Yule-Walker Equations Research Articles

Massive MIMO Channel Prediction: Kalman Filtering Vs. Machine Learning

This paper focuses on channel prediction techniques for massive multiple-input multiple-output (MIMO) systems. Previous channel predictors are based on theoretical channel models, which would be deviated from realistic channels. In this paper, we develop and compare a vector Kalman filter (VKF)-based channel predictor and a machine learning (ML)-based channel predictor using the realistic channels from the spatial channel model (SCM), which has been adopted in the 3GPP standard for years. First, we propose a low-complexity mobility estimator based on the spatial average using a large number of antennas in massive MIMO. The mobility estimate can be used to determine the complexity order of developed predictors. The VKF-based channel predictor developed in this paper exploits the autoregressive (AR) parameters estimated from the SCM channels based on the Yule-Walker equations. Then, the ML-based channel predictor using the linear minimum mean square error (LMMSE)-based noise pre-processed data is developed. Numerical results reveal that both channel predictors have substantial gain over the outdated channel in terms of the channel prediction accuracy and data rate. The ML-based predictor has larger overall computational complexity than the VKF-based predictor, but once trained, the operational complexity of ML-based predictor becomes smaller than that of VKF-based predictor.

An Analysis of Time-Series Models for Age-Specific Mortality Rates in India

Mortality is a continuous force of attrition, tending to reduce the population, a prime negative force in the balance of vital processes (Bhasin and Nag, 2004). Sample Registration System (SRS) serves as the only source of annual data on vital events on a full scale from 1969-70 in India. Few studies have examined the trends and patterns of mortality across time and regions in India (Preston and Bhat, 1984). The Under 5 Mortality Rates (U5MR) can be seen to decrease by more than half from 1970 to 2017 but in contrast little is known about the mortality patterns of the older children (5-9) and young adolescents (10-14), and not many studies have been done on their changing trends (Masquelier et al., 2018). Using the annual data for the 5-14 age, the trend of decline in the mortality patterns is studied from 1970 to 2013. The linear trend in the time series plot suggests analysis using time series models AR(p), MA(q), ARMA(p,q), Box- Jenkins ARIMA(p,d,q) and Random Walk with drift models to get the best fit to the trend of the data. The order of the time series models have been calculated by studying the ACF, PACF plots and the coefficients have been derived using the Yule-Walker equation matrix. An in-sample forecast of the years 2014-17 are taken. The Mean Squared Error (MSE) and the Mean Absolute Percentage Error (MAPE) as a measure of accuracy is used to determine the best fit model. ARIMA(3,1,1) produced lower values making it the best-fit model. Out-of-sample forecasting was done for 2018-2025. The forecast value shows that at the current trend, India would have 0.03 deaths per 1000 population in the 5-14 age group in 2025 showing that the government’s policies and health care interventions towards realization of the MDG4 goal is working positively.

A semi-parametric estimation method for the quantile spectrum with an application to earthquake classification using convolutional neural network

In this paper, a new estimation method is introduced for the quantile spectrum, which uses a parametric form of the autoregressive (AR) spectrum coupled with nonparametric smoothing. The method begins with quantile periodograms which are constructed by trigonometric quantile regression at different quantile levels, to represent the serial dependence of time series at various quantiles. At each quantile level, we approximate the quantile spectrum by a function in the form of an ordinary AR spectrum. In this model, we first compute what we call the quantile autocovariance function (QACF) by the inverse Fourier transformation of the quantile periodogram at each quantile level. Then, we solve the Yule–Walker equations formed by the QACF to obtain the quantile partial autocorrelation function (QPACF) and the scale parameter. Finally, we smooth QPACF and the scale parameter across the quantile levels using a nonparametric smoother, convert the smoothed QPACF to AR coefficients, and obtain the AR spectral density function. Numerical results show that the proposed method outperforms other conventional smoothing techniques. We take advantage of the two-dimensional property of the estimators and train a convolutional neural network (CNN) to classify smoothed quantile periodogram of earthquake data and achieve a higher accuracy than a similar classifier using ordinary periodograms.

Estimation of UFMC Fading Channels Using H∞ Filter

Universal filtered multi-carrier (UFMC) modulation is a very powerful candidate to be employed for future 5G mobile systems. It overcomes the limitations and restrictions in current modulation techniques employed in 4G mobile systems and supports future applications, such as machineto-machine (M2M), device-to-device (D2D), and vehicle-tovehicle (V2V) communications. In this paper, we address the estimation of UFMC fading channels based on the comb-type pilot arrangement in the frequency domain. The basic solution is to estimate the fading channel based on the mean square error (MSE) or least square (LS) criteria with adaptive implementation using least mean square (LMS) or recursive least square (RLS) algorithms. However, these adaptive filters seem not to be effective, as they cannot fully exploit fading channel statistics, particularly at high Doppler rates. To take advantage of these statistics, time-variations of the fading channel are modeled by an autoregressive process (AR), and are tracked by an H∞ filter. This, however, requires that AR model parameters be known, which are estimated by solving the Yule-Walker equation (YWE), based on the Bessel autocorrelation function (ACF) of the fading channel with a known Doppler rate. Results of Matlab simulations show that the proposed H∞ filter-based channel estimator is more effective when compared with existing estimators

A New Integer-Valued AR(1) Process Based on Power Series Thinning Operator

In this paper, we introduce the first-order integer-valued autoregressive (INAR(1)) model, with Poisson-Lindley innovations based on power series thinning operator. Some mathematical features of this process are given and estimating the parameters is discussed by three methods; conditional least squares, Yule-Walker equations and conditional maximum likelihood.Then the results are studied for three special cases of power series operators. Finally, some numerical results are presented with a discussion to the obtained results and Four real data sets are used to show the potentially of the new process.

New estimation methods for autoregressive process in the presence of white observation noise

This paper presents four new methods for estimating the parameters of an autoregressive (AR) process based on observations corrupted by white noise. The first three methods are iterative, while the last one is non-iterative. One method is designed to achieve an unbiased estimation of the AR parameters by undermining the destructive impact of observation noise in terms of utilizing the null space of the AR parameter vector. Another one uses both low- and high-order Yule-Walker equations to construct a constrained least squares optimization problem, in which the variance of observation noise is estimated by alternating between two equations. One more method exploits an approximation which leads to reducing the problem of estimating the AR parameters with arbitrary order p to estimating just two parameters, while the last one estimates the variance of the observation noise using the minimum eigenvalue of the enlarged autocorrelation matrix. The performance of the proposed methods is evaluated in terms of various numerical examples, which demonstrate their superiority in terms of accuracy and robustness against the observation noise compared to state-of-the-art existing methods in most simulation examples. It makes the proposed methods a good fit for practical analysis of data contaminated by observation noise, when AR modeling is applicable, and gives a range of choices of methods for different data analysis situations.

Functional ARCH and GARCH models: A Yule-Walker approach

Conditional heteroskedastic financial time series are commonly modelled by (G)ARCH processes. ARCH$(1)$ and GARCH were recently established in $C[0,1]$ and $L^{2}[0,1]$. This article provides sufficient conditions for the existence of strictly stationary solutions, weak dependence and finite moments of (G)ARCH processes for any order in $C[0,1]$ and $L^{p}[0,1]$. It deduces explicit asymptotic upper bounds of estimation errors for the shift term, the complete (G)ARCH operators and the projections of ARCH operators on finite-dimensional subspaces. The operator estimaton is based on Yule-Walker equations, and estimating the GARCH operators also involves a result estimating operators in invertible linear processes being valid beyond the scope of (G)ARCH. Moreover, our results regarding (G)ARCH can be transferred to functional AR(MA).

Maximum Entropy Method without False Peaks with Exact Numerical Equation

The standard numerical maximum entropy method (MEM) still uses the Yule-Walker equation which contains rough approximation by Walker. The commonly used numerical equation contains additional modifications to reduce calculation cost. Nowadays, we have powerful computers and there is no reason to use the modified equation. We show that the drawbacks of MEM such as false peaks and peak splittings are from the modifications. They do not appear when using the exact numerical equation, even a given time series is fractional.

Identification of noisy input-output FIR models with colored output noise

This paper deals with the identification of FIR models corrupted by white input noise and colored output noise. An identification algorithm that exploits the properties of both the dynamic Frisch scheme and the high-order Yule-Walker (HOYW) equations is proposed. It is shown how the HOYW equations allow to define a selection criterion for identifying the input noise variance (and then the FIR coefficients) within the Frisch locus of solutions. The proposed approach does not require any a priori knowledge about the input and output noise variances. The algorithm performance is assessed by means of Monte Carlo simulations.

Generalized Binary Time Series Models

The serial dependence of categorical data is commonly described using Markovian models. Such models are very flexible, but they can suffer from a huge number of parameters if the state space or the model order becomes large. To address the problem of a large number of model parameters, the class of (new) discrete autoregressive moving-average (NDARMA) models has been proposed as a parsimonious alternative to Markov models. However, NDARMA models do not allow any negative model parameters, which might be a severe drawback in practical applications. In particular, this model class cannot capture any negative serial correlation. For the special case of binary data, we propose an extension of the NDARMA model class that allows for negative model parameters, and, hence, autocorrelations leading to the considerably larger and more flexible model class of generalized binary ARMA (gbARMA) processes. We provide stationary conditions, give the stationary solution, and derive stochastic properties of gbARMA processes. For the purely autoregressive case, classical Yule–Walker equations hold that facilitate parameter estimation of gbAR models. Yule–Walker type equations are also derived for gbARMA processes.

Аccounting for autocorrelation in a linear regression problem on an example of analysis of atmospheric column NO2 content

A method is proposed for taking into account a serial correlation (an autocorrelation) of data in a linear regression problem, which allows accounting for the autocorrelation on long scales. A residual series is presented as an autoregressive process of an order, k, that can be much larger than 1, and the autocorrelation function of the processes is calculated by solving the system of the Yule–Walker equations. Given the autocorrelation function, the autocorrelation matrix is constructed which enters the formulas for estimates of regression coefficients and their errors. The efficiency of the method is demonstrated on the base of the multiple regression analysis of data of 26-year measurements of the column NO2 contents at the Zvenigorod Research Station of the Institute of Atmospheric Physics. Estimates of regression coefficients and their errors depend on the autoregression order k. At first the error increases with increasing k. Then it approaches its maximum and thereafter begins to decrease. In the case of NO2 at the Zvenigorod Station the error more than doubled in its maximum compared to the beginning value. The decrease in the error after approaching the maximum stops if k approaches the value such that the autoregressive process of this order allows accounting for important features of the autocorrelation function of the residual series. Estimates have been obtained of seasonally dependent linear trends and effects on NO2 of nature factors such that the 11-year solar cycle, the quasi-biennial oscillation, the North Atlantic Oscillation and other.

Empirical study of robust estimation methods for PAR models with application to the air quality area

This paper compares three estimators for periodic autoregressive (PAR) models. The first is the classical periodic Yule-Walker estimator (YWE). The second is a robust version of YWE (RYWE) which uses the robust autocovariance function in the periodic Yule-Walker equations, and the third is the robust least squares estimator (RLSE) based on iterative least squares with robust versions of the original time series. The daily mean particulate matter concentration (PM10) data is used to illustrate the methodologies in a real application, that is, in the Air Quality area.

Accounting for long-term serial correlation in a linear regression problem

The method is proposed for accounting for the long-term serial correlation (autocorrelation) of data in a linear regression problem. A serially correlated residual series is presented as an autoregressive (AR) process of an order, k, that can be much larger than 1, and the autocorrelation (AK) function of the processes is calculated by solving the system of the Yule-Walker equations. Given the AK function, the corresponding AK matrix is constructed which enters the formulas for estimates of regression coefficients and their errors. The efficiency of the method is demonstrated on the base of the multiple regression analysis of data of ground-based measurements of the column NO2 content at the Zvenigorod Research Station, Russia, and overpass satellite total ozone measurements over the station. Estimates of regression coefficients and their errors depend on the AR order, k. At first the error increases with increasing k. Then it can approach its maximum and thereafter begin to decrease. In the case of NO2 and O3 at Zvenigorod the errors more than double in their maxima compared to the beginning values. The decrease in the error stops at larger k if k approaches the value such that the AR process of this order is able to account for important features of the AK function of the residual series. The multiple linear regression model for NO2 and O3 observations includes the seasonally dependent linear trends and the effects of the solar cycle, the quasi-biennial oscillation, the North Atlantic and Southern Oscillations, and the Pinatubo volcanic eruption. Annual and seasonally dependent estimates of these effects in NO2 and O3 have been obtained taking into account the serial correlation as long as 50 months.

An extended Levinson-Durbin algorithm and its application in mixed excitation linear prediction

Ten order all-pole model is used in the 2400 bit/second mixed excitation linear prediction to describe human vocal tract. Traditional Levinson-Durbin algorithm is one of the methods to solve the Yule-Walker equations conducted by the ten order linear prediction model. Taking the iteration step of traditional Levinson-Durbin algorithm as 1, an extended algorithm with any positive integer iteration step which is no larger than the order of Teoplitz matrix is proposed. The extended algorithm considers interaction between the adjacent subtracts. A hybrid algorithm of the extended algorithm and traditional algorithm has been applied to solve the 2400 bit/second mixed excitation linear prediction under some conditions. The perceptual evaluation of speech quality mean opinion score of nasal syllable is improved in some degree.

Banded spatio-temporal autoregressions

We propose a new class of spatio-temporal models with unknown and banded autoregressive coefficient matrices. The setting represents a sparse structure for high-dimensional spatial panel dynamic models when panel members represent economic (or other type) individuals at many different locations. The structure is practically meaningful when the order of panel members is arranged appropriately. Note that the implied autocovariance matrices are unlikely to be banded, and therefore, the proposal is radically different from the existing literature on the inference for high-dimensional banded covariance matrices. Due to the innate endogeneity, we apply the least squares method based on a Yule–Walker equation to estimate autoregressive coefficient matrices. The estimators based on multiple Yule–Walker equations are also studied. A ratio-based method for determining the bandwidth of autoregressive matrices is also proposed. Some asymptotic properties of the inference methods are established. The proposed methodology is further illustrated using both simulated and real data sets.

Subjectively correlated estimation of noise due to blurriness distortion based on auto‐regressive model using the Yule–Walker equations

In this study, a block-based estimation of noise due to blurriness distortion is proposed based on auto-regressive (AR) modelling. In the proposed method; a de-correlated, low-energy version of the blurred image is auto regressively modelled. To this end, AR parameters are estimated using the Yule–Walker equations. As these equations include auto-correlation function (ACF) coefficients, ACF estimation is also required. The Yule–Walker equations are solved making use of Durbin–Levinson algorithm. Finally, noise energy is mathematically defined and computed for each block. Since blurriness is a signal-dependent image distortion, estimating and describing its characteristics via a noise like that of the AR model input, is significant. In fact, extracting features of such ‘noise’ can lead to the design and development of a new method of image quality metrics. Inspired by the ‘stem cells’ concept in medical science that is convertible to other cell types, the AR model input is called ‘stem noise’. To visualise contribution of the ‘Stem Noise’ in the reconstruction of blurriness image distortion, a map called stem noise energy map is created. It is shown that the characteristics of the estimated noise energy are well correlated with the human subjective scores.

Banded Spatio-Temporal Autoregressions

Abstract We propose a new class of spatio-temporal models with unknown and banded autoregressive coefficient matrices. The setting represents a sparse structure for high-dimensional spatial panel dynamic models when panel members represent economic (or other type) individuals at many different locations. The structure is practically meaningful when the order of panel members is arranged appropriately. Note that the implied autocovariance matrices are unlikely to be banded, and therefore, the proposal is radically different from the existing literature on the inference for high-dimensional banded covariance matrices. Due to the innate endogeneity, we apply the least squares method based on a Yule–Walker equation to estimate autoregressive coefficient matrices. The estimators based on multiple Yule–Walker equations are also studied. A ratio-based method for determining the bandwidth of autoregressive matrices is also proposed. Some asymptotic properties of the inference methods are established. The proposed methodology is further illustrated using both simulated and real data sets.

Two-dimensional noisy autoregressive estimation with application to joint frequency and direction of arrival estimation

The present study addresses the problem of two-dimensional autoregressive estimation in the presence of additive white noise. The estimation method is based on combining the low-order and high-order Yule-Walker equations. The noise-compensated YW equations are solved using an iterative algorithm. The proposed method is also applied to joint frequency and direction of arrival estimation in uniform linear arrays. Using simulation study, the performance of the proposed algorithm is evaluated and compared with other methods.

Proof of Local Convergence of a New Algorithm for Covariance Matching of Periodic ARMA Models

In a companion paper, the authors have proposed a new algorithm for the partial stochastic realization of vector discrete-time processes from finite covariance data, based on a nonlinear generalization of the classical Yule-Walker equations. In particular the algorithm provides solutions of the covariance matching problem for periodic ARMA models on a finite interval. In this letter, we provide a proof of convergence of the algorithm for scalar periodic ARMA models based on Lyapunov stability theory.

Approximation of Vector Processes by Covariance Matching With Applications to Smoothing

We propose a new algorithm for the partial stochastic realization of vector-valued periodic processes from finite covariance data, based on a nonlinear generalization of the classical Yule-Walker equations. We discuss an application to finite-interval smoothing of a linear time-invariant system.