Asymptotic normality of error density estimator in stationary and explosive autoregressive models

In this paper,we consider the limit distribution of the error density function estima-tor in the rst-order autoregressive models with negatively associated and positively associated random errors.Under mild regularity assumptions,some asymptotic normality results of the residual density estimator are obtained when the autoregressive models are stationary process and explosive process.In order to illustrate these results,some simulations such as con dence intervals and mean integrated square errors are provided in this paper.It shows that the residual density estimator can replace the density\estimator"which contains errors.

JUMP DETECTION BY WAVELET IN NONLINEAR AUTOREGRESSIVE MODELS

Wavelets are applied to detection of the jump points of a regression function in nonlinear autoregressive model x(t) = T(x(t-1)) + epsilon t. By checking the empirical wavelet coefficients of the data,which have significantly large absolute values across fine scale levels, the number of the jump points and locations where the jumps occur are estimated. The jump heights are also estimated. All estimators are shown to be consistent. Wavelet method ia also applied to the threshold AR(1) model(TAR(1)). The simple estimators of the thresholds are given,which are shown to be consistent.

Polynomial network autoregressive models with divergent orders

We propose a novel polynomial network autoregressive model by incorporating higher-order connected relationships to simultaneously model the effects of both direct and indirect connections. A quasimaximum likelihood estimation method is proposed to estimate the unknown influence parameters, and we demonstrate its consistency and asymptotic normality without imposing any distribution assumption. Moreover,an extended Bayesian information criterion is set for order selection with a divergent upper order. The application of the proposed polynomial network autoregressive model is demonstrated through both the simulation and the real data analysis.

Constructing Confidence Regions for Autoregressive-Model Parameters

We discuss formulas and techniques for finding maximum-likelihood estimators of parameters of autoregressive (with particular emphasis on Markov and Yule) models, computing their asymptotic variance-covariance matrix and displaying the resulting confidence regions;Monte Carlo simulation is then used to establish the accuracy of the corresponding level of confidence. The results indicate that a direct application of the Central Limit Theorem yields errors too large to be acceptable;instead, we recommend using a technique based directly on the natural logarithm of the likelihood function, verifying its substantially higher accuracy. Our study is then extended to the case of estimating only a subset of a model’s parameters, when the remaining ones (called nuisance) are of no interest to us.

A Score Type Test for General Autoregressive Models in Time Series

This paper is devoted to the goodness-of-fit test for the general autoregressive models in time series. By averaging for the weighted residuals, we construct a score type test which is asymptotically standard chi-squared under the null and has some desirable power properties under the alternatives. Specifically, the test is sensitive to alternatives and can detect the alternatives approaching, along a direction, the null at a rate that is arbitrarily close to n-1/2. Furthermore, when the alternatives are not directional, we construct asymptotically distribution-free maximin tests for a large class of alternatives. The performance of the tests is evaluated through simulation studies.

Inference for Spatial Autoregressive Models with Infinite Variance Noises

A self-weighted quantile procedure is proposed to study the inference for a spatial unilateral autoregressive model with independent and identically distributed innovations belonging to the domain of attraction of a stable law with index of stabilityα,α∈(0,2).It is shown that when the model is stationary,the self-weighted quantile estimate of the parameter has a closed form and converges to a normal limiting distribution,which avoids the difficulty of Roknossadati and Zarepour(2010)in deriving their limiting distribution for an M-estimate.On the contrary,we show that when the model is not stationary,the proposed estimates have the same limiting distributions as those of Roknossadati and Zarepour.Furthermore,a Wald test statistic is proposed to consider the test for a linear restriction on the parameter,and it is shown that under a local alternative,the Wald statistic has a non-central chisquared distribution.Simulations and a real data example are also reported to assess the performance of the proposed method.

Empirical Likelihood-Based Subset Selection for Partially Linear Autoregressive Models

Based on the empirical likelihood method, the subset selection and hypothesis test for parameters in a partially linear autoregressive model are investigated. We show that the empirical log-likelihood ratio at the true parameters converges to the standard chi-square distribution. We then present the definitions of the empirical likelihood-based Bayes information criteria （EBIC） and Akaike information criteria （EAIC）. The results show that EBIC is consistent at selecting subset variables while EAIC is not. Simulation studies demonstrate that the proposed empirical likelihood confidence regions have better coverage probabilities than the least square method, while EBIC has a higher chance to select the true model than EAIC.

Empirical likelihood inference in autoregressive models with time-varying variances

This paper develops the empirical likelihood(EL)inference procedure for parameters in autore-gressive models with the error variances scaled by an unknown nonparametric time-varying function.Compared with existing methods based on non-parametric and semi-parametric esti-mation,the proposed test statistic avoids estimating the variance function,while maintaining the asymptotic chi-square distribution under the null.Simulation studies demonstrate that the proposed EL procedure(a)is more stable,i.e.,depending less on the change points in the error variances,and(b)gets closer to the desired confidence level,than the traditional test statistic.

Noise reduction of acoustic Doppler velocimeter data based on Kalman filtering and autoregressive moving average models

Oceanic turbulence measurements made by an acoustic Doppler velocimeter(ADV)suffer from noise that potentially affects the estimates of turbulence statistics.This study examines the abilities of Kalman filtering and autoregressive moving average models to eliminate noise in ADV velocity datasets of laboratory experiments and offshore observations.Results show that the two methods have similar performance in ADV de-noising,and both effectively reduce noise in ADV velocities,even in cases of high noise.They eliminate the noise floor at high frequencies of the velocity spectra,leading to a longer range that effectively fits the Kolmogorov-5/3 slope at midrange frequencies.After de-noising adopting the two methods,the values of the mean velocity are almost unchanged,while the root-mean-square horizontal velocities and thus turbulent kinetic energy decrease appreciably in these experiments.The Reynolds stress is also affected by high noise levels,and de-noising thus reduces uncertainties in estimating the Reynolds stress.

Trend Autoregressive Model Exact Run Length Evaluation on a Two-Sided Extended EWMA Chart

The Extended Exponentially Weighted Moving Average(extended EWMA)control chart is one of the control charts and can be used to quickly detect a small shift.The performance of control charts can be evaluated with the average run length(ARL).Due to the deriving explicit formulas for the ARL on a two-sided extended EWMA control chart for trend autoregressive or trend AR(p)model has not been reported previously.The aim of this study is to derive the explicit formulas for the ARL on a two-sided extended EWMA con-trol chart for the trend AR(p)model as well as the trend AR(1)and trend AR(2)models with exponential white noise.The analytical solution accuracy was obtained with the extended EWMA control chart and was compared to the numer-ical integral equation(NIE)method.The results show that the ARL obtained by the explicit formula and the NIE method is hardly different,but the explicit for-mula can help decrease the computational(CPU)time.Furthermore,this is also expanded to comparative performance with the Exponentially Weighted Moving Average(EWMA)control chart.The performance of the extended EWMA control chart is better than the EWMA control chart for all situations,both the trend AR(1)and trend AR(2)models.Finally,the analytical solution of ARL is applied to real-world data in the healthfield,such as COVID-19 data in the United Kingdom and Sweden,to demonstrate the efficacy of the proposed method.

Influencing Factors and Prediction of Risk of Returning to Ecological Poverty in Liupan Mountain Region,China

China has resolved its overall regional poverty in 2020 by attaining moderate societal prosperity.The country has entered a new development stage designed to achieve its second centenary goal.However,ecological fragility and risk susceptibility have increased the risk of returning to ecological poverty.In this paper,the Liupan Mountain Region of China was used as a case study,and the counties were used as the scale to reveal the spatiotempora differentiation and influcing factors of the risk of returning to poverty in study area.The indicator data for returning to ecological poverty from 2011-2020 were collected and summarized in three dimensions:ecological,economic and social.The autoregressive integrated moving average model(ARIMA)time series and exponential smoothing method(ES)were used to predict the multidimensional indicators of returning to ecological poverty for 61 counties(districts)in the Liupan Mountain Region for 2021-2030.The back propagation neural network(BPNN)and geographic information system(GIS)were used to generate the spatial distribution and time variation for the index of the risk of returning to ecological poverty(RREP index).The results show that 1)ecological factors were the main factors in the risk of returning to ecological poverty in Liupan Mountain Region.2)The RREP index for the 61 counties(districts)exhibited a downward trend from 2021-2030.The RREP index declined more in medium-and high-risk areas than in low-risk areas.From 2021 to 2025,the RREP index exhibited a slight downward trend.From 2026 to2030,the RREP index was expected to decline faster,especially from 2029-2030.3)Based on the RREP index,it can be roughly divided into three types,namely,the high-risk areas,the medium-risk areas,and the low-risk areas.The natural resource conditions in lowrisk areas of returning to ecological poverty,were better than those in medium-and high-risk areas.

AUTOREGRESSIVE MODEL AND POWER SPECTRUM CHARATERISTICS OF CURRENT SIGNAL IN HIGH FREQUENCY GROUP PULSE MICRO-ELECTROCHEMICAL MACHINING

The identification of the inter-electrode gap size in the high frequency group pulse micro-electrochemical machining （HGPECM） is mainly discussed. The auto-regressive（AR） model of group pulse current flowing across the cathode and the anode are created under different situations with different processing parameters and inter-electrode gap size. The AR model based on the current signals indicates that the order of the AR model is obviously different relating to the different processing conditions and the inter-electrode gap size; Moreover, it is different about the stability of the dynamic system, i.e. the white noise response of the Green＇s function of the dynamic system is diverse. In addition, power spectrum method is used in the analysis of the dynamic time series about the current signals with different inter-electrode gap size, the results show that there exists a strongest power spectrum peak, characteristic power spectrum（CPS）, to the current signals related to the different inter-electrode gap size in the range of 0～5 kHz. Therefore, the CPS of current signals can implement the identification of the inter-electrode gap.

Empirical likelihood for first-order mixed integer-valued autoregressive model

In this paper, we not only construct the confidence region for parameters in a mixed integer-valued autoregressive process using the empirical likelihood method, but also establish the empirical log-likelihood ratio statistic and obtain its limiting distribution. And then, via simulation studies we give coverage probabilities for the parameters of interest. The results show that the empirical likelihood method performs very well.

PARTICLE FILTERING BASED AUTOREGRESSIVE CHANNEL PREDICTION MODEL

A particle filtering based AutoRegressive (AR) channel prediction model is presented for cognitive radio systems. Firstly, this paper introduces the particle filtering and the system model. Secondly, the AR model of order p is used to approximate the flat Rayleigh fading channels; its stability is discussed, and an algorithm for solving the AR model parameters is also given. Finally, an AR channel prediction model based on particle filtering and second-order AR model is presented. Simulation results show that the performance of the proposed AR channel prediction model based on particle filtering is better than that of Kalman filtering.

Multivariate Generalized Autoregressive Conditional Heteroscedastic Model

In this paper, by making use of the Hadamard product of matrices, a natural and reasonable generalization of the univariate GARCH (Generalized Autoregressive Conditional heteroscedastic) process introduced by Bollerslev (J. Econometrics 31(1986), 307-327) to the multivariate case is proposed. The conditions for the existence of strictly stationary and ergodic solutions and the existence of higher-order moments for this class of parametric models are derived.

Threshold autoregression models for forecasting El Nino events

-In this paper, monthly mean SST data in a large area are used. After the spacial average of the data is carried out and the secular monthly means are substracted, a time series (Jan. 1951-Dec. 1985) of SST anomalies of the cold tongue water area in the eastern tropical Pacific Ocean is obtained. On the basis of the time series, an autoregression model, a self-exciting threshold autoregression model and an open loop autoregression model are developed respectively. The interannual variations are simulated by means of those models. The simulation results show that all the three models have made very good hindcasting for the nine El Nino events since 1951. In order to test the reliability of the open loop threshold model, extrapolated forecast was made for the period of Jan. 1986-Feb. 1987. It can be seen from the forecasting that the model could forecast well the beginning and strengthening stages of the recent El Nino event (1986-1987). Correlation coefficients of the estimations to observations are respectively 0. 84, 0. 88 and 0. 89. It is obvious that all the models work well and the open loop threshold one is the best. So the open loop threshold autoregression model is a useful tool for monitoring the SSTinterannual variation of the cold tongue water area in the Eastern Equatorial Pacific Ocean and for estimating the El Nino strength.

Modeling groundwater nitrate concentrations using spatial and non-spatial regression models in a semi-arid environment

Nitrate nitrogen(NO_(3)^(-)N)from agricultural activities and in industrial wastewater has become the main source of groundwater pollution,which has raised widespread concerns,particularly in arid and semi-arid river basins with little water that meets relevant standards.This study aimed to investigate the performance of spatial and non-spatial regression models in modeling nitrate pollution in a semi-intensive farming region of Iran.To perform the modeling of the groundwater's NO_(3)^(-)N concentration,both natural and anthropogenic factors affecting groundwater NO_(3)^(-)N were selected.The results of Moran's I test showed that groundwater nitrate concentration had a significant spatial dependence on the density of wells,distance from streams,total annual precipitation,and distance from roads in the study area.This study provided a way to estimate nitrate pollution using both natural and anthropogenic factors in arid and semi-arid areas where only a few factors are available.Spatial regression methods with spatial correlation structures are effective tools to support spatial decision-making in water pollution control.

The m-delay Autoregressive Model with Application

The classical autoregressive(AR)model has been widely applied to predict future data usingmpast observations over five decades.As the classical AR model required m unknown parameters,this paper implements the AR model by reducing m parameters to two parameters to obtain a new model with an optimal delay called as the m-delay AR model.We derive the m-delay AR formula for approximating two unknown parameters based on the least squares method and develop an algorithm to determine optimal delay based on a brute-force technique.The performance of them-delay AR model was tested by comparing with the classical AR model.The results,obtained from Monte Carlo simulation using the monthly mean minimum temperature in PerthWestern Australia from the Bureau of Meteorology,are no significant difference compared to those obtained from the classical AR model.This confirms that the m-delay AR model is an effective model for time series analysis.

Bootstrap Approaches to Autoregressive Model on Exchange Rates Currency

The use of historical data is important in making the predictions, for instance in the exchange rate. However, in the construction of a model, extreme data or dirtiness of data is inevitable. In this study, AR model is used with the exchange rate historical data (January 2007 until December 2007) for USD/MYR and is divided into 1-, 3- and 6-horizontal months respectively. Since the presence of extreme data will affect the accuracy of the results obtained in a prediction. Therefore, to obtain a more accurate prediction results, the bootstrap approach was implemented by hybrid with AR model coins as the Bootstrap Autoregressive model (BAR). The effectiveness of the proposed model is investigated by comparing the existing and the proposed model through the statistical performance methods which are RMSE, MAE and MAD. The comparison involves 1%, 5% and 10% for each horizontal month. The results showed that the BAR model performed better than the AR model in terms of sensitivity to extreme data, the accuracy of forecasting models, efficiency and predictability of the model prediction. In conclusion, bootstrap method can alleviate the sensitivity of the model to the extreme data, thereby improving the accuracy of forecasting model which also have high prediction efficiency and that can increase the predictability of the model.

Adaptive Time-Frequency Distribution Based on Time-Varying Autoregressive and Its Application to Machine Fault Diagnosis

The time-varying autoregressive （TVAR） modeling of a non-stationary signal is studied. In the proposed method, time-varying parametric identification of a non-stationary signal can be translated into a linear time-invariant problem by introducing a set of basic functions. Then, the parameters are estimated by using a recursive least square algorithm with a forgetting factor and an adaptive time-frequency distribution is achieved. The simulation results show that the proposed approach is superior to the short-time Fourier transform and Wigner distribution. And finally, the proposed method is applied to the fault diagnosis of a bearing , and the experiment result shows that the proposed method is effective in feature extraction.