Rohan Hemasinha

Uncertainty measures for evidential reasoning I: A review☆

Abstract This paper is divided into two parts. Part I discusses limitations of the measures of global uncertainty of Lamata and Moral and total uncertainty of Klir and Ramer. We prove several properties of different nonspecificity measures. The computational complexity of different total uncertainty measures is discussed. The need for a new measure of total uncertainty is established in Part I. In Part II, we propose a set of intuitively desirable axioms for a measure of total uncertainty and then derive an expression for the same. Several theorems are proved about the new measure. The proposed measure is additive, and unlike other measures, has a unique maximum. This new measure reduces to Shannons probabilistic entropy when the basic probability assignment focuses only on singletons. On the other hand, complete ignorance—basic assignment focusing only on the entire set, as a whole—reduces it to Hartleys measure of information. The computational complexity of the proposed measure is O(N), whereas the previous measures are O(N 2 ).

Uncertainty measures for evidential reasoning II: A new measure of total uncertainty

Abstract In Part I we discussed limitations of two measures of global (non-fuzzy) uncertainty of Lamata and Moral, and a measure of total (non-fuzzy) uncertainty due to Klir and Ramer and established the need for a new measure. In this paper we propose a set of intuitively desirable axioms for a measure of total uncertainty (TU) associated with a basic assignment m(A), and then derive an expression for a (unique) function that satisfies these requirements. Several theorems are proved about the new measure. Our measure is additive, and unlike other TU measures, has a unique maximum. The new measure reduces to Shannons probabilistic entropy when the basic probability assignment focuses only on singletons. On the other hand, complete ignorance—basic assignment focusing only on the entire set, as a whole—reduces it to Hartleys measure of information. We show that the computational complexity of the new measure is O(N), whereas previous measures of TU are O(N 2 ). Finally, we compare the new measure to its predecessors by extending the numerical example of Part I so that it includes values of the new measure.

The switching behavior of charts with variable sampling intervals

In a variable sampling interval control scheme the time interval between successive samples is allowed to vary depending on what is being observed from the data. It has been the practice to compare the average time to signal and the average run length of variable sampling interval schemes and the corresponding control procedures with fixed sampling intervals. Thus far design tables for control charts with variable sampling intervals have not considered the effect of the design parameters on the switching behavior of such control procedures. Frequent switches between the different sampling intervals can be a complicating factor in the application of control charts with variable sampling intervals. The problem of switches is addressed in this article and improved switching rules are presented and evaluated for Shewhart control procedures. Expressions for the average number of switches and the average time to signal are obtained. The proposed runs rules considerably reduce the average number of switches betw...

On Almost Regular Tournament Matrices

Abstract Spectral and determinantal properties of a special class M n of 2n×2n almost regular tournament matrices are studied. In particular, the maximum Perron value of the matrices in this class is determined and shown to be achieved by the Brualdi–Li matrix, which has been conjectured to have the largest Perron value among all tournament matrices of even order. We also establish some determinantal inequalities for matrices in M n and describe the structure of their associated walk spaces.

Properties of the Brualdi-Li tournament matrix

The Brualdi–Li tournament matrix is conjectured to have the largest spectral radius among all tournament matrices of even order. In this paper two forms of the characteristic polynomial of the Brualdi–Li tournament matrix are found. Using the first form it is shown that the roots of the characteristic polynomial are simple and that the Brualdi–Li tournament matrix is diagonalizable. Using the second form an expression is found for the coefficients of the powers of the variable λ in the characteristic polynomial. These coefficients give information about the cycle structure of the cycles of length 1–5 of the directed graph associated with the Brualdi–Li tournament matrix.

Properties of Tournaments Among Well-Matched Players

1. TOURNAMENTS. In an n-player round robin tournament, each player plays one match against each of the other n - 1 players. The win-loss outcomes of these matches can be conveniently recorded in a tournament matrix A = [aij] as follows: First label the players in any order as 1, 2,.. ., n. For each pair i and j, set ai1 = 1 if player i defeats player j, and set aij = 0 otherwise. If i j j, then exactly one of aij and aji is nonzero; when i = j, aii = 0. What properties of the matrix A are related to the strengths of the players? The simplest measure of strength is the number of matches that the player wins, and the row sums of A count the number of matches won by each player. We are interested in understanding tournaments among players who are well matched in the sense that each player wins about half of the matches played. If the number of players is odd, many properties of A are very well understood. If the number of players is even, however, the properties of A are far less well understood. Indeed, there are many easily stated questions that lead to hard, open problems. Some of these problems are the focus of this paper. We let I denote the identity matrix, we let J denote the square matrix all of whose entries are ones, and we let e denote the column vector whose entries are all ones. A matrix A whose entries are zeros and ones is a tournament matrix exactly when

The determinant of a fuzzy matrix with respect to t and co-t norms

Abstract We study determinants of square matrices over the interval [0,1] when ordinary multiplication is replaced by a triangular norm and ordinary addition is replaced by a triangular conorm.

Iterates of fuzzy circulant matrices

Abstract The iterates of circulant matrices under the max-min product is investigated in this article. It is shown that if the first row of a fuzzy circulant matrix is in decreasing order, then the iterates of the circulant converge and if the first row is in increasing order, then the iterates oscillate. In both cases a complete description of the iterates is obtained. As an application, this yields an O(n) algorithm for computing the transitive closure of a certain type of fuzzy relation; viz. relations whose adjacency matrices are fuzzy circulants with decreasing first row.

Norms induced by symmetric guage functions

Let F n be the set of all n×1 column vectors over F, where F=R, A norm ∥ċ∥ on F n is permutationally invariant if and it is an absolute norm if A permutationally invariant absolute norm on F n is called a symmetric gauge function. Given a norm ∥ċ∥ on F n and a nonsingular matrix HeF n×n , one can define a norm ∥ċ∥H by The purpose of this note is to study the conditons on H for which the norm ∥ċ∥H is an absolute norm, a permutationally invariant norm, and a symmetric gauge function, respectively, if ∥ċ∥is a symmetric gauge function.

The sign invariance of certain norms on

Let denote the set of all n×1 real column vectors and let Δn denote the group of all n×n diagonal matrices over whose diagonal entries have absolute value one. A norm ∥⋅∥ on is said to be sign invariant of Given a norm ∥⋅∥ on and an m×n real matrix H of rank n on can define a norm ∥⋅∥ H on by In this note we study the following problem. If ∥⋅∥ p is the lp norm on then what conditions on H will entail that ∥⋅∥ H,p is sign invariant.