Thirty-seven programs that summarize data with histograms and other graphics, calculate measures of spatial continuity, provide smooth least-squares-type maps, and perform stochastic spatial simulation.

GSLIB: Geostatistical Software Library and User's Guide

This article focuses on the construction, directly in physical space, of simply parametrized covariance functions for data-assimilation applications. A self-contained, rigorous mathematical summary of relevant topics from correlation theory is provided as a foundation for this construction. Covariance and correlation functions are defined, and common notions of homogeneity and isotropy are clarified. Classical results are stated, and proven where instructive. Included are smoothness properties relevant to multivariate statistical-analysis algorithms where wind/wind and wind/mass correlation models are obtained by differentiating the correlation model of a mass variable. the Convolution Theorem is introduced as the primary tool used to construct classes of covariance and cross-covariance functions on three-dimensional Euclidean space R3. Among these are classes of compactly supported functions that restrict to covariance and cross-covariance functions on the unit sphere S2, and that vanish identically on subsets of positive measure on S2. It is shown that these covariance and cross-covariance functions on S2, referred to as being space-limited, cannot be obtained using truncated spectral expansions. Compactly supported and space-limited covariance functions determine sparse covariance matrices when evaluated on a grid, thereby easing computational burdens in atmospheric data-analysis algorithms.

Convolution integrals leading to practical examples of compactly supported covariance and cross-covariance functions on R3 are reduced and evaluated. More specifically, suppose that gi and gj are radially symmetric functions defined on R3 such that gi(x) = 0 for |x| > di and gj(x) = 0 for |xv > dj, O di + dj and |x - y| > 2di, respectively, Additional covariance functions on R3 are constructed using convolutions over the real numbers R, rather than R3. Families of compactly supported approximants to standard second- and third-order autoregressive functions are constructed as illustrative examples. Compactly supported covariance functions of the form C(x,y) := Co(|x - y|), x,y ∈ R3, where the functions Co(r) for r ∈ R are 5th-order piecewise rational functions, are also constructed. These functions are used to develop space-limited product covariance functions B(x, y) C(x, y), x, y ∈ S2, approximating given covariance functions B(x, y) supported on all of S2 × S2.

/pdf/construction-of-correlation-functions-in-two-and-three-5v9yuj9d1h.pdf

Construction of correlation functions in two and three dimensions

Preface.Acknowledgments.1 Introduction.1.1 Why Spatial Data in Public Health?1.2 Why Statistical Methods for Spatial Data?1.3 Intersection of Three Fields of Study.1.4 Organization of the Book.2 Analyzing Public Health Data.2.1 Observational vs. Experimental Data.2.2 Risk and Rates.2.2.1 Incidence and Prevalence.2.2.2 Risk.2.2.3 Estimating Risk: Rates and Proportions.2.2.4 Relative and Attributable Risks.2.3 Making Rates Comparable: Standardized Rates.2.3.1 Direct Standardization.2.3.2 Indirect Standardization.2.3.3 Direct or Indirect?2.3.4 Standardizing to What Standard?2.3.5 Cautions with Standardized Rates.2.4 Basic Epidemiological Study Designs.2.4.1 Prospective Cohort Studies.2.4.2 Retrospective Case-Control Studies.2.4.3 Other Types of Epidemiological Studies.2.5 Basic Analytic Tool: The Odds Ratio.2.6 Modeling Counts and Rates.2.6.1 Generalized Linear Models.2.6.2 Logistic Regression.2.6.3 Poisson Regression.2.7 Challenges in the Analysis of Observational Data.2.7.1 Bias.2.7.2 Confounding.2.7.3 Effect Modification.2.7.4 Ecological Inference and the Ecological Fallacy.2.8 Additional Topics and Further Reading.2.9 Exercises.3 Spatial Data.3.1 Components of Spatial Data.3.2 An Odyssey into Geodesy.3.2.1 Measuring Location: Geographical Coordinates.3.2.2 Flattening the Globe: Map Projections and Coordinate Systems.3.2.3 Mathematics of Location: Vector and Polygon Geometry.3.3 Sources of Spatial Data.3.3.1 Health Data.3.3.2 Census-Related Data.3.3.3 Geocoding.3.3.4 Digital Cartographic Data.3.3.5 Environmental and Natural Resource Data.3.3.6 Remotely Sensed Data.3.3.7 Digitizing.3.3.8 Collect Your Own!3.4 Geographic Information Systems.3.4.1 Vector and Raster GISs.3.4.2 Basic GIS Operations.3.4.3 Spatial Analysis within GIS.3.5 Problems with Spatial Data and GIS.3.5.1 Inaccurate and Incomplete Databases.3.5.2 Confidentiality.3.5.3 Use of ZIP Codes.3.5.4 Geocoding Issues.3.5.5 Location Uncertainty.4 Visualizing Spatial Data.4.1 Cartography: The Art and Science of Mapmaking.4.2 Types of Statistical Maps.MAP STUDY: Very Low Birth Weights in Georgia Health Care District 9.4.2.1 Maps for Point Features.4.2.2 Maps for Areal Features.4.3 Symbolization.4.3.1 Map Generalization.4.3.2 Visual Variables.4.3.3 Color.4.4 Mapping Smoothed Rates and Probabilities.4.4.1 Locally Weighted Averages.4.4.2 Nonparametric Regression.4.4.3 Empirical Bayes Smoothing.4.4.4 Probability Mapping.4.4.5 Practical Notes and Recommendations.CASE STUDY: Smoothing New York Leukemia Data.4.5 Modifiable Areal Unit Problem.4.6 Additional Topics and Further Reading.4.6.1 Visualization.4.6.2 Additional Types of Maps.4.6.3 Exploratory Spatial Data Analysis.4.6.4 Other Smoothing Approaches.4.6.5 Edge Effects.4.7 Exercises.5 Analysis of Spatial Point Patterns.5.1 Types of Patterns.5.2 Spatial Point Processes.5.2.1 Stationarity and Isotropy.5.2.2 Spatial Poisson Processes and CSR.5.2.3 Hypothesis Tests of CSR via Monte Carlo Methods.5.2.4 Heterogeneous Poisson Processes.5.2.5 Estimating Intensity Functions.DATA BREAK: Early Medieval Grave Sites.5.3 K Function.5.3.1 Estimating the K Function.5.3.2 Diagnostic Plots Based on the K Function.5.3.3 Monte Carlo Assessments of CSR Based on the K Function.DATA BREAK: Early Medieval Grave Sites.5.3.4 Roles of First- and Second-Order Properties.5.4 Other Spatial Point Processes.5.4.1 Poisson Cluster Processes.5.4.2 Contagion/Inhibition Processes.5.4.3 Cox Processes.5.4.4 Distinguishing Processes.5.5 Additional Topics and Further Reading.5.6 Exercises.6 Spatial Clusters of Health Events: Point Data for Cases and Controls.6.1 What Do We Have? Data Types and Related Issues.6.2 What Do We Want? Null and Alternative Hypotheses.6.3 Categorization of Methods.6.4 Comparing Point Process Summaries.6.4.1 Goals.6.4.2 Assumptions and Typical Output.6.4.3 Method: Ratio of Kernel Intensity Estimates.DATA BREAK: Early Medieval Grave Sites.6.4.4 Method: Difference between K Functions.DATA BREAK: Early Medieval Grave Sites.6.5 Scanning Local Rates.6.5.1 Goals.6.5.2 Assumptions and Typical Output.6.5.3 Method: Geographical Analysis Machine.6.5.4 Method: Overlapping Local Case Proportions.DATA BREAK: Early Medieval Grave Sites.6.5.5 Method: Spatial Scan Statistics.DATA BREAK: Early Medieval Grave Sites.6.6 Nearest-Neighbor Statistics.6.6.1 Goals.6.6.2 Assumptions and Typical Output.6.6.3 Method: q Nearest Neighbors of Cases.CASE STUDY: San Diego Asthma.6.7 Further Reading.6.8 Exercises.7 Spatial Clustering of Health Events: Regional Count Data.7.1 What Do We Have and What Do We Want?7.1.1 Data Structure.7.1.2 Null Hypotheses.7.1.3 Alternative Hypotheses.7.2 Categorization of Methods.7.3 Scanning Local Rates.7.3.1 Goals.7.3.2 Assumptions.7.3.3 Method: Overlapping Local Rates.DATA BREAK: New York Leukemia Data.7.3.4 Method: Turnbull et al.'s CEPP.7.3.5 Method: Besag and Newell Approach.7.3.6 Method: Spatial Scan Statistics.7.4 Global Indexes of Spatial Autocorrelation.7.4.1 Goals.7.4.2 Assumptions and Typical Output.7.4.3 Method: Moran's I .7.4.4 Method: Geary's c.7.5 Local Indicators of Spatial Association.7.5.1 Goals.7.5.2 Assumptions and Typical Output.7.5.3 Method: Local Moran's I.7.6 Goodness-of-Fit Statistics.7.6.1 Goals.7.6.2 Assumptions and Typical Output.7.6.3 Method: Pearson's chi2.7.6.4 Method: Tango's Index.7.6.5 Method: Focused Score Tests of Trend.7.7 Statistical Power and Related Considerations.7.7.1 Power Depends on the Alternative Hypothesis.7.7.2 Power Depends on the Data Structure.7.7.3 Theoretical Assessment of Power.7.7.4 Monte Carlo Assessment of Power.7.7.5 Benchmark Data and Conditional Power Assessments.7.8 Additional Topics and Further Reading.7.8.1 Related Research Regarding Indexes of Spatial Association.7.8.2 Additional Approaches for Detecting Clusters and/or Clustering.7.8.3 Space-Time Clustering and Disease Surveillance.7.9 Exercises.8 Spatial Exposure Data.8.1 Random Fields and Stationarity.8.2 Semivariograms.8.2.1 Relationship to Covariance Function and Correlogram.8.2.2 Parametric Isotropic Semivariogram Models.8.2.3 Estimating the Semivariogram.DATA BREAK: Smoky Mountain pH Data.8.2.4 Fitting Semivariogram Models.8.2.5 Anisotropic Semivariogram Modeling.8.3 Interpolation and Spatial Prediction.8.3.1 Inverse-Distance Interpolation.8.3.2 Kriging.CASE STUDY: Hazardous Waste Site Remediation.8.4 Additional Topics and Further Reading.8.4.1 Erratic Experimental Semivariograms.8.4.2 Sampling Distribution of the Classical Semivariogram Estimator.8.4.3 Nonparametric Semivariogram Models.8.4.4 Kriging Non-Gaussian Data.8.4.5 Geostatistical Simulation.8.4.6 Use of Non-Euclidean Distances in Geostatistics.8.4.7 Spatial Sampling and Network Design.8.5 Exercises.9 Linking Spatial Exposure Data to Health Events.9.1 Linear Regression Models for Independent Data.9.1.1 Estimation and Inference.9.1.2 Interpretation and Use with Spatial Data.DATA BREAK: Raccoon Rabies in Connecticut.9.2 Linear Regression Models for Spatially Autocorrelated Data.9.2.1 Estimation and Inference.9.2.2 Interpretation and Use with Spatial Data.9.2.3 Predicting New Observations: Universal Kriging.DATA BREAK: New York Leukemia Data.9.3 Spatial Autoregressive Models.9.3.1 Simultaneous Autoregressive Models.9.3.2 Conditional Autoregressive Models.9.3.3 Concluding Remarks on Conditional Autoregressions.9.3.4 Concluding Remarks on Spatial Autoregressions.9.4 Generalized Linear Models.9.4.1 Fixed Effects and the Marginal Specification.9.4.2 Mixed Models and Conditional Specification.9.4.3 Estimation in Spatial GLMs and GLMMs.DATA BREAK: Modeling Lip Cancer Morbidity in Scotland.9.4.4 Additional Considerations in Spatial GLMs.CASE STUDY: Very Low Birth Weights in Georgia Health Care District 9.9.5 Bayesian Models for Disease Mapping.9.5.1 Hierarchical Structure.9.5.2 Estimation and Inference.9.5.3 Interpretation and Use with Spatial Data.9.6 Parting Thoughts.9.7 Additional Topics and Further Reading.9.7.1 General References.9.7.2 Restricted Maximum Likelihood Estimation.9.7.3 Residual Analysis with Spatially Correlated Error Terms.9.7.4 Two-Parameter Autoregressive Models.9.7.5 Non-Gaussian Spatial Autoregressive Models.9.7.6 Classical/Bayesian GLMMs.9.7.7 Prediction with GLMs.9.7.8 Bayesian Hierarchical Models for Spatial Data.9.8 Exercises.References.Author Index.Subject Index.

/pdf/applied-spatial-statistics-for-public-health-data-5e3go3weys.pdf

Applied Spatial Statistics for Public Health Data

The method of weighted least squares is shown to be an appropriate way of fitting variogram models. The weighting scheme automatically gives most weight to early lags and down-weights those lags with a small number of pairs. Although weights are derived assuming the data are Gaussian (normal), they are shown to be still appropriate in the setting where data are a (smooth) transform of the Gaussian case. The method of (iterated) generalized least squares, which takes into account correlation between variogram estimators at different lags, offer more statistical efficiency at the price of more complexity. Weighted least squares for the robust estimator, based on square root differences, is less of a compromise.

Fitting variogram models by weighted least squares

From observations of wind and of water surface elevation at 14 wave staffs in an array in Lake Ontario and in a large laboratory tank, the directional spectrum of wind-generated waves on deep water is determined by using a modification of Barber's (1963) method. Systematic investigations reveal the following: (a) the frequency spectrum in the rear face is inversely proportional to the fourth power of the frequency $\omega $, with the equilibrium range parameter and the peak enhancement factor clearly dependent on the ratio of wind speed to peak wave speed; (b) the angular spreading $\theta $ of the wave energy is of the form sech$^{2}$ ($\beta \theta $), where $\beta $ is a function of frequency relative to the peak; (c) depending on the gradient of the fetch, the direction of the waves at the spectral peak may differ from the mean wind direction by up to 50 degrees, but this observed difference is predictable by a similarity analysis; (d) under conditions of strong wind forcing, significant effects on the phase velocity caused by amplitude dispersion and the presence of bound harmonics are clearly observed and are in accordance with the Stokes theory, whereas (e) the waves under natural wind conditions show amplitude dispersion, but bound harmonics are too weak to be detected among the background of free waves.

Directional spectra of wind-generated waves

One of the fundamental problems of interpreting the sedimentary record is reconstructing the original heights of palaeotopographical features such as bedforms or river channels. This requires an understanding of the relationship between topography and set thickness, but at present an exact theory exists only for periodic topography of uniform height. The applicability of this simple theory is severely limited by the random variability characteristic of many sedimentary systems. In this paper, we develop an exact theory for the probability-density function (PDF) of sets generated from topography of random height. We focus on the limiting case of zero net deposition in order to provide a lower bound for the set thickness, and derive an analytical set-thickness PDF that is determined by one parameter of the PDF for topographical height. This parameter, β, measures the breadth of the tail of the topographical PDF. The mean set thickness is 0.8225 β for bedforms and 1.645 β for river channels. If the topographical height is gamma distributed, the preservation ratio, defined as (mean thickness of preserved sets)/(mean topographical height), is 0.8225 r2 for bedforms and 1.645 r2 for river channels, where r is the coefficient of variation (standard deviation/mean) of the generating topography.



In a comparison with data from laboratory current ripples, our analytical predictions compare well with observations of both mean set thickness and thickness distribution. The preservation ratio for the ripples is about 0.5, whilst measured dune heights give a preservation ratio of about 0.12. Depth data from two modern braided streams yield preservation ratios ranging from 0.4 to 0.75. As more data on the distributions of topographical height in modern environments become available, calculations such as these should help provide reliable error bounds for quantitative topographical reconstruction.

Reconstructing random topography from preserved stratification

Statistical Models for Ocean Waves and Wave Forces

The function, which is called the extremal Rayleigh distribution, is found to be an accurate approximation to the probability distribution function for the maximum pro a hurricane with time-varying intensity A formula is developed for the maximum wave distribution function in time-varying storms and computed for a large number of historical hurricanes The extremal Rayleigh function was found to be a very accurate approximation in all cases Two parameters are shown to be interpretable in terms of an equivalent uniform intensity storm as the reciprocal of the logarithm square wave height and the logarithm of the number of waves in the storm, respectively Data from Hurricane Carla was analyzed to determine the parameters which best described the probability behavior of the larger waves in each 20 min record Statistical formulas and tables are given to facilitate probability use of the extremal Rayleigh probability law in coastal engineering and hurricane research

Probabilities for Highest Wave in Hurricane

Layered three-dimensional, frequency-domain computer simulations have many advantages an an alternative to the turning band method, for certain geological problems. Computer speed and even, accurate duplication of statistical properties are the main advantages. Limitations in the permitted extent in the third dimension are the primary disadvantages of the method.

Three-Dimensional, Frequency-Domain Simulations of Geological Variables

This paper describes theory and application of the empirical simulation technique (EST), a statistical procedure for simulating time series and frequency-of-occurrence relationships for nondeterministic multiparameter systems. Procedures described are applied to a storm surge analysis for tropical events along the coast of Delaware. The approach involves the numerical simulation of historical events, the analysis of the parameters associated with each event, the application of the EST to that data, and the generation of frequency-of-occurrence relationships for 40 locations within the study area. The development of the input data set of descriptive storm parameters and their respective responses require the use of planetary boundary layer and very large domain hydrodynamic modeling techniques. Development of this input database is described herein. Implementation of the EST is based on N repetitions of T -year simulations; therefore, mean-value frequency relationships are computed and assigned confidence ...

Leon E. Borgman

Papers

Reconstructing random topography from preserved stratification

Statistical Models for Ocean Waves and Wave Forces

Probabilities for Highest Wave in Hurricane

Three-Dimensional, Frequency-Domain Simulations of Geological Variables

Empirical simulation technique based storm surge frequency analyses