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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.
Interpolating and Forecasting Chinese Temperature
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.
Interpolating and Forecasting Chinese Temperature
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).
Interpolating and Forecasting Chinese Temperature
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
Interpolating and Forecasting Chinese Temperature
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.
Interpolating and Forecasting Chinese Temperature
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.
Interpolating and Forecasting Chinese Temperature
Spatial Model 4-5
Empirical Results of Time Series Model
Figure 4: Fitted volatility model in station 1.
Interpolating and Forecasting Chinese Temperature
Spatial Model 4-6
Empirical Results of Kriging
Figure 5: Spatial description map of εt for t = 1
Interpolating and Forecasting Chinese Temperature
Spatial Model 4-7
Empirical Results of Kriging
Figure 6: Directional variogram in four directions for t = 1.
Interpolating and Forecasting Chinese Temperature
Spatial Model 4-8
Empirical Results of Kriging
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 .
Interpolating and Forecasting Chinese Temperature
Dynamic Semiparametric Factor Model 5-2
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.
Interpolating and Forecasting Chinese Temperature
Dynamic Semiparametric Factor Model 5-3
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.
Interpolating and Forecasting Chinese Temperature
Dynamic Semiparametric Factor Model 5-4
Empirical Results
Figure 8: Basis risk functions m.
Interpolating and Forecasting Chinese Temperature
Dynamic Semiparametric Factor Model 5-5
Empirical Results
Figure 9: Time series of weights Z .
Interpolating and Forecasting Chinese Temperature
Dynamic Semiparametric Factor Model 5-6
Empirical Results
Figure 10: Forecasting at station 1: true observation, forecasting withconsidering of Z vector volatility modeling.
Interpolating and Forecasting Chinese Temperature
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.
Interpolating and Forecasting Chinese Temperature
Model Comparison 6-2
Model Comparison
Figure 12: Root mean square errors from Model A and Model B for all 100stations.
Interpolating and Forecasting Chinese Temperature
Model Comparison 7-1
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.
Interpolating and Forecasting Chinese Temperature
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.
Interpolating and Forecasting Chinese Temperature
Appendix 9-1
Empirical Results
Figure 13: Simulation results of Model A and Model B in station 1.
Interpolating and Forecasting Chinese Temperature
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.
Interpolating and Forecasting Chinese Temperature
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.
Interpolating and Forecasting Chinese Temperature