Estimation theory

E. J. Baum, Chemical Property Estimation Theory and Applications Lewis, Boca Raton (1998). 

Grimble, M.J. and Johnson, M.A. (1988) Optimal Control and Stochastic Estimation Theory and Application, Vols 1 and 2, John Wiley Sons, Chichester, UK. 

Irwan, R., Lane, R.G., 1999, Analysis of optimal centroid estimation applied to Shack-Hartmann sensing. Applied Optics 38, 6737 Kay, S.M., 1993, Fundamentals of statistical signal processing estimation theory, Prentice-Hall, New Jersey, pp 27-81... 

ESTIMABILITY AND REDUNDANCY WITHIN THE FRAMEWORK OF THE GENERAL ESTIMATION THEORY... 

In this chapter, the mathematical tools and fundamental concepts utilized in the development and application of modem estimation theory are considered. This includes the mathematical formulation of the problem and the important concepts of redundancy and estimability in particular, their usefulness in the decomposition of the general optimal estimation problem. A brief discussion of the structural aspects of these concepts is included. 

Throughout this chapter, we will refer to estimation in a very general sense. We will see later that data reconciliation is only a particular case within the framework of the optimal estimation theory. 

Deutsch, R. (1973). Estimation Theory. Prentice-Hall, Englewood Cliffs, NJ. 

Scott, D. W. (1992). Multivariate Density Estimation Theory Practice and Visualisation. Wiley, New... 

This chapter is structured as follows. Section 3.2 provides a refresher on some principles of distribution theory and estimation theory. The approach is didactic, and practical issues are put off until Section 3.3. Concepts such as skewness andkurtosis are reviewed, useful for characterizing and comparing different distribution types. Some special distributions are mentioned, which are possibly useful in enviromnen-tal risk assessment. 

One often encounters a distinction between precision and accuracy. Accuracy relates to systematic deviation between parameter estimates and actual parameter values precision relates to the spread in the distribution of estimates. This terminology is not often used explicitly in the estimation theory literature, but the concepts are often implicit. 

Baum EJ (1998) Chemical property estimation. Theory and application, CRC, Boca Ratton... 

Parameter estimation is rooted in several scientific areas with their own preferences and approaches. While linear estimation theory is a nice chapter of mathematical statistics (refs. 1-3), practical considerations are equally important in nonlinear parameter estimation. As emphasised by Bard (ref. 4), in spite of its statistical basis, nonlinear estimation is mainly a variety of computational algorithms which perform well on a class of problems but may fail on some others. In addition, most statistical tests and estimates of... 

Kay, 1993] Kay, S. M. (1993). Fundamentals of statistical signal processing Estimation theory. PM signal processing series. Prentice-Hall, Englewood Cliffs, NJ. 

Statistical estimation uses sample data to obtain the best possible estimate of population parameters. The p value of the Binomial distribution, the p value in Poison s distribution, or the p and a values in the normal distribution are called parameters. Accordingly, to stress it once again, the part of mathematical statistics dealing with parameter distribution estimate of the probabilities of population, based on sample statistics, is called estimation theory. In addition, estimation furnishes a quantitative measure of the probable error involved in the estimate. As a result, the engineer not only has made the best use of this data, but he has a numerical estimate of the accuracy of these results. 

The proposed approach combines macroscopic and elemental balances on the reactor with state-of-the-art adaptive estimation theories. Experiments and simulations show that estimates in excellent agreement with the true values can be obtained without using any growth models and both under transient and steady state operating conditions. 

In the parameter estimation theory it is generally assumed that the experimental errors arc normally distributed with zero mean and a constant variance parameter values can then be estimated by max-... 

Caracotsios, M., Model Parametric Sensitivity Analysis and Nonlinear Parameter Estimation Theory and Applications, Ph. D. thesis. University of Wisconsin-Madison (1986). 

Maximum entropy method is a powerful numerical technique, which is based on Bayesian estimation theory and is often applied to derive the most... 

Baum EJ. Chemical property estimation Theory and application. Boca Raton, FL CRC Press, 1998. 

Baum EJ. Chemical Property Estimation. Theory and Application, ASPEN MAX Release 1.1, Aspen Tech Europe. Boca Raton, FL Lewis Publishers, 1998. 

DW Scott. Multivariate Density Estimation Theory, Practice and Visualization. John Wiley Sons, New York, NY, 1992. 

There are several concepts for the description of phase in quantum theory at present. Some of them are accenting the theoretical aspects, other the experimental ones. Quantization based on the correspondence principle leads to the formulation of operational quantum phase concepts. For example, the well-known operational approach formulated by Noh et al. [63,64] is motivated by the correspondence principle in classical wave theory. Further generalization may be given in the framework of quantum estimation theory. The prediction may be improved using the maximum-likelihood estimation. The optimization of phase inference will be pursued in the following. 

First, let us show that the operational phase concepts can naturally be embedded in the general scheme of quantum estimation theory [66,67] as was done by Hradil, Zawisky, and others [68-71]. Let us consider the eight port homodyne detection scheme [63,72] with four output channels numbered by indices 3,4,5,6, where the actual values of intensities are registered in each run. Assume that these values fluctuate in accordance with some statistics. The mean intensities are modulated by a phase parameter 0... 

Figure 16. Asymptotic dispersions of the NFM estimator theory (solid line) and experimentally obtained values (squares). Asymptotic dispersions of the unconstrained ML estimator theory (dashed line) and experimentally obtained values (triangles). Experimentally obtained dispersions of the ML estimation on the physical space of parameters (circles). The corresponding input mean number of photons and the estimated visibility were N — 160 and V = 99.2%, respectively.

C. W. Helstrom, Quantum Detection and Estimation Theory, Academic, New York, 1976. 

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