Transferable Belief Model Framework Research Articles

Approximating probability distribution of circuit performance function for parametric yield estimation using transferable belief model

This paper applies the transferable belief model (TBM) interpretation of the Dempster-Shafer theory of evidence to approximate distribution of circuit performance function for parametric yield estimation. Treating input parameters of performance function as credal variables defined on a continuous frame of real numbers, the suggested approach constructs a random set-type evidence for these parameters. The corresponding random set of the function output is obtained by extension principle of random set. Within the TBM framework, the random set of the function output in the credal state can be transformed to a pignistic state where it is represented by the pignistic cumulative distribution. As an approximation to the actual cumulative distribution, it can be used to estimate yield according to circuit response specifications. The advantage of the proposed method over Monte Carlo (MC) methods lies in its ability to implement just once simulation process to obtain an available approximate value of yield which has a deterministic estimation error. Given the same error, the new method needs less number of calculations than MC methods. A track circuit of high-speed railway and a numerical eight-dimensional quadratic function examples are included to demonstrate the efficiency of this technique.

Constructing consonant belief functions from sample data using confidence sets of pignistic probabilities

A new method is proposed for building a predictive belief function from statistical data in the transferable belief model framework. The starting point of this method is the assumption that, if the probability distribution P X of a random variable X is known, then the belief function quantifying our belief regarding a future realization of X should have its pignistic probability distribution equal to P X . When P X is unknown but a random sample of X is available, it is possible to build a set P of probability distributions containing P X with some confidence level. Following the least commitment principle, we then look for a belief function less committed than all belief functions with pignistic probability distribution in P . Our method selects the most committed consonant belief function verifying this property. This general principle is applied to arbitrary discrete distributions as well as exponential and normal distributions. The efficiency of this approach is demonstrated using a simulated multi-sensor classification problem.

Estimating parameter distributions in structural reliability assessment using the Transferable Belief Model

This paper applies the Transferable Belief Model (TBM) interpretation of the Dempster–Shafer theory of evidence to estimate parameter distributions for probabilistic structural reliability assessment based on information from previous analyses, expert opinion, or qualitative assessments (i.e., evidence). Treating model parameters as credal variables, the suggested approach constructs a set of least-committed belief functions for each parameter defined on a continuous frame of real numbers that represent beliefs induced by the evidence in the credal state, discounts them based on the relevance and reliability of the supporting evidence, and combines them to obtain belief functions that represent the aggregate state of belief in the true value of each parameter. Within the TBM framework, beliefs held in the credal state can then be transformed to a pignistic state where they are represented by pignistic probability distributions. The value of this approach lies in its ability to leverage results from previous analyses to estimate distributions for use within a probabilistic reliability and risk assessment framework. The proposed methodology is demonstrated in an example problem that estimates the physical vulnerability of a notional office building to blast loading.

Human action recognition in videos based on the Transferable Belief Model

This paper focuses on human behavior recognition where the main problem is to bridge the semantic gap between the analogue observations of the real world and the symbolic world of human interpretation. For that, a fusion architecture based on the Transferable Belief Model framework is proposed and applied to action recognition of an athlete in video sequences of athletics meeting with moving camera. Relevant features are extracted from videos, based on both the camera motion analysis and the tracking of particular points on the athlete’s silhouette. Some models of interpretation are used to link the numerical features to the symbols to be recognized, which are running, jumping and falling actions. A Temporal Belief Filter is then used to improve the robustness of action recognition. The proposed approach demonstrates good performance when tested on real videos of athletics sports videos (high jumps, pole vaults, triple jumps and long jumps) acquired by a moving camera and different view angles. The proposed system is also compared to Bayesian Networks.

Global cost of assignment in the TBM framework for association of uncertain ID reports

Suppose we are given multiple independent reports on the count and identification (class, type, allegiance) of objects that are present in a certain volume of interest. The reports are uncertain (random and non-specific) and collected with the probability of detection P d < 1 and the probability of false alarm P f > 0 . The problem is to estimate the actual number of objects and their true identity. Adopting the formalism of the belief function theory as interpreted by the transferable belief model (TBM), a solution is proposed which maximises the plausibility of the global assignment cost of identification reports. The proposed global cost of assignment is shown by Monte Carlo simulations to outperform its ad hoc alternatives.

Kalman filter and joint tracking and classification based on belief functions in the TBM framework

The paper develops an approach to joint tracking and classification based on belief functions as understood in the transferable belief model (TBM). The TBM model is identical to the classical model except all probability functions are replaced by belief functions, which are more flexible for representing uncertainty. It is felt that the tracking phase is well handled by the classical Kalman filter but that the classification phase deserves amelioration. For the tracking phase, we derive a minimal set of assumptions needed in the TBM approach in order to recover the classical relations. For the classification phase, we distinguish between the observed target behaviors and the underlying target classes which are usually not in one-to-one correspondence. We feel the results obtained with the TBM approach are more reasonable than those obtained with the corresponding Bayesian classifiers.

Application of the transferable belief model to diagnostic problems

Short presentation of the most relevant elements of the transferable belief model and its use for problems related to the diagnostic process. These examples illustrate the use of the transferable belief model and in particular of the Generalized Bayesian Theorem. Uncertainty is classically represented by probability functions, and diagnostic in an environment poised by uncertainty is usually handled through the application of the Bayesian theorem that permits the computation of the posterior probability over the diagnostic categories given the observed data from the prior probability over the same categories. We show here that the whole problem admits a similar solution when uncertainty is quantified by belief functions as in the transferable belief model. The classical Bayesian theorem admits a generalization within the transferable belief model (TBM) that we called the Generalized Bayesian Theorem (Smets, 1978, 1981, 1993a). This theorem seems to have been often overlooked, and the use of conditional belief functions for diagnostic problems neglected. The Generalized Bayesian Theorem (GBT) permits the computation of the conditional belief over the diagnostic classes given an observed data from the knowledge of the set of the conditional beliefs about which data will be observed when the case belongs to a given diagnostic category. Loosely expressed, this inversion theorem permits to pass from a belief on the symptoms given the diseases to a belief on the diseases given the symptoms. We present hereafter four examples of diagnostic process within the TBM, and compared the TBM solution with its obvious contender, the probability solution. The examples are analyzed in detail in order to give a clear understanding of the exact use of the TBM and its GBT. We restrict ourself to ‘simple’ examples, cases of complex systems and common or dependent causes are not tackled. Our aim is in showing how the classical Bayesian theorem can be extended and applied within the TBM framework.

Belief functions: The disjunctive rule of combination and the generalized Bayesian theorem

We generalize the Bayes' theorem within the transferable belief model framework. The generalized Bayesian theorem (GBT) allows us to compute the belief over a space θ given an observation x ⊆ X when one knows only the beliefs over X for every θ i ∈ Θ. We also discuss the disjunctive rule of combination (DRC) for distinct pieces of evidence. This rule allows us to compute the belief over X from the beliefs induced by two distinct pieces of evidence when one knows only that one of the pieces of evidence holds. The properties of the DRC and GBT and their uses for belief propagation in directed belief networks are analyzed. The use of the discounting factors is justified. The application of these rules is illustrated by an example of medical diagnosis.