FORECASTING TELECOMMUNICATIONS DATA WITH AUTOREGRESSIVE INTEGRATED MOVING AVERAGE MODELS

Transcription

1 FORECASTING TEECOMMUNICATIONS DATA WITH AUTOREGRESSIVE INTEGRATED MOVING AVERAGE MODES Niesh Subhash naawade a, Mrs. Meenakshi Pawar b a SVERI's Coege of Engineering, Pandharpur. nieshsubhash15@gmai.com Abstract The abiity to perform forecasting cacuations is an important thing for forecasting practitioners as it heps them in panning and determining their networks. Accurate forecasting heps forecasting practitioners to provide guidance to business peopes to take important decisions about their business in advance. The Internationa Teecommunication Union (ITU) Recommendation E.507 expains about many techniques for evauating the forecasting modes and choice of the mode. But, it does not provide any guidance to seect which modes are more proper for forecasting the teecommunications series. This paper deas with Autoregressive Integrated Moving Average mode for forecasting teecommunication data. Evauation metrics such as Sum of Squared Regression, Root Mean Square Error, Mean Absoute Deviation, Mean Absoute Percentage Error and Maximum Absoute Error shows that the ARIMA modes provides good forecasting performance. Abstract must be of Time New Roman Front of size 10 and must be justified aignment. Keywords Teecommunication forecasting; ITU Recommendations; ARIMA mode; Time series; M-3 Competition data; autocorreation; Future trend estimation. INTRODUCTION In the growing teecommunications industry the abiity to determine the future trends is an important endeavor [12]. The boundaries of the teecommunication industries are increasing very fast and the competition among the industries is aso growing as we. Teecommunications companies are in need to deveop new strategies to face chaenges in providing the best communication services. Many industries depend on data monitoring to improve their business by anayzing the future trends and marketing vaue for their products. Data forecasting has a major roe in network traffic management, optimizing infrastructures and in panning process. [1, 4]. The forecasting abiity of a time series data is an important thing for the forecasting practitioners. The forecasters can determine the outcome of any event with the priori knowedge about the event. Whie forecasting teecommunication data, many errors occurs and it motivates to find a better forecasting approach which minimizes the forecasting errors. If a time series data is not forecasted accuratey, then reducing the forecasting error wi be ineffective. So, it is necessary to find new ways to minimize the effects of poor forecasts [2]. The forecastabiity of a time series depends on the reguarity of the data. If the time series is ess reguar, then achieving good eve of forecasting accuracy is very difficut [3, 9-11, 17]. There are severa techniques for forecasting such as Random wak (simpe forecasting method), exponentia smoothing methods such as simpe exponentia smoothing (SES), Hot, Hot-winters, Robust trend etc. In the random wak mode, the vaue of the variabe foows a random step for each time interva. It assumes that the current observation is ony important and the previous observations provide no information [15]. Simpe exponentia smoothing is a suitabe method for forecasting seasona data. But it is not a suitabe method when there is a trend [16]. Hot method is an extended form of simpe exponentia smoothing method and it aows forecasting data with trends. It can be improved to dea with both trend and seasona variations. Hot-winter method overcomes Hot method by considering the seasona variations [18]. Robust trend methods are suitabe for forecasting univariate time series data in the presence of outiers. When using this method, incorrect abeing of sampes as outiers may occur. In this paper, ARIMA mode for forecasting teecommunication data is proposed. M3-Copetitive data is used here and the performance is measured in terms of Sum of Squared Regression (SSR), Root Mean Square Error (RMSE), Mean Absoute Deviation (MAD), Mean Absoute Percentage Error (MAPE), Maximum Absoute Error (MAE). The paper is structured as foows. Section 2 describes the M3-Competition teecommunication data. Section 3 describes about the ARIMA based forecast mode. Section 4 evauates forecasting performance. Forecast resuts are given in section 5 and section 6 concudes the paper

2 M3-Competition Data As the forecasting avaiabiity is growing up, there is a need for anayzing the adequacy of forecasting methods. Makridakis and Hibon deveoped the M-Competition which is considered to be one of the important researches in this fied [5, 13]. Data used here are obtained from the Institute of Forecasters. The ast edition (M3) is referred to year 2000 and it incudes 3003 time series cassified in to various types such as micro (828), industry (519), macro (731), finance (308), demographic (413) and other (204) and the different time intervas between the successive observations are gives by yeary, quartery, monthy and of unknown periodicity( others ). Minimum number of observations is required for each type of data to ensure that enough data can deveop an appropriate forecasting mode. This minimum was set as 14 observations for yeary series, 16 for quartery, 48 for monthy and 60 for other series. Tabe 1 show the cassification of the 3003 series according to the two major grouping discussed above. The tabe shows the number of time series based on both time interva and domain. Tabe 1 The cassification of the 3003 time series used in the M3-Competition Time interva between Types of time series data successive observations Micro Industry Macro Finance Demographic Other Tota Yeary Quartery Monthy Other Tota ARIMA-based forecast mode Box and Jenkins introduced the Autoregressive integrated moving average (ARIMA) mode. It is used as one of the popuar method for forecasting data [6, 14]. The ARIMA mode is used for time series forecasting where the future vaue of a variabe is a inear function of past observations and random errors and is expressed as, y t 0 1 yt 1 2 yt2 p yt p t 1 t1 2 t2 q tq Where, y t is the actua vaue and ε t is the random error at time t, and φ i (i = 1, 2,..., p) and θ j (j= 0, 1, 2,..., q) are mode parameters. Integers, p and q are the order of the mode. The random errors, ε t are assumed to be independent and identicay distributed with a zero mean a constant variance of σ 2. ARIMA mode invoves the foowing three iterative steps [7]. (i) Mode Identification: ARIMA mode has the assumption that the time series is stationary. So, data transformation is done to produce a stationary time series. For stationary time series, the mean and autocorreation structure are constant over time. So, differentiation and power transformation are needed to change the time series to be stationary. To identify the appropriate mode form, autocorreation and partia autocorreation are cacuated from the data and it is compared to the theoretica autocorreation and partia autocorreation for the various ARIMA modes. Steps (ii) and (iii) wi determine whether the mode is appropriate [8]. (ii) Parameter Estimation: Parameters in ARIMA mode is estimated using the noninear east square procedure. (iii) Diagnostic Checking: Severa diagnostic statistics and pots such as Histogram, norma probabiity pot and time sequence pot are used to check the fitness of the mode estimated in step (i). Chi-square test can be used to test the mode adequacy. Once a satisfactory mode is obtained, the seected mode wi be used for forecasting. Evauation metrics Forecast accuracy is measured using Sum of Squared Regression (SSR), Root Mean Square Error (RMSE), Mean Absoute Deviation (MAD), Mean Absoute Percentage Error (MAPE), Maximum Absoute Error (MAE). (i) SSR: The Sum of Squares Regression (SSR) is the sum of the squared differences between the prediction for each observation and the popuation mean

3 SSR 1 y y (ii) MSE: Mean Square Error (MSE) averages the squared prediction error at the same prediction horizon. A derivation of MSE is Root mean Square Error (RMSE). 1 2 MSE( i) (iii) MAD: Mean Absoute Deviation from the sampe median (MAD) is the resistant estimator of the dispersion /spread of the prediction error. It is intented to be used when the error pots do not resembe those of a norma distribution. where, 1 ADi i n 1 n 1 M median i and median is the th 2 1 order statistic. (iv) MAPE: Mean Absoute Percentage Error (MAPE) averages the absoute percentage errors in the predictions of mutipe UUTd at the same prediction horizon. Instead of the mean, median can be used to campute Median absoute percentage error (MdAPE) in a simiar fashion. 1 MAPE ( i) 1 i 100 r i i v) MAE : Mean Absoute Error (MAE) averages the absoute prediction error for mutipe UUTs at the same prediction horizon. Using median instead of mean gives absoute error (MdAE). 1 Resuts and discussion MAE i M This section presents the performance evauation of the ARIMA mode in M3C data. The ARIMA is impemented using Matab (R2013b) with a system configuration of 2GB RAM Inte processor and 32 bit OS. To determine the forecasting abiity of the ARIMA mode, it is appied to the yeary, quartery, monthy and other data. The Graphica User Interface (GUI) for the ARIMA mode for forecasting teecommunication data is shown in figure 1. 1 i 241

4 Figure 1. GUI The autocorreation pots are used in the mode identification stage of the ARIMA time series modes. Autocorreation pots are used to check the randomness in a data set. To find the randomness, the autocorreations for data vaues at varying time ags are computed. If the autocorreation is near to zero for a the time-ag separations, then the data is random. If the autocorreations are not zero, then the data is not random. Partia autocorreation are used to find the order of the autoregressive mode. The sampe autocorreation pots are given in figure 2. Tabe 2 shows the statistics of ARIMA (1,0,0) Mode extracted from the input data. Figure 2. Autocorreation pot * Tabe 2. ARIMA(1,0,0) Mode ARIMA(1,0,0) Mode: Conditiona Probabiity Distribution: Gaussian Parameter Vaue Standard Error t-statistic Constant AR{1} Variance e The time-series graph iustrates the data points at successive time intervas. The time is measured on the horizonta axis and the variabe is measured on the vertica axis. The time series graph for the comparison between the origina data observed and the observed forecasting data and respective performance measure for the yeary data, quartery data, monthy data and other data are shown in figure 3, 4, 5 and 6 respectivey

5 Figure 3. Yeary data Figure 4. Quartery data Figure 5. Monthy data 243

6 6. Concusion Figure 6. Other data In this study, we have appied the ARIMA mode to the M3-competition data set of 3003 time series to determine its forecasting abiity. It is observed that the ARIMA mode provides better forecasting even when itte detai about the data is avaiabe. The forecasting accuracy of the ARIMA mode is measured using parameters such as SSR, RMSE, MAD, MAPE, and MAE for the yeary, quartery monthy and other data and it is compared with the resuts given in [19] using MAPE. It showed that the performance of ARIMA mode is better than the other modes. We therefore concude that the ARIMA mode is the better forecasting method for teecommunication data. REFERENCES: [1] Boyan, J.E. Towards a more precise definition of forecastabiity. Foresight: the Internationa Journa of Appied Forecasting 2009, [2] Koassa, S. How to assess forecastabiity. Foresight: the Internationa Journa of Appied Forecasting 2009, [3] Tashman,.J. Specia feature on forecastabiity: Preview. Foresight: the Internationa Journa of Appied Forecasting 2009, 23. [4] Gary Madden, Joachim Tan, "Forecasting teecommunications data with inear modes", Teecommunications Poicy, vo. 31, pp , [5] Makridakis, S. and Hibon, M. (2000): The M-3 Competition: resuts, concusions and impications, Internationa Journa of Forecasting, 16, pp [6] P. F. Pai and C. S. in, A hybrid ARIMA and support vector machines mode in stock price forecasting, Omega, vo. 33, no. 6, pp , [7] G. P. Zhang, Time series forecasting using a hybrid ARIMA and neura network mode, Neurocomputing, vo. 50, pp , [8] J. E. Hanke and D. W. Wichern, Business Forecasting, Prentice Ha, Engewood Ciffs, NJ, USA, 2009 [9] Armstrong, J., & Coopy, F. (1992). Error measures for generaizing about forecast methods: Empirica comparisons. Internationa Journa of Forecasting, 8, [10] Fides, R. (1992). The evauation of extrapoative forecasting methods. Internationa Journa of Forecasting, 8, [11] Fides, R., Hibon, M., Makridakis, S., & Meade, N. (1998). Generaising about univariate forecasting methods: Further empirica evidence. Internationa Journa of Forecasting, 14, [12] Grubesic, T., & Murray, A. (2005). Geographies of imperfection in teecommunication anaysis. Teecommunications Poicy, 29, [13] Makridakis, S., Chatfied, C., Hibon, M., awrence, M., Mis, T., Ord, K., et a. (1993). The M-2 competition: A rea-time judgmentay based forecasting study. Internationa Journa of Forecasting, 9, [14] Parzen, E. (1982). ARARMA modes for time series anaysis and forecasting. Journa of Forecasting, 1, [15] Simmons,. F. (1986). M-Competition A coser ook at Naıve2 and median APE: a note. Internationa Journa of Forecasting 4, [16] Gardner, E., Exponentia smoothing: The state of the artpart ii. Internationa Journa of Forecasting 22, [17] Kotsiaos, A., Papageorgiou, M., Pouimenos, A., ong-term saes forecasting using Hot-Winters and neura network methods. Journa of Forecasting 24, [18] Fides, R. & Petropouos F. (2013). An evauation of simpe forecasting mode seection rues (UMS Working Paper 2013:2). ancaster University: The Department of Management Science. [19] Makridakis, S., & Hibon, M. (2000).The M-3 competition: resuts, concusion and impications. Internationa Journa of Forecasting, 16,