Order Autoregressive Process Research Articles

Blind signal separation based on widely linear complex autoregressive process of order one

In this paper, the blind signal separation problem of complex baseband signal is addressed. A widely linear complex autoregressive process of order one is employed to represent the temporal structure of complex sources. We formulate a new contrast function by a convex combination of generalized autocorrelations and the statistics of the innovation. And the proposed contrast function is optimized by gradient method. Simulation results show that the proposed algorithm is better than the comparison algorithm in convergence speed and convergence accuracy.

Analysis of Variance Amplification and Service Level in a Supply Chain with Correlated Demand

The bullwhip effect reflects the variance amplification of demand as they are moving upstream in a supply chain, and leading to the distortion of demand information that hinders supply chain performance sustainability. Extensive research has been undertaken to model, measure, and analyze the bullwhip effect while assuming stationary independent and identically distributed (i.i.d) demand, employing the classical order-up-to (OUT) policy and allowing return orders. On the contrary, correlated demand where a period’s demand is related to previous periods’ demands is evident in several real-life situations, such as demand patterns that exhibit trends or seasonality. This paper assumes correlated demand and aims to investigate the order variance ratio (OVR), net stock amplification ratio (NSA), and average fill rate/service level (AFR). Moreover, the impact of correlated demand on the supply chain performance under various operational parameters, such as lead-time, forecasting parameter, and ordering policy parameters, is analyzed. A simulation modeling approach is adopted to analyze the response of a single-echelon supply chain model that restricts return orders and faces a first order autoregressive demand process AR(1). A generalized order-up-to policy that allows order smoothing through the proper tuning of its smoothing parameters is applied. The characterization results confirm that the correlated demand affects the three performance measures and interacts with the operating conditions. The results also indicate that the generalized OUT inventory policy should be adopted with the correlated demand, as its smoothing parameters can be adapted to utilize the demand characteristics such that OVR and NSA can be reduced without affecting the service level (AFR), implying sustainable supply chain operations. Furthermore, the results of a factorial design have confirmed that the ordering policy parameters and their interactions have the largest impact on the three performance measures. Based on the above characterization, the paper provides management with means to sustain good performance of a supply chain whenever a correlated demand pattern is realized through selecting the control parameters that decrease the bullwhip effect.

PREDICTING SYSTEMATIC RISK WITH MACROECONOMIC AND FINANCIAL VARIABLES

AbstractWe analyze the predictive power of several macroeconomic and financial indicators in forecasting quarterly realized betas of 30 industry and 25 size and book‐to‐market portfolios. We model realized betas as autoregressive processes of order 1 and include lagged values of macroeconomic and financial indicators as exogenous predictor variables. In out‐of‐sample forecasting exercises, forecasts using bond market variables as exogenous predictors statistically outperform forecasts from a benchmark model without any exogenous predictors. These forecasts based on bond market variables also economically outperform benchmark forecasts by providing better performance in hedging the market risk of portfolios.

A first order autoregressive process with a change point: A bayesian approach based on model selection

The change points have considerable effects in different areas of applied research. We will use in this work the pseudo-bayes factor in three autoregressive models of order (1); this method permits to analyse the impact of choice between models and allows the use of a simpler technique with model selection in time series. For application, the monthly fluctuations of the DOW-JONES series between January 1999 and September 2009 have been used; we try to detect the financial crisis between 2007 and 2008 to evaluate the model selection method.

Best Linear Predictors in a Stationary Second Order Autoregressive process by means of near and far observations

Best Linear Predictors in a Stationary Second Order Autoregressive process by means of near and far observations

A non‐stationary bivariate INAR(1) process with a simple cross‐dependence: Estimation with some properties

SummaryThis paper considers modelling of a non‐stationary bivariate integer‐valued autoregressive process of order 1 (BINAR(1)) where the cross‐dependence between the counting series is formed through the relationship of the current series with the previous‐lagged count series observations while the pair of innovations is independent and marginally Poisson. In addition, this paper proposes a generalised quasi‐likelihood (GQL) estimating equation based on the exact specification of the mean score and the auto‐covariance structure. The proposed approach is also compared with other popular techniques such as conditional maximum likelihood (CML), generalised least squares (GLS) and generalised method of moment (GMM) based on simulated data from the proposed BINAR(1). Moreover, the model is applied to weekly series of day and night road accidents arising in some regions of Mauritius and is compared with other existing BINAR(1) models.

Time Stable Small Area Estimates of General Parameters Under a Unit-Level Model

Longitudinal surveys collecting information on certain phenomena at several time points are very popular because they allow to analyze the changes over time. Data coming from those surveys often present correlation over time that should be accounted for by the considered statistical procedures. In fact, methods that account for the existing time correlation are expected to yield more stable small area estimates over time. Temporal stability is a desirable property of statistics that are published regularly, specially in certain applications like in poverty mapping, where poverty estimates for the same area with big jumps from one period to the next are rarely credible. This paper considers a unit-level temporal linear mixed model for small area estimation that includes random time effects nested within the usual area effects, following an auto-regressive process of order 1, AR(1). Based on the proposed model, we obtain empirical best predictors of general area parameters, giving explicit expressions for some common poverty indicators. We also propose a parametric bootstrap method for estimating their mean square errors under the model. The proposed methods are studied through simulation experiments and illustrated with an application to poverty mapping in Spanish provinces using survey data from 2004-2006.

Recent and future manifestations of a contingent global mean sea level acceleration

Abstract We analyzed globally averaged satellite altimetry mean sea level time series during 1993 – 2018 and their future manifestations for the following 25 years using a kinematic model, which consists of a trend, a contingent uniform acceleration, and a random error model. The analysis of variance results shows that the model explains 71.7% of the total variation in global mean sea level for which 70.6% is by the secular trend, and 1.07% is due to a contingent uniform acceleration. The remaining 28.3% unexplained variation is due to the random errors, which are dominated by a first order autoregressive process driven mostly by oceanic and atmospheric variations over time. These numbers indicate more bumps and jumps for the future manifestations of the global mean sea level anomalies as illustrated using a one-step ahead predictor in this study. Our findings suggest preponderant random errors are poised to further confound and negatively impact the certitude of published estimates of the uniform global sea level acceleration as well as its prediction under an increasingly warmer Earth.

On the BIC for determining the number of control points in B-spline surface approximation in case of correlated observations

Abstract B-spline curves are a linear combination of control points (CP) and B-spline basis functions. They satisfy the strong convex hull property and have a fine and local shape control as changing one CP affects the curve locally, whereas the total number of CP has a more general effect on the control polygon of the spline. Information criteria (IC), such as Akaike IC (AIC) and Bayesian IC (BIC), provide a way to determine an optimal number of CP so that the B-spline approximation fits optimally in a least-squares (LS) sense with scattered and noisy observations. These criteria are based on the log-likelihood of the models and assume often that the error term is independent and identically distributed. This assumption is strong and accounts neither for heteroscedasticity nor for correlations. Thus, such effects have to be considered to avoid under-or overfitting of the observations in the LS adjustment, i.e. bad approximation or noise approximation, respectively. In this contribution, we introduce generalized versions of the BIC derived using the concept of quasi- likelihood estimator (QLE). Our own extensions of the generalized BIC criteria account (i) explicitly for model misspecifications and complexity (ii) and additionally for the correlations of the residuals. To that aim, the correlation model of the residuals is assumed to correspond to a first order autoregressive process AR(1). We apply our general derivations to the specific case of B-spline approximations of curves and surfaces, and couple the information given by the different IC together. Consecutively, a didactical yet simple procedure to interpret the results given by the IC is provided in order to identify an optimal number of parameters to estimate in case of correlated observations. A concrete case study using observations from a bridge scanned with a Terrestrial Laser Scanner (TLS) highlights the proposed procedure.

The Marshall-Olkin Weibull Truncated Negative Binomial Distribution and its Applications

The Weibull distribution is one of the widely known lifetime distribution that has been extensively used for modelling data in reliability and survival analysis. A generalization of both the Marshall-OlkinWeibull distribution and the Weibull truncated negative binomial distribution is introduced and studied in this article. Various distributional properties of the new distribution are derived. Estimation of model parameters using the method of maximum likelihood is discussed. Applications to a real data set is provided to show the flexibility and potentiality of the new distribution over other Weibull models. The first order autoregressive minification process with the new distribution as marginal is also developed. We hope that the new model will serve as a good alternative to other models available in the literature for modeling positive real data in several areas.

State-Discretization of V-Geometrically Ergodic Markov Chains and Convergence to the Stationary Distribution

Let $(X_n)_{n \in\mathbb{N}}$ be a $V$-geometrically ergodic Markov chain on a measurable space $\mathbb{X}$ with invariant probability distribution $\pi$. In this paper, we propose a discretization scheme providing a computable sequence $(\widehat\pi_k)_{k\ge 1}$ of probability measures which approximates $\pi$ as $k$ growths to infinity. The probability measure $\widehat\pi_k$ is computed from the invariant probability distribution of a finite Markov chain. The convergence rate in total variation of $(\widehat\pi_k)_{k\ge 1}$ to $\pi$ is given. As a result, the specific case of first order autoregressive processes with linear and non-linear errors is studied. Finally, illustrations of the procedure for such autoregressive processes are provided, in particular when no explicit formula for $\pi$ is known.

An automated system for predicting detection limit and precision profile from a chromatogram

This paper presents a basic model of an automated system for predicting the detection limit and precision profile (plot of relative standard deviation (RSD) of measurements against concentration) in chromatography. The fundamental assumption is that the major source of response errors at low sample concentrations is background noise and at high concentrations, it is the volumes injected into an HPLC system by a sample injector. The noise is approximated by the mixed random processes of the first order autoregressive process AR(1) and white noise. The research procedures are: (1) the description of the standard deviation (SD) of measurements in terms of the parameters of the mixed random processes; (2) the algorithm for the parameter estimation of the mixed processes from actual background noise; (3) the mathematical distinction between noise and signal in a chromatogram. When compounds are chromatographically separated, each obtained signal is given the detection limit and precision profile on laboratory-made software. A file of a chromatogram is the only requirement for the theoretical prediction of measurement uncertainty and therefore the repeated measurements of real samples can be dispensed with. The theoretically predicted RSDs are verified by comparing them with the statistical RSDs obtained by repeated measurements. Signal shapes on noise are illustrated at the detection limit and quantitation limit, the signal-to-noise ratios of which are close to the widely adopted values, 3 and 10, respectively.

Chemometric Evaluation of Repeatability Using the Autocorrelation Method in High-Performance Liquid Chromatography with Ultraviolet Detection.

The mixed random processes of the first order autoregressive process (AR(1)) and white noise have been proved to provide a good approximation of baseline noise in a variety of analytical instruments, and may therefore be useful for estimating precision profiles. This study aims to examine a recently proposed autocorrelation method for estimating three noise parameters involved in the mixed processes (two for AR(1) and one for white noise) of HPLC, which can then be used to calculate the precision profile. This chemometric method was applied to repeatability evaluations of estriol determination using HPLC with UV detection (HPLC-UV). The relative standard deviations (RSDs) of peak area measurements for 5.0 mg/L estriol were observed to be 1.42% for the autocorrelation method and 1.63% for actual repeated measurements of real samples (n = 6). The theoretical RSDs of the autocorrelation method fell within the 95% confidence intervals of the repeated measurements. It is found that the noise parameters are obtained from real chromatographic baseline via the autocorrelation method. Moreover, the instrumental detection limit of estriol based on ISO 11843 was obtained from the precision profile (plot of RSD of measurements against concentration). This is the first paper to describe the autocorrelation method is a practically useful technique for evaluating the precision profile of HPLC-UV analyses without recourse to the repeated measurements of real samples.

Feedback Capacity of Stationary Gaussian Channels Further Examined

It is well known that the problem of computing the feedback capacity of a stationary Gaussian channel can be recast as an infinite-dimensional optimization problem; moreover, necessary and sufficient conditions for the optimality of a solution to this optimization problem have been characterized, and based on this characterization, an explicit formula for the feedback capacity has been given for the case that the noise is a first-order autoregressive moving-average Gaussian process. In this paper, we further examine the above-mentioned infinite-dimensional optimization problem. We prove that unless the Gaussian noise is white, its optimal solution is unique, and we propose an algorithm to recursively compute the unique optimal solution, which is guaranteed to converge in theory and features an efficient implementation for a suboptimal solution in practice. Furthermore, for the case that the noise is a k-th order autoregressive moving-average Gaussian process, we give a relatively more explicit formula for the feedback capacity; more specifically, the feedback capacity is expressed as a simple function evaluated at a solution to a system of polynomial equations, which is amenable to numerical computation for the cases k=1, 2 and possibly beyond.

Parameter change test for periodic integer-valued autoregressive process

This paper studies a parameter change testing problem in periodic integer-valued autoregressive process of order with period (). The process is transformed into a multivariate process for which basic probabilistic and statistical properties are researched. The cumulative sum test based on conditional least-squares estimators is employed to test the parameter change of process. That returns to tests for the parameter change of - models. The power of test is evaluated through simulation results.

Acceptance sampling plans based on EWMA yield index for the first order autoregressive process

Acceptance-sampling plan plays an important role in quality control. Four new sampling plans based on the yield index are proposed to deal with lot sentencing for a first-order autoregressive process. The first plan is based on exponentially weighted moving average (EWMA) model. The other three plans are based on resubmitted, repetitive group sampling (RGS), and multiple dependent state repetitive (MDSR), respectively. The EWMA and MDSR models use the quality information of the current lot and previous lots. The resubmitted and repetitive group sampling plans are allowed resampling under a certain condition. We found that the sample size required for lot sentencing is the most economical for the EWMA model. Moreover, the RGS and MDSR plans are much more efficient than the traditional single sampling plan. The resubmitted scheme has the least efficiency. Considering the acceptable quality level at the producer’s risk and the lot tolerance percent defective at the consumer’s risk, a nonlinear optimisation models are proposed to determine the plan parameters. Two examples are provided to show the applicability of the proposed sampling plans.

Note on AR(1)-characterisation of stationary processes and model fitting

It was recently proved that any strictly stationary stochastic process can be viewed as an autoregressive process of order one with coloured noise. Furthermore, it was proved that, using this characterisation, one can define closed form estimators for the model parameter based on autocovariance estimators for several different lags. However, this estimation procedure may fail in some special cases. In this article we provide a detailed analysis of these special cases. In particular, we prove that these cases correspond to degenerate processes.

Bayesian Analysis of Functional Magnetic Resonance Imaging Data with Spatially Varying Auto-Regressive Orders

Summary Statistical modelling of functional magnetic resonance imaging data is challenging as the data are both spatially and temporally correlated. Spatially, measurements are taken at thousands of contiguous regions, called voxels, and temporally measurements are taken at hundreds of time points at each voxel. Recent advances in Bayesian hierarchical modelling have addressed the challenges of spatiotemporal structure in functional magnetic resonance imaging data with models incorporating both spatial and temporal priors for signal and noise. Whereas there has been extensive research on modelling the functional magnetic resonance imaging signal (i.e. the convolution of the experimental design with the functional choice for the haemodynamic response function) and its spatial variability, less attention has been paid to realistic modelling of the temporal dependence that typically exists within the functional magnetic resonance imaging noise, where a low order auto-regressive process is typically adopted. Furthermore, the auto-regressive order is held constant across voxels (e.g. AR(1) at each voxel). Motivated by an event-related functional magnetic resonance imaging experiment, we propose a novel hierarchical Bayesian model with automatic selection of the auto-regressive orders of the noise process that vary spatially over the brain. With simulation studies we show that our model is more statistically efficient and we apply it to our motivating example.

Sequential fixed accuracy estimation for nonstationary autoregressive processes

For an autoregressive process of order p, the paper proposes new sequential estimates for the unknown parameters based on the least squares (LS) method. The sequential estimates use p stopping rules for collecting the data and presumes a special modification the sample Fisher information matrix in the LS estimates. In case of Gaussian disturbances, the proposed estimates have non-asymptotic normal joint distribution for any values of unknown autoregressive parameters. It is shown that in the i.i.d. case with unspecified error distributions, the new estimates have the property of uniform asymptotic normality for unstable autoregressive processes under some general condition on the parameters. Examples of unstable autoregressive models satisfying this condition are considered.

Modeling Active Cell Movement With the Potts Model

In the last decade, the cellular Potts model has been extensively used to model interacting cell systems at the tissue-level. However, in {\bf early} applications of this model, cell movement was taken as a consequence of membrane fluctuations due to cell-cell interactions, or as a response to an external chemotactic gradient. Recent findings have shown that eukaryotic cells can exhibit persistent displacements across scales larger than cell size, even in the absence of external signals. Persistent cell motion has been incorporated to the cellular Potts model by many authors in the context of collective motion, chemotaxis and morphogenesis. In this paper, we use the cellular Potts model in combination with a random field applied over each cell. This field promotes a uniform cell motion in a given direction during a certain time interval, after which the movement direction changes. The dynamics of the direction is coupled to a first order autoregressive process. We investigated statistical properties, such as the mean-squared displacement and spatio-temporal correlations, associated to these self-propelled {\it in silico} cells in different conditions. The proposed model emulates many properties observed in different experimental setups. By studying low and high density cultures, we find that cell-cell interactions decrease the effective persistent time.