non stationarity, non linearity and boundednesspagesperso.univ-brest.fr/~ailliot/roscoff/Parey.pdf · using the simulations if the law is known or bootstrap otherwise) ... Choose

Statistical simulation model for air

temperatures: non stationarity, non

linearity and boundedness

Thi Thu huong HOANG (EDF R&D), Sylvie PAREY(EDF R&D), Didier DACUNHA-CASTELLE (Université Paris Sud)

1st of June , 2012

1- Simulation model for air temperatures- 1st June 2012

Outline

Introduction

Pre-processing

Model and estimation procedure

Application to air temperature

Conclusion and perspectives


INTRODUCTION


ContextEDF is interested in the impact of climate change on energyThis will help outline the offer/demand balance of the 21st century and envisage a decision-making procedures related to adaptation strategies.We concentrate on temperature, a key parameter influencing energy:

Reduce or increase the demandhot or cold waves can affect overhead linesExtreme temperatures can have adverse effects on renewable energy sources that are sensitive to climate variables and on thermal production.

ObjectivesBuild a simulation model

for (maximum or minimum) daily air temperature for a fixed locationfor a month or a seasonwith good qualities for the bulk and the tails of distribution

able to easily produce a great number of realistic trajectories


Very large models, complexity, heavy computationNon global chaos, chaotic sub-models

Coupled PDE : fluid mechanics, atmosphere, ocean radiation scheme (solar variations) water cycle, greenhouse gases (CO2, …), aerosols

physical parameterizationsRemaining uncertainties : clouds, carbon cycle, ice melting

Uncertainties How to estimate them? No stochasticityPseudo simulation, sensitivity to small variations on initial conditions ⇒ limited number of trajectoriesStill difficulties to reproduce variability and extremes

X o o x

MODEL

Initial conditions

Trajectories

Some features of numerical climate models


Important number of stations. Complicated spatial correlations

Non stationary, non linear

Two periodicities (in mean and in variance), maybe non constant for long periods

Boundedness : only a very accurate application of extremes theory allows to prove that every model has to take into account this feature.

Continuous-time process versus discrete measurements: Temperature is a continuous-time process but observed at the discrete time steps How to apply the properties of continuous-time model to discrete observations?

Difficulties in the stochastic modelling of temperature


PRE-PROCESSING FROM NON STATIONARITY TOWARDS STATIONARITY


Pre-processingThe aim is to remove the trends in mean and in variance and the (additive et multiplicative) seasonalities to obtain reduced series as stationary as possible

The processing treatment uses both the nonparametric and parametric approaches

: mean, scale function, : seasonalities in mean and in scale

Estimation procedure: estimate by loess, by a trigonometric function from the series , then

by loess and by a trigonometric function from the series

For , the modified partitioned cross validation(1) is used , for , the

Akaike criteria are used

The reduced series:

(1) Modified partitioned CV: new algorithm for correlated data (thesis of Hoang, 2010)

)()()()()()( tZtStstStmtX V++=

)(),( tstm )(),( tStS V

( ) )ˆˆ/(ˆˆ Vtttttt SsSmXZ −−=


[ ] 2)(ˆ)(ˆ)( tStmtX −−

)(tm )(tS )(tX

)(),( tstm

)(ts

)(2 tSV

)(),( 2 tStS V

MODEL AND ESTIMATION PROCEDURE FOR THE REDUCED SERIES


Characteristics of the reduced series short memory seasonality remains in the correlation and in the volatility cyclo-stationary bounded in the tails non linearityvolatility depends on the state Studies:

tests of trend(2) on the basic statistics of (mean, variance, skewness, kurtosis)test of cyclo-stationarity(3) on the extremes of (see Parey et al., 2012)study the trend and the seasonality in the series


kttZZ −

tZtZ

(2), (3): A new test proposed in Hoang (2010), the principe of test is described in the next slide

Principle of our test of trend (or of stationarity)

Situation: Xk = X(tk) , tk ∈ (0,1) ∼ G(x, θ) known or unknown distribution, θ (t) ∈ C compact in R

Hypothesis of test: θ is constant

Let be the constant estimator of θ, the non-parametric (spline or loess) estimator of θ

Idea: compare these two estimators by use of the distance :

The asymptotic of the test is proven (Hoang, 2010)

In practice: build a test on ∆ (build the distribution of ∆ under H0 hypothesis by using the simulations if the law is known or bootstrap otherwise)


nc nθˆ

nn cˆ −=∆ θ

Asymptotic of the test

Consider that the law of ε is known. Let θ0 be the true value of θ

We have: when (always)

when (only if θ0 is constant !)

We proved that (Hoang, 2010) (t is supposed to belong to (0,1): t = tk = k/n)

The theorem can be proven for an unknown law of ε(based on least squares estimation)

12- Simulation model for air temperatures- 31st May 2012

Theorem (m.l.e theory and Le Cam ‘s point of view)

enoughhighnforacsurelyalmosthave

weFccathatsoantstaconnotisWhen

nn 02ˆˆ),,min(, 00

>>−

∈−≤∀

θ

θθ

0ˆ θθ →n

∞→n

0ˆ θ→nc ∞→n

Theory of the diffusion process with inaccessible boundaries

Temperature has Markov properties and is bounded → Diffusion process with inaccessible boundaries r1 and r2 (r1 < ∞, r2 < ∞):

where b is the drift, a is the diffusion coefficient and W is a Brownian motion.

The invariant marginal density of the continuous-time process can be estimated from the discrete observations

Moreover, we proved (Hoang, 2010) that if ξ < 0, the domain of max attraction of the continuous-time process is the same as that of an i.i.d discrete sample with the same marginal density

Conclusion: we can apply the properties of extreme value theory of the bounded diffusion process to a discrete-time sample. We don’t need to use the exact formula of the discrete Markov chain


tttt dWZaZbdZ )()( +=

The SFHAR(seasonal functional heteroscedastic autoregressive) model

)1,0(

),()1(3652sin

3652cos)( 1

1,2,1,0

1

N

ZtatZtjtjtZ

t

tt

p

j

jk

jkk

∝

+−

++= −

=∑

ε

επθπθθ

∀>

+−−= −

= =−

∧

∑ ∑

ttatrCtrC

ZtjtjrttrZta kt

k

p

j

jk

jkt

0)²(ˆ),ˆ(),,ˆ(

3652cos

3652cos)ˆ)(ˆ(),(²

21

1

5

0 1,2,1121

2 παπα

Estimate with constraints:),( 12

−tZta Zero outside the boundaries positive constraints C on the first derivative from the continuous-time diffusion process (see thesis of Hoang, 2010):

Form of a:


( ) ( )2

22

2

1

11

2

/11),(2)('

/11),(2)('

ξξ −=

−= trbraettrbra

First order Euler scheme of a discrete diffusion:Rq: This is a discrete approximation of the bounded diffusion. With a gaussian noise, the model is not bounded but ‘almost’ bounded

Extension: SFHAR model

)1,0(,)()( 11 NZaZbZ ttttt ∝+= −− εε

Estimation procedure and optimisation

Estimation of the autoregressive part (AR(1))

Choose the number of cosine and sine terms by a Akaike criterion

Estimation of the volatility through maximum likelihood with constraintsFind the initial values using least squares estimation: problem of least squares with equality and inequality constraints → transform to the quadratic programming problem and use the algorithm of Goldfarb and Idnani (1982,1983) Maximum likelihood estimation with constraints: use the results of least square estimation as the initial values and use the Nelder and Mead algorithm (1965)


APPLICATION TO AIR TEMPERATURE


Observation data Maximum or Minimum daily temperature for one month or one season at one location


Europe: ECA&D 1950-2009Homogenous (« useful »)Tx, Tn<3 years of missing data ⇒106 Tx series and 120 Tn series

United States: NCDC, Global Historical Climatology Network

Series with <4 years of missing dataBeginning before 1966 and ending after 200886 Tx series, 85 Tn series

Validation criteria

Residuals: Whiteness of the residuals and of the squared residuals Tests of normality of the residuals

Comparison to the observations: Basic statistics: mean, variance, skewness, kurtosis Marginal law: density function, test of homogeneity Quantiles Temperature of a fixed date for X Extreme parameters for Z Proportion of outliers for X: we expect that there are x% observations lower than the estimated x-percentile of the simulations


Results: Europe (Déols in France) Model for a month and for a season a estimated non parametrically by splines

19- Simulation model for air temperatures- 04 May 2012

a estimated parametrically

Tmax: a tends to increase with Z(t-1) Tmin: a tends to decrease with Z(t-1)

Tmax

-10 -5 0

0.0

0.5

1.0

1.5

2.0

8 Jan

Z(t-1)

a

nonparapara_without constrpara+constr

Tminlower bound is unrealistic because ξ≈0

Results: Europe (Déols in France)


The residuals: normality is not rejected by Komogorov-Smirnov test but is one time over two (especially for winter and summer) rejected by Shaprio-Wilk test or Anderson-Darling test. They take more account of the tails.

Tmax, August

Tmin, January



X ZObservation

sSimulations Observation

sSimulations

Mean 25.247 25.263(24.947,25.632) 0.003 0.007 ( -0.066 , 0.093 )Variance 18.084 17.826 (15.88 ,

19.682)0.957 0.935 ( 0.843 , 1.019 )

Skewness 0.297 0.434 ( 0.31 , 0.551 ) 0.365 0.439 ( 0.308 , 0.576 )Kurtosis -0.466 0.412 ( 0.133 , 0.712 ) -0.404 0.446 ( 0.165 , 0.807 )

X ZObservation


sSimulations

Mean 24.235 24.232(24.046,24.401) -0.001 -0.002 ( -0.046 , 0.04 )Variance 20.097 19.245(18.188,20.255) 1.01 0.981 ( 0.931 , 1.04 )Skewness 0.32 0.435 ( 0.339 , 0.512 ) 0.39 0.575 ( 0.488 , 0.656 )Kurtosis -0.195 0.534 ( 0.331 , 0.779 ) -0.317 0.624 ( 0.407 , 0.875 )

X ZObservation


sSimulations

Mean 0.944 0.987 ( 0.554 , 1.37 ) -0.004 0.002 ( -0.09 , 0.081 )Variance 23.24 22.014( 19.79 ,

23.951)0.976 0.949 ( 0.846 , 1.038 )

Skewness -0.571 -0.308 (-0.523 , -0.137)

-0.522 -0.324( -0.541 , -0.157 )

Kurtosis 0.836 0.275 ( -0.189 , 1.259 ) -0.269 0.283 ( -0.181 , 1.119 )X Z

Observations

Simulations Observations

Simulations

Mean 1.289 1.242 ( 0.986 , 1.513 ) 0.007 -0.006 ( -0.06 , 0.052 )Variance 21.398 25.191(23.574,26.714) 0.99 1.181 ( 1.106 , 1.249 )Skewness -0.402 -0.319 ( -0.434,

-0.202)-0.311 -0.271 ( -0.378 , -0.16 )

Kurtosis 0.457 0.553 ( 0.279 , 0.812 ) 0.089 0.43 ( 0.204 , 0.682 )

Tmax, July

Tmax, summer

Tmin, January

Tmin, winter

Simulations represent correctly mean and variance, but not skewness and kurtosis

The results seem better for a month than for a season



Tmax, July Tmax, summer

Tmax, October Tmax, autumn

The results are better for individual months than for seasonsThe results are better for the inter-seasons (spring, fall) whose distributions are more symmetric



The observed temperatures are usually in the confidence interval of the simulations

except for the very special cases



Tmax, July• 0.2% observations higher than estimated Q99% from simulations• 1% observations higher than estimated Q98% from simulations

Tmax, summer• 0.45% observations higher than estimated Q99% from simulations• 1.4% observations higher than estimated Q98% from simulations

Tmin, January•1.5% observations lower than estimated Q1% from simulations• 3.0% observations lower than estimated Q2% from simulations

Tmin, winter• 0.6% observations lower than estimated Q1% from simulations• 1.4% observations lower than estimated Q2% from simulations

Problem when ξ ≈ 0, simulations more bounded

Simulations less bounded


Results: United-States (Minneapolis) Model for a month and for a season a estimated non parametrically by splines: the same behavior as in Europe


a estimated parametrically

Tmax

Tmax: a tends to increase with Z(t-1) Tmin: a tends to decrease with Z(t-1)

TmaxTmin

Results: United-States (Minneapolis)


The residuals: normality is more often rejected by the normality tests: Komogorov-Smirnov test Shaprio-Wilk test and Anderson-Darling test than for Deols

Tmax, July

Tmin, January

Results: United-States (Minneapolis)


Tmax, July

Tmax, summer

Tmin, January

Tmin, winter

The simulations represent rather well all basic statistics

These results are better compared to those of Deols

The simulations represent better the observations when the observations are less asymmetric and their (normalized) kurtosis is close to that of a gaussian law (0)

X ZObservation


sSimulations

Mean 28.598 28.45 (28.191 , 28.655)

0.042 0.003 ( -0.065 , 0.057 )

Variance 14.93 14.515 (13.57 , 15.681)

0.993 0.99 ( 0.928 , 1.065 )

Skewness -0.104 0.025 ( -0.101 , 0.103 ) -0.129 0.023 ( -0.087 , 0.108 )Kurtosis 0.08 0.102 ( -0.088 , 0.317 ) 0.056 0.094 ( -0.062 , 0.286 )X Z

Observations


Simulations

Mean 27.192 27.221(27.053,27.414) -0.006 0.001 ( -0.039 , 0.049 )Variance 19.526 19.498(18.686,20.443) 1.002 0.992 ( 0.961 , 1.033 )Skewness -0.303 -0.136 ( -0.216,

-0.058)-0.131 0.068 ( -0.007 , 0.124 )

Kurtosis 0.179 0.209 ( 0.046 , 0.361 ) -0.1 0.041 ( -0.079 , 0.156 )X Z

Observations


Simulations

Mean -15.022 -14.86(-15.33, -14.407) 0.042 -0.001 ( -0.06 , 0.057 )Variance 62.904 60.028(54.635,65.691) 0.993 0.976 ( 0.884 , 1.065 )Skewness -0.129 -0.184 ( -0.284 , -0.09 ) -0.129 -0.17 ( -0.268 , -0.076 )Kurtosis -0.76 -0.064 ( -0.215 , 0.151) 0.056 -0.105 ( -0.255 , 0.08 )

X ZObservation


sSimulations

Mean -12.856 -12.90 (-13.28 ,12.559) 0.008 0 ( -0.052 , 0.044 )Variance 61.257 59.037(55.691,62.453) 1.002 0.983 ( 0.937 , 1.031 )Skewness -0.239 -0.287 ( -0.353, -0.229) -0.186 -0.257 ( -0.321, -0.207 )Kurtosis -0.659 -0.079 ( -0.179 , 0.024) -0.731 -0.076 ( -0.162 , 0.011 )

Results: US (Minneapolis)


The results are better for individual months than for seasonsThe simulations perform better in the bulk than for Deols because the skewness and kurtosis of the observations (X) in Minneapolis are closer to those of a Gaussian law

Tmax, July Tmax, summer

Tmin, January Tmin, winter

Results: US (Minneapolis)


Tmax, July• 0.8% observations higher than estimated Q99% from simulations• 2.1% observations higher than estimated Q98% from simulations

Tmax, summer• 0.45% observations higher than estimated Q99% from simulations• 1.3% observations higher than estimated Q98% from simulations

Tmin, January• 0.35% observations lower than estimated Q1% from simulations• 1.3% observations lower than estimated Q2% from simulations

Tmin, winter• 0.4% observations lower than estimated Q1% from simulations• 1.2% observations lower than estimated Q2% from simulations


Simulations rather good for extremes



CONCLUSION & PERSPECTIVES


ConclusionVolatility is not constant but linear in the centreTemperature is generally boundedOur model performs better in the bulk when the skewness and kurtosis of the observations are close to those of a Gaussian law (because of the use of a gaussian law for the residuals)Our model gives in general better results for individual months than for seasonsTaking constraints at the boundaries, improves the simulation of the extremes (see Hoang et al., 2011)The model (with constraints at the boundaries) is not adapted when the data is not bounded (ξ ≥ 0)

Perspectives


Use transformations to make the series more symmetric Bootstrap residuals instead of simulating the gaussian law N(0,1) to solve the problem of skewed data

References

Hoang T.T.H., Modélisation de séries chronologiques non stationnaires, non linéaires. Application à la définition des tendances sur la moyenne, la variabilité et les extrêmes de la température de l'air en Europe, 2010

Parey S., Hoang T.T.H and Dacunha-Castelle D., The role of variance in the evolution of observed temperature extremes in Europe and in the United States, submitted to Climatic Change, 2012, under revision

D. Goldfarb and A. Idnani. Dual and Primal-Dual Methods for Solving Strictly Convex Quadratic Programs. In J. P. Hennart (ed.), Numerical Analysis, Springer-Verlag, Berlin, 226–239, 1982

D. Goldfarb and A. Idnani. A numerically stable dual method for solving strictly convex quadratic programs. Mathematical Programming, 27, 1–33, 1983

Nelder, J. A. and Mead, R. A simplex algorithm for function minimization. Computer Journal 7, 308–313, 1965

Hoang T.T.H., Dacunha-Castelle D. and Benmenzer G., Estimation of a diffusion model with trends taking into account the extremes. Application to temperature in France, Evironmetrics, 22 (3), 464-479, 2011


Principle of our test of trend (or of stationarity)

The considered model: X(t)= θ(t)+ ε(t), the law of ε is known or unknown

Hypothesis of test: θ is constant / θ is not constant

Let: the constant estimator of θ (by m.l.e if the law of ε is known, by least squares if not) the nonparametric estimator of θ (by splines if the law of ε is known, by loess if not)

Idea: compaire these two estimators by the L² distance :

The asymptotic of test is proven (Hoang, 2010)

In practice: build a test on ∆ (build the empirical distribution of ∆ under H0 hypothesis by using the simulations if the law is known or the boostrap samples if not )

33- Simulation model for air temperatures- 31st May 2012

ncnθˆ

nn cˆ −=∆ θ

Documents

non stationarity, non linearity and boundednesspagesperso.univ-brest.fr/~ailliot/roscoff/Parey.pdf · using the simulations if the law is known or bootstrap otherwise) ... Choose