Estimating and testing zones of abrupt change for spatial data

We propose a method for detecting the zones where a variable irregularly sampled in the plane changes abruptly. Our general model is that under the null hypothesis the variable is the realisation of a stationary Gaussian process with constant expectation. The alternative is that the mean function presents abrupt changes. We define potential Zones of Abrupt Change (ZACs) by the points where the gradient, estimated under the null hypothesis, exceeds a determined threshold. We then design a global test to assess the global significance of the potential ZACs, an issue missing in all existing methods. The theory that links the threshold and the global level is based on asymptotic distributions of excursion sets of non-stationary χ2 fields for which we provide new results. The method is evaluated by a simulation study and applied to a soil data set in the context of precision agriculture.

[1]  H. O. Posten.  Multidimensional Gaussian Distributions , 1964 .

[2]  Sudipto Banerjee,et al.  Bayesian Wombling , 2006, Journal of the American Statistical Association.

[3]  Donald Geman,et al.  Stochastic Relaxation, Gibbs Distributions, and the Bayesian Restoration of Images , 1984, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[4]  R. Adler,et al.  Random Fields and Geometry , 2007 .

[5]  Robert Haining,et al.  Statistics for spatial data: by Noel Cressie, 1991, John Wiley & Sons, New York, 900 p., ISBN 0-471-84336-9, US $89.95 , 1993 .

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

[7]  Edith Gabriel.  Détection de zones de changement abrupt dans des données spatiales et application à l'agriculture de précision. (Detecting zones of abrupt change for spatial data and application to precision agriculture) , 2004 .

[8]  J. Marron,et al.  SCALE SPACE VIEW OF CURVE ESTIMATION , 2000 .

[9]  J. Chilès,et al.  Geostatistics: Modeling Spatial Uncertainty , 1999 .

[10]  Donald Geman,et al.  Stochastic relaxation, Gibbs distributions, and the Bayesian restoration of images , 1984 .

[11]  Denis Allard,et al.  Evaluating the sampling pattern when detecting Zones of Abrupt Change , 2007, Environmental and Ecological Statistics.

[12]  Xiaotong Shen,et al.  Nonparametric Hypothesis Testing for a Spatial Signal , 2002 .

[13]  A. Gelfand,et al.  Gaussian predictive process models for large spatial data sets , 2008, Journal of the Royal Statistical Society. Series B, Statistical methodology.

[14]  R. Adler,et al.  The Geometry of Random Fields , 1982 .

[15]  M. Guérif,et al.  Detecting zones of abrupt change in soil data, with an application to an agricultural field , 2007 .

[16]  Andrew B. Lawson,et al.  Spatial cluster modelling , 2002 .

[17]  R. Adler,et al.  Sample path behaviour of Χ 2 surfaces at extrema , 1988, Advances in Applied Probability.

[18]  J. Besag.  On the Statistical Analysis of Dirty Pictures , 1986 .

[19]  W. H. Womble,et al.  Differential systematics. , 1951, Science.

[20]  A. Yaglom.  Correlation Theory of Stationary and Related Random Functions I: Basic Results , 1987 .

[21]  J. Marron,et al.  SiZer for Exploration of Structures in Curves , 1999 .

[22]  Yazhen Wang,et al.  Change Curve Estimation via Wavelets , 1998 .

[23]  K. Worsley,et al.  Local Maxima and the Expected Euler Characteristic of Excursion Sets of χ 2, F and t Fields , 1994, Advances in Applied Probability.

[24]  Liang Peng,et al.  Local likelihood tracking of fault lines and boundaries , 2001 .

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

[26]  Marie-Josée Fortin,et al.  Edge detection algorithms for two-dimensional ecological data , 1993 .

[27]  Peihua Qiu,et al.  Discontinuous regression surfaces fitting , 1998 .

[28]  J. Cao.  The size of the connected components of excursion sets of χ2, t and F fields , 1999, Advances in Applied Probability.

[29]  R. Adler.  On excursion sets, tube formulas and maxima of random fields , 2000 .

[30]  Robert J. Adler,et al.  Extrema and level crossings of χ2 processes , 1986, Advances in Applied Probability.

[31]  R. Sokal,et al.  Detecting Regions of Abrupt Change in Maps of Biological Variables , 1989 .

[32]  Bradley P Carlin,et al.  Bayesian Wombling for Spatial Point Processes , 2009, Biometrics.

[33]  M. Fortin,et al.  Delineation of Ecological Boundaries: Comparison of Approaches and Significance Tests , 1995 .

[34]  C. F. Sirmans,et al.  Directional Rates of Change Under Spatial Process Models , 2003 .

[35]  Fred Godtliebsen,et al.  Significance in scale-space for clustering , 2002 .

[36]  R. Adler.  The Geometry of Random Fields , 2009 .

[37]  Noel A Cressie,et al.  Statistics for Spatial Data, Revised Edition. , 1994 .

[38]  Keith J. Worsley,et al.  Testing for signals with unknown location and scale in a χ2 random field, with an application to fMRI , 2001, Advances in Applied Probability.

[39]  D. Aldous.  Probability Approximations via the Poisson Clumping Heuristic , 1988 .

[40]  D. Rasch.  Discussion in: Precision Agriculture: Spatial and Temporal Variability of Environmental Quality. , 1997 .