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Modelling spatio-temporal variability of temperature


Forecasting temperature in time and space is an important precondition for both, the design of weather derivatives and the assessment of the hedging effectiveness of index based weather insurance. In this article, we show how this task can be accomplished by means of Kriging techniques. Moreover, we compare Kriging with a dynamic semiparametric factor model (DSFM) that has been recently developed for the analysis of high dimensional financial data. We apply both methods to comprehensive temperature data covering a large area of China and assess their performance in terms of predicting a temperature index at an unobserved location. The results show that the DSFM performs worse than standard Kriging techniques. Moreover, we show how geographic basis risk inherent to weather derivatives can be mitigated by regional diversification.

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  1. The procedure could also be conducted in the opposite way: first, daily temperature models are fitted to the historical data at all locations to forecast the temporal scale. Then, Kriging is used to interpolate the forecasted data for the left-out location on the spatial scale. Although the RMSE results show no significant difference between the two sub-models, the computational effort is much higher.

  2. Note that the temperature model and all coefficients depend on the specific location. To simplify notation, however, we omit the location index \(i\) in this section.

  3. Unfortunately, there is no automatic selection procedure so far to select \(K\) and \(L\). According to Song et al. (2013), either the classic “90 %” rule in principal component analysis could be used to select \(K\) and \(L\) or the “explained variance” method. Alternatively, one could also sequentially test the size of the eigenvalues. These procedures, however, are becoming very costly and time-consuming and a theory for those in the context of DSFM has not yet been developed.


  • Balzer N, Hess U (2010) Climate change and weather risk management: evidence from index-based insurance schemes in China and Ethiopia. In: World Food Programme, Omamo SW, Gentilini U, Sandström S (eds) Revolution: from food aid to food assistance—innovations in overcoming hunger, pp 103–124

  • Barnett BJ, Mahul O (2007) Weather index insurance for agricultural and rural areas in lower-income countries. Am J Agric Econ 89(5):1241–1247. doi:10.1111/j.1467-8276.2007.01091.x

    Article  Google Scholar 

  • Benth FE, Šaltytė-Benth J (2005) Stochastic modelling of temperature variations with a view towards weather derivatives. Appl Math Finance 12(1):53–85. doi:10.1155/2011/576791

    Article  MATH  Google Scholar 

  • Campbell SD, Diebold FX (2005) Weather forecasting for weather derivatives. J Am Stat Assoc 100(469):6–16. doi:10.1198/016214504000001051

    Article  MATH  MathSciNet  Google Scholar 

  • Caruso C, Quarta F (1998) Interpolation methods comparison. Comput Math Appl 35(12):109–126. doi:10.1016/S0898-1221(98)00101-1

    Article  MATH  MathSciNet  Google Scholar 

  • Choroś-Tomczyk B, Härdle WK, Okhrin O (2013) CDO surfaces dynamics. SFB 649 discussion paper 2013-032, Humboldt Universität zu Berlin

  • Cressie NAC (1993) Statistics for spatial data. Revised edition. Wiley series in probability and mathematical statics. Wiley, Hoboken

    Google Scholar 

  • Cressie NAC, Wikle CK (2011) Statistics for spatio-temporal data. Wiley series in probability and mathematical statics. Wiley, Hoboken

    Google Scholar 

  • Deng X, Barnett BJ, Vedenov DV, West JW (2007) Hedging dairy production losses using weather-based index insurance. Agric Econ 36(2):271–280. doi:10.1111/j.1574-0862.2007.00204.x

    Article  Google Scholar 

  • Isaaks Edward H, Mohan Srivastava R (1989) An introduction to applied geostatistics. Oxford University Press, Oxford

    Google Scholar 

  • Fengler MR, Härdle WK, Mammen E (2007) A semiparametric factor model for implied volatility surface dynamics. J Financ Econom 5(2):189–218. doi:10.1093/jjfinec/nbm005

    Article  Google Scholar 

  • Gething PW, Atkinson PM, Noor AM, Gikandi PW, Hay SI, Nixon MS (2007) A local space–time Kriging approach applied to a national outpatient malaria data set. Comput Geosci 33(10):1337–1350. doi:10.1016/j.cageo.2007.05.006

    Article  Google Scholar 

  • Giacomini E, Härdle WK, Krätschmer V (2009) Dynamic semiparametric factor models in risk neutral density estimation. AStA 93(4):387–402. doi:10.1007/s10182-009-0115-4

    Article  MathSciNet  Google Scholar 

  • Härdle WK, López Cabrera B (2011) Localising temperature risk. SFB 649 discussion paper 2011-001, Humboldt-Universität zu Berlin

  • Hellmuth ME, Osgood DE, Hess U, Moorhead A, Bhojwani H (eds) (2009) Index insurance and climate risk: Prospects for development and disaster management. Climate and Society No. 2. International Research Institute for Climate and Society (IRI), Columbia University, New York

  • Holdaway MR (1996) Spatial modeling and interpolation of monthly temperature using kriging. Clim Res 6(3):215–225

    Article  Google Scholar 

  • Krige DG (1951) A statistical approach to some basic mine valuation problems on the Witwatersrand. J Chem Metall Min Soc South Afr

  • Matheron G (1965) Les variables régionalisées et leur estimation. Dissertation, Masson, Paris

  • Miranda M, Vedenov DV (2001) Innovations in agricultural and natural disaster insurance. Am J Agric Econ 83(3):650–655

    Article  Google Scholar 

  • Mußhoff O, Odening M, Xu W (2011) Management of climate risks in agriculture—will weather derivatives permeate? Appl Econ 43(9):1067–1077. doi:10.1080/00036840802600210

    Article  Google Scholar 

  • Okhrin O, Odening M, Xu W (2012) Systemic weather risk and crop insurance: the case of China. J Risk Insur 80(2):351–372. doi:10.1111/j.1539-6975.2012.01476.x

    Article  Google Scholar 

  • Olea RA (ed) (1999) Geostatistics for engineers and earth scientists. Kluwer, Boston

  • Park BU, Mammen E, Härdle WK, Borak S (2009) Time series modelling with semiparametric factor dynamics. J Am Stat Assoc 104(485):284–298. doi:10.1198/jasa.2009.0105

    Article  Google Scholar 

  • Paulson ND, Hart CE (2006) A spatial approach to addressing weather derivative basis risk: a drought insurance example. Annual meeting of the American Agricultural Economics Association, California.

  • Paulson ND, Hart CE, Hayes DJ (2010) A spatial Bayesian approach to weather derivatives. Agric Finance Rev 70(1):79–96. doi:10.1108/00021461011042657

    Article  Google Scholar 

  • People’s Bank of China (2013) China financial stability report 2013. Chapter 4: insurance sector. China Financial Publishing House

  • Ritter M, Mußhoff O, Odening M (2014) Minimizing geographical basis risk of weather derivatives using a multi-site rainfall model. Comput Econ 44(1):67–86. doi:10.1007/s10614-013-9410-y

    Article  Google Scholar 

  • Šaltytė-Benth J, Benth FE (2012) A critical view on temperature modeling for application in weather derivatives markets. Energy Econ 34(2):592–602. doi:10.1016/j.eneco.2011.09.012

    Article  Google Scholar 

  • Skees JR, Barnett BJ, Murphy AG (2008) Creating insurance markets for natural disaster risk in lower income countries: the potential role for securitization. Agric Finance Rev 68(1):151–167. doi:10.1108/00214660880001224

    Article  Google Scholar 

  • Sluiter R (2009) Interpolation methods for climate data. KNMI IR 2009-04. Royal Netherlands Meteorological Institute, De Bilt

    Google Scholar 

  • Song S, Härdle WH, Ritov Y (2010) High dimensional nonstationary time series modelling with generalized dynamic semiparametric factor model. SFB 649 discussion paper 2010-039, Humboldt-Universität zu Berlin

  • Song S, Härdle WH, Ritov Y (2013) Generalized dynamic semiparametric factor models for high dimensional nonstationary time series. Econom J. doi:10.1111/ectj.12024

  • Stahl K, Moore RD, Floyer JA, Asplin MG, McKendry IG (2006) Comparison of approaches for spatial interpolation of daily air temperature in a large region with complex topography and highly variable station density. Agric For Meteorol 139(3–4):224–236. doi:10.1016/j.agrformet.2006.07.004

    Article  Google Scholar 

  • Stroud JR, Mueller P, Sanso B (2001) Dynamic models for spatiotemporal data. J R Stat Soc B Stat Methodol 63(4):673–689. doi:10.1111/1467-9868.00305

    Article  MATH  Google Scholar 

  • Woodard JD, Garcia P (2008) Basis risk and weather hedging effectiveness. Agric Finance Rev 68(1):99–117. doi:10.1108/00021461211277295

    Article  Google Scholar 

  • World Bank (2005) Managing agricultural production risk: innovations in developing countries. Washington, DC.

  • Ye Y, Yang S, Huang Y, Liu F (2007) Development and prospect of wheat production in Henan Province. Chin Agric Sci Bull 23(1):199–203

    Google Scholar 

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Financial support from the German Research Foundation via CRC 649 “Economic Risk” is gratefully acknowledged.

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Correspondence to Matthias Ritter.



See Figs. 910 and 11.

Fig. 9
figure 9

Ten years of temperature data (black) with the fitted conditional mean function (red) in station Quzhou (color figure online)

Fig. 10
figure 10

Estimated model residuals in station Quzhou after removing the trend, seasonality, and the AR components

Fig. 11
figure 11

Empirical and fitted volatility function in station Quzhou

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Cao, X., Okhrin, O., Odening, M. et al. Modelling spatio-temporal variability of temperature. Comput Stat 30, 745–766 (2015).

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