Propagating error in land-cover-change analyses: impact of temporal dependence under increased thematic complexity

We examined the impact of temporal dependence between patterns of error in classified time-series imagery through a simulation modeling approach. This research extended the land-cover-change simulation model we previously developed to investigate: (1) the assumption of temporal independence between patterns of error in classified time-series imagery; and (2) the interaction of patterns of change and patterns of error in a post-classification change analysis. In this research, the thematic complexity of the classified land-cover maps was increased by increasing the number of simulated land-cover classes. Simulating maps with increased categorical resolution permitted the incorporation of: (1) higher-order, more complex spatial and temporal interactions between land-cover classes; and (2) patterns of error that better reproduce the complex error interactions that often occur in time-series classified imagery. The overall modeling framework was divided into two primary components: (1) generation of a map representing true change; and (2) generation of a suite of change maps that had been perturbed by specific patterns of error. All component maps in the model were produced using simulated annealing, which enabled us to create a series of map realizations with user-defined spatial and temporal patterns. Comparing the true map of change to the error-perturbed maps of change using accuracy assessment statistics showed that increasing the temporal dependence between classification errors did not improve the accuracy of resulting maps of change when the categorical scale of the land-cover classified maps was increased. The increased structural complexity within the time series of maps effectively inhibited the impact of temporal dependence. However, results demonstrated that there are interactions between patterns of error and patterns of change in a post-classification change analysis. These interactions played a major role in determining the accuracy associated with the maps of change.     

Spatial orthogonality of the principal components computed from coregionalized variables

Within the frame of the linear model of coregionalization, this paper sets up equations relating the variogram matrix of the principal components extracted from the variance-covariance matrix to the diagonal variogram matrices of the regionalized factors. The spatial orthogonality of the principal components is investigated in three situations: the intrinsic correlation, two basic structures with independent nugget components, three basic structures with independent nugget components and uncorrelated subsets of variables. Two examples point out that the correlation between the principal components may be nonnegligible at short distances, especially if the correlation structure changes according to the spatial scale considered. For one of the two case studies, an orthogonal varimax rotation of the first principal components is found to greatly reduce the spatial correlation between some of them.     

On a controversial method for modeling a coregionalization

This paper reviews two alternative approaches for modeling the (cross) variograms in a coregionalization: (1) fitting the traditional linear model of coregionalization. or (2) deducing the crossvariogram model as a linear combination of prior direct (auto) variogram models while checking the Cauchy-Schwarz inequalities. We show that the second approach has no practical advantage over the traditional one and may not be valid if more than two variables are involved. In such case. Cauchy-Schwarz inequalities are necessary but not sufficient conditions for validity of a coregionalization model.     

Speleothem-based chronology and environmental context of deposits from the Mishin Kamik Cave,NW Bulgaria – A contribution to the archaeological study of the Late Pleistocene human occupation in the Balkans

The Balkan Peninsula represents one of the most important human pathways into and out of Europe during the Pleistocene. Mishin Kamik cave, located in the karst region of Western Stara Planina, has a rich faunal content and shows promising features indicating a human occupation site with the discovery of potential bone artefacts and an intriguing accumulation of bear skulls and bones. Petrographic study and U-series dating of a stalagmite and other calcite deposits in the cave provide an absolute chronological frame for the detrital infillings and their archaeological content and inform the environmental and climatic context of the cave evolution. Most detrital deposits in the cave were probably deposited before Marine Isotope Stage (MIS) 5 and the cave morphology and sedimentary deposits display current morphologies since ~135 ka. Consequently, the palaeontological and archaeological findings are older than ~135 ka. Calcite dated on and under the accumulation of bear skulls and bones suggests deposition during MIS 7. A first depositional contextualization of the bone accumulation does not allow us to discriminate between a natural or anthropogenic origin. The study emphasizes the added value of speleothem studies in archaeological sites and particularly in bringing a well-constrained chronological and environmental framework.     

Kriging and Semivariogram Deconvolution in the Presence of Irregular Geographical Units

This paper presents a methodology to conduct geostatistical variography and interpolation on areal data measured over geographical units (or blocks) with different sizes and shapes, while accounting for heterogeneous weight or kernel functions within those units. The deconvolution method is iterative and seeks the point-support model that minimizes the difference between the theoretically regularized semivariogram model and the model fitted to areal data. This model is then used in area-to-point (ATP) kriging to map the spatial distribution of the attribute of interest within each geographical unit. The coherence constraint ensures that the weighted average of kriged estimates equals the areal datum.This approach is illustrated using health data (cancer rates aggregated at the county level) and population density surface as a kernel function. Simulations are conducted over two regions with contrasting county geographies: the state of Indiana and four states in the Western United States. In both regions, the deconvolution approach yields a point support semivariogram model that is reasonably close to the semivariogram of simulated point values. The use of this model in ATP kriging yields a more accurate prediction than a naïve point kriging of areal data that simply collapses each county into its geographic centroid. ATP kriging reduces the smoothing effect and is robust with respect to small differences in the point support semivariogram model. Important features of the point-support semivariogram, such as the nugget effect, can never be fully validated from areal data. The user may want to narrow down the set of solutions based on his knowledge of the phenomenon (e.g., set the nugget effect to zero). The approach presented avoids the visual bias associated with the interpretation of choropleth maps and should facilitate the analysis of relationships between variables measured over different spatial supports.     

Ordinary Cokriging Revisited

This paper sets up the relations between simple cokriging and ordinary cokriging with one or several unbiasedness constraints. Differences between cokriging variants are related to differences between models adopted for the means of primary and secondary variables. Because it is not necessary for the secondary data weights to sum to zero, ordinary cokriging with a single unbiasedness constraint gives a larger weight to the secondary information while reducing the occurrence of negative weights. Also the weights provided by such cokriging systems written in terms of covariances or correlograms are not related linearly, hence the estimates are different. The prediction performances of cokriging estimators are assessed using an environmental dataset that includes concentrations of five heavy metals at 359 locations. Analysis of reestimation scores at 100 test locations shows that kriging and cokriging perform equally when the primary and secondary variables are sampled at the same locations. When the secondary information is available at the estimated location, one gains little by retaining other distant secondary data in the estimation.     

Ordinary Cokriging Revisited

This paper sets up the relations between simple cokriging and ordinary cokriging with one or several unbiasedness constraints. Differences between cokriging variants are related to differences between models adopted for the means of primary and secondary variables. Because it is not necessary for the secondary data weights to sum to zero, ordinary cokriging with a single unbiasedness constraint gives a larger weight to the secondary information while reducing the occurrence of negative weights. Also the weights provided by such cokriging systems written in terms of covariances or correlograms are not related linearly, hence the estimates are different. The prediction performances of cokriging estimators are assessed using an environmental dataset that includes concentrations of five heavy metals at 359 locations. Analysis of reestimation scores at 100 test locations shows that kriging and cokriging perform equally when the primary and secondary variables are sampled at the same locations. When the secondary information is available at the estimated location, one gains little by retaining other distant secondary data in the estimation.     

Impact of the simulation algorithm, magnitude of ergodic fluctuations and number of realizations on the spaces of uncertainty of flow properties

 Geostatistical simulation algorithms are routinely used to generate conditional realizations of the spatial distribution of petrophysical properties, which are then fed into complex transfer functions, e.g. a flow simulator, to yield a distribution of responses, such as the time to recover a given proportion of the oil. This latter distribution, often referred to as the space of uncertainty, cannot be defined analytically because of the complexity (non-linearity) of transfer functions, but it can be characterized algorithmically through the generation of many realizations. This paper compares the space of uncertainty generated by four of the most commonly used algorithms: sequential Gaussian simulation, sequential indicator simulation, p-field simulation and simulated annealing. Conditional to 80 sample permeability values randomly drawn from an exhaustive 40×40 image, 100 realizations of the spatial distribution of permeability values are generated using each algorithm and fed into a pressure solver and a flow simulator. Principal component analysis is used to display the sets of realizations into the joint space of uncertainty of the response variables (effective permeability, times to reach 5% and 95% water cuts and to recover 10% and 50% of the oil). The attenuation of ergodic fluctuations through a rank-preserving transform of permeability values reduces substantially the extent of the space of uncertainty for sequential indicator simulation and p-field simulation, while improving the prediction of the response variable by the mean of the output distribution. Differences between simulation algorithms are the most pronounced for long-term responses (95% water cut and 50% oil recovery), with sequential Gaussian simulation yielding the most accurate prediction. In this example, utilizing more than 20 realizations generally increases only slightly the size of the space of uncertainty.     

Detection of temporal changes in the spatial distribution of cancer rates using local Moran’s I and geostatistically simulated spatial neutral models

This paper presents the first application of spatially correlated neutral models to the detection of changes in mortality rates across space and time using the local Morans I statistic. Sequential Gaussian simulation is used to generate realizations of the spatial distribution of mortality rates under increasingly stringent conditions: 1) reproduction of the sample histogram, 2) reproduction of the pattern of spatial autocorrelation modeled from the data, 3) incorporation of regional background obtained by geostatistical smoothing of observed mortality rates, and 4) incorporation of smooth regional background observed at a prior time interval. The simulated neutral models are then processed using two new spatio-temporal variants of the Morans I statistic, which allow one to identify significant changes in mortality rates above and beyond past spatial patterns. Last, the results are displayed using an original classification of clusters/outliers tailored to the space-time nature of the data. Using this new methodology the space-time distribution of cervix cancer mortality rates recorded over all US State Economic Areas (SEA) is explored for 9 time periods of 5 years each. Incorporation of spatial autocorrelation leads to fewer significant SEA units than obtained under the traditional assumption of spatial independence, confirming earlier claims that Type I errors may increase when tests using the assumption of independence are applied to spatially correlated data. Integration of regional background into the neutral models yields substantially different spatial clusters and outliers, highlighting local patterns which were blurred when local Morans I was applied under the null hypothesis of constant risk.This research was funded by grants R01 CA92669 and 1R43CA105819-01 from the National Cancer Institute and R43CA92807 under the Innovation in Biomedical Information Science and Technology Initiative at the National Institute of Health. The views stated in this publication are those of the authors and do not necessarily represent the official views of the NCI. The authors also thank three anonymous reviewers for their comments that helped improve the presentation of the methodology.