A kernel spatial density estimation allowing for the analysis of spatial clustering. Application to Monsoon Asia Drought Atlas data

A nonparametric density estimate that incorporates spatial dependency has not been studied in the literature. In this article, we propose a new spatial density estimator that depends on two kernels: one controls the distance between observations while the other controls the spatial dependence structure. The uniform almost sure convergence of the density estimate is established with the rate of convergence. The consistency of the mode of this kernel density is also studied. Then a spatial hierarchical unsupervised clustering algorithm based on the mode estimate is presented. Some simulations as well as an application to the Monsoon Asia Drought Atlas data illustrate the efficiency of our algorithm, and a comparison of the spatial structures of these data detected by the density estimate and clustering algorithm are done.

[1]  S. Dabo‐Niang,et al.  Kernel regression estimation for continuous spatial processes , 2007 .

[2]  Denis Bosq,et al.  Nonparametric statistics for stochastic processes , 1996 .

[3]  Bo Huang,et al.  Prediction for spatio-temporal models with autoregression in errors , 2012 .

[4]  Noel A Cressie,et al.  Statistics for Spatio-Temporal Data , 2011 .

[5]  Jussi Klemelä.  Density estimation with locally identically distributed data and with locally stationary data , 2007 .

[6]  L. Baxter.  Random Fields on a Network: Modeling, Statistics, and Applications , 1996 .

[7]  C. Francq,et al.  Kernel regression estimation for random fields , 2007 .

[8]  Brian D. Ripley,et al.  Spatial Statistics: Ripley/Spatial Statistics , 2005 .

[9]  Hans Wackernagel,et al.  Multivariate Geostatistics: An Introduction with Applications , 1996 .

[10]  L. Tran,et al.  Kernel density estimation for random fields (density estimation for random fields) , 1997 .

[11]  Sophie Dabo-Niang,et al.  Spatial mode estimation for functional random fields with application to bioturbation problem , 2010 .

[12]  Z. Q. John Lu,et al.  Nonlinear Time Series: Nonparametric and Parametric Methods , 2004, Technometrics.

[13]  E. Ziegel,et al.  Proceedings in Computational Statistics , 1998 .

[14]  W. C. Mitchell.  The number of lattice points in a $k$-dimensional hypersphere , 1966 .

[15]  Frédéric Ferraty,et al.  Nonparametric Functional Data Analysis: Theory and Practice (Springer Series in Statistics) , 2006 .

[16]  Edward R. Cook,et al.  Asian Monsoon Failure and Megadrought During the Last Millennium , 2010, Science.

[17]  Mike Rees,et al.  5. Statistics for Spatial Data , 1993 .

[18]  P. Robinson,et al.  Asymptotic theory for nonparametric regression with spatial data , 2011 .

[19]  J. Fort,et al.  Statistics and Applications , 1991 .

[20]  H. Wackernagle,et al.  Multivariate geostatistics: an introduction with applications , 1998 .

[21]  S. Dabo‐Niang,et al.  Kernel spatial density estimation in infinite dimension space , 2011, Metrika.

[22]  Luc Anselin,et al.  New Directions in Spatial Econometrics , 2011 .

[23]  Martin L. Hazelton,et al.  sparr: Analyzing Spatial Relative Risk Using Fixed and Adaptive Kernel Density Estimation in R , 2011 .

[24]  Annika Meyer.  On the number of lattice points in a small sphere and a recursive lattice decoding algorithm , 2013, Des. Codes Cryptogr..

[25]  S. Yakowitz,et al.  Nearest neighbor estimators for random fields , 1993 .

[26]  R. Menezes,et al.  Nonparametric spatial prediction under stochastic sampling design , 2010 .

[27]  Ping-Shi Wu,et al.  Real-Time Density and Mode Estimation With Application to Time-Dynamic Mode Tracking , 2006 .

[28]  Marc Hallin,et al.  Kernel density estimation for spatial processes: the L 1 theory , 2004 .

[29]  Raquel Menezes,et al.  A Kernel Variogram Estimator for Clustered Data , 2008 .

[30]  Mohamed El Machkouri,et al.  Asymptotic normality of the Parzen–Rosenblatt density estimator for strongly mixing random fields , 2010, 1008.1342.

[31]  Roger Woodard,et al.  Interpolation of Spatial Data: Some Theory for Kriging , 1999, Technometrics.

[32]  Z. Q. John Lu,et al.  Nonparametric Functional Data Analysis: Theory And Practice , 2007, Technometrics.

[33]  Denis Allard,et al.  CART algorithm for spatial data: Application to environmental and ecological data , 2009, Comput. Stat. Data Anal..

[34]  Henryk Iwaniec,et al.  On the Sphere Problem , 1995 .

[35]  S. Dabo‐Niang,et al.  KERNEL REGRESSION ESTIMATION FOR SPATIAL FUNCTIONAL RANDOM VARIABLES , 2010 .

[36]  K. Tsang,et al.  Counting Lattice Points in The Sphere , 2000 .

[37]  Christoph F. Eick,et al.  On supervised density estimation techniques and their application to spatial data mining , 2007, GIS.

[38]  John Odenckantz,et al.  Nonparametric Statistics for Stochastic Processes: Estimation and Prediction , 2000, Technometrics.

[39]  Noel A Cressie,et al.  Statistics for Spatial Data. , 1992 .

[40]  Estimation of the spatial distribution through the kernel indicator variogram , 2012 .

[41]  P. Newbold,et al.  Estimation and Prediction , 1985 .

[42]  Zhe Jiang,et al.  Spatial Statistics , 2013 .

[43]  Polygone des fréquences pour des champs aléatoires , 2006 .

[44]  Arne Kovac.  Proceedings in Computational Statistics , 1976 .

[45]  Michael L. Stein,et al.  Interpolation of spatial data , 1999 .