Statistical methods for understanding hydrologic change€¦ · Statistical modeling of extreme values has ﬂourished since about the mid-1980s. Application to estimating both the

Statistical methods for understandinghydrologic change

C. Bocci1 E. Caporali2 A. Petrucci1

1Dept. of Statistics “G. Parenti”University of Florence

2Dept. of Civil and Environmental EngineeringUniversity of Florence

58th World Statistics Congress of theInternational Statistical Institute - ISI 2011

C. Bocci, et al. (University of Florence) Understanding hydrologic change ISI 2011 1 / 51

Outline

1 IntroductionFramework of the studyMotivation of the work

2 DataDescriptionPreliminary Analysis

3 ModelDefinitionModel Implementation

4 Results

5 Concluding remarks and open questionsConcluding remarksOpen questions


Introduction Framework of the study

Framework .

Protection of water



Framework ..

Protection from water



Framework ...

Environmental extreme events such as floods, earthquakes, hurricanes have a massiveimpact on everyday life for the consequences and damage that they cause.Extreme value models and techniques are widely applied in environmental studies to defineprotection systems against the effects of extreme levels of environmental processes.Considerable attention is showed in studying, understanding and predicting the nature of suchphenomena and the problems caused by them, not least because of the possible link betweenextreme climate events and climate change.A certain importance is covered by the implication of changes in the hydrological cycle.


Introduction Motivation of the work

Motivation .

In this framework, in the past two decades there has been an increasing interest for statisticalmethods that model rare events.Statistical modeling of extreme values has flourished since about the mid-1980s.Application to estimating both the rate and magnitude of rare events for planning for theirimpact and mitigate their effects.



Motivation ..

The Generalized Extreme Value distribution (GEV) is widely adopted model for extremeevents in the univariate context.Its motivation derives from asymptotic arguments that are based on reasonably wide classesof stationary processes.



Motivation ...

For modeling extremes of non-stationary sequences it is commonplace to still use the GEV asa basic model, but to handle the issue of non-stationarity by regression modeling of the GEVparameters. Traditionally this has been done using parametric models.Recent interest was laid about nonparametric or semiparametric modeling of extreme valuemodel parameters.Padoan and Wand in 2008 have proposed the use of mixed model-based splines for extremalmodels developing nonparametric estimation for a smoothly varying location parameter withinthe GEV model.A compelling feature of this approach is that the smoothing parameters correspond tovariance components, so maximum likelihood or Bayesian techniques can be applied formodel fitting, assessment and inference.



Aim

The aim is to implement a geoadditive mixed model for rainfall extremes.We assume that the observations follow generalized extreme value distributions whoselocations are spatially dependent where the dependence is captured using the geoadditivemodel.We discuss also the addition of a temporal random effect.The analyzed territory is the catchment area of Arno River in Tuscany in Central Italy.


Data Description

Study area characterization .

The investigation is developed on the catchment area of Arno River almost entirely situatedwithin Tuscany, Central Italy. The area is expected to suffer from global climate change.The area is characterized by a climate that ranges from temperate to Mediterranean maritime,and by a complex physical topography. It presents plain areas near the sea and around themain metropolitan areas, hilly internal zones, and the mountainous area of the Apennines.The river is 241 km long and the catchment area is of about 9000 (8830) km2 and has a meanelevation of 353 m a.m.s.l..


Data Description

Study area characterization ..

Figure: Geographical location and orography (i.e. Digital Terrain Model) of the catchment area of Arno Riverin Central Italy.


Data Description

Study area characterization ...

The precipitation regime is greatly influenced by the topography. Total annual precipitationranges from 720 mm to 1690 mm.Heavy storms mainly occur in autumn following dry summers. Most of the territory of ArnoRiver basin have suffered in the past from many severe hydro-geological events, with highlevels of risk due to the vulnerability of a unique artistic and cultural heritage.


Data Description

Study area characterization ....

Figure: Average Total Annual Precipitation and rain gauges distribution of Arno River basin (daily precipitationdataset; recorded period 1916-2008).


Data Description

Study area characterization .....

The time series of annual maxima of daily rainfall recorded in 415 rain gauges are analysed.The registrations cover the period 1916-2008 and the available rain gauges series lengthranges from 1 to 81 years.Using the time series minimum length suggested by WMO (1983), only stations with at least30 hydrologic years of data, even not consecutive, were considered.In addition, in order to have enough rain gauges observations to estimate each year specificeffect we reduce the time series length to the post Second World War period. The finaldataset is composed by the data recorded from 1951 to 2000 at 118 rain gauges for a total of4903 observations.


Data Preliminary Analysis

Rain gauges time series .



Rain gauges time series ..

(a) Camaldoli (b) Arezzo (c) Badia Agnano



Rain gauges time series ...

(d) Vallombrosa (e) Firenze Ximeniano (f) Marliana



Rain gauges time series ....

(g) San Giovanni alla Vena (h) Pisa (Facoltà Agraria) (i) San Rossore



Frequency Analysis of Hydrological Extreme Values .

Extreme value theory begins with a sequence Y1,Y2, . . . of independent and identicallydistributed random variables and, for a given n asks about parametric models forMn = maxY1, . . . ,Yn.If the distribution of the Yi is specified, the exact distribution of Mn is known. In the absence ofsuch specification, extreme value theory considers the existence oflimn→∞ P

[Mn−bn

an≤ y

]≡ F (y) for two sequences of real numbers an > 0, bn.

If F (y) is a non-degenerate distribution function, it belongs to either the Gumbel, the Fréchetor the Weibull class of distributions, which can all be usefully expressed under the umbrella ofthe GEV(µ, ψ, ξ) .

F (y ;µ, ψ, ξ) = exp

{−[1 + ξ

(y − µψ

)] 1ξ

}, −∞ < µ, ξ <∞, ψ > 0 (1)

for y : 1 + ξ (y−µ)ψ > 0 and µ, ψ and ξ are respectively location, scale and shape parameters.



Frequency Analysis of Hydrological Extreme Values ..

The GEV distribution is heavy-tailed and its probability density function decreases at a slowrate when the shape parameter ξ is positive.On the other hand, the GEV distribution has a bounded upper tail for a negative shapeparameter. Note that n is not specified; the GEV is viewed as an approximate distribution tomodel the maximum of a sufficiently long sequence of random variables.Now suppose we observe n sample maxima y1, . . . , yn as well as corresponding covariatevectors x1, . . . ,xn.The yi are obtained from approximately equi-sized samples of a variable of interest. Acommon situation is yi corresponding to the annual maximum of a daily measurement, suchas rainfall in a particular town, for year i (1 ≤ i ≤ n).



Frequency Analysis of Hydrological Extreme Values ...

General GEV regression models take the form

yi |xi ∼ GEV(µ (xi) , ψ (xi) , ξ (xi))

where, for example, µ (xi) = g ([Xβ]i), g is a link function, β is a vector of regressioncoefficients and X is a design matrix associated with the xis.Similar structures may be imposed upon ψ (xi) and ξ (xi).The regression coefficients can be estimated via maximum likelihood.The classic literature illustrate GEV regression with parametric models, however recent workspresent more flexible non-parametric approaches.



Estimated GEV parameters for each rain gauge

Figure: Location parameter µ̂ Figure: Relation between coordinates and µ̂



Estimated GEV parameters for each rain gauge ..

Figure: Scale parameter ψ̂ Figure: Shape parameter ξ̂


Model Definition

Mixed model-based additive models

Padoan and Wand (2008) discuss how generalized additive models (GAM) with penalizedsplines can be carried out in a mixed model framework for the GEV family. Assuming that thelocation parameter in the GEV distribution is smooth on an interval [a,b] in the xi domain thenthe simplest time-nonhomogeneous nonparametric regression model is given by

yi |xi ∼ GEV(µ (xi) , ψ, ξ)

with a mixed model-based penalised spline model for µ

η(x) = g(µ(x)) = β0 + β1x +K∑

k=1

uk zk (x), u1, . . . ,uK i.i.d. N(0, σ2u)

where g is a link function and z1, . . . , zK is an appropriate set of spline basis functions.


Model Definition

Geoadditive Model 1 .

Geoadditive models, introduced by Kammand and Wand (2003), are a particular specificationof GAM that models the spatial distribution of y with a bivariate penalized spline on the spatialcoordinates. Suppose to observe n sample maxima yij at spatial location sij , s ∈ R2,j = 1, . . . ,p and at time i = 1, . . . , t :

yij |sij ∼ GEV(µ (sij) , ψ, ξ)

µ (sij) = β0 + sTij βs +

K∑k=1

uk btps(sij ,κk ),(2)

where btps are the low-rank thin plate spline basis functions with K knots.Geoadditive models analyze the spatial distribution of the study variable while accounting forpossible non-linear covariate effects.Obtained by merging an additive model and a kriging model and by expressing both as alinear mixed model.


Model Definition

Geoadditive Model 2 .

In order to model both the spatial and the temporal influence on the annual rainfall maxima,we consider also a geoadditive mixed model for extremes with a temporal random effect:

yij |sij ∼ GEV(µ (sij) , ψ, ξ)

µ (sij) = β0 + sTij βs +

K∑k=1

uk btps(sij ,κk ) + γi ,(3)

where γi is the time specific random effect.


Model Definition

Geoadditive Model 2 ..

The model - in this case (3) - can be written as a mixed model

y| (u,γ) ∼ GEV(Xβ + Zu + Dγ, ψ, ξ). (4)

with

E[uγ

]=

[00

], Cov

[uγ

]=

[σ2

u IK 00 σ2

γ It

].

where

β =[β0,β

Ts

],

u = [u1, ...,uK ] ,

γ = [γ1, ..., γt ] ,

X =[1,sT

ij]

1≤ij≤n,

D = [dij ]1≤ij≤n ,


Model Definition

Geoadditive Model 2 ...

with dij an indicator taking value 1 if we observe a rainfall maxima at rain gauge j in year i and0 otherwise, and Z is the matrix containing the spline basis functions, that is

Z = [btps(sij ,κk )]1≤ij≤n,1≤k≤K = [C (sij − κk )]1≤ij≤n,1≤k≤K · [C (κh − κk )]−1/21≤h,k≤K ,

where C(v) = ‖v‖2 log ‖v‖ and κ1, ...,κK are the spline knots locations.


Model Model Implementation

Model estimation.

The geoadditive mixed models for extremes can be naturally formulated as a hierarchicalBayesian model and estimated under the Bayesian paradigm.



Model estimation ..The complete hierarchical Bayesian formulation is

1st level yi | (u,γ)ind∼ GEV( [Xβ + Zu + Dγ]i , ψ, ξ),

2st level

u|σ2u ∼ N(0, σ2

γ IK ),

γ|σ2γ ∼ N(0, σ2

γ It),

β ∼ N(0,104I)ξ ∼ Unif(−5,5)

ψ ∼ InvGamma(10−4,10−4)

3st levelσ2

u ∼ InvGamma(10−4,10−4)

σ2γ ∼ InvGamma(10−4,10−4).

where the parameters setting of the priors distributions for ξ, ψ, β, σ2u , σ2

γ , corresponds tonon-informative priors.



Model estimation ...

Given the complexity of the proposed hierarchical models, we employ OpenBUGS BayesianMCMC inference package to do the model fitting. We access OpenBUGS using the packageBRugs in the R computing environment.We implement the MCMC analysis with a burn-in period of 40000 iterations and then we retain10000 iterations, that are thinned by a factor of 5, resulting in a sample of size 2000 collectedfor inference.Finally, the last setting concern the thin plate spline knots that are selected setting K = 30 andusing the clara space filling algorithm of Kaufman and Rousseeuw (1990), available in the Rpackage cluster.



Model estimation ....

Figure: Knots location (in red) for the spline component. Black dots indicate the rain gauges sites.


Results

Estimated parameters of GEV - Model 1, without Year component

Table: Estimated parameters of the GEV geoadditive model for the annual maxima of daily rainfall.

Parameter* Posterior Mean 95% Credible Intervalβ0 -117.65 (-233.18 ; 19.61)βs1 4.64 ( 1.28 ; 9.69)βs2 2.70 (-0.37 ; 5.27)ξ 0.12 ( 0.10 ; 0.14)ψ 16.79 (16.40 ; 17.21)σγ 25.55 (19.07 ; 34.08)

*Intercept and coordinates coefficients are required by model structure.


Results

Spatial component - Model 1, without Year component

Figure: Estimated spatial component of µ (sij). Blackdots indicate the rain gauges.

Figure: Average Total Annual Precipitation and raingauges distribution of Arno River basin (dailyprecipitation dataset; recorded period 1916-2008).


Results

Estimated parameters of GEV - Model 2, with Year component

Table: Estimated parameters of the GEV geoadditive mixed model for the annual maxima of daily rainfall.

Parameter* Posterior Mean 95% Credible Intervalβ0 11.31 ( 8.75 ; 13.82)βs1 -1.74 (-4.39 ; 1.21)βs2 1.02 ( 0.62 ; 1.38)ξ 0.11 ( 0.09 ; 0.12)ψ 15.13 (14.79 ; 15.45)σγ 7.75 ( 6.35 ; 9.51)σu 27.24 (20.66 ; 35.86)

*Intercept and coordinates coefficients are required by model structure.


Results

The estimated parameters are presented in Table, that provides their posterior means alongwith the corresponding 95% credible intervals. The posterior mean of the ξ takes value of 0.11with 95% credible interval (0.09, 0.12), indicating the GEV distributions of annual maximumrainfalls in the Arno catchment belong to the Gumbel family and have heavy upper tails.


Results

Spatial component - Model 2, with Year component

Figure: Estimated spatial component of µ (sij). Blackdots indicate the rain gauges.

Figure: Average Total Annual Precipitation and raingauges distribution of Arno River basin (dailyprecipitation dataset; recorded period 1916-2008).


Results

Temporal component - Model 2, with Year component

Figure: Estimated year specific random effects of µ (sij) (in red). Black dots indicate the observed values.


Results

Outcome - Model 2, with Year component

Figure: Predicted values of E(y|u,γ) (in red). Black dots indicate the observed values.


Results

Comments

Observing the map, it is evident the presence of a spatial trend in the rainfall extremedynamic, even after controlling for the year effect.The time influence is pointed out by the estimated year specific random effects, that present astrong variability through years.Finally, in order to asses the usefulness of our model we plot the predicted values of E(y|u,γ)against the observed values. The results show a good prediction performance.


Concluding remarks and open questions Concluding remarks

Concluding Remarks

We have implemented a geoadditive modeling approach for explaining a collection of spatiallyreferenced time series of extreme values. We assume that the observations follow generalizedextreme value distributions whose locations are spatially dependent.The results show that this model allows us to capture both the spatial and the temporaldynamics of the rainfall extreme.Under this approach we expect to reach a better understand of the occurrence of extremeevents which are of practical interest in climate change studies particularly when related tointense rainfalls and floods, and hydraulic risk management.


Concluding remarks and open questions Open questions

Open Questions

Providing a reliable analysis of changes in rainfall extreme values is quite a complex task.Complexity of central Italy climatic system.High demand of computational resources and time-consuming.


Appendix Main references

Main References

Coles, S. G.An Introduction to Statistical Modeling of Extreme Values.Springer, London, 2001.

Hastie T.J., Tibshirani R.Generalized Additive Models.Chapman & Hall, New York, 1990.

Ruppert D., Wand M., Carrol R.J.Semiparametric Regression.Cambridge university press, Cambridge, 2003.

Fatichi S. and Caporali E.

A comprehensive analysis of changes in precipitation regime in Tuscany.International Journal of Climatology,29(13):1883-1893, 2009.

Kammann E.E., Wand M.P.

Geoadditive models.Applied Statistics,52:1–18, 2003.

Padoan, S.A. and Wand, M.P.

Mixed model-based additive models for sample extremes.Statistics and Probability Letters,78:2850–2858, 2008.


Documents

Statistical methods for understanding hydrologic change€¦ · Statistical modeling of extreme values has ﬂourished since about the mid-1980s. Application to estimating both the