Geostatistics with infinite dimensional data: a generalization of cokriging and multivariable spatial prediction

Abstract

We  extend  cokriging  analysis  and  multivariable  spatial  prediction  to  the  case  where  the  observations  at  each  sampling  location  consist  of  samples  of  random  functions,  that  is,  we  extend  two  classical  multivariable  geostatistical  methods  to  the  functional  context.  Our  cokriging method predicts one variable at a time as in a classical multivariable sense, but considering as auxiliary information curves instead of vectors. We also propose an extension of multivariable kriging to the functional context by defining a predictor of a whole curve based on samples  of  curves  located  at  a  neighborhood  of  the  prediction  site.  In  both  cases  a  non-parametric  approach  based  on  basis  function expansion  is  used  to  estimate  the  parameters,  and  we  prove  that  both  proposals  coincide  when  using  such  an  approach.  A  linear  model  of  coregionalization  is  used  to  define  the  spatial  dependence  among  the  coeficients  of  the  basis  functions,  and  therefore  for  estimating  the  functional  parameters.  As  an  illustration  the  methodological  proposals  are  applied  to  analyze  two  real  data  sets  corresponding  to  average  daily temperatures measured at 35 weather stations located in the Canadian Maritime Provinces, and penetration resistance data collected at 32 sampling sites of an experimental plot.