递推局部线性回归估计及其应用

在非参数统计中,局部多项式回归是重要的工具,然而以往研究的算法基本都是非递推的.本文研究递推的局部线性回归估计及其应用.首先推导出递推算法,给出了回归函数及其导函数的非参数估计.在一定的条件下,证明了算法的强一致性.并且通过仿真例子研究了它在非线性条件异方差模型的回归函数估计和非线性ARX(nonlinear autoregressive system with exogenous inputs,NARX)系统辨识中的应用.     

Multi-objective parameter estimation via minimal correlation criterion

The paper deals with the problem of parameter estimation using two different sources of information, namely a time series with dynamic data and steady-state data. The new estimator is based on a two-step procedure: first a multi-objective optimization is performed, leading to a set of Pareto-optimal vectors of parameter estimates and, second, a single model is chosen based on the free-run simulation error which is required to be minimally correlated with the model output. The procedure is general in nature and can be applied to any model representation, but for the sake of simplicity, the new procedure is illustrated using NARX polynomial models for which closed formulae for generating the Pareto-set are readily available. Monte Carlo simulation studies suggest that the new estimator, which does not assume any particular noise model, is fairly unbiased even when the conventional least-squares estimator is biased.     

An identification algorithm for polynomial NARX models based on simulation error minimization

Classical prediction error approaches for the identification of non-linear polynomial NARX/NARMAX models often yield unsatisfactory results for long-range prediction or simulation purposes, mainly due to incorrect or redundant model structure selection. The paper discusses some limitations of the standard approach and suggests two modifications: namely, a new index, based on the simulation error, is employed as the regressor selection criterion and a pruning mechanism is introduced in the model selection algorithm. The resulting algorithm is shown to be effective in the identification of compact and robust models, generally yielding model structures closer to the correct ones. Computational issues are also discussed. Finally, the identification algorithm is tested on a long-range prediction benchmark application.     

An approach to polynomial NARX/NARMAX systems identification in a closed-loop with variable structure control

Many physical processes have nonlinear behavior which can be well represented by a polynomial NARX or NARMAX model. The identification of such models has been widely explored in literature. The majority of these approaches are for the open-loop identification. However, for reasons such as safety and production restrictions, open-loop identification cannot always be done. In such cases, closed-loop identification is necessary. This paper presents a two-step approach to closed-loop identification of the polynomial NARX/NARMAX systems with variable structure control （VSC）. First, a genetic algorithm （GA） is used to maximize the similarity of VSC signal to white noise by tuning the switching function parameters. Second, the system is simulated again and its parameters are estimated by an algorithm of the least square （LS） family. Finally, simulation examples are given to show the validity of the proposed approach.     

Identification of the rainfall–runoff relationship in urban drainage networks

The calibration of conceptual models for the design of urban drainage networks is an important and well-known problem in hydraulic engineering. In this paper the problem is analysed and the use of black-box identification methods is proposed and applied to experimental data. Both linear (ARX and state space) and nonlinear (polynomial and neural NARX) models are considered and their performance in the simulation and prediction of the network flow from rainfall measurements is evaluated.     

Real-time design of an adaptive nonlinear predictive controller

Based on real-time identification and using the concept of NARX (Nonlinear AutoRegressive with exogenous inputs) models, a new adaptive nonlinear predictive controller (ANPC) design is proposed. NARX models represent a natural way to describe the input-output relationship of severely nonlinear systems. From an initial batch of input-output data, a parsimonious NARX model is obtained using the Modified Gram-Schmidt (MGS) orthogonalization algorithm. Following this initial off-line identification and model reduction procedure, the control loop is closed. The ANPC directly uses the obtained structure and initial parameter estimates, which are updated each time step using recursive identification. The controller is designed similar to a typical linear predictive controller based on solving a nonlinear programming (NLP) problem. This paper shows how to solve this NLP problem on-line without the knowledge of the NARX model structure. The design is given for the multi-input multi-output (MIMO) case.     

基于NARX神经网络航空发动机参数动态辨识模型

针对航空发动机参数非线性动态特性,提出一种基于外部输入非线性自回归（NARX）神经网络的发动机参数动态辨识模型。主要思路是根据NARX网络的非线性时序预测特性,结合发动机参数的稳态和动态参数,提出一种基于偏稳态差值预测的NARX参数动态模型结构。设计了SP-P辨识结构,整定了模型内部结构参数并建立N1（低压转子转速）、N2（高压转子转速）、EGT（涡轮后排气温度）参数非线性差分预测模型。最后依据某发动机试车样本,对推杆加减速时N1、N2、EGT动态辨模型进行仿真。仿真结果表明,N2相对误差小于0.2%,N1相对误差小于0.3%,EGT相对误差小于[1℃],满足发动机试车仿真需要。最后,将所建模型应用于某A320机务维修训练器的发动机仿真系统。     

New bound characteristics of NARX model in the frequency domain

New results about the bound characteristics of both the generalized frequency response functions (GFRFs) and the output frequency response for the NARX (Non-linear AutoRegressive model with eXogenous input) model are established. It is shown that the magnitudes of the GFRFs and the system output spectrum can all be bounded by a polynomial function of the magnitude bound of the first order GFRF, and the coefficients of the polynomial are functions of the NARX model parameters. These new bound characteristics of the NARX model provide an important insight into the relationship between the model parameters and the magnitudes of the system frequency response functions, reveal the effect of the model parameters on the stability of the NARX model to a certain extent, and provide a useful technique for the magnitude based analysis of nonlinear systems in the frequency domain, for example, evaluation of the truncation error in a volterra series expression of non-linear systems and the highest order needed in the volterra series approximation. A numerical example is given to demonstrate the effectiveness of the theoretical results.     

Uncertainty quantification and apportionment in air quality models using the polynomial chaos method

Current air quality models generate deterministic forecasts by assuming perfect model, perfectly known parameters, and exact input data. However, our knowledge of the physics is imperfect. It is of interest to extend the deterministic simulation results with “error bars” that quantify the degree of uncertainty, and analyze the impact of the uncertainty input on the simulation results. This added information provides a confidence level for the forecast results. Monte Carlo (MC) method is a popular approach for air quality model uncertainty analysis, but it converges slowly. This work discusses the polynomial chaos (PC) method that is more suitable for uncertainty quantification (UQ) in large-scale models. We propose a new approach for uncertainty apportionment (UA), i.e., we develop a PC approach to attribute the uncertainties in model results to different uncertainty inputs. The UQ and UA techniques are implemented in the Sulfur Transport Eulerian Model (STEM-III). A typical scenario of air pollution in the northeast region of the USA is considered. The UQ and UA results allow us to assess the combined effects of different input uncertainties on the forecast uncertainty. They also enable to quantify the contribution of input uncertainties to the uncertainty in the predicted ozone and PAN concentrations.     

Wavelet based non-parametric NARX models for nonlinear input–output system identification

Wavelet based non-parametric additive NARX models are proposed for nonlinear input–output system identification. By expanding each functional component of the non-parametric NARX model into wavelet multiresolution expansions, the non-parametric estimation problem becomes a linear-in-the-parameters problem, and least-squares-based methods such as the orthogonal forward regression (OFR) approach, coupled with model size determination criteria, can be used to select the model terms and estimate the parameters. Wavelet based additive models, combined with model order determination and variable selection approaches, are capable of handling problems of high dimensionality.     

Nonlinear identification and control of a high-purity distillation column: a case study

Identification and control of ill-conditioned, interactive and highly nonlinear processes pose a challenging problem to the process industry. In the absence of a reasonably accurate model, these processes are fairly difficult to control. Using a high-purity distillation column as an example, model identification and control issues are addressed in this paper. The structure of the identified models is that of the polynomial type nonlinear autoregressive models with exogenous inputs (NARX). While most of the work in this area has concentrated on linear models (one-time scale and two-time scale models), this work is aimed at identifying the inherent nonlinearities. Comparisons are drawn between the identified models based on statistical criteria (AIC etc.) and other validation tests. Simulation results are provided to demonstrate the closed-loop performance of the nonlinear ARX models in the control of the distillation column. The controller employed is based on a nonlinear model predictive scheme with state and parameter estimation.     

Dynamic Ensemble Multivariate Time Series Forecasting Model for PM2.5

In forecasting real time environmental factors, large data is needed to analyse the pattern behind the data values. Air pollution is a major threat towards developing countries and it is proliferating every year. Many methods in time series prediction and deep learning models to estimate the severity of air pollution. Each independent variable contributing towards pollution is necessary to analyse the trend behind the air pollution in that particular locality. This approach selects multivariate time series and coalesce a real time updatable autoregressive model to forecast Particulate matter (PM) PM2.5. To perform experimental analysis the data from the Central Pollution Control Board (CPCB) is used. Prediction is carried out for Chennai with seven locations and estimated PM’s using the weighted ensemble method. Proposed method for air pollution prediction unveiled effective and moored performance in long term prediction. Dynamic budge with high weighted k-models are used simultaneously and devising an ensemble helps to achieve stable forecasting. Computational time of ensemble decreases with parallel processing in each sub model. Weighted ensemble model shows high performance in long term prediction when compared to the traditional time series models like Vector Auto-Regression (VAR), Autoregressive Integrated with Moving Average (ARIMA), Autoregressive Moving Average with Extended terms (ARMEX). Evaluation metrics like Root Mean Square Error (RMSE), Mean Absolute Error (MAE) and the time to achieve the time series are compared.     

带自相关约束的NARX动态软测量模型

非线性带外输入自回归模型(NARX)在进行预测估计时依赖于主导变量的实时测量,因此在实际工业过程中存在一定的实施难度.针对该问题,利用神经网络构造一种新型NARX动态软测量模型,当工业过程无法及时提供上时刻主导变量测量值时,能通过多步预测方法来确保主导变量的实时预测,通过设计模型结构来降低预测序列的自相关性,从而抑制由多步估计造成的累积误差,以适当降低单步预测精度为代价,使模型在主导变量检测时间长、采样周期长、测量存在噪声的工业场合下得到更好的预测效果.通过数学分析和脱丁烷塔数据仿真实验验证了所构建模型的有效性.     

An iterative algorithm for simulation error based identification of polynomial input–output models using multi-step prediction

Effective identification of polynomial input–output models for applications requiring long-range prediction or simulation performance relies on both careful model selection and accurate parameter estimation. The simulation error minimisation (SEM) approach has been shown to provide significant advantages in the model selection phase by ruling out candidate models with good short-term prediction capabilities but unsuitable long-term dynamics. However, SEM-based parameter estimation has been generally avoided due to excessive computational effort. This article extends to the nonlinear case a computationally efficient approach for this task, that was previously developed for linear models, based on the iterative estimation of predictors with increasing prediction horizon. Conditions for the applicability of the approach to various model classes are also discussed. Finally, some examples are provided to show the effectiveness and computational convenience of the proposed algorithm for polynomial input–output identification, as well as the improvements achievable by enforcing SEM parameter estimation. A benchmark for nonlinear identification is also analysed, with encouraging results.     

EMD与NARX神经网络的风电场总功率组合预测

探索构建对风电场总功率进行直接预测的高精度组合预测算法。考虑到风速的非平稳性导致风电总功率表现为非平稳时间序列,采用NARX神经网络作为初步预测模型,提出了经验模态分解与NARX神经网络相结合的混合预测模型。对风电场总功率非平稳时间序列进行经验模态分解,得到不同频带本征模式分量的平稳序列。对不同频带的平稳分量建立相应的NARX神经网络预测模型,并将各分量模型的预测值进行等权求和得到最终预测值。此外,为研究不同时间间隔对预测结果的影响,采用某大型风电场时间间隔为5 min与15 min的数据进行实验。预测结果表明,提出的组合预测模型适合于总功率预测,其预测效果比传统模型的效果更佳,且时间间隔为5 min的数据比时间间隔为15 min的数据预测精度更高。     

多自由度非线性系统动态参数化模型建模方法研究

针对多自由度非线性系统的动态模型辨识问题,基于NARX(Non-linear Autoregressive with Exogenous inputs)模型的建模方法,考虑系统的物理设计参数,建立非线性系统动态参数化模型.首先,根据系统输入、输出数据建立系统不同参数下的NARX模型,并通过EFOR(Extended Forward Orthogonal Regression)算法对不同参数下NARX模型进行修正,以统一辨识得到的系统模型结构.随后,建立NARX模型系数与物理设计参数间的函数关系,得到多自由度非线性系统的动态参数化模型.以单输入、单输出两自由度非线性系统为例,根据数值仿真结果,对系统的动态参数化模型建模过程进行说明.最后,以带非线性涂层阻尼的悬臂梁作为试验对象,建立其动态参数化模型以反映其动力学特性.试验结果表明,非线性系统动态参数化模型能准确预测多自由度非线性系统的输出响应,为非线性系统的分析与优化设计提供了理论基础.     

Determination of model order for NARX models directly from input-output data

Nonlinear auto-regressive models with exogenous inputs (NARX models) have proved to be versatile and useful empirical models for industrial processes. There are a wide variety of identification methods and model structures from which to choose; in all methods, however, key parameters are the model orders, which are the number of past outputs and inputs used in the model. In this paper the methods of Lipschitz numbers and false nearest neighbors are evaluated as a means of estimating the model orders of dynamic, discrete-time NARX models. No specific model structure is assumed and the model orders are estimated directly from input-output data using only geometric concerns and the continuity property. The two methods are applied to several dynamic systems, including realistic process simulations and experimental data from the UCSB pH neutralization process, and the consistency and accuracy of these methods are examined. The usefulness of these methods of model order determination for radial basis function network (RBFN) identification is examined.     

A fuzzy ensemble of parallel polynomial neural networks with information granules formed by fuzzy clustering

In this paper, we introduce a new category of fuzzy models called a fuzzy ensemble of parallel polynomial neural network (FEP²N²), which consist of a series of polynomial neural networks weighted by activation levels of information granules formed with the use of fuzzy clustering. The two underlying design mechanisms of the proposed networks rely on information granules resulting from the use of fuzzy C-means clustering (FCM) and take advantage of polynomial neural networks (PNNs).The resulting model comes in the form of parallel polynomial neural networks. In the design procedure, in order to estimate the optimal values of the coefficients of polynomial neural networks we use a weighted least square estimation algorithm. We incorporate various types of structures as the consequent part of the fuzzy model when using the learning algorithm. Among the diverse structures being available, we consider polynomial neural networks, which exhibit highly nonlinear characteristics when being viewed as local learning models.We use FCM to form information granules and to overcome the high dimensionality problem. We adopt PNNs to find the optimal local models, which can describe the relationship between the input variables and output variable within some local region of the input space.We show that the generalization capabilities as well as the approximation abilities of the proposed model are improved as a result of using polynomial neural networks. The performance of the network is quantified through experimentation in which we use a number of benchmarks already exploited within the realm of fuzzy or neurofuzzy modeling.     

Regressor and structure selection in NARX models using a structured ANOVA approach

Regressor selection can be viewed as the first step in the system identification process. The benefits of finding good regressors before estimating complex models are especially clear for nonlinear systems, where the class of possible models is huge. In this article, a structured way of using the tool analysis of variance (ANOVA) is presented and used for NARX model (nonlinear autoregressive model with exogenous input) identification with many candidate regressors.     

Autoregressive forecast of monthly total ozone concentration: A neurocomputing approach

The present study endeavors to generate autoregressive neural network (AR-NN) models to forecast the monthly total ozone concentration over Kolkata (22°34′, 88°22′), India. The issues associated with the applicability of neural network to geophysical processes are discussed. The autocorrelation structure of the monthly total ozone time series is investigated, and stationarity of the time series is established through the periodogram. From various autoregressive moving average (ARMA) and autoregressive models fit to the time series, the autoregressive model of order 10 is identified as the best. Subsequently, 10 autoregressive neural network (AR-NN) models are generated; the 10th order autoregressive neural network model with extensive input variable selection performs the best among all the competitive models in forecasting the monthly total ozone concentration over the study zone.