Hidden Markov Models With Set-Valued Parameters

Hidden Markov models (HMMs) are widely used probabilistic models of sequential data. As with other probabilistic models, they require the specification of local conditional probability distributions, whose assessment can be too difficult and error-prone, especially when data are scarce or costly to acquire. The imprecise HMM (iHMM) generalizes HMMs by allowing the quantification to be done by sets of, instead of single, probability distributions. iHMMs have the ability to suspend judgment when there is not enough statistical evidence, and can serve as a sensitivity analysis tool for standard non-stationary HMMs. In this paper, we consider iHMMs under the strong independence interpretation, for which we develop efficient inference algorithms to address standard HMM usage such as the computation of likelihoods and most probable explanations, as well as performing filtering and predictive inference. Experiments with real data show that iHMMs produce more reliable inferences without compromising the computational efficiency.

Vision-based target tracking and surveillance with robust set-valued state estimation

Tracking a target from a video stream (or a sequence of image frames) involves nonlinear measurements in Cartesian coordinates. However, the target dynamics, modeled in Cartesian coordinates, result in a linear system. We present a robust linear filter based on an analytical nonlinear to linear measurement conversion algorithm. Using ideas from robust control theory, a rigorous theoretical analysis is given which guarantees that the state estimation error for the filter is bounded, i.e., a measure against filter divergence is obtained. In fact, an ellipsoidal set-valued estimate is obtained which is guaranteed to contain the true target location with an arbitrarily high probability. The algorithm is particularly suited to visual surveillance and tracking applications involving targets moving on a plane.

A robust set-valued state estimation approach to the problem of vision based SLAM for mobile robots

The problem of visual simultaneous localization and mapping (SLAM) is examined in this paper using ideas and algorithms from robust control and estimation theory. Using a stereo-vision based sensor, a nonlinear measurement model is derived which leads to nonlinear measurements of the landmark coordinates along with optical flow based measurements of the relative robot-landmark velocity. Using a novel analytical measurement transformation, the nonlinear SLAM problem is converted into the linear filter is guaranteed stable and the ALAM state estimation error is bounded within an ellipsoidal set. No similar results are available for the commonly employed extended Kalman filter which is known to exhibit divergent and inconsistency characteristics in practice.

Fusion based 3D tracking of mobile transmitters via robust set-valued state estimation with RSS measurements

This paper investigates the problem of location and velocity detection of a mobile agent using Received Signal Strength (RSS) measurements captured by geographically distributed seed nodes. With inherently nonlinear power measurements, we derive a powerful linear measurement scheme using an analytical measurement conversion technique which can readily be used with RSS measuring sensors. We also employ the concept of sensor fusion in conjunction for the case of redundant measurements to further enhance the estimation accuracy.

Dense Continuity and Selections of Set-Valued Mappings

∗ The first and third author were partially supported by National Fund for Scientific Research at the Bulgarian Ministry of Science and Education under grant MM-701/97.

Subdifferentials of Performance Functions and Calculus of Coderivatives of Set-Valued Mappings

The paper contains calculus rules for coderivatives of compositions, sums and intersections of set-valued mappings. The types of coderivatives considered correspond to Dini-Hadamard and limiting Dini-Hadamard subdifferentials in Gˆateaux differentiable spaces, Fréchet and limiting Fréchet subdifferentials in Asplund spaces and approximate subdifferentials in arbitrary Banach spaces. The key element of the unified approach to obtaining various calculus rules for various types of derivatives presented in the paper are simple formulas for subdifferentials of marginal, or performance functions.

Averaging operators and set-valued maps

MSC 2010: 54C35, 54C60.

Approximation of Univariate Set-Valued Functions - an Overview

2000 Mathematics Subject Classification: 26E25, 41A35, 41A36, 47H04, 54C65.

Uniform Convergence of the Newton Method for Aubin Continuous Maps

* This work was supported by National Science Foundation grant DMS 9404431.

Acceleration of Convergence in Dontchev’s Iterative Method for Solving Variational Inclusions

2000 Mathematics Subject Classification: 47H04, 65K10.

Applications of the Fréchet subdifferential

2000 Mathematics Subject Classification: 46A30, 54C60, 90C26.

An Iterative Procedure for Solving Nonsmooth Generalized Equation

2000 Mathematics Subject Classification: 47H04, 65K10.

Corrigendum to “The envelope theorem for multiobjective convex programming via contingent derivatives” [J. Math. Anal. Appl. 372 (1) (2010) 197–207]

The aim of this note is to formulate an envelope theorem for vector convex programs. This version corrects an earlier work, “The envelope theorem for multiobjective convex programming via contingent derivatives” by Jiménez Guerra et al. (2010) [3]. We first propose a necessary and sufficient condition allowing to restate the main result proved in the alluded paper. Second, we introduce a new Lagrange multiplier in order to obtain an envelope theorem avoiding the aforementioned error.

Sensitivity Analysis in Convex Optimization through the Circatangent Derivative

The main goal of this paper is to analyse the sensitivity of a vector convex optimization problem according to variations in the right-hand side. We measure the quantitative behavior of a certain set of Pareto optimal points characterized to become minimum when the objective function is composed with a positive function. Its behavior is analysed quantitatively using the circatangent derivative for set-valued maps. Particularly, it is shown that the sensitivity is closely related to a Lagrange multiplier solution of a dual program.

A natural extension of the classical envelope theorem in vector differential programming

The aim of this paper is to extend the classical envelope theorem from scalar to vector differential programming. The obtained result allows us to measure the quantitative behaviour of a certain set of optimal values (not necessarily a singleton) characterized to become minimum when the objective function is composed with a positive function, according to changes of any of the parameters which appear in the constraints. We show that the sensitivity of the program depends on a Lagrange multiplier and its sensitivity.