Statistics for Spatial Data.

An overview of model-based geostatistics.- Gaussian models for geostatistical data.- Generalized linear models for geostatistical data.- Classical parameter estimation.- Spatial prediction.- Bayesian inference.- Geostatistical design.

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Model-based Geostatistics

Continuously indexed Gaussian fields (GFs) are the most important ingredient in spatial statistical modelling and geostatistics. The specification through the covariance function gives an intuitive interpretation of the field properties. On the computational side, GFs are hampered with the big n problem, since the cost of factorizing dense matrices is cubic in the dimension. Although computational power today is at an all time high, this fact seems still to be a computational bottleneck in many applications. Along with GFs, there is the class of Gaussian Markov random fields (GMRFs) which are discretely indexed. The Markov property makes the precision matrix involved sparse, which enables the use of numerical algorithms for sparse matrices, that for fields in R-2 only use the square root of the time required by general algorithms. The specification of a GMRF is through its full conditional distributions but its marginal properties are not transparent in such a parameterization. We show that, using an approximate stochastic weak solution to (linear) stochastic partial differential equations, we can, for some GFs in the Matern class, provide an explicit link, for any triangulation of R-d, between GFs and GMRFs, formulated as a basis function representation. The consequence is that we can take the best from the two worlds and do the modelling by using GFs but do the computations by using GMRFs. Perhaps more importantly, our approach generalizes to other covariance functions generated by SPDEs, including oscillating and non-stationary GFs, as well as GFs on manifolds. We illustrate our approach by analysing global temperature data with a non-stationary model defined on a sphere. (Less)

https://rss.onlinelibrary.wiley.com/doi/epdf/10.1111/j.1467-9868.2011.00777.x

An explicit link between Gaussian fields and Gaussian Markov random fields: the stochastic partial differential equation approach

We review 74 experiments with no, low, or high performance-based financial incentives. The modal result has no effect on mean performance (though variance is usually reduced by higher payment). Higher incentive does improve performance often, typically judgment tasks that are responsive to better effort. Incentives also reduce “presentation” effects (e.g., generosity and risk-seeking). Incentive effects are comparable to effects of other variables, particularly “cognitive capital” and task “production” demands, and interact with those variables, so a narrow-minded focus on incentives alone is misguided. We also note that no replicated study has made rationality violations disappear purely by raising incentives.

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The Effects of Financial Incentives in Experiments: A Review and Capital-Labor-Production Framework

This paper surveys locally weighted learning, a form of lazy learning and memory-based learning, and focuses on locally weighted linear regression. The survey discusses distance functions, smoothing parameters, weighting functions, local model structures, regularization of the estimates and bias, assessing predictions, handling noisy data and outliers, improving the quality of predictions by tuning fit parameters, interference between old and new data, implementing locally weighted learning efficiently, and applications of locally weighted learning. A companion paper surveys how locally weighted learning can be used in robot learning and control.

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Locally Weighted Learning

When setting up a computer experiment, it has become a standard practice to select the inputs spread out uniformly across the available space. These so-called space-filling designs are now ubiquitous in corresponding publications and conferences. The statistical folklore is that such designs have superior properties when it comes to prediction and estimation of emulator functions. In this paper we want to review the circumstances under which this superiority holds, provide some new arguments and clarify the motives to go beyond space-filling. An overview over the state of the art of space-filling is introducing and complementing these results.

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Design of computer experiments: space filling and beyond

Fundamentals of Spatial Statistics.- Fundamentals of Experimental Design.- Exploratory Designs.- Designs for Spatial Trend Estimation.- Design and Dependence.- Multipurpose Designs.

Collecting Spatial Data: Optimum Design of Experiments for Random Fields

Badinger H., Muller W. G. and Tondl G. (2004) Regional convergence in the European Union, 1985- 1999: a spatial dynamic panel analysis, Reg. Studies38, 241-253. We estimate the speed of income convergence for a sample of 196 European NUTS 2 regions over the period 1985-1999. So far there is no direct estimator available for dynamic panels with spatial dependencies. We propose a two-step procedure, which involves first spatial filtering of the variables to isolate the spatial dependence, and application of standard GMM estimators for dynamic panels in a second step. Our results show that ignorance of the spatial correlation leads to potentially misleading results. Applying a system GMM estimator on the filtered variables, we obtain a speed of convergence of 7%. Badinger H., Muller W. G. et Tondl G. (2004) Convergence regionale dans l'Union europeenne, 1985-1999: une analyse en panel spatial dynamique, Reg. Studies38, 241-253. Nous estimons le taux de convergence des revenus regionaux dans un echantillon de...

/pdf/regional-convergence-in-the-european-union-1985-1999-a-3944h6v771.pdf

Regional Convergence in the European Union (1985-1999): A Spatial Dynamic Panel Analysis

We estimate the speed of income convergence for a sample of 196 European NUTS 2 regions over the period 1985-1999. So far there is no direct estimator available for dynamic panels with strong spatial dependencies. We propose a two-step procedure, which involves first spatial filtering of the variables to remove the spatial correlation, and application of standard GMM estimators for dynamic panels in a second step. Our results show that ignorance of the spatial correlation leads to potentially misleading results. Applying a system GMM estimator on the filtered variables, we obtain a speed of convergence of 6.9 per cent and a capital elasticity of 0.43.

Regional convergence in the European Union (1985 - 1999): A spatial dynamic panel analysis

The variogram plays a central role in the analysis of geostatistical data. A valid variogram model is selected and the parameters of that model are estimated before kriging (spatial prediction) is performed. These inference procedures are generally based upon examination of the empirical variogram, which consists of average squared differences of data taken at sites lagged the same distance apart in the same direction. The ability of the analyst to estimate variogram parameters efficiently is affected significantly by the sampling design, i.e., the spatial configuration of sites where measurements are taken. In this paper, we propose design criteria that, in contrast to some previously proposed criteria oriented towards kriging with a known variogram, emphasize the accurate estimation of the variogram. These criteria are modifications of design criteria that are popular in the context of (nonlinear) regression models. The two main distinguishing features of the present context are that the addition of a single site to the design produces as many new lags as there are existing sites and hence also produces that many new squared differences from which the variograrn is estimated. Secondly, those squared differences are generally correlated, which inhibits the use of many standard design methods that rest upon the assumption of uncorrelated errors. Several approaches to design construction which account for these features are described and illustrated with two examples. We compare their efficiency to simple random sampling and regular and space-filling designs and find considerable improvements. (author's abstract)

http://mmc2.geofisica.unam.mx/cursos///geoest/Articulos/Geostatistics/Optimal%20designs%20for%20variogram%20estimation.pdf

Werner G. Müller

Papers

Design of computer experiments: space filling and beyond

Collecting Spatial Data: Optimum Design of Experiments for Random Fields

Regional Convergence in the European Union (1985-1999): A Spatial Dynamic Panel Analysis

Regional convergence in the European Union (1985 - 1999): A spatial dynamic panel analysis

Optimal designs for variogram estimation