Comparison Between ARIMA And Fourier ARIMA Model To Forecast The Demand Of Electricity In Sulaimani Governorate

Electric energy is accounted as one of the major goods in human life, and also have a great role in progressing and developing the several sectors as economics, manufactures and any other sector related to daily use. In this study the monthly demand of electricity in Sulaimani governorate have been used, the main goal of the study is to choose appropriate model to forecast the monthly demand of electric in Sulaimani governorate for 12 months in 2020, the analyzing,results and comparison shows that FSARIMA(0,0,0)x(2,1,0) 4 is appropriate model


Introduction
Decision making is so sensitive when it is based on the forecasting methods, therefore the researcher should be careful during selection of forecasting methods, these methods are depending on the number of observation and its run-time.
The methodology of this paperis dependent on Seasonal Autoregressive Integrated Moving Average (SARIMA) model because there is no doupt that the using of electric contain seasonal effect, and the four season in our country are distingushed. The form of SARIMA model is (p,d,q)x(P,D,Q,s)where the first part (p,d,q) is non-seasonal ARIMA model and the second part (P,D,Q,s) is the seasonal. The seasonal part of the model consists of terms that are similar to the non-seasonal components of the model, but involve backshifts of the seasonal period. The application section contain the data that was used in this paper that comes from directory of electricity in Sulaimani city for monthly demand, whilefitted six different models which are presented in table 3.3, and the maximum likelihood estimation method have been used to estimate the parameter of the best model that is adequate the data under consideration.

Time series
A range of data variables is said to be time series data, which is consisting of sequential observations on a quantifiable variable(s), which is making more than one interval time. Usually, the observations are arranged according to time and taken at orderly intervals (days, months, years). There are many use of time series data in daily life including; (Economics, Finance, Environmental, and Medicine), whilea set of Doi: 10.25212/lfu.qzj. 5.3.36 for this mission of forecasting which has minimum AIC among the other candidate models that equal to 0.28 [1] [4] The processes and expansion of modern organizations are completely need time series data also public and personal foundation are using time series data to manage the networks, more use this type of data to understanding of thousands of time series data that consist of economic and financial information so in any field of life time series are necessary.

Time series Analysis
The reality about data information dots that accepted through the time, use time series analysis and also it must be acceptable inside template like(AR,trend or seasonal variation).as previously defined time series analysis acclimated expectation in statistics and other quality of data so that is mean use time series to drawing suitableinference, so the main purpose of time series analysis is forecasting. Forecasting is used to predict model for future values based on formerly observed time series values [8].
Data type and patterns display from the time series plots is a fundamental step in choosing a good model and forecasting procedure to a timeseries.Classical approaches of time series analysis are mainly affected with decaying the variation in series in terms of patterns in the time series data. The sources of variation are mostly classified into four main components. The components are: 2.3.1Trend [13] A trend pattern exists when there is a long-term secular increase or decrease in the data the sales of many companies, the gross national production, and many other business or economic indicators follow a trend pattern in their movement over time. [18] Is a timeseries the variations of seasonal are a short-term fluctuation that occurs in a year periodically, then it repeats year after year. The responsible for a pattern is customs of people or conditions of repetitive of seasonal variations as a major factor. In general,when a series is affected by seasonal factors the seasonality is occur [22] A cyclical pattern exists when the data are influenced by longer term fluctuations in economic which is related with the business cycle. The products sell such as automobiles and main devicedisplay this type of pattern.the major distinction between a seasonal and a cyclical pattern is firstconstant length and repeat on a regular periodic basis while the latter varies in length and magnitude [4] .

Irregular Variation [19]
The short duration fluctuations in a time series that is named by Irregular variation, which means it happened unsystematically variation at occurrence; these variations are also referred to as residual variations, cyclical and seasonal variations. Irregular

2.3.5Stationary and Non-Stationary Series [11]:
constant mean level iscalled stationary seriesdecreasing or increasing regullary according to time with fixed variance. Non-stationary series have regullar trends, such as linear, quadratic, and so on. Differencing can use for change non-stationary series to stationary and is called "non-stationary in the homogenous sense." Stationary is used as a tool in time series analysis, where the raw data are often changed to become stationary. For example, economic data dependent on a non-stationary price level and they are often seasonal. unreliable and dummy results product from non-stationary time series re caus whe e understanding and less forecasting.while the problem solution occurs when the data connected by time become stationary,so for the random walk the process is nonstationary with or without a drift,so the process convert to stationary by differencing.Differencing the grade is the easiest way to produce a non-stationary mean stationary (flat). The times number you have to difference the grade to create the stationary process chooses the value of d. If d equal to 0, the model is stationary previously and does not have any trend. When the series is differenced once, d=1 and linear trend is removed. When d=2 and both linear and quadratic trend are removed. For non-stationary series, d values of 1 or 2 are usually adequate to produce the mean stationary. If the time series data analyzed exhibits a deterministic trend, the spurious results can be avoided by detruding. Sometimes merge the stochastic and deterministic trend at the same time cause non-stationary series and to avoid obtaining misleading results both differencing and detruding should be applied, as differencing will remove the trend in the variance and detruding will remove the deterministic trend [2] [3] .
A process stationary has the property that the mean, variance and autocorrelation don't change over time. The mathematical term of Stationary can be defined in: 2. The variance  2 (t) = Var(y(t)) = γ(0) There are twokinds of stationary:

2.4Non-Stationary around Variance
In the case fluctuation of time series about the contrast and this discrepancy is not fixed, it means that the series is stationary about the contrast, and there are transferations to convert the string non stationary to a series of stationary, including the conversion logarithmic and transfers of power and the square root of the absence of stationary, about the contrast non-fixed and turn it into a series of fixed and stationary contrast by applying the following formula

Non-Stationary around the Mean [6][9]
The basic conditional in being a stationary time series about mean and middle hard as the changes that occur in the qualities and characteristics of chains with time makes it unstable so you must remove the property not stability, these chains are used difference method (Difference) to convert the string unstable to a series stable in terms of time difference and take her first and be in the following format: In 1970 the George-Box and Gwilyn Jenkinsapplied his model in time series data. It's model show that if the data stationary or not and solve the non-stationary data by differencing one or more than to become stationary data with the "I" standing for "Integrated" of an ARIMA model,also the box-Jenkins is used to solve many time series problem.This methodology depends on parts of procedure which is [autoregressive (AR), moving average (MA) and autoregressive moving average (ARMA)] that can be explained as follow:

Autoregressive (AR) model [9][12]
The order of AR model can be determined based onthe statistically significant partial autocorrelation function (PACF), the (P) stands for the order of AR model, the AR model can be written as follow: (2) By using backshift operator equation (3) can be rewrite as follow: : is the origin series.
To calculate Variance-Covariance the equation (3) should be multiplied by (Xt-k) and taking expectation so we get: Note: Then: To get the ACF the equation (5) should be divided by the variance of the series (ϒ0). Note: Then the PACF for the AR(P) model can be estimated by using Yule-Walker equations

Vol. (5), No (3), Summer 2020 ISSN 2518-6566 (Online) -ISSN 2518-6558 (Print)
916 the number of significant ACF is defined order of moving average model and -ʘ1 is the coefficient of dependency of observations (Xt) on the error term et and the previous error term at-1, the MA(q) model can be written as follow: Equation (8) can be reformulate with back shift operator as follow: The Var-Cov of MA(q) model is: And the ACF is:

Autoregressive Moving Average Model (ARMA) [14][17]
There is a large family of models which is named "Autoregressive-Moving Average Models" and abbreviated by ARMA. Many of researchers in different

Vol. (5), No (3), Summer 2020 ISSN 2518-6566 (Online) -ISSN 2518-6558 (Print)
917 application fields prove that ARMA models fits more than other traditional methods for forecasting [10] . The more general model is ARMA model because it is mix model between AR(p) and MA(q) models and it is called an ARMA model of order (p,q). The ARMA(p,q) is given by [19] : Where: We write equation (13) as:

The Autoregressive Integrated Moving Average Models (ARIMA) [19][15]
When you have not stationary time series, by difference operator can be change non-stationary to stationary, the time series data after differencing is called adjusted data and the fitted model is called integrated model which combines both autoregressive and moving average models.
Generally, all AR(p) and MA(q) models can be figured as ARIMA(1,0,0) this tells there is no differencing and no MA part. The general form of ARIMA model is written as (p,d,q) whilea degree of AR part is P and the differencing degree is D also MA for q [11] .
The general ARIMA process is of the form [13] Seasonal model perhaps used in a manner similar to that described thus far for no seasonalseries. For mix models, the exact order of the AEMA process -p and qand the length of seasonality must be specified. The sequential ARMA models are much simpler to apply.one starts by specifying as many parameters as the length of seasonality and then inspecting the autocorrelation of residual. In the seasonal model the form of seasonality shown by the significant ACF and PACF patterns. The number of times over year where seasonal revision happen is further importantfor yearly seasonality the ACF spikes are raise patterns at seasonal lags according the regular non-seasonal variation once a year. If the seasonality is quarterly, there will be eminent ACF spikes four times a year [5] .

Seasonal Autoregressive Integrated Moving Average (Seasonal ARIMA) Models [21][22]
The seasonal difference is an important tool in non-stationary seasonal modeling. The difference of seasonality of period s for the series {X } is symbol by ∇ X and is expressed as: The difference of seasonality will be n-s if the time series of size n while s is data values and lose because of the difference of seasonality so a time series Xt is called a seasonalmultiple ARIMA by non-seasonal order p,d and q and seasonal periods . if the differenced series expressed as below: the ARMA model (p×q)(P×Q)s with seasonal period s. leads toXtbe an ARIMA (p,d,q)(P,D,Q)s model with seasonality in a period s. the non-seasonal part is equal with the seasonal part of any seasonal ARIMA model because they have an ARIMA

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factor and differencing order. Togeneralize ARIMA model to agree with seasonality use the test box and Jenkins, also the popular multiply SARIMA model in the form: (18) Where the backward shift operator denoted by B, Φ, , Θ, ʘare polynomials with p, I, q, degreesand Q so israndom process with mean (0) and constant variance 2 . Consider a Seasonal ARIMA (0, 1, 1) × (0, 1, 1)4 model for instance. The model specification is: By expansion, we have: The predicts of the parameters of the model in (19) may be obtained using the method of maximum likelihood [5], [15] .

2.9
Time series analysis steps: [18] build any model of Box -Jenkins step by step to forecast and can represent the stages of the scheme follows (1)

Augmented Dickey-Fuller test
Dickey-Fuller is used to test the hypothesis when exista not stationary series, and can be tested in regression equation [9] .
Where a random walk,a = a −1 + a is allowed [9] .

2.11Autocorrelation Function (ACF) [4]
In a time series the order of correlation between neighboring observations measures by ACF. The autocorrelation coefficient is estimated from sample observation using the formula [10] : Thus, the autocorrelation function at lag k is characterized as:

Akaike Information Criterion AIC [2]
Akaike Information Criterion AIC is defined as where L: maximized likelihood function p: number of emotional parameters.
smallest AIC is the best. The likelihood function part reverses the appropriate of fit of the model to the data, while 2p is described as a penalty. Since L usually increases with p, AIC reaches the minimum at a certain p. AIC is depend on the information theory.

Estimating the Parameters of an ARMA Model [9]
Estimating the parameters of the ARMA model used Iterative method. At every point sum square residual should be calculated of suitable grid of the parameter values, and the sufficient values are given minimum sum of squared residuals. For an ARMA (1,1) the model is given by

Models Forecasts [22]
After building model the main goal in this type of data is create forecast for the series.it also show an important role in assessing the forecasts accuracy. The power or ability to forecast is conclusive test of an ARIMA model. In order to obtain a forecast with a least error, there are seven features of a good ARIMA model taken into account [13], [15] . First,it has the minimum coefficients number which show in the data set. Secondly, a sufficient AR model must not be nonstationary. Thirdly, the MA must be invertible. Fourth, insufficient model the residuals must be independent. Fifth, the distribution of residuals of a good model must be distributed normal. From the exist series according time t, namely, 1 , 2 , 3 , … , −1 , , we can forecast +ℎ , that will use h time units ahead. In this case, time t is the forecast source and the lead time forecast. This forecast is denoted and estimated as ̂( ) = ( +ℎ | 1 , 2 , … … , )………………. (25)

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Once an enough and satisfactory model is fitted to the series of benefit, forecasts can be produced using the model [13] .
The one-step-before forecast fort + 1 is shownas follow:

Fourier Residual Modifications (FR) [20][22]
The firstconceptionof the Fourier model is derived from Tien. Meanwhile, the forecasting of primary series trends originally uses ARIMA,Then the ARIMA model is adaptive using the Fourier series. The main goal behind using Fourier series is to gain a better forecasting with a high precision. To achieve the accuracy of forecasting models, the Fourier series has been widely and profitably applied in adapting the ) sin (   2 2 ) … cos ( 2 ) sin (  2 ) ] se least square method where: (1) , (1) , (2) , (2) , … , ( ) , (

Data Description
The dataset used for the analysis in this study came from the direcory of electricity of Sulaimani citywhich is contained one variable and deals with monthly demand of electricity since January 2015 up to December 2019. These data measurement is Gigahertz.

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The time series plots are display observations on the y-axis against equally spaced time intervals on the x-axis. They are used to evaluate patterns, knowledge of the general trend and behaviors in data over time. The time series plot of monthly Demand of Electricityin Sulaimani city displayed in Figure 3.1 below:

Figure 3.3: Partial Autocorrelation Function for the monthly Demand of Electricityin
Sulaimani city All the above results and plots support that the series is not random at the level, which needs some treatments to be transformed to a random series. Also, the differencing of a series by suitable transformation. We see that the time series for the first-Seasonal differenced series in Figure 3.4 show that the data series is stationary.  This result indicates that the non-stationarity hypotheses of the seasonal differenced monthly Demand of Electricityin Sulaimani city is rejected and this demonstrates by estimating the autocorrelation and partial autocorrelation function (ACF and PACF) for the first-differenced series in Figure 3.5 and 3.6 The results above demonstrate the success of differencing of the time series data of the monthly Demand of Electricityin Sulaimani city. Thus, the data series became stationarity.

Model Identification
This section shows how we determine the order of the seasonal ARIMA model. We  It is shown in table (3.3) that the SARIMA(0,0,0)x(2,1,0) 4 model produced the value of AIC with the smallest values. This means that the SARIMA (0,0,0)x(2,1,0) 4 model is the best among all the other models, which is the most suitable model that can be obtained for the monthly demand of electricity in Sulaimani city.

Comparison
Depending on the residuals of the SARIMA (0,0,0)x(2,1,0) 4 model we computed the Fourier SARIMA (0,0,0)x(2,1,0) 4 model and then comparing them to select the best model between them which has minimum AIC to forecast the demand of electric in Sulaimani city for the comes period of time, the comparative it is shown in table (3.5) Then the appropriate model to represent the monthly demand of electricity in Sulaimani city is FARIMA(0,0,0)x(2,1,0) 4 with minimum AIC.
therefore, the mentioned model is used to forecast the monthly demand of electricity in Sulaimani city as it is shown below

3.6Forecasting for weight of imported equipment:
After getting the final model FSARIMA (0,0,0)x(2,1,0) 4 of the data of the monthly demand of electricity in Sulaimani city that has been expressed above. We forecasted the monthly demand of electricity in Sulaimani city in 2020 for 12 months. The forecasting of time series for monthly demand of electricity in Sulaimani city have been plotted as in figure (3.8).    The results of the forecasted values in table (3.8) for the year 2020 are all falleninto the boundaries of the 95% confidence intervals. This confirms that the forecasting is very efficient.