Konstantin Krivoruchko

Statistical tools for regional data analysis using GIS

A GIS provides a powerful collection of tools for the management, visualization and analysis of spatial data. These tools can be even more powerful when they are integrated with statistical methods for spatial data analysis and many GIS users are requesting this integration. The Geostatistical Analyst extension to ArcGIS was developed to integrate statistical methods with GIS tools for mapping and modeling spatially continuous data such as temperature or pollution. However, many GIS applications involve data that are aggregated over geographic regions and the analysis of this type of spatial data poses additional challenges. In this paper, we illustrate several different analytical goals that commonly arise in applications based on regional data. Many of these require a measure of local spatial dependence and this is commonly based on Morans I index of spatial association. However, as we describe in this paper, other measures that more explicitly take into account the aggregated nature of the data may be preferred. Using county-level crime data in California we show how many different statistical methods for regional data analysis can be implemented within a GIS to provide a powerful set of interactive, analytical tools uniquely suited to the goals of regional analysis.

Geostatistical Mapping with Continuous Moving Neighborhood

An issue that often arises in such GIS applications as digital elevation modeling (DEM) is how to create a continuous surface using a limited number of point observations. In hydrological applications, such as estimating drainage areas, direction of water flow is easier to detect from a smooth DEM than from a grid created using standard interpolation programs. Another reason for continuous mapping is esthetic; like a picture, a map should be visually appealing, and for some GIS users this is more important than map accuracy. There are many methods for local smoothing. Spline algorithms are usually used to create a continuous map, because they minimize curvature of the surface. Geostatistical models are commonly used approaches to spatial prediction and mapping in many scientific disciplines, but classical kriging models produce noncontinuous surfaces when local neighborhood is used. This motivated us to develop a continuous version of kriging. We propose a modification of kriging that produces continuous prediction and prediction standard error surfaces. The idea is to modify kriging systems so that data outside a specified distance from the prediction location have zero weights. We discuss simple kriging and conditional geostatistical simulation, models that essentially use information about mean value or trend surface. We also discuss how to modify ordinary and universal kriging models to produce continuous predictions, and limitations using the proposed models.

New Flexible Non-parametric Data Transformation for Trans-Gaussian Kriging

This paper proposes a new flexible non-parametric data transformation to Gaussian distribution. This option is often required because kriging is the best predictor under squared-error minimization criterion only if the data follow multivariate Gaussian distribution, while environmental data are often best described by skewed distributions with non-negative values and a heavy right tail. We assume that the modeling random field is the result of some nonlinear transformation of a Gaussian random field. In this case, the researchers commonly use a certain parametric monotone (for example, power or logarithmic) or variants of normal score transformation. We discuss drawbacks of these methods and propose a new flexible non-parametric transformation. We compare the performance of simple kriging with the proposed data transformation to several other data transformation methods, including transformation based on a mixture of Gaussian kernels and multiplicative skewing with several base distributions. Our method is flexible, and it can be used for automatic data transformation, for example, in black-box kriging models in emergency situations.

Pragmatic Bayesian Kriging for Non-Stationary and Moderately Non-Gaussian Data

We discuss two flexible and fast empirical Bayesian kriging models: (1) intrinsic random function of order zero and one and (2) kriging with local data transformation to a Gaussian distribution. In the case of large datasets, all calculations are made in the data subsets, and predictions are made using weighted sums of predictions from different subsets, possibly overlapping. The methodology is illustrated using 1.35 billion samples collected using LiDAR technology.

GIS, Users, Developers, and Spatial Statistics: On Monarchs and Their Clothing

The development and documentation of software for the analysis of geographical data is maturing, and the needs and desires of varying user communities are becoming clearer. Certainly today there are more users in more communities, and in general much more data than before, even though data is more accessible in some countries than in others. Many more users are now meeting geographical data through geographical information systems software (GIS). GIS are general-purpose environments for handling geographical data, and do not assume that the user will need to make predictions or draw inferences from the data, or error propagation in geographical data analysis. Indeed, much of current progress in GIS is in making it easier for users to construct maps at the front end and in providing open and consistent data base support at the back end. Neither of these two areas lie close to the central concerns of statistical data analysts, such as making predictions with associated uncertainties, but can be of great value to them. In meeting and undertaking dialogues with users and developers, it seems both valid and important to attempt to explore some of the assumptions the different communities hold themselves, have about each other, and the tasks they undertake separately and jointly. Some of the points to be made will draw in the ontology discourse in geographical information science (GIScience), which may be helpful in throwing light of different assumptions made by different communities, not just technical/motivational, but also related to the sociology of organizations and of scientific disciplines. The paper3 discusses these issues in general terms, but more specifically touching on tools and methods that may propagate between communities of users, and on difficulties associated with the use of inference in inappropriate settings. In particular, we will present and discuss selected examples of analytical practice that are

Modeling the Semivariogram: New Approach, Methods Comparison, and Simulation Study

This chapter proposes some new methods for computing empirical semivariograms and covariances and for fitting semivariogram and covariance models to empirical data. Grid-based empirical semivariograms and covariances are described, in which the grid values are smoothed using triangular kernels. A model-fitting procedure using modified iterative weighted least squares is presented. This algorithm is shown to be reliable for a wide range of data types and conditions, and its implementation in commercial software is discussed. Comparisons to restricted maximum likelihood estimation are also discussed.

A New Method for Handling the Nugget Effect in Kriging

This chapter discusses methods for estimating the nugget in semivariogram models. Commonly used exact and filtered kriging methods are compared with an alternative method, which predicts a new value at the sampled location. Using the alternative method, estimation at a location where data have been collected involves predicting the smooth underlying value plus a new observation from measurement error. This is exactly what is necessary for validation and cross-validation diagnostics. Three examples of using new value kriging are presented that involve comparison of simulated results, porosity estimation for the North Cowden unit in west Texas, and analysis of radiocesium soil-contamination data collected in Belarus after the Chernobyl accident.