Vol.11 (2021) No. 1 ISSN: 2088-5334 Employment the State Space and Kalman Filter Using ARMA models Najlaa Saad Ibrahim a,* , Heyam A.A. Hayawi a a Department of Statistics and Informatics, University of Mosul, Mosul, Iraq Corresponding author: * najlaa.s.a@uomosul.edu.iq Abstract—The research is interested in studying a modern mathematical topic of great importance in contemporary applications known as the representation of the state space for mathematical models of time series represented by ARMA models and the discussion of a Kalman filter such as the one who has very general characteristics and of the utmost importance and depends on the representation of the state space. Raw data on electrical energy consumption in Mosul city have been used for the period from (15/6/2003 to 25/9/2003), and after examining these data as to whether they are stationary or not, it was found that there is no stationary for the series behavior in the arithmetic mean, variance and after conversion. The state-space model is characterized by being an efficient scale in all states that are not observed or controlled, and for this, the state-space model can be used to estimate states that cannot be observed. It can also express the state-space model simply for complex operations and is characterized by the flexible model. The series into a stationary time series with variance and mean. The autocorrelation function (ACF) and the partial autocorrelation function (PACF) have been calculated, and observation of the propagation behavior of these two functions shows that the best model for representing data is ARMA (2,1) model. And then, the parameters of the model were estimated using the matrix system for the state-space model and then taking advantage of the state-space model in estimating the observation equation for a Kalman filter such as the security and it was found that a Kalman filter such as security is very efficient in purifying the series from noise. Keywords— Time series; state-space; Kalman filter; recursive particularity. Manuscript received 7 Jul. 2020; revised 24 Dec. 2020; accepted 7 Jan. 2021. Date of publication 28 Feb. 2021. IJASEIT is licensed under a Creative Commons Attribution-Share Alike 4.0 International License. I. INTRODUCTION Since the beginning of the seventh decade of the twentieth century, the subject of Time Series Analysis has emerged as one of the vital topics at various levels. Applications of this subject have expanded, so we do not find a scientific, technical, or literary field free of it. Usually, we are interested in the subject of time series analysis to study phenomena or variables that change with time change such as the number of heartbeats per minute, the temperature during hours on a particular day, the daily closing price of a specific company’s shares, as well as fluctuations in currency exchange rates and financial stock markets, among others. Sometimes it is important to use filters to extract important patterns in time series data, as filters are used to show some of the time series properties as the general direction. The primary purpose of filters is to obtain the optimum estimator. The first pioneering studies on the issue of filters appeared in the early forties of the last century, as (Kolmogorov) in 1941 and Wiener in 1942 independently developed a technique to find a linear estimate with linear minimum mean-square error estimation, which received great attention and subsequently had a significant impact in developing the idea of a Kalman filter. [1], then [2] culminated that study with other results resulting in the development of a Recursive Algorithm to find the optimum linear estimator, and this algorithm was constrained by conditions such that the studied view has a single dimension (Scalar) as well as the parameter. The data is almost endless, and the stationary process is stable. The results of (Wiener) developed by making it more general to make the data expired and cover the non-stationary processes [3]. Other studies followed in the same field, including but not limited to the study undertaken by Kim [4], which did not exceed the results of Sharaf et al [5]. II. MATERIALS AND METHODS It is a very general method of mathematical representation developed by Salmi et al. [6]. Through the state-space method, the relationships between Inputs and Outputs dynamic systems can be represented. It is known that the outputs of the determent dynamic system depend on both the inputs as well as the previous outputs. To know the number of previous outputs that we need to know the current output of the system. If we have a dynamic system described by a 145