On the parameters estimation of continuous-time ARMA processes from noisy observations

Recently, an iterative algorithm has been presented for estimating the parameters of partially observed continuous-time processes [1]. In this note we concentrate on continuous-time ARMA processes observed in white noise. A maximum a-posteriori (MAP) estimator is defined for the trajectory of the parameters' random process. This approach enables the MAP estimation of randomly slowly varying parameters, and extends the conventional treatment of time-invariant parameters. The iterative algorithm derived for the MAP estimation, increases the posterior probability of the parameters in each iteration, and converges to a stationary point of the posterior probability functional. Each iteration involves a standard linear smoother followed by a finite-dimensional linear system, and thus is easily implemented.     

Linear estimation of ARMA processes

The method of estimating ARMA parameters first discussed by Mayne and Firoozan is investigated. It is shown that if, at the first stage, the order of the fitted autoregression is allowed to depend on the number of time points, in a reasonable manner, then it will still be true that the final estimate of the parameter vector will converge, almost surely, to the true value. This is to be compared to the result in the original paper where the order is fixed and it is shown that, as the sample size increases, the estimate converges to a value which, if the order of the autoregression is high enough, will be arbitrarily near to the true value. Some comments are made on other extensions, on the law of the iterated logarithm, on the central limit theorem and on the choice of the order of the fitted autoregression. The innovation sequence need not be Gaussian and for the convergence result only a natural condition relating to prediction needs be imposed.     

A high-order Yule-Walker method for estimation of the AR parameters of an ARMA model

The high-order Yule-Walker (HOYW) method is often used to estimate the frequencies of sinusoidal signals from noisy measurements. A modification of the HOYW method is proposed for estimating the parameters of the autoregressive part of general ARMA processes. The prime application of the HOYW method introduced in this paper is the estimation of the parameters of autoregressions from noisy measurements. The performance of the proposed technique is illustrated by some numerical examples.     

A new algorithm for recursive estimation of parameters in controlled ARMA processes

A recursive algorithm for estimating the constant but unknown parameters of a controlled ARMA process is presented. The algorithm is a recursive version of an off-line algorithm using three stages of standard least-squares. In the first stage the parameters of a controlled AR model of degree p are estimated. The residuals used in this stage are employed in the second stage to estimate the parameters of a controlled ARMA process. The first two stages constitute a recursive version of Durbin's algorithm. The model obtained in the second stage is used to filter the input, output and residuals and these filtered variables are used in the final stage to obtain improved estimates of the controlled ARMA process. It is shown that the estimate is (globally) p-consistent, i.e. that the estimate converges a.s. as the number of data tends to infinity, to a vector which, in turn, converges to the true parameter vector as the degree p of the AR model tends to infinity.     

ARMA model parameter estimation based on the equivalent MA approach

The paper investigates the relation between the parameters of an autoregressive moving average (ARMA) model and its equivalent moving average (EMA) model. On the basis of this relation, a new method is proposed for determining the ARMA model parameters from the coefficients of a finite-order EMA model. This method is a three-step approach: in the first step, a simple recursion relating the EMA model parameters and the cepstral coefficients of an ARMA process is derived to estimate the EMA model parameters; in the second step, the AR parameters are estimated by solving the linear equation set composed of EMA parameters; then, the MA parameters are obtained via simple computations using the estimated EMA and AR parameters. Simulations including both low- and high-order ARMA processes are given to demonstrate the performance of the new method. The end results are compared with the existing method in the literature over some performance criteria. It is observed from the simulations that our new algorithm produces the satisfactory and acceptable results.     

A method for adaptive estimation of ARMA processes

When the noise process in adaptive identification of linear stochastic systems is correlated, and can be represented by a moving average model, extended least squares algorithms are commonly used, and converge under a strictly positive real (SPR) condition on the noise model. In this paper, we present an adaptive algorithm for the estimation of autoregressive moving average (ARMA) processes, and show that it is convergent without any SPR condition, and has a convergence rate of O({loglog t)/t}^1/2).     

Adaptive prediction for ARMA processes with Markov switching parameters

The problem of prediction for ARMA processes with switching parameters modelled as a finite-state Markov chain is considered. The Markov transition probability matrix is assumed to be unknown but constant and can take values only from a finite collection which contains the true transition matrix. A multiple-model prediction method is presented. The digital simulation shows a good performance of the proposed predictor.     

Large-sample estimation of the AR parameters of an ARMA model

The purpose of this note is to show that the optimal instrumental-variable estimate of the AR parameters of an ARMA model, recently proposed in [1] can also be interpreted as a large-sample approximation of a maximum-likelihood estimate.     

Approximate estimation methods for the moments of stochastic processes

This paper studies estimation methods for the statistical characteristics of stochastic processes with measuring the number of crossings of a certain level by a stochastic process and overshoot durations. The authors present approximate computing formulas for mathematical expectation and variance.     

On the martingale approximation of the estimation error of ARMA parameters

The aim of this paper is to prove a theorem which is instrumental in verifying Rissanen's tail condition for the estimation error of the parameters of a Gaussian ARMA process. We get an improved error bound for the martingale approximation of the estimation error for a wide class of ARMA processes.     

Specific optimal estimation of special Markov jump processes

A solution to the filtering problem of states of special Markov jump processes that is optimal in the mean-square sense at the class of polynomial observation functions is presented. A comparison of the proposed estimates with the known estimates of optimal linear and nonlinear filtering is given.     

Linear identification of ARMA processes

A new method for estimating the parameters of an ARMA process is presented. The method consists of three linear least-squares estimations. In the first an autoregressive model is fitted to the observation sequence, yielding an estimate of the values of the driving white noise sequence. Linear least squares is then used to fit an ARMA model to the observation and estimated white noise sequences. This model is used to filter the observation and estimated white noise sequences. Finally an ARMA model is fitted to the filtered sequences. It is shown that the resultant estimator is ‘p-consistent’ (the asymptotic bias tends to zero as the degree p of the autoregressive model tends to infinity) and is ‘p-efficient’ (the asymptotic efficiency approaches the theoretical maximum as p tends to infinity).     

确定ARMA模型MA阶数的一种方法

本文提出一种ARMA模型MA定阶的新方法.其基本思想是,将阶数确定转化为一上三角阵的秩的确定.仿真例子表明,该方法在数值上是鲁棒的.     

Optimal instrumental variable multistep algorithms for estimation of the AR parameters of an ARMA process

Multistep implementations are derived for the optimal instrumental variable (OIV) estimators introduced by Stoica et al. in 1985. The proposed algorithms provide asymptotically efficient estimates of the AR parameters of an ARMA process. The computational complexity of these algorithms is modest compared with the maximum likelihood estimator. The performance of the OIV algorithms is illustrated by some numerical examples.     

A simple characterization of optimal ARMA predictors

The coefficients of the optimal steady state k-step ahead predictor for an ARMA process in general depend on k. It is shown that a simple formula completely characterizes all these coefficients.     

Aggregation and systematic sampling of periodic ARMA processes

The aim of this work is to investigate the effects of temporal aggregation and systematic sampling on periodic autoregressive moving average (PARMA) time series. Firstly, it is shown that the class of weak PARMA processes, i.e. with uncorrelated but possibly dependent errors, is closed under a particular class of linear transformations that include both temporal aggregation and systematic sampling. This extends a similar result for autoregressive moving average processes; see [Wei, W.W.S., 2006. Time Series Analysis: Univariate and Multivariate Methods, second ed. Addison-Wesley, New York (Chapter 20)] for a review on the subject. Secondly, the properties of the noise of the transformed process are investigated. A sufficient condition is given under which aggregation and systematic sampling of a strong PARMA process, i.e. with independent errors, give rise in general to a weak PARMA process. Under that condition, the noise of the transformed process is neither strong nor a martingale difference. This result points out that the assumption of strong PARMA should not be used without careful considerations when analyzing aggregated time series that naturally occur in many scientific fields. The sufficient condition for non-independent errors is illustrated with the PARMA(1,1) model. A simulation study underlines the practical relevance of our findings and the importance of taking into account the dependence of the errors when fitting a PARMA model to an aggregated time series.     

参数未知的离散系统Q-学习优化状态估计与控制

控制系统的应用中存在状态不能直接测量或测量成本高的实际问题,给模型参数未知的系统完全利用状态数据学习最优控制器带来挑战性难题.为解决这一问题,首先构建具有状态观测器且系统矩阵中存在未知参数的离散线性增广系统,定义性能优化指标;然后基于分离定理、动态规划以及Q-学习方法,给出一种具有未知模型参数的非策略Q-学习算法,并设计近似最优观测器,得到完全利用可测量的系统输出和控制输入数据的非策略Q-学习算法,实现基于观测器状态反馈的系统优化控制策略,该算法的优点在于不要求系统模型参数全部已知,不要求系统状态直接可测,利用可测量数据实现指定性能指标的优化;最后,通过仿真实验验证所提出方法的有效性.     

Chaos, estimation and optimal control of habitat destruction model with uncertain parameters

The present paper discusses chaos, estimation and optimal control of the habitat destruction model with unknown parameters. The linear stability analysis of the steady states of the model will be discussed. Further, the chaotic behavior of this system will be investigated. The dynamic estimators of the unknown parameters and their updating rules are derived. Using Pontryagin principle, the optimal control inputs are derived with respect to a selected measure. Finally, a numerical simulation study for various parameters and different initial densities is presented.     

Asymptotically efficient estimation of the conditional expected shortfall

A procedure for efficient estimation of the trimmed mean of a random variable conditional on a set of covariates is proposed. For concreteness, the focus is on a financial application where the trimmed mean of interest corresponds to the conditional expected shortfall, which is known to be a coherent risk measure. The proposed class of estimators is based on representing the estimator as an integral of the conditional quantile function. Relative to the simple analog estimator that weights all conditional quantiles equally, asymptotic efficiency gains may be attained by giving different weights to the different conditional quantiles while penalizing excessive departures from uniform weighting. The approach presented here allows for either parametric or nonparametric modeling of the conditional quantiles and the weights, but is essentially nonparametric in spirit. The asymptotic properties of the proposed class of estimators are established. Their finite sample properties are illustrated through a set of Monte Carlo experiments and an empirical application¹.     

Efficient computation of high-order Meixner moments for large-size signals and images analysis

In this paper, we present an in-depth study on the computational aspects of high-order discrete orthogonal Meixner polynomials (MPs) and Meixner moments (MMs). This study highlights two major problems related to the computation of MPs. The first problem is the overflow and the underflow of MPs values (“Nan” and “infinity”). To overcome this problem, we propose two new recursive Algorithms for MPs computation with respect to the polynomial order n and with respect to the variable x. These Algorithms are independent of all functions that are the origin the numerical overflow and underflow problem. The second problem is the propagation of rounding errors that lead to the loss of the orthogonality property of high-order MPs. To fix this problem, we implement MPs based on the following orthogonalization methods: modified Gram-Schmidt process (MGS), Householder method, and Givens rotation method. The proposed Algorithms for the stable computation of MPs are efficiently applied for the reconstruction and localization of the region of interest (ROI) of large-sized 1D signals and 2D/3D images. We also propose a new fast method for the reconstruction of large-size 1D signal. This method involves the conversion of 1D signal into 2D matrix, then the reconstruction is performed in the 2D domain, and a 2D to 1D conversion is performed to recover the reconstructed 1D signal. The results of the performed simulations and comparisons clearly justify the efficiency of the proposed Algorithms for the stable analysis of large-size signals and 2D/3D images.