Web-based Supplementary Material for the manuscript

titled “Modeling Spatio-Temporal Trends in the

Productivity of North Pacific Salmon” by Oksana

Chkrebtii and Jiguo Cao

1

1 Details on the form of the penalty functions

The form of the penalty used for estimating the ith smooth function (i = 1, . . . ,m) for model M1is given in Wood (2006). It has the form λic

>i Sici, where ci = (ci1, . . . , ciqi)

> is a vector of basis

coefficients for the ith smooth function. Here m is the total number of smooth functions to beestimated in the model. Let c be the vector of all unknown coefficients in the model and defineD (λ) as follows, for use hereafter in defining the model penalty D (λ).

M1 : c =

a0δc1c2

D (λ) =

0 0 0 00 0 0 00 0 λ1S1 00 0 0 λ2S2

The penatly for the bivariate smooth term in model M2 is

S12 =

∫ ∫ [λ1

(∂2a

∂s2

)2

+ λ2

(∂2a

∂t2

)2]ds dt,

so we define,

M2 : c =

a0δ

c12

D (λ) =

0 0 00 0 00 0 λ12S12

Now, we may write the overall model penatly matrix for each of M1 and M2 as

J (y|λ) = c>D (λ) c.

The value of the m-dimensional smoothing parameter vector λ controls the tradeoff between fit tothe data and smoothness of the resulting functions must be specified by the analyst.

2 Derivation of the pBIC for Gaussian additive models

For a Gaussian additive model with coefficients c, denote by M (λ) the model indexed by any givensmoothing parameter vector λ associated with pdf f (y|c,λ). The posterior probability of the modelgiven observations y (see, for example, Raftery 1996) is,

P (M (λ) |y) =P (M (λ)) f (y|λ)∑r

α=1 P (M (α)) f (y|λα), (1)

where P (M (·)) represents the prior probability of model M (·). Assign a prior distribution π (c|λ)to the GAM coefficients and write f (y|λ) =

∫f (y|c,λ)π (c|λ) dc. The goal is to select among

M (λ1) , . . . ,M (λr) the model with the largest posterior probability by maximizing the numerator of(1) by choice of λ. Thus, we seek to maximize,

P (M (λ)) f (y|λ) = P (M (λ))

∫f (y|c,λ)π (c|λ) dc,

2

or, if all models are a priori equally probable,

f (y|λ) =

∫f (y|c,λ)π (c|λ) dc, (2)

equivalently minimizing the pseudo BIC,

pBIC ≡ −2 log f (y|λ) ,

for model M by choice of smoothing parameter λ. The integral in (2) can be estimated by a Laplaceapproximation (see, for example, Kass and Raftery 1995). Following Raftery (1996), define thefunction,

h (c|y,λ) = n−1 {log f (y|c) + log π (c|λ)} .

By rewriting the likelihood (2) of y given the model, in terms of this function, the integral can beapproximated by Laplace’s method (Tierney and Kadane 1986).

f (y|λ) =

∫f (y|c,λ)π (c|λ) dc,

=

∫exp {n h (c|y,λ)} dc

=(2π)p/2

np/2|Hλ (c) |1/2exp {n h (c|y,λ)}

{1 +Op

(n−1

)},

where,

Hλ (c) = −∂2h (c|y,λ)

∂c∂c>

∣∣∣∣c=c

,

and c is the value of c that maximizes h (c|y,λ). Equivalently, c is the mode of the posteriordistribution of c.

An appropriate prior distribution for c given λ incorporates the belief that values of c that leadto a smoother model are more likely than values that lead to a model that is less smooth (notethat before observing the data, we cannot have any prior knowledge regarding the fit of the modelto the data). This property is associated with a penalty of the form J (y|λ) = c>D (λ) c, whichsuggests a prior distribution proportional to exp

{−1

2c>D (λ) c}

(see, for example, Silverman 1985).As discussed in the next section, the prior π (c|λ) is partially improper in the case of a GAM withat least one parametric component.

log π (c|λ) = −p− d2

log (2π) +1

2log |D (λ) |+ −

1

2c>D (λ) c (3)

∝ −1

2c>D (λ) c.

The log likelihood of the data, assuming constant unknown variance Var (y) = σ2I, is,

log f(y|c, σ2

)= −n

2log (2πn)− n

2log(σ2)− 1

2σ−2 (y −Φc)> (y −Φc) (4)

3

And thus,

h(c, σ2|y,λ

)= n−1

{log f

(y|c, σ2

)+ log π (c|λ)

}∝ log f

(y|c, σ2

)+ J (y|λ) , (5)

is proportional to criterion (2) given in the paper. Therefore its minimizers,(c, σ2

)= argmin

c,σ2

h(c, σ2|y,λ

)(6)

= argminc,σ2

[log f

(y|c, σ2

)+ J (y|λ)

]=

([Φ>WΦ + D (λ)

]−1Φ>Wy, ||y − y||2/n

)is already available from the model fitting procedure.

Next, substitute the prior (3) and the likelihood (4) evaluated at c into equation (1) to obtainthe apporoximate value of pBIC,

pBIC ≡ −2 log f∗ (y|λ)

= −2 log f (y|c)− 2 log π∗ (c∗|λ)− p log (2π) + p log n+ log |Hλ (c) |+ Op(n−1

)= n log (2πn) + n log

(σ2)

+ σ−2 (y −Φc)> (y −Φc)

+ (p− d) log (2π)− log |D (λ) |+ + c>D (λ) c

− p log (2π) + p log n+ log |Hλ (c) |+ Op(n−1

),

where,

Hλ (c) = −∂2h (c|y,λ)

∂c∂c>

∣∣∣∣c=c

= − ∂2

∂c∂c>n−1 {log f (y|c) + log π (c|λ)}

∣∣∣∣c=c

= − ∂2

∂c∂c>n−1

{−n

2log (2πn)− n

2log σ2 − 1

2σ−2 (y −Φc)> (y −Φc)

} ∣∣∣∣c=c

− ∂2

∂c∂c>n−1

{−p− d

2log (2π) +

1

2log |D|+ −

1

2c>D c

} ∣∣∣∣c=c

= −n−1 ∂∂c

{∂

∂c>

[−1

2σ−2 (y −Φc)> (y −Φc)− 1

2c>Dc

]} ∣∣∣∣c=c

= −n−1 ∂∂c

{σ−2Φ>y − σ−2Φ>Φc− 1

2

(D + D>

)c

} ∣∣∣∣c=c

=1

nσ2Φ>Φ +

1

2n

(D> + D

).

Therefore, the problem of finding an appropriate value of λ is equivalent to finding k such thatM (λk) is optimal among M (λ1) , . . . ,M (λr) in the sense that it minimizes the pBIC. Note that,although we may use this result to choose among r smoothing parameters for a particular model,it is not possible to compare non-nested models using the pBIC in the presence of improper priors.This fact is discussed in, for example, Carlin and Louis (2000).

4

2.1 Partially improper priors

This section explains why the prior distribution π (c|λ) of the coefficients from a GAM with a para-metric component is partially improper. Consider a p × 1 vector of parameters c consisting of dparametric coefficients and p− d basis coefficients for the m smooth functions being estimated. Anappropriate prior distribution for c is proportional to exp

{−1

2c>D (λ) c}

(see, for example, Silver-man 1985). Note that the block-diagonal matrix D (λ) can be written as the weighted sum of p− dnon-negative semi-definite p×p matrices and is thus non-negative semi-definite with d zero eigenval-ues (Wood 2006). Therefore the prior for c is always partially improper by the following argument:since the eigenvalues of the penalty matrix D (λ) are real and non-negative, this matrix can bedecomposed (Horn and Johnson 1985, p. 204) into D (λ) = UΛU>. Here Λ is a (p− d) × (p− d)real nonnegative diagonal matrix containing the non-zero eigenvalues of D (λ) ordered from highestto lowest and U is a p × (p− d) matrix with orthonormal columns of corresponding eigenvectors.Therefore, the parameter vector c contains (p− d) elements c∗ = U>c, which are characterized bya proper prior,

log π∗ (c∗|λ) = log

((2π)−

p−d2 |Λ|1/2 exp

{−1

2c∗>Λ c∗

})= −p− d

2log (2π) +

1

2log |D (λ) |+ −

1

2c>D (λ) c

∝ −1

2c>D (λ) c.

Combining this with the above assumption that log π (c|λ) ∝ −12c>D (λ) c and assuming conditional

prior independence π (c|λ) ≡ π ((c∗, cu) |λ) = π∗ (c∗|λ)πu (cu|λ), the remaining d elements cu havea purely noninformative prior πu (cu|λ) ∝ 1.

The following argument justifies the use of this improper prior in the calculation of the posteriormodel probability for a given model M. We have π (c|λ) = π∗ (c∗|λ)πu (cu|λ) where πu (cu|λ) =C · 1 ∝ 1 where C is some normalizing constant that does not depend on the value of λ (as pointedout by O’Hagan 1995, C does not actually exist in the case of the uniform prior and is used onlyfor the sake of demonstrating the following argument). Therefore, we may re-write the posteriorprobability of the model as,

P (M (λ) |y) =P (M (λ))

∫f (y|c,λ)π∗ (c∗|λ)πu (cu|λ) dc∑r

α=1 P (Mα)∫f (y|cα,λα)π∗ (c∗α|λα)πu (cuα|λα) dcα

=CP (M (λ))

∫f (y|c,λ)π∗ (c∗|λ) dc

C∑r

α=1 P (Mα)∫f (y|cα,λα)π∗ (c∗α|λα) dcα

=P (M (λ))

∫f (y|c,λ)π∗ (c∗|λ) dc∑r

α=1 P (Mα)∫f (y|cα,λα)π∗ (c∗α|λα) dcα

.

3 Details on smoothing parameter selection procedure

For smoothing parameter selection we evaluate and minimize pBIC with respect to λ via grid searchusing the R software. For each model, pBIC was minimized over a grid of values on a scale of logbase 10, from −10 to 10 with an increment of 0.05.

5

4 Figures and Tables

0 1000 2000 3000 4000

1960

1965

1970

1975

1980

1985

1990

1995

Bro

od Y

ear

0 1000 2000 3000 4000

1950

1960

1970

1980

1990

2000

Bro

od Y

ear

0 1000 2000 3000 4000

1950

1960

1970

1980

1990

Bro

od Y

ear

1000 2000 3000 4000

1950

1960

1970

1980

1990

Bro

od Y

ear

Figure S1: Design plots showing the distributions of measurements over brood year and along-shorelocation for odd year pink, even year pink, chum, and sockeye salmon with the intensity of shadingproportional to the log ratio of recruits to spawners observed at that location and brood year. Thevertical lines represent the clusters of spawning sites examined in the conditional plots.

6

4.1 Pink Salmon - Odd Year Runs

0 1000 2000 3000 4000

Location (s)

a 1(s

)−

1−

0.5

00.

51

1950 1960 1970 1980 1990

Brood Year (t)

a 2(t)

0 1000 2000 3000 4000

Location (s)

a 1(s

)−

1−

0.5

00.

51

1950 1960 1970 1980 1990

Brood Year (t)

a 2(t)

−1

−0.

50

0.5

1

Figure S2: Estimated one dimensional component functions for model M1 and confidence bands (twostandard deviations above and below) for odd year runs of pink salmon with smoothing parametervector selected by minimization of the GCV criterion (above) and pseudo BIC (below).

References

Carlin, B. P. and T. A. Louis (2000). Bayes and empirical Bayes methods for data analysis. BocaRaton, Fla: Chapman and Hall/CRC.

Horn, R. A. and C. R. Johnson (1985). Matrix analysis. Cambridge, New York: Cambridge Uni-versity Press.

Kass, R. E. and A. E. Raftery (1995). Bayes factors. Journal of the American Statistical Associa-tion 90 (430), 773–795.

7

−3 −2 −1 0 1 2 3

−2

02

4

theoretical quantiles

devi

ance

res

idua

ls

−3 −2 −1 0 1 2

−2

02

4

Resids vs. linear pred.

linear predictor

resi

dual

s

Histogram of residuals

Residuals

Fre

quen

cy

−4 −2 0 2 4

050

100

150

200

−3 −2 −1 0 1 2

−4

−2

02

4

Response vs. Fitted Values

Fitted Values

Res

pons

e

0 1000 2000 3000 4000

−2

02

4

Location (s)

resi

dual

s

1950 1960 1970 1980 1990

−2

02

4

Brood Year (t)

resi

dual

s

Figure S3: Diagnostics (above) and residuals over time and spatial location (below) for GAM modelM1 of productivity of odd year runs of pink salmon with smoothing parameter vector selected byminimization of the pBIC criterion.

O’Hagan, A. (1995). Fractional bayes factors for model comparison. Journal of the Royal StatisticalSociety. Series B (Methodological) 57 (1), 99–138.

Raftery, A. E. (1996). Approximate Bayes factors and accounting for model uncertainty in gener-alised linear models. Biometrika 83 (2), 251–266.

8

Parametric CoefficientsParameter Estimate Std. Error t value Pr(>|t|)(Intercept) 1.3 0.065 20 2e-68as.factor(SNIndex)1:X -0.032 0.012 -2.6 9e-03as.factor(SNIndex)3:X -0.0042 0.0013 -3.2 1e-03as.factor(SNIndex)6:X -0.0017 0.00051 -3.4 8e-04as.factor(SNIndex)7:X -0.0064 0.0017 -3.8 2e-04as.factor(SNIndex)9:X -0.008 0.0024 -3.3 1e-03as.factor(SNIndex)10:X -0.0089 0.0032 -2.8 6e-03as.factor(SNIndex)11:X -5.9e-05 4.8e-05 -1.2 2e-01as.factor(SNIndex)13:X -0.037 0.0097 -3.9 1e-04as.factor(SNIndex)14:X -0.0067 0.0029 -2.3 2e-02as.factor(SNIndex)15:X -0.00079 0.00022 -3.6 3e-04as.factor(SNIndex)16:X -0.0012 0.00042 -2.8 5e-03as.factor(SNIndex)17:X -0.021 0.0041 -5 7e-07as.factor(SNIndex)18:X -2.9e-05 1.8e-05 -1.6 1e-01as.factor(SNIndex)19:X -7.7e-05 4e-05 -1.9 5e-02as.factor(SNIndex)20:X -0.00032 9.6e-05 -3.3 9e-04as.factor(SNIndex)21:X -0.0061 0.0012 -5 1e-06as.factor(SNIndex)22:X -0.016 0.0031 -5 6e-07as.factor(SNIndex)23:X -0.0007 0.00068 -1 3e-01as.factor(SNIndex)24:X 0.001 0.0006 1.7 9e-02as.factor(SNIndex)25:X -0.0039 0.0014 -2.9 5e-03as.factor(SNIndex)26:X -0.0024 0.0017 -1.4 2e-01as.factor(SNIndex)27:X -0.0068 0.002 -3.4 6e-04as.factor(SNIndex)28:X -0.00024 0.00023 -1 3e-01as.factor(SNIndex)29:X -0.0012 0.00074 -1.7 1e-01as.factor(SNIndex)30:X -0.0059 0.0012 -4.8 2e-06as.factor(SNIndex)31:X -0.00044 0.00021 -2.1 4e-02as.factor(SNIndex)32:X -0.00055 0.00024 -2.3 2e-02as.factor(SNIndex)33:X -7e-05 0.00019 -0.37 7e-01as.factor(SNIndex)34:X -0.0018 0.00059 -3.2 2e-03as.factor(SNIndex)35:X -0.0025 0.0012 -2.1 3e-02as.factor(SNIndex)36:X -0.0084 0.0057 -1.5 1e-01as.factor(SNIndex)37:X -0.0031 0.0011 -2.8 5e-03as.factor(SNIndex)38:X -0.0027 0.0011 -2.5 1e-02as.factor(SNIndex)39:X -0.00024 0.00015 -1.6 1e-01as.factor(SNIndex)40:X -0.29 0.059 -4.9 1e-06as.factor(SNIndex)41:X -0.00076 0.0004 -1.9 6e-02as.factor(SNIndex)43:X -0.021 0.0083 -2.5 1e-02as.factor(SNIndex)44:X -0.016 0.0031 -5.1 4e-07as.factor(SNIndex)45:X -0.019 0.0044 -4.3 2e-05as.factor(SNIndex)46:X -0.096 0.022 -4.4 1e-05Approximate Significance of Smooth TermsFunction edf Ref.df F p-values(asd) 1 1 2.6 1e-01s(yr) 7.1 8.1 2.7 7e-03Smoothing Parameter Estimates UsedSmooth function lambdas(asd) 9e+06s(yr) 1e-03Model DiagnosticsAIC 1831BIC 6306GCV score 1.126Log-Likelihood -865.5Effective Degrees of Freedom 570.9Scale Parameter Estimate 1.037Observations 620

Table S1: Model M1– Odd-year Pink Salmon (smoothing parameter selected by GCV)

Silverman, B. W. (1985). Some aspects of the spline smoothing approach to non-parametric re-gression curve fitting. Journal of the Royal Statistical Society. Series B (Methodological) 47 (1),1–52.

Tierney, L. and J. B. Kadane (1986). Accurate approximations for posterior moments and marginaldensities. Journal of the American Statistical Association 81 (393), 82–86.

Wood, S. N. (2006). Generalized additive models : an introduction with R. Boca Raton, FL:Chapman and Hall.

9

Parametric CoefficientsParameter Estimate Std. Error t value Pr(>|t|)(Intercept) 1.3 0.065 20 1e-68as.factor(SNIndex)1:X -0.032 0.012 -2.6 9e-03as.factor(SNIndex)3:X -0.0042 0.0013 -3.3 1e-03as.factor(SNIndex)6:X -0.0017 0.00051 -3.4 7e-04as.factor(SNIndex)7:X -0.0064 0.0017 -3.8 2e-04as.factor(SNIndex)9:X -0.008 0.0024 -3.3 1e-03as.factor(SNIndex)10:X -0.0089 0.0032 -2.8 5e-03as.factor(SNIndex)11:X -5.9e-05 4.8e-05 -1.2 2e-01as.factor(SNIndex)13:X -0.037 0.0097 -3.9 1e-04as.factor(SNIndex)14:X -0.007 0.0029 -2.4 1e-02as.factor(SNIndex)15:X -0.00079 0.00022 -3.6 3e-04as.factor(SNIndex)16:X -0.0012 0.00042 -2.8 5e-03as.factor(SNIndex)17:X -0.02 0.0041 -5 8e-07as.factor(SNIndex)18:X -2.9e-05 1.8e-05 -1.6 1e-01as.factor(SNIndex)19:X -7.8e-05 4e-05 -1.9 5e-02as.factor(SNIndex)20:X -0.00032 9.6e-05 -3.3 9e-04as.factor(SNIndex)21:X -0.0061 0.0012 -4.9 1e-06as.factor(SNIndex)22:X -0.016 0.0031 -5 6e-07as.factor(SNIndex)23:X -0.00069 0.00068 -1 3e-01as.factor(SNIndex)24:X 0.001 0.0006 1.7 9e-02as.factor(SNIndex)25:X -0.0039 0.0014 -2.9 4e-03as.factor(SNIndex)26:X -0.0023 0.0017 -1.3 2e-01as.factor(SNIndex)27:X -0.0067 0.002 -3.4 7e-04as.factor(SNIndex)28:X -0.00024 0.00023 -1 3e-01as.factor(SNIndex)29:X -0.0012 0.00074 -1.7 1e-01as.factor(SNIndex)30:X -0.0059 0.0012 -4.8 2e-06as.factor(SNIndex)31:X -0.00045 0.00021 -2.1 4e-02as.factor(SNIndex)32:X -0.00054 0.00024 -2.3 2e-02as.factor(SNIndex)33:X -6.8e-05 0.00019 -0.36 7e-01as.factor(SNIndex)34:X -0.0019 0.00059 -3.2 2e-03as.factor(SNIndex)35:X -0.0025 0.0012 -2.1 3e-02as.factor(SNIndex)36:X -0.0086 0.0057 -1.5 1e-01as.factor(SNIndex)37:X -0.0031 0.0011 -2.8 5e-03as.factor(SNIndex)38:X -0.0027 0.0011 -2.5 1e-02as.factor(SNIndex)39:X -0.00023 0.00015 -1.6 1e-01as.factor(SNIndex)40:X -0.29 0.059 -4.9 1e-06as.factor(SNIndex)41:X -0.00075 0.0004 -1.9 6e-02as.factor(SNIndex)43:X -0.02 0.0083 -2.4 2e-02as.factor(SNIndex)44:X -0.016 0.0031 -5.1 4e-07as.factor(SNIndex)45:X -0.019 0.0044 -4.3 2e-05as.factor(SNIndex)46:X -0.095 0.022 -4.3 2e-05Approximate Significance of Smooth TermsFunction edf Ref.df F p-values(asd) 1 1 2.3 1e-01s(yr) 5.9 7 2.4 2e-02Smoothing Parameter Estimates UsedSmooth function lambdas(asd) 1e+08s(yr) 4e-03Model DiagnosticsAIC 1832BIC 6300GCV score 1.128Log-Likelihood -867.3Effective Degrees of Freedom 572.1Scale Parameter Estimate 1.041Observations 620

Table S2: Model M1 – Odd-year Pink Salmon (smoothing parameter selected by BIC)

10

Locatio

n (s)

0

10002000

30004000

Brood Year (t)1960

19701980

1990

a(s,t)

−2

−1

0

1

2

0 1000 2000 3000 400019

6019

7019

8019

90

Location (s)

Bro

od y

ear

(t)

−1.5

−1.5

−1.5

−1

−1

−1

−1

−1

−0.5

−0.5

−0.5

0

0

0.5

1 1.5

Locatio

n (s)

0

10002000

30004000

Brood Year (t)1960

19701980

1990

a(s,t)

−1.5

−1.0

−0.5

0 1000 2000 3000 4000

1960

1970

1980

1990

Location (s)

Bro

od y

ear

(t)

−1.6

−1.4

−1.4

−1.2

−1.2

−1.2

−1

−1

−1

−0.8

−0.8

−0.

8

−0.6

−0.6

−0.

6

−0.4

−0.

4

−0.2

Figure S4: Estimated bivariate component functions for model M2 for odd year runs of pink salmonwith smoothing parameter vector selected by minimization of the GCV criterion (above) and pseudoBIC (below).

11

−3 −2 −1 0 1 2 3

−2

02

4

theoretical quantiles

devi

ance

res

idua

ls

−3 −2 −1 0 1 2

−2

02

4

Resids vs. linear pred.

linear predictor

resi

dual

s

Histogram of residuals

Residuals

Fre

quen

cy

−2 0 2 4

020

4060

8010

014

0

−3 −2 −1 0 1 2

−4

−2

02

4

Response vs. Fitted Values

Fitted Values

Res

pons

e

0 1000 2000 3000 4000

−2

02

4

Location (s)

resi

dual

s

1950 1960 1970 1980 1990

−2

02

4

Brood Year (t)

resi

dual

s

Figure S5: Diagnostics (above) and residuals over time and spatial location (below) for GAM modelM2 of productivity of odd year runs of pink salmon with smoothing parameter vector selected byminimization of the pBIC criterion.

12

Parametric CoefficientsParameter Estimate Std. Error t value Pr(>|t|)(Intercept) 1.4 0.067 21 2e-71as.factor(SNIndex)1:X -0.028 0.013 -2.2 3e-02as.factor(SNIndex)3:X -0.0048 0.0013 -3.6 3e-04as.factor(SNIndex)6:X -0.0017 0.00052 -3.3 1e-03as.factor(SNIndex)7:X -0.0075 0.0017 -4.4 1e-05as.factor(SNIndex)9:X -0.0095 0.0024 -3.9 9e-05as.factor(SNIndex)10:X -0.0083 0.0032 -2.6 1e-02as.factor(SNIndex)11:X -3.3e-05 4.8e-05 -0.68 5e-01as.factor(SNIndex)13:X -0.035 0.0098 -3.6 4e-04as.factor(SNIndex)14:X -0.0069 0.0029 -2.4 2e-02as.factor(SNIndex)15:X -0.00074 0.00022 -3.3 9e-04as.factor(SNIndex)16:X -0.0012 0.00043 -2.7 7e-03as.factor(SNIndex)17:X -0.021 0.0042 -5 9e-07as.factor(SNIndex)18:X -2.4e-05 2e-05 -1.2 2e-01as.factor(SNIndex)19:X -7.4e-05 4.4e-05 -1.7 9e-02as.factor(SNIndex)20:X -0.00031 0.0001 -3 3e-03as.factor(SNIndex)21:X -0.0065 0.0013 -4.9 1e-06as.factor(SNIndex)22:X -0.016 0.0032 -5 7e-07as.factor(SNIndex)23:X -0.0011 0.00075 -1.5 1e-01as.factor(SNIndex)24:X 0.00061 0.00067 0.91 4e-01as.factor(SNIndex)25:X -0.005 0.0015 -3.3 1e-03as.factor(SNIndex)26:X -0.0036 0.0018 -1.9 5e-02as.factor(SNIndex)27:X -0.0081 0.0021 -3.9 1e-04as.factor(SNIndex)28:X -0.00049 0.00024 -2.1 4e-02as.factor(SNIndex)29:X -0.002 0.00075 -2.6 1e-02as.factor(SNIndex)30:X -0.0072 0.0012 -5.8 1e-08as.factor(SNIndex)31:X -0.0007 0.00021 -3.3 9e-04as.factor(SNIndex)32:X -0.00073 0.00024 -3.1 2e-03as.factor(SNIndex)33:X -0.00028 0.00019 -1.5 1e-01as.factor(SNIndex)34:X -0.0027 0.00059 -4.5 7e-06as.factor(SNIndex)35:X -0.0041 0.0012 -3.4 7e-04as.factor(SNIndex)36:X -0.015 0.0056 -2.7 7e-03as.factor(SNIndex)37:X -0.0037 0.0011 -3.4 6e-04as.factor(SNIndex)38:X -0.0042 0.0011 -3.8 2e-04as.factor(SNIndex)39:X -0.0004 0.00015 -2.7 8e-03as.factor(SNIndex)40:X -0.3 0.059 -5.1 5e-07as.factor(SNIndex)41:X -0.0012 0.00041 -2.9 4e-03as.factor(SNIndex)43:X -0.018 0.0085 -2.2 3e-02as.factor(SNIndex)44:X -0.015 0.0033 -4.6 5e-06as.factor(SNIndex)45:X -0.019 0.0048 -3.9 1e-04as.factor(SNIndex)46:X -0.08 0.027 -2.9 4e-03Approximate Significance of Smooth TermsFunction edf Ref.df F p-valuete(asd,yr) 18 20 3.4 1e-06Smoothing Parameter Estimates UsedSmooth function lambdate(asd,yr)1 2e-01te(asd,yr)2 6e-03Model DiagnosticsAIC 1804BIC 6359GCV score 1.081Log-Likelihood -842.1Effective Degrees of Freedom 561.3Scale Parameter Estimate 0.9784Observations 620

Table S3: Model M2 – Odd-year Pink Salmon (smoothing parameter selected by GCV)

13

Parametric CoefficientsParameter Estimate Std. Error t value Pr(>|t|)(Intercept) 1.4 0.066 21 9e-71as.factor(SNIndex)1:X -0.023 0.013 -1.8 7e-02as.factor(SNIndex)3:X -0.0045 0.0013 -3.4 6e-04as.factor(SNIndex)6:X -0.0016 0.00052 -3.1 2e-03as.factor(SNIndex)7:X -0.0071 0.0017 -4.2 3e-05as.factor(SNIndex)9:X -0.0091 0.0024 -3.8 2e-04as.factor(SNIndex)10:X -0.0083 0.0032 -2.6 1e-02as.factor(SNIndex)11:X -2.8e-05 4.9e-05 -0.58 6e-01as.factor(SNIndex)13:X -0.032 0.0097 -3.3 9e-04as.factor(SNIndex)14:X -0.0063 0.0028 -2.2 3e-02as.factor(SNIndex)15:X -0.00069 0.00022 -3.2 2e-03as.factor(SNIndex)16:X -0.0011 0.00043 -2.5 1e-02as.factor(SNIndex)17:X -0.02 0.0042 -4.8 2e-06as.factor(SNIndex)18:X -2.1e-05 2e-05 -1.1 3e-01as.factor(SNIndex)19:X -6.8e-05 4.3e-05 -1.6 1e-01as.factor(SNIndex)20:X -0.0003 0.0001 -2.9 4e-03as.factor(SNIndex)21:X -0.0061 0.0013 -4.7 3e-06as.factor(SNIndex)22:X -0.016 0.0032 -4.9 1e-06as.factor(SNIndex)23:X -0.00094 0.00074 -1.3 2e-01as.factor(SNIndex)24:X 0.00079 0.00066 1.2 2e-01as.factor(SNIndex)25:X -0.0046 0.0015 -3.1 2e-03as.factor(SNIndex)26:X -0.003 0.0018 -1.7 1e-01as.factor(SNIndex)27:X -0.0076 0.0021 -3.7 3e-04as.factor(SNIndex)28:X -0.00048 0.00024 -2 5e-02as.factor(SNIndex)29:X -0.002 0.00076 -2.6 1e-02as.factor(SNIndex)30:X -0.007 0.0012 -5.6 3e-08as.factor(SNIndex)31:X -0.00068 0.00021 -3.2 1e-03as.factor(SNIndex)32:X -0.00074 0.00024 -3.1 2e-03as.factor(SNIndex)33:X -0.00027 0.00019 -1.4 2e-01as.factor(SNIndex)34:X -0.0026 0.00059 -4.3 2e-05as.factor(SNIndex)35:X -0.0037 0.0012 -3.1 2e-03as.factor(SNIndex)36:X -0.014 0.0056 -2.5 1e-02as.factor(SNIndex)37:X -0.0035 0.0011 -3.2 1e-03as.factor(SNIndex)38:X -0.004 0.0011 -3.6 3e-04as.factor(SNIndex)39:X -0.00037 0.00015 -2.5 1e-02as.factor(SNIndex)40:X -0.3 0.059 -5 9e-07as.factor(SNIndex)41:X -0.0011 0.00041 -2.8 5e-03as.factor(SNIndex)43:X -0.019 0.0085 -2.2 3e-02as.factor(SNIndex)44:X -0.015 0.0033 -4.5 9e-06as.factor(SNIndex)45:X -0.02 0.0046 -4.3 2e-05as.factor(SNIndex)46:X -0.076 0.025 -3 3e-03Approximate Significance of Smooth TermsFunction edf Ref.df F p-valuete(asd,yr) 8.5 11 3.7 4e-05Smoothing Parameter Estimates UsedSmooth function lambdate(asd,yr)1 3e-01te(asd,yr)2 1e+01Model DiagnosticsAIC 1812BIC 6332GCV score 1.092Log-Likelihood -855.3Effective Degrees of Freedom 570.5Scale Parameter Estimate 1.005Observations 620

Table S4: Model M2 – Odd-year Pink Salmon (smoothing parameter selected by BIC)

14

1000 2000 3000 4000

Location (s)

a 1(s

)−

1−

0.5

00.

51

1950 1970 1990

Brood Year (t)

a 2(t)

1000 2000 3000 4000

Location (s)

a 1(s

)−

1−

0.5

00.

51

1950 1970 1990

Brood Year (t)

a 2(t)

−1

−0.

50

0.5

1

Figure S6: Estimated one dimensional component functions for model M1 and confidence bands (twostandard deviations above and below) for even year runs of pink salmon with smoothing parametervector selected by minimization of the GCV criterion (above) and pseudo BIC (below).

15

−3 −2 −1 0 1 2 3

−3

−2

−1

01

23

theoretical quantiles

devi

ance

res

idua

ls

−3 −2 −1 0 1 2

−3

−2

−1

01

23

Resids vs. linear pred.

linear predictor

resi

dual

s

Histogram of residuals

Residuals

Fre

quen

cy

−3 −2 −1 0 1 2 3

020

4060

8010

012

0

−3 −2 −1 0 1 2

−2

02

4

Response vs. Fitted Values

Fitted Values

Res

pons

e

1000 2000 3000 4000

−3

−1

12

3

Location (s)

resi

dual

s

1950 1970 1990

−3

−1

12

3

Brood Year (t)

resi

dual

s

Figure S7: Diagnostics (above) and residuals over time and spatial location (below) for GAM modelM1 of productivity of even year runs of pink salmon with smoothing parameter vector selected byminimization of the pBIC criterion.

16

Parametric CoefficientsParameter Estimate Std. Error t value Pr(>|t|)(Intercept) 1.5 0.075 19 2e-63as.factor(SNIndex)12:X -0.00064 0.00028 -2.3 2e-02as.factor(SNIndex)13:X -0.054 0.012 -4.6 6e-06as.factor(SNIndex)14:X -0.0053 0.0013 -4.1 5e-05as.factor(SNIndex)15:X -0.00054 0.00011 -4.8 2e-06as.factor(SNIndex)16:X -0.00072 0.00025 -2.8 5e-03as.factor(SNIndex)17:X -0.0021 0.00051 -4.2 3e-05as.factor(SNIndex)18:X -2.7e-05 1.8e-05 -1.5 1e-01as.factor(SNIndex)19:X -2e-05 5.1e-05 -0.4 7e-01as.factor(SNIndex)20:X -0.00022 0.00016 -1.4 2e-01as.factor(SNIndex)21:X -0.0084 0.0016 -5.4 1e-07as.factor(SNIndex)22:X -0.033 0.014 -2.3 2e-02as.factor(SNIndex)23:X -0.0028 0.0011 -2.6 1e-02as.factor(SNIndex)24:X 0.00025 0.00066 0.38 7e-01as.factor(SNIndex)25:X -0.011 0.0038 -3 3e-03as.factor(SNIndex)26:X -0.00044 0.0024 -0.19 9e-01as.factor(SNIndex)27:X -0.004 0.0031 -1.3 2e-01as.factor(SNIndex)28:X -0.00054 0.00054 -0.99 3e-01as.factor(SNIndex)29:X -0.00011 0.0009 -0.12 9e-01as.factor(SNIndex)30:X -0.0029 0.00072 -4 6e-05as.factor(SNIndex)31:X -0.00093 0.00094 -0.99 3e-01as.factor(SNIndex)32:X -0.0012 0.00065 -1.8 8e-02as.factor(SNIndex)33:X -0.00014 0.00013 -1.1 3e-01as.factor(SNIndex)34:X -0.0021 0.00046 -4.6 5e-06as.factor(SNIndex)35:X -0.012 0.0024 -4.8 2e-06as.factor(SNIndex)36:X -0.017 0.0055 -3 2e-03as.factor(SNIndex)37:X -0.008 0.0018 -4.6 6e-06as.factor(SNIndex)38:X -0.01 0.0024 -4.4 2e-05as.factor(SNIndex)39:X -0.00056 0.00019 -2.9 4e-03as.factor(SNIndex)40:X -0.011 0.0039 -2.9 4e-03as.factor(SNIndex)41:X -0.00078 0.00033 -2.4 2e-02as.factor(SNIndex)42:X -0.00034 0.0001 -3.3 1e-03as.factor(SNIndex)43:X -0.0077 0.0021 -3.7 3e-04as.factor(SNIndex)44:X -0.0026 0.00071 -3.6 3e-04as.factor(SNIndex)45:X -0.0062 0.0015 -4.2 3e-05as.factor(SNIndex)46:X -0.002 0.00042 -4.7 4e-06Approximate Significance of Smooth TermsFunction edf Ref.df F p-values(asd) 8.4 8.9 3.5 3e-04s(yr) 8.1 8.8 5.2 1e-06Smoothing Parameter Estimates UsedSmooth function lambdas(asd) 5e-05s(yr) 5e-04Model DiagnosticsAIC 1606BIC 5671GCV score 1.048Log-Likelihood -749.8Effective Degrees of Freedom 505.5Scale Parameter Estimate 0.9495Observations 558

Table S5: Model M1 – Even-year Pink Salmon (smoothing parameter selected by GCV)

17

Parametric CoefficientsParameter Estimate Std. Error t value Pr(>|t|)(Intercept) 1.4 0.075 19 2e-61as.factor(SNIndex)12:X -0.00054 0.00021 -2.6 9e-03as.factor(SNIndex)13:X -0.042 0.012 -3.6 3e-04as.factor(SNIndex)14:X -0.0036 0.0012 -3 3e-03as.factor(SNIndex)15:X -0.0004 0.00011 -3.7 2e-04as.factor(SNIndex)16:X -0.00044 0.00022 -2 5e-02as.factor(SNIndex)17:X -0.002 0.00042 -4.7 3e-06as.factor(SNIndex)18:X -3.4e-05 1.5e-05 -2.2 3e-02as.factor(SNIndex)19:X -0.0001 4.3e-05 -2.4 2e-02as.factor(SNIndex)20:X -0.00045 0.00013 -3.4 8e-04as.factor(SNIndex)21:X -0.0089 0.0013 -6.9 1e-11as.factor(SNIndex)22:X -0.039 0.012 -3.3 9e-04as.factor(SNIndex)23:X -0.003 0.00089 -3.4 7e-04as.factor(SNIndex)24:X 7.3e-05 0.00052 0.14 9e-01as.factor(SNIndex)25:X -0.014 0.0032 -4.3 2e-05as.factor(SNIndex)26:X -0.005 0.002 -2.4 2e-02as.factor(SNIndex)27:X -0.011 0.0026 -4 8e-05as.factor(SNIndex)28:X -0.0013 0.0005 -2.7 8e-03as.factor(SNIndex)29:X -0.0014 0.00085 -1.6 1e-01as.factor(SNIndex)30:X -0.0033 0.00072 -4.6 6e-06as.factor(SNIndex)31:X -0.0014 0.0009 -1.6 1e-01as.factor(SNIndex)32:X -0.0014 0.00062 -2.3 2e-02as.factor(SNIndex)33:X -0.00018 0.00012 -1.5 1e-01as.factor(SNIndex)34:X -0.0016 0.00043 -3.8 2e-04as.factor(SNIndex)35:X -0.0072 0.0022 -3.3 1e-03as.factor(SNIndex)36:X -0.0085 0.0052 -1.6 1e-01as.factor(SNIndex)37:X -0.0047 0.0016 -2.9 3e-03as.factor(SNIndex)38:X -0.0058 0.0021 -2.7 7e-03as.factor(SNIndex)39:X -0.00031 0.00018 -1.7 8e-02as.factor(SNIndex)40:X -0.011 0.0037 -3.1 2e-03as.factor(SNIndex)41:X -0.0008 0.00029 -2.7 6e-03as.factor(SNIndex)42:X -0.00037 9.6e-05 -3.9 1e-04as.factor(SNIndex)43:X -0.0061 0.0018 -3.4 7e-04as.factor(SNIndex)44:X -0.0025 0.00067 -3.7 3e-04as.factor(SNIndex)45:X -0.0065 0.0014 -4.7 3e-06as.factor(SNIndex)46:X -0.0021 0.00039 -5.5 5e-08Approximate Significance of Smooth TermsFunction edf Ref.df F p-values(asd) 1 1 0.063 8e-01s(yr) 6.9 7.9 4.4 4e-05Smoothing Parameter Estimates UsedSmooth function lambdas(asd) 1e+01s(yr) 2e-03Model DiagnosticsAIC 1631BIC 5648GCV score 1.091Log-Likelihood -770.4Effective Degrees of Freedom 514.1Scale Parameter Estimate 1.005Observations 558

Table S6: Model M1 – Even-year Pink Salmon (smoothing parameter selected by BIC)

18

Locatio

n (s)

1000

20003000

4000

Brood Year (t) 1950

19601970

19801990

a(s,t)

0

5

10

15

1000 2000 3000 400019

5019

6019

7019

8019

90

Location (s)

Bro

od y

ear

(t)

0

0

2

4

6 8

10

12

14 16

Locatio

n (s)

1000

20003000

4000

Brood Year (t) 1950

19601970

19801990

a(s,t)

0

5

10

15

1000 2000 3000 4000

1950

1960

1970

1980

1990

Location (s)

Bro

od y

ear

(t)

0

0

0

2 4

6 8

10

12 14

Figure S8: Estimated bivariate component functions for model M2 for even year runs of pink salmonwith smoothing parameter vector selected by minimization of the GCV criterion (above) and pseudoBIC (below).

19

−3 −2 −1 0 1 2 3

−3

−2

−1

01

23

theoretical quantiles

devi

ance

res

idua

ls

−3 −2 −1 0 1 2

−3

−2

−1

01

23

Resids vs. linear pred.

linear predictor

resi

dual

s

Histogram of residuals

Residuals

Fre

quen

cy

−3 −2 −1 0 1 2 3

020

4060

8010

014

0

−3 −2 −1 0 1 2

−2

02

4

Response vs. Fitted Values

Fitted Values

Res

pons

e

1000 2000 3000 4000

−3

−1

12

3

Location (s)

resi

dual

s

1950 1970 1990

−3

−1

12

3

Brood Year (t)

resi

dual

s

Figure S9: Diagnostics (above) and residuals over time and spatial location (below) for GAM modelM2 of productivity of even year runs of pink salmon with smoothing parameter vector selected byminimization of the pBIC criterion.

20

Parametric CoefficientsParameter Estimate Std. Error t value Pr(>|t|)(Intercept) 1.4 0.075 19 3e-61as.factor(SNIndex)12:X -0.00063 0.00023 -2.8 6e-03as.factor(SNIndex)13:X -0.042 0.012 -3.6 4e-04as.factor(SNIndex)14:X -0.004 0.0013 -3.1 2e-03as.factor(SNIndex)15:X -0.00043 0.00011 -3.9 1e-04as.factor(SNIndex)16:X -0.00053 0.00022 -2.4 2e-02as.factor(SNIndex)17:X -0.0022 0.00043 -5.2 3e-07as.factor(SNIndex)18:X -2.9e-05 1.6e-05 -1.8 7e-02as.factor(SNIndex)19:X -8.5e-05 4.7e-05 -1.8 7e-02as.factor(SNIndex)20:X -0.00039 0.00014 -2.7 7e-03as.factor(SNIndex)21:X -0.0078 0.0013 -5.9 9e-09as.factor(SNIndex)22:X -0.032 0.012 -2.5 1e-02as.factor(SNIndex)23:X -0.0026 0.00093 -2.8 6e-03as.factor(SNIndex)24:X 0.00038 0.00054 0.7 5e-01as.factor(SNIndex)25:X -0.012 0.0033 -3.7 2e-04as.factor(SNIndex)26:X -0.0038 0.0022 -1.7 8e-02as.factor(SNIndex)27:X -0.0086 0.0028 -3.1 2e-03as.factor(SNIndex)28:X -0.0013 0.00052 -2.5 1e-02as.factor(SNIndex)29:X -0.0013 0.00087 -1.5 1e-01as.factor(SNIndex)30:X -0.0033 0.00072 -4.6 5e-06as.factor(SNIndex)31:X -0.0014 0.0009 -1.6 1e-01as.factor(SNIndex)32:X -0.0015 0.00063 -2.3 2e-02as.factor(SNIndex)33:X -0.00019 0.00012 -1.6 1e-01as.factor(SNIndex)34:X -0.0016 0.00044 -3.7 3e-04as.factor(SNIndex)35:X -0.0074 0.0022 -3.3 1e-03as.factor(SNIndex)36:X -0.008 0.0052 -1.5 1e-01as.factor(SNIndex)37:X -0.0046 0.0016 -2.9 4e-03as.factor(SNIndex)38:X -0.0061 0.0022 -2.8 5e-03as.factor(SNIndex)39:X -0.00036 0.00018 -2 4e-02as.factor(SNIndex)40:X -0.012 0.0037 -3.3 1e-03as.factor(SNIndex)41:X -0.00093 0.0003 -3.1 2e-03as.factor(SNIndex)42:X -0.0004 9.5e-05 -4.2 3e-05as.factor(SNIndex)43:X -0.0068 0.0019 -3.7 3e-04as.factor(SNIndex)44:X -0.0026 0.00073 -3.5 4e-04as.factor(SNIndex)45:X -0.0067 0.0015 -4.6 6e-06as.factor(SNIndex)46:X -0.0021 0.00042 -5.1 5e-07Approximate Significance of Smooth TermsFunction edf Ref.df F p-valuete(asd,yr) 16 18 3.6 6e-07Smoothing Parameter Estimates UsedSmooth function lambdate(asd,yr)1 6e-01te(asd,yr)2 2e-04Model DiagnosticsAIC 1617BIC 5694GCV score 1.068Log-Likelihood -755.1Effective Degrees of Freedom 505.6Scale Parameter Estimate 0.9675Observations 558

Table S7: Model M2 – Even-year Pink Salmon (smoothing parameter selected by GCV)

21

Parametric CoefficientsParameter Estimate Std. Error t value Pr(>|t|)(Intercept) 1.4 0.075 19 1e-61as.factor(SNIndex)12:X -0.00051 0.00021 -2.4 2e-02as.factor(SNIndex)13:X -0.038 0.011 -3.3 1e-03as.factor(SNIndex)14:X -0.0035 0.0012 -2.9 5e-03as.factor(SNIndex)15:X -0.00039 0.00011 -3.7 3e-04as.factor(SNIndex)16:X -0.0005 0.00022 -2.3 2e-02as.factor(SNIndex)17:X -0.0023 0.00042 -5.3 1e-07as.factor(SNIndex)18:X -3e-05 1.6e-05 -1.9 6e-02as.factor(SNIndex)19:X -9.5e-05 4.3e-05 -2.2 3e-02as.factor(SNIndex)20:X -0.00042 0.00013 -3.1 2e-03as.factor(SNIndex)21:X -0.0086 0.0013 -6.7 4e-11as.factor(SNIndex)22:X -0.037 0.012 -3.2 2e-03as.factor(SNIndex)23:X -0.003 0.00089 -3.4 8e-04as.factor(SNIndex)24:X 0.0001 0.00052 0.2 8e-01as.factor(SNIndex)25:X -0.013 0.0032 -4.3 2e-05as.factor(SNIndex)26:X -0.0047 0.002 -2.3 2e-02as.factor(SNIndex)27:X -0.01 0.0026 -3.8 2e-04as.factor(SNIndex)28:X -0.0014 0.0005 -2.7 7e-03as.factor(SNIndex)29:X -0.0014 0.00085 -1.7 9e-02as.factor(SNIndex)30:X -0.0034 0.00072 -4.7 4e-06as.factor(SNIndex)31:X -0.0015 0.00089 -1.6 1e-01as.factor(SNIndex)32:X -0.0015 0.00062 -2.4 2e-02as.factor(SNIndex)33:X -0.00019 0.00012 -1.6 1e-01as.factor(SNIndex)34:X -0.0016 0.00043 -3.6 3e-04as.factor(SNIndex)35:X -0.0069 0.0022 -3.2 2e-03as.factor(SNIndex)36:X -0.0073 0.0052 -1.4 2e-01as.factor(SNIndex)37:X -0.0041 0.0016 -2.6 9e-03as.factor(SNIndex)38:X -0.0054 0.0021 -2.6 1e-02as.factor(SNIndex)39:X -0.00029 0.00017 -1.7 1e-01as.factor(SNIndex)40:X -0.011 0.0036 -3 3e-03as.factor(SNIndex)41:X -0.0008 0.00029 -2.7 7e-03as.factor(SNIndex)42:X -0.00037 9.5e-05 -3.9 1e-04as.factor(SNIndex)43:X -0.0069 0.0018 -3.8 1e-04as.factor(SNIndex)44:X -0.0027 0.00068 -4 6e-05as.factor(SNIndex)45:X -0.0069 0.0014 -5 7e-07as.factor(SNIndex)46:X -0.0023 0.0004 -5.8 1e-08Approximate Significance of Smooth TermsFunction edf Ref.df F p-valuete(asd,yr) 9.4 9.8 5.2 4e-07Smoothing Parameter Estimates UsedSmooth function lambdate(asd,yr)1 1e+02te(asd,yr)2 1e-03Model DiagnosticsAIC 1623BIC 5669GCV score 1.077Log-Likelihood -765.2Effective Degrees of Freedom 512.6Scale Parameter Estimate 0.9895Observations 558

Table S8: Model M2 – Even-year Pink Salmon (smoothing parameter selected by BIC)

22

0 1000 3000

Location (s)

a 1(s

)−

1−

0.5

00.

51

1960 1970 1980 1990

Brood Year (t)

a 2(t)

0 1000 3000

Location (s)

a 1(s

)−

1−

0.5

00.

51

1960 1970 1980 1990

Brood Year (t)

a 2(t)

−1

−0.

50

0.5

1

Figure S10: Estimated one dimensional component functions for model M1 and confidence bands (twostandard deviations above and below) for chum salmon with smoothing parameter vector selectedby minimization of the GCV criterion (above) and pseudo BIC (below).

23

−2 −1 0 1 2

−2

−1

01

23

theoretical quantiles

devi

ance

res

idua

ls

−2 −1 0 1

−2

−1

01

23

Resids vs. linear pred.

linear predictor

resi

dual

s

Histogram of residuals

Residuals

Fre

quen

cy

−2 −1 0 1 2 3

050

100

150

200

250

−2 −1 0 1

−2

−1

01

23

4

Response vs. Fitted Values

Fitted Values

Res

pons

e

0 1000 3000

−2

−1

01

23

Location (s)

resi

dual

s

1960 1970 1980 1990

−2

−1

01

23

Brood Year (t)

resi

dual

s

Figure S11: Diagnostics (above) and residuals over time and spatial location (below) for GAM modelM1 of productivity of chum salmon with smoothing parameter vector selected by minimization ofthe pBIC criterion.

24

Parametric CoefficientsParameter Estimate Std. Error t value Pr(>|t|)(Intercept) 1.4 0.041 36 3e-178as.factor(SNIndex)1:X -0.0022 0.002 -1.1 3e-01as.factor(SNIndex)2:X -0.027 0.0094 -2.9 4e-03as.factor(SNIndex)3:X -0.026 0.0052 -5 7e-07as.factor(SNIndex)4:X -0.0079 0.0022 -3.6 3e-04as.factor(SNIndex)5:X -0.024 0.0046 -5.3 2e-07as.factor(SNIndex)6:X -0.0024 0.0036 -0.66 5e-01as.factor(SNIndex)7:X -0.0079 0.0018 -4.3 2e-05as.factor(SNIndex)8:X -0.032 0.0068 -4.8 2e-06as.factor(SNIndex)9:X -0.018 0.0052 -3.4 7e-04as.factor(SNIndex)10:X -0.0013 0.00028 -4.8 2e-06as.factor(SNIndex)11:X -0.0015 0.00023 -6.3 3e-10as.factor(SNIndex)12:X -0.033 0.0041 -8.2 8e-16as.factor(SNIndex)13:X -0.019 0.0027 -7.2 1e-12as.factor(SNIndex)14:X -0.0027 0.00087 -3.1 2e-03as.factor(SNIndex)15:X -0.0057 0.00088 -6.5 1e-10as.factor(SNIndex)16:X -0.053 0.0092 -5.8 1e-08as.factor(SNIndex)17:X -0.078 0.016 -4.9 9e-07as.factor(SNIndex)18:X -0.0025 0.00084 -2.9 3e-03as.factor(SNIndex)19:X -0.0011 0.00015 -7.1 2e-12as.factor(SNIndex)20:X -0.045 0.0069 -6.5 1e-10as.factor(SNIndex)21:X -0.13 0.019 -6.6 6e-11as.factor(SNIndex)22:X -0.018 0.0022 -8.1 2e-15as.factor(SNIndex)23:X -0.0015 0.00027 -5.5 5e-08as.factor(SNIndex)24:X -0.013 0.0014 -9.2 2e-19as.factor(SNIndex)25:X -0.033 0.0055 -5.9 4e-09as.factor(SNIndex)26:X -0.014 0.0027 -5.2 3e-07as.factor(SNIndex)27:X -0.0082 0.002 -4.1 4e-05as.factor(SNIndex)28:X -0.0015 0.00058 -2.6 1e-02as.factor(SNIndex)29:X -0.0049 0.00083 -5.9 4e-09as.factor(SNIndex)30:X -0.014 0.0053 -2.7 7e-03as.factor(SNIndex)31:X -0.0033 0.00092 -3.6 3e-04as.factor(SNIndex)33:X -0.0018 0.00067 -2.7 7e-03as.factor(SNIndex)34:X -0.0039 0.00097 -4.1 5e-05as.factor(SNIndex)35:X -0.00067 0.00033 -2.1 4e-02as.factor(SNIndex)36:X -0.00083 0.00027 -3.1 2e-03as.factor(SNIndex)37:X -0.0031 0.00063 -4.8 2e-06as.factor(SNIndex)38:X -0.024 0.0039 -6.2 9e-10as.factor(SNIndex)39:X -0.019 0.0029 -6.4 2e-10as.factor(SNIndex)40:X -0.0099 0.0025 -3.9 9e-05Approximate Significance of Smooth TermsFunction edf Ref.df F p-values(asd) 8.5 8.9 3.7 1e-04s(yr) 8.9 9 17 1e-26Smoothing Parameter Estimates UsedSmooth function lambdas(asd) 2e-05s(yr) 3e-05Model DiagnosticsAIC 2204BIC 9969GCV score 0.4901Log-Likelihood -1044Effective Degrees of Freedom 980.6Scale Parameter Estimate 0.463Observations 1038

Table S9: Model M1 – Chum Salmon (smoothing parameter selected by GCV)

25

Parametric CoefficientsParameter Estimate Std. Error t value Pr(>|t|)(Intercept) 1.5 0.041 35 2e-178as.factor(SNIndex)1:X -0.0044 0.0018 -2.4 2e-02as.factor(SNIndex)2:X -0.037 0.009 -4.1 4e-05as.factor(SNIndex)3:X -0.032 0.0049 -6.6 7e-11as.factor(SNIndex)4:X -0.0097 0.0022 -4.5 7e-06as.factor(SNIndex)5:X -0.029 0.0045 -6.3 4e-10as.factor(SNIndex)6:X -0.0055 0.0035 -1.6 1e-01as.factor(SNIndex)7:X -0.0093 0.0018 -5.2 2e-07as.factor(SNIndex)8:X -0.036 0.0067 -5.4 1e-07as.factor(SNIndex)9:X -0.021 0.0051 -4.1 5e-05as.factor(SNIndex)10:X -0.0014 0.00026 -5.5 6e-08as.factor(SNIndex)11:X -0.0013 0.00017 -7.9 6e-15as.factor(SNIndex)12:X -0.031 0.0036 -8.7 9e-18as.factor(SNIndex)13:X -0.018 0.0025 -7.5 2e-13as.factor(SNIndex)14:X -0.0023 0.00075 -3 2e-03as.factor(SNIndex)15:X -0.0053 0.00064 -8.2 6e-16as.factor(SNIndex)16:X -0.039 0.0061 -6.3 4e-10as.factor(SNIndex)17:X -0.06 0.012 -5 6e-07as.factor(SNIndex)18:X -0.0024 0.00062 -3.8 1e-04as.factor(SNIndex)19:X -0.00098 0.00012 -8.1 2e-15as.factor(SNIndex)20:X -0.04 0.0056 -7.1 2e-12as.factor(SNIndex)21:X -0.11 0.016 -6.8 1e-11as.factor(SNIndex)22:X -0.014 0.0019 -7.5 2e-13as.factor(SNIndex)23:X -0.00099 0.00023 -4.3 2e-05as.factor(SNIndex)24:X -0.011 0.0012 -8.8 5e-18as.factor(SNIndex)25:X -0.027 0.0049 -5.5 5e-08as.factor(SNIndex)26:X -0.012 0.0025 -5 8e-07as.factor(SNIndex)27:X -0.0077 0.0019 -4 8e-05as.factor(SNIndex)28:X -0.0018 0.00051 -3.5 5e-04as.factor(SNIndex)29:X -0.0056 0.00071 -7.9 8e-15as.factor(SNIndex)30:X -0.019 0.0044 -4.3 2e-05as.factor(SNIndex)31:X -0.0041 0.00076 -5.4 7e-08as.factor(SNIndex)33:X -0.0018 0.00049 -3.7 2e-04as.factor(SNIndex)34:X -0.0042 0.00069 -6.1 2e-09as.factor(SNIndex)35:X -0.0015 0.00027 -5.4 7e-08as.factor(SNIndex)36:X -0.0015 0.00022 -6.6 6e-11as.factor(SNIndex)37:X -0.0046 0.00053 -8.8 5e-18as.factor(SNIndex)38:X -0.026 0.0035 -7.4 2e-13as.factor(SNIndex)39:X -0.021 0.0025 -8.2 5e-16as.factor(SNIndex)40:X -0.005 0.0015 -3.3 1e-03Approximate Significance of Smooth TermsFunction edf Ref.df F p-values(asd) 1 1 0.049 8e-01s(yr) 8.5 8.9 16 5e-24Smoothing Parameter Estimates UsedSmooth function lambdas(asd) 1e+01s(yr) 2e-04Model DiagnosticsAIC 2229BIC 9930GCV score 0.5013Log-Likelihood -1064Effective Degrees of Freedom 988.4Scale Parameter Estimate 0.4774Observations 1038

Table S10: Model M1 – Chum Salmon (smoothing parameter selected by BIC)

26

Locatio

n (s)

0

10002000

30004000

Brood Year (t)1960

19701980

1990

a(s,t)

−2

−1

0

0 1000 2000 3000 400019

6019

7019

8019

90

Location (s)

Bro

od y

ear

(t)

−2.5

−2

−2

−2

−1.5

−1.5

−1.5

−1.5

−1

−1

−1

−0.5

0

0.5

Locatio

n (s)

0

10002000

30004000

Brood Year (t)1960

19701980

1990

a(s,t)

−2.0

−1.5

−1.0

0 1000 2000 3000 4000

1960

1970

1980

1990

Location (s)

Bro

od y

ear

(t)

−2

−2

−1.8

−1.8

−1.8

−1.6

−1.6

−1.6

−1.6

−1.4

−1.4

−1.4

−1.4

−1.2

−1.2

−1.2

−1.2 −1

−1

−1

−0.8

−0.6

Figure S12: Estimated bivariate component functions for model M2 for Chun salmon with smoothingparameter vector selected by minimization of the GCV criterion (above) and pseudo BIC (below).

27

−2 −1 0 1 2

−2

−1

01

23

theoretical quantiles

devi

ance

res

idua

ls

−2 −1 0 1 2

−2

−1

01

23

Resids vs. linear pred.

linear predictor

resi

dual

s

Histogram of residuals

Residuals

Fre

quen

cy

−2 −1 0 1 2 3

050

100

150

200

250

300

−2 −1 0 1 2

−2

−1

01

23

4

Response vs. Fitted Values

Fitted Values

Res

pons

e

0 1000 3000

−2

−1

01

23

Location (s)

resi

dual

s

1960 1970 1980 1990

−2

−1

01

23

Brood Year (t)

resi

dual

s

Figure S13: Diagnostics (above) and residuals over time and spatial location (below) for GAM modelM2 of productivity of chum salmon with smoothing parameter vector selected by minimization ofthe pBIC criterion.

28

Parametric CoefficientsParameter Estimate Std. Error t value Pr(>|t|)(Intercept) 1.5 0.041 37 1e-186as.factor(SNIndex)1:X -0.0038 0.002 -1.9 6e-02as.factor(SNIndex)2:X -0.033 0.0094 -3.5 5e-04as.factor(SNIndex)3:X -0.031 0.0052 -5.9 4e-09as.factor(SNIndex)4:X -0.0098 0.0022 -4.4 1e-05as.factor(SNIndex)5:X -0.028 0.0047 -5.9 4e-09as.factor(SNIndex)6:X -0.0034 0.0037 -0.93 4e-01as.factor(SNIndex)7:X -0.0087 0.0018 -4.7 2e-06as.factor(SNIndex)8:X -0.037 0.0069 -5.4 1e-07as.factor(SNIndex)9:X -0.02 0.0052 -3.8 1e-04as.factor(SNIndex)10:X -0.0014 0.00026 -5.3 2e-07as.factor(SNIndex)11:X -0.0014 0.00018 -8 4e-15as.factor(SNIndex)12:X -0.036 0.004 -8.9 4e-18as.factor(SNIndex)13:X -0.022 0.0027 -8 5e-15as.factor(SNIndex)14:X -0.0034 0.00087 -4 8e-05as.factor(SNIndex)15:X -0.006 0.00076 -7.9 1e-14as.factor(SNIndex)16:X -0.054 0.0068 -8 5e-15as.factor(SNIndex)17:X -0.079 0.013 -5.9 4e-09as.factor(SNIndex)18:X -0.003 0.00075 -4.1 5e-05as.factor(SNIndex)19:X -0.0011 0.00014 -8.1 2e-15as.factor(SNIndex)20:X -0.049 0.0065 -7.6 8e-14as.factor(SNIndex)21:X -0.13 0.018 -7.4 3e-13as.factor(SNIndex)22:X -0.016 0.0021 -7.8 1e-14as.factor(SNIndex)23:X -0.0012 0.00025 -5 7e-07as.factor(SNIndex)24:X -0.012 0.0013 -9.6 8e-21as.factor(SNIndex)25:X -0.032 0.0052 -6.3 5e-10as.factor(SNIndex)26:X -0.014 0.0025 -5.6 2e-08as.factor(SNIndex)27:X -0.0087 0.002 -4.4 1e-05as.factor(SNIndex)28:X -0.002 0.00055 -3.7 2e-04as.factor(SNIndex)29:X -0.0061 0.00078 -7.8 1e-14as.factor(SNIndex)30:X -0.022 0.0049 -4.5 8e-06as.factor(SNIndex)31:X -0.0047 0.00085 -5.5 4e-08as.factor(SNIndex)33:X -0.0021 0.00052 -4 8e-05as.factor(SNIndex)34:X -0.0045 0.00073 -6.1 1e-09as.factor(SNIndex)35:X -0.0015 0.00029 -5.2 3e-07as.factor(SNIndex)36:X -0.0013 0.00024 -5.6 3e-08as.factor(SNIndex)37:X -0.0047 0.00056 -8.4 2e-16as.factor(SNIndex)38:X -0.022 0.0039 -5.5 4e-08as.factor(SNIndex)39:X -0.021 0.0029 -7.2 1e-12as.factor(SNIndex)40:X -0.0075 0.0021 -3.6 3e-04Approximate Significance of Smooth TermsFunction edf Ref.df F p-valuete(asd,yr) 23 24 7.5 2e-23Smoothing Parameter Estimates UsedSmooth function lambdate(asd,yr)1 3e-03te(asd,yr)2 3e-02Model DiagnosticsAIC 2220BIC 9997GCV score 0.498Log-Likelihood -1046Effective Degrees of Freedom 975.3Scale Parameter Estimate 0.4679Observations 1038

Table S11: Model M2 – Chum Salmon (smoothing parameter selected by GCV)

29

Parametric CoefficientsParameter Estimate Std. Error t value Pr(>|t|)(Intercept) 1.5 0.041 37 3e-185as.factor(SNIndex)1:X -0.0039 0.0019 -2 4e-02as.factor(SNIndex)2:X -0.034 0.0093 -3.6 3e-04as.factor(SNIndex)3:X -0.031 0.0051 -6 2e-09as.factor(SNIndex)4:X -0.0098 0.0022 -4.4 1e-05as.factor(SNIndex)5:X -0.028 0.0047 -6 3e-09as.factor(SNIndex)6:X -0.0037 0.0037 -1 3e-01as.factor(SNIndex)7:X -0.0087 0.0018 -4.8 2e-06as.factor(SNIndex)8:X -0.037 0.0069 -5.3 1e-07as.factor(SNIndex)9:X -0.02 0.0052 -3.8 1e-04as.factor(SNIndex)10:X -0.0014 0.00026 -5.2 2e-07as.factor(SNIndex)11:X -0.0014 0.00017 -8.1 2e-15as.factor(SNIndex)12:X -0.035 0.0038 -9.1 6e-19as.factor(SNIndex)13:X -0.021 0.0026 -8 4e-15as.factor(SNIndex)14:X -0.0032 0.00082 -3.9 8e-05as.factor(SNIndex)15:X -0.0058 0.00072 -8.1 2e-15as.factor(SNIndex)16:X -0.053 0.0067 -8 5e-15as.factor(SNIndex)17:X -0.075 0.013 -5.8 1e-08as.factor(SNIndex)18:X -0.0031 0.0007 -4.4 1e-05as.factor(SNIndex)19:X -0.0011 0.00014 -8.4 2e-16as.factor(SNIndex)20:X -0.047 0.0062 -7.6 6e-14as.factor(SNIndex)21:X -0.13 0.017 -7.3 6e-13as.factor(SNIndex)22:X -0.016 0.0021 -7.9 1e-14as.factor(SNIndex)23:X -0.0012 0.00024 -4.9 1e-06as.factor(SNIndex)24:X -0.012 0.0013 -9.4 3e-20as.factor(SNIndex)25:X -0.032 0.0051 -6.2 1e-09as.factor(SNIndex)26:X -0.014 0.0025 -5.7 2e-08as.factor(SNIndex)27:X -0.0084 0.002 -4.3 2e-05as.factor(SNIndex)28:X -0.002 0.00054 -3.8 1e-04as.factor(SNIndex)29:X -0.0061 0.00075 -8.1 2e-15as.factor(SNIndex)30:X -0.022 0.0047 -4.7 3e-06as.factor(SNIndex)31:X -0.0047 0.00081 -5.8 7e-09as.factor(SNIndex)33:X -0.002 0.00051 -3.9 9e-05as.factor(SNIndex)34:X -0.0044 0.00072 -6.2 9e-10as.factor(SNIndex)35:X -0.0015 0.00029 -5.2 3e-07as.factor(SNIndex)36:X -0.0013 0.00024 -5.6 3e-08as.factor(SNIndex)37:X -0.0047 0.00056 -8.4 2e-16as.factor(SNIndex)38:X -0.022 0.0039 -5.7 1e-08as.factor(SNIndex)39:X -0.021 0.0029 -7.2 1e-12as.factor(SNIndex)40:X -0.0066 0.002 -3.3 1e-03Approximate Significance of Smooth TermsFunction edf Ref.df F p-valuete(asd,yr) 17 20 7.1 1e-18Smoothing Parameter Estimates UsedSmooth function lambdate(asd,yr)1 2e-01te(asd,yr)2 1e-01Model DiagnosticsAIC 2228BIC 9962GCV score 0.5014Log-Likelihood -1056Effective Degrees of Freedom 981.2Scale Parameter Estimate 0.474Observations 1038

Table S12: Model M2 – Chum Salmon (smoothing parameter selected by BIC)

30

0 1000 2000 3000 4000

Location (s)

a 1(s

)−

1−

0.5

00.

51

1950 1970 1990

Brood Year (t)

a 2(t)

0 1000 2000 3000 4000

Location (s)

a 1(s

)−

1−

0.5

00.

51

1950 1970 1990

Brood Year (t)

a 2(t)

−1

−0.

50

0.5

1

Figure S14: Estimated one dimensional component functions for model M1 and confidence bands(two standard deviations above and below) for sockeye salmon with smoothing parameter vectorselected by minimization of the GCV criterion (above) and pseudo BIC (below).

31

−2 −1 0 1 2

−3

−2

−1

01

23

theoretical quantiles

devi

ance

res

idua

ls

−1 0 1 2

−3

−2

−1

01

23

Resids vs. linear pred.

linear predictor

resi

dual

s

Histogram of residuals

Residuals

Fre

quen

cy

−4 −2 0 2

010

020

030

040

0

−1 0 1 2

−2

02

4

Response vs. Fitted Values

Fitted Values

Res

pons

e

0 1000 2000 3000 4000

−3

−1

12

3

Location (s)

resi

dual

s

1950 1970 1990

−3

−1

12

3

Brood Year (t)

resi

dual

s

Figure S15: Diagnostics (above) and residuals over time and spatial location (below) for GAM modelM1 of productivity of sockeye salmon with smoothing parameter vector selected by minimization ofthe pBIC criterion.

32

Parametric CoefficientsParameter Estimate Std. Error t value Pr(>|t|)(Intercept) 1.6 0.037 44 8e-266as.factor(SNIndex)1:X -0.0016 0.00098 -1.7 1e-01as.factor(SNIndex)2:X -0.0032 0.00078 -4.1 4e-05as.factor(SNIndex)3:X -0.039 0.0085 -4.5 7e-06as.factor(SNIndex)4:X -0.015 0.0062 -2.4 2e-02as.factor(SNIndex)5:X -0.018 0.0066 -2.7 7e-03as.factor(SNIndex)6:X -0.038 0.0053 -7.2 9e-13as.factor(SNIndex)7:X -0.049 0.012 -4.1 4e-05as.factor(SNIndex)8:X -0.0097 0.0021 -4.7 3e-06as.factor(SNIndex)9:X -0.0012 0.00029 -4.2 3e-05as.factor(SNIndex)10:X -0.00012 0.00027 -0.45 7e-01as.factor(SNIndex)11:X -0.00086 0.00046 -1.9 6e-02as.factor(SNIndex)12:X -0.0034 0.00099 -3.4 6e-04as.factor(SNIndex)13:X -0.0048 0.0012 -4 6e-05as.factor(SNIndex)14:X -0.00053 0.00012 -4.4 1e-05as.factor(SNIndex)15:X -0.029 0.007 -4.1 4e-05as.factor(SNIndex)16:X -0.022 0.014 -1.6 1e-01as.factor(SNIndex)17:X -0.0048 0.002 -2.4 2e-02as.factor(SNIndex)18:X 0.00049 0.0034 0.15 9e-01as.factor(SNIndex)19:X -4.5e-05 0.00024 -0.19 8e-01as.factor(SNIndex)20:X -0.0023 0.0016 -1.4 1e-01as.factor(SNIndex)21:X -0.0017 0.00059 -2.9 4e-03as.factor(SNIndex)22:X -0.0018 0.0007 -2.5 1e-02as.factor(SNIndex)23:X -0.0097 0.003 -3.2 1e-03as.factor(SNIndex)24:X -0.00077 0.001 -0.75 5e-01as.factor(SNIndex)25:X -0.0036 0.00087 -4.2 3e-05as.factor(SNIndex)26:X -0.0015 0.00061 -2.5 1e-02as.factor(SNIndex)27:X -0.00013 0.00016 -0.85 4e-01as.factor(SNIndex)28:X 0.00031 0.00013 2.4 2e-02as.factor(SNIndex)29:X -0.0013 0.00029 -4.6 4e-06as.factor(SNIndex)30:X -0.00024 0.0001 -2.3 2e-02as.factor(SNIndex)31:X -6.8e-05 1.7e-05 -4 7e-05as.factor(SNIndex)32:X -0.0017 0.00043 -4 6e-05as.factor(SNIndex)33:X -0.0008 0.00024 -3.3 9e-04as.factor(SNIndex)34:X -0.00038 0.00012 -3.2 2e-03as.factor(SNIndex)35:X -0.0024 0.00092 -2.7 8e-03as.factor(SNIndex)36:X -0.00091 0.00088 -1 3e-01as.factor(SNIndex)37:X -0.00016 0.00032 -0.48 6e-01Approximate Significance of Smooth TermsFunction edf Ref.df F p-values(asd) 8.8 9 14 1e-22s(yr) 8.8 9 6.4 5e-09Smoothing Parameter Estimates UsedSmooth function lambdas(asd) 5e-06s(yr) 5e-05Model DiagnosticsAIC 3533BIC 1.475e+04GCV score 0.6608Log-Likelihood -1710Effective Degrees of Freedom 1402Scale Parameter Estimate 0.6356Observations 1458

Table S13: Model M1 – Sockeye Salmon (smoothing parameter selected by GCV)

33

Parametric CoefficientsParameter Estimate Std. Error t value Pr(>|t|)(Intercept) 1.7 0.034 48 1e-300as.factor(SNIndex)1:X -0.0057 0.0006 -9.6 3e-21as.factor(SNIndex)2:X -0.003 0.0008 -3.8 2e-04as.factor(SNIndex)3:X -0.035 0.0087 -4.1 5e-05as.factor(SNIndex)4:X -0.012 0.0063 -1.9 6e-02as.factor(SNIndex)5:X -0.015 0.0068 -2.3 2e-02as.factor(SNIndex)6:X -0.035 0.0054 -6.5 9e-11as.factor(SNIndex)7:X -0.044 0.012 -3.6 4e-04as.factor(SNIndex)8:X -0.009 0.0021 -4.3 2e-05as.factor(SNIndex)9:X -0.0011 0.0003 -3.7 2e-04as.factor(SNIndex)10:X -8.2e-05 0.00027 -0.3 8e-01as.factor(SNIndex)11:X -0.00082 0.00047 -1.8 8e-02as.factor(SNIndex)12:X -0.0029 0.001 -2.8 5e-03as.factor(SNIndex)13:X -0.0043 0.0012 -3.6 4e-04as.factor(SNIndex)14:X -0.0005 0.00012 -4.1 5e-05as.factor(SNIndex)15:X -0.026 0.0071 -3.7 2e-04as.factor(SNIndex)16:X -0.018 0.014 -1.3 2e-01as.factor(SNIndex)17:X -0.0042 0.002 -2.1 4e-02as.factor(SNIndex)18:X -0.009 0.0011 -8.1 2e-15as.factor(SNIndex)19:X -0.00084 0.00013 -6.5 8e-11as.factor(SNIndex)20:X -0.0035 0.00082 -4.2 3e-05as.factor(SNIndex)21:X -0.0012 0.00034 -3.5 4e-04as.factor(SNIndex)22:X -0.0012 0.0004 -3 3e-03as.factor(SNIndex)23:X -0.013 0.0026 -5 6e-07as.factor(SNIndex)24:X -0.0019 0.00088 -2.2 3e-02as.factor(SNIndex)25:X -0.0046 0.00077 -6 3e-09as.factor(SNIndex)26:X -0.0022 0.00052 -4.3 2e-05as.factor(SNIndex)27:X -0.00025 0.00012 -2.1 4e-02as.factor(SNIndex)28:X 0.00025 0.00012 2.1 3e-02as.factor(SNIndex)29:X -0.0013 0.00029 -4.5 7e-06as.factor(SNIndex)30:X -0.00022 0.0001 -2.1 3e-02as.factor(SNIndex)31:X -6.3e-05 1.7e-05 -3.7 2e-04as.factor(SNIndex)32:X -0.0016 0.00043 -3.9 1e-04as.factor(SNIndex)33:X -0.00077 0.00024 -3.2 2e-03as.factor(SNIndex)34:X -0.00033 0.00012 -2.8 5e-03as.factor(SNIndex)35:X -0.0015 0.00072 -2.1 4e-02as.factor(SNIndex)36:X -0.001 0.00031 -3.4 7e-04as.factor(SNIndex)37:X 4e-05 0.00016 0.25 8e-01Approximate Significance of Smooth TermsFunction edf Ref.df F p-values(asd) 1 1.1 74 3e-18s(yr) 4.8 5.8 4.4 2e-04Smoothing Parameter Estimates UsedSmooth function lambdas(asd) 1e+01s(yr) 1e-02Model DiagnosticsAIC 3591BIC 1.471e+04GCV score 0.6871Log-Likelihood -1751Effective Degrees of Freedom 1414Scale Parameter Estimate 0.6665Observations 1458

Table S14: Model M1 – Sockeye Salmon (smoothing parameter selected by BIC)

34

Locatio

n (s)

0

10002000

30004000

Brood Year (t)1950

19601970

19801990

a(s,t)

0.0

0.5

1.0

1.5

0 1000 2000 3000 400019

5019

6019

7019

8019

90

Location (s)

Bro

od y

ear

(t)

0

0.2

0.4

0.4

0.6

0.6

0.6

0.8

0.8

0.8

1

1

1

1.2

1.2

1.4 1.4

1.6

1.6

1.8

Locatio

n (s)

0

10002000

30004000

Brood Year (t)1950

19601970

19801990

a(s,t)

0.6

0.8

1.0

1.2

1.4

1.6

0 1000 2000 3000 4000

1950

1960

1970

1980

1990

Location (s)

Bro

od y

ear

(t)

0.6

0.7

0.8

0.9

0.9

1

1

1.1

1.1

1.2

1.2

1.3

1.4

1.5

1.6

1.7

1.7

Figure S16: Estimated bivariate component functions for model M2 for sockeye salmon with smooth-ing parameter vector selected by minimization of the GCV criterion (above) and pseudo BIC (below).

35

−2 −1 0 1 2

−3

−2

−1

01

23

theoretical quantiles

devi

ance

res

idua

ls

−1.5 −1.0 −0.5 0.0 0.5 1.0 1.5 2.0

−3

−2

−1

01

23

Resids vs. linear pred.

linear predictor

resi

dual

s

Histogram of residuals

Residuals

Fre

quen

cy

−4 −2 0 2

010

020

030

040

0

−1.5 −1.0 −0.5 0.0 0.5 1.0 1.5 2.0

−2

02

4

Response vs. Fitted Values

Fitted Values

Res

pons

e

0 1000 2000 3000 4000

−3

−1

12

3

Location (s)

resi

dual

s

1950 1970 1990

−3

−1

12

3

Brood Year (t)

resi

dual

s

Figure S17: Diagnostics (above) and residuals over time and spatial location (below) for GAM modelM2 of productivity of sockeye salmon with smoothing parameter vector selected by minimization ofthe pBIC criterion.

36

Parametric CoefficientsParameter Estimate Std. Error t value Pr(>|t|)(Intercept) 1.7 0.037 46 5e-282as.factor(SNIndex)1:X -0.0058 0.00062 -9.3 4e-20as.factor(SNIndex)2:X -0.0028 0.00079 -3.5 4e-04as.factor(SNIndex)3:X -0.036 0.0086 -4.3 2e-05as.factor(SNIndex)4:X -0.0095 0.0062 -1.5 1e-01as.factor(SNIndex)5:X -0.013 0.0066 -2 4e-02as.factor(SNIndex)6:X -0.035 0.0053 -6.6 7e-11as.factor(SNIndex)7:X -0.044 0.012 -3.6 3e-04as.factor(SNIndex)8:X -0.0083 0.0021 -4 6e-05as.factor(SNIndex)9:X -0.001 0.00029 -3.6 4e-04as.factor(SNIndex)10:X 2.7e-05 0.00027 0.1 9e-01as.factor(SNIndex)11:X -0.00074 0.00046 -1.6 1e-01as.factor(SNIndex)12:X -0.0026 0.001 -2.6 9e-03as.factor(SNIndex)13:X -0.0037 0.0012 -3.1 2e-03as.factor(SNIndex)14:X -0.00049 0.00012 -4.1 5e-05as.factor(SNIndex)15:X -0.029 0.007 -4.1 5e-05as.factor(SNIndex)16:X -0.014 0.014 -1 3e-01as.factor(SNIndex)17:X -0.0035 0.002 -1.7 8e-02as.factor(SNIndex)18:X -0.0087 0.0013 -6.4 2e-10as.factor(SNIndex)19:X -0.00074 0.00024 -3.1 2e-03as.factor(SNIndex)20:X -0.003 0.0014 -2.2 3e-02as.factor(SNIndex)21:X -0.0015 0.00044 -3.4 7e-04as.factor(SNIndex)22:X -0.0013 0.00052 -2.6 1e-02as.factor(SNIndex)23:X -0.015 0.0031 -4.8 2e-06as.factor(SNIndex)24:X -0.0026 0.0011 -2.5 1e-02as.factor(SNIndex)25:X -0.0051 0.00091 -5.6 3e-08as.factor(SNIndex)26:X -0.0026 0.00062 -4.2 3e-05as.factor(SNIndex)27:X -0.00043 0.00014 -3 2e-03as.factor(SNIndex)28:X 0.00013 0.00012 1 3e-01as.factor(SNIndex)29:X -0.0015 0.00029 -5.3 1e-07as.factor(SNIndex)30:X -0.00031 0.0001 -3 3e-03as.factor(SNIndex)31:X -7.4e-05 1.7e-05 -4.4 1e-05as.factor(SNIndex)32:X -0.0018 0.00043 -4.3 2e-05as.factor(SNIndex)33:X -0.0009 0.00024 -3.7 2e-04as.factor(SNIndex)34:X -0.00044 0.00012 -3.6 3e-04as.factor(SNIndex)35:X -0.002 0.00085 -2.3 2e-02as.factor(SNIndex)36:X -0.00087 0.00047 -1.8 7e-02as.factor(SNIndex)37:X 2.9e-06 0.00021 0.014 1e+00Approximate Significance of Smooth TermsFunction edf Ref.df F p-valuete(asd,yr) 17 20 9.1 2e-26Smoothing Parameter Estimates UsedSmooth function lambdate(asd,yr)1 9e-02te(asd,yr)2 8e-02Model DiagnosticsAIC 3543BIC 1.472e+04GCV score 0.6652Log-Likelihood -1715Effective Degrees of Freedom 1403Scale Parameter Estimate 0.6399Observations 1458

Table S15: Model M2 – Sockeye Salmon (smoothing parameter selected by GCV)

37

Parametric CoefficientsParameter Estimate Std. Error t value Pr(>|t|)(Intercept) 1.7 0.035 48 9e-296as.factor(SNIndex)1:X -0.0057 0.0006 -9.5 6e-21as.factor(SNIndex)2:X -0.0028 0.00079 -3.5 5e-04as.factor(SNIndex)3:X -0.036 0.0086 -4.2 3e-05as.factor(SNIndex)4:X -0.0096 0.0063 -1.5 1e-01as.factor(SNIndex)5:X -0.013 0.0067 -2 5e-02as.factor(SNIndex)6:X -0.034 0.0053 -6.5 1e-10as.factor(SNIndex)7:X -0.044 0.012 -3.6 3e-04as.factor(SNIndex)8:X -0.0083 0.0021 -4 7e-05as.factor(SNIndex)9:X -0.001 0.00029 -3.5 4e-04as.factor(SNIndex)10:X 2.5e-05 0.00027 0.094 9e-01as.factor(SNIndex)11:X -0.00074 0.00046 -1.6 1e-01as.factor(SNIndex)12:X -0.0026 0.001 -2.6 1e-02as.factor(SNIndex)13:X -0.0037 0.0012 -3.1 2e-03as.factor(SNIndex)14:X -0.00049 0.00012 -4 6e-05as.factor(SNIndex)15:X -0.028 0.007 -4 6e-05as.factor(SNIndex)16:X -0.014 0.014 -0.99 3e-01as.factor(SNIndex)17:X -0.0034 0.002 -1.7 8e-02as.factor(SNIndex)18:X -0.0087 0.0011 -7.6 8e-14as.factor(SNIndex)19:X -0.00083 0.00015 -5.6 3e-08as.factor(SNIndex)20:X -0.0032 0.00095 -3.4 7e-04as.factor(SNIndex)21:X -0.0013 0.00037 -3.5 5e-04as.factor(SNIndex)22:X -0.0011 0.00044 -2.6 1e-02as.factor(SNIndex)23:X -0.015 0.0029 -5.3 2e-07as.factor(SNIndex)24:X -0.0028 0.00098 -2.8 5e-03as.factor(SNIndex)25:X -0.0052 0.00085 -6.2 9e-10as.factor(SNIndex)26:X -0.0027 0.00057 -4.6 4e-06as.factor(SNIndex)27:X -0.00037 0.00012 -3 3e-03as.factor(SNIndex)28:X 0.00014 0.00012 1.2 2e-01as.factor(SNIndex)29:X -0.0015 0.00029 -5.3 1e-07as.factor(SNIndex)30:X -0.00031 0.0001 -3 3e-03as.factor(SNIndex)31:X -7.4e-05 1.7e-05 -4.4 1e-05as.factor(SNIndex)32:X -0.0018 0.00043 -4.2 2e-05as.factor(SNIndex)33:X -0.0009 0.00024 -3.7 2e-04as.factor(SNIndex)34:X -0.00044 0.00012 -3.7 2e-04as.factor(SNIndex)35:X -0.0022 0.00074 -3 2e-03as.factor(SNIndex)36:X -0.0011 0.00037 -2.9 3e-03as.factor(SNIndex)37:X -9.3e-05 0.00019 -0.5 6e-01Approximate Significance of Smooth TermsFunction edf Ref.df F p-valuete(asd,yr) 11 13 12 4e-26Smoothing Parameter Estimates UsedSmooth function lambdate(asd,yr)1 1e+01te(asd,yr)2 1e-01Model DiagnosticsAIC 3548BIC 1.47e+04GCV score 0.6673Log-Likelihood -1724Effective Degrees of Freedom 1409Scale Parameter Estimate 0.6449Observations 1458

Table S16: Model M2 – Sockeye Salmon (smoothing parameter selected by BIC)

38