HYDROLOGICAL PROCESSES Hydrol. Process. 24, 673– 685 (2010) Published online 24 November 2009 in Wiley InterScience (www.interscience.wiley.com) DOI: 10.1002/hyp.7506
A flexible nonlinear modelling framework for nonstationary generalized extreme value analysis in hydroclimatology Alex J. Cannon* Meteorological Service of Canada, Environment Canada, Vancouver, British Columbia, Canada V6C 3S5
Abstract: Parameters in a generalized extreme value (GEV) distribution are specified as a function of covariates using a conditional density network (CDN), which is a probabilistic extension of the multilayer perceptron neural network. If the covariate is time or is dependent on time, then the GEV-CDN model can be used to perform nonlinear, nonstationary GEV analysis of hydrological or climatological time series. Owing to the flexibility of the neural network architecture, the model is capable of representing a wide range of nonstationary relationships. Model parameters are estimated by generalized maximum likelihood, an approach that is tailored to the estimation of GEV parameters from geophysical time series. Model complexity is identified using the Bayesian information criterion and the Akaike information criterion with small sample size correction. Monte Carlo simulations are used to validate GEV-CDN performance on four simple synthetic problems. The model is then demonstrated on precipitation data from southern California, a series that exhibits nonstationarity due to interannual/interdecadal climatic variability. Copyright 2009 Her Majesty the Queen in right of Canada. Published by John Wiley & Sons. Ltd KEY WORDS
extreme value analysis; nonstationary; statistical modelling; neural network; nonlinear hydroclimatology
Received 4 February 2009; Accepted 15 September 2009
INTRODUCTION The distribution of a series of extreme values computed from long sequences of data asymptotically approaches the generalized extreme value (GEV) distribution (Jenkinson, 1955) as the number of samples becomes large. The extreme value theorem, which is the extreme value analogue of the central limit theorem (Coles, 2001), forms the basis for extreme value analysis of meteorological and hydrological series, for example, annual maxima of rainfall or streamflow observations, and, in turn, the estimation of design criteria for engineering structures (Maidment, 1993). The main assumptions are (i) that the series of extremes is suitably long, (ii) that the elements of the series are independent and identically distributed, and (iii) that the series is stationary, meaning that its statistical properties are independent of time. There is ample evidence that the hydroclimatic system is nonstationary on time scales relevant to applied extreme value analysis, whether due to natural climate variability or anthropogenic climate change (Jain and Lall, 2001; Rial et al., 2004; Milly et al., 2008). The assumption of stationarity in extreme value analysis is therefore questionable, and new methods that explicitly allow for nonstationarity in the GEV distribution parameters are required (Coles, 2001). For example, Kharin and Zwiers (2004) allowed for linear trends in the location and shape and a log-linear trend in the scale parameter of the GEV distribution * Correspondence to: Alex J. Cannon, Meteorological Service of Canada, Environment Canada, 201-401 Burrard Street, Vancouver, British Columbia, Canada V6C 3S5. E-mail:
[email protected]
with time, whereas Wang et al. (2004) considered models with covariate-dependent changes in the location and scale parameters. Parameter estimates were made via the principle of maximum likelihood (ML). El Adlouni et al. (2007) extended this approach by fitting parameters via the generalized maximum likelihood (GML) approach of Martins and Stedinger (2000), which specifies a geophysically realistic prior distribution for the shape parameter within a Bayesian framework. The GML method performed better than the standard ML method for small sample sizes. In each of these studies, simple parametric models (all linear or log-linear in the parameters) were specified a priori for the nonstationary dependence of the GEV parameters on the covariates. In practice, an assumption of linearity may not be appropriate. For example, Kharin and Zwiers (2004) allowed for nonlinear trends in precipitation and temperature extremes over a 110-year transient global climate model simulation by estimating linear trends in GEV parameters based on a series of overlapping 51-year time windows. In a GEV analysis of winter precipitation in the United States, Schubert et al. (2008) fit separate distributions to La Ni˜na, neutral, and El Ni˜no years to account for a nonlinear relationship between precipitation extremes in the southwest and El Ni˜no-Southern oscillation (ENSO) conditions. As an alternative, nonparametric and semi-parametric approaches to nonstationary extreme value analysis have been developed to overcome the linearity assumption of the conditional GEV models described above. For example, Koenker and Schorfheide (1994) used quantile
Copyright 2009 Her Majesty the Queen in right of Canada. Published by John Wiley & Sons. Ltd
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regression splines to assess nonlinear trends in quantiles of a global temperature series, Hall and Tajvidi (2000) fit local-linear trend models to wind storm intensity and temperature series, and Gaetan and Grigoletto (2004) used dynamic smoothing models to assess trends in a monthly maximum temperature extremes. Such models are, however, generally limited to analysing nonlinear relationships involving a single covariate. Methods based on generalized additive models, for example, those by Chavez-Demoulin and Davison (2005), who analysed relationships between extremes in alpine temperatures and altitude/North Atlantic Oscillation, and Yee and Stephenson (2007), who considered the dependence between annual maximum sea-levels and time/ENSO, allow multiple covariates, but cannot model interactions between covariates without a priori specification of the form of the interactions by the modeller. In this study, a more flexible nonlinear model for nonstationary extreme value analysis is proposed. Parameters of the GEV distribution are specified as a function of covariates using a conditional density network (CDN) (Neuneier et al., 1994; Bishop, 2006; Cawley et al., 2007; Cannon, 2008), which is a probabilistic extension of the commonly used multilayer perceptron (MLP) neural network (Gardner and Dorling, 1998; Hsieh and Tang, 1998; Dawson and Wilby, 2001). The MLP is a universal function approximator that can model nonlinear relationships, including ones involving unspecified interactions between multiple covariates. If one of the covariates is time (or is dependent on time), then the GEV-CDN model performs nonstationary GEV analysis. Model parameters are estimated following the GML approach of Martins and Stedinger (2000), and model complexity is identified using two model selection criteria. Confidence limits for parameters and estimated quantiles are estimated by the bootstrap. Monte Carlo simulations are used to validate performance on synthetic test problems introduced by El Adlouni et al. (2007). Finally, models are applied to precipitation data from southern California, a time series that exhibits nonstationarity due to the influence of climate variability on interannual/interdecadal time scales.
GEV DISTRIBUTION The GEV distribution (Jenkinson, 1955) is specified by three parameters: the location , the scale ˛ (˛ > 0), and the shape . The shape parameter is the main determinant of the behaviour of the tails of the distribution. Following the hydroclimatological convention, negative values of correspond to positive skewness. The cumulative density function (cdf) of a random variable y drawn from a GEV distribution is given by y 1/ Fy; , ˛, D exp 1 , ˛ 6D 0, 1
y >0 ˛
1
y Fy; , ˛ D exp exp , D 0 2 ˛
and the probability density function (pdf) is given by 1 y 1/ fy; , ˛, D 1 exp ˛ ˛ y 1/ 1 , 6D 0 3 ˛ y fy; , ˛ D exp exp ˛ y exp , D 0 4 ˛ Parameters in the GEV distribution can be estimated via the method of moments or L moments (Hosking et al., 1985; Hosking, 1990), or by the principle of ML (Coles and Dixon, 1999), where the goal is to identify the most ‘likely’ set of parameters by maximizing the likelihood function L, ˛, jy D
N
fyt; , ˛,
5
tD1
where y D fyt, t D 1, . . . , Ng is a series of N independent observations. For convenience, one often minimizes the negative of the log-likelihood function instead. For small samples, ML estimates can be unstable relative to the method of moments (which restricts values of to be greater than 1/3) due to the generation of physically unrealistic values (Coles and Dixon, 1999; Martins and Stedinger, 2000). To remedy this problem, Coles and Dixon (1999) and Martins and Stedinger, 2000 modified the ML approach so that is forced to take more realistic values. In the penalized ML approach of Coles and Dixon (1999), a penalty term is added to the likelihood function, leading to Lp , ˛, jy D L, ˛, jy
where the recommended penalty is of the form if ½ 0 1 1 if 0 > > 1 p D exp 1 1 C 0 if 1.
6
7
The shape parameter is forced to be greater than 1, with values close to 1 penalized more than larger values. Martins and Stedinger (2000) developed the GML estimator, which applies a more restrictive penalty (or prior distribution) on g D Beta C 0Ð5; c1 , c2 ,
8
in which the shape parameter is limited to the range 0Ð5 0Ð5 and where Beta denotes the pdf of a beta distribution with shape parameters c1 and c2 . Martins and Stedinger (2000) recommended that c1 and c2 be set equal to 6 and 9, respectively, resulting in a pdf
Copyright 2009 Her Majesty the Queen in right of Canada. Published by John Wiley & Sons. Ltd
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π
1.0
1.5
2.0
2.5
3.0
which assumes that all parameters are independent of time (ˇ2 D 0, ˇ3 D 0, and υ2 D 0), GEV1, which assumes a linear change in the location parameter (ˇ3 D 0 and υ2 D 0), GEV2, which assumes a quadratic change in the location parameter (υ2 D 0), and GEV11, which assumes a linear change in the location and a log-linear change in the scale parameter (ˇ3 D 0). The shape is assumed to be constant with time in all models.
GEV-CDN
0.0
0.5
General framework
–1.0
–0.5
0.0 κ
0.5
1.0
Figure 1. Penalties for , including p from Coles and Dixon (1999) (dotted line), g with c1 D 6 and c2 D 9 from Martins and Stedinger (2000) (solid line), and g with c1 D 2 and c2 D 3Ð3 used in this study (dashed line)
with a mode at 0Ð1 and ¾90% of its probability mass concentrated over values between 0Ð3 and C0Ð1. In this study, c1 and c2 are instead set equal to 2 and 3Ð3, respectively, which results in a broader pdf with a mode of approximately 0Ð2 and approximately 90% of its probability mass concentrated between 0Ð4 and C0Ð2. As shown in Figure 1, this penalty is intermediate between those recommended by Coles and Dixon (1999) and Martins and Stedinger (2000). The result is a prior distribution for that fits many natural processes well. For example, Kysel´y and Picek (2007) analysed daily precipitation extremes at 78 stations in the Czech Republic and found values of ranging from 0Ð37 to C0Ð16, which corresponds to slightly more than 80% of the probability mass of g with c1 D 2 and c2 D 3Ð3.
NONSTATIONARY GEV El Adlouni et al. (2007) extended the GML estimator to cases where the GEV parameters are functions of covariates. Parameters are estimated by the Markov Chain Monte Carlo method rather than the numerical optimization approach of Martins and Stedinger (2000). If the covariate is time, or is dependent on time, then this amounts to nonstationary GEV estimation within a Bayesian framework. El Adlouni et al. (2007) considered nonstationary relationships in the location and scale ˛ parameters with time of the form t D ˇ1 C ˇ2 t C ˇ3 t2 ˛t D expυ1 C υ2 t
In the GEV-CDN, parameters of the GEV distribution are modelled as a function of covariates using an MLP architecture, which is shown schematically in Figure 2a. The model has K D 3 outputs, corresponding to the three GEV parameters (, ˛, and ). Details on the CDN, which is a probabilistic variant of the standard MLP, are given in Bishop (2006, [Ch. 5Ð6]) and Cawley et al. (2007). The reader is referred to Gardner and Dorling (1998), Hsieh and Tang (1998), and Dawson and Wilby (2001) for reviews of the MLP in the context of meteorological, climatological, and hydrological prediction. Given covariates at time t, xt D fxi t, i D 1, . . . , Ig, outputs from a GEV-CDN model with J hidden-layer nodes are evaluated as follows. First, the output from the jth hidden-layer node hj is given by applying the hidden-layer activation function mÐ to the inner product between the covariates and the input-hidden layer weights 1 wji plus the bias bj1
I 1 1 hj t D m xi t wji C bj . 11 iD1
If the GEV-CDN mapping is to be nonlinear, then mÐ is taken to be a sigmoidal function, e.g. the hyperbolic tangent function tanhÐ. The identity function is adopted if the GEV-CDN mapping is to be strictly linear. The value of the kth output from the network ok is then given by J 2 ok t D hj t wkj C bk2 , 12 jD1 2 where wkj are the hidden-output layer weights and 2 bk are the hidden-output layer biases. Finally, GEV parameters are obtained by applying the output-layer activation functions gk Ð,
t D g1 o1 t D o1 t
13
˛t D g2 o2 t D expo2 t
14
Ł
9
t D g3 o3 t D tanho3 t.
10
The function g2 Ð forces the scale parameter ˛ to take positive values. The function g3 Ð constrains the shape parameter to the interval [Ł , Ł ]. In this study, GML (Martins and Stedinger, 2000) is adopted, with the value of Ł set to 0Ð5 to limit the search space
where ˇ1 , ˇ2 , ˇ3 , υ1 , and υ2 are parameters that must be estimated from the data. Based on Equations (9) and (10), four models of differing complexity were defined: GEV0,
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(a) Fully-connected (2) J wkj
I
(c) Nonstationary m, a linear
(b) Stationary m, a, k
w(1)
CDN-CON (P=3)
m(t ) = o1 (t )
m
m(t )
CDN-LIN (P=5)
ji
x 1( t ) a(t ) = exp(o2 (t ))
a
a(t )
k(t ) = k* tanh(o3 (t ))
k
k
xI ( t )
b(1) j
b(2) k
(e) Nonstationary m, a nonlinear (2 hidden)
(d) Nonstationary m, a nonlinear (1 hidden)
CDN-NLIN1 (P=7)
(f)
CDN-NLIN2 (P=11)
Nonstationary m, a nonlinear (3 hidden) CDN-NLIN3 (P=15)
m(t )
m(t )
m(t )
a(t )
a(t )
a(t )
k
k
k
active parameter
identity function
redundant parameter
exponential function
inactive parameter
hyperbolic tangent function
Figure 2. (a) Neural network architecture of the fully connected GEV-CDN model; (b) CDN-CON model; (c) CDN-LIN model with constant ; (d) CDN-NLIN1 model with constant ; (e) CDN-NLIN2 model with constant ; (f) CDN-NLIN3 model with constant
of during optimization to the support of the shifted beta distribution prior. The conditional pdf is now given by Equations (3–4) with time-dependent (rather than constant) GEV parameters t, ˛t, and t, with the corresponding likelihood, Lm, a, kjy D
N
fyt; t, ˛t, t.
16
tD1
A hierarchy of models can be defined by adjusting three aspects of the GEV-CDN model architecture: (i) by specifying either a linear or a nonlinear hidden-layer activation function mÐ; (ii) by adjusting the number of hidden-layer nodes J; or (iii) by disconnecting weights leading to output-layer nodes ok . To illustrate, five GEV-CDN models, shown in order of increasing complexity in Figures 2b–2f, are specified here for the case of a single covariate (I D 1), which, in this case, is assumed to be time x1 t D t. Following El Adlouni et al. (2007), the shape parameter is assumed to be constant in all models, although, as shown above, this is not a requirement of the GEV-CDN framework. A description of the hierarchy of GEV-CDN models for this particular case follows. CDN-CON The CDN-CON model (Figure 2b) assumes that the GEV distribution parameters are constant, i.e. the data
are stationary. This can be accomplished in the GEVCDN framework by disconnecting (i.e. setting to zero) all parameters except for the hidden-output layer biases bk2 . From Equations (11)–(15), this reduces to a stationary model in which the GEV parameters are decoupled from the covariates and hidden-layer nodes. The total number of adjustable parameters is thus P D 3. The CDN-CON model is equivalent to the GEV0 model of El Adlouni et al. (2007). CDN-LIN The CDN-LIN model (Figure 2c) assumes that the network outputs o1 and o2 associated with t and ˛t, respectively, are a linear function of time, which can be accomplished by setting the hidden-layer activation function mÐ to the identity function and, for simplicity, J D 1. In this example, nonstationary parameters t and ˛t are given by Equations (13) and (14). To accom2 modate constant , the hidden-output layer weight w31 is disconnected from the model. To show that this leads to a simple linear model, consider the net effect of these choices on Equations (11)– (13) for t, 1 2 2 t D o1 t D w11 w x1 t C b11 w11 C b12 . 11
Copyright 2009 Her Majesty the Queen in right of Canada. Published by John Wiley & Sons. Ltd
ˇ2
17
ˇ1
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The weights and biases can be collapsed down to two effective parameters. A similar equation results for ˛t, whereas is specified by the hidden-output layer bias b32 . Discounting redundancies, the total number of adjustable model parameters is thus P D 5. The representational capacity of the model does not depend on the number of hidden nodes J, as increasing J instead results in additional redundant parameters in the equations rather than a true increase in model flexibility. As shown in Figure 2c, the CDN-LIN model is equivalent to the GEV11 model of El Adlouni et al. (2007). If weights associated with the scale parameter ˛ are also disconnected from the network, then the CDN-LIN model is equivalent to the GEV1 model of El Adlouni et al. (2007). CDN-NLIN1, CDN-NLIN2, and CDN-NLIN3 CDN-NLIN models allow for general nonlinear relationships between the covariates and the GEV parameters. The hidden-layer activation function mÐ is set to the hyperbolic tangent rather than the identity function. For a model with I covariates, the total number of parameters, assuming constant , therefore depends on the number of hidden-layer nodes as P D JI C 3 C 3. The MLP is a universal function approximator; the model can approximate any smooth function to an arbitrary degree of accuracy because J increases to infinity (Hornik et al., 1989). However, given the small number of samples available in hydrological and climatological extreme value analyses (usually around 30–100 years of annual extremes), J is limited to values between one and three in this paper. For the CDN-NLIN models specified in Figure 2d, each involving a single covariate, P thus ranges between 7 and 15. Models contain not more than half the number of parameters as the number of years (30) in a standard climate normal period. The CDN-NLIN1 model takes the same form as CDN-LIN, except, as noted above, mÐ is set to the hyperbolic tangent function. With a single hidden-layer node, J D 1, this is the simplest form of nonlinear GEVCDN model. The hyperbolic tangent can approximate linear [tanhwx ' wx as w ! 0] and step functions (as w ! 1). The CDN-NLIN2 model (Figure 2e) adds a second hidden-layer node to CDN-NLIN1. The GEV2 model of El Adlouni et al. (2007) would be subsumed by the CDN-NLIN2 model, although CDN-NLIN2 is capable of approximating more complicated functions than a secondorder polynomial. A Z-shaped continuous curve can, for example, be described by a neural network with two hidden nodes (Christiansen, 2005). The CDN-NLIN3 model (Figure 2f) adds a third hidden-layer node to CDN-NLIN2, and, as a result, can describe yet more complicated nonlinear relationships. Further insight is given by Carpenter and Barthelemy (1993), who compared approximations made by polynomials and MLP neural networks with similar numbers of parameters.
677
MONTE CARLO SIMULATIONS Simulation procedure Monte Carlo simulations based on the nonstationary GEV1, GEV2, and GEV11 test cases of El Adlouni et al. (2007) are used to evaluate the performance of the GEV-CDN model. A fourth test case, GEVstep, is added to illustrate model performance when faced with a step change in GEV parameters. Following El Adlouni et al. (2007), Monte Carlo simulations involve fitting GEV-CDN models to 50 samples of a single covariate, t D 1, . . . , 50, representing time, with time-dependent GEV parameters as specified in Figure 3. One thousand trials are run for each test case and value of ( D 0Ð1, 0Ð2, 0Ð3). In each trial, samples are randomly generated based on the specified nonstationary GEV distributions, and GEV-CDN models are fit to the random samples. The bias and root mean squared error (rmse) of predicted 0Ð5, 0Ð8, 0Ð9, 0Ð99, and 0Ð999 -quantiles, which, in a stationary context correspond to return periods of 2, 5, 10, 100, and 1000 years, respectively, are calculated with respect to the true quantiles. For the GEV1, GEV2, and GEVstep test cases, CDN-LIN and CDN-NLIN models are fit with stationary ˛ and ; only is dependent on time. For the GEV11 test case, both and ˛ are allowed to be time dependent, whereas is stationary. For all GEV-CDN models, neural network weights and biases are estimated by minimizing the GML cost function using a Broyden–Fletcher–Goldfarb–Shanno– quasi-Newton optimization algorithm as implemented by Nash (1990). To avoid convergence to a shallow local minimum of the error surface, the optimization algorithm is run 100 times, each time starting from different initial weights and biases. Parameters associated with the maximum GML over the 100 random restarts are selected as the final parameters. Model selection The appropriate GEV-CDN model architecture for a given dataset is selected by fitting increasingly complicated models and choosing the one that minimizes the Akaike information criterion with small sample size correction (AICc) (Akaike, 1974; Hurvich and Tsai, 1989) or the Bayesian information criterion (BIC) (Schwarz, 1978). Both AICc and BIC are cost-complexity model selection criteria that penalize the negative log-likelihood as a function of the number of model parameters P. The objective then is to choose the most parsimonious model that is capable of accounting for the true (but unknown) deterministic function responsible for generating the N observations. Overfitting, that is fitting to noise in the finite dataset rather than the underlying signal, is thus avoided. Theoretical justifications for AICc and BIC in the context of model selection can be motivated either by information-theoretic or Bayesian arguments. The reader is referred to Burnham and Anderson (2004) for more details.
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10
20
30
40
40 30 20 10
Quantiles τ={0.5, 0.8, 0.9, 0.99, 0.999}
15 10 5 0 –5
Quantiles τ={0.5, 0.8, 0.9, 0.99, 0.999}
–10 0
GEV11 (β1=0, β2=0.1, β3=0, δ1=1, δ2=0.02)
(b)
0
GEV1 (β1=0, β2=0.2, β3=0, δ1=0, δ2=0)
(a)
50
0
10
20
t
10
20
50
30
40
50
10
15
20
25
GEVstep (β1=(0, 10), β2=0, β3=0, δ1=0, δ2=0)
5
(d) Quantiles τ={0.5, 0.8, 0.9, 0.99, 0.999}
20 15 10 5 0
Quantiles τ={0.5, 0.8, 0.9, 0.99, 0.999}
0
40
0
GEV2 (β1=0, β2=0.3, β3=0.005, δ1=0, δ2=0)
(c)
30 t
0
10
20
30
40
50
t
t
Figure 3. Each panel shows, from bottom to top, time series of 0Ð5, 0Ð8, 0Ð9, 0Ð99, and 0Ð999 -quantiles for (a) GEV1, (b) GEV11, (c) GEV2, and (d) GEVstep test cases (with D 0Ð2) defined by Equations (9) and (10). Equation parameters are given in the plot titles
AICc and BIC are given, respectively, by AICc D 2 logL C 2P C
2PP C 1 NP1
18
and BIC D 2 logL C P logN
19
where BIC penalizes more complicated models more than does AICc. As an alternative to AICc and BIC, both of which are estimated directly from the training dataset, more computationally intensive split-sample methods, such as cross-validation, can also be used for model selection. Although this paper relies on information criteria for this purpose, the reader is referred to Smyth (2000) for a comparison between models selected via cross-validation and BIC in a ML context. The utility of AICc and BIC for nonstationary model selection is assessed in the following sections. GEV1 test case For the GEV1 test case, where the nonstationarity is linear, a CDN-LIN model with constant ˛ and is functionally equivalent to the linear model evaluated by El Adlouni et al. (2007). Performance intercomparisons between the neural network and the corresponding quantiles from the fitted GEV1 model in
terms of bias and rmse are therefore not instructive. In these cases, the Monte Carlo simulations will instead focus on issues related to model selection, namely, (i) the ability of AICc and BIC to select the appropriate level of model complexity and (ii) the cost of model misspecification. Average values of BIC, AICc, bias, and rmse are given in Table I for each of the five GEV-CDN models (CDNCON, CDN-LIN, CDN-NLIN1, CDN-NLIN2, and CDNNLIN3) and three values of . In all cases, both BIC and AICc correctly recommend the CDN-LIN model. As expected given the linear nonstationarity of the GEV1 test case, values of bias and rmse are typically best for the CDN-LIN model. Quantile estimation by the CDNLIN model is effectively unbiased for all but the highest -quantile, which is consistent with the results found by El Adlouni et al. (2007) for the GML estimator. Similarly, values of rmse are relatively low. Although results for the standard ML estimator were also computed during the study, they are not reported because they support the findings of El Adlouni et al. (2007). Namely, GML quantile estimates are less biased and have lower rmse relative to ML estimates. The cost of model misspecification is generally high. For example, choosing a stationary CDN-CON model
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Hydrol. Process. 24, 673–685 (2010)
Statistic BIC AICc bias D 0.5 0.8 0.9 0.99 0.999 rmse D 0.5 0.8 0.9 0.99 0.999
D
0Ð21 0Ð29 0Ð38 1Ð24 3Ð41
2Ð90 3Ð35 3Ð78 4Ð34 4Ð02
0Ð27 0Ð35 0Ð44 1Ð42 4Ð06
0Ð02 0Ð06 0Ð05 0Ð35 1Ð83
−0Ð01 −0Ð03 −0Ð04 0Ð19 1Ð13
0Ð05 1Ð69 2Ð40 3Ð09 1Ð75
NLIN1
181Ð5 172Ð0
LIN
175Ð0 168Ð2
269Ð4 264Ð2
CON
0.1
0Ð44 0Ð50 0Ð57 1Ð54 5Ð03
0Ð05 0Ð15 0Ð15 0Ð56 3Ð34
186Ð5 173Ð8
NLIN2
0Ð60 0Ð68 0Ð73 1Ð76 6Ð83
0Ð09 0Ð26 0Ð26 0Ð89 5Ð62
191Ð0 176Ð5
NLIN3
2Ð90 3Ð40 3Ð82 4Ð05 4Ð83
0Ð11 1Ð77 2Ð45 2Ð39 1Ð39
272Ð7 267Ð4
CON
0Ð20 0Ð31 0Ð44 1Ð63 5Ð02
−0Ð01 −0Ð06 −0Ð09 −0Ð05 0Ð74
180Ð4 173Ð7
LIN
0Ð27 0Ð37 0Ð50 1Ð72 5Ð32
0Ð02 0Ð08 0Ð11 0Ð10 1Ð47
187Ð0 177Ð5
NLIN1
0.2
0Ð45 0Ð54 0Ð65 1Ð78 6Ð27
0Ð04 0Ð19 0Ð23 0Ð32 3Ð50
191Ð8 179Ð1
NLIN2
0Ð61 0Ð73 0Ð83 1Ð70 6Ð77
0Ð08 0Ð31 0Ð40 0Ð24 4Ð48
195Ð9 181Ð4
NLIN3
2Ð90 3Ð46 3Ð85 3Ð90 9Ð16
0Ð17 1Ð86 2Ð47 1Ð14 7Ð50
276Ð3 271Ð1
CON
0Ð20 0Ð34 0Ð53 2Ð11 6Ð86
0Ð00 −0Ð10 −0Ð22 0Ð97 2Ð51
185Ð5 178Ð7
LIN
0Ð27 0Ð40 0Ð58 2Ð05 6Ð69
0Ð00 0Ð12 0Ð24 0Ð83 1Ð67
192Ð2 182Ð7
NLIN1
0.3
0Ð43 0Ð57 0Ð73 1Ð96 6Ð36
0Ð03 0Ð23 0Ð37 −0Ð64 0Ð41
198Ð4 185Ð7
NLIN2
0Ð63 0Ð78 0Ð97 2Ð02 5Ð83
0Ð05 0Ð36 0Ð58 1Ð03 0Ð42
201Ð7 187Ð2
NLIN3
Table I. Model performance of CDN-CON, CDN-LIN, CDN-NLIN1, CDN-NLIN2, and CDN-NLIN3 models on the Monte Carlo simulation for the GEV1 test case. Given the linearity of the GEV1 model, the a priori expectation is for CDN-LIN to yield the best performance; the column header associated with the CDN-LIN model is shown in bold to reflect this fact. For each combination of performance statistic and shape parameter , the value corresponding to the model that actually performed best (e.g., the minimum value for BIC, AICc, and rmse, and the value closest to zero for bias) is shown in bold italics
FLEXIBLE MODELLING FRAMEWORK FOR NONSTATIONARY GEV
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1Ð22 2Ð29 3Ð45 10Ð57 28Ð57 1Ð00 1Ð82 2Ð70 8Ð31 22Ð76 0Ð66 1Ð15 1Ð67 5Ð27 15Ð39 0Ð41 0Ð86 1Ð34 4Ð81 14Ð52 1Ð66 2Ð48 3Ð25 7Ð75 19Ð09 1Ð06 1Ð85 2Ð70 7Ð80 20Ð52 0Ð94 1Ð55 2Ð19 6Ð06 15Ð78 0Ð63 1Ð04 1Ð45 4Ð18 11Ð39 0Ð40 0Ð74 1Ð09 3Ð52 9Ð85 1Ð66 2Ð41 3Ð09 6Ð68 14Ð23 1Ð01 1Ð66 2Ð33 6Ð29 15Ð49 0Ð64 0Ð98 1Ð31 3Ð35 8Ð08 0Ð39 0Ð66 0Ð94 2Ð74 6Ð79 1Ð66 2Ð31 2Ð88 5Ð74 11Ð23
0Ð90 1Ð38 1Ð88 4Ð88 11Ð77
0Ð11 0Ð15 0Ð36 0Ð95 0Ð26 0Ð07 0Ð19 0Ð42 1Ð37 1Ð68 0Ð02 0Ð17 0Ð36 1Ð41 2Ð88 0Ð02 −0Ð14 −0Ð32 1Ð36 3Ð19 0Ð05 0Ð42 0Ð63 0Ð37 2Ð75 0Ð02 0Ð23 0Ð36 0Ð06 4Ð26 −0Ð01 0Ð14 0Ð20 0Ð05 1Ð88 0Ð04 0Ð13 0Ð14 0Ð36 2Ð50 0Ð00 −0Ð06 −0Ð06 0Ð33 1Ð92 0Ð17 0Ð34 0Ð75 2Ð73 6Ð52
0Ð02 0Ð13 0Ð16 0Ð55 3Ð87
0Ð02 0Ð17 0Ð20 0Ð77 5Ð28
0Ð11 0Ð43 0Ð81 2Ð32 4Ð96
−0Ð01 −0Ð12 −0Ð19 0Ð20 0Ð92
0Ð04 0Ð16 0Ð26 −0Ð01 2Ð91
245Ð5 231Ð4 244Ð2 233Ð5 238Ð4 230Ð2 260Ð0 254Ð7 240Ð8 226Ð3 239Ð5 225Ð4 237Ð7 226Ð9 232Ð1 223Ð9 255Ð1 249Ð9 235Ð2 220Ð7 233Ð7 219Ð6 232Ð1 221Ð4 227Ð2 218Ð9
LIN CON
248Ð3 243Ð0
Statistic BIC AICc bias D 0.5 0.8 0.9 0.99 0.999 rmse D 0.5 0.8 0.9 0.99 0.999
NLIN3 NLIN2 NLIN1 NLIN2
NLIN3
CON
LIN
0.2 0.1 D
For the GEV2 test case, where the location parameter has a nonlinear dependence on time, the main objective is to compare the performance of the CDN-NLIN models against the GEV2 model of El Adlouni et al. (2007). In contrast to the GEV1 and GEV11 test cases, where CDN-LIN models are functionally equivalent to the corresponding GEV1 or GEV11 models, the CDNNLIN model approximates the quadratic signal specified by the GEV2 model via the nonlinearity introduced by the sigmoidal activation functions in the neural network’s hidden layer. As indicated by Christiansen (2005), accurate approximation of a quadratic function should be possible using an MLP with two hidden nodes, i.e. the CDN-NLIN2 model. Results from the GEV2 model and the GEV-CDN models are shown in Table III. Among the GEV-CDN models, the CDN-NLIN2 model, as expected, minimizes both BIC and AICc for all values of . Differences in BIC and AICc between the CDN-NLIN2 model and the GEV2 model are small, meaning that the information loss resulting from the use of the CDN-NLIN2 model instead of the GEV2 model is likely to be minimal (Burnham and Anderson, 2004). Comparing bias and rmse values between the two models supports this conclusion. Bias and rmse values are comparable for < 0Ð999, whereas performance statistics for D 0Ð999 tend to favour the
Table II. As in Table I but for the GEV11 test case
GEV2 test case
CON
GEV11 test case In the GEV11 test case, the location parameter is a function of time, and the scale parameter is a log-linear function of time. Similar to the previous test case, a CDN-LIN model with constant is thus functionally equivalent to the corresponding GEV11 model. The a priori expectation then is for the CDN-LIN model to be selected by the BIC and AICc cost-complexity model selection criteria. As shown in Table II, this is indeed the case for all three values of . In terms of model performance, the pattern of results is similar to that reported in the previous section for the GEV1 test case. As expected, the CDN-LIN model is relatively unbiased and gives the lowest values rmse for all and . The cost of model misspecification is, again, quite high. Specifying either stationary or nonlinear models can lead to large drops in model performance.
NLIN1 LIN
0.3
NLIN2
NLIN3
increases the absolute bias and rmse by a factor of 2 or more (and, in most cases, by an order of magnitude) for almost all combinations of and < 0Ð999. Also, with < 0Ð999 the increase in rmse and (absolute) bias associated with selection of a nonlinear model (CDN-NLIN1, CDN-NLIN2, or CDN-NLIN3) is typically lower than the increase associated with selection of the stationary model (CDN-CON), which means that overfitting has less effect on model performance than does improperly by assuming stationarity. With D 0Ð999, however, performance of the nonlinear models can be slightly worse than the stationary model, in particular for > 0Ð3.
248Ð8 234Ð3
A. J. CANNON
NLIN1
680
Copyright 2009 Her Majesty the Queen in right of Canada. Published by John Wiley & Sons. Ltd
Hydrol. Process. 24, 673–685 (2010)
681
FLEXIBLE MODELLING FRAMEWORK FOR NONSTATIONARY GEV
Table III. Model performance of CDN-NLIN1, CDN-NLIN2, and CDN-NLIN3 models on the Monte Carlo simulation for the GEV2 test case. For comparison, results from GEV2 model fits to Equations (9) and (10) are also shown. Among the GEV-CDN models, the a priori expectation is for CDN-NLIN2 to yield the best performance; the column header associated with this model is shown in bold to reflect this fact. For each combination of performance statistic and shape parameter , the value corresponding to the GEV-CDN model that actually performed best (e.g. the minimum value for BIC, AICc, and rmse, and the value closest to zero for bias) is shown in bold italics D
Statistic BIC AICc bias D 0.5 0.8 0.9 0.99 0.999 rmse D 0.5 0.8 0.9 0.99 0.999
0.1
0.2
0.3
GEV2
NLIN1
NLIN2
NLIN3
GEV2
NLIN1
NLIN2
NLIN3
GEV2
NLIN1
NLIN2
NLIN3
181Ð3 173Ð1
192Ð6 183Ð1
187Ð0 174Ð3
190Ð8 176Ð3
187Ð5 179Ð3
199Ð0 189Ð5
192Ð3 179Ð6
196Ð2 181Ð6
194Ð2 186Ð0
207Ð0 197Ð5
198Ð9 186Ð2
201Ð1 186Ð6
0Ð06 0Ð10 0Ð15 0Ð50 1Ð52
0Ð05 0Ð11 0Ð10 0Ð17 0Ð72
−0Ð05 0Ð14 0Ð14 0Ð54 3Ð23
0Ð07 0Ð24 0Ð25 0Ð77 5Ð08
0Ð10 0Ð15 0Ð16 0Ð15 0Ð21
0Ð07 0Ð13 0Ð09 0Ð53 −2Ð04
−0Ð04 0Ð18 0Ð24 0Ð13 2Ð64
0Ð05 0Ð29 0Ð38 0Ð20 4Ð22
0Ð15 0Ð17 0Ð10 0Ð92 4Ð23
0Ð11 0Ð18 0Ð10 1Ð26 5Ð64
−0Ð02 0Ð22 0Ð36 −0Ð73 0Ð06
0Ð04 0Ð36 0Ð59 1Ð14 −0Ð01
0Ð30 0Ð38 0Ð52 1Ð54 3Ð89
0Ð59 0Ð64 0Ð70 1Ð33 3Ð00
0Ð44 0Ð51 0Ð57 1Ð54 5Ð00
0Ð65 0Ð72 0Ð77 1Ð77 6Ð54
0Ð31 0Ð44 0Ð60 1Ð88 5Ð06
0Ð59 0Ð66 0Ð76 1Ð94 5Ð36
0Ð44 0Ð53 0Ð64 1Ð74 5Ð97
0Ð67 0Ð77 0Ð86 1Ð64 6Ð43
0Ð33 0Ð47 0Ð67 2Ð50 8Ð05
0Ð60 0Ð70 0Ð82 2Ð60 8Ð78
0Ð43 0Ð57 0Ð73 2Ð08 6Ð81
0Ð71 0Ð86 1Ð03 2Ð06 5Ð78
GEV2 model for D 0Ð1 and D 0Ð2 but the CDNNLIN2 model for D 0Ð3. GEVstep test case A neural network with a single hidden node provides a continuous approximation to the step function. For the GEVstep test case, where nonstationary in the location parameter follows a step function, the CDN-NLIN1 model should, therefore, be selected by BIC and AICc. As shown in Table IV, this is indeed the case. Corresponding values of bias and rmse for the CDN-NLIN1 model are low for all and < 0Ð999, with moderate increases noted for D 0Ð999. Failure to specify a nonstationary model results in large increases in (absolute) bias and rmse. Comparing CDN-NLIN1 results with those from CDN-NLIN2 and CDN-NLIN3 suggests that overfitting lowers model performance, but that increases in (absolute) bias and rmse are small relative to those that result from using a stationary model. Confidence intervals Confidence intervals for GEV distribution parameters and associated quantiles can be estimated by bootstrapbased methods. For nonstationary GEV models, Kharin and Zwiers (2004) and Khaliq et al. (2006) recommend the residual bootstrap. Following Khaliq et al. (2006), the method proceeds by (i) fitting a nonstationary GEV model to the observed data; (ii) transforming residuals from the fitted model to be identically distributed yt t 1/ εt D 1 t 20 ˛t where εt is the tth transformed residual, and t, ˛t, and t are GEV parameters from the fitted model; (iii)
resampling the transformed residuals with replacement to form a bootstrapped set of residuals fεb t, t D 1, . . . , Ng; (iv) rescaling the bootstrapped residuals by inverting the transformation y b t D t ˛t
εb t 1 ; t
21
(v) fitting a nonstationary GEV model to the bootstrapped samples; (vi) estimating parameters and quantiles from the fitted model; and (vii) repeating steps (i) to (vi) for a large number of times. In a comparison of bootstrap-based methods for estimating confidence intervals of stationary extreme value models, Kysel´y (2008) founds the parametric bootstrap to outperform a resampling-based bootstrap. In the parametric bootstrap, bootstrap samples [steps (ii) to (iv) in previous paragraph] are generated by randomly sampling directly from the fitted distribution, rather than by resampling and rescaling model residuals. In either case, confidence intervals can be formed by calculating percentiles (e.g. the 5th and 95th percentiles for the 90% confidence interval) of the bootstrapped parameter/quantile estimates. Bias-corrected alternatives to the percentile method are reviewed by Kysel´y (2008), but their application is beyond the scope of this study. Monte Carlo simulations are used to evaluate whether empirical coverage probabilities of residual and parametric bootstrapped confidence intervals for the GEV-CDN model parameters and quantiles match those expected. GEV-CDN models, selected based on minimum AICc and BIC statistics from Tables I–4, are fit to 50 samples with time-dependent GEV parameters as specified in Figure 3. One thousand trials are run for each test case. For each Monte Carlo trial, confidence intervals are
Copyright 2009 Her Majesty the Queen in right of Canada. Published by John Wiley & Sons. Ltd
Hydrol. Process. 24, 673– 685 (2010)
682
1Ð06 1Ð17 1Ð33 2Ð49 6Ð30 0Ð71 0Ð81 0Ð96 2Ð25 6Ð82 0Ð32 0Ð45 0Ð63 2Ð19 7Ð17 2Ð52 3Ð02 3Ð34 3Ð65 9Ð99 5Ð06 5Ð95 7Ð40 13Ð06 22Ð37 0Ð90 0Ð99 1Ð08 1Ð88 5Ð71 0Ð30 0Ð40 0Ð53 1Ð69 5Ð14 2Ð52 2Ð94 3Ð25 3Ð34 5Ð08
0Ð62 0Ð70 0Ð79 1Ð80 5Ð59
0Ð02 0Ð32 0Ð59 1Ð50 1Ð62 0Ð01 0Ð22 0Ð40 1Ð10 −1Ð45 0Ð01 −0Ð11 −0Ð22 −0Ð77 1Ð46 0Ð20 1Ð62 2Ð11 0Ð59 7Ð92 0Ð42 3Ð21 5Ð40 11Ð41 16Ð01 0Ð01 0Ð27 0Ð41 0Ð33 1Ð99 0Ð00 0Ð00 −0Ð10 0Ð00 1Ð06
5Ð07 5Ð89 7Ð32 13Ð39 22Ð13 0Ð77 0Ð84 0Ð91 1Ð69 5Ð29 0Ð56 0Ð62 0Ð68 1Ð48 4Ð20 0Ð30 0Ð37 0Ð46 1Ð35 3Ð75 2Ð52 2Ð90 3Ð22 3Ð54 3Ð44 5Ð08 5Ð87 7Ð46 15Ð17 27Ð34
0Ð03 0Ð15 0Ð18 0Ð22 2Ð07 0Ð09 1Ð43 1Ð99 2Ð28 0Ð70 0Ð62 3Ð07 5Ð50 14Ð05 26Ð28
−0Ð01 −0Ð06 −0Ð06 0Ð25 1Ð49
0Ð04 0Ð24 0Ð31 0Ð23 3Ð13
0Ð49 3Ð10 5Ð30 12Ð02 19Ð92
0Ð14 1Ð50 2Ð00 1Ð47 2Ð65
0Ð02 0Ð18 0Ð27 0Ð18 1Ð50
198Ð9 184Ð4 196Ð8 184Ð1 192Ð8 183Ð3 270Ð5 263Ð7 320Ð8 315Ð6 194Ð1 179Ð6 186Ð9 177Ð4 188Ð7 174Ð2 185Ð7 173Ð0 181Ð4 171Ð9 262Ð0 255Ð2 317Ð6 312Ð4
constructed by fitting GEV-CDN models to 500 bootstrapped datasets and calculating percentiles of the resulting parameter/quantile distributions. Empirical coverage proportions for 90% confidence intervals of the 0Ð5, 0Ð8, 0Ð9, 0Ð99, and 0Ð999 -quantiles, along with , , and parameters, are calculated and compared with the nominal value of 0Ð9. Results are shown in Table V for GEV1, GEV11, GEV2, and GEVstep test cases. Bootstrapped confidence intervals for the location parameter and scale parameter ˛ tend to be too narrow for both the residual and parametric bootstrap. Coverage proportions for ˛, in particular, are much lower than 0Ð9 for all test cases. Conversely, the parametric bootstrap leads to confidence intervals for the shape parameter that are too broad, with coverage proportions exceeding 0Ð9 in 11 of 12 test cases. In terms of confidence intervals for quantiles, coverage proportions for low to intermediate quantiles ( D 0Ð5, 0Ð8, and 0Ð9) are, again, narrower than expected for both the residual and parametric bootstrap. Results are mixed when moving to higher quantiles ( D 0Ð99 and 0Ð999). The residual bootstrap continues to generate confidence intervals that are too narrow, whereas the parametric bootstrap generates broader than expected confidence intervals. Aggregated over all test cases and values of , the parametric bootstrap performs better than the residual bootstrap; coverage proportions are within 0Ð05 of the nominal value in 34 out of 60 combinations of test case and for the parametric bootstrap, but just 23 of 60 combinations for the residual bootstrap.
PRECIPITATION DATA
Statistic BIC AICc bias D 0.5 0.8 0.9 0.99 0.999 rmse D 0.5 0.8 0.9 0.99 0.999
NLIN1
318Ð4 313Ð2
265Ð4 258Ð7
190Ð6 177Ð9
NLIN1 NLIN1 LIN
NLIN2 D
CON
LIN
0.1
NLIN2
NLIN3
CON
0.2
Table IV. As in Table I but for the GEVstep test case
NLIN3
CON
LIN
0.3
NLIN2
NLIN3
A. J. CANNON
Following El Adlouni et al. (2007), nonstationary GEVCDN models are demonstrated on annual precipitation data recorded at Randsburg, California (station number 047253, latitude 35Ð22, longitude 117Ð39, elevation 36 m). In the analysis by El Adlouni et al. (2007), stationary (GEV0) and nonstationary (GEV1 and GEV2) models were fit to the precipitation series. The southern oscillation index (SOI) was used as a covariate describing ENSO conditions in the two nonstationary models. The GEV2 model, representing a quadratic dependence between the GEV location parameter and SOI, was found to best describe the relationship between annual precipitation and ENSO at the station of interest. The analysis is replicated here using the GEV-CDN modelling framework. In addition to the SOI, the Pacific decadal oscillation (PDO) index is included as a second covariate, as the PDO, which varies on an interdecadal rather than interannual time scale, has been found to modulate the effect of ENSO teleconnections on precipitation in the western United States (Gershunov and Barnett, 1998; McCabe and Dettinger, 1999; Gutzler et al., 2002). Precipitation data from 1937 to 2007 are obtained from the Western Regional Climate Centre of the National Oceanographic and Atmospheric Administration. Concurrent PDO and SOI indices for the spring season are
Copyright 2009 Her Majesty the Queen in right of Canada. Published by John Wiley & Sons. Ltd
Hydrol. Process. 24, 673–685 (2010)
0Ð93 0Ð94 0Ð89 0Ð86 0Ð87 0Ð81 0Ð92 0Ð90 0Ð84 0Ð85 0Ð86 0Ð77 0Ð87 0Ð83 0Ð75 0Ð83 0Ð84 0Ð76 0Ð82 0Ð83 0Ð77 0Ð83 0Ð84 0Ð80 0Ð85 0Ð85 0Ð83 0Ð84 0Ð86 0Ð86 0Ð92 0Ð96 0Ð94 0Ð85 0Ð87 0Ð88 0Ð72 0Ð78 0Ð76 0Ð86 0Ð86 0Ð85
0Ð74 0Ð74 0Ð73
0Ð96 0Ð95 0Ð91 0Ð90 0Ð90 0Ð85 0Ð95 0Ð90 0Ð80 0Ð89 0Ð87 0Ð75 0Ð84 0Ð77 0Ð72 0Ð80 0Ð80 0Ð76 0Ð80 0Ð77 0Ð75 0Ð80 0Ð80 0Ð80 0Ð82 0Ð80 0Ð82 0Ð82 0Ð82 0Ð82 0Ð92 0Ð97 0Ð98 0Ð90 0Ð92 0Ð92 0Ð79 0Ð82 0Ð86 0Ð83 0Ð81 0Ð83
0Ð74 0Ð73 0Ð75
0Ð92 0Ð91 0Ð84 0Ð87 0Ð85 0Ð80 0Ð90 0Ð88 0Ð80 0Ð85 0Ð83 0Ð77 0Ð85 0Ð83 0Ð78 0Ð82 0Ð81 0Ð81 0Ð85 0Ð83 0Ð81 0Ð84 0Ð82 0Ð84 0Ð88 0Ð86 0Ð85 0Ð87 0Ð87 0Ð89 0Ð90 0Ð95 0Ð92 0Ð88 0Ð87 0Ð84 0Ð80 0Ð80 0Ð82 0Ð88 0Ð86 0Ð85
0Ð78 0Ð80 0Ð80
0Ð93 0Ð95 0Ð85 0Ð86 0Ð87 0Ð82 0Ð91 0Ð91 0Ð80 0Ð84 0Ð83 0Ð80 0Ð86 0Ð85 0Ð76 0Ð83 0Ð82 0Ð80 0Ð85 0Ð85 0Ð80 0Ð84 0Ð85 0Ð84 0Ð86 0Ð86 0Ð85 0Ð86 0Ð88 0Ð88 0Ð93 0Ð96 0Ð94 0Ð86 0Ð89 0Ð87 0Ð77 0Ð80 0Ð83 0Ð86 0Ð85 0Ð84
0Ð80 0Ð80 0Ð76
Parametric
683
obtained from the Joint Institute for the Study of the Atmosphere and Ocean at the University of Washington. For reference, annual time series of SOI, PDO, and precipitation data are shown in Figure 4. The SOI is negatively correlated with precipitation, whereas the PDO is positively correlated with precipitation. Negative values of the SOI (e.g. El Ni˜no years) and positive values of the PDO (e.g. warm PDO phase years) tend to coincide with wetter years; the opposite tendency occurs during La Ni˜na and cool PDO phase years. Combinations of PDO and SOI indices are entered as covariates in CDN-LIN, CDN-NLIN1, CDN-NLIN2, and CDN-NLIN3 models. For each combination, models with (i) nonstationary , ˛, and parameters; (ii) models with nonstationary and ˛ parameters; and (iii) models with a nonstationary parameter are fit to precipitation data separately. The top five covariate/model combinations recommended by AICc and BIC are listed in Table VI. All are nonlinear, with the top four involving both SOI and PDO as covariates. The CDN-NLIN2 model with SOI and PDO as predictors of the location and scale ˛ parameters is recommended as the best model. Contour plots of CDN-NLIN2 modelled relationships between the covariates and the GEV parameters are shown in Figure 5, along with a sample plot of the D 0Ð90 quantile (10-year return period). Threshold behaviour is present, resulting in two main regions of SOI/PDO space, each with near constant values of , ˛, and the precipitation quantile. The primary influence on precipitation appears to be SOI: largest values tend to occur when SOI anomalies are more than one standard deviation below the mean (e.g. El Ni˜no years), whereas lowest values occur when the SOI index is near zero or positive (e.g. neutral or La Ni˜na years). The PDO modifies the El Ni˜no signal. When the PDO index exceeds ¾1Ð5 standard deviations from the mean during El Ni˜no events, the GEV location parameter increases while the scale parameter ˛ decreases. As noted previously, neural network models are capable of modelling interactions between covariates without a priori specification of the form of the interaction. In this example, the modification of ENSO impacts by PDO phase would be difficult to capture with a nonstationary linear model. One would, for example, need to explicitly include an interaction term in the regression equation. Conversely, the CDN-NLIN2 model detects the nonlinearity automatically from the training data.
0Ð1 0Ð2 0Ð3 GEVstep
0Ð84 0Ð85 0Ð86
0Ð1 0Ð2 0Ð3 GEV2
0Ð82 0Ð82 0Ð81
0Ð1 0Ð2 0Ð3 GEV11
0Ð87 0Ð86 0Ð88
0Ð1 0Ð2 0Ð3
0Ð86 0Ð88 0Ð88
DISCUSSION AND CONCLUSION
GEV1
0.999
Residual Parametric
0.99
Residual Parametric
0.9
Residual Parametric
0.8
Residual Parametric
D 0Ð5
Residual Parametric
Residual Parametric
˛
Residual Parametric Residual
Test case
Table V. Empirical coverage proportions for residual and parametric bootstrapped 90% confidence intervals of GEV parameters and -quantiles from GEV-CDN models applied to GEV1, GEV11, GEV2, and GEVstep test cases. Coverage proportions that differ by more than 0Ð05 from the nominal value of 0Ð90 are italicized. Those that are greater than or equal to 0Ð90 are marked in bold
FLEXIBLE MODELLING FRAMEWORK FOR NONSTATIONARY GEV
Parameters in a GEV distribution are specified as a function of covariates using a probabilistic variant of the MLP neural network model. If the covariate is time, or is dependent on time, then the resulting GEV-CDN model can be used to perform nonstationary GEV analysis. The use of a neural network architecture means that the model is capable of representing a wide range of linear and nonlinear relationships among covariates and the
Copyright 2009 Her Majesty the Queen in right of Canada. Published by John Wiley & Sons. Ltd
Hydrol. Process. 24, 673– 685 (2010)
684
4
40
–4 –2
0
60
2
r = –0.52
0
20
Precipitation (mm)
80
6
8
r = +0.48
SOI, PDO+5 (sdev)
100
A. J. CANNON
1940
1950
1960
1970
1980
1990
2000
Year
Figure 4. Time series of annual precipitation at Randsburg (vertical bars), along with standardized SOI (dashed line) and PDO (dotted line) anomalies. Values of r are Spearman rank correlation coefficients between SOI/PDO and precipitation
Table VI. Top five models in terms of AICc and BIC for Randsburg precipitation series Model
Nonstationary
Covariate(s)
AICc
BIC
CDN-NLIN2 CDN-NLIN3 CDN-NLIN2 CDN-NLIN3 CDN-NLIN2
, , , , ,
SOI, SOI, SOI, SOI, SOI
453Ð7 454Ð8 457Ð3 465Ð5 479Ð5
476Ð5 481Ð9 482Ð1 493Ð4 499Ð7
PDO PDO PDO PDO
location, scale, and shape parameters of the GEV distribution. For example, the neural network can exactly replicate the GEV0, GEV1, and GEV11 models of El Adlouni et al. (2007), and can approximate the GEV2 model with good accuracy. Other forms of nonlinearity, for example, step changes, interactions between covariates, and higher-order polynomial relationships, can also be modelled with the same architecture. Two model selection criteria, AICc and BIC, correctly identified the generating model for four weakly nonstationary synthetic test datasets. Although this paper focused on results from the single ‘best’ model, model averaging, which involves taking a weighted average of multiple models, has been recommended as a means of Location µ
2 SOI
1
2 1 –1
0 PDO
1
2
–1 –2
–1 –2 –2
τ = 0.90
(c)
0
SOI
1 0 –2
–1
SOI
Scale α
(b)
2
(a)
0
˛ ˛ ˛, ˛, ˛
improving estimation performance Burnham and Anderson, 2004. This approach has been applied successfully in the context of CDN models by Carney et al. (2005) and is worth exploring for GEV-CDN models. Confidence intervals calculated via the residual and parametric bootstrap led to overly optimistic estimates of uncertainty for low to moderate -quantiles. This pattern continued for high -quantiles with the residual bootstrap, whereas the parametric bootstrap led to broader than expected confidence intervals. Aggregated over all test cases, the parametric bootstrap outperformed the residual bootstrap. In general, results are similar to those found by Kharin and Zwiers (2004) for nonstationary models and Kysel´y (2008) for stationary models. It is possible that alternative bootstrap approaches, for example, the bias-adjusted percentile estimators evaluated by Kysely (2008), might yield better calibrated confidence intervals, although improvements were modest for stationary GEV models. Application of the GEV-CDN models to precipitation data from southern California identified a nonlinear relationship among PDO, SOI, and parameters of the GEV distribution. Results are consistent with other work linking interannual/interdecadal modes of climate variability
–2
–1
0
1
2
PDO
–2
–1
0
1
2
PDO
Figure 5. Contour plots of CDN-NLIN2 relationships among SOI, PDO, and (a) GEV location parameter , (b) GEV scale parameter ˛, and (c) D 0Ð90 precipitation quantile. The GEV shape parameter D 0Ð02. Black dots indicate observed SOI and PDO index values Copyright 2009 Her Majesty the Queen in right of Canada. Published by John Wiley & Sons. Ltd
Hydrol. Process. 24, 673–685 (2010)
FLEXIBLE MODELLING FRAMEWORK FOR NONSTATIONARY GEV
to local hydroclimatological datasets, for example, work by Gershunov and Barnett (1998) among others. Caution is required in the interpretation of the GEV-CDN model due to the small number of samples in some regions of the SOI/PDO phase space. The result does, however, demonstrate the ability of the GEV-CDN model to account for interactions between covariates without their a priori specification. One perceived limitation of MLP neural networks and, by extension CDN models, is that they are ‘black boxes’ in which modelled relationships are difficult to interpret. It bears noting that sensitivity analysis methods, for example, the one used by Cannon and McKendry (2002), are applicable to CDN models and could be used to identify the form of nonlinear relationships between covariates and GEV distribution parameters or quantiles.
ACKNOWLEDGEMENTS
The author would like to thank three anonymous reviewers for their helpful and constructive comments. Portions of this work were conducted while visiting the Climate Prediction Group in the Department of Earth and Ocean Sciences at The University of British Columbia.
REFERENCES Akaike H. 1974. A new look at the statistical model identification. IEEE Transactions on Automatic Control 19: 716– 723. Bishop CM. 2006. Pattern Recognition and Machine Learning. SpringerVerlag: New York. Burnham KP, Anderson DR. 2004. Multimodel inference: understanding AIC and BIC in model selection. Sociological Methods and Research 33: 261– 304. Cannon AJ. 2008. Probabilistic multisite precipitation downscaling by an expanded Bernoulli-gamma density network. Journal of Hydrometeorology 9: 1284– 1300. Cannon AJ, McKendry IG. 2002. A graphical sensitivity analysis for statistical climate models: Application to Indian monsoon rainfall prediction by artificial neural networks and multiple linear regression models. International Journal of Climatology 22: 1687– 1708. Carney M, Cunningham P, Dowling J, Lee C. 2005. Predicting probability distributions for surf height using an ensemble of mixture density networks. ICML ‘05: Proceedings of the 22nd International Conference on Machine Learning, ACM, New York, 113–120. Carpenter WC, Barthelemy JFM. 1993. A comparison of polynomial approximations and artificial neural nets as response surfaces. Structural and Multidisciplinary Optimization 5: 166–174. Cawley GC, Janacek GJ, Haylock MR, Dorling SR. 2007. Predictive uncertainty in environmental modelling. Neural Networks 20: 537– 549. Chavez-Demoulin V, Davison A. 2005. Generalized additive modelling of sample extremes. Applied Statistics 54: 207–222. Christiansen B. 2005. The shortcomings of nonlinear principal component analysis in identifying circulation regimes. Journal of Climate 18: 4814– 4823. Coles S. 2001. An Introduction to Statistical Modeling of Extreme Values. Springer: London. Coles SG, Dixon MJ. 1999. Likelihood-based inference for extreme value models. Extremes 2: 5–23. Dawson CW, Wilby RL. 2001. Hydrological modelling using artificial neural networks. Progress in Physical Geography 25: 80– 108. El Adlouni S, Ouarda TBMJ, Zhang X, Roy R, Bobee B. 2007. Generalized maximum likelihood estimators for the nonstationary generalized extreme value model. Water Resources Research 43: W03410. Gaetan C, Grigoletto M. 2004. Smoothing sample extremes with dynamic models. Extremes 7: 221–236.
685
Gardner MW, Dorling SR. 1998. Artificial neural networks (the multilayer perceptron)– a review of applications in the atmospheric sciences. Atmospheric Environment 32: 2627– 2636. Gershunov A, Barnett TP. 1998. Interdecadal modulation of ENSO teleconnections. Bulletin of the American Meteorological Society 79: 2715– 2725. Gutzler DS, Kann DM, Thornbrugh C. 2002. Modulation of ENSO-based long-lead outlooks of Southwestern US winter precipitation by the Pacific decadal oscillation. Weather and Forecasting 17: 1163– 1172. Hall P, Tajvidi N. 2000. Nonparametric analysis of temporal trend when fitting parametric models to extreme-value data. Statistical Science 15: 153– 167. Hornik K, Stinchcombe M, White H. 1989. Multilayer feedforward networks are universal approximators. Neural Networks 2: 359– 366. Hosking J. 1990. L-moments: analysis and estimation of distributions using linear combinations of order statistics. Journal of the Royal Statistical Society, Series B 52: 105–124. Hosking J, Wallis J, Wood E. 1985. Estimation of the generalized extreme value distribution by the method of probability-weighted moments. Technometrics 27: 251– 261. Hsieh WW, Tang B. 1998. Applying neural network models to prediction and data analysis in meteorology and oceanography. Bulletin of the American Meteorological Society 79: 1855– 1870. Hurvich CM, Tsai C. 1989. Regression and time series model selection in small samples. Biometrika 76: 297– 307. Jain S, Lall U. 2001. Floods in a changing climate: does the past represent the future? Water Resources Research 37: 3193– 3205. Jenkinson AF. 1955. The frequency distribution of the annual maximum (or minimum) values of meteorological elements. Quarterly Journal of the Royal Meteorological Society 81: 158–171. Khaliq M, Ouarda T, Ondo J-C, Gachon P, Bob´ee B. 2006. Frequency analysis of a sequence of dependent and/or non-stationary hydrometeorological observations: A review. Journal of Hydrology 329: 534– 552. Kharin VV, Zwiers FW. 2004. Estimating extremes in transient climate change simulations. Journal of Climate 18: 1156– 1173. Koenker R, Schorfheide F. 1994. Quantile spline models for global temperature change. Climatic Change 28: 395–404. Kysel´y J. 2008. A cautionary note on the use of nonparametric bootstrap for estimating uncertainties in extreme-value models. Journal of Applied Meteorology and Climatology 47: 3236– 3251. Kysel´y J, Picek J. 2007. Regional growth curves and improved design value estimates of extreme precipitation events in the Czech Republic. Climate Research 33: 243– 255. Maidment DR. 1993. Handbook of Hydrology. McGraw-Hill Professional: New York. Martins ES, Stedinger JR. 2000. Generalized maximum-likelihood generalized extreme-value quantile estimators for hydrologic data. Water Resources Research 36: 737– 744. McCabe GJ, Dettinger MD. 1999. Decadal variations in the strength of ENSO teleconnections with precipitation in the western United States. International Journal of Climatology 19: 1399– 1410. Milly PCD, Betancourt J, Falkenmark M, Hirsch RM, Kundzewicz ZW, Lettenmaier DP, Stouffer RJ. 2008. Stationarity Is Dead: Whither Water Management? Science 319: 573– 574. Nash JC. 1990. Compact Numerical Methods for Computers. Linear Algebra and Function Minimisation. Adam Hilger: Bristol. Neuneier R, Hergert F, Finnoff W, Ormoneit D. 1994. Estimation of conditional densities: a comparison of neural network approaches. In Proceedings of ICANN 94 , Marinaro M, Morasso P (eds). Springer: Berlin; 689–692. Rial JA, Pielke RA, Beniston M, Claussen M, Canadell J, Cox P, Held H, De Noblet-Ducoudre N, Prinn R, Reynolds JF, Salas JD. 2004. Nonlinearities, feedbacks and critical thresholds within the Earth’s climate system. Climatic Change 65: 11–38. Schwarz G. 1978. Estimating the dimension of a model. The Annals of Statistics 6: 461– 464. Shubert SD, Chang Y, Suarez MJ, Pegion PJ. 2008. Enso and wintertime extreme precipitation events over the contiguous united states. Journal of Climate 21: 22–39. Smyth P. 2000. Model selection for probabilistic clustering using crossvalidated likelihood. Statistics and Computing 9: 63–72. Wang XL, Zwiers FW, Swail VR. 2004. North Atlantic ocean wave climate change scenarios for the twenty-first century. Journal of Climate 17: 2368– 2383. Yee TW, Stephenson AG. 2007. Vector generalized linear and additive extreme value models. Extremes 10: 1–19.
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Hydrol. Process. 24, 673– 685 (2010)
