First-order Autoregressive Process Research Articles

On Discrete Poisson–Mirra Distribution: Regression, INAR(1) Process and Applications

Several pieces of research have spotlighted the importance of count data modelling and its applications in real-world phenomena. In light of this, a novel two-parameter compound-Poisson distribution is developed in this paper. Its mathematical functionalities are investigated. The two unknown parameters are estimated using both maximum likelihood and Bayesian approaches. We also offer a parametric regression model for the count datasets based on the proposed distribution. Furthermore, the first-order integer-valued autoregressive process, or INAR(1) process, is also used to demonstrate the utility of the suggested distribution in time series analysis. The unknown parameters of the proposed INAR(1) model are estimated using the conditional maximum likelihood, conditional least squares, and Yule–Walker techniques. Simulation studies for the suggested distribution and the INAR(1) model based on this innovative distribution are also undertaken as an assessment of the long-term performance of the estimators. Finally, we utilized three real datasets to depict the new model’s real-world applicability.

Bias-correction of some estimators in the INAR(1) process

A class of estimators in the first-order nonnegative integer-valued autoregressive process is considered, which contains the Yule–Walker, Burg, and method of moment estimators. Bias-correction and higher-order mean squared error comparison are studied. Some simulations demonstrate that the bias-correction works well.

Memory Behaviors of Air Pollutions and Their Spatial Patterns in China

Particulate matter (PM2.5 and PM10) and ozone (O3) are the two major air pollutants in China in recent years. The fluctuations of PM2.5, PM10 and O3 strongly depend on the weather processes and anthropogenic emission. These processes may lead to the existence of short- and long-term memory behaviors in air pollutants. Hence, here we use the autoregressive parameter a of the first-order autoregressive process [AR (1)] to characterize the short-term memory effects of pollutants. We estimate the scaling exponent α using detrended fluctuation analysis (DFA) for the long-term memory effects of air pollutants (PM2.5, PM10, and O3) in summer and winter for different cities in China. Our results show that PM2.5, PM10, and O3 have strong short-term and long-term memory characteristics both in summer and winter. Furthermore, both the short- and long-term memory effects are stronger in winter than summer for most cities associated with stronger and longer persistent weather systems in winter. In general, the scaling exponent α of PM2.5 and PM10 are smaller for northern cities than those of southern cities in China. The long-term memory patterns of O3 are stronger in northern cities and weaker in southern cities in relative to those of PM2.5 and PM10 in winter. Our results show that the short- and long-term memory behaviors of air pollutions are dominated by the weather systems with different time scales.

Left–Right Ambiguity Resolution Methods for Closely Spaced Arrays

One linear array with omnidirectional sensors cannot discriminate between the azimuth angle and its complementary angle due to axisymmetry about the array axis. This article examines solutions to the bearing ambiguity problem known as left–right (LR) ambiguity resolution using two closely spaced parallel arrays. We design a second-order nulling (SoN) beamformer to produce a cardioid-type spatial response with a null electronically steered to the complementary angle of the direction of interest. Three methods—namely, SoN, null constrained beamformer (NCB), and 2-D direct beamformer (2-DDB), are studied. The performance is evaluated in correlated noise conditions modeled by a first-order auto-regressive process. This work establishes analytical results for the bound on the detection gain (DG) of the two parallel arrays in terms of the Rayleigh quotient of the circulant noise correlation matrix. A closed-form expression is developed for estimating the loss due to cardioid processing at closer inter-array spacing. The expressions are validated based on the deflection criterion. Twin-array processing using the 2-DDB method results in a higher DG, while cardioid-based beamformers give a higher rejection ratio (RR). Data from an at-sea experiment using a towed array are analyzed. Field results show that the deeper null achieved by the cardioid method results in a higher RR compared to 2-D direct beamformer. From the field data, approximately 19-dB LR discrimination is demonstrated at 1650 Hz for the broadside. Experimental results from a towed array corroborate theory and simulations.

Left-Right ambiguity resolution for coprime sparse arrays.

Coprime Sparse Arrays (CSA) with two collocated uniform linear arrays are used to estimate O(MN) direction of arrivals for a narrow band source with only O(M+N) sensors. The CSA cannot discriminate the azimuth and its complementary angles due to the symmetry along the array-axis. The bearing ambiguity, also referred to as a Left-Right (LR) ambiguity, is addressed in this paper using CSA. We design Left-Right resolved Coprime Sparse Arrays (LRCSA) to produce a null that is electronically steered to the complementary angle of the desired direction. LRCSA twin array achieves higher spatial resolution than the conventional fully populated LR array with same number of sensors, much like the collocated CSA. Three methods, namely, Null Constrained Beamformer (NCB), two-dimensional Direct Beamformer (2DDB), and LRCSA are studied. The bound on the detection gain is analytically established for the LRCSA under correlated noise scenarios modeled by the first-order autoregressive process. Twin array processing using the LRCSA method results in narrow mainlobe beam width along with a higher rejection ratio (RR) compared to 2DDB and NCB. Data obtained from a sea experiment using a towed array is analyzed. The results confirm that the CSA based cardioid processing resolved LR ambiguity along with a higher RR and narrower beams.

Optimal skip-lot sampling plan based on the yield index for auto-correlated data

Acceptance sampling is one of the most important tools in quality control, and it has been extensively used in the industry for inspecting received lots. As different situations and purposes could be encountered in real life, numerous approaches for sampling plans have been established. When the quality of the process is good and stable, Skip-lot sampling plans (SkSPs) are useful strategies of sampling and could be an advantageous examination scheme. Therefore, this paper aims to incorporate the process yield index for a first-order autoregressive process S p k ; A R ( 1 ) and the SkSP, when the quality characteristic follows a normal distribution with bilateral specification limits, to produce a variables Skip lot sampling plan of type 2 (SkSP-2) with a single sampling plan (SSP) based on S p k ; A R ( 1 ) as the reference plan. A non-linear optimization model is solved to decide the plan parameters, by fulfiling the quality and risk obligations identified by the manufacturer and buyer and minimizing the average sample number (ASN). We investigated the efficiency and performance of the suggested SkSP-2 S p k ; A R ( 1 ) -based, associated with the variables SSP in terms of average sample number curves and the operating characteristic. Furthermore, the plan parameters with particular conditions are delivered for reference as tables, and an industrial application is offered to prove the applicability of the proposed plan.

River levels from multi mission altimetry, a statistical approach

Satellite altimetry is a key technique to measure water level change in continental water bodies. Altimetry-based water level time series of rivers are typically constructed at locations where the satellite ground tracks intersect the rivers, the so-called virtual stations. The relatively low sampling frequency (10–27 days) of the repeat missions may result in an under-sampling of the hydrological regime in rivers with sub-monthly to weekly events. We are currently in a unique position with more than a handful satellite altimetry missions simultaneously mapping the surface elevation of the Earth. In combination, these missions contain an unexploited potential to obtain a more detailed picture of the hydrological regime of many of the Earth's rivers. The task of combining water levels measured at different locations and/or by satellites with different orbits is however, challenging due to e.g. topography, intermission biases, variation in river morphology, and other unidentified causes.In this work, we present a new method to combine multi-mission altimetry-based water levels from a river reach. This will also enable the use of geodetic missions like CryoSat-2 and SARAL/AltiKa (after June 2016) in water level time series. To combine the data we set up a state-space model where the process part is a first-order autoregressive process. The observations as a function of time and distance along the reach are described as a sum of the water level at a given time scaled by a distance-dependent factor, the mean water level at the given distance, and an error term. The scale factor and the mean water level are modeled with spline functions. We employ the model for the six rivers Lena, Solimões, Mississippi, Danube, Po, and Red, which range in width from 3 km to a few hundred meters. For each river, we consider a reach of 200–300 km and apply water levels from the satellite altimetry missions CryoSat-2, Sentinel-3A/3B, and SARAL/AltiKa. The selected reach must have a continuous elevation profile and preferable no major tributaries, which might alter the hydrological regime considerably. The length of the reach is a compromise of ensuring enough data but not violating the aforementioned criteria.When validated against in situ data we find a root mean square error ranging from 0.34 m (Solimões River) to 2.53 m (Lena River) and a correlation ranging from 0.83 (Danube River) to 0.99 (Solimões River). These summary statistics are based on approximately 2000–3000 pairs of in situ and modeled water levels. We find the largest increase in detail for the reconstructed water levels for the Danube, Po, and Red Rivers, where the water level variations are under-sampled at the virtual stations. For the Po River, we can detect sub-weekly events with the model and for the Lena River, the spring flood related to ice and snowmelt is better captured when combining the data. An additional advantage of the approach is that the water level time series can be reconstructed at all locations along the considered reach.

On one family of first passage times of a Markov random walk described by an autoregressive process AR(1) for nonlinear boundaries

The consider one family of first passage times of a Markov random walk described by a first-order autoregressive process AR(1) for nonlinear boundaries. Limit theorems are proved for a Markov random walk and a family of first passage times of this walk for nonlinear boundaries.

A matrix approach to get the variance of the square of the fluctuation function of the DFA

The fluctuation analysis (FA) and the detrended fluctuation analysis (DFA) make it possible to estimate the Hurst exponent H, which characterizes the self-similarity of a signal. Both are based on the fact that the so-called fluctuation function, which can be seen as an approximation of the standard deviation of the process scaled in time by multiplying the time variable by a positive constant, depends on H. The main novelty of the paper is to provide the expression of the variance of the square of the fluctuation function, by using a matrix formulation. We show that it depends on the correlation function of the signal under study when it is zero-mean and Gaussian. Illustrations are given when dealing with a zero-mean white Gaussian noise. Moving average processes and first-order autoregressive processes are also addressed.

Effect of the COVID-19 pandemic on the number of deaths in Peru: 2017.03-2020.07

The objective of the research is to estimate the excess deaths during the COVID-19 epidemic in Peru. The methodology used for the ARMA models with structural with a structural change in mean. In the first place, the results show that the behavior of the number of deaths before the pandemic follows a firstorder auto-regressive process (AR (1)) with a change in the mean in the month of March 2020; while by including the period of the pandemic, the behavior of the number of deaths follows the ARMA (1,1) process with two changes in the mean (March 2018 and March 2020). Second, the results show that the excess mortality during the COVID-19 pandemic between March and July 2020 is 58 885 deaths compared to years under normal conditions.

Discrete Pseudo Lindley Distribution: Properties, Estimation and Application on INAR(1) Process

In this paper, we introduce a discrete version of the Pseudo Lindley (PsL) distribution, namely, the discrete Pseudo Lindley (DPsL) distribution, and systematically study its mathematical properties. Explicit forms gathered for the properties such as the probability generating function, moments, skewness, kurtosis and stress–strength reliability made the distribution favourable. Two different methods are considered for the estimation of unknown parameters and, hence, compared with a broad simulation study. The practicality of the proposed distribution is illustrated in the first-order integer-valued autoregressive process. Its empirical importance is proved through three real datasets.

Temporal analysis of the heavy metal concentration in road sediment and dust using statistical models

The objective of this paper is to show a temporal analysis using autoregressive integrated moving average models of the heavy metal concentration in road sediment and dust of Soacha, a Colombian locality. The representative size fractions in the road sediment and dust were <250 μm and ⩽10 μm, respectively. The results suggest that lead is the best metallic element to study the relationship between the heavy metal concentration in the road sediment and dust (r-Pearson = 0.90). Univariate models (R2 ⩾ 0.58) suggest that the time series of lead concentrations in road sediment and dust have the same temporal structure. Namely, because they are first-order autoregressive processes, concentrations at a given moment of time are influenced by the immediately preceding concentrations. The transfer function model (R2 = 0.91) suggests that there is no delay in impulse transfer from road dust concentration to lead concentration in the road sediment. The effect is immediate for a sampling interval of 3 days. The results show that modeling has a better fit during the rainy season compared to the dry season. In the context of the simulation of physical phenomena in engineering, this study is relevant to deepen knowledge in relation to the use of autoregressive integrated moving average models.

Two-stage procedure in a first-order autoregressive process and comparison with a purely sequential procedure

A two-stage procedure in a first-order autoregressive model is considered that investigates the point and the interval estimation of parameters based on the least squares estimator. The two-stage procedure is shown to be as effective as the best fixed-sample-size procedure. In this regard, the significant properties of the procedure, such as asymptotic risk efficiency, asymptotic efficiency, and asymptotic consistency, are established. A Monte Carlo simulation study is conducted to compare the performance of the two-stage procedure and the purely sequential procedure. Finally, real-time series data are considered to illustrate the applicability of the two-stage procedure.

Ambient backscatter communication with mobile RF source for IoT-based applications

Ambient backscatter communication (AmBc) has been recently transpired as a promising communication methodology for battery-free internet of things (IoT)-based applications. This paper investigates an AmBc system with a mobile ambient radio-frequency (RF) source such as mobile wireless fidelity (Wi-Fi) access point. Mobility of the source causes the channels between source to tag and source to the reader to experience time-selective fading. This dynamic nature of channels is modeled using a first-order autoregressive (AR-1) process. The closed-form expressions for the bit-error-rate (BER) and thresholds are obtained for maximum-likelihood (ML) and energy detector (ED) with imperfect channel state information (CSI) at the reader. Further, the closed-form expressions for outage probability and channel capacity are also obtained. It is found that the overall system performance worsens with increase in mobility of the nodes. The results are validated using Monte-Carlo (MC) simulations.

Bayes factor asymptotics for variable selection in the Gaussian process framework

We investigate Bayesian variable selection in models driven by Gaussian processes, which allows us to treat linear, nonlinear and nonparametric models, in conjunction with even dependent setups, in the same vein. We consider the Bayes factor route to variable selection, and develop a general asymptotic theory for the Gaussian process framework in the “large p, large n” settings even with \(p\gg n\), establishing almost sure exponential convergence of the Bayes factor under appropriately mild conditions. The fixed p setup is included as a special case. To illustrate, we apply our result to variable selection in linear regression, Gaussian process model with squared exponential covariance function accommodating the covariates, and a first-order autoregressive process with time-varying covariates. We also follow up our theoretical investigations with ample simulation experiments in the above regression contexts and variable selection in a real, riboflavin data consisting of 71 observations and 4088 covariates. For implementation of variable selection using Bayes factors, we develop a novel and effective general-purpose transdimensional, transformation-based Markov chain Monte Carlo algorithm, which has played a crucial role in simulated and real data applications.

Probabilistic Projection of Subnational Life Expectancy

Abstract Projecting mortality for subnational units, or regions, is of great interest to practicing demographers. We seek a probabilistic method for projecting subnational life expectancy that is based on the national Bayesian hierarchical model used by the United Nations, and at the same time is easy to use. We propose three methods of this kind. Two of them are variants of simple scaling methods. The third method models life expectancy for a region as equal to national life expectancy plus a region-specific stochastic process which is a heteroskedastic first-order autoregressive process (AR(1)), with a variance that declines to a constant as life expectancy increases. We apply our models to data from 29 countries. In an out-of-sample comparison, the proposed methods outperformed other comparative methods and were well calibrated for individual regions. The AR (1) method performed best in terms of crossover patterns between regions. Although the methods work well for individual regions, there are some limitations when evaluating within-country variation. We identified four countries for which the AR(1) method either underestimated or overestimated the predictive between-region within-country standard deviation. However, none of the competing methods work better in this regard than the AR(1) method. In addition to providing the full distribution of subnational life expectancy, the methods can be used to obtain probabilistic forecasts of age-specific mortality rates.

The effect of autocorrelation on the diagnostic procedures

Apart from quick detection of abnormal changes in a multivariate process, it is also critical to accurately identify the factors of abnormal changes after an out-of-control signal in multivariate statistical process control. Some diagnostic methods for fault identification were proposed by scholars on which one of the basic assumptions is that the process data are independent. The independent assumption is reasonable for many applications. However, with the development of industrial automation, the process data usually meet the phenomenon of autocorrelation. As it is well known, the autocorrelation affects detection ability of control chart. It is a natural question that is whether the autocorrelation affects the diagnostic performance of the diagnostic procedures. In the article, we propose the method adopted the analogy residual sequences and compare it with the performance of existing diagnostic methods for different shift sizes and various autocorrelation and cross-correlation structures. To limit the complexity, our discussion employs a first-order vector autoregressive process and focuses mainly on bivariate data in this article. The results of the simulation show the performances of diagnostic procedures were affected by autocorrelation and the impacts become greater as autocorrelation increases. On the other hand, the proposed method in the article can be free from the impacts of autocorrelation. Finally, a real example is also presented to demonstrate the implementation of the proposed method.

NEW EXPONENTIAL PROBABILITY INEQUALITY AND COMPLETE CONVERGENCE FOR F-LNQD RANDOM VARIABLES SEQUENCE WITH APPLICATION TO AR(1)MODEL GENERATED BY F-LNQD ERRORS

The exponential probability inequalities have been important tools in probability and statistics. In this paper, we prove a new tail probability inequality for the distri-butions of sums of conditionally linearly negative quadrant dependent (F-LNQD , in short) random variables, and obtain a result dealing with conditionally complete con-vergence of first-order autoregressive processes with identically distributed (F-LNQD) innovations.

Tracking an Auto-Regressive Process with Limited Communication per Unit Time.

Samples from a high-dimensional first-order auto-regressive process generated by an independently and identically distributed random innovation sequence are observed by a sender which can communicate only finitely many bits per unit time to a receiver. The receiver seeks to form an estimate of the process value at every time instant in real-time. We consider a time-slotted communication model in a slow-sampling regime where multiple communication slots occur between two sampling instants. We propose a successive update scheme which uses communication between sampling instants to refine estimates of the latest sample and study the following question: Is it better to collect communication of multiple slots to send better refined estimates, making the receiver wait more for every refinement, or to be fast but loose and send new information in every communication opportunity? We show that the fast but loose successive update scheme with ideal spherical codes is universally optimal asymptotically for a large dimension. However, most practical quantization codes for fixed dimensions do not meet the ideal performance required for this optimality, and they typically will have a bias in the form of a fixed additive error. Interestingly, our analysis shows that the fast but loose scheme is not an optimal choice in the presence of such errors, and a judiciously chosen frequency of updates outperforms it.

Modeling The Amount of Insurance Claim using Gamma Linear Mixed Model with AR (1) random effect

The amount of insurance claims is continuous data and is positive so it is usually assumed to have gamma distribution. Usually, Generalized Linear Model (GLM)’s approach is used since the gamma distribution is a member of the exponential family. In case there is a random effect in modeling, then GLM can be extended to Generalized Linear Mixed Model (GLMM). This study models the amount of insurance claims with the most GLMM’s approach using two random effects, namely the region and time of the occurrence which is assumed to follow a first-order autoregressive process. The h-likelihood method is used to estimate the regression parameters and the variance parameters. A simulation study is carried out with an evaluation using the average relative bias and the average MSE. An application study which is conducted to model the amount of insurance claims in a certain region and time based on the 2014 profile of risk and loss of motor vehicle insurance in Indonesia is also carried out.