First-order Autoregressive Model Research Articles

The effect of autocorrelation on control charts performance and process capability indices calculation

Chemical processes as well as many non-industrial processes exhibit autocorrelation, for which the above-mentioned control procedures are not suitable. This paper considers the problem of monitoring a process in which the observations can be represented as a first-order autoregressive model following a heavy tailed distribution. It also presents practical usage of time series control charts on the chemical process of monitoring concentrations. The second part discusses the effect of autocorrelation on the process capability analysis when the observations are made by an autoregressive model of the first order. The process capability indices provide a measure of how a process fits within the specification limits. When calculating indices, it is usual to assume that the process data are independent. The simulation study in this part discusses the effect of a higher autocorrelation order on process capability indices.

Forecasting the number of confirmed new cases of COVID-19 in Italy for the period from 19 May to 2 June 2020.

In this paper we forecast the spread of the coronavirus disease 2019 outbreak in Italy in the time window from May 19 to June 2, 2020. In particular, we consider the forecast of the number of new daily confirmed cases. A forecast procedure combining a log-polynomial model together with a first-order integer-valued autoregressive model is proposed. An out-of-sample comparison with forecasts from an autoregressive integrated moving average (ARIMA) model is considered. This comparison indicates that our procedure outperforms the ARIMA model. The Root Mean Square Error (RMSE) of the ARIMA is always greater than that of the our procedure and generally more than twice as high as the our procedure RMSE. We have also conducted Diebold and Mariano (1995) tests of equal mean square error (MSE). The tests results confirm that forecasts from our procedure are significantly more accurate at all horizons. We think that the advantage of our approach comes from the fact that it explicitly takes into account the number of swabs.

Forecasting the annual flow rate of the Tigris river using stochastic modelling

The monthly and annual time series of the flow of the Tigris River at Al-Amara barrage was analyzed in order to forecast the annual and monthly future values of discharge of that river. The monthly series from 1980-2010 was used as data for this analysis. Non-homogeneity was detected in the series with significant positive jump’s observed in the periods 1987-1988 and 1992-1997. The non-homogeneity was removed using the method suggested by Reference [1]. A Box and Cox [2] transformation was then used to normalize the homogeneous series. The dependent stochastic component was obtained from the series after removing the periodic component, which was observed using harmonic analysis after fitting the normalized series. A first-order auto-regressive model (Markovian chain) was then used to model the last obtained component. The data in the period 1980-2000 was used to conduct that analysis. For generation of a future series, the model was verified using the remaining 9-years, 2001-2010. Some statistical properties were obtained for both the forecasted and the observed series and then used to compare the two series. The comparison indicates that the model is capable of forecasting acceptable future data.

A new mixed first-order integer-valued autoregressive process with Poisson innovations

Integer-valued time series, seen as a collection of observations measured sequentially over time, have been studied with deep notoriety in recent years, with applications and new proposals of autoregressive models that broaden the field of study. This work proposes a new mixed integer-valued first-order autoregressive model with Poisson innovations, denoted POMINAR(1), mixing two operators known as binomial thinning and Poisson thinning. The proposed process presents some advantages in relation to the most common Poisson innovation processes: (1) this new process allows to capture structural changes in the data; (2) if there are no structural changes, the most common processes with Poisson innovations are particular cases of POMINAR(1). Another important contribution of this work is the establishment of the POMINAR(1) theoretical results, such as the marginal expectation, marginal variance, conditional expectation, conditional variance, transition probabilities. Moreover, the Conditional Maximum Likelihood (CML) and Yule-Walker (YW) estimators for the process parameters are studied. We also present three techniques for one-step-ahead forecasting, the nearest integer of the conditional expectation, conditional median and mode. A simulation study of the forecasting procedures, considering the two estimators, CML and YW methods, is performed, and prediction intervals are presented. Finally, we show an application of the proposed process to a real dataset, referred here as larceny data, including a residual analysis.

EWMA control charts for monitoring correlated counts with finite range

In this work, we develop and study upper and lower one-sided EWMA control charts for monitoring correlated counts with finite range. Often in practice, data of that kind can be adequately described by a first-order binomial or beta-binomial autoregressive model. Especially, when there is evidence that data demonstrate extra-binomial variation, the latter model is preferable than the former. The proposed charts can be used for detecting upward or downward shifts in process mean level. Practical guidelines concerning the statistical design of the proposed charts are given, while the effect of the extra-binomial variation is investigated as well. Comparisons with existing control charting procedures are also provided. Finally, an illustrative real-data example is also given.

Life expectancy and lifespan disparity forecasting: a long short-term memory approach

After the World War II, developed countries experienced a constant decline in mortality. As a result, life expectancy has never stopped increasing, despite an evident deceleration in developed countries, e.g. England, USA and Denmark. In this paper, we propose a new approach for forecasting life expectancy and lifespan disparity based on the recurrent neural networks with a long short-term memory. This type of neural network leads to predicting future values of longevity indexes while maintaining the significant influence of the past trend, but at the same time adequately reproducing the recent trend into forecasting. The model was applied to five countries for two fitting periods focusing on the forecasting life expectancy and lifespan disparity, both independently and simultaneously at birth and age 65. The results were compared to the projections obtained by four different models, namely, the Double Gap, ARIMA, CoDa and Lee-Carter in the independent case and the first-order Vector Autoregression model in the simultaneous case. Our predictions seem to be coherent with historical trends and biologically reasonable, providing a more accurate portrait of the future life expectancy and lifespan disparity.

A dependent counting INAR model with serially dependent innovation

ABSTRACT To provide a more flexible model of count data, we extend the first-order integer-valued autoregressive model with serially dependent innovations based on the dependent thinning operator. This model is appropriate for modelling the number of dependent random events affecting each other when the number of new cases depend on the previous count through a linear functional relationship. Several statistical properties of the model are determined, parameters are estimated by some methods and their properties are studied via simulations. This study was carried out to investigate the efficiency of the new model by two real count data sets, the number of contagious diseases and robbery.

Asymptotic theory for time series with changing mean and variance

The paper develops point estimation and asymptotic theory with respect to a semiparametric model for time series with moving mean and unconditional heteroscedasticity. These two features are modelled nonparametrically, whereas autocorrelations are described by a short memory stationary parametric time series model. We first study the usual least squares estimate of the coefficient of the first-order autoregressive model based on constant but unknown mean and variance. Allowing for both the latter to vary over time in a general way we establish its probability limit and a central limit theorem for a suitably normed and centred statistic, giving explicit bias and variance formulae. As expected mean variation is the main source of inconsistency and heteroscedasticity the main source of inefficiency, though we discuss circumstances in which the estimate is consistent for, and asymptotically normal about, the autoregressive coefficient, albeit inefficient. We then consider standard implicitly-defined Whittle estimates of a more general class of short memory parametric time series model, under otherwise more restrictive conditions. When the mean is correctly assumed to be constant, estimates that ignore the heteroscedasticity are again found to be asymptotically normal but inefficient. Allowing a slowly time-varying mean we resort to trimming out of low frequencies to achieve the same outcome. Returning to finite order autoregression, nonparametric estimates of the varying mean and variance are given asymptotic justification, and forecasting formulae developed. Finite sample properties are studied by a small Monte Carlo simulation, and an empirical example is also included.

Confirmed detection of Palaeogene and Jurassic orbitally-forced sedimentary cycles in the depth domain using False Discovery Rates and Bayesian probability spectra

It has been common practice to assume that power spectral backgrounds in cyclostratigraphy conform to a first-order autoregressive (AR1) model. Vaughan et al. (2011, Paleoceanography) argued that an unbiased approach to fitting the spectral backgrounds, as well as adjustment of confidence levels for multiple frequency testing, should be mandatory during the search in the depth domain for significant spectral peaks. To address these requirements Smoothed Window Averaging to find spectral backgrounds are combined with False Discovery Rates (FDR) for setting confidence levels and were applied to time series from seven Oligocene and Jurassic formations. Bayesian probability spectra provide an alternative method for detecting regular cyclicity. Pre-whitening the linearly detrended time series prior to calculation of Bayesian probabilities avoids confounding effects due to red noise. In all seven formations there are sub-sections associated with spectral peaks exceeding the 5% FDR, in four formations they even exceed the 0.01% FDR. Elevated Bayesian probability at the same frequencies as these significant power spectral peaks, supports the detection of regular cyclicity. This prevalence of detections conflicts with the assertion of Vaughan et al. (2011) that “almost certainly the vast majority of cycle detections … in the stratigraphy literature are false.” In previous publications spectral peaks exceeding the standard 95% level were considered significant so very high confidence levels were not reported. Nevertheless, the examples re-studied demonstrate that pre-Neogene cyclostratigraphic time series do indeed contain regular cycles most likely linked to orbital-forcing.

The new synthetic and runs-rules schemes to monitor the process mean of autocorrelated observations with measurement errors

The use of the 2-of-(H + 1) runs-rules and synthetic schemes to improve the performance of the currently available schemes in monitoring the process mean under the combined effect of measurement errors and autocorrelation is proposed. To maximize the detection ability of the 2-of-(H + 1) runs-rules and synthetic schemes, we implement the modified side-sensitive (MSS) design approach for the charting regions as we show it yields the best possible performance out of all the available designs. These new monitoring schemes incorporate the additive model with a constant standard deviation and a first-order autoregressive model to the computation of the control limits in order to account for measurement errors and autocorrelation, respectively. Moreover, to construct a dedicated Markov chain matrix, the abovementioned models and some sampling methods are incorporated into the values of probability elements which are then used to derive closed-form expressions for the zero- and steady-state run-length distribution. A real life example is used to illustrate the practical implementation of the proposed schemes.

Assessment of geocenter motion estimates from the IGS second reprocessing

We investigate geocenter motion time series derived from the combined solutions and six individual analysis center (AC) solutions of the International GNSS Service (IGS) second reprocessing campaign using the network shift approach, in terms of noise content, long-term trends, periodic and aperiodic variations. We assess these GNSS geocenter motion estimates by comparison with independent estimates from satellite laser ranging (SLR). The GNSS geocenter time series exhibit correlated noise which is better represented by a white plus power–law noise model in the X and Y directions, and by a white plus first-order autoregressive (or generalized Gauss–Markov) noise model in the Z direction. The GNSS geocenter time series include expected seasonal variations, but also spurious draconitic signals, particularly in the Z direction. GNSS annual geocenter motion estimates are in reasonable agreement with SLR estimates in the X and Y directions. In the Z direction, however, the annual signals derived from the IGS solutions disagree with SLR estimates, except for three particular ACs. This suggests that the different orbit modeling strategies used by these ACs may constitute an improvement over the conventional strategy employed by the other ACs. The background noise in GNSS and SLR geocenter time series finally appears to be correlated, suggesting that it might partly reflect real, aperiodic geocenter motion.

Choosing various likelihood functions on uncertainty assessment in groundwater simulation-optimization model

Abstract The main goal in this research is study of impacts of various likelihood functions on DREAM(zs) (Differential Evolution Adaptive Metropolis) method results in simulation-optimization model of aquifer. In this study, DREAM(zs) algorithm has been developed to study aquifer simulation-optimization model uncertainties. DREAM(zs) is used to investigate uncertainty of parameters of the simulation-optimization model in Isfahan-Barkhar aquifer, Isfehan province, Iran. This study is carried out on an aquifer simulation model of MODFLOW that is coupled with MOPSO (multi-objective particle swarm optimization) optimization. Three likelihood functions, L1, L2, and L3, are considered as informal and the remaining (L4 and L5) are represented as formal categories. Likelihood function L1 is Nash–Sutcliffe efficiency and L2 is based on minimum mean square error. L3 uses estimation of model error variance and L4 focuses on the relationship between the traditional least squares fitting and the Bayesian inference. In likelihood function L5 the serial dependence of residual errors is calculated using a first-order autoregressive model of the residuals. Results suggested that the parameters sensitivity depend on the likelihood function selection, and sensitivity of all parameters is not similar in different likelihood functions. MOPSO algorithm outputs indicated that likelihood function No. 5 has a higher speed in reaching convergence and this function also showed that objective functions had a better performance compared to the other likelihood functions.

УМОВИ ТОРГІВЛІ ТА ІНДУСТРІАЛІЗАЦІЯ: РЕАЛІЇ В УКРАЇНІ

he paper explores the impact of terms of trade on the industrialization and economic growth in Ukraine due to significant vulnerability of national economy to foreign economic shocks, its openness and mainly commodity structure of exports. In this research we have chosen manufacturing value added as percent in GDP to identify periods of industrialization, as its growth corresponds to periods of accelerated industrial development and vice versa. Also we considered GDP per capita as indicator of national economy’s performance. The changes in terms of trade were investigated based on the analysis of terms of trade adjustments, which are determined by the ability to import goods and services minus exports at constant prices. As an empirical research tool vector autoregressive models have been chosen to explore the relationship between endogenous and exogenous variables in dynamics. Thus, the endogenous variables in the model are the annual growth rate of GDP per capita, manufacturing value added and terms of trade adjustments in 1991-2018. Therefore, the first-order vector autoregression model was constructed to examine this relationship. According to the results of research, acceleration of terms of trade adjustments growth rate (deterioration of terms of trade) in Ukraine leads to fluctuations in the manufacturing value added growth with an increase of almost 3% in the second period and further declining. It indicates an increase in industrial production in response to the deterioration of terms of trade in the short run and possible intensification of innovative economic growth triggers. Fluctuations in manufacturing value added account for from almost 7% to 14% in fluctuations of GDP per capita growth. In turn, fluctuations in terms of trade adjustments account for only from 3% to almost 5%. At the same time, fluctuations in manufacturing value added from 8% to 13% are explained by fluctuations in terms of trade adjustments.

Trading with Path-Dependent Effect: A Continuous-Time Cointegration Perspective

Continuous-time pairs-trading rules are often developed based on the diffusion limit of first-order autoregressive cointegration models. Empirical identification of cointegration effects is generally made according to discrete-time error correction representation of general vector autoregressive moving average (VARMA($p,q$)) processes. We show that the diffusion limit of a VARMA($p,q$) process appears as a stochastic delayed differential equation. Motivated by this, we investigate the dynamic portfolio problem under a class of path-dependent models embracing path-dependent cointegration models in continuous time as special cases. Under certain reasonable conditions, we prove the existence of the optimal strategy and show that it is related to a system of differential equations. The proof is developed by means of functional It\^{o}'s calculus. When the process satisfies cointegration conditions, our results lead to the optimal dynamic pairs-trading rule. By contrasting our strategy with a pairs-trading strategy that ignores the path-dependent effect (or serial dependency), we numerically demonstrate the superiority of our strategy.

Fractional differenced first-order autoregressive model for long-range correlations and biosignal

Fractional differenced first-order autoregressive model for long-range correlations and biosignal

Parameter Estimation Effect of the Homogeneously Weighted Moving Average Chart to Monitor the Mean of Autocorrelated Observations With Measurement Errors

In statistical process monitoring, the usual assumption when designing monitoring schemes is that process parameters are known and have perfect measurements with independent and identically distributed observations. However, in real-life situation, these assumptions rarely hold. Hence, in this paper, the Phase II performance of the homogenously weighted moving average (HWMA) $\bar {X}$ monitoring scheme under the combined effect of autocorrelation and measurement errors is investigated when the unknown process parameters are estimated from an in-control Phase I dataset. Two models are considered, i.e. the first-order autoregressive model for within-sample autocorrelation and the linear covariate model for (constant and linearly increasing variance) measurement system error. Sampling strategies based on skipping some observations as well as mixing different subgroup samples and taking multiple measurements are implemented to reduce the negative effect of autocorrelation and measurement errors. Since the latter sampling strategies incur costs, as an alternative, increasing the slope coefficient of the linear covariate model compensate the negative effect of measurement errors. The new HWMA $\bar {X}$ scheme is shown to have some interesting detection abilities as compared to its competitors. A real-life example is used to illustrate the implementation of the proposed monitoring scheme.

Estimation for random coefficient integer-valued autoregressive model under random environment

A first-order random coefficient integer-valued autoregressive model based on the negative binomial thinning operator under r states random environment is introduced. This paper derives numerical characteristics of the proposed model, establishes Yule–Walker estimators of model parameters, and discusses the strong consistency of the obtained estimators. Finally, a simulation is carried out to verify the feasibility of parameter estimation.

Flexible modelling of spatial variation in agricultural field trials with the R package INLA

Key messageEstablished spatial models improve the analysis of agricultural field trials with or without genomic data and can be fitted with the open-source R package INLA.The objective of this paper was to fit different established spatial models for analysing agricultural field trials using the open-source R package INLA. Spatial variation is common in field trials, and accounting for it increases the accuracy of estimated genetic effects. However, this is still hindered by the lack of available software implementations. We compare some established spatial models and show possibilities for flexible modelling with respect to field trial design and joint modelling over multiple years and locations. We use a Bayesian framework and for statistical inference the integrated nested Laplace approximations (INLA) implemented in the R package INLA. The spatial models we use are the well-known independent row and column effects, separable first-order autoregressive (mathrm{AR1} otimes mathrm{AR1}) models and a Gaussian random field (Matérn) model that is approximated via the stochastic partial differential equation approach. The Matérn model can accommodate flexible field trial designs and yields interpretable parameters. We test the models in a simulation study imitating a wheat breeding programme with different levels of spatial variation, with and without genome-wide markers and with combining data over two locations, modelling spatial and genetic effects jointly. The results show comparable predictive performance for both the mathrm{AR1} otimes mathrm{AR1} and the Matérn models. We also present an example of fitting the models to a real wheat breeding data and simulated tree breeding data with the Nelder wheel design to show the flexibility of the Matérn model and the R package INLA.

Efficient monitoring of autocorrelated Poisson counts

Statistical surveillance for autocorrelated Poisson counts has drawn considerable attention recently. These works are usually based on a first-order integer-valued autoregressive model and focus on monitoring separately either the marginal mean or the autocorrelation coefficient. Inspired by multivariate statistical process control, this article transforms autocorrelated Poisson counts into a bivariate representation and proposes an efficient control chart. By borrowing the power of the likelihood ratio test, albeit surprisingly, this chart demonstrates almost uniformly stronger power than the existing alternatives in simultaneously detecting shifts in both the marginal mean and the autocorrelation coefficient. In addition, the robustness of the proposed chart against overdispersion encountered often in counts is also verified. It is shown that this chart also has superiority in monitoring autocorrelated overdispersed counts.

South Pacific Decadal Climate Variability and Potential Predictability

Abstract The South Pacific decadal oscillation (SPDO) characterizes the Southern Hemisphere contribution to the Pacific-wide interdecadal Pacific oscillation (IPO) and is analogous to the Pacific decadal oscillation (PDO) centered in the North Pacific. In this study, upper ocean variability and potential predictability of the SPDO is examined in HadISST data and an atmosphere-forced ocean general circulation model. The potential predictability of the IPO-related variability is investigated in terms of both the fractional contribution made by the decadal component in the South, tropical and North Pacific Oceans and in terms of a doubly integrated first-order autoregressive (AR1) model. Despite explaining a smaller fraction of the total variance, we find larger potential predictability of the SPDO relative to the PDO. We identify distinct local drivers in the western subtropical South Pacific, where nonlinear baroclinic Rossby wave–topographic interactions act to low-pass filter decadal variability. In particular, we show that the Kermadec Ridge in the southwest Pacific enhances the decadal signature more prominently than anywhere else in the Pacific basin. Applying the doubly integrated AR1 model, we demonstrate that variability associated with the Pacific–South American pattern is a critically important atmospheric driver of the SPDO via a reddening process analogous to the relationship between the Aleutian low and PDO in the North Pacific—albeit that the relationship in the South Pacific appears to be even stronger. Our results point to the largely unrecognized importance of South Pacific processes as a key source of decadal variability and predictability.