A class of upper bounds on the probability of error for the general multihypotheses pattern recognition problem is obtained. In particular, an upper bound in the class is shown to be a linear functional of the pairwise Bhattacharya coefficients. Evaluation of the bounds requires knowledge of a priori probabilities and of the hypothesis-conditional probability density functions. A further bound is obtained that is independent of a priori probabilities. For the case of unknown a priori probabilities and conditional probability densities, an estimate of the latter upper bound is derived using a sequence of classified samples and Kernel functions to estimate the unknown densities.

A class of upper bounds on probability of error for multihypotheses pattern recognition (Corresp.)

Generalized partitioned solutions (GPS) of Riccati equations (RE) are presented in terms of forward and backward time differential equations that are theoretically interesting, possibly computationally advantageous, as well as provide interesting interpretations of these resuits, e.g., in terms of generalized partial observability and controllability matrices. The GPS are the natural framework for the effective change of initial conditions, and the transformation of backward RE to forward RE and vice-versa. The GPS are given in terms of families of forward or backward RE, and constitute generalizations to time-varying RE of well-known solution algorithms such as the X - Y or Chandrasekhar algorithms. Most importantly, based on the GPS, computationally effective algorithms are obtained for the numerical solution of RE. These partitioned numerical algorithms (PNA) have a decomposed or "partitioned" structure, namely, they are given exactly in terms of a set of elemental solutions which are completely decoupled, and as such computable in either a parallel or serial processing mode. Further, the overall solution is given exactly in terms of a simple recursive operation on the elemental solutions. Finally, the PNA for a large class of RE, namely those with periodic or constant matrices, are completely integration-free, other than for a subinterval of the total computation interval, whose length, moreover, can be chosen arbitrarily. Also based on the GPS, a computationally attractive numerical algorithm is obtained for the computation of the steady-state solution of time-invariant RE. This algorithm results from doubling the length of the partitioning interval, and straightforward use of the GPS. The resulting "doubling" PNA is fast and is also essentially, integration-free requiring integration only in an initial subinterval, whose length is arbitrary, and subsequently consisting of simple iterative operations at the end of each time-interval which is twice as long as the interval in the previous iterations, i.e., doubling.

http://ieeexplore.ieee.org/iel5/9/24156/01101321.pdf

Partitioned Ricatti solutions and integration-free doubling algorithms

: Simplified upper and lower bounds to the probability of error for general M-ary hypotheses pattern recognition are obtained. The bounds, applicable to general non-gaussian densities and especially mixture densities encountered in adaptive pattern recognition, are simple to calculate and hence valuable for on-line performance evaluation of pattern recognition system. Computer evaluation of the bounds, established their tight nature and computational simplicity. Based on the bounds, feature extraction criteria are derived for supervised as well as parametric adaptive pattern recognition. (Author)

Probability of Error Bounds

A class of upper bounds on the probability of error for the general multihypotheses pattern recognition problem is obtained. In particular, an upper bound in the class is shown to be a linear functional of the pairwise Bhattacharya coefficients. Evaluation of the bounds requires knowledge of a-priori probabilities and of the hypothesis-conditional probability density functions. A further bound is obtained that is independent of apriori probabilities. For the case of unknown apriori probabilities and conditional probability densities, an estimate of the latter upper bound is derived using a sequence of classified samples and Kernel functions to estimate the unknown densities.

A class of upper-bounds on probability of error for multi-hypotheses pattern recognition

A general and fundamental relationship between estimation, detection/pattern recognition, and the Bhattacharyya coefficient is established. Specifically, it is shown that a functional of the likelihood ratio necessary for Bayes optimal detection/pattern recognition is the Bayes-optimal mean-square estimate of the indicator variable \beta , where \beta = 1, 0 , for hypotheses H_{1}, H_{0} , respectively. The Bhattacharyya coefficient is then shown to be the expected value of the estimation error standard deviation.

On a general relationship between estimation, detection, and the Bhattacharyya coefficient (Corresp.)

Optimal continuous linear feature extraction for the binary Gaussian pattern recognition or detection problem necessitates finding the double orthogonal expansion of the observable random process under hypothesis H i , i = 1, 2. State variable techniques are utilized here to yield efficient computer-implementable procedures for obtaining the double orthogonal expansion.

D.G. Lainiotis

Papers

Application of state variable techniques to optimal feature extraction