interpolating and forecasting chinese...

Interpolating and Forecasting Chinese Tem-perature: A Comparison of Statistical Ap-proaches.

Xiaofeng Cao, Prof. Okhrin, Prof. OdeningEnergy Finance WorkshopApril 2013, Stolberg

Humboldt-University of BerlinDepartment of Agricultural EconomicsFarm Management Group

Motivation 1-1

Motivation

Figure 1: Source:www.xn121.com.

Interpolating and Forecasting Chinese Temperature

Motivation 1-2

Motivation

� Chinese farmers are vulnerable to weather risk;� The weather insurance market grows in China: pilot study of

weather index-based insurance (WII);� Low density of historical weather data..

� Objective:Find a statistical model to forecast future temperature (index)and interpolate at unobservable locations, as fundamentalcalculation to price WII and qualify geographical basis risk.


Motivation 1-3

Procedure

� Model A: interpolate by Kriging → time series analysis toforecast;

� Model B: time series analysis to forecast → interpolate byKriging;

� Model C: dynamic semiparametric factor model.

� Cross validation:1, One station out of the sample as unobserved location eachtime.2, Repeat the procedure for all stations and compare resultswith true observations.


Motivation 1-4

Key Model References

� Benth et al (2005, 2011): stochastic time series model with trend,seasonality, AR process and volatility patterns.

� Hofstra et al (2008): Kriging is the best method based on thecomparison of current best-of-class interpolation methods.

� Park/Härdle/Borak (2009): time series model with semiparametricfactor dynamics (DSFM).


Outline 2-1

Outline

1. Motivation X2. Modeling Temperature on Temporal and Spatial Level

2.1 Univariate Time Series Model on Temporal Level2.2 Kriging on Spatial Level

3. Dynamic Semiparametric Factor Model4. Model Comparison5. References


Modeling Temperature on Temporal and Spatial Level 3-1

Stochastic Time Series Model

1.Detrend and deseasonalize using truncated Fourier SeriesYt = Λt +Xt

Λt = β0 + β1d(t) + ∑Pp=1

[as,p sin

{2πp d(t)

365

}+ac,p cos

{2πp d(t)

365

}]2.Eliminate the AR part

Xt = ∑Ll=1 δlXt−l + ηt

ηt = σtεt εt ∼ iid(0,1)

3.Capture conditional variance dynamicsσ2

t = b0 + ∑Qq=1

[bs,q sin 2πq d(t)

365 +bc,q cos 2πq d(t)365

]Interpolating and Forecasting Chinese Temperature

Spatial Model 4-1

The Experimental Variogram

Assumption: "intrinsic stationarity" E [Z (Si +h)−Z (Si )] = 0

The variogram estimator: 2γ(h) = 1|N(h)| ∑N(h) [Z (Si +h)−Z (Si )]2

where Z(Si ) is the value at locations Si , N(h) is the number of experimental pairsseparated by the distance h between each pair.

The Stein-Matern Variogram: ρ(h) = π12 φ

2κ−1Γ(κ+ 12 )a2κ

(a |h|)κKκ (a |h|)where Kκ is a modified Bessel function, κ is a additional smooth parameter, and a is

the range.


Spatial Model 4-2

The Ordinary Kriging

The estimation requirements:

(1) Linearity: Z (S0) = ∑ni=1ωiZ (Si )

(2) Unbiasedness: ∑ni=1ωi = 1

(3) Minimize the mean-square prediction error:E [∑n

i=1ωiZ (Si )−Z (S0)]2−2m(∑ni=1ωi −1)

Z(S0): estimator at new location S0, ωi weights, m is the Lagrange multiplier. The

mean square predict error (Kriging variance or Kriging Standard error) is

σ2e = E

[Z(Si )− Z(S0)

]2Interpolating and Forecasting Chinese Temperature

Spatial Model 4-3

Data and Empirical Results

Figure 2: : Map of Chinese weather stations. Source: cdc.cma.gov.cn.

Daily temperature 1957 - 2008, 100 selected stations in China, data in2009 as benchmarkInterpolating and Forecasting Chinese Temperature

Spatial Model 4-4

Empirical Results of Time Series Model

Figure 3: Fitted time series model in station 1.


Spatial Model 4-5

Empirical Results of Time Series Model

Figure 4: Fitted volatility model in station 1.


Spatial Model 4-6

Empirical Results of Kriging

Figure 5: Spatial description map of εt for t = 1


Spatial Model 4-7


Figure 6: Directional variogram in four directions for t = 1.


Spatial Model 4-8


Figure 7: Fitted variogram and Ordinary Kriging model for t = 1 .Interpolating and Forecasting Chinese Temperature

Dynamic Semiparametric Factor Model 5-1

DSFM

The dynamic semiparametric factor model has a form:Yt,j = m0(Xt,j) + ∑

Ll=1Zt,lml (Xt,j) + εt,j = ZT

t ml (Xt,j) + εt,j

where:

� m(•) is a tuple of functions (m0,m1, . . . ,ml )T , and reflects the time

invariant (factor) structure;

� Zt = (1,Zt,1, . . . ,Zt,L) is a multivariate time series with a certaindynamics structure;

� t = 1, . . . ,T , j = 1, . . . ,J,Xt,j ∈ [0,1]d .



Series Estimator

Use a series estimator to estimate m:ZT

t ml (Xt,j) = ∑Ll=0Zt,l ∑

Kk=0 al ,kΨk(X ) = ZT

t AΨ(X )

To minimize:(ZtA) = argmin

Zt ,A∑

Ll=1∑

Kk=1

{Yt,j − ZtAΨ(Xt,j)

}2

Smoothing parameters:� L - dimension of the time series;� Ψ - known basis functions (here B-splines);� K - number of the series expansion functions, and plays a role

of the bandwidth in Kernel estimation.

� consider Zt as the same time series process in Model A.



Empirical Results

RV (L) =∑

Ll=1 ∑

Kk=1{Yt,j−Zt AΨ(Xt,j )}2

∑Ll=1 ∑

Kk=1(Yt,j−Y )2

No. of Factors 1−RV (L)L=2 0.954L=3 0.959L=4 0.965L=5 0.968

We estimate L = 3 basis functions, and choose numbers of K = 6and order of splines P = 2.



Empirical Results

Figure 8: Basis risk functions m.



Empirical Results

Figure 9: Time series of weights Z .



Empirical Results

Figure 10: Forecasting at station 1: true observation, forecasting withconsidering of Z vector volatility modeling.


Model Comparison 6-1

Model Comparison

Figure 11: Absolute deviations between Y and Y from Model A (withµ = 3.20,sd = 1.20) and Model B (with µ = 3.53,sd = 2.28) at station 1.



Model Comparison

Figure 12: Root mean square errors from Model A and Model B for all 100stations.



Test forecast accuracy

� Set true observations in 2009 as benchmark;� Test forecast errors of both models by Diebold-Mariano test.

H0 α = 0.05 α = 0.1DM test Equal predictive accuracy 66 72t test No difference in mean 21 32

Table 1: Number of stations not to accept H0.


Further Work 8-1

Further Work

� The simulation results of Model A and B are similarly accurate;� Possible reasons for forecasting problem in DSFM?� Apply Kriging and DSFM directly to weather indexes (GDD

and FI) for whole period;� Complete the model comparison.


Appendix 9-1

Empirical Results

Figure 13: Simulation results of Model A and Model B in station 1.


Reference 10-1

Reference

S.D. Campbell and F.X. Diebold, Weather Forecasting for Weather Derivatives,Journal of the American Statistical Association, March 2005, Vol.100, No.469.

S.D. Campbell and F.X. Diebold, Weather Forecasting for Weather Derivatives,Working paper, Dec. 2002.

W. Haerdle, B. Cabrera, O. Okhrin and W. Wang, Localising temperature risk,SFB 649 Discussion Paper 2011-001.

F. E. Benth, J. S. Benth, Stochastic Modelling of Temperature Variations with aView Towards Weather Derivatives, Applied Mathematical Finance, March 2005,Vol.12, No.1.

F. E. Benth, J. S. Benth, A critical view on temperature modelling forapplication in weather derivatives markets, Energy Economics, doi:10.1016/j.eneco.2011.09.012.

N. Cressie and C.K.Wikle, Statistics for Spatio-Temporal Data, Wiley, 2011.


Reference 10-2

Reference

Park, B. Mammen, E. Härdle, W. and Borak, S., Time series modelling withsemipaprametric factor dynamics , Journal of the American StatisticalAssociation, 104:485.

A.B.Koehler and E.S.Murphree, A Comparison of the Akaike and SchwarzCriteria for Selecting Model Order, Journal of the Royal Statistical Society.Series C(Applied Statistics), 1988, Vol.37, No.2, Pages 187-195.

T. Hengl, A Practical Guide to Geostatistical Mapping , University ofAmsterdam Press, 2nd edition, 2011.

R.A.Olea, Geostatistics for Engineers and Earth Scientists, Kluwer AcademicPublishers, 1999.

M.L.Stein, Interpolation of Spatial Data, Springer, 1999.

N.A.C. Cressie, Statistics for Spatial Data, Wiley-Interscience, 1993.


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