Modeling nonstationary temperature maxima based on extremal dependence changing with event magnitude
Item Type Article
Authors Zhong, Peng;Huser, Raphaël;Opitz, Thomas
Citation Zhong, P., Huser, R., & Opitz, T. (2022). Modeling nonstationary temperature maxima based on extremal dependence changing with event magnitude. The Annals of Applied Statistics, 16(1).
https://doi.org/10.1214/21-aoas1504 Eprint version Publisher's Version/PDF
DOI
10.1214/21-aoas1504Publisher Institute of Mathematical Statistics Journal The Annals of Applied Statistics
Rights Archived with thanks to The Annals of Applied Statistics Download date 2023-11-29 17:30:19
Link to Item
http://hdl.handle.net/10754/663646SUPPLEMENTARY MATERIAL FOR
“MODELING NONSTATIONARY TEMPERATURE MAXIMA BASED ON EXTREMAL DEPENDENCE CHANGING WITH EVENT MAGNITUDE"
BYPENGZHONG*, RAPHAËLHUSER*ANDTHOMASOPITZ† Computer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division,
King Abdullah University of Science and Technology (KAUST)* [email protected];[email protected]
BioSP, INRAE, Avignon, 84914, France† [email protected]
1. Further simulation results. In this section, we complement the simulation study of
§4.3 in the main manuscript by considering alternative parameter settings. We first consider the same setting as in the main manuscript (see the details therein), and report results for the casesν= 0.5andν= 1; see Figures1and2, respectively.
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α β λ0 λ1 ν
−2−101
β = 0 ν = 0.5 λ1= −0.5
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α β λ0 λ1 ν
−20123
β = 0.5 ν = 0.5 λ1= −0.5
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α β λ0 λ1 ν
−11234
β = 1 ν = 0.5 λ1= −0.5
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α β λ0 λ1 ν
−1012
β = 0 ν = 0.5 λ1= −0.25
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α β λ0 λ1 ν 0246β = 0.5 ν = 0.5 λ1= −0.25
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α β λ0 λ1 ν 02468 β = 1 ν = 0.5 λ1= −0.25
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α β λ0 λ1 ν
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α β λ0 λ1 ν
0246
β = 0.5 ν = 0.5 λ1= 0
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α β λ0 λ1 ν 0246 β = 1 ν = 0.5 λ1= 0
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FIGURE1. Boxplots of estimated parameters for the simulation study described in §4.3 of the main manuscript.
Each panel corresponds to a different simulation scenario withλ1=−0.5,−0.25,0(top to bottom),β= 0,0.5,1 (left to right) andν= 0.5(see details in the main manuscript), and shows boxplots for each of the 5 parameters based on 200 experiments. Red dots indicate the true values.
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α β λ0 λ1 ν
−10123
β = 0 ν = 1 λ1= −0.5
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α β λ0 λ1 ν
−11234
β = 0.5 ν = 1 λ1= −0.5
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α β λ0 λ1 ν
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α β λ0 λ1 ν
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α β λ0 λ1 ν
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β = 0.5 ν = 1 λ1= −0.25
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α β λ0 λ1 ν 0246 β = 1 ν = 1 λ1= −0.25
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α β λ0 λ1 ν
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α β λ0 λ1 ν
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FIGURE2. Boxplots of estimated parameters for the simulation study described in §4.3 of the main manuscript.
Each panel corresponds to a different simulation scenario withλ1=−0.5,−0.25,0(top to bottom),β= 0,0.5,1 (left to right) andν= 1(see details in the main manuscript), and shows boxplots for each of the 5 parameters based on 200 experiments. Red dots indicate the true values.
In all the considered simulation settings, the true parameter values are contained within the boxplots and are close to the boxplots’ medians. This indicates that the estimation procedure works quite well and that all parameters (in particular β andν, which both have a direct impact on the joint tail dependence structure) can be identified.
To explore the performance of the proposed estimator in a simulation study that resem- bles the real data application, we here perform another experiment. We consider 44 stations located as in the real data application, and then simulate 300 max-id datasets according to Model 6 in the application with parameters chosen according to the estimates in Table 2 of the main manuscript. We make sure the number of temporal replicates and the missing values are the same as in our real data application, and we then estimate the model parameters for each of the 300 datasets. The boxplots are reported in Figure3. Our results show that param- eters can be quite well estimated and are identifiable in this more realistic scenario, as well, although the variability ofαandβ is quite large.
In order to further explore the correlation between model parameters, we then display bivariate scatterplots of estimated parameters for the simulation study withβ= 0.5,ν= 0.5 andλ=−0.25, and the simulation study that mimics the real data application. See Figures4 and5, respectively.
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0246810
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FIGURE3. Boxplots of parameter estimates for the simulation study mimicking the real data application. In par- ticular, 44 locations are chosen as in the application, the same number of time replicates and missing values are used, and parameters are chosen according to parameter estimates for Model 6 in Table 2 of the main manuscript.
The red dots represents the true values (i.e., parameter estimates obtained from our application).
α
0.01.02.0−0.6−0.2
0.0 1.5 3.0
0.0 1.0 2.0
β
λ0
−1.0 0.5
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λ1
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0.0 0.4 0.8
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ν β = 0.5, ν = 0.25, λ1= −0.25
FIGURE4. Bivariate scatterplots of estimated parameters for the simulation study withβ= 0.5,ν= 0.5and λ=−0.25, obtained from the 200 simulated datasets.
α
2468−0.40.00.4
2 4 6 8
23456
2 4 6 8
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λ0
1.5 2.5 3.5
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λt
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24681.52.53.5−0.50.5
ν
FIGURE5. Bivariate scatterplots of estimated parameters for the simulation study that mimics the real data application (see the text for detailed), obtained from the 300 simulated datasets.
No noticeable correlation can be detected between β andν. However, in the first simu- lation study, there exist fairly strong positive correlations between the parametersλ0 andα, and betweenλ0andν. This strong dependence means that these parameters have an opposite effect on the dependence structure: while increasingλ0implies stronger dependence, increas- ingαorνimply weaker dependence, and vice versa. Nevertheless, from Figure1, it is clear that all parameters can still be estimated satisfactorily, despite these correlations. Moreover, interestingly, Figure5shows that these strong correlations mostly disappear when simulating datasets that mimic our real data application. In practice, this means that the model parame- ters from Model 6, our most complex max-id model, can be well identified in our application, both theoretically and numerically.
2. Further inference details in the context of the real data application. We here pro- vide further details on the performance of our proposed parametric bootstrap, in the context of our real data application, first in terms of coverage probabilities, then by comparing the results against the jackknife, a natural alternative approach.
2.1. Coverage analysis. Unfortunately, fitting our most complex max-id process (Model 6) is extremely intensive, and obtaining the results from 300 bootstrap fits for a single observed dataset and a single parameter combination takes several days when using a supercomputer
and efficiently running all bootstrap fits in parallel. Therefore, studying the coverage prob- abilities of our parametric bootstrap in this context is out of reach with our computational resources. To illustrate the performance of the parametric bootstrap, we here consider three simpler cases of increasing complexity as detailed below.
2.1.1. Univariate GEV case. We start with a much simpler case, whereunivariatedata are generated using a generalized extreme-value (GEV) distribution with location parameter µ∈R, scale parameterσ >0and shape parameterξ∈R. Specifically, to mimic the real data application, we generate 1000datasets with100independent time replicates each from the GEV distribution with µ= 32, σ= 17.7 andξ=−0.20. We view the 100time replicates as 100annual maxima. The chosen parameter values are very similar to station-wise esti- mates obtained in our real data application. For each of these1000datasets, we estimate the model parameters by maximum likelihood, and then perform a parametric bootstrap with300 bootstrap replicates as in our real data application, in order to derive(1−α)-confidence in- tervals based on the percentile method, withα= 0.05,0.10, . . . ,0.90,0.95. For each nominal probability 1−α, we then compute the empirical coverage probability, i.e., the proportion of times that the corresponding confidence interval contains the true value among the1000 experiments. Figure 6 reports the results. From this figure, we can see that the coverage probability is slightly underestimated in some cases for σ andξ, but overall the parametric bootstrap works very well. In particular, considering95%-confidence intervals, the coverage probabilities are93.9%forµ,91.7%forσ, and93.6%forξ. Thus95%-confidence intervals have about the correct coverage (thus correct width), though they are all slightly below the nominal probability and the situation is slightly worse forσ.
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µ
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ξ
Nominal probability
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FIGURE6. Empirical coverage probabilities (based on1000simulations) plotted against nominal probabilities, i.e.,1−α, for parametric bootstrap-based confidence intervals calculated for three GEV parametersµ(left),σ (middle) andξ(right). We here considered the true parametersµ= 32,σ= 17.7andξ=−0.20, which are very similar to parameter values estimated in our real data application.
2.1.2. Bivariate logistic case. We then consider a slightly more complex bivariate exam- ple, where the data are generated from the max-stable logistic model, with distribution
G(z1, z2) = exp[−V{t(z1;µ1, σ1, ξ1), t(z2;µ2, σ2, ξ2)}], where the exponent functionV is
V(z1, z2) = (z1−1/ψ+z−1/ψ2 )ψ,
for some dependence parameterψ∈(0,1]andt(·;µ, σ, ξ)is the marginal transformation t(z;µ, σ, ξ) ={1 +ξ(z−µ)/σ}1/ξ+ ,
wherea+= max(0, a). This model has GEV margins with parameters given by(µj, σj, ξj)>, j = 1,2, and has an exchangeable dependence structure controlled by the parameter ψ, interpolating from independence (ψ= 1) to perfect dependence (ψ→0). We here take ψ= 0.3,0.6,0.9 (with extremal coefficients θ2 = 2ψ ≈1.23,1.52,1.87, respectively), i.e., very strong to weak dependence. Moreover, to mimic the real data application and the GEV simulation above, we assume thatµ1= 28,µ2= 36,σ1=σ2= 17.7,ξ1=ξ2=−0.20, and we simulate datasets with100independent time replicates. We consider four different infer- ence approaches:
1. Known margins(ideal case): here, the marginal distributions are assumed to be known exactly, and we only estimate the dependence parameterψ by maximum likelihood. A parametric bootstrap with300bootstrap experiments is then run to assess the estimation uncertainty.
2. One-step: here we estimate marginal and dependence parameters jointly in one single step, by maximum likelihood. A parametric bootstrap with300bootstrap experiments is then run to assess the overall estimation uncertainty (accounting both for marginal and dependence uncertainty).
3. Two-step: here we estimate marginal and dependence parameters separately in two con- secutive steps: we first estimate all marginal parameters (jointly) using an independence likelihood, and then we estimate the dependence parameterψin a second step with mar- gins fixed to their estimates. A parametric bootstrap with 300bootstrap experiments is then run to assess the overall estimation uncertainty (accounting both for marginal and dependence uncertainty).
4. Two-step, dependence only: here we estimate parameters like in the two-step approach above, but the parametric bootstrap with300bootstrap experiments is run to assess only the dependence estimation uncertainty (ignoring marginal uncertainty).
Empirical coverage probabilities for the dependence parameterψ, calculated from1000sim- ulations, are displayed in Figure7. In all inference approaches that properly account for the overall—both marginal and dependence—estimation uncertainty (i.e., the “known margins”,
“one-step” and “two-step” cases), the coverage probabilities forψare strikingly good. Even the two-step approach, which estimates marginal parameters in a first step using a misspeci- fied likelihood that assumes independence across variables, performs very well. Specifically, when considering 95%confidence intervals, the coverage probabilities are equal to95.0%, 95.2%and93.9%for “known margins”, “one-step” and “two-step” approaches, respectively, in the case ψ= 0.3, while they are equal to 93.4%, 94.3% and 94.4% for ψ= 0.6, and 97.0%,96.8%and96.2%forψ= 0.9. There even seems to be a slight advantage of the two- step approach in mild and weak dependence cases. This provides a strong support for our two-step estimation approach, combined with our parametric bootstrap method. Regarding the “two-step, dependence only” approach, we can see, as expected, that coverage proba- bilities are slightly underestimated, because the marginal uncertainty is here not taken into consideration. However, we believe that these results are still surprisingly good. For 95%
confidence intervals, the coverage is underestimated by about 9%forψ= 0.3(very strong dependence case), and only about5%forψ= 0.6(mild dependence case), while it is over- estimated by just2%forψ= 0.9(weak dependence case). Considering the large uncertainty typically associated with the estimation of the GEV shape parameter, these results are more than decent. It is also worth noting that in our real data application, we keep the scale σ and shapeξparameters constant across theD= 44stations, which implies that the marginal
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FIGURE7. Empirical coverage probabilities (based on1000simulations) plotted against nominal probabilities for parametric bootstrap-based confidence intervals calculated for the dependence parameterψin the bivariate logistic model (see details in the text), forψ= 0.3,0.6,0.9(top to bottom) and the different inference approaches:
known margins (left), one-step estimator (second column), two-step estimator (third column), two-step estimator with dependence uncertainty only (right).
uncertainty should be largely reduced relative to the overall uncertainty, and thus that the
“two-step, dependence only” approach (which ignores marginal uncertainty) should perform quite similarly to the “two-step” approach (which accounts for marginal uncertainty). Finally, in the real data application, the data suggest asymptotic independence, which is a weak form of dependence, and based on the pairwise extremal dependence plot in Figure 9 of the main paper, the average extremal coefficient across all pairs of sites is roughly about 1.7 (which corresponds approximately toψ= 0.77). Therefore, based on the results of Figure7, the cov- erage probability of the confidence intervals reported in Table 2 of the main paper (which are based on the “two-step, dependence only” approach) should be only slightly underestimated (probably about5%or less). This is indeed confirmed by our simulations done in the spatial max-stable case reported below.
2.1.3. Spatial max-stable case. We finish our coverage analysis by mimicking our real data application in a simplified setting. We here simulate data according to the non-stationary extremal-tmax-stable process (Model 2 in the application), for which the exponent function and its derivatives can be obtained in closed form. The data are simulated at the same loca- tions as in the application, and the true dependence parameters are chosen according to the
estimates reported in Table 2 of the main paper. We consider the same number of time repli- cates and insert missing values as in the real annual temperature data considered in the main paper. We simulate 1000 such max-stable datasets, fit the extremal-tmodel to each of these datasets by pairwise likelihood as in the application, and then perform a parametric bootstrap with300bootstrap replicates as explained in the main paper, for each of these fitted models.
Like in the bivariate logistic simulations reported above, we here consider different inference procedures: (i) “known margins”, where we assume the marginal distributions to be known exactly (so coverage computations do not involve any marginal uncertainty); (ii) “two-step”
estimator, where the simulated data are back-transformed to the GEV scale (with parameters chosen according to the fitted parameters in the real data application, with the GEV location varying across space and time) and where the marginal parameters are then re-estimated in a first step like in the data application (with splines involved in the location µ, and com- mon scale σ and shape ξ), before the dependence structure is estimated in a second step;
(iii) “two-step, dependence only” approach, where the parameters are estimated in two steps (margins first, dependence second) like above, but unlike the “two-step” procedure, the boot- strap here only accounts for the dependence estimation uncertainty, while ignoring marginal uncertainty. We do not consider the “one-step” estimator in this spatial simulation, because it is much more tricky to implement, while being also much more computer-intensive with the large number of parameters to estimate, and it is difficult to come up with some code that runs smoothly and robustly when estimating all marginal and dependence parameters at once. Figure8displays the coverage probabilities, computed from the 1000 simulations, for the degrees of freedom parameterα, the correlation parameterλ0 and the regression co- efficients λ1 (altitude) and λ2 (time), based on the three inference approaches. The results are excellent overall in all inference procedures (“known margins”, “two-step”, “two-step, dependence only”), except perhaps for the degrees of freedom parameterα, which is more difficult to estimate. In general, the results are quite similar to the bivariate logistic model in the “known margins” or “two-step” cases (recall Figure7), but the “two-step, dependence only” case shows an improved performance overall. The reason for this is that marginal un- certainty is here moderate relative to the overall uncertainty due to the scale σ and shapeξ being kept constant across space and to the large numberD= 44of stations. This makes the confidence intervals for dependence parameters quite accurate despite ignoring marginal un- certainty. Specifically, for95%confidence intervals, the coverage probabilities in the ‘known margins” case are99.6%forα,95.8%forλ0,93.0%forλ1and94.3%forλ2, while they are 91.5%forα,95.0%forλ0,92.0%forλ1and94.2%forλ2in the “two-step” case, and98.0%
forα,97.0%forλ0,93.8%forλ1and93.7%forλ2in the “two-step, dependence only” case.
Therefore, the coverage is a bit off forαin some cases, but it is very accurate for the other parameters. By analogy, we expect that the parametric bootstrap will also yield reasonable results in the more complex spatial max-id setting (Model 6), though this would in princi- ple need to be validated more formally if enough computational resources were available.
Overall, all the simulation results presented above for the univariate GEV, bivariate max- stable logistic, and spatial extremal-tmodels suggest that the confidence intervals reported in Table 2 of the main paper are reliable.
2.2. Comparison with jackknife estimates. To assess whether confidence intervals from our proposed parametric bootstrap scheme are indeed reasonable, we now compare standard errors based on the bootstrap with standard errors obtained from the jackknife, which is a nat- ural alternative approach. The jackknife originates fromQuenouille(1949,1956) andTukey (1958) and consists in fitting the model to the original data with thei-th replicate left out, fori= 1, . . . , n, and then computing the (rescaled) sample variance of thesenparameter es- timates. In this sense, it is related to leave-one-out cross-validation. The jackknife has been
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FIGURE8. Empirical coverage probabilities (based on1000simulations) plotted against nominal probabilities for parametric bootstrap-based confidence intervals calculated for the parametersα, λ0, λ1, λ2(left to right) in Model 2 of the main paper, a non-stationary max-stable extremal-tmodel (see details in the main paper). We here consider three difference inference schemes: “known margins” (top), “two-step” (middle), and “two-step, dependence only” (bottom); see the definition of these inference schemes in the text.
shown to be a linear approximation to the (non-parametric) bootstrap (Efron,1979), so both methods should in principle give similar results. However, the parametric bootstrap is more widely applicable, and it better handles missing values (seeDavison and Hinkley,1997, page 88). Standard errors based on the parametric bootstrap and the jackknife are compared in Ta- ble1for Models 1–6 fitted in our data application. Overall, the estimated standard errors are of the same order of magnitude for both methods, although some differences do exist (e.g., standard errors ofβ seem to be consistently underestimated with the jackknife, perhaps be- cause the distribution for this parameter is more skewed than for the other parameters, recall Figure 5). Nevertheless, both methods give generally consistent results, such that the main conclusions drawn in the paper (e.g., about the asymptotic dependence class or the signifi- cance of covariate effects) are not affected when considering our preferred model (Model 6).
3. Further results in our real data application.
3.1. Marginal homogeneity analysis. In addition to the basic visual diagnostics and two- sided Kolmogorov–Smirnov tests done to check that the marginal distributions fit well (see
