An upper bound for the recursive least squares estimation error

A new upper bound for the convergence rate of recursive least squares (RLS) errors is presented. The bound is free of some deficiencies of a cell-known RLS upper bound and allows a realistic assessment of factors influencing convergence rate, such as input-output data scaling, disturbances, signal-to-noise ratio, number of estimated parameters, data discounting, and excitation properties of plant inputs. Some of the properties of the new bound are discussed     

A lower bound on the estimation error for Markov processes

A lower bound on the minimal mean-square error in estimating nonlinear Markov processes is presented. The bound holds for causal and uncausal filtering. The derivation is based on the Van Trees' version of the Cramér-Rao inequality.     

A combinatorial error bound for t-point-based sampling

The method of two-point-based sampling using orthogonal arrays (Inform. Process. Lett. 60 (1996) 91) is extended to consider t-wise independent sampling using orthogonal arrays of higher strength t. Using combinatorial considerations, an error bound is calculated which agrees with the previously known result when t=2, and has the advantage of exponentially decreasing in t. The result is shown to be strictly sharper than that arising from the generalized Chebyshev inequality. Finally, the behavior of the family of error bounds we obtain for increasing values of t is analyzed.     

A tight upper bound on the Bayesian probability of error

In this paper, we present a new upper bound on the minimum probability of error of Bayesian decision systems for statistical pattern recognition. This new bound is continuous everywhere and is shown to be tighter than several existing bounds such as the Bhattacharyya and the Bayesian bounds. Numerical results are also presented     

Comments on "A least upper bound on quantization error"

A summary of methods proposed by Cavin and Bertram for bounding quantization errors in hybrid control systems is presented. Their methods are compared and similarities in the methods are noted. An apparent error by Cavin is demonstrated.     

Implicit upper bound error estimates for combined expansive model and discretization adaptivity

A computational methodology for goal-oriented combined discretization and expansive (refined) model adaptivity by overall implicit error control of quantities of interest is presented, requiring estimators of primal and dual discretization and model errors. In the case of dimensional within model adaptivity, prolongations of coarse model solutions into the solution space of a fine model for defining a consistent model error are necessary, which can be achieved at the element level by two strategies. The first one is an orthogonalized kinematic prolongation of nodal displacements, whereas the second one uses prolongations of the external loads which are then used to solve additional local variational problems thus yielding prolongated solutions which a priori fulfill the required orthogonality relations at the element level. Finally, a numerical example of an elastic continuous T-beam is presented with comparative results where goal-oriented error estimation is applied to linear elasticity with a $2\frac{1}{2}\text{D}$ discrete Reissner–Mindlin plate model as the coarse model and the 3D theory as the fine model.     

An improved upper bound on the queuenumber of the hypercube

In this note, we show that the n-dimensional hypercube Q(n) can be laid out using n−2 queues for all n?5. Our result improves the previously known result for the case n?5.     

An upper bound for the gain of stabilizing proportional controllers

A stable linear system controlled by a proportional controller is closed-loop stable provided the controller has sufficiently small gain. If the system has an unstable zero then any proportional controller with sufficiently large gain is destabilizing.In this note we give an upper bound for the gain of stabilizing proportional controllers of stable systems that have one or more unstable zeros.     

Toward a tight upper bound for the error probability of the binary Gaussian classification problem

It is well known that the error probability, of the binary Gaussian classification problem with different class covariance matrices, cannot be generally evaluated exactly because of the lack of closed-form expression. This fact pointed out the need to find a tight upper bound for the error probability. This issue has been for more than 50 years ago and is still of interest. All derived upper-bounds are not free of flaws. They might be loose, computationally inefficient particularly in highly dimensional situations, or excessively time consuming if high degree of accuracy is desired. In this paper, a new technique is developed to estimate a tight upper bound for the error probability of the well-known binary Gaussian classification problem with different covariance matrices. The basic idea of the proposed technique is to replace the optimal Bayes decision boundary with suboptimal boundaries which provide an easy-to-calculate upper bound for the error probability. In particular, three types of decision boundaries are investigated: planes, elliptic cylinders, and cones. The new decision boundaries are selected in such a way as to provide the tightest possible upper bound. The proposed technique is found to provide an upper bound, tighter than many of the often used bounds such as the Chernoff bound and the Bayesian-distance bound. In addition, the computation time of the proposed bound is much less than that required by the Monte-Carlo simulation technique. When applied to real world classification problems, obtained from the UCI repository [H. Chernoff, A measure for asymptotic efficiency of a hypothesis based on a sum of observations, Ann. Math. Statist. 23 (1952) 493–507.], the proposed bound was found to provide a tight bound for the analytical error probability of the quadratic discriminant analysis (QDA) classifier and a good approximation to its empirical error probability.     

An upper bound on the time complexity of iterative-deepening-A*

This paper establishes an upper bound on the time complexity of iterative-deepening-A^* (IDA^*) in terms of the number of states that are surely-expanded by A^* during a state space tree search. It is shown that given an admissible evaluation function, IDA^* surely-expands in the worst caseN(N+1)/2 states, whereN is the number of states that are surely-expanded by A^*. The conditions that give rise to the worst case performance of IDA^* on any state space tree are described. Worst case examples are also given for uniform and non-uniform state space trees.This work was supported in part by the Canadian Natural Sciences and Engineering Research Council Grant NSERC3599.     

A distribution-free geometric upper bound for the probability of error of a minimum distance classifier

An upperbound to the probability of error per class in a multivariate pattern classification is derived. The bound, given by

$\text{P(E|class w}{}_{\mathrm{i}}\text{)≤}\text{N}\text{R}{}^2{}_{\mathrm{i}}$

is derived with minimal assumptions; specifically the mean vectors exist and are distinct and the covariance matrices exist and are non-singular. No other assumptions are made about the nature of the distributions of the classes. In equation (i) N is the number of features in the feature (vector) space and R_i is a measure of the “radial neighbourhood” of a class. An expression for R_i is developed. A comparison to the multivariate Gaussian hypothesis is presented.     

An upper bound of structured singular value

An improved upper bound of structured singular value for mixed uncertainties with purely real uncertainty blocks is proposed.     

Improving feature extraction by replacing the Fisher criterion by an upper error bound

A lot of alternatives and constraints have been proposed in order to improve the Fisher criterion. But most of them are not linked to the error rate, the primary interest in many applications of classification. By introducing an upper bound for the error rate a criterion is developed which can improve the classification performance.     

An upper bound for minimizing coefficients of dimension Kolchin polynomial

In this paper, an upper bound for minimizing coefficients of the dimension Kolchin polynomial for a subset E ⊂ ℕ₀ ^m , which depends on the maximal order of elements in E, is obtained. The minimizing coefficients are always positive; some of them are invariant and play an important role in differential algebra. As an example of application of the result obtained, an estimate for a typical differential dimension of a system of partial differential equations is obtained in the case where the orders and degrees of the equations are bounded.