Bayesian Data Fusion Approach Research Articles

Post-fire seed dispersal of a wind-dispersed shrub declined with distance to seed source, yet had high levels of unexplained variation.

Plant-population recovery across large disturbance areas is often seed-limited. An understanding of seed dispersal patterns is fundamental for determining natural-regeneration potential. However, forecasting seed dispersal rates across heterogeneous landscapes remains a challenge. Our objectives were to determine (i) the landscape patterning of post-disturbance seed dispersal, and underlying sources of variation and the scale at which they operate, and (ii) how the natural seed dispersal patterns relate to a seed augmentation strategy. Vertical seed trapping experiments were replicated across 2 years and five burned and/or managed landscapes in sagebrush steppe. Multi-scale sampling and hierarchical Bayesian models were used to determine the scale of spatial variation in seed dispersal. We then integrated an empirical and mechanistic dispersal kernel for wind-dispersed species to project rates of seed dispersal and compared natural seed arrival to typical post-fire aerial seeding rates. Seeds were captured across the range of tested dispersal distances, up to a maximum distance of 26 m from seed-source plants, although dispersal to the furthest traps was variable. Seed dispersal was better explained by transect heterogeneity than by patch or site heterogeneity (transects were nested within patch within site). The number of seeds captured varied from a modelled mean of ~13 m-2 adjacent to patches of seed-producing plants, to nearly none at 10 m from patches, standardized over a 49-day period. Maximum seed dispersal distances on average were estimated to be 16 m according to a novel modelling approach using a 'latent' variable for dispersal distance based on seed trapping heights. Surprisingly, statistical representation of wind did not improve model fit and seed rain was not related to the large variation in total available seed of adjacent patches. The models predicted severe seed limitations were likely on typical burned areas, especially compared to the mean 95-250 seeds per m2 that previous literature suggested were required to generate sagebrush recovery. More broadly, our Bayesian data fusion approach could be applied to other cases that require quantitative estimates of long-distance seed dispersal across heterogeneous landscapes.

A Data Fusion Approach for Interpurchase Timing Models Using Incomplete Purchase Histories

Interpurchase timing models are important for understanding the effects of marketing variables. However, a company's database cannot record purchase histories in competing stores.Its database records only the sum of durations for its own and competing stores. Moreover, it is difficult for companies to know the number of purchases in competing stores between two purchases observed in their own stores. Therefore, a simple application of interpurchase timing models would be incorrect and generate biased results. In this study, we construct a model that predicts the missing number of purchases in competing stores and a gamma proportional hazard model in which parameters are estimated from the sum of the durations. It is difficult to predict the missing number of purchases in competing stores and estimate parameters suitably from the summed durations. Therefore, we propose a Bayesian data fusion approach that combines two datasets and incorporates macro-level information in the model. We show the performance of the proposed method through a simulation study and application using a real purchase history dataset. The results show that the proposed method can correctly estimate the effects of the marketing variables, while the sign of the estimated parameter is reversed in the simple analysis.

Bayesian data fusion for combining maps of predicted soil classes: A case study using legacy soil profiles and DEM covariates in Iran

This paper tackles the issue of spatially predicting soil classes by combining at best soil information coming directly from legacy soil profiles with soil information indirectly obtained from spatial covariates. Based on Multinomial Logistic Regression (MLR) and Bayesian Maximum Entropy (BME) models, we first show that prediction models easily lead to very different soil maps while having at the same time quite comparable global performances. By relying afterwards on a Bayesian data fusion (BDF) approach, we emphasize the benefit of combining the output of these two prediction models in order to get a single final map that combines the major spatial features of the MLR and BME maps while at the same time improving the quality of the predictions.The advocated methodology is illustrated with the mapping of World Reference Base (WRB) soil classes over a 10,480 km2 area located in Iran. A set of 390 soil profiles allowed us to assign the WRB soil classes at these locations. In parallel, a set of potentially related covariates were computed from a 90 m resolution digital elevation model. Using MLR and BME models, predictions were obtained separately at the nodes of a 90 m resolution grid. Even if the performances of the MLR and BME models compare well, it is shown that a BDF procedure that combines both results yields improved performances, with spatial features that are a balanced combination of those found separately on the MLR and BME maps.These results emphasize the benefit of data fusion in order to improve the quality of the final map. Though the study was conducted here using MLR and BME models for predicting WRB soil classes, we believe this methodology and the corresponding findings are relevant when it comes to handle the results of spatial prediction models that are making use of distinct information sources in other soil science mapping contexts.

Spatial filtering and Bayesian data fusion for mapping soil properties: A case study combining legacy and remotely sensed data in Iran

In a spatial mapping context, we address in this paper the issue of combining at best soil data coming from legacy soil surveys with soil information that is indirectly obtained from remote sensing (RS). We show first how spatial scale issues need to beproperly addressed for soil mapping in order to increase the predictive performances of spatial prediction models, where the two prediction techniques that are compared here are kriging and random forests (RF). By relying afterwards on a Bayesian data fusion approach, we then emphasize the benefit of combining the output of these two prediction models in order to get a single prediction result that leads to an improved final map.The advocated methodology is illustrated with the mapping of soil saturation percentage (SP) over a 10,480 km2 area located in Iran, were SP is a key soil parameter for sustainable development and land management. A set of 396 soil profiles were obtained from legacy soil surveys and were used to compute vertically averaged SP values. In parallel, a set of RS covariates were obtained from a 90 meters resolution Landsat image and a digital elevation model. Based on the modeling of the spatialdependence both for soil SP data and for RS covariates, it is shown how the relationship between them is improved by using a filtered kriging technique that allows us to focus on their long range component only. Using these filtered SP value and RS covariates, predictions were obtained separately from kriging and from a RF model at the nodes of a 90 meters resolution grid. Even if the performances of these two models are almost identical with R2 values equal to 0.66–0.67, it is shown afterwards that a Bayesian data fusion procedure that combines both predictions yields improved performance with a R2 value equal to 0.86. These results clearly emphasize the importance of properly addressing scale issues and the benefit of data fusion inorder to improve the quality of the final map. Though the study was conducted here for SP values, we believe this methodology and these findings are relevant when it comes to handle other soil properties in a spatial mapping context that involves several data sources.

Modified Bayesian data fusion model for travel time estimation considering spurious data and traffic conditions

This paper presents a framework for the development of the travel time estimation model using multiple sources of data with consideration of spurious data and traffic conditions. A modified Bayesian data fusion approach, combined with the Gaussian mixture model, is used to fuse the travel time data, which are estimated from different types of sensors to improve accuracy, precision, as well as completeness of data, in terms of spatial and temporal distribution. Two additional features are added into existing models including the difference of traffic conditions classified by the Gaussian mixture model and the bias estimation from individual sensor by introducing a non-zero mean Gaussian distribution which learned from the training dataset. The methodology and computational procedure are presented. The Gaussian mixture model is used to classify states of traffic into predefined number of traffic regimes. Once a traffic condition is classified, the modified Bayesian data fusion approach is used to estimate travel time. The proposed model provides explicit advantages over the basic Bayesian approach, such as being robust to noisy data, reducing biases of an individual estimation, and producing a more precise estimation of travel time. Two different real-world datasets and one simulated dataset are used to evaluate the performance of the proposed model under three different traffic regimes: free flow, transitional flow and congested flow regimes. The results when compared with the results from benchmark models show significant improvement in the accuracy of travel time estimation in terms of mean absolute percentage errors (MAPE) in the range of 3.46% to 16.3%.

A Bayesian Data Fusion Approach to Spatio-Temporal Fusion of Remotely Sensed Images

Remote sensing provides rich sources of data for the monitoring of land surface dynamics. However, single-sensor systems are constrained from providing spatially high-resolution images with high revisit frequency due to the inherent sensor design limitation. To obtain images high in both spatial and temporal resolutions, a number of image fusion algorithms, such as spatial and temporal adaptive reflectance fusion model (STARFM) and enhanced STARFM (ESTARFM), have been recently developed. To capitalize on information available in a fusion process, we propose a Bayesian data fusion approach that incorporates the temporal correlation information in the image time series and casts the fusion problem as an estimation problem in which the fused image is obtained by the Maximum A Posterior (MAP) estimator. The proposed approach provides a formal framework for the fusion of remotely sensed images with a rigorous statistical basis; it imposes no requirements on the number of input image pairs; and it is suitable for heterogeneous landscapes. The approach is empirically tested with both simulated and real-life acquired Landsat and Moderate Resolution Imaging Spectroradiometer (MODIS) images. Experimental results demonstrate that the proposed method outperforms STARFM and ESTARFM, especially for heterogeneous landscapes. It produces surface reflectances highly correlated with those of the reference Landsat images. It gives spatio-temporal fusion of remotely sensed images a solid theoretical and empirical foundation that may be extended to solve more complicated image fusion problems.

Combining land cover products using a minimum divergence and a Bayesian data fusion approach

ABSTRACTLand cover mapping plays an important role for a wide spectrum of applications that are ranging from climate modeling to food security. However, it is a common case that several and partially conflicting land cover products are available at the same time over a same area, where each product suffers from specific limitations and lack of accuracy. In order to take advantage of the best features of each product while at the same time attenuating their respective weaknesses, this paper is proposing a methodology that allows the user to combine these products together based on a general framework involving maximum entropy/minimum divergence principles, Bayesian data fusion and Bayesian updating. First, information brought by each land cover product is coded in terms of inequality constraints so that a first estimation of their quality can be computed based on a maximum entropy/minimum divergence principle. Information from these various land cover products can then be fused afterwards in a Bayesian framework, leading to a single map with an associated measure of uncertainty. Finally, it is shown how the additional information brought by control data can help improving this fused map through a Bayesian updating procedure. The first part of the paper is briefly presenting the most important theoretical results, while the second part is illustrating the use of this suggested approach for a specific area in Belgium, where five different land cover products are at hand. The benefits and limitations of this approach are finally discussed by the light of the results for this case study.

Bayesian maximum entropy and data fusion for processing qualitative data: theory and application for crowdsourced cropland occurrences in Ethiopia

Categorical data play an important role in a wide variety of spatial applications, while modeling and predicting this type of statistical variable has proved to be complex in many cases. Among other possible approaches, the Bayesian maximum entropy methodology has been developed and advocated for this goal and has been successfully applied in various spatial prediction problems. This approach aims at building a multivariate probability table from bivariate probability functions used as constraints that need to be fulfilled, in order to compute a posterior conditional distribution that accounts for hard or soft information sources. In this paper, our goal is to generalize further the theoretical results in order to account for a much wider type of information source, such as probability inequalities. We first show how the maximum entropy principle can be implemented efficiently using a linear iterative approximation based on a minimum norm criterion, where the minimum norm solution is obtained at each step from simple matrix operations that converges to the requested maximum entropy solution. Based on this result, we show then how the maximum entropy problem can be related to the more general minimum divergence problem, which might involve equality and inequality constraints and which can be solved based on iterated minimum norm solutions. This allows us to account for a much larger panel of information types, where more qualitative information, such as probability inequalities can be used. When combined with a Bayesian data fusion approach, this approach deals with the case of potentially conflicting information that is available. Although the theoretical results presented in this paper can be applied to any study (spatial or non-spatial) involving categorical data in general, the results are illustrated in a spatial context where the goal is to predict at best the occurrence of cultivated land in Ethiopia based on crowdsourced information. The results emphasize the benefit of the methodology, which integrates conflicting information and provides a spatially exhaustive map of these occurrence classes over the whole country.

Assessment of Data Fusion Algorithms for Earth Observation Change Detection Processes.

In this work a parametric multi-sensor Bayesian data fusion approach and a Support Vector Machine (SVM) are used for a Change Detection problem. For this purpose two sets of SPOT5-PAN images have been used, which are in turn used for Change Detection Indices (CDIs) calculation. For minimizing radiometric differences, a methodology based on zonal “invariant features” is suggested. The choice of one or the other CDI for a change detection process is a subjective task as each CDI is probably more or less sensitive to certain types of changes. Likewise, this idea might be employed to create and improve a “change map”, which can be accomplished by means of the CDI’s informational content. For this purpose, information metrics such as the Shannon Entropy and “Specific Information” have been used to weight the changes and no-changes categories contained in a certain CDI and thus introduced in the Bayesian information fusion algorithm. Furthermore, the parameters of the probability density functions (pdf’s) that best fit the involved categories have also been estimated. Conversely, these considerations are not necessary for mapping procedures based on the discriminant functions of a SVM. This work has confirmed the capabilities of probabilistic information fusion procedure under these circumstances.

Integrating Crowdsourced Data with a Land Cover Product: A Bayesian Data Fusion Approach

For many environmental applications, an accurate spatial mapping of land cover is a major concern. Currently, land cover products derived from satellite data are expected to offer a fast and inexpensive way of mapping large areas. However, the quality of these products may also largely depend on the area under study. As a result, it is common that various products disagree with each other, and the assessment of their respective quality still relies on ground validation datasets. Recently, crowdsourced data have been suggested as an alternate source of information that might help overcome this problem. However, crowdsourced data still remain largely discarded in scientific studies due to their inherent poor quality assurance. The aim of this paper is to present an efficient methodology that allows the user to code information brought by crowdsourced data even if no prior quality estimation is at hand and possibly to fuse this information with existing land cover products in order to improve their accuracy. It is first suggested that information brought by volunteers can be coded as a set of inequality constraints about the probabilities of the various land use classes at the visited places. This in turn allows estimating optimal probabilities based on a maximum entropy principle and to proceed afterwards with a spatial interpolation of these volunteers’ information. Finally, a Bayesian data fusion approach can be used for fusing multiple volunteers’ contributions with a remotely-sensed land cover product. This methodology is illustrated in this paper by focusing on the mapping of croplands in Ethiopia, where the aim is to improve the mapping of cropland as coming out from a land cover product with mitigated performances. It is shown how crowdsourced information can seriously improve the quality of the final product. The corresponding results also suggest that a prior assessing of remotely-sensed data quality can seriously improve the benefit of crowdsourcing campaigns, so that both sources of information need to be accounted together in order to optimize the sampling efforts.

Bayesian Data Fusion Applied to Soil Drainage Classes Spatial Mapping

Soil drainage classes spatial mapping is of great interest since drainage has direct effects on crop productivity and hydrological modelling. However, the prediction of this categorical variable often requires a laborious and expensive sampling over large areas. There is thus a need for a methodology that is able to combine several sources of information to improve the prediction. Bayesian maximum entropy (BME) has become a complete framework in the context of space–time prediction. This method proposes solutions to combine several sources of data no matter what the nature of information is. However, the various attempts that were made for adapting the BME methodology to categorical variables and mixed random fields faced some limitations, as a high computational burden. The main objective of this paper is to overcome this limitation by generalizing the Bayesian data fusion (BDF) theoretical framework to categorical variables, which is a simplification of the BME method through the conditional independence hypothesis. The BDF methodology for categorical variables is first described and then applied to a practical case study: the estimation of soil drainage classes using a soil map and point observations around Mechelen (Belgium). The BDF approach is compared to BME along with more classical approaches, as indicator cokriging (ICK) and logistic regression. Estimators are compared using various indicators, namely the percentage of correctly classified locations and the average highest probability. Although BDF methodology for categorical variables is a simplification of BME approach, both methods lead to very close results and have strong advantages compared to ICK and logistic regression.

Sensor Fusion on Structural Monitoring Data Analysis: Application to a Cable-Stayed Bridge

This paper illustrates an application of Bayesian logic to monitoring data analysis and structural condition state inference. The case study is a 260 m long cable-stayed bridge spanning the Adige River 10 km north of the town of Trento, Italy. This is a statically indeterminate structure, having a composite steel-concrete deck, supported by 12 stay cables. Structural redundancy, possible relaxation losses and an as-built condition differing from design, suggest that long-term load redistribution between cables can be expected. To monitor load redistribution, the owner decided to install a monitoring system which combines built-on-site elasto-magnetic and fiber-optic sensors. In this note, we discuss a rational way to improve the accuracy of the load estimate from the EM sensors taking advantage of the FOS information. More specifically, we use a multi-sensor Bayesian data fusion approach which combines the information from the two sensing systems with the prior knowledge, including design information and the outcomes of laboratory calibration. Using the data acquired to date, we demonstrate that combining the two measurements allows a more accurate estimate of the cable load, to better than 50 kN.

Fusing Aggregate and Disaggregate Data with an Application to Multiplatform Media Consumption

As firms collect greater amounts of data about their customers from an ever broader set of “touchpoints,” a new set of methodological challenges arises. Companies often collect data from these various platforms at differing levels of aggregation, and it is not clear how to merge these data sources to draw meaningful inferences about customer-level behavior patterns. In this article, the authors provide a method that firms can use, based on readily available data, to gauge and monitor multiplatform media usage. The key innovation in the method is a Bayesian data-fusion approach that enables researchers to combine individual-level usage data (readily available for most digital platforms) with aggregated data on usage over time (typically available for traditional platforms). This method enables the authors to disentangle the intraday correlations between platforms (i.e., the usage of one platform vs. another on a given day) from longer-term correlations across users (i.e., heavy/light usage of multiple platforms over time). The authors conclude with a discussion of how this method can be used in a variety of marketing contexts for which data have become readily available, such as gauging the interplay between online and brick-and-mortar purchasing behavior.

Bayesian Data Fusion of Multiview Synthetic Aperture Sonar Imagery for Seabed Classification

A Bayesian data fusion approach for seabed classification using multiview synthetic aperture sonar (SAS) imagery is proposed. The principled approach exploits all available information and results in probabilistic predictions. Each data point, corresponding to a unique 10 m x 10 m area of seabed, is represented by a vector of wavelet-based features. For each seabed type, the distribution of these features is then modeled by a unique Gaussian mixture model. When multiple views of the same data point (i.e., area of seabed) are available, the views are combined via a joint likelihood calculation. The end result of this Bayesian formulation is the posterior probability that a given data point belongs to each seabed type. It is also shown how these posterior probabilities can be exploited in a form of entropy-based active-learning to determine the most useful additional data to acquire. Experimental results of the proposed multiview classification framework are shown on a large data set of real, multiview SAS imagery spanning more than 2 km (2) of seabed.

Support-Based Implementation of Bayesian Data Fusion for Spatial Enhancement: Applications to ASTER Thermal Images

In this letter, a general Bayesian data fusion (BDF) approach is proposed and applied to the spatial enhancement of ASTER thermal images. This method fuses information coming from the visible or near-infrared bands (15 times 15 m pixels) with the thermal infrared bands (90 times 90 m pixels) by explicitly accounting for the change of support. By relying on linear multivariate regression assumptions, differences of support size for input images can be explicitly accounted for. Due to the use of locally varying variances, it also avoids producing artifacts on the fused images. Based on a set of ASTER images over the region of Lausanne, Switzerland, the advantages of this support-based approach are assessed and compared to the downscaling cokriging approach recently proposed in the literature. Results show that improvements are substantial with respect to both visual and quantitative criteria. Although the method is illustrated here with a specific case study, it is versatile enough to be applied to the spatial enhancement problem in general. It thus opens new avenues in the context of remotely sensed images.