Using Analysis of Time Series to Forecast numbers of The Patients with Malignant Tumors in Anbar Provinc /. ) ( ) / (Box & Jenkins).(.(2010-2006) ARIMA(2,1,0). Abstract: The aim of this research is to analysis time series with using (Box & Jenkins) method (Identification, Estimation, Diagnostic Checking of Model, Forecasting) to find the beast forecasting model to the number of patients with Malignant Tumors in Anbar Province by using the monthly data for the period (2006-2010). The result of data analysis show that the proper and suitable model is Integrated Autoregressive model of order (2) ARIMA (2, 1, 0). According to this model the Research forecast the numbers of patients with Malignant Tumors the next two years in monthly bass, so the forecasting values represented the scours time series data that deal to the efficiency of the model... 371
(2010-2006). (Time Series).(2012-2011) (2010-2006) ).( Box and Jenkins(B-J) ( ) :.(SPSS V.10 ) (Minitab) : (Wheel Wright & Markedis, 1978:60-64) MSE (1,2,1,2, ) (MSE) MSE (Dent and Min,1978 ). (A-F) (Box Jenkins) ARMA ( ) 372
MLE (100) (Ahtol, MSE J. and Tiao,1987:15-19) (p) (Wei:1990) ( 9-4:2000 ) ARIMA ARMA -990) (97-94) (993 1973 (Kaiser and Maravall:2001) ARIMA ARMA ARIMA (2009) ARMA (2,1) ARIMA (1,1,1) (59 :2009 ) (2008) (2004).VARIMA(2,1,0) -1 (B-J, 1976). Fundamental Concepts -1-1 Time Series : ) t 1, t 2,, t n (Z t1, Z t2,, Z tn ).( 373
(Statistical Series) n :(12-11 :2007 ) :. :.( ) : (Deterministic).. Sinusoidal ) (Periodic) : (Data : S : (Stationary in Variance). (Stationary in Mean).. Autocorrelation (AC) (k) :(Box&Pierce,1970,1509-1519 ) (-1 1) (1-1) : : ( 1/n) =0,1,2,3, : n (k=1,2,3, ) (k). (ACF) 374
(ACF) (k=q) (k) ( ) : : (ACF).(11-1989:9 ) (RACF) Residual Autocorrelation Function : Partial Autocorrelation (PACF) AR(P) 1996 ) (Autocorrelation Function) :(8: (PACF). (PACF). (k) (Anderson,1942:113-129) Stationary Time Series ) ( (Trend) Non stationary ). 375
(Homogenous :(20: 2007 ) ( ) (Back Shift Differences Operator) : (d).(nuno,1996:87) Box & Jenkins (B J) 2-1 Autoregressive Model (AR) 1-2-1 :(13 :2010 ) : (p) (1-2) )3(1- : : i= 1, 2, 3,.P : : : p : : (Back shift operator) : B (exponential damping) AR(p).(17: 1981 ) p (PACF) (k) AR(p) AR(2) AR(1). (33-27:1989:27 ) : (1-2) (p=1) 376
. AR(1) : : : ( ACF) : ( =0 ) ) ( (exponentially) AR(1). ) ( : AR(1) (PACF) AR (2) (PACF) (1-2) (p=2) : ) AR(2) :,, : AR(2) : (k=1,2) (exponential AR(2) (B-J,1976:58-60) : damping) : 377
.(Wave damping sine) (ACF) : AR(2) ) (.) ( AR(2) (PACF) (B) Moving Average Model (MA) 2-2-1 (q) :(17 :2010 ) :. : q (q) (MA). (17 :1981 ) (PACF) ( - ) 3-2-1 Mixed Autoregressive Moving Average Model (ARMA) )6(1- (B-J,1976) (p, q) )7(1- :(B).(.( (,.. ) (B) :,, ) (B) : ).(19: 2010 ) 378
4-2-1 Autoregressive Integrated Moving Average Models (ARIMA). (ARIMA). AR(p) I(d) MA(q) (Auto Regressive Integrated Moving Average (Stationary) : (p,d,q) ARIMA Models). AR(p) : P. MA(q) : q. : d :2007 ), (Dent& Min,1978:33-45): (B) (30 : : ARIMA(p, d, q) :.(Kaiser,2001:34) ARMA ARIMA (1-8) 3-1 : (B-J, 1976 :240-243) 379
Identification 1-3-1 1976 Jenkins Box. (d=1). (d=2) ARIMA q d p. (B-J,1976:245) (1-7) (ACF) (PACF) (ACF) (PACF) (PACF) (ACF) :(18-16:1981 ) (PACF) (ACF).AR(P) (P) (PACF) (q) (ACF)).MA (q) (PACF) (ACF) ARMA (p,q) Estimation 2-3-1 ( :Pirce,1971:299-312) Method of Ordinary Least Squares (O.L.S.E.).1. Maximum Likelihood Method.2. 380
Diagnostic Checking of Model 3-3-1 : (t-student) MA AR : (28-29 :2010 ) : Q (Ljung & Box) -1 : (Q). )9(1- : Q. =s =k =m : : -2 ( ) (Residuals) (0.95) ( ).(1\n) S.Makridakis, ) (AIC) (Akaike's Information Criterion) ) (1998:75 : (AIC ) ( (1-10). (p + q) : : (1-11) : (Schwartz) 381
(1-12) Forecasting 4-4-1 (L) (Dougls, 1977:78-91) : (L) (1-13) : (L) AR(1) : (L) AR(2) : (L) MA(q) : (L) ARMA(p, q) -2 1-2 (60) 2010 2006.(1) (1) * Year Month 2006 2007 2008 2009 2010 January February March April May June July 17 18 18 20 19 25 29 33 35 36 37 38 37 38 44 45 47 46 46 49 47 61 60 63 62 64 66 67 80 88 85 79 90 92 93 382
August September October November December 30 33 33 32 33 39 40 42 43 46 53 69 93 51 69 97 53 68 95 55 75 97 60 74 99 : * 2-2 (1) (Minitab) (2) (15) ( ) : (%95) : (Q.stat) Ljung & Box (2) ) (4) ( (B-J,1976). 383
(2010-2006) (1) 384
(2) (3) 385
(4) 3-2 (Identification) (ACF) (PACF) (ACF) ( 4) (k) (PACF) (Cut-off) ARIMA(2,1,0) ARIMA(2,1,0) 386
ARIMA(1,1,0). (2) (2) Model order (p) Adj.SSE Risiduals.V AIC MAIC SBC ARIMA(1,1,0) 414.7877 7.4069 286.5302 4.8564 290.6852 ARIMA(2,1,0) 377.9074 6.7483 283.0703 4.7978 289.3029 : ARIMA(2, 1, 0) Or : Estimation 2-4 Ordinary ) (SPSS V.10) (Least Squares. Final Estimation of Parameters: T StDev Ceoff. Type -2.9243 0.1272-0.3715 AR 1-2.3136 0.1272-0.2943 AR 2 0.2045 6.7994 1.3907 Constant Differencing : 1 regular differenc Number of observation : original series 60,after defferencing 59 Analysis of Variance : DF Adj.sum of squares Residuals variance Residuals 56 373.9074 6.7484 Standard error = 2.5919 Log Liklehood= -138.53514 283.0703 AIC = 4.7978 MAIC = 289.3029 SBC = 387
5-2 (5).( ): (=5.5) [(Q.stat) (Ljung & Box)] (White Noise) (18.307) ( ). ARIMA(2,1,0) (6). (5) 388
(6) 6-2 (3-2) (3) (2012-2011). (7) (3) (2012-2011) Year Month 2011 2012 January 100 117 February 101 118 March 103 120 April 104 121 May 106 122 June 107 124 July 108 125 August 110 126 September 111 128 October 113 129 November 114 131 December 115 132 389
(2012-2011) (7) -3 1-3.1 2009..2 (K). ).3 (SBC) (AIC) ).( 390
ARIMA (2,.1,0).4-2011). (2012.5 2-3 :.1..2. 391
" (2010).1 " " (1996).2.. " " (1981).3. " "ARIMA " (2007).4. " (1989).5. " " (2000).6 ".(1) (3) 7. Ahtola, J. and Tiao, G. C. (1987), A note on asymptotic inference in autoregressive models with roots on the unit circle, Journal of time series analysis, Vol. 8 No. 1 pp. 15-19. 8. Anderson, R.l, (1942),"Distribution oftheseries Analysis Correlation Coeffficient", Ann, Mat.Statistic, Vol.13. P (113-129). 9. Box, G.E.P and Jenkins, G.M. (1976), "Time Series Analysis Forecasting and control", Holden day, London. 10. Box, G.E.P. & Pierce, D.A., (1970), "Distribution of the Residual Autocorrelation in Autoregressive-integrated moving Average Time Series Models", JASA, VOL.65, P. (1520-1526). 11. Douglas,C.M & Contreas, J.G.,(1976),"Note On Forecasting With Adaptive Filtering, O.P.Q,Vol.24,No.4,P(87-90). 12. Kaiser, R. and Maravall, A. (2001), Notes on Time Series Analysis ARIMA Models and Signal Extraction, Benco de Esponaservicio Estudios 13. Nuno Crato, (1996), Some Results on the Spectral Analysis of stationary Time Series Portugal Mathematic Vol. 53. Fasc. 2-1996. 392
14. Pirece,A.D.(1971),"Least Squares Estimation in the Regression Model with Autoregresseion-Moving Average Errors",Biomatrika,Vol.58,P(299-321). 15. S. Makridakis, S. C. Wheelwright and R. J. Hyndman (1998), Forecasting Methods and Applications ", Third Edition, John Wiley & Sons, Inc. Publication, New York, USA. 16. Wheel Wright, S.G. & Markedis, S. (1973), "An Examination of the Use of Adaptive filtering in Forecasting O.P.Q", vol.24, No.1, P (60-64). 393