Advances in Water Resources 78 (2015) 145–154

Contents lists available at ScienceDirect

Advances in Water Resources

journal homepage: www.elsevier .com/ locate/advwatres

A frailty-contagion model for multi-site hourly precipitation drivenby atmospheric covariates

http://dx.doi.org/10.1016/j.advwatres.2015.01.0010309-1708/� 2015 Elsevier Ltd. All rights reserved.

⇑ Corresponding author.E-mail addresses: erwan.koch@math.ethz.ch (E. Koch), philippe.naveau@lsce.

ipsl.fr (P. Naveau).

Erwan Koch a,⇑, Philippe Naveau b

a ISFA, CREST and ETH Zurich (Department of Mathematics, RiskLab), Switzerlandb LSCE (CNRS), France

a r t i c l e i n f o a b s t r a c t

Article history:Received 9 April 2014Received in revised form 27 November 2014Accepted 2 January 2015Available online 9 January 2015

Keywords:Common factorContagionPrecipitation simulatorsSpatial–temporal dependenceWeather generators

Accurate stochastic simulations of hourly precipitation are needed for impact studies at local spatialscales. Statistically, hourly precipitation data represent a difficult challenge. They are non-negative,skewed, heavy tailed, contain a lot of zeros (dry hours) and they have complex temporal structures(e.g., long persistence of dry episodes). Inspired by frailty-contagion approaches used in finance andinsurance, we propose a multi-site precipitation simulator that, given appropriate regional atmosphericvariables, can simultaneously handle dry events and heavy rainfall periods. One advantage of our modelis its conceptual simplicity in its dynamical structure. In particular, the temporal variability is represent-ed by a common factor based on a few classical atmospheric covariates like temperatures, pressures andothers. Our inference approach is tested on simulated data and applied on measurements made in thenorthern part of French Brittany.

� 2015 Elsevier Ltd. All rights reserved.

1. Introduction

Stochastic weather generators (WG) are statistical models thataim at simulating quickly and realistically random sequences ofatmospheric variables like temperatures, precipitation and windspeeds. Historically, weather generators started in hydrologicalsciences in the sixties and seventies. In 1962, [13] proposed aMarkov model for daily precipitation occurrences. Since then, astrong research effort in modeling precipitation distributions hassustained in the hydrological and statistical communities. Thisattention towards precipitation WGs can be explained by, at least,two different reasons. These stochastic simulations can be used inassessment studies, especially these linked to water resourcesmanagements. As one of the initial drivers in impact studies,simulated times series of precipitation can play a fundamental rolein exploring some part of the sensitivity of floods, erosion andcrops models. The second reason, very attractive to applied statis-ticians, is that modeling accurately precipitation distributions inspace and time has always been an intriguing mathematical chal-lenge. From a probabilistic point of view, a sequence of precipita-tion mixes two types of events (dry or wet) and the rainfallintensity represents a strong departure from the Gaussian territory

(positive, skewed and sometime heavy tailed). Spatially and tem-porally, events can be strongly correlated within dry or wet epi-sodes. Still today, the spatio-temporal dynamics of precipitationis difficult to model statistically. Numerous elaborate parametricapproaches have been proposed in recent years (see, e.g.[3,12,23,30]) (non-parametric approaches also exist but we willnot discuss them here).

From a probabilistic point of view, most of daily precipitationgenerators decouple the occurrence and intensity processes. Forexample, [22] first generated spatially and temporally correlatedrainfall occurrences and then, at locations with positive precipita-tion, simulated their intensities. This strategy is classical in the WGliterature (see, e.g. [20,26]). It is mathematically convenient, interms of inference, to frame the estimation scheme into a cleartwo step algorithm. But the hypothesis that the occurrence processcan be simulated before the intensity process, i.e. independently ofpast, present or future rainfall amounts, could be challenged, espe-cially at the hourly scale. If very large (small) rainfall amounts havebeen observed at time t, it seems more likely that a wet (dry) hourwill follow at time t þ 1. To quantify this naive reasoning, we needto introduce the example that will lead all our discussions in thisarticle. We will work with winter (DJF) hourly precipitation record-ed in the northern part of Brittany in France at three stations from1995 to 2011, see Fig. 1. In a nutshell, winter precipitation in thisregion are generally due to large scale perturbations coming fromthe Atlantic ocean.

Table 1Rain occurrence probability at time t þ 1 given the rainfall intensity at t. Each rowrepresents a specific range of hourly rainfall intensity. The Chi-Square p-values testthe independence hypothesis.

Intensity range (mm) Weather stations in Fig. 1

Brest-Guipavas Landivisiau Pleyber-Christ

0.2–0.8 0.63 0.65 0.680.8–1.4 0.74 0.75 0.791.4–2 0.81 0.82 0.822–4 0.90 0.91 0.91> 4 0.94 0.94 0.94

Chi-Square test p-value 8:40� 10�64 4:57� 10�59 4:39� 10�43

146 E. Koch, P. Naveau / Advances in Water Resources 78 (2015) 145–154

Coming back to the question of simulating the occurrence pro-cess independently of the intensity, Table 1 shows that the prob-ability of rainfall occurrence at time t þ 1 increases in function ofthe rainfall intensity at date t for each station. To couple the occur-rence and intensity processes, a possible approach (see e.g. [2,22]who discussed this issue on p. 4) is to first generate a Gaussian ran-dom field, say Zt . Secondly, a censoring mechanism applied to thisfield produces rainfall occurrences, i.e. wet events occur wheneverZt is above a given threshold. Then, rainfall intensities are obtainedby transforming the Gaussian values above the threshold into posi-tive quantities (e.g., see [4]). The resulting rainfall intensity distri-bution depends on the choice of the transform function that mayhave different flavors (see, e.g. [3,12,23,30,31]). Concerning thecorrelation structure, the hidden process Zt , by construction, drivesboth the intensity and occurrence processes that have becomedependent. Another strategy [27] is to directly fit a discrete–con-tinuous at-site mixture distribution with two components (zeroor rainfall amount). The parameters of these two components arethen modeled by spatio-temporal processes (General AdditiveModel (GAM) or General Linear Model (GLM)).

This idea of working with an unobserved ‘‘seed’’ process ismathematically appealing but it adds some difficulties. In termsof statistical inference, it makes the estimation difficult.Concerning the interpretation, the latent process and its associatedparameters are not necessary simple to explain. Although it hasbeen a classical strategy to introduce hidden quantities in weathergenerators, for example see the abundant WG literature concern-ing hidden Markov structures like [1,28,34], it would be a clearimprovement, in terms of interpretation, if multi-site precipitationgenerators could bypass the hypothesis of hidden processes anddirectly model raw observations. This implies that additional infor-mation characterizing the dynamical structure of the weather sys-tem has to be integrated in a different way.

A well-maintained weather station recording hourly precipita-tion is rarely climatologically isolated in the sense that atmospher-ic variables like temperatures, pressures or others should beavailable in the same region. Most likely, the weather station itselfmay monitor this type of data. In this context, our plan is to devel-op the following two simple ideas in order to propose a new multi-site precipitation generator:

Fig. 1. Three weather stations in northern Brittany (France) wi

(A) The main driver of current precipitation at a single locationare precipitation recorded one hour early at all sites.

(B) The second source of information comes from atmosphericvariables like temperatures and others that jointly impactall sites by dynamically driving the variability of the resi-duals obtained from step (A).

Although resembling to existing concepts, this two step road-map differs from the aforementioned studies by a few but impor-tant points. We will not represent the precipitation range by afinite mixture of distributions based on numerous weather types,each one representing different atmospheric conditions. Neitheran hidden Gaussian process nor a function to transform aGaussian variable into a Gamma type rainfall intensity are needed.

Our point (A) is not novel and it corresponds to the well-knownidea of an auto-regressive process (see, for example the Markovchain in [20]). Here, we will use a basic multivariate auto-regres-sive representation (e.g., see [9,17]). Point (B), modeling the resi-duals as a dynamical function of atmospheric variables, could beconsidered as more innovative in the WG literature. It is closelylinked to the research developed in the statistical downscaling lit-erature (e.g., see [24,32,33]). Downscaling approaches have mainlyfocused redon how to make the connection between circulationpatterns and local atmospheric variables at the daily scale (e.g.,see [10,29]). Besides a different temporal scale (hourly in our case),

th hourly winter precipitation records from 1995 to 2011.

E. Koch, P. Naveau / Advances in Water Resources 78 (2015) 145–154 147

our ‘‘spatial’’ component differs from downscaling. Our ‘‘common’’signal, ordinary atmospheric variables averaged over our threesites in northern Brittany, corresponds to an information sharedby all the sites, but it does not represent a regional feature overhundred of kilometers. The common point with downscaling israther the idea that precipitation simulators performances can beimproved if some appropriate explanatory variables can be inject-ed in the statistical model.

The most interesting point of our article resides in combiningideas (A) and (B). From this coupling, a very simple model canhandle precipitation in northern Brittany at the hourly scale, notonly in terms of moderate rainfalls but also in terms of intensedry periods and even heavy precipitation. Before describing indetail our model, see Section 2, we need to say that our modelingstrategy has been inspired by tools coming from two researchfields not directly linked to geosciences: systemic risk in insuranceand epidemic models.

Analyzing systemic risk in the financial system involves model-ing the dependence between institutions. A first kind of depen-dence comes from common risk factors. These are calledsystematic factors, or frailties. For instance, the risk of life insur-ance contracts depends on the general uncertain increase ofhuman lifetime, called longevity risk. As highlighted by the regula-tion since 2008, the second main source of dependence stems fromcontagion phenomenon between institutions. The contagion effectis due to the interconnections between banks and insurance com-panies by means of their debt and shares participations. Typically,the failure of a company will imply losses for its lenders, and may-be the failure of some of them. Then failure of some lenders of thelenders can occur, and so on. Originally, contagion models infinance come from the epidemic model introduced by Bailey[6,7], Kendall [21] and reintroduced in the so-called infectiousmodel used in a static framework by Davis and Lo [8].Specifications including both frailty and contagion effects areintroduced among others in [5,11,14–16].

Coming back to hourly precipitation data, our weather systemin Brittany also contains both a common factor (frailty term) anda contagion term. The analogy with the financial system is thefollowing. Each weather station corresponds to an institution.Our common factor represents large scale conditions that are, insome sense, exogenous to our system. For example, a storm frontcoming from the Altantic ocean is not directly produced by theweather system observed in Brittany. It is similar to an exogenousshock affecting the whole financial system. Then once large scaleprocesses are set then local physical processes are involved. Theseprocesses depend on the local characteristics. For instance, pres-ence of mountains or flows can originate a privilegiate directionfor thunder-storms propagation. It is some kind of contagion fromone site to another. These local interactions between sites are theequivalent of cross-holdings at the origin of contagion in thefinancial system. Contrary to the usual assumptions used infinance, the common factor in climatology can be observable,covariates as temperatures, winds, pressures, and so on are alsorecorded at the weather stations and they represent valuableinformation that we have to take into account. Another link withfinancial statistics resides in the assumption that rainfall vari-ability will be modeled by an heteroscedastic variance, a classicalfeature in econometrics. See e.g. the ARCH/GARCH modelliterature.

The remaining of the paper is organized as follows. Ourheteroscedastic multi-site rainfall generator is detailed inSection 2. Then Section 3 presents our inference method basedon maximum likelihood. Section 4 assesses our model’s perfor-mance on simulations and summarizes our case study results deal-ing with the northern part of Brittany. Finally a discussion is givenin Section 5.

2. A heteroscedastic multi-site rainfall generator

Denote M the number of weather stations and Pm;t the precipita-tion amount at station m recorded during the tth hour. Our multi-site model based on (A) and (B) is defined as follows

Pm;t ¼B0mPt�1 þ em;t if B0mPt�1 þ em;t P u;

0 if B0mPt�1 þ em;t < u;

(ð1Þ

where the scalar product

B0mPt�1 ¼ bm;1P1;t�1 þ � � � þ bm;MPM;t�1

between the vector Pt�1 ¼ ðP1;t�1; . . . ; PM;t�1Þ0 and Bm ¼ ðbm;1; . . . ;

bm;MÞ0 represents the multivariate auto-regressive vector of order

one that captures the dynamical local-scale effect, i.e. how theneighborhood behavior affects station m one hour later. TheM �M unknown auto-regressive coefficients bi;j can be concatenat-ed into a matrix, say B, composed of the rows B0m. For our Brittanyexample, large scale perturbations coming from the Atlantic oceanshould make the matrix B asymmetric. The Brest weather stationshould influence the two stations eastward, while the converseshould not be true.

The random variable em;t in (1) corresponds to the local vari-ability driven by a few atmospheric variables and em;t is simplymodeled by a sequence of zero-mean Gaussian independent ran-dom variables. The non-stationarity comes from the standarddeviation of em;t ;rt > 0, that varies with time t in a log linearfashion

lnrt ¼ h0Ft ; ð2Þ

where Ft represents a vector of d atmospheric explanatory variablesat time t. The vector of scalars h corresponds to the unknownregression terms of the log-linear model in Eq. (2). Concerning thespatio-temporal features of Eq. (2), neither h and Ft change from siteto site, they only contain a spatially pooled information shared byall sites, and only Ft varies with time. With a financial vocabulary,an economist to describe (1) and (2) will speak of a heteroscedasticmodel with a volatility driven by the ‘‘frailty term’’ (exogenouscommon factor) Ft .

To ensure non-negative precipitation and produce dry events,Eq. (1) contains the condition that rainfall only occur if the quanti-ty B0mPt�1 þ em;t is greater than some positive fixed threshold u. Thiscould be loosely interpreted as saying that the rainfall generated attime t, B0mPt�1 þ em;t , has to be large enough to be recorded by themth weather station. This view emphasizes the fundamental role ofhaving a temporally varying rt .

Imagine that all stations are dry at time t � 1. All elements ofthe vector Pt�1 would then be equal to zero. In this case, the onlyway to produce large (small) rainfall at time t is to randomly gen-erate a large (small) em;t . This can be done by choosing a large(small) volatility, rt , or equivalently a large (small) frailty termFt . In other words, large scale conditions are captured by Ft thatdrives the volatility of the system, and ultimately drives precipita-tion occurrences and strengths. This implies that heavy rainfallbehavior, as well as dry period persistence, can be reproduced onlyif an adequate vector Ft is chosen.

Compared to [22], the dependence between occurrence andintensity is directly built in Eq. (1). If it had rained a lot at timet � 1, then Pt�1 is large, and it is very likely to have a wet hour attime t because of (1). In the opposite case, if Pt�1 is equal to zero,it is rather unlikely to have a wet hour at time t, unless the volati-lity is large. So, current rainfall occurrences are correlated with theintensity value observed one hour early.

Concerning the rain intensity distribution itself, our hypothesisthat all em;t are normally distributed does not imply that our

148 E. Koch, P. Naveau / Advances in Water Resources 78 (2015) 145–154

simulated rainfall will follow a truncated Gaussian density, andconsequently this does not mean that heavy tailed behavior cannotbe simulated. The non-stationarity of rt explains this phe-nomenon. At any time, rt can take a large value and thereforesimulated precipitation resembles to a complex infinite mixtureof truncated Gaussian random variables with a wide range of stan-dard deviation. A consequence of this is that we do not need, atleast in our example, to apply a power transform to go back andforth between the Gaussian world and the rainfall values like in[4,2]. Another reason is that em;t does not represent raw precipita-tion but a type of increment between two consecutive hours, seeEq. (1).

As emphasized in the previous paragraphs, the choice of Ft isparamount in the overall capacity of our model to accurately simu-late hourly precipitation. In order to limit overfitting, the dimen-sion of d should not be too high and, to make our model useful,the type of covariates within Ft should be easy to obtain in mostrainfall applications. For example, the components of Ft in ournorthern Brittany case are hourly temperatures, pressures at thesea level, and humidity, respectively. For each atmospheric vari-able, we spatially average hourly values recorded over our threesites to get Ft . Before closing this section, we would like to empha-size that the choice of the threshold u plays an important role inaccurately modeling the length of dry periods. The coming two sec-tions will illustrate this point.

3. Inference

Given the vector Ft and the threshold u, our inference scheme isbased on maximizing the likelihood (ML) of model (1) with respectto the matrix of auto-regressive parameters B and the regressioncoefficients ðh0; . . . ; hdÞ where h0 represents the intercept. Thelog-likelihood function denoted LuðB; h0; . . . ; hdÞ for a given u canbe written as (see the proof in Appendix A)

LuðB; h0; . . . ; hdÞ ¼XT

t¼2

XM

m¼1

IfPm;tPug ln1rt

/Pm;t � B0mPt�1

rt

� �� ��

þ IfPm;t¼0g ln Uu� B0mPt�1

rt

� �� ��; ð3Þ

where Uð:Þ and /ð:Þ represent the cumulative distribution functionand the probability distribution function of the standardized nor-mal distribution, respectively. The vector ðh0; . . . ; hdÞ0 appearsthroughout rt , see Eq. (2). The indicator function IfPm;tPug, equal toone if Pm;t P u and zero otherwise, corresponds to our conditionin (1) that generates either a dry or wet hour at station m. Theseindicator functions and the absence of a closed form for Uð:Þ makethe derivation of explicit ML estimates impossible, these values canonly be obtained numerically. Our confidence intervals are derivedby using a profile log-likelihood approach ([25]).

The above ML approach assumed that the threshold u has beencorrectly chosen. As our weather station precision is 0.2 mm, thescalar u is iteratively set to small values close to this instrumentprecision limit, say 0.2, 0.3, 0.4, 0.5, 0.6 and 0.7. To find the optimalu from this set, we arbitrarily chose a first guest, say 0.5 mm, andimplement our ML inference that provides estimates forðB; h0; . . . ; hdÞ. For this set of ML estimates, we can simulate hourlyprecipitation for different values of u and then, choose the value ofu minimizing the difference between the observed mean length ofdry periods and the mean length of dry periods of the simulatedprecipitation. This process can be repeated with the new value uand we stop when the estimated threshold remains unchanged.To conclude on the choice of u, we note that, by construction, mod-el (1) will never simulate values above zero but smaller than u.Hence, if our fixed u is little bit greater than the instrument

precision of 0.2 mm, then very small but positive precipitation inBrittany are not considered in our ML optimization scheme, seeEq. (3). This technical detail does not play an important role inour overall modeling strategy but allowing u to be greater than0.2 mm improves substantially our capacity to reproduce dry peri-od lengths.

4. Application

For our simulations and Brittany example, a learning set of10;000 hours and a validation set of 26;816 hours are used toassess the performance of our estimation scheme. Fig. 2 summa-rizes our inference and validation scheme.

4.1. Simulations

To test our inference scheme, we simulate 100 independentreplicas for our model (1) with parameters that are chosen to besimilar to our Brittany example, see the first column of Table 2.For two sample sizes (100 and 1000), the parameter estimatesare derived by a classical ML approach based on Eq. (3) and indi-cates that inferring 13 parameters with a small sample may bedone with caution. The empirical mean, standard deviation andrelative error derived from our 100 replicas are shown in Table 2.As expected, the sample length plays an important role in the infer-ence. For a size of 100, the estimation can be difficult. In contrast,working with a sample size of 1000 provides accurate estimates. Inpractice, the sample size for hourly data is often long, i.e. morethan 1000 hours. For our Brittany example, we have inferred ourparameters with 10,000 observations. In our simulations, the infer-ence accuracy with a sample size of 10,000 points is slightly betterto the one with 1000 observations, except for h0 ¼ 30:63 which ismuch improved (the bias and stdev become �0.005 and 0.59,respectively).

4.2. Hourly precipitation in northern Brittany

The estimated auto-regressive coefficients bij of the matrix Bwith the 95% confidence intervals and coverage probability are dis-played in Table 3. The first column makes it clear that the weatherstation of Brest has a strong influence in the two eastward stations.In particular, the station of Landivisiau has a larger coefficient withBrest (.47) than with itself (.25). In contrast, the two stations ofLandivisiau and Pleyber-Christ, which are close to each other, donot have an impact on Brest. This pattern corresponds to theexpected behavior for this region where rainfall comes from theAtlantic side, see Fig. 1. This implies that Brest would be the firstone to be impacted by a westerly front, and then the two eastwardstations will be hit by the storm later on.

Concerning the three atmospheric variables in Ft (temperatures,pressures at the sea level, and humidity) that drive our variabilityrt , Table 4 provides the respective estimated regression coeffi-cients from (2) with their associated 95% confidence intervalsand coverage probability. The values of hj for j ¼ 1;2;3 are smallbut their 95% confidence intervals do not contain zero. So, we keepthese three explanatory variables in our rainfall generator.

To visualize the predictive capacity of our model that has beenfitted on the training set, we can generate synthetic hourly pre-cipitation trajectories and compare them to the one kept in ourvalidation period. For each weather station shown in Fig. 1, Fig. 3compares observed rainfall (dark gray) with the simulated one(light gray) during 500 hours of the validation period, see Fig. 2.Note that we have chosen a relative short period for the sake of vis-ibility. However, the conclusions that can be drawn here are truewhatever the period considered in the validation set (figures

Fig. 2. Inference and validation scheme: a learning set of 10;000 hours is used to infer the parameters of our model. Then, assuming that the covariates are known in thefuture, we predict during the next 26;816 hours and compare forecasted rainfall (so-called out-of-sample predictions) with precipitation recorded in Brittany. A few visualsummaries of this forecast are shown in Figs. 3–8.

Table 2Inference assessment by simulations for two sample sizes of 100 and 1000. The empirical bias, standard deviation and relative error are derived from 100 independent replicassimulated from (1) with parameters given in the first column.

Sample size 100 1000

Parameter values Bias Stdev Relative error Bias Stdev Relative error

b11 ¼ 0:65 �2.08 17.71 �3.19 0.00 0.05 0.00b12 ¼ �0:08 �8.74 22.97 114.37 �0.01 0.06 0.07b13 ¼ 0:11 �13.30 30.62 �122.71 0.00 0.06 0.03b21 ¼ 0:47 �3.44 17.94 �7.37 0.01 0.05 0.02b22 ¼ 0:25 �5.57 13.83 �22.36 0.00 0.04 �0.01b23 ¼ 0:02 �7.76 20.43 �435.58 0.00 0.06 �0.09b31 ¼ 0:22 �8.64 27.02 �39.95 0.00 0.05 �0.01b32 ¼ 0:10 �7.17 21.43 �75.40 0.00 0.05 0.02b33 ¼ 0:36 �8.18 24.75 �22.65 �0.01 0.05 �0.03

h0 ¼ 30:63 �12.36 146.13 �0.40 0.19 1.54 0.01h1 ¼ 0:07 0.07 0.36 0.96 0.00 0.01 �0.01h2 ¼ 0:03 �0.06 0.14 �0.27 �0.07 0.00 0.01h3 ¼ 0:03 0.03 0.08 0.92 0.00 0.00 0.02

Table 3Estimated auto-regressive coefficients bij in (1) for the three weather stations plotted in Fig. 1. The intervals represent the 95% confidenceintervals and the number between brackets corresponds to the 95% coverage probability.

Matrix B Brest-Guipavas Landivisiau Pleyber-Christ

Brest-Guipavas 0.65 �0.08 0.11[0.59, 0.74] (87) [�0.15, 0.01] (86) [0.02, 0.19] (89)

Landivisiau 0.47 0.25 0.02[0.41, 0.53] (81) [0.17, 0.32] (92) [�0.06, 0.09] (91)

Pleyber-Christ 0.22 0.10 0.36[0.16, 0.27] (77) [0.02, 0.17] (82) [0.30, 0.43] (80)

Table 4Estimated regression coefficients in (2) and corresponding 95% confidence intervals in function of our threeexplanatory atmospheric variables.

Atmospheric variables Estimates Confidence interval Coverage probability

Regression intercept h0 ¼ 30:626 [28.02, 32.32] 80

Temperature h1 ¼ 0:070 [0.064, 0.076] 38

Seal level pressure h2 ¼ �0:034 [-0.037, �0.031] 94

Humidity h3 ¼ 0:028 [0.022, 0.036] 94

E. Koch, P. Naveau / Advances in Water Resources 78 (2015) 145–154 149

Fig. 3. Each panel corresponds to one of our three weather stations shown in Fig. 1. The dark and light gray color represent recorded precipitation and forecasted rainfall,respectively. Our predicted values are obtained on the validation period according to the scheme shown in Fig. 2. The regional factor rt in (2) is the only quantity supposed tobe known for the forecast. In particular, we do not use the precipitation recorded at time t to forecast the following hour or the following week.

Fig. 4. Out-sample quantile–quantile plot between observed rainfall amount (x-axis) and simulated one (y-axis) from model (1). Each panel corresponds to one of our threeweather stations shown in Fig. 1. The gray color corresponds to the 98% confidence interval and the solid line to the median.

150 E. Koch, P. Naveau / Advances in Water Resources 78 (2015) 145–154

available upon request). In this exercise, we assume that the threeatmospheric variables in Ft , temperature, sea level pressure andhumidity, are known at the regional scale.

The simulated rainfall appear to reproduce well the dynamicalstructure of real precipitation in the sense that clusters of simulat-ed rainfall seem to be temporally and spatially synchronized withthe observations. This indicates that combining the spatial factor Ft

with a multivariate auto-regressive structure drives accurately thedynamics of the system. Consequently, our model, conditionally onthe common factor, cannot only be used as a rainfall generator butit can also carry out predictions. One can also notice in Fig. 3 thatreal rainfall (left panel) have a jigsaw pattern. This is due to therecording device that is discrete in nature. Recorded precipitationamounts are added by increment of 0.2 mm (the precision of theinstrument). This mechanical feature is not built in our statisticalmodel defined by (1) because we view this as an undesirable

characteristic of measured precipitation. This discrepancy explainsthe difference of ‘‘granularity’’ between the two panels. Thisphenomenon is particularly important for small and mediumprecipitation amounts. To illustrate this, we show the quantile–quantile plot between recorded and simulated rainfall intensitiesin Fig. 4. The median and the 98% confidence interval in gray,obtained by parametric bootstrap (over 1000 simulated out-sample series), have been added. Rainfall amounts above 3 mmare well captured by our statistical model, this is particularly truefor extreme intensities. Precipitation under 3 mm appear to bemore difficult to reproduce and their intensities are slightly overes-timated. This may be due to the nature of the recording process,the jigsaw pattern due to the instrument precision, and also tobe the choice of our threshold u in model (1). The latter was opti-mized (u ¼ 0:7 mm) to capture accurately dry episodes. Fig. 5shows that this objective has been basically reached. Both long

Fig. 5. Out-sample quantile–quantile plot between observed dry periods length (x-axis) and simulated one (y-axis) from model (1). The gray color corresponds to the 98%confidence interval and the solid line to the median.

Fig. 6. Each row represents a weather station. The left and right panels display the out-sample probability of having a wet hour given that the preceding hour was wet (left)and dry (right). The black points represent the observed estimates whereas the curves correspond to the density of estimates over 1000 out-sample simulated series.

E. Koch, P. Naveau / Advances in Water Resources 78 (2015) 145–154 151

and short dry episodes appear to be simulated accurately by ourstatistical model, especially at the Brest-Guipavas station.

To go one step further in analyzing our predictive out-sample,the probability of having a wet hour given that the preceding hourwas also wet is displayed on the left panels of Fig. 6, as well as theprobability of moving from a dry hour to a wet one (right panels).Overall, these transition probabilities appear to give reasonablevalues, the black point corresponding to the observed estimateand the density to the distribution from simulated out-samples.This is particularly true for the dry to wet transition. Let usnote that the other transition probabilities stem directly fromthese due to the relations PðWetjWetÞ þ PðDryjWetÞ ¼ 1 andPðWetjDryÞ þ PðDryjDryÞ ¼ 1.

Concerning the temporal memory, the top panel of Fig. 7shows the out-sample auto-correlations at different time lagsof one hour for the Brest-Guipavas station. This graph indicatesthat this local short term persistence (from one to five hours)is very well reproduced. The lower two panels focus on theout-sample cross-correlations between two pairs of stations,Landivisiau-Brest and Pleyber-Brest, respectively. These correla-tions are more difficult to reproduce: the observed ones (dottedlines) are above the ones computed on our out-sample. However,we capture the temporal asymmetry. The underestimation ofcorrelations at lag �1;0 and 1 stems partly from the fact thatthe fem;tgm¼1;...;M are i.i.d. This point will be further discussed inconclusion.

Fig. 7. These panels correspond to out-sample correlations at different lags. On top, the auto-correlation at Brest-Guipavas. On the middle, the cross-correlation betweenLandivisiau and Brest-Guipavas and on bottom the cross-correlation between Pleyber-Christ and Brest-Guipavas. The gray color corresponds to the 98% confidence interval,the solid line to the median and the dashed one to the observed correlations.

Fig. 8. Each panel corresponds to one of our three weather stations shown in Fig. 1. In solid are displayed the estimated probabilities to predict a dry hour while this hour iseffectively dry. The constant line corresponds to the estimate obtained with our model. The curve corresponds to the model predicting for a given horizon the same value asnow. In dashed, the same concerns the probabilities to predict a wet hour while this hour is effectively wet.

152 E. Koch, P. Naveau / Advances in Water Resources 78 (2015) 145–154

In order to analyze the prediction ability of our model, we havecomputed the following odd-ratios: the out-sample probability ofpredicting a dry hour while this hour is indeed dry and theout-sample probability of predicting a wet hour while this hour

is indeed wet. In order to carry out a comparison, we have consid-ered as a reference the simple model that gives as a prediction for afuture horizon the value observed now. In Fig. 8, we observe thatfrom a certain value of the horizon, our model outperforms the

E. Koch, P. Naveau / Advances in Water Resources 78 (2015) 145–154 153

naive prediction scheme. In some sense, this test allows to seeapproximately at which moment the frailty factor takes over thecontagion term.

In our example, the frailty factor plays an important role inreproducing accurately temporal dynamics, rainfall intensity anddry period persistence. To test this (figures available upon request),our model was fitted without the frailty component and the afore-mentioned features were not adequately reproduced in such aninstance. We have also fitted our model without the contagionterm (figures available upon request). The performance is quitegood, although the persistence of wet periods seems to beunderestimated.

We would like to emphasize that Figs. 3–8 have been obtainedon a validation set, totally different from the training one used tofit the model. This is reassuring and, albeit a few adjustments orextensions, we can expect that the basic ideas developed herecould be relevant for other regions. These aspects are discussedin our conclusion.

Finally, we have also applied our model on daily precipitationsmeasured at the same stations (see Fig. 1). Our model’s perfor-mance is at least as good as in the hourly case. Corresponding fig-ures are available upon request. To give perspective, we comparedour model with the standard model by [31]. Our model gives simi-lar results concerning the general statistical properties (distribu-tions of intensity and dry periods lengths, transition probabilities,cross-correlations, . . .). Due to the absence of covariates, Wilks’model does not capture the dynamic structure, and especially thesynchronicity between observed and simulated rainfalls. Asalready mentioned, these features are well reproduced by ourmodel.

5. Concluding remarks

Our main objective was to show that a ‘‘simple’’ multivariateauto-regressive model with a heteroscedastic variance driven bya few well-chosen atmospheric covariates can provide an interest-ing blueprint to generate dry episodes, medium and heavy rainfall.From there, it is easy to extend this work by adding or modifying afew elements of our model. For example, we could imagine toreplace the spatially independent random Gaussian noise em;t bya multivariate Gaussian vector with a covariance matrix that couldrepresent some spatial dependence among the different weatherstations. One can even think about a multivariate student or ellip-tical distribution in order to provide heavier tails. This may beneeded in regions with very heavy rainfall. Our northern Brittanyexample is known for frequent rainfall episodes but with ratherlow intensities, say compared to the South of France.

Another possible road for improvements would be to replacethe linearity assumption in our regression model (2) by a non-parametric relationship (like a GAM, [e.g., 27]) that will allow tocapture some non-linear behavior. Although it was not really need-ed for our Brittany example, it is likely that the link between rain-fall variability and temperatures could be more complex than ourlinear model for other regions.

It is also true that our multi-site statistical model is not a pureprecipitation weather generator in the sense that we need morethan precipitation data to fit and run our model. A few atmosphericcovariates are necessary to drive our rainfall variability (for simula-tion of temperature time series, see e.g. [19,18]). This limitationcould be viewed as an advantage in the context of climate changestudies. By letting our spatially averaged atmospheric covariatesbeing driven by a numerical model, we could explore how pre-cipitation change under different forcings. This is obviously relatedto downscaling themes and it would be interesting to pursue thispath in future work. In this context, selecting appropriate

covariates could be even more relevant, especially for large regionswhere scale indices like the North Atlantic Oscillation (NAO) Indexor the El Nino Southern Oscillation (ENSO) Index could be useful.

One possible drawback of our approach may reside in our smallnumber of sites. Modeling tens or even hundreds of stations,instead of three, will lead to computationally difficulties becauseof the M �M size of the matrix B. This could be solved by imposinga parametric structure on the bij, for example they could decreasewith the distance with respect to other stations or even set to zerofor far apart stations. Finding a suitable distance is not trivial andshould depend on orographic and other physical features.

To conclude, there are many different ways to fine tune andextend our approach. Basically, this will strongly depend on theapplication at hand and more research is needed to clearly see ifthis framework can be generalized.

Acknowledgements

We thank three referees, Pierre Ribereau, Christian Y. Robertand the participants of the 12th International Meeting onStatistical Climatology and the ISI World Statistics Congress 2013for their useful comments and suggestions. The authors have beenfinancially supported by the different projects: ANR McSim, ANRDADA, MIRACCLE-GICC, LEFE-Multirisk and Extremoscope. ErwanKoch would like to thank RiskLab at ETH Zurich and the SwissFinance Institute for financial support.

Appendix A. Likelihood computation

Proof. We consider that the common factor’s path Ft ðt ¼ 1; . . . ; TÞis given. Therefore we omit Ft in the following. Let us denote by It�1

the information available at time t � 1,It�1 ¼ ðP1;t�1; . . . ; PM;t�1; P1;t�2; . . . ; PM;t�2; . . .Þ and by h the den-

sity function. We can write by using a sequential argument that

hðP1;t; . . . ; PM;t; t 2 1; . . . ; TÞ ¼YT

t¼2

hðP1;t ; . . . ; PM;t jP1;t�1; . . . ; PM;t�1Þ:

ð4Þ

Since the em;t ;m ¼ 1; . . . ;M are i.i.d. (conditionally on Ft), thevariables

B0mPt�1 þ em;t; m ¼ 1; . . . ;M

are independent. Thus it is the same for the variables

IB0mPt�1þem;t>u;m ¼ 1; . . . ;M

and therefore for the product

Pm;t ¼ B0mPt�1 þ em;t�

IB0mPt�1þem;t>u; m ¼ 1; . . . ;M:

Therefore the measurement equation given by model (1) gives thatconditionally on ðIt�1Þ, the variables P1;t; . . . ; PM;t are independent,yielding

hðP1;t; . . . ; PM;tjP1;t�1; . . . ; PM;t�1Þ ¼YMm¼1

hðPm;t jP1;t�1; . . . ; PM;t�1Þ: ð5Þ

Combining Eqs. (4) and (5), we obtain

hðP1;t; . . . ; PM;t; t 2 1; . . . ; TÞ ¼YT

t¼2

YMm¼1

hðPm;t jP1;t�1; . . . ; PM;t�1Þ: ð6Þ

Let us recall that

Pm;t ¼B0mPt�1 þ em;t if B0mPt�1 þ em;t P u;

0 if B0mPt�1 þ em;t < u:

(

154 E. Koch, P. Naveau / Advances in Water Resources 78 (2015) 145–154

Due to this threshold mechanism, Pm;t is a mixture of a discrete ran-dom variable and a continuous one. Let us denote by hd the densityof the discrete part and by hc the density of the continuous one. Wehave

hcðPm;tjP1;t�1; . . . ; PM;t�1Þ ¼1rt

/Pm;t � B0mPt�1

rt

� �

and

hdðPm;t jP1;t�1; . . . ; PM;t�1Þ ¼ P Pm;t ¼ 0 jP1;t�1; . . . ; PM;t�1½ �¼ P B0mPt�1 þ em;t < u

�¼ P em;t < u� B0mPt�1

�¼ U

u� B0mPt�1

rt

� �:

Using Eq. (6), the log-density is written

ln hðP1;t ; . . . ;PM;t ;t¼1; . . . ;TÞ½ � ¼XT

t¼2

XM

m¼1

ln hcðPm;t jP1;t�1; . . . ;PM;t�1Þ IfPm;t Pug

þhdðPm;t jP1;t�1; . . . ;PM;t�1Þ IfPm;t¼0g�

¼XT

t¼2

XM

m¼1

IfPm;t Pug ln hcðPm;t jP1;t�1; . . . ;PM;t�1Þ½ �

þ IfPm;t¼0g ln hdðPm;t jP1;t�1; . . . ;PM;t�1Þ½ �:

That finally yields the log-likelihood function

LuðB; h0; . . . ; hdÞ ¼XT

t¼2

XM

m¼1

IfPm;tPug ln1rt

/Pm;t � B0mPt�1

rt

� �� ��

þ IfPm;t¼0g ln Uu� B0mPt�1

rt

� �� ��;

completing the proof. h

References

[1] Ailliot P, Allard D, Monbet V, Naveau P. Stochastic weather generators: anoverview of weather type models, submitted for publication.

[2] Ailliot P, Thompson C, Thomson P. Space time modeling of precipitation using ahidden Markov model and censored Gaussian distributions. J R Stat Soc Ser C(Appl Stat) 2009;58(3):405–26. http://dx.doi.org/10.1111/j.1467-9876.2008.00654.x.

[3] Allard D. Modeling spatial and spatio-temporal non Gaussian processes. In:Porcu E, Montero J-M, Schlather M, editors. Advances and challenges in space-time modelling of natural events. Lecture notes in statistics. BerlinHeidelberg: Springer; 2012. p. 141–64. http://dx.doi.org/10.1007/978-3-642-17086-7_7.

[4] Allard D, Bourotte M. Disaggregating daily precipitations into hourly valueswith a transformed censored latent Gaussian process. Stochastic Environ ResRisk Assess, 2013, submitted for publication, http://dx.doi.org/10.1007/s00477-014-0913-4.

[5] Azizpour S, Giesecke K, et al. Self-exciting corporate defaults: contagion vs.frailty. Manuscript, Stanford University; 2008.

[6] Bailey NT. The total size of a general stochastic epidemic. Biometrika1953;40(2):177–85.

[7] Bailey NT. The mathematical theory of epidemics. Griffin London: 1957.[8] Davis M, Lo V. Modelling default correlation in bond portfolios. Master Risk

2001;2:141–51.[9] Davis RA, Zang P, Zheng T. Sparse vector autoregressive modeling, submitted

for publication.[10] Flecher C, Naveau P, Allard D, Brisson N. A stochastic daily weather generator

for skewed data. Water Resour Res 2010;46(7). http://dx.doi.org/10.1029/2009WR008098.

[11] Frey R, Backhaus J. Interacting defaults and counterparty risk: a Markovianapproach. Leipzig, Germany: University of Leipzig; 2003.

[12] Furrer E, Katz R. Improving the simulation of extreme precipitation events bystochastic weather generators. Water Resour Res 2008;44(12). http://dx.doi.org/10.1029/2008WR007316.

[13] Gabriel K, Neumann J. A Markov chain model for rainfall occurrence at TelAviv. Q J R Meteorolog Soc 1962;88:90–5. http://dx.doi.org/10.1002/qj.49708837511.

[14] Gagliardini P, Gouriéroux C. Correlated risks vs contagion in stochastictransition models. J Econ Dyn Control 2013;37(11):2241–69. http://dx.doi.org/10.1016/j.jedc.2013.05.016.

[15] Giesecke K, Weber S. Cyclical correlations, credit contagion, and portfoliolosses. J Bank Finan 2004;28(12):3009–36. http://dx.doi.org/10.1016/j.jbankfin.2003.11.002.

[16] Giesecke K, Weber S. Credit contagion and aggregate losses. J Econ Dyn Control2006;30(5):741–67. http://dx.doi.org/10.1016/j.jedc.2005.01.004.

[17] Grimaldi S, Serinaldi F, Tallerini C. Multivariate linear parametric modelsapplied to daily rainfall time series. Adv Geosci 2005;2(2):87–92. http://dx.doi.org/10.5194/adgeo-2-87-2005.

[18] Hoang T, Dacunha-Castelle D, Benmenzer G. Estimation of a diffusion modelwith trends taking into account the extremes. Application to temperature inFrance. Environmetrics 2011;22(3):464–79. http://dx.doi.org/10.1002/env.1100.

[19] Huong Hoang T, Parey S, Dacunha-Castelle D. Multidimensional trends: theexample of temperature. Eur Phys J Spec Top 2009;174(1):113–24. http://dx.doi.org/10.1140/epjst/e2009-01094-6.

[20] Katz R. Precipitation as a chain-dependant process. J Appl Meteorol1977;16:671–6. http://dx.doi.org/10.1175/1520-0450(1977)016<0671:PAACDP>2.0.CO;2.

[21] Kendall DG. Deterministic and stochastic epidemics in closed populations. In:Proceedings of the third berkeley symposium on mathematical statistics andprobability, vol. 4, 1956. p. 149–65.

[22] Kleiber W, Katz R, Rajagopalan B. Daily spatiotemporal precipitationsimulation using latent and transformed Gaussian processes. Water ResourRes 2012;48(1). http://dx.doi.org/10.1029/2011WR011105.

[23] Lennartsson J, Baxevani A, Chen D. Modelling precipitation in Sweden usingmultiple step Markov chains and a composite model. J Hydrol2008;363(1):42–59. http://dx.doi.org/10.1016/j.jhydrol.2008.10.003.

[24] Maraun D, Wetterhall F, Ireson A, Chandler R, Kendon E, Widmann M, BrienenS, Rust H, Sauter T, Themel M, et al. Precipitation downscaling under climatechange: recent developments to bridge the gap between dynamical modelsand the end user. Rev Geophys 2010;48(3). http://dx.doi.org/10.1029/2009RG000314.

[25] Pawitan Y. All likelihood: statistical modelling and inference usinglikelihood. Oxford University Press; 2001.

[26] Richardson C. Stochastic simulation of daily precipitation, temperature, andsolar radiation. Water Resour Res 1981;17(1):182–90. http://dx.doi.org/10.1029/WR017i001p00182.

[27] Serinaldi F, Kilsby CG. Simulating daily rainfall fields over large areas forcollective risk estimation. J Hydrol 2014;512:285–302. http://dx.doi.org/10.1016/j.jhydrol.2014.02.043.

[28] Srikanthan R, McMahon T, et al. Stochastic generation of annual, monthly anddaily climate data: a review. Hydrol Earth Syst Sci Discuss 2001;5(4):653–70.

[29] Vrac M, Naveau P. Stochastic downscaling of precipitation: from dry events toheavy rainfalls. Water Resour Res 2007;43(7). http://dx.doi.org/10.1029/2006WR005308.

[30] Vrac M, Stein M, Hayhoe K. Statistical downscaling of precipitation throughnon homogeneous stochastic weather typing. Clim Res 2007;34:169–84.http://dx.doi.org/10.3354/cr00696.

[31] Wilks D. Multisite generalization of a daily stochastic precipitation generationmodel. J Hydrol 1998;210(1):178–91. http://dx.doi.org/10.1016/S0022-1694(98)00186-3.

[32] Wilks D. Use of stochastic weather generators for precipitation downscaling.Wiley Interdiscip Rev Clim Change 2010;1(6):898–907. http://dx.doi.org/10.1002/wcc.85.

[33] Wilks D. Stochastic weather generators for climate-change downscaling, partII: multivariable and spatially coherent multisite downscaling. WileyInterdiscip Rev Clim Change 2012;3(3):267–78. http://dx.doi.org/10.1002/wcc.167.

[34] Wilks D, Wilby R. The weather generation game: a review of stochasticweather models. Prog Phys Geog 1999;23(3):329–57. http://dx.doi.org/10.1177/030913339902300302.