Conditional Mean Estimator Research Articles

Comparison of Gaussian Conditional Mean and Kriging Estimation in the Geostatistical Solution of the Inverse Problem

Two separate applications of the geostatistical solution to the inverse problem in groundwater modeling are presented. Both applications estimate the transmissivity field for a two‐dimensional model of a confined aquifer under steady flow conditions. The estimates are based on point observations of transmissivity and hydraulic head and also on a model of the aquifer which includes prescribed head boundaries, leakage, and steady state pumping. The model used to describe the spatial variability of the log‐transmissivity describes large‐scale fluctuations through a linear mean or drift intermediate and small‐scale fluctuations through a two‐parameter covariance function. The first application presented estimates the log‐transmissivities using Gaussian conditional mean estimation. The second application uses an extended form of cokriging. The two methods are compared and their relative merits discussed. The extended cokriging application is applied to the Jordan Aquifer of Iowa. A comparison is also made between the conditional mean application and an analytical approach.

Optimum mean-square error use of convolutional codes

The k input and output digits of a rate (k/n) linear convolutional code over a finite field GF (q) are related to a finite set of integers by a q -ary expansion. The mean-square error criterion is used to simultaneously select the optimum encoder and decoder rules. This optimization is performed over all one-to-one generalized encoding rules and all decoding functions that map into the real numbers. The optimal design procedure relies upon generalized Fourier transforms, and it is shown that the encoder part of the optimum pair of rules can be taken as a linear function when the input space of symbols is viewed in a natural algebraic setting. The decoder part is a conditional mean estimator coupled with a rounding operation. One method of implementing the decoder uses the nonlinear combination of filter functions defined in the generalized frequency domain.

Accumulation of distortion in tandem communication links

The mean-square errors (mse) of individual links of a communication network generally do not add up to the overall mse because of mutual correlations among the errors on the individual links. This makes it difficult to interpret the magnitude of contribution of each link to the mse. It is shown that if the mse of each link is defined alternatively as the mse between conditional mean estimators of the original signal at the input and output of that link, then the overall mse is in fact the sum of the mse's on individual links. Several examples illustrate the usefulness of this principle, particularly in digital communications, in understanding the source of system impairments.

Optimum symbol-by-symbol mean-square error channel coding

An <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">(n,k)</tex> linear block code is used to convey <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k</tex> information digits, each having a numerical value attached, in such a way as to minimize the mean-square error of the individual information digits regardless of the other <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">(k-1)</tex> positions. This criterion is used because it provides a graceful degradation in the coding performance when the channel temporarily becomes less reliable. Unambiguous encoding is employed, and the channel is assumed to obey a realistic additive property. An optimum pair of symbol encoding and decoding functions are derived, and it is shown that the optimum symbol encoding can be performed in a linear fashion. The related decoding role is the conditional mean estimator. Fourier transforms are central to the optimal design of both rules. The optimum symbol rules tire identical with the properly isolated parts of the optimum mean-square error rules when the code is used to carry the k Information digits considered as a single complete numerical entity, The same channel coding design is therefore optimum for a wide variety of Information configurations. Techniques for implementing the optimum symbol decoders are examined with special emphasis on memoryless channels where the storage requirements for the necessary weighting coefficients are drastically reduced.

Optimal nonlinear estimation for a class of discrete-time stochastic systems

Recursive estimation for nonlinear discrete-time stochastic systems with additive white Gaussian observation noise is investigated. It is proved that for certain classes of systems, described either by finite Volterra series expansions or by state-linear realizations under certain algebraic conditions, the optimal conditional mean estimator is recursive and of fixed finite dimension. An example is presented to illustrate the structure of the estimators.

A generalized likelihood ratio formula: Arbitrary noise statistics for doubly composite hypotheses (Corresp.)

A relationship between the likelihood ratio and a generalized causal conditional mean estimator is presented for the doubly composite hypotheses problem. The observation statistics are arbitrary and need not be Gaussian. The relationship parallelsthe well-known relationship for the Gaussian noise and composite signal hypothesis case, and the likelihood ratio can still be viewed as an estimator-correlator operation.

The estimator-correlator for discrete-time problems

A class of discrete-time detection problems is formulated and a general recursive formula for the likelihood ratio is obtained. The result is analogous to the general likelihood-ratio formula or estimator-correlator receiver which has been derived in continuous time for a number of detection problems. As in the continuous-time versions, a conditional-mean estimate again plays a central role in the receiver structure.

Conditional mean estimates and Bayesian hypothesis testing (Corresp.)

For conditional probability density functions (pdf's) drawn from the exponential family, it is shown that the marginal pdf is completely determined by a posterior conditional mean estimate (CME). This result implies that likelihood ratios involving these marginals have the estimator-correlator structure in the following sense: if the noise is drawn from an exponential pdf, then independent of the signal (prior pdf), the optimum detector correlates the estimate with the data. A generalization of Esposito's result on "pseudoestimates" is also given.

Stochastic control with continuously variable observation costs for a class of discrete nonlinear systems

The discrete stochastic linear servomechanism problem is reformulated to include the costs of observations. The optimal control solution to this modified problem is identical to that of the standard servomechanism problem, but the optimal observations policy requires the solution of a TPBVP which may be computed a priori. The problem is further extended to include the class of nonlinear systems suggested by dynamic populations. The solution presented for the nonlinear problem is an iterative solution involving the second-order conditional-mean estimation and prediction algorithms and the linear control algorithms which are applied about the linearized predicted trajectories. A nonlinear example problem is included.

Approximate estimation for systems with quantized data

Estimation of the state of a nonlinear discrete-time system using quantized data is considered. An exact solution for the maximum likelihood estimate is expressed as the solution of a nonlinear two-point boundary-value problem. Approximate recursive solutions for both the maximum likelihood and the conditional-mean estimates are obtained. The results of Monte-Carlo simulations are presented in which the performance of these two algorithms is compared with that of a Kalman filter in which the quantization error is approximated by white noise.

Nonlinear sequential algorithms for estimation under uncertainty

In many realistic estimation problems, the presence of the state vector to be estimated is uncertain. In this instance, use of the linear sequential conditional mean filter algorithms (the Kalman filter) results in suboptimum performance. This paper derives nonlinear sequential filter algorithms for conditional mean estimation of a Gauss Markov process when there is uncertainty as to the presence of the Gauss Markov process (signal process) in the observation. Associated estimation error variances are determined and a simple example is considered.

Optimal minimal-order observer-estimators for discrete linear time-varying systems

This paper generalizes and unifies the concepts developed by Kalman and Luenberger pertaining to the design of discrete linear systems which estimate the state of a linear plant on the basis of both noise-free and noisy measurements of the output variables. Classes of minimal-order optimum observer-estimators are obtained which yield the conditional mean estimate of the state of the dynamical system. One explicit minimal-order optimal observer-estimator is constructed which generates one version of the conditional mean state estimate.

On the Calculation of Mutual Information

Abstract : Calculating the amount of information about a random function contained in another random function has important uses in communication theory. An expression for the mutual information for continuous time random processes has been given by Gelfand and Yaglom, Chiang, and Perez by generalizing Shannon's result in a natural way. Under a condition of absolute continuity of measures the continuous time expression has the same form as Shannon's result. For two Gaussian processes Gelfand and Yaglom express the mutual information in terms of a mean square estimation error. We generalize this result to diffusion processes and express the solution in a different form which is more naturally related to a corresponding filtering problem. We also use these results to calculate some information rates.

Computational algorithms for discrete detection and likelihood ratio computation

A pseudo Bayes approach to likelihood ratio determination for discrete time random processes imbedded in Gaussian noise is presented. The resulting computationally feasible algorithm for the likelihood ratio is expressed as a function of the one-step prediction conditional mean estimate of the message, suggesting an estimator-correlator type structure for the optimum digital detector. The use of smoothed or iterated conditional mean estimates in the pseudo Bayes detection scheme is also considered. The likelihood ratio formulas for continuous random signals in Gaussian noise are derived from the discrete results as a limiting case. Examples indicate the efficacy of the method.