On robust estimation of the location parameter

Linear models

Being newly immersed in the upstream part of the oil business, I just recently had my first work session with data in ARC–GIS®. The project involves subsurface geographical modeling. Obviously I had considerable interest in discovering if the methodology in this book would enhance my modeling capabilities. The book is for social scientists, but I had no difficulty imagining my own important oil exploration application within the framework of geographically weighted regression (GWR). The first chapter nicely explains what is unique in this book. A standard regression model using geographically oriented data (the example is housing prices across all of England) is a global representation of a spatial relationship, an average that does not account for any local differences. In y = f (x), imagine a whole family of f ’s that are indexed by spatial location. That is the focus of this book. It is about one form of local spatial modeling, which is GWR. A more general resource for this topic is the earlier book by Fotheringham and Wegener (2000), which escaped the notice of Technometrics. Imagine a display of model parameters in a geographical information system (GIS) and you will understand the focus for this book. The authors note, “only where there is no significant spatial variation in a measured relationship can global models be accepted” (p. 10). The second chapter develops the basis of GWR. It analyzes the housing sales prices versus the 33 boroughs in London and begins by fitting a conventional multiple regression model versus housing characteristics. The GWR is motivated by differences in the regression models fitted separately by borough. The GWR is a spatial moving-window approach with all data distances weighted versus a specific data point using a weighting function and a bandwidth. A GIS can then be used to evaluate the spatial dependency of the parameters. As in kriging, local standard errors also are calculated. The chapter also provides all the math. Chapter 3 comprises several further considerations: parameters that are globally constant, outliers, and spatial heteroscedasticity. The first issue leads to hypothesis tests for model comparison using an Akaike information criterion (AIC). Local outliers are hard to detect. Studentized (deletion) residuals are recommended. The outliers can be plotted geographically. Robust regression is suggested as a less computationally intensive alternative. Hetereoscedasticity is harder to handle. Chapter 4 adds statistical inference to the capabilities of GWR: both a confidence interval approach using local likelihood and an AIC method. Four additional methodology chapters present various extensions of GWR. Chapter 5 considers the relationship between GWR and spatial autocorrelation, and includes a combined version of GWR and spatial regression using some complex hybrid models. Chapter 6 examines the relationship of scale and zoning problems in spatial analysis to GWR. Chapter 7 introduces the use of initial exploratory data analysis using geographically weighted statistics, which are based on the idea of using a kernel around each data point to create weights. Univariate statistics and correlation coefficients are defined for exploring local patterns in data. A final set of extensions in Chapter 8 discusses regression models with non-Gaussian errors, logistic regression, local principal components analysis, and local probability density estimation. The methods all use some kind of distributional model. The million-dollar question for me is always, “What about software?” The authors have a stand-alone program, GWR 3, available in CD–ROM by contacting the authors. Basically the drill with GWR 3 is to gather your data, use Excel to transform and reformat the data for GWR 3, use GWR 3 to produce a set of coefficients, and feed those coefficients to your favorite GIS to produce your maps. Forty pages of discussion about using the software are provided. A final epilogue chapter also discusses embedding GWR in R or Matlab and includes some references to people who have done that type of work. I probably would not have read this book if I had not happened to have had it in my briefcase on a visit with the exploration technologists. Though inclusive of appropriate mathematical development, this material is readily approachable because of the many illustrations and the pages and pages of GIS displays. The authors unabashedly present much of the material as their developmental work, so GWR offers a lot of opportunity for research and further development through novel applications and extensions.

Geographically Weighted Regression

THE interpretation of science to the lay reader—and we may quite candidly admit that ‘lay reader’ must be interpreted as including practically all men of science who do not happen to be professionally interested in the particular branch of science under consideration—is a task both important and difficult. Of books which might be described as being suitable for some “ Wonders of the Universe ” series there is an excellent supply, and very good some of them are. The universe is a wonderful place, and to explore it under the guidance of a distinguished cicerone is both pleasant and profitable. Too often, however, one lays the book down with an uneasy feeling that something is missing. It reads like a detective story from which all the middle pages are missing. The first chapter, with its neatly arranged pile of corpses, is there, and the last page or so in which the name of the villain is disclosed ; but the account of the search, and in particular, that thrilling chapter, near the end, where the 'little grey cells' come iftto play, is wanting, and the real interest of the book has vanished with it. While it is, no doubt, edifying to learn that a lump of matter is nothing but a whirl of flying electric charges, what the reader would like to know, and what it is important that he should know, is how, and by what stages, the physicist arrives at this astounding conclusion. If, as Prof. Einstein and his co-author claim, “Physics is a creation of the human mind, with freely invented ideas and concepts ”, it is this intellectual content which gives to physics one of its chief claims to cultural significance, and provides for the thoughtful non-technical reader his main source of interest in it. It is with this aspect of the subject that the authors are concerned in this very distinguished book. The Evolution of Physics: the Growth of Ideas from the Early Concepts to Relativity and Quanta. By Albert Einstein and Leopold Infeld. (Cambridge Library of Modern Science.) Pp. x+319+3 plates. (Cambridge: At the University Press, 1938.) 8s. 6d. net.

The Evolution of Physics

Statistics for spatial data: by Noel Cressie, 1991, John Wiley & Sons, New York, 900 p., ISBN 0-471-84336-9, US $89.95

Tick infestation and prophylaxis of dogs in northeastern Germany: A prospective study☆

The effective use of spatial information in a regressionâ€ based approach to small area estimation is an important practical issue. One approach to account for geographic information is by extending the linear mixed model to allow for spatially correlated random area effects. An alternative is to include the spatial information by a nonâ€ parametric mixed models. Another option is geographic weighted regression where the model coefficients vary spatially across the geography of interest. Although these approaches are useful for estimating small area means efficiently under strict parametric assumptions, they can be sensitive to outliers. In this paper, we propose robust extensions of the geographically weighted empirical best linear unbiased predictor. In particular, we introduce robust projective and predictive estimators under spatial nonâ€ stationarity. Mean squared error estimation is performed by two analytic approaches that account for the spatial structure in the data. Modelâ€ based simulations show that the methodology proposed often leads to more efficient estimators. Furthermore, the analytic mean squared error estimators introduced have appealing properties in terms of stability and bias. Finally, we demonstrate in the application that the new methodology is a good choice for producing estimates for average rent prices of apartments in urban planning areas in Berlin.

/pdf/robust-small-area-estimation-under-spatial-non-stationarity-24nnl67phk.pdf

Robust Small Area Estimation under Spatial Non‐stationarity

Ticks can transmit different pathogens to humans and animals. Dogs are frequently exposed to tick infestation, which underscores the importance of tick control measures. The objective of this study was to examine the awareness of dog owners regarding tick infestation and tick prophylaxis by a questionnaire survey. During the period from March to December 2010 a total of 616 owners of 670 dogs completed the questionnaire. According to the questionnaire results, 92% of the dogs were previously infested by ticks; 31% of these showed a moderate tick infestation (1-2 ticks a month), almost one in ten dogs was infested by eight or more ticks a month. 17% of the dogs were examined for ticks by the respective owner not at all or only at irregular intervals, 61% of the dogs were examined at least once a day. A tick prophylaxis was performed in 469 dogs (71%). In 353 dogs (53%), registered pharmaceutical products with appropriate label claims were employed. Spot-on products were used most frequently (93%), followed by collars (5%) and sprays (1%).These products were not used as recommended in 56% of the dogs. For further 33% of the dogs, it was not possible to decide if the products were used correctly or not. According to the dog owner statements, tick borne diseases were diagnosed in approximately 2% of the dogs. Dog specific characters, such as coat length, size, age, and walking habits were significant factors influencing the frequency of tick infestation. In summary it can be concluded that nearly every dog in the area of Berlin/Brandenburg is infested by ticks. In the majority of cases the prophylactic and/or therapeutic measures to prevent infestation are not performed correctly.

[Tick infestation and tick prophylaxis in dogs in the area of Berlin/Brandenburg--results of a questionnaire study].

Geographically weighted small area methods have been studied in literature for small area estimation. Although these approaches are useful for the estimation of small area means efficiently under strict parametric assumptions, they can be very sensitive to outliers in the data. In this paper, we propose a robust extension of the geographically weighted empirical best linear unbiased predictor (GWEBLUP). In particular, we introduce robust projective and predictive small area estimators under spatial non-stationarity. Mean squared error estimation is performed by two different analytic approaches that account for the spatial structure in the data. The results from the model-based simulations indicate that the proposed approach may lead to gains in terms of efficiency. Finally, the methodology is demonstrated in an illustrative application for estimating the average total cash costs for farms in Australia.

Claudia Baldermann

Papers

Tick infestation and prophylaxis of dogs in northeastern Germany: A prospective study☆

Robust Small Area Estimation under Spatial Non‐stationarity

[Tick infestation and tick prophylaxis in dogs in the area of Berlin/Brandenburg--results of a questionnaire study].

Robust small area estimation under spatial non-stationarity