Deterministic Variation Research Articles

Correction for nonstationarity and window broadening in Doppler spectrum estimation

The width of the time-varying spectrum of signals from the pulsed Doppler ultrasonic blood flow detector is overestimated as a result of window and nonstationarity broadening when analyzed using the conventional short-term Fourier transform (STFT). By deriving the broadening function width from a consideration of signal phase variation arising from the deterministic spectral center-frequency variation during each window, the increase In the root-mean-squared (RMS) spectral width can be calculated and the spectral width estimation corrected. Using this procedure on simulated Doppler signals of known characteristics, a significant reduction in spectral width estimation error can be demonstrated.

Retrofocusing of a time reversing acoustic array in shallow-water random media

A time reversing or phase conjugate array operating in a random medium has been shown to focus acoustic radiation better than an equivalent array operating in a nonrefractive medium, making such arrays attractive for a large class of underwater acoustic problems. The present work extends this result to include the lifetime of the array’s focus, and shallow-water random refraction. Analytic predictions of focal properties, based on the parabolic approximation to the second moment of the Green’s function connecting the source with the array, are made for acoustic propagation through an unbounded dynamic random medium. The necessary projected autocorrelation function for the spatial and temporal evolution of the acoustic medium has been constructed from available oceanographic shallow-water fluctuation statistics. Focus size and focus lifetime are predicted to depend on acoustic frequency, source–array range, and array size. Focus lifetime predictions suggest that there is substantial opportunity for retrofocusing application of this technology. Influence of the ocean surface and deterministic water column variations will be addressed in future research.

Scheduling of batch processes with operational uncertainties

In this work, a mathematical programming model for optimal scheduling of the operations of a batch processing chemical plant is developed. The model is capable to handle all possible deterministic variations in the set-up and operation times of batch operations, and the model is sufficiently general to include the uncertainties introduced by the probabilistic behavior of set-up and operation times of batches when such variations can be mathematically defined. It is shown that the probabilistic model can be reduced to a Mixed Integer Non-Linear Program (MINLP), once the probability distribution functions used to model the random variations are defined. The resulting MINLP is then linearized and solved for small examples using CPLEX 3.0 in order to obtain a schedule which maximizes the net expected operational profit.

A robust metric for measuring within-wafer uniformity

Within-wafer uniformity is traditionally measured by the signal-to-noise ratio (SNR), defined as the estimated standard-deviation of within-wafer measurements over the mean of those measurements, Unfortunately, in the presence of deterministic variations of the response over the wafer (such as the bull's eye effect of some processes), the SNR is sensitive to both the location and the number of the measurements taken, A robust metric for describing within-wafer uniformity is developed and compared with the SNR method. The new metric, termed the integration statistic (I) is shown to be robust to both the location and number of measurements taken on the wafer and has lower variance than the SNR metric. The implications of this robust behavior are that fewer measurements can be taken to achieve a given accuracy in the uniformity estimate and that uniformity estimates are consistent with respect to variations in the orientation of the uniformity pattern to the measurement pattern.

Dynamic operating policies for periodic processes subject to equipment failures

The role of storage in buffering a process train from the effects of periodically reoccuring equipment failure is studied. The effects of deterministic and stochastic variations in the failure frequency and recovery time on the required size of intermediate storage are investigated. A dynamic operating policy is proposed to accommodate process variations in concern with the use of intermediate storage. Dynamic operation offers an increase in the average production rate of the process while using only modest levels of intermediate storage.

Spurious deterministic seasonality

It is sometimes assumed that the R 2 of a regression of a first-order differenced time series on seasonal dummy variables reflects the amount of seasonal fluctuations that can be explained by deterministic variation in the series. In this paper we show that neglecting the presence of seasonal unit roots may yield spuriously high values of this coefficient.

荷重の確定的変動に対する複合材料の最適設計

荷重の確定的変動に対する複合材料の最適設計

Wave propagation in a randomly layered medium : Kohler, W; Papanicolaou, G; White, B Proc 6th ASCE Speciality Conference on Probabilistic Mechanics and Structural and Geotechnical Reliability, Denver, 8–10 July 1992P381–383. Publ New York: ASCE, 1992

The authors briefly outline a problem discussed at length in a paper called Frequency content of randomly scattered signals, by M. Ash, W. Kohler, G. Papanicolaou, M Postel and B. White. A randomly layered half space is illuminated by a pulsed acoustic source. The constitutive parameters of the layered region are assumed to vary on two very different spatial scales, a slow deterministic variation that models large-scale refractive effects and a rapid zero mean random variation that models (generally unknown) constitutive parameter fine structure.

The strong law of large numbers for martingales with deterministic quadratic variation

The strong law of large numbers is proved for multivariate martingales with deterministic quadratic variation, along the same lines as in Lai, Wei and Robbins (1979), though the setting here is more general

Nonquadratic cost and nonlinear feedback control

AbstractNonlinear controllers offer significant advantages over linear controllers in a variety of circumstances. Hence there has been significant interest in extending linear‐quadratic synthesis to nonlinear‐nonquadratic problems. The purpose of this paper is to review the current status of such efforts and to present, in a simplified and tutorial manner, some of the basic ideas underlying these results. Our approach focuses on the role of the Lyapunov function in guaranteeing stability for autonomous systems on an infinite horizon. Sufficient conditions for optimality are given in a form that corresponds to a steady‐state version of the Hamilton‐Jacobi‐Bellman equation. These results are used to provide a simplified derivation of the nonlinear feedback controller obtained by Bass and Webber (1966)38 and to obtain a deterministic variation of the stochastic nonlinear feedback controller developed by Speyer (1976).45.

Simple method to start and maintain self-mode-locking of a Ti:sapphire laser

By periodically moving one of the end mirrors of a Ti:sapphire laser using a shaker, we have been able to start and maintain self-mode-locking of the laser. The resulting laser is stable over a long period with low-amplitude noise (3.3%) and low random timing jitter (4.8 ps). In addition, there is a deterministic timing variation on the order of subnanoseconds, caused by the moving mirror, which can be reduced by employing a feedback system. Pulses as short as 43 fs have been obtained.

Chronological Influences of the Variance of Electric Power Production Costs

The cost of producing electric power at a given time depends on the demand and the set of generating units that are available. We present a Markovian model of the generation system together with a deterministic, time-varying demand function that yields a stochastic integral for the production cost over a time interval. The variance of this integral may be computed exactly by enumerating states. An expression for the integrand is developed in which the deterministic time variation is decoupled from the stochastic variation. This expression is amenable to an asymptotic approximation that reduces the computation. When system state transition rates are high, little accuracy is lost. The exact and approximate results are compared for several parameter values in a small example system.

Multiscale solutions for the high‐frequency propagators in an inhomogeneous background random medium

Ray trajectories, as has been shown in the recently formulated stochastic geometrical theory of diffraction (SGTD), play an important role in determining the propagation properties of high‐frequency wave fields and their paired measures. One of the major analytical building blocks in SGTD is the choice of an appropriate propagator to transport these measures. This work describes how a multiscale expansion asymptotic procedure is applied to solve the partial differential equation governing the propagation of the paired field function along geometrical ray trajectories in an inhomogeneous background medium. A similar strategy is applied in the presence of a weak random refractive index component perturbing the deterministic background variation. Assuming a single realization of the random part of the refractive index, the multiscale expansion can be applied in the same way as in the purely deterministic case. This leads to a three‐dimensional solution for the high‐frequency propagator of a paired field function, which, apart from the spectral information required for uniformization in the transition regions, contains the information about the random refractive index variation along the propagation path. The last feature makes them suitable for application to the double passage problems in which the forward–backward correlation has to be taken into account. For a homogeneous background medium, ray trajectories are straight lines and the corresponding propagators take a simple analytic form. Analysis of intensity enhancement effects is performed for several source–reflector configurations.

A nonstatistical approach to estimating confidence intervals about model parameters: application to respiratory mechanics.

Estimates of parameters obtained by fitting models to physiologic data are of little use unless accompanied by confidence intervals. The standard methods for estimating confidence intervals are statistical, and make the assumption that the fitted model accounts for all the deterministic variation in the data while the residuals between the fitted model and the data reflect only stochastic noise. In practice, this is frequently not the case, as one often finds the residuals to be systematically distributed about zero. In this paper, we develop an approach for assessing confidence in a parameter estimate when the order of the model is clearly less than that of the system being modeled. Our approach does not require a parameter to have a single value located within a region of confidence. Instead, we let the parameter value vary over the data set in such a way as to provide a good fit to the entire data set. We apply our approach to the estimation of the resistance of the respiratory system in which a simple model is fitted to measurements of tracheal pressure and flow by recursive multiple linear regression. The values of resistance required to achieve a good fit are represented as a modified histogram in which the contribution of a particular resistance value to the histogram is weighted by the amount of information used in its determination. Our approach provides parameter frequency distribution functions that convey the degree of confidence one may have in the parameter, while not being based on erroneous statistical assumptions.

Reflection of pulsed electromagnetic waves from a randomly stratified half-space

We consider the reflection of temporally pulsed electromagnetic waves from a weakly dispersive and dissipative, randomly stratified half-space. The incident pulse width is chosen to be short relative to deterministic background variations in the constitutive parameters but broad relative to the rapid, fine-scale random fluctuations in these parameter values. We study both plane-wave excitation (normal and oblique incidences) and excitation by a point current source. We are interested in the coherent reflected field and the mutual coherence function of the reflected field at the surface. Explicit results are given in certain special cases. We also compare the theory with numerical simulations.

Comparison of deterministic and stochastic kinetics for nonlinear systems

The deterministic kinetics of chemical reactions is compared with a stochastic description for the cubic Schlögl model with a single stable steady state, which has a nonlinear reaction mechanism. We solve numerically the birth-death master equation for this system for various numbers of particles (N=20–160). For small systems with tens of particles the deviation of the first moment of the stochastic distribution from the deterministic temporal variation of concentration can be substantial in the initial relaxation towards a stationary state. The relaxation of the master equation is faster than that of the deterministic equation. The maximum deviation in trajectories decreases as the parameters in the kinetic model are altered towards a linear mechanism. The maximum deviation differs from N1/2 as N decreases, but approaches N1/2 as N increases. Deviations from deterministic temporal evolution due to fluctuations depend on the extent of nonlinearity of the reaction. The variance of a stationary distribution of the master equation is shown to be significantly larger than the average for a nonlinear system.

Using spatial information to analyze correlations between test structure data (semiconductor IC manufacture)

A modeling strategy is presented that captures the dependence of performance on the spatial position of the chips on the water. The information from this model can be used to determine whether the variance and correlation of parameters are due to either random variation or deterministic function of wafer position. The modeling strategy covers deterministic variations of the mean as a function of wafer position. The authors provide a method to determine the amount of correlation which is due to the common spatial dependence (referred to as spatial correlation). When coupled with knowledge about the manufacturing process, the diagnosis system can determine the physical reasons for the yield loss. The problem is formalized and a solution is developed. Extensions to this model are discussed.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Interpretation of cyclic flow variations in motored internal combustion engines

Cyclic variations and turbulence characterisation are considered in the context of in-cylinder flows. Three methods of data analysis, namely ensemble averaging, high pass filtering and cycle-by-cycle smoothing, are applied to velocity data obtained by laser Doppler velocimetry in a model axisymmetric and a realistic engine configuration. The interpretation of the results is supported by similar analysis of a steady-state simulation of in-cylinder flow with emphasis on the discrimination between random and deterministic flow variations. The results suggest that it is not always possible to identify cyclic variations as deterministic processes and that turbulence estimates based on filtering techniques may be misleading. The differences in the estimates of mean values are, however, small.

Intermediate storage and operation of batch processes under batch failure

The role of intermediate storage in buffering a process train from the effects of periodically reoccuring losses of batches of an intermediate material caused by failure to meet quality specifications is studed. For the building block 1-1 system, simple analytical expressions are obtained for the limiting volume of intermediate storage as a function of the frequency of batch failure by using Fourier series expressions. The effects of both deterministic and stochastic variations in the failure frequency and recovery time on the required size of intermediate storage are investigated. A dynamic operating policy is proposed which reduces the need for excess storage by switching the discharge rate between the nominal or a throttled value depending upon system conditions.

Designs for Disc Permeameters

AbstractDisc permeameters are designed to measure hydraulic properties of field soils containing macropores and preferential flow paths and are particularly useful in soil management studies. We present here designs for disc permeameters for both positive and negative water supply heads. The effects of the water supply membrane and soil contact material on permeameter performance are examined using approximate quasi‐analytic solutions to the flow equation. This analysis provides approximate criteria for the selection of membrane and soil contact materials. Limitations to performance caused by restricted air entry are considered and design criteria are given also. We present in situ tests of the disc permeameter for the early stages of one‐dimensional infiltration and an example of the deterministic variation of sorptivity of a field soil with supply potential. Finally, we use ponded and unsaturated sorptivities measured in situ with disc permeameters to find the saturated hydraulic conductivity and flow‐weighted mean characteristic pore dimension of a field soil.