Dynamic Predictive Modeling Under Measured and Unmeasured Continuous-Time Stochastic Input Behavior

Many input variables of chemical processes have a continuous-time stochastic (CTS) behavior. The nature of these variables is a persistent, time-correlated variation that manifests as process variation as the variables deviate in time from their nominal levels. This work introduces methodologies in process identification for improving the modeling of process outputs by exploiting CTS input modeling under cases where the input is measured and unmeasured. In the measured input case, the output variable is measured offline, infrequently, and at a varying sampling rate. A method is proposed for estimating CTS parameters from the measured input by exploiting statistical properties of its CTS model. The proposed approach is evaluated based on both output accuracy and predictive ability several steps ahead of the current input measurement. Two parameter estimation techniques are proposed when the input is unmeasured. The first is a derivative-free approach that uses sample moments and analytical expressions for population moments to estimate the CTS model parameters. The second exploits the CTS input model and uses the analytical solution of the dynamic model to estimate these parameters. The predictive ability of the latter approach is evaluated in the same way as the measured input case. All of the data in this work were artificially generated under the probabilistic CTS model.Reprinted (adapted) with permission from Industrial and Engineering Chemistry Research 51 (2012): 5469, doi: 10.1021/ie201998b. Copyright 2012 American Chemical Society.</p

Recursive Identification of Wiener Systems

A Wiener system, i.e. a cascade system consisting of a linear dynamic subsystem and a nonlinear memoryless subsystem is identified. The a priori information is nonparametric, i.e. neither the functional form of the nonlinear characteristic nor the order of the dynamic part are known. Both the input signal and the disturbance are Gaussian white random processes. Recursive algorithms to estimate the nonlinear characteristic are proposed and their convergence is shown. Results of numerical simulation are also given. A known algorithm recovering the impulse response of the dynamic part is presented in a recursive form

Hermite series estimates of a probability density and its derivatives

The following estimate of the pth derivative of a probability density function is examined: [Sigma]k = 0Nâkhk(x), where hk is the kth Hermite function and âk = ((-1)p/n)[Sigma]i = 1n hk(p)(Xi) is calculated from a sequence X1,..., Xn of independent random variables having the common unknown density. If the density has r derivatives the integrated square error converges to zero in the mean and almost completely as rapidly as O(n-[alpha]) and O(n-[alpha] log n), respectively, where [alpha] = 2(r - p)/(2r + 1). Rates for the uniform convergence both in the mean square and almost complete are also given. For any finite interval they are O(n-[beta]) and O(n-[beta]/2 log n), respectively, where [beta] = (2(r - p) - 1)/(2r + 1).Hermite series orthogonal series density estimate