scholarly journals New estimation method for periodic autoregressive time series of order 1 with additive noise

2021 ◽  
Author(s):  
Wojciech Żuławiński ◽  
Agnieszka Wyłomańska
Keyword(s):  
Time Series ◽  
Additive Noise ◽  
Moving Average ◽  
Estimation Method ◽  
Real Data ◽  
Finite Variance ◽  
Arma Process ◽  
Periodic Behavior ◽  
Autocovariance Function ◽  
External Forces

AbstractThe periodic behavior of real data can be manifested in the time series or in its characteristics. One of the characteristics that often manifests the periodic behavior is the sample autocovariance function. In this case, the periodically correlated (PC) behavior is considered. One of the main models that exhibits PC property is the periodic autoregressive (PARMA) model that is considered as the generalization of the classical autoregressive moving average (ARMA) process. However, when one considers the real data, practically the observed trajectory corresponds to the “pure” model with the additional noise which is a result of the noise of the measurement device or other external forces. Thus, in this paper we consider the model that is a sum of the periodic autoregressive (PAR) time series and the additive noise with finite-variance distribution. We present the main properties of the considered model indicating its PC property. One of the main goals of this paper is to introduce the new estimation method for the considered model’s parameters. The novel algorithm takes under consideration the additive noise in the model and can be considered as the modification of the classical Yule–Walker algorithm that utilizes the autocovariance function. Here, we propose two versions of the new method, namely the classical and the robust ones. The effectiveness of the proposed methodology is verified by Monte Carlo simulations. The comparison with the classical Yule–Walker method is presented. The approach proposed in this paper is universal and can be applied to any finite-variance models with the additive noise.

ARMA processes have maximal entropy among time series with prescribed autocovariances and impulse responses

1985 ◽  
Vol 17 (04) ◽  
pp. 810-840 ◽  
Author(s):  
Jürgen Franke
Keyword(s):  
Time Series ◽  
Impulse Response ◽  
Moving Average ◽  
Autoregressive Process ◽  
Natural Generalization ◽  
Maximal Entropy ◽  
Autoregressive Moving Average ◽  
Moving Average Process ◽  
Arma Process ◽  
Arma Processes

The maximum-entropy approach to the estimation of the spectral density of a time series has become quite popular during the last decade. It is closely related to the fact that an autoregressive process of order p has maximal entropy among all time series sharing the same autocovariances up to lag p. We give a natural generalization of this result by proving that a mixed autoregressive-moving-average process (ARMA process) of order (p, q) has maximal entropy among all time series sharing the same autocovariances up to lag p and the same impulse response coefficients up to lag q. The latter may be estimated from a finite record of the time series, for example by using a method proposed by Bhansali (1976). By the way, we give a result on the existence of ARMA processes with prescribed autocovariances up to lag p and impulse response coefficients up to lag q.

Continuous Autoregressive Moving Average Models

2021 ◽  
pp. 118-141
Author(s):  
Yakup Ari
Keyword(s):  
Time Series ◽  
Parameter Estimation ◽  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Discrete Time ◽  
Moving Average ◽  
Likelihood Estimation ◽  
Real Data ◽  
Autoregressive Moving Average ◽  
The Difference

The financial time series have a high frequency and the difference between their observations is not regular. Therefore, continuous models can be used instead of discrete-time series models. The purpose of this chapter is to define Lévy-driven continuous autoregressive moving average (CARMA) models and their applications. The CARMA model is an explicit solution to stochastic differential equations, and also, it is analogue to the discrete ARMA models. In order to form a basis for CARMA processes, the structures of discrete-time processes models are examined. Then stochastic differential equations, Lévy processes, compound Poisson processes, and variance gamma processes are defined. Finally, the parameter estimation of CARMA(2,1) is discussed as an example. The most common method for the parameter estimation of the CARMA process is the pseudo maximum likelihood estimation (PMLE) method by mapping the ARMA coefficients to the corresponding estimates of the CARMA coefficients. Furthermore, a simulation study and a real data application are given as examples.

scholarly journals A New Strategy of Hybrid Models using ARIMA, ANN, and DWT in Time Series Modelling

2021 ◽  
Vol 25 (1) ◽  
pp. 27-50
Author(s):  
Tsung-Lin Li ◽  
◽  
Chen-An Tsai ◽  
Keyword(s):  
Time Series ◽  
Moving Average ◽  
Arima Model ◽  
Real Data ◽  
Hybrid Models ◽  
Discrete Wavelet ◽  
Data Sets ◽  
Model Combining ◽  
Time Series Modelling ◽  
Significant Difference

Time series forecasting is a challenging task of interest in many disciplines. A variety of techniques have been developed to deal with the problem through a combination of different disciplines. Although various researches have proved successful for hybrid models, none of them carried out the comparisons with solid statistical test. This paper proposes a new stepwise model determination method for artificial neural network (ANN) and a novel hybrid model combining autoregressive integrated moving average (ARIMA) model, ANN and discrete wavelet transformation (DWT). Simulation studies are conducted to compare the performance of different models, including ARIMA, ANN, ARIMA-ANN, DWT-ARIMA-ANN and the proposed method, ARIMA-DWT-ANN. Also, two real data sets, Lynx data and cabbage data, are used to demonstrate the applications. Our proposed method, ARIMA-DWT-ANN, outperforms other methods in both simulated datasets and Lynx data, while ANN shows a better performance in the cabbage data. We conducted a two-way ANOVA test to compare the performances of methods. The results showed a significant difference between methods. As a brief conclusion, it is suggested to try on ANN and ARIMA-DWT-ANN due to their robustness and high accuracy. Since the performance of hybrid models may vary across data sets based on their ARIMA alike or ANN alike natures, they should all be considered when encountering a new data to reach an optimal performance.

scholarly journals COVID-19 pandemic control using restrictions and vaccination

2021 ◽  
Vol 19 (2) ◽  
pp. 1355-1372
Author(s):  
Vinicius Piccirillo ◽  
Keyword(s):  
Time Series ◽  
Moving Average ◽  
Real Data ◽  
The United States ◽  
Prevention Policy ◽  
Autoregressive Integrated Moving Average ◽  
Infected People ◽  
Marquardt Algorithm ◽  
The Impact ◽  
Methods Numerical

<abstract><p>This work deals with the impact of the vaccination in combination with a restriction parameter that represents non-pharmaceutical interventions measures applied to the compartmental SEIR model in order to control the COVID-19 epidemic. This restriction parameter is used as a control parameter, and the univariate autoregressive integrated moving average (ARIMA) is used to forecast the time series of vaccination of all individuals of a specific country. Having in hand the time series of the population fully vaccinated (real data + forecast), the Levenberg–Marquardt algorithm is used to fit an analytic function that models this evolution over time. Here, it is used two time series of real data that refer to a slow vaccination obtained from India and Brazil, and two faster vaccination as observed in Israel and the United States of America. Together with vaccination, two different control approaches are presented in this paper, which enable reduces the infected people successfully: namely, the feedback and nonfeedback control methods. Numerical results predict that vaccination can reduce the peaks of infections and the duration of the pandemic, however, a better result is achieved when the vaccination is combined with any restrictions or prevention policy.</p></abstract>

scholarly journals Time Series Modelling of The Volume of Overland Timber Export in The Bolgatanga Municipality

2020 ◽  
pp. 124-138
Author(s):  
Henry M. Kpamma ◽  
Silverius K. Bruku ◽  
John A. Awaab
Keyword(s):  
Time Series ◽  
Moving Average ◽  
Preliminary Investigation ◽  
Arima Model ◽  
Estimation Method ◽  
Long Run ◽  
Time Series Modelling ◽  
The Future ◽  
Equilibrium Relationships ◽  
The Right

Aims/ Objectives: This research was carried out with the intention of using time series to model the volume of overland timber exported within Bolgatanga municipalityPlace and Duration of Study: Study of the time series was based on a historical data of the volume of timber exported for twenty consecutive years, from 1999 to 2019 within Bolgatanga municipality.Methodology: The three-stage iterative modeling approach for Box Jenkins was used to match an ARIMA model and to forecast both the amount of timber export and the confiscated lumber. ARIMA method incorporates a cycle of autoregressive and a moving average. The three-stage iterative modeling technique of Box Jenkins which were used are model recognition, parameter estimation and/or diagnostic checks were also made. Results: From the preliminary investigation, the study showed that the amount of timber exported in municipality is skewed to the right, suggesting that much of the amount of timber exported is below the average. This, together with the high volatility in the volume of timber exported, indicates that the amount of timber exported within the municipalities during the twenty-year period was low. The plots from the trends also showed robust variations in the volume of timber exported indicating that timber exporters do not have better grips with the concepts and applications of export technology, hence the erratic nature of the volume of timber exported over the period. The quadratic pattern and the ARIMA (1,1,1) model best represented the amount of timber exported.The analysis further indicated that there will be a further decrease in the amount of timber export from the five years projection into the future. Over the last two decades the Bayesian approach to VAR has gained ground. For a future report, this estimation method will be followed to examine the ”long-run equilibrium relationships” between timber export volumes and climate change.Conclusion: The quadratic pattern and the ARIMA (1,1,1) model best represented the amount of timber exported. There will be a further decrease in the amount of timber export from the five years projection into the future.

ARMA processes have maximal entropy among time series with prescribed autocovariances and impulse responses

1985 ◽  
Vol 17 (4) ◽  
pp. 810-840 ◽  
Author(s):  
Jürgen Franke
Keyword(s):  
Time Series ◽  
Impulse Response ◽  
Moving Average ◽  
Autoregressive Process ◽  
Natural Generalization ◽  
Maximal Entropy ◽  
Autoregressive Moving Average ◽  
Moving Average Process ◽  
Arma Process ◽  
Arma Processes

The maximum-entropy approach to the estimation of the spectral density of a time series has become quite popular during the last decade. It is closely related to the fact that an autoregressive process of order p has maximal entropy among all time series sharing the same autocovariances up to lag p. We give a natural generalization of this result by proving that a mixed autoregressive-moving-average process (ARMA process) of order (p, q) has maximal entropy among all time series sharing the same autocovariances up to lag p and the same impulse response coefficients up to lag q. The latter may be estimated from a finite record of the time series, for example by using a method proposed by Bhansali (1976). By the way, we give a result on the existence of ARMA processes with prescribed autocovariances up to lag p and impulse response coefficients up to lag q.

MODELING POLIO DATA USING THE FIRST ORDER NON-NEGATIVE INTEGER-VALUED AUTOREGRESSIVE, INAR(1), MODEL

2012 ◽  
Vol 09 ◽  
pp. 232-239 ◽  
Author(s):  
TURAJ VAZIFEDAN ◽  
MAHENDRAN SHITAN
Keyword(s):  
Time Series ◽  
Time Series Data ◽  
Moving Average ◽  
Negative Integer ◽  
Series Data ◽  
Autoregressive Moving Average ◽  
Road Accidents ◽  
Arma Process ◽  
Number Of Patients ◽  
Number Of Customers

Time series data may consists of counts, such as the number of road accidents, the number of patients in a certain hospital, the number of customers waiting for service at a certain time and etc. When the value of the observations are large it is usual to use Gaussian Autoregressive Moving Average (ARMA) process to model the time series. However if the observed counts are small, it is not appropriate to use ARMA process to model the observed phenomenon. In such cases we need to model the time series data by using Non-Negative Integer valued Autoregressive (INAR) process. The modeling of counts data is based on the binomial thinning operator. In this paper we illustrate the modeling of counts data using the monthly number of Poliomyelitis data in United States between January 1970 until December 1983. We applied the AR(1), Poisson regression model and INAR(1) model and the suitability of these models were assessed by using the Index of Agreement(I.A.). We found that INAR(1) model is more appropriate in the sense it had a better I.A. and it is natural since the data are counts.

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