Richard J. Caron

Estimating the Prediction Function and the Number of Unseen Species in Sampling with Replacement

Abstract A sample of N units is taken from a population consisting of an unknown number of species. We are interested in estimating the number of species and the prediction function for future sampling. The prediction function is defined as the expected number of new species that will be found if an additional sample of size tN is taken for any positive real number t. In this paper we point out that an estimator suggested by Efron and Thisted lacks some essential properties of the true prediction function, for example, the property of alternating copositivity. As a result it cannot be used for large values of t. We propose an alternative estimator that possesses the essential properties and is easily obtained. We illustrate our estimator with two numerical examples and a simulation study.

Grounding co-occurrence: Identifying features in a lexical co-occurrence model of semantic memory

Lexical co-occurrence models of semantic memory represent word meaning by vectors in a high-dimensional space. These vectors are derived from word usage, as found in a large corpus of written text. Typically, these models are fully automated, an advantage over models that represent semantics that are based on human judgments (e.g., feature-based models). A common criticism of co-occurrence models is that the representations are not grounded: Concepts exist only relative to each other in the space produced by the model. It has been claimed that feature-based models offer an advantage in this regard. In this article, we take a step toward grounding a cooccurrence model. A feed-forward neural network is trained using back propagation to provide a mapping from co-occurrence vectors to feature norms collected from subjects. We show that this network is able to retrieve the features of a concept from its co-occurrence vector with high accuracy and is able to generalize this ability to produce an appropriate list of features from the co-occurrence vector of a novel concept.

Constraint classification in mathematical programming

Consider a set of algebraic inequality constraints defining either an empty or a nonempty feasible region. It is known that each constraint can be classified as either absolutely strongly redundant, relatively strongly redundant, absolutely weakly redundant, relatively weakly redundant, or necessary. We show that is is worth making a distinction between weakly necessary constraints and strongly necessary constraints. We also present afeasible set cover method which can detect both weakly and strongly necessary constraints.The main interest in constraint classification is due to the advantages gained by the removal of redundant constraints. Since classification errors are likely to occur, we examine how the removal of a single constraint can affect the classification of the remaining constraints.

An algorithm to determine boundedness of quadratically constrained convex quadratic programmes

Abstract In this paper we present an algorithm which can be used to determine whether or not a convex quadratic objective function is bounded from below over a feasible region defined by convex quadratic constraints. If there are m constraints and n variables, the algorithm terminates after at most min{ m − 1, n − 1} iterations. Each iteration requires the identification of implicit equality constraints in a system of homogeneous linear inequality and equality constraints, and the identification can be completed by the solution of linear programmes. In addition, the algorithm has the advantage of providing a mechanism to reduce both the number of constraints and the dimension of the problem.

Unboundedness of a convex quadratic function subject to concave and convex quadratic constraints

Abstract We present conditions for the existence of upper and lower bounds on convex quadratic objective functions subject to concave and convex quadratic constraints. We also present techniques for determining whether or not the conditions are satisfied.

Minimal representation of quadratically constrained convex feasible regions

In this paper we are concerned with characterizing minimal representations of feasible regions defined by both linear and convex quadratic constraints. We say that representation is minimal if every other representation has either more quadratic constraints, or has the same number of quadratic constraints and at least as many linear constraints. We will prove that a representation is minimal if and only if it contains no redundant constraints, no pseudo-quadratic constraints and no implicit equality constraints. We define a pseudo-quadratic constraint as a quadratic constraint that can be replaced by a finite number of linear constraints. In order to prove the minimal representation theorem, we also prove that if the surfaces of two quadratic constraints match on a ball, then they match everywhere.In this paper we also provide algorithms that can be used to detect implicit equalities and pseudoquadratic constraints. The redundant constraints can be identified using the hypersphere directions (HD) method.

On the best case performance of hit and run methods for detecting necessary constraints

The hit and run methods are probabilistic algorithms that can be used to detect necessary (nonredundant) constraints in systems of linear constraints. These methods construct random sequences of lines that pass through the feasible region. These lines intersect the boundary of the region at twohit-points, each identifying a necessary constraint. In order to study the statistical performance of such methods it is assumed that the probabilities of hitting particular constraints are the same for every iteration. An indication of the best case performance of these methods can be determined by minimizing, with respect to the hit probabilities, the expected value of the number of iterations required to detect all necessary constraints. We give a set of isolated strong local minimizers and prove that for two, three and four necessary constraints the set of local minimizers is the complete set of global minimizers. We conjecture that this is also the case for any number of necessary constraints. The results in this paper also apply to sampling problems (e.g., balls from an urn) and to the coupon collectors problem.

Convergence properties of hit–and–run samplers

Hit–and–Run algorithms are probabilistic methods for generating points at random according to some prescribed distribution π on a subset A of R d . Given a current point, say , a direction vector, say , is chosen at random according to some prescribed random mechanism. The next point , is then chosen at random according to the conditionalization of π on the line defined by Xk and . Under appropriate conditions, the sequence will be a Markov chain converging in total variation to the target distribution π. This paper introduces a new class of Hit–and–Run algorithms. A general convergence theorem is obtained and the existence, within this class, of particular Hit–and–Run algorithms with desirable asymptotic properties is established

A new approach to the analysis of random methods for detecting necessary linear inequality constraints

A new approach is given for the analysis of random methods for detecting necessary constraints in systems of linear inequality constraints. This new approach directly accounts for the fact that two constraints are detected as necessary (hit) at each iteration of a random method. The significance of this two-hit analysis is demonstrated by comparing it with the usual one-hit analysis.

Feasibility and Constraint Analysis of Sets of Linear Matrix Inequalities

We present a constraint analysis methodology for linear matrix inequality constraints. If the constraint set is found to be feasible, we search for a minimal representation; otherwise, we search for an irreducible infeasible system. The work is based on the solution of a set-covering problem where each row corresponds to a sample point and is determined by constraint satisfaction at the sampled point. Thus, an implementation requires a method to collect points in the ambient space and a constraint oracle. Much of this paper will be devoted to the development of a hit-and-run sampling methodology. Test results confirm that our approach not only provides information required for constraint analysis but will also, if the feasible region has a nonvoid interior, with probability one, find a feasible point.