repeated 500 times, with the final CV being the average of all runs.

The estimated threshold value of a model with Kola water temperature and current velocity fell at the lower extreme of the examined Kola temperature range (Fig. B1). This resulted in an uneven distribution of samples among regimes (Fig. 7 of main text). Here (Fig. B2), we show the results of a model with the lowest possible temperature threshold that allowed a more homogeneous spread of samples. The so found model R² and GCV are 48.2% and 4.14. The results are essentially similar to the one obtained from the best model, however, they allow a comparison of survival values in the eastern portion of the grid during high current velocity regimes. Even in this case, the distribution map of older cod shows an eastward expansion of catches during warm environmental regimes (Fig. B3).

LITERATURE CITED

Green, P. J., and B. W. Silverman. 1994. Nonparametric regression and generalized linear models: A Roughness Penalty Approach. Chapman and Hall, New York, New York, USA.

FIG. B1. GCV profile for each of the examined threshold formulations. The vertical solid line indicates the selected threshold value, corresponding to the lowest GCV. The threshold for Kola temperature fell at the lower extreme of the examined range (bottom right panel), thus we also examined the results from a model with the threshold value closer to the center of the range (i.e., 3.54oC, indicated by the dashed line).

FIG. B2. Spatial pattern of age-0 cod survival over four environmental regimes, defined by average current velocity along the 30o longitude line and by the Kola temperature indicated by the dashed line in Fig. B1.

FIG. B3. Distribution of adult cod from the winter bottom trawl survey, partitioned according to water temperature along the Kola section and to the average current velocity along the 30o longitude line. The environmental regimes are the same as the one estimated in the age-0 survival analysis and shown in Fig. B2. The area of the circles is proportional to the log-transformed and effort-standardized catches. Dots indicate stations with zero catch.