Analysis of Some Linear Dynamic Systems with Bivariate Wavelets

Abstract

There are many statistical methods related to the forecasting of time series without any input variables such as autoregressive integrated moving average (ARIMA models). In this research, some linear dynamic systems, represented by ARIMA with exogenous input variables (ARIMAX models) were used to forecast crude oil prices (considered as output variable) for OPEC organization with the help of crude oil production (considered as input variable) depending on the data starting from the period of 1973 until 2018. Using traditional ARIMAX method and proposed method (Bivariate Wavelet Filtering) for the time series data in order to select one of them for forecasting through comparing some measures of accuracy, such as MSE, FPE, and AIC. Then, applying crude oil prices for OPEC using the traditional ARIMAX models and ARIMAX models with applying the bivariate wavelet filtering, especially bivariate Haar wavelet. The main conclusions of the research were that the success of bivariate wavelet filtering in forecasting of crude oil prices using proposed model was more appropriate than traditional models, and the forecasting of crude oil prices using proposed method in 2020 will be fairly less than 2019. This is an open access article under the CC BY 4.0 license http://creativecommons.org/licenses/by/4.0/). \ ‫الثنائية‬ ‫المويجات‬ ‫مع‬ ‫الخطية‬ ‫الديناميكية‬ ‫النماذج‬ ‫بعض‬ ‫تحليل‬ ‫الملخص‬ ‫اىطش‬ ٍِ ‫اىعذٌذ‬ ‫هْاك‬ ‫ائ‬ ‫باىسيسيت‬ ‫اىخْبؤ‬ ً‫ف‬ ‫اىَخعيقت‬ ‫اإلدصائٍت‬ ‫ق‬ ‫اى‬ ٌ‫أ‬ ُ‫دو‬ ‫زٍٍْت‬ ‫ت‬ ‫ٍخغٍشاث‬ ‫ٍثو‬ ‫داخيت‬ ‫أ‬ ‫اال‬ ‫َّىرج‬ ‫اىَخذشمت‬ ‫ىألوساط‬ ‫اىَخناٍو‬ ً‫اىزاح‬ ‫ّذذاس‬ ‫َّارج‬ (ARIMA .) ‫و‬ ‫هزا‬ ً‫ف‬ ،‫اىبذث‬ ً‫إسخخذا‬ ٌ‫ح‬ ‫اىخطٍت‬ ‫اىذٌْاٍٍنٍت‬ ‫األّظَت‬ ‫بعض‬ ‫اىَخَثيت‬ ‫ب‬ ‫ئ‬ ‫َّىرج‬ ARIMA ‫ٍخغٍشاث‬ ‫ٍع‬ ‫(َّارج‬ ‫اىخاسجٍت‬ ‫اىَذخالث‬ ARIMAX ،) ‫فقذ‬ ‫ا‬ ٌ(ً‫اىخا‬ ‫اىْفط‬ ‫بأسعاس‬ ‫ىيخْبؤ‬ ‫سخخذٍج‬ ‫عذ‬ ‫مَخغٍش‬ ‫(ٌع‬ ً‫اىخا‬ ‫اىْفط‬ ‫إّخاج‬ ‫بَساعذة‬ ‫أوبل‬ ‫ىَْظَت‬)‫حابع‬ ‫ذ‬)‫ٍسخقو‬ ‫مَخغٍش‬ ‫و‬ ‫حبذأ‬ ً‫اىخ‬ ‫اىبٍاّاث‬ ‫عيى‬ ‫ًا‬ ‫اعخَاد‬ ‫اىفخشة‬ ٍِ 1793 ‫دخى‬ 2012 ‫اىخقيٍذٌت‬ ‫اىطشٌقت‬ ً‫باسخخذا‬. ARIMAX ‫(ٍششخ‬ ‫اىَقخشح‬ ‫واألسيىب‬)‫اىثْائٍت‬ ‫اىَىٌجاث‬ ‫أجو‬ ٍِ ‫اىزٍٍْت‬ ‫اىسيسيت‬ ‫ىبٍاّاث‬ ‫ا‬ ‫خخٍاس‬ َ‫إدذاه‬ ‫ا‬ ‫بعض‬ ‫ٍقاسّت‬ ‫خاله‬ ٍِ ‫ىيخْبؤ‬ ‫ٍثو‬ ‫اىذقت‬ ‫ٍقاٌٍس‬ MSE ، FPE ، AIC. ] 68 Iraqi Journal of Statistical Science (30) 2019 [ ‫حطبٍق‬ ٌ‫ث‬ ‫ا‬ ‫َّارج‬ ً‫باسخخذا‬ ‫ألوبل‬ ً‫اىخا‬ ‫اىْفط‬ ‫سعاس‬ ARIMAX ‫اىخقيٍذٌت‬ ‫وَّارج‬ ARIMAX ‫حطبٍق‬ ‫ٍع‬ ‫اىَخغٍش‬ ‫ثْائٍت‬ ‫اىَىٌجاث‬ ‫ٍششخ‬ ، ً ‫وخصىصا‬ ‫اىثْائٍت‬ ‫هاس‬ ‫ٍىٌجت‬ (Haar .) ‫و‬ ٍِ ً ‫أ‬ ٌ‫ه‬ ‫االس‬ ‫اىذساس‬ ‫اىٍها‬ ‫حىصيج‬ ً‫اىخ‬ ‫خْخاجاث‬ ‫ت‬ ُ ٌ ‫أ‬ ‫ثْائٍت‬ ‫اىَىٌجاث‬ ‫ٍششخ‬ ‫ّجاح‬ ‫باسعاس‬ ‫اىخْبؤ‬ ً‫ف‬ ‫اىَخغٍش‬ ‫ٍال‬ ‫أمثش‬ ‫ماّج‬ ‫اىَقخشح‬ ‫اىَْىرج‬ ً‫باسخخذا‬ ً‫اىخا‬ ‫اىْفط‬ ‫ئ‬ ‫اىَْارج‬ ٍِ ‫َت‬ ‫سْت‬ ً‫ف‬ ‫اىَقخشدت‬ ‫اىطشٌقت‬ ً‫باسخخذا‬ ً‫اىخا‬ ‫اىْفط‬ ‫بأسعاس‬ ‫اىخْبؤ‬ ُ‫وسٍنى‬ ،‫اىخقيٍذٌت‬ 2020 ‫ّى‬ ‫أقو‬ ً ‫ًا‬ ‫ع‬ ‫ٍا‬ ٍِ 2017. ‫اة‬ ‫اليئي:ا‬ ‫اات‬ ‫الكلما‬ ‫لت‬ ‫اىزٍٍْل‬ ‫لو‬ ‫اىسالسل‬ ، ‫لت‬ ‫اىخطٍل‬ ‫لت‬ ‫اىذٌْاٍٍنٍل‬ ‫لت‬ ‫األّظَل‬ ، ‫لارج‬ ‫َّل‬ ARIMAX ، ‫لو‬ ‫حذىٌل‬ ‫اىَىٌجاث‬ ، ً‫اىثْائ‬ ‫اىَىٌجت‬ ‫ٍششخ‬. 1. Introduction An important field of Statistics is Time Series Analysis which could be used in many other scientific fields in order to make future planning and management for governmental and non-governmental institutions. It deals with the methods and theory involved in analyzing datasets which are collected over time and helps to understand the past behavior of phenomena. It can be used for comparison between two or more times series concerning the type of growth, for instance growth in crude oil prices, consumption of a product, etc. Forecasting of Linearly time series analysis is a famous developed method that researchers use in their forecast, but in the actual world most data systems are nonlinear, which demonstrates itself in a much larger extents of possible dynamical behaviors (Brockwell and Davis, 2016). Also, it is clear that models based on the time series only, without taking any related input variable into consideration, could not be accurate. Therefore, more accurate models are taken by integrating exogenous input variables into the model which is called Linear Dynamic Systems (Nelles, 2001). The linear dynamic systems (LDS) model is the most commonly used time series model for economic, engineering, financial, and medical applications because of its respective simplicity, mathematically predictable behavior, inference, and predictions could be done effectively for the models. It is possible to identify LDS models using parametric and nonparametric models, and both are divided into time domain models and frequency domain models. Time domain models consist of equation error models and output error models. Equation error models are the study"s goal Thus, the equation error models is divided into ARX, ARMAX, and ARARX (Nelles, 2001; Saaed , 2015; ،‫أبىىبذة‬ 2019). The ARIMAX models can be utilized to enhance the forecasting of output series (Y), through applying the past data of both the Y series and the exogenous input series (X). That is especially right if the input series is an important indicator (Wei, 2006).