基于岭估计的化工过程测量置信水平研究(英文)

Ordinary least squares （OLS） algorithm is widely applied in process measurement, because the sensor model used to estimate unknown parameters can be approximated through multivariate linear model. However, with few or noisy data or multi-collinearity, unbiased OLS leads to large variance. Biased estimators, especially ridge es-timator, have been introduced to improve OLS by trading bias for variance. Ridge estimator is feasible as an esti-mator with smaller variance. At the same confidence level, with additive noise as the normal random variable, the less variance one estimator has, the shorter the two-sided symmetric confidence interval is. However, this finding is limited to the unbiased estimator and few studies analyze and compare the confidence levels between ridge estima-tor and OLS. This paper derives the matrix of ridge parameters under necessary and sufficient conditions based on which ridge estimator is superior to OLS in terms of mean squares error matrix, rather than mean squares error. Then the confidence levels between ridge estimator and OLS are compared under the condition of OLS fixed sym-metric confidence interval, rather than the criteria for evaluating the validity of different unbiased estimators. We conclude that the confidence level of ridge estimator can not be directly compared with that of OLS based on the criteria available for unbiased estimators, which is verified by a simulation and a laboratory scale experiment on a single parameter measurement.     

Exactly/Nearly Unbiased Estimation of Autocovariances of a Univariate Time Series With Unknown Mean

This article proposes an exactly/nearly unbiased estimator of the autocovariance function of a univariate time series with unknown mean. The estimator is a linear function of the usual sample autocovariances computed using the observed demeaned data. The idea is to stack the usual sample autocovariances into a vector and show that the expectation of this vector is a linear combination of population autocovariances. A matrix that we label, A , collects the weights in these linear combinations. When the population autocovariances of high lags are zero (small), exactly (nearly) unbiased estimators of the remaining autocovariances can be obtained using the inverse of upper blocks of the A matrix. The A ‐matrix estimators are shown to be asymptotically equivalent to the usual sample autocovariance estimators. The A ‐matrix estimators can be used to construct estimators of the autocorrelation function that have less bias than the usual estimators. Simulations show that the A ‐matrix estimators can substantially reduce bias while not necessarily increasing mean square error. More powerful tests for the null hypothesis of white noise are obtained using the A ‐matrix estimators.     

基于有偏估计的数据融合提高双传感器测量可靠性

为了提高测量数据可靠性,过程控制领域广泛采用双冗余传感器测量生产状态信息,而目前处理双传感器测量数据常用的方法为线性无偏估计融合理论。通过估计理论得到的测量数据同时涉及方差和偏差两个统计特性。由于无偏测量数据具有无偏性的优良性质,所以测量方差能够全面地描述无偏测量数据的可靠性,然而不能由此认为无偏测量数据一定具有高可靠性。为了进一步提高双传感器测量数据的可靠性,提出有偏估计数据融合方法。首先,证明了有偏测量能够改善单传感器测量数据的可靠性;其次,采用凸组合方法推导了双传感器有偏估计融合表达式;最后,证明了有偏估计融合的均方误差小于任意单传感器的均方误差。仿真分析与实例应用均表明有偏估计数据融合可以有效地提高双传感器测量数据的可靠性。     

Sequential estimation in the group testing problem

Estimation using pooled sampling has long been an area of interest in the group testing literature. Such research has focused primarily on the assumed use of fixed sampling plans (i), although some recent papers have suggested alternative sequential designs that sample until a predetermined number of positive tests (ii). One major consideration, including in the new work on sequential plans, is the construction of debiased estimators that either reduce or keep the mean square error from inflating. However, whether under the above or other sampling designs unbiased estimation is in fact possible has yet to be established in the literature. In this article, we introduce a design that samples until a fixed number of negatives (iii), and show that an unbiased estimator exists under this model, whereas unbiased estimation is not possible for either of the preceding designs (i) and (ii). We present new estimators under the different sampling plans that are either unbiased or that have reduced bias relative to those already in use as well as generally improve on the mean square error. Numerical studies are done in order to compare designs in terms of bias and mean square error under practical situations with small and medium sample sizes.     

Estimating reaction model parameter uncertainty with Markov Chain Monte Carlo

Predicting the performance of chemical reactions with a mechanistic model is desired during the development of pharmaceutical and other high value chemical syntheses. Model parameters usually must be regressed to experimental observations. However, experimental error may not follow conventional distributions and the validity of common statistical assumptions used for regression should be examined when fitting mechanistic models.This paper compares different techniques to estimate parameter confidence for reaction models encountered in pharmaceutical manufacturing, simulated with either normally distributed or experimentally measured noise. Confidence intervals were calculated following standard linear approaches and two Markov Chain Monte Carlo algorithms utilizing a Bayesian approach to parameter estimation: one assuming a normal error distribution, and a new non-parametric likelihood function. While standard frequentist approaches work well for simpler nonlinear models and normal distributions, only MCMC accurately estimates uncertainty when the system is highly nonlinear, and can account for any measurement bias via customized likelihood functions.     

Error Propagation Biases in the Calculation of Indentation Fracture Toughness for Ceramics

Indentation fracture toughness models generally share the derived parameter Pc ^−3/2, where P is the indentation load and c the measured crack length. Biases, inherent to error propagation through this nonlinear transformation ( c to c ^3/2), can be introduced into calculated values for K _{I C} , depending upon the amount of averaging of crack length data performed prior to the transformation. This work utilizes Monte Carlo simulations to evaluate the bias in K _{I C} calculated using both mean and linear regression methods. Significant positive biases were demonstrated when using mean-based calculations where coefficients of variation (cv) in c exceeded 10%. Regression methods produced significantly less bias. With cv < 10% or when c is averaged per load, both methods produce essentially unbiased estimates for K _{I C} .     

Simultaneous Measurement Bias Correction and Dynamic Data Reconciliation

The presence of measurement bias and random noise significantly deteriorates the information quality of plant data. Data reconciliation techniques for steady-state processes have been widely applied to processing industries to improve the accuracy and precision of the raw measurements. This paper develops an algorithm for simultaneous bias correction and data reconciliation for dynamic processes. The algorithm considers process model error as an important contributing factor in the estimation of the measurement bias and process state variables. It employs black-box models for the process as would be done when phenomenological models are difficult or impractical to obtain. Simulation results of a distillation column demonstrated that this algorithm effectively compensates constant and non-constant measurement biases yielding much improved reconciled values of process variables. It has computational advantages over previously proposed algorithms based on non-linear dynamic data reconciliation because an analytical solution is available when using linear process models to approximate the process.     

Randomization-based nonparametric methods for the analysis of multicentre trials

Multicentre trials offer several advantages over single centre trials in clinical research, including the ability to recruit patients at a faster rate over the course of the study, increased generalizability through the use of a broader patient population, and the ability to shed light on the replication of findings at multiple centres in a single study. A nonparametric approach to the analysis of multicentre trial data provides a convenient way for addressing the role of centres as well as baseline covariables during data analysis. With the use of randomization-based nonparametric methods, the strategy for evaluating the null hypothesis of no treatment effect can be prespecified during study planning without requiring a specific structure for the relationship of response criteria (or endpoints) to centres, covariables, or potential interaction terms. Further, the basis of inference for the application of these methods is the randomization mechanism, and the population to which inference can be directly made is the study population itself. No assumptions about underlying distributions, data structures, likelihood functions, or samples from super populations of inference are required. A three-step approach is proposed for handling centres via randomization-based nonparametric methods. In Step 1, a test of overall treatment effect is carried out using data from all centres simultaneously, without any assumption about treatment by centre interaction. In Step 2, the question of treatment by centre interaction is addressed, usually through the use of parametric multiple regression methods. In cases with suggestion of such interaction, Step 3 is conducted to evaluate different weighting schemes in forming pairwise treatment comparisons averaged across centres to assess the robustness of treatment effects observed in Step 1. An attractive inferential feature of this three-step approach is that the Type I error for the test of treatment effect is controlled by requiring statistical significance at each step to proceed to the next step. Extended Mantel-Haenszel methods with stratification adjustment for centre can be used to provide a nonparametric assessment of treatment effect. When adjustment for other covariates, such as baseline values, is desired, the more recent nonparametric analysis of covariance methods are available. Both methods are easy to use, require no assumptions beyond that of a valid randomization mechanism, and can be applied in a similar manner to dichotomous, ordinal, failure time, or continuous response criteria (endpoints). The methods are illustrated using data from a confirmatory clinical trial of a therapeutic agent for the treatment of dry eye disease.     

A Nonparametric Prewhitened Covariance Estimator

This paper proposes a new nonparametric spectral density estimator for time series models with general autocorrelation. The conventional nonparametric estimator that uses a positive kernel has mean squared error no better than n^?4/5. We show that the best implementation of our estimator has mean squared error of order n^?8/9, provided there is sufficient smoothness present in the spectral density. This is, of course, achieved by bias reduction; however, unlike most other bias reduction methods, like the kernel method with higher‐order kernels, our procedure ensures a positive definite estimate. Our method is a generalization of the well‐known prewhitening method of spectral estimation; we argue that this can best be interpreted as multiplicative bias reduction. Higher‐order expansions for the proposed estimator are derived, providing an improved bandwidth choice that minimizes the mean squared error to the second order. A simulation study shows that the recommended prewhitened kernel estimator reduces bias and mean squared error in spectral density estimation.     

The Use of Simplified or Misspecified Models: Linear Case

Simplified models have many appealing properties and sometimes give better parameter estimates and model predictions, in sense of mean‐squared‐error, than extended models, especially when the data are not informative. In this paper, we summarize extensive quantitative and qualitative results in the literature concerned with using simplified or misspecified models. Based on confidence intervals and hypothesis tests, we develop a practical strategy to help modellers decide whether a simplified model should be used, and point out the difficulty in making such a decision. We also evaluate several methods for statistical inference for simplified or misspecified models.     

ON THE UNBIASEDNESS PROPERTY OF AIC FOR EXACT OR APPROXIMATING LINEAR STOCHASTIC TIME SERIES MODELS

Abstract. A rigorous analysis is given of the asymptotic bias of the log maximum likelihood as an estimate of the expected log likelihood of the maximum likelihood model, when a linear model, such as an invertible, gaussian ARMA ( p, q ) model, with or without parameter constraints, is fit to stationary, possibly non-gaussian observations. It is assumed that these data arise from a model whose spectral density function either (i) coincides with that of a member of the class of models being fit, or, that failing, (ii) can be well-approximated by invertible ARMA ( p, q ) model spectral density functions in the class, whose ARMA coefficients are parameterized separately from the innovations variance. Our analysis shows that, for the purpose of comparing maximum likelihood models from different model classes, Akaike's AIC is asymptotically unbiased, in case (i), under gaussian or separate parametrization assumptions, but is not necessarily unbiased otherwise. In case (ii), its asymptotic bias is shown to be of the order of a number less than unity raised to the power max { p, q } and so is negligible if max { p, q } is not too small. These results extend and complete the somewhat heuristic analysis given by Ogata (1980) for exact or approximating autoregressive models.     

Improved prediction limits for a general class of Gaussian models

In this article we consider the problem of prediction for a general class of Gaussian models, which includes, among others, autoregressive moving average time‐series models, linear Gaussian state space models and Gaussian Markov random fields. Using an idea presented in Sjöstedt‐De Luna and Young (2003) , in the context of spatial statistics, we discuss a method for obtaining prediction limits for a future random variable of interest, taking into account the uncertainty introduced by estimating the unknown parameters. The proposed prediction limits can be viewed as a modification of the estimative prediction limit, with unconditional, and eventually conditional, coverage error of smaller asymptotic order. The modifying term has a quite simple form and it involves the bias and the mean square error of the plug‐in estimators for the conditional expectation and the conditional variance of the future observation. Applications of the results to Gaussian time‐series models are presented.     

Selection of weak VARMA models by modified Akaike's information criteria

This article considers the problem of order selection of the vector autoregressive moving‐average (VARMA) models under the assumption that the errors are uncorrelated but not necessarily independent. These models are called weak VARMA by opposition to the standard VARMA models, also called strong VARMA models, in which the error terms are supposed to be i.i.d. We relax the standard independence assumption to extend the range of application of the VARMA models, allowing us to treat linear representations of general nonlinear processes. We propose a modified version of the Akaike information criterion for identifying the orders of weak VARMA models.     

A method of calculating densities of polymers

Molar volumes of substantially amorphous polymers may be calculated by using additive constants derived from constitutive atomic and structural parachor contributions. For 34 polymers, ranging in density from 0.83 to 2.03 g/cm³, the greatest error in the calculated molar volume compared with the measured value was 13.9%, while for 30 of the polymers the error was less than ±7%, with a mean error of ±3.9%.     

Artificial and Hybrid Fuzzy Linear Neural Network‐Based Estimation of Seed Oil Content of Safflower

Inexpensive and rapid methods for measurement of seed oil content by near infrared reflectance spectroscopy (NIRS) are useful for developing new oil seed cultivars. Adopting default multiple linear regression (MLR), the predictions of safflower oil content were made by 20–140 samples using a Perten Inframatic 8620 NIR spectrometer. Although the obtained interpolation results of MLR had desired accuracy, the extrapolation was extremely poor. The extrapolation determination coefficient (R²) and standard error (SE) of cross validation for MLR models were 0.63–0.78 and 3.71–4.44, respectively. In order to overcome the accuracy limitation of linear MLR models, a common suggestion is to use a nonlinear artificial neural network (ANN); however, it needs a large number of data to yield significant accurate results. We developed a novel robust hybrid fuzzy linear neural (HFLN) network to capture simultaneously linear and nonlinear patterns of data with a limited number of safflower samples. Empirical extrapolation results showed that the HFLN had higher R² (=0.85) and lower SE (=1.83) compared to those obtained by MLR and ANN models. It is concluded that hybrid methodologies could be used to construct efficient and appropriate models for estimation of seed oil content set up on NIR system.     

Multivariate prediction of eight kerosene properties employing vapour-phase mid-infrared spectrometry

Eight physico-chemical properties of kerosene (aviation jet fuel) are predicted employing vapour-phase generation, Fourier transform mid-infrared (FT-MIR) spectra and partial least squares regression (PLS). Two devices were implemented and studied in order to generate the kerosene vapour from 100 liquid samples from a Spanish refinery. One of them is very simple whilst the other one requires thermostatic and gas flow controls. The FT-MIR spectra are recorded and used to deploy PLS models for each property (distillation curve, flash point, freezing point, percentage of aromatics and viscosity) and each device. In general, the simplest device yields the more satisfactory models. Several criteria are used to evaluate their performance: the average prediction error (corrected to take into account the error in the reference values), the F-test to assess the absence of bias in the predictions, repeatability and reproducibility. In general, all the models provide unbiased predictions, with low average errors and good precision.     

On the Optimal Segment Length for Parameter Estimates for Locally Stationary Time Series

We discuss the behaviour of parameter estimates when stationary time series models are fitted locally to non-stationary processes which have an evolutionary spectral representation. A particular example is the estimation for an autoregressive process with time-varying coefficients by local Yule–Walker estimates. The bias and the mean squared error for the parameter estimates are calculated and the optimal length of the data segment is determined.     

Non‐Linear Model Predictive Control: A Personal Retrospective

An overview of non‐linear model predictive control (NMPC) is presented, with an extreme bias towards the author's experiences and published results. Challenges include multiple solutions (from non‐convex optimization problems), and divergence of the model and plant outputs when the constant additive output disturbance (the approach of dynamic matrix control, DMC) is used. Experiences with the use of fundamental models, multiple linear models (MMPC), and neural networks are reviewed. Ongoing work in unmeasured disturbance estimation, prediction and rejection is also discussed.     

Control of the linear density of fibres

The error of measurement of the linear density of cord PCA fibre by a standard method of measurement and using a system of the ASKN type was estimated and analyzed. Control of the linear density of a fibre using the ASKN system is more accurate, effective, and informative than the standard method of measurement. Recommendations are made on calibration and testing the ASKN system using simulators and a production control variant using a portable instrument with system functions is proposed. The proposed method of estimating and analyzing errors of measurement can also be used for controlling other properties of PCA fibres and operating conditions, as well as for other types of fibres. __________ Translated from Khimicheskie Volokna, No. 1, pp. 57–60, January–February, 2006.