Representation of maxitive measures: an overview

Idempotent integration is an analogue of Lebesgue integration where $\sigma$-maxitive measures replace $\sigma$-additive measures. In addition to reviewing and unifying several Radon--Nikodym like theorems proven in the literature for the idempotent integral, we also prove new results of the same kind.Comment: 40 page

Dempster-Shafer Theory Applications in Structural Damage Assessment and Social Vulnerability Ranking

This thesis explores the different mathematical frameworks that have been used in risk assessment, with an emphasis on Dempster-Shafer Evidence Theory and the applicability in cases of post seismic structural damage assessments and social vulnerability ranking. Evidence Theory allows the combination of multiple expert beliefs while considering uncertainties that are often inherent in such evaluations. In cases such as seismic hazards, for which structural vulnerability and structural damage are evaluated in a case-by-case scenario, subjective assessments are not only useful but necessary. The results of experimentation and survey distribution suggest that probability may not be the most natural framework in which to quantitatively incorporate the involved uncertainty. Ignorance and evidence-based assessments may be better represented using Evidence Theory.</p

Multiple Instance Choquet Integral for multiresolution sensor fusion

Imagine you are traveling to Columbia, MO for the first time. On your flight to Columbia, the woman sitting next to you recommended a bakery by a large park with a big yellow umbrella outside. After you land, you need directions to the hotel from the airport. Suppose you are driving a rental car, you will need to park your car at a parking lot or a parking structure. After a good night's sleep in the hotel, you may decide to go for a run in the morning on the closest trail and stop by that recommended bakery under a big yellow umbrella. It would be helpful in the course of completing all these tasks to accurately distinguish the proper car route and walking trail, find a parking lot, and pinpoint the yellow umbrella. Satellite imagery and other geo-tagged data such as Open Street Maps provide effective information for this goal. Open Street Maps can provide road information and suggest bakery within a five-mile radius. The yellow umbrella is a distinctive color and, perhaps, is made of a distinctive material that can be identified from a hyperspectral camera. Open Street Maps polygons are tagged with information such as "parking lot" and "sidewalk." All these information can and should be fused to help identify and offer better guidance on the tasks you are completing. Supervised learning methods generally require precise labels for each training data point. It is hard (and probably at an extra cost) to manually go through and label each pixel in the training imagery. GPS coordinates cannot always be fully trusted as a GPS device may only be accurate to the level of several pixels. In many cases, it is practically infeasible to obtain accurate pixel-level training labels to perform fusion for all the imagery and maps available. Besides, the training data may come in a variety of data types, such as imagery or as a 3D point cloud. The imagery may have different resolutions, scales and, even, coordinate systems. Previous fusion methods are generally only limited to data mapped to the same pixel grid, with accurate labels. Furthermore, most fusion methods are restricted to only two sources, even if certain methods, such as pan-sharpening, can deal with different geo-spatial types or data of different resolution. It is, therefore, necessary and important, to come up with a way to perform fusion on multiple sources of imagery and map data, possibly with different resolutions and of different geo-spatial types with consideration of uncertain labels. I propose a Multiple Instance Choquet Integral framework for multi-resolution multisensor fusion with uncertain training labels. The Multiple Instance Choquet Integral (MICI) framework addresses uncertain training labels and performs both classification and regression. Three classifier fusion models, i.e. the noisy-or, min-max, and generalized-mean models, are derived under MICI. The Multi-Resolution Multiple Instance Choquet Integral (MR-MICI) framework is built upon the MICI framework and further addresses multiresolution in the fusion sources in addition to the uncertainty in training labels. For both MICI and MR-MICI, a monotonic normalized fuzzy measure is learned to be used with the Choquet integral to perform two-class classifier fusion given bag-level training labels. An optimization scheme based on the evolutionary algorithm is used to optimize the models proposed. For regression problems where the desired prediction is real-valued, the primary instance assumption is adopted. The algorithms are applied to target detection, regression and scene understanding applications. Experiments are conducted on the fusion of remote sensing data (hyperspectral and LiDAR) over the campus of University of Southern Mississippi - Gulfpark. Clothpanel sub-pixel and super-pixel targets were placed on campus with varying levels of occlusion and the proposed algorithms can successfully detect the targets in the scene. A semi-supervised approach is developed to automatically generate training labels based on data from Google Maps, Google Earth and Open Street Map. Based on such training labels with uncertainty, the proposed algorithms can also identify materials on campus for scene understanding, such as road, buildings, sidewalks, etc. In addition, the algorithms are used for weed detection and real-valued crop yield prediction experiments based on remote sensing data that can provide information for agricultural applications.Includes biblographical reference

Counterfactuals 2.0 Logic, Truth Conditions, and Probability

The present thesis focuses on counterfactuals. Specifically, we will address new questions and open problems that arise for the standard semantic accounts of counterfactual conditionals. The first four chapters deal with the Lewisian semantic account of counterfactuals. On a technical level, we contribute by providing an equivalent algebraic semantics for Lewis' variably strict conditional logics, which is notably absent in the literature. We introduce a new kind of algebra and differentiate between local and global versions of each of Lewis' variably strict conditional logics. We study the algebraic properties of Lewis' logics and the structure theory of our newly introduced algebras. Additionally, we employ a new algebraic construction, based on the framework of Boolean algebras of conditionals, to provide an alternative semantics for Lewisian counterfactual conditionals. This semantic account allows us to establish new truth conditions for Lewisian counterfactuals, implying that Lewisian counterfactuals are definable conditionals, and each counterfactual can be characterized as a modality of a corresponding probabilistic conditional. We further extend these results by demonstrating that each Lewisian counterfactual can also be characterized as a modality of the corresponding Stalnaker conditional. The resulting formal semantic framework is much more expressive than the standard one and, in addition to providing new truth conditions for counterfactuals, it also allows us to define a new class of conditional logics falling into the broader framework of weak logics. On the philosophical side, we argue that our results shed new light on the understanding of Lewisian counterfactuals and prompt a conceptual shift in this field: Lewisian counterfactual dependence can be understood as a modality of probabilistic conditional dependence or Stalnakerian conditional dependence. In other words, whether a counterfactual connection occurs between A and B depends on whether it is "necessary" for a Stalnakerian/probabilistic dependence to occur between A and B. We also propose some ways to interpret the kind of necessity involved in this interpretation. The remaining two chapters deal with the probability of counterfactuals. We provide an answer to the question of how we can characterize the probability that a Lewisian counterfactual is true, which is an open problem in the literature. We show that the probability of a Lewisian counterfactual can be characterized in terms of belief functions from Dempster-Shafer theory of evidence, which are a super-additive generalization of standard probability. We define an updating procedure for belief functions based on the imaging procedure and show that the probability of a counterfactual A &gt; B amounts to the belief function of B imaged on A. This characterization strongly relies on the logical results we proved in the previous chapters. Moreover, we also solve an open problem concerning the procedure to assign a probability to complex counterfactuals in the framework of causal modelling semantics. A limitation of causal modelling semantics is that it cannot account for the probability of counterfactuals with disjunctive antecedents. Drawing on the same previous works, we define a new procedure to assign a probability to counterfactuals with disjunctive antecedents in the framework of causal modelling semantics. We also argue that our procedure is satisfactory in that it yields meaningful results and adheres to some conceptually intuitive constraints one may want to impose when computing the probability of counterfactuals