There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

The Theory of Stochastic Processes. By D. R. Cox and H. D. Miller. Pp. x, 398. 70s. (Methuen)

The theory and applications of integral equations is an very important subject within applied mathematics. Integral equations are used as mathematical models for many and varied physical situations but the integral equations also occur as reformulations of other mathematical problems.

https://link.springer.com/content/pdf/10.1007%2F978-94-011-5338-6_4.pdf

Numerical Treatment of the Integral Equations

Calculation of definite integrals by the fundamental formula of integral calculus, 

$$\int_{a}^{b}{f\left( x \right)dx\,=\,F\left( b \right)\,-\,F\left( a \right)},$$

where f(x), let us say, is a continuous function on [a, b] and F(x) is its primitive function, is made difficult by the fact that the actual determination of F(x) is possible only in rare cases. For this reason, formulas for the approximate calculation of integrals are of great significance. In this chapter, we shall become acquainted with the most important of them. Many formulas for the approximate calculation of definite integrals have the form 

$$\int_{a}^{b}{p\left( x \right)f\left( x \right)dx\,\cong \,\sum\limits_{k\,=\,1}^{n}{A_{k}^{\left( n \right)}f\left( x_{k}^{\left( n \right)} \right)}}$$

(1.1)

and are called mechanical quadrature formulas. The sum on the right hand side of (1.1) is called the quadrature sum. The numbers x k (n) belong to the interval [a, b], and are called the knots of the quadrature formula, and the numbers A k (n) are the coefficients of the quadrature formula. We shall always consider the knots of the quadrature formula to be numbered in increasing order: 

$$x_{1}^{\left( n \right)}\,<\,x_{2}^{\left( n \right)}\,<\,\ldots \,<\,x_{n}^{\left( n \right)}.$$

The interval of integration [a, b] can also be infinite. The integrand is written in the form of the product of the two functions p(x) and f(x). The first of these, p(x), is assumed to be fixed for the given formula (1.1) and is called the weight function. The function f(x) belongs to some sufficiently wide class of functions, for example, continuous and such that the integral on the left hand side of (1.1) exists.

Approximate Calculation of Integrals

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Handbook Of Integral Equations

Joint decision and estimation (JDE) is for solving problems involving inter-dependent decision and estimation and was proposed recently based on a generalized Bayes risk. The currently available JDE algorithm processes data in a batch manner. This batch method is computationally inefficient or infeasible for many dynamic JDE problems where measurements are made available sequentially. Therefore, this paper follows the same JDE framework based on the generalized Bayes risk and proposes a recursive version of the JDE algorithm, which fits the dynamic JDE problems more naturally and inherits JDE's theoretical superiorities. Further, a joint performance measure in the measurement space is proposed for dynamic JDE problems. To the authors' knowledge, this is the only performance evaluation framework currently available for JDE evaluation. Finally, an illustrative example of the recursive JDE algorithm is elaborated and numerical simulations comparing it with the traditional two-stage algorithms (i.e., decide-then-estimate and estimate-then-decide) are presented.

Recursive joint decision and estimation based on generalized Bayes risk

Nonlinearity, among other factors, is often the root cause of difficulties in nonlinear problems. It is important to quantify a problem's degree of nonlinearity to decide a proper solution. For example, a full-blown nonlinear filter is needed in general if the estimation problem is highly nonlinear, but a quasi-linear filter (e.g., an extended Kalman filter) is sufficient for a weakly nonlinear case. This paper first surveys various measures of nonlinearity (MoNs) for different applications. For nonlinear estimation, we conclude that these MoNs are not suitable and a better measure is needed. In view of this, we propose a general MoN for estimation. It measures the mean-square closeness between a point and a subspace in a functional space. Properties and computation of this measure are studied. Numerical examples of static models for parameter estimation and dynamic models for process estimation are given to illustrate our measure.

Measure of Nonlinearity for Estimation

The operating characteristic (OC) and average sample number (ASN) of the sequential probability ratio test (SPRT) and multi-hypothesis SPRT (MSPRT) are studied. We consider the case where the observation sequence is independent but not necessarily identically distributed. Also, the thresholds for the test can be time varying. Based on the governing equations for OC and ASN of the SPRT developed in our previous work, a solution for the general case is proposed. The governing equations for OC and ASN of the MSPRT are also obtained. Numerical solutions for MSPRT are developed. Basically, the solutions rely on approximating the original test by truncation, that is, truncating the test at some finite time $K$ . We show that under some mild conditions, the approximation error diminishes as $K$ increases, at the cost of increased computation. Numerical examples are provided to demonstrate our solutions by comparing with Monte Carlo simulations, Simon’s lower bound, and Dragalin’s method (if available) for ASN.

Operating Characteristic and Average Sample Number of Binary and Multi-Hypothesis Sequential Probability Ratio Test

The linear minimum mean-square error (LMMSE) estimation plays an important role in nonlinear estimation. Generalized LMMSE (GLMMSE) estimation is proposed in this work. LMMSE estimation finds the best estimator in the set of all estimators that are linear in the data. We extend this candidate set in GLMMSE estimation by employing a vector-valued function of the data and hence find the best one among all estimators that are linear in this function, rather than the data itself. The estimation performance may be enhanced since linear functions may not be adequate to provide good accuracy for a highly nonlinear problem. Theoretically speaking, GLMMSE estimation should perform at least as well as LMMSE estimation if the moments involved can be evaluated exactly. Unfortunately, similar to LMMSE estimation, those moments are difficult to evaluate analytically in general. Many numerical approximations for LMMSE estimation are also applicable to GLMMSE estimation. Computation of GLMMSE estimation based on the Gaussian-Hermite quadrature is presented, and its superior performance, compared with the unscented filter and the Gaussian filter, is demonstrated by several numerical examples.

Generalized linear minimum mean-square error estimation

Although methods for target maneuver onset detection (MOD) abound, few have been proposed for target maneuver termination detection (MTD), since MOD is easier and considered more important. Without MTD, however, decision based maneuvering target tracking is not complete and may suffer from performance degradation and computational waste. This paper proposes sequential testing methods for MTD in the multiple model framework. The celebrated CUSUM and SSPRT are adopted to detect maneuver termination, and the likelihood functions involved in these two tests are approximated by the outputs of the interacting multiple model (IMM) estimator, which is one of the most widely used algorithms for maneuvering target tracking. An Illustrative example is provided and the simulation results demonstrate the applicability of our algorithms.

Yu Liu

Papers

Recursive joint decision and estimation based on generalized Bayes risk

Measure of Nonlinearity for Estimation

Operating Characteristic and Average Sample Number of Binary and Multi-Hypothesis Sequential Probability Ratio Test

Generalized linear minimum mean-square error estimation

Sequential multiple-model detection of target maneuver termination