IMPROVEMENT FOR EXPONENTIAL SMOOTHING

Transcription

1 DOKUZ EYLÜL UNIVERSITY GRADUATE SCHOOL OF NATURAL AND APPLIED SCIENCES IMPROVEMENT FOR EXPONENTIAL SMOOTHING by Seda ÇAPAR Ocober, 2009 İZMİR

2 IMPROVEMENT FOR EXPONENTIAL SMOOTHING A Thesis Submied o he Graduae School of Naural and Applied Sciences of Dokuz Eylül Universiy In Parial Fulfillmen of he Requiremens for he Degree of Docor of Philosophy in Saisics, Saisics Program by Seda ÇAPAR Ocober, 2009 İZMİR

3 Ph.D. THESIS EXAMINATION RESULT FORM We have read he hesis eniled IMPROVEMENT FOR EXPONENTIAL SMOOTHING compleed by SEDAT ÇAPAR under supervision of ASSOC. PROF. DR. GÜÇKAN YAPAR and we cerify ha in our opinion i is fully adequae, in scope and in qualiy, as a hesis for he degree of Docor of Philosophy. Assoc. Prof. Dr. Güçkan YAPAR Supervisor Prof. Dr. Serdar KURT Assis. Prof. Dr. Adil ALPKOÇAK Thesis Commiee Member Thesis Commiee Member Prof. Dr. Gülay KIROĞLU Assoc. Prof. Dr. C. Cengiz ÇELİKOĞLU Examining Commiee Member Examining Commiee Member Prof. Dr. Cahi HELVACI Direcor Graduae School of Naural and Applied Sciences ii

4 ACKNOWLEDGMENTS I would like o express my deep and sincere graiude o my supervisor Assoc. Prof. Dr. Güçkan YAPAR for his helpful suggesions, imporan advice and consan encouragemen. I wish o express my warm and sincere hanks o Prof. Dr. Serdar KURT and Assis. Prof. Dr. Adil ALPKOÇAK for heir valuable conribuions. I hank o my parens, my moher Ayen ÇAPAR, my faher Musafa ÇAPAR and my siser Sevim ERGAN for heir suppor. And finally my deepes hank o my wife Mine for her paience and encouragemen. Seda ÇAPAR iii

5 IMPROVEMENT FOR EXPONENTIAL SMOOTHING ABSTRACT Exponenial smoohing mehods have been employed since 1950s and hey are mos popular and used mehods in business and indusry for forecasing. However here are wo main problems abou choosing he smoohing consan and saring value. In his hesis a new mehod is inroduced for smoohing consan and saring value. Modified mehod gives even more weighs han he classical mehod o mos recen observaions. A sofware ool developed o compare he modified mehod wih he original. And real ime series from M-compeiion are used o compare he mehods empirically. Keywords : Forecas, Exponenial Smoohing, Smoohing Consan, Saring Value iv

6 ÜSTEL DÜZELTME İÇİN KATKILAR ÖZ İlk 1950 li yıllarda oraya çıkan üsel düzelme yönemleri bugün iş ve endüsri dünyasında en çok bilinen ve kullanılan zaman serisi ahmin yönemleri arasında yer almakadır. Ancak üsel düzelme yönemlerinin düzelme erimi ve başlangıç değerinin belirlenmesi gibi iki önemli problemi bulunmakadır. Bu ezde düzelme erimi ve başlangıç değeri için yeni bir yönem gelişirilmişir. Yeni yönemde son gözlemlere verilen ağırlık klasik yönemde verilen ağırlıklardan daha da fazladır. Yeni yönemin eorik olarak klasik yönemin emel özelliklerine sahip olduğu ispa edilmiş, meoların karşılaşırılması için gelişirilen yazılımla M-Compeiion olarak bilinen çalışmalara ai zaman serileri kullanılarak deneysel karşılaşırmalar yapılmışır. Anahar sözcükler : Zaman Serileri, Tahminleme, Üsel Düzelme, Düzelme Terimi, Başlangıç Değeri v

7 CONTENTS Page PH.D. THESIS EXAMINATION RESULT FORM... ii ACKNOWLEDGEMENTS...iii ABSTRACT... iv ÖZ... v CHAPTER ONE - TIME SERIES Inroducion Displaying Time Series Daa Forecasing Time Series Componens of a Time Series Trend Componen Cycle Componen Seasonal Componen Irregular Componen Time Series Models Algebraic Models Transcendenal Models Composie Models Regression Models Errors in Forecasing Measuring Forecas Errors CHAPTER TWO - SMOOTHING Moving Average Exponenial Smoohing vi

8 2.3 Hisory of Exponenial Smoohing Simple Exponenial Smoohing Smoohing Consan and Saring Value Double Exponenial Smoohing Triple Exponenial Smoohing CHAPTER THREE - MODIFIED EXPONENTIAL SMOOTHING Modified Simple Exponenial Smoohing Modified Double Exponenial Smoohing Modified Triple Exponenial Smoohing CHAPTER FOUR - APPLICATION Daa Operaions Analysis Operaions Run Mehods Run Mehods on Daases Run Mehods Auomaically CHAPTER FIVE - EMPIRICAL COMPARISONS Modified Simple Exponenial Smoohing vs. Simple Exponenial Smoohing In-sample Performance Ou-of-sample Performance Modified Double Exponenial Smoohing vs Double Exponenial Smoohing In-sample Performance Ou-of-sample Performance CHAPTER SIX - CONCLUSION vii

9 REFERENCES APPENDIX A APPENDIX B APPENDIX C APPENDIX D APPENDIX E viii

10 CHAPTER ONE TIME SERIES 1.1 Inroducion A ime series is a collecion of daa values measured a regular inervals of ime. In fac i consiss of wo variables: he measuremen and he ime a which he measuremen was aken. So, a ime series is usually sored as a pair of wo daa ses. Firs daa se represens he ime while he second daa se represens he observaions. However, i is also possible o form a ime series as only one daa se of observaions ordered by ime. Formally, a ime series is defined as a se of random variables indexed in ime, X 1, X 2,, X } and an observed ime series is denoed by { x 1, x2,, xt }. { T There are wo ypes of ime series called as coninuous and discree ime series. Time componen deermines he ype of a ime series i is no imporan ha he measured variable may be coninuous or discree. If he measuremens are observed a every insan of ime hen i is called coninuous ime series, e.g. elecro diagrams. If he measuremens are observed a regularly spaced inervals hen i is called discree ime series. Usually a coninuous ime series is also analyzed like a discree ime series by sampling he coninuous series a equal inervals of ime o obain a discree ime series. Time series analysis is a saisical mehod or model which is rying o find a paern inheren in a ime series. There are wo main goals of a ime series analysis: idenifying a paern and forecasing. Time series analysis is based on he premise ha by knowing he pas, he fuure can be forecas. Therefore, he primary assumpion of a ime series analysis is ha he near fuure will depend on he pas and ha any pas paerns will coninue in he fuure. 1

11 2 1.2 Displaying Time Series Daa A line graph is he mos used ype of graph o display a ime series. The measuremen is ploed on he verical (y) axis and ime is ploed on he horizonal (x) axis. Line graph may easily illusrae he paern of a ime series. I will give a visual represenaion of he daa over ime. For example, he following able includes he number of marriages ha ook place each quarer beween 2001 and 2003 in England and Wales (Marriage, 2008). Table 1.1 Marriages ha ook place each quarer beween 2001 and 2003 Year Quarer Marriages , , , , , , , , , , , ,063 Figure 1.1 shows he above ime series as a ime char. The horizonal axis represens he quarers beween 2001 and 2003 and he verical axis represens he number of marriages ha varies over ime. The ime char displays he ime series such ha he paern of he daa is immediaely apparen.

12 3 Marriages beween 2001 and , ,000 80,000 60,000 40,000 20, Q Q Q Q Q Q Q Q Q Q Q Q4 Marriages Quarer Figure 1.1 Time char for marriages ha ook place each quarer beween 2001 and Forecasing Time Series As originally described by Brown (1964) and George (George, Gwilym and Gregory, 1994), forecass are usually needed over a period of ime known as lead ime, which varies wih each problem. The observaions available up o a ime are used o forecas is value a some fuure ime +l where l is he lead ime (or someime i is also called forecas horizon). In generaing forecass of evens ha will occur in he fuure, a forecaser mus rely on informaion concerning evens ha occurred in he pas (Bruce and Richard, 1979). Therefore, he forecaser mus analyze he observed daa and mus base he forecas on he resul of his analysis. Firs, he daa is analyzed o idenify a paern hen his paern can be used in he fuure o make a forecas. We mus agree wih he assumpion ha he paern ha has been idenified will coninue in he fuure o use he forecas obained from he idenified paern. I is also menioned by (Bruce and Richard, 1979), a forecasing echnique canno be expeced o give good predicions

13 4 unless his assumpion is valid. If he daa paern ha has been idenified does no persis in he fuure, he forecasing echnique being used will likely produce inaccurae predicions. 1.4 Componens of a Time Series A ime series is a combinaion of four componens; Trend, Cycle, Seasonal and Irregular (error) componens. These componens do no always have o occur alone. They can occur in any combinaion herefore here is no single bes forecasing echnique exiss. So, he mos imporan hing is o selec mos appropriae forecasing echnique o he paern of he ime series daa Trend Componen Trend refers o a long-erm movemen in he ime series. I is he resul of influences such as populaion growh, echnological progress or general economic changes. Trend may be upward or downward. Thus, rend reflecs he long-run growh or decline in he ime series. For mos ime series i evolves smoohly and gradually. I is possible o deec a rend in a ime series simply by aking averages of i over a cerain ime period. If hese averages are changing wih ime hen i is possible o say ha here is a rend in his ime series. A visual represenaion will also be helpful o deermine he rend componen of a ime series. Figure 1.2 displays an example of rend componen.

14 Employees Year Figure 1.2 Trend componen of a ime series Cycle Componen Cycle refers o regular or periodic up and down movemens around he rend. There is a repeaing paern wih some regulariy bu he flucuaions in he series are longer han 1 year. Someimes he cycle and he rend are esimaed joinly because mos ime series are oo shor for he idenificaion of a rend. A cycle consiss of an expansion phase followed by a recession phase. This sequence is recurren bu no sricly periodic. Figure 1.3 displays an example of cycle componen.

15 Sales ($ millions) Year Figure 1.3 Cycle componen of a ime series Seasonal Componen Seasonal componen is a periodic change in he ime series ha occurs in a shor erm. There are periodic flucuaions and hese periods occur wihin one year (e.g., 12 mohs per year, or 7 days per week). The seasonal cycle is he period of ime ha elapses before he periodic paern repeas iself. Figure 1.4 displays an example of seasonal componen.

16 Winer 1996 Summer 1997 Winer 1997 Summer 1998 Winer 1998 Summer 1999 Winer 1999 Summer 2000 Winer 2000 Summer Sales ($ millions) Figure 1.4 Seasonal componen of a Time Series Irregular Componen Irregular componen is anyhing lef over in a ime series afer he rend, cycle and seasonal componens. These are erraic movemens ha follow no recognizable or regular paern. These flucuaions may be caused by unusual evens or may conain noisy or random componen of he daa and in a highly irregular series hese flucuaions will preven he deecion of he rend and seasonaliy. Figure 1.5 displays an example of irregular componen.

17 Unemploymen (percen) Year Figure 1.5 Irregular componen of a ime series Figure 1.6 displays four componens in a ime series. Figure 1.6 Four componens of a ime series

18 9 1.5 Time Series Models There are many forecasing mehods ha can be used o predic fuure evens. These mehods can be divided in o wo basic ypes; qualiaive and quaniaive mehods. Time series models are quaniaive mehods ha can have many forms. In such models, hisorical daa is analyzed o idenify a daa paern. Then, assuming ha i will coninue in he fuure, his daa paerns is exrapolaed in order o produce forecass. In all models, here is an underlying process ha generaes he observaions in erms of a se of significan paern in ime, plus an unpredicable random elemen which can be described by a probabiliy disribuion having zero mean (Brown, 1964) Algebraic Models Consan Models In consan models he observaions are random samples from some disribuion and he mean of he disribuion doesn change significanly wih ime. So, underlying process doesn change a where a is he rue value which we shall never know. The observaions some random error X include X a I is always assumed ha he expeced value of he error is zero, i has consan variance and usually he disribuion of i is Gaussian. The rue value of he average is no known bu i can be esimaed from recen observaions. Then he forecas of he mean of he disribuion for fuure samples will be represened by

19 10 Xˆ m aˆ Linear Models If here is a significan rend hen he underlying process will be a b where a is he average when he ime is zero and b is he rend. Again rue values of a and b are no known bu hey mus be esimaed from he daa in he recen pas. Afer esimaing hese values he mean of he disribuion from which fuure observaions will be aken is forecas as Xˆ ˆ ˆ m a b m Polynomial Models In general, any degree of polynomial can be used o represen he process by adding erms 2, 3,, N o he model. The highes exponen in he model deermines he degree of he polynomial. The number of coefficiens which mus be esimaed is always one more han he degree of he polynomial. For example, for a second-degree polynomial he following equaion can be wrien as follows a b c 2 Afer esimaing he coefficiens a ˆ, bˆ and cˆ he forecas will hen be Xˆ ˆ ˆ ˆ m a b m c m 2

20 Transcendenal Models Exponenial Models An exponenial funcion will describe he process where he rae of growh is proporional. The change in value from one observaion o he nex can be expressed as a consan percenage of he curren value. A model of he process may log log k log a where k is consan of proporionaliy and a is he raio of one observaion o he previous observaion. A more complicaed model would be log log k log a 2 log b and for he simple exponenial funcion i would be ka In general form k 1 a 0 k 2 a 1 1 b k n a n 1 n1 b n Trigonomeric Models When he process o be forecased is periodic i is appropriae o describe i in erms of sines and cosines. A model would be a cos 6 or 2 a cos ( p p0 ) c p

21 Composie Models I is possible o use algebraic and ranscendenal models ogeher. Models ha combine algebraic and ranscendenal models are called composie models. For example; a 0 a 1 a 2 a4sin Regression Models The algebraic and ranscendenal models and heir combinaion may exhaus o model he process. There is a very wide class of linear forecas models, in which he process is described by ( ) ( ) a1 f 1 an fn where he funcions f i () can be any arbirary funcions. 1.6 Errors in Forecasing Unforunaely, all forecasing mehods will include some degree of uncerainy (Bowerman & O Connel, 1987). This is recognized by including an irregular componen in he descripion of a ime series. The presence of he irregular componen means ha some error in forecasing mus be expeced. However, here are oher sources of errors ake place in forecasing. Prediced rend, seasonal and cyclical componens may influence he magniude of error in forecass. So, large forecasing error may indicae ha forecasing echnique being used in no capable of accuraely deermine he rend, seasonal and cyclical componens and, herefore, he echnique being used is inappropriae.

22 Measuring Forecas Errors If he acual value of he variable of ineres a ime period is x and he prediced value of and prediced value e x xˆ x is xˆ hen he forecas error e is difference of he acual value I is possible o sum forecas errors o deermine wheher accurae forecasing is possible. n 1 ( x xˆ ) Summaion of he difference beween he prediced and acual values from ime period =1 hrough ime period =n, where n is he oal number of observed ime periods. However, his quaniy is no appropriae since some errors will be posiive while ohers are negaive. If he errors display a random paern hen sum of he forecas errors will be close o zero. One way o solve his problem is o use absolue values of forecas errors where Absolue Error = e x xˆ Using absolue values mean absolue error (MAE) defined as he average of he absolue deviaions n e ˆ x x 1 1 MAE = n n n Anoher way is o use square of he forecas errors 2 2 Squared Error = e x xˆ Then using squared errors, Mean Squared Error (MSE) is defined as he average of he squared errors

23 14 MSE = n 2 n 2 e x xˆ 1 1 n n These wo measures MAE and MSE can be used he measure he magniude of forecas errors. These measures can be used in he process of selecing a forecasing model. Hisorical daa can be simulaed o produce predicions and comparing hese predicions wih he acual values MAE and MSE can be calculaed o measure accuracy of he seleced model. For example, suppose we have wo forecasing mehods and from hisorical daa given in Table 1.1, predicions, forecas errors, MAE and MSE are calculaed (Table 1.2). Table 1.2 Comparisons of he errors produced by wo differen forecasing mehods Acual y Prediced y A Error e A Absolue Error Squared Error Prediced y B Error e B Absolue Error Squared Error Sum From Table MAE A , 27 MSE A and 11 MAE B , 13 MSE B I is possible o say ha mehod B is more accurae han mehod A according o accuracy measures MAE and MSE.

24 15 In addiion o comparing differen mehods, MAE and MSE can also be used o monior a forecasing sysem. Forecass canno be expeced o be accurae unless he hisorical daa paern ha idenified coninues in he fuure. If here exiss sudden changes in ha paern for an exended period of ime hen forecasing mehod used o forecas he variable of ineres migh now be expeced o become inaccurae because of his change. A his siuaion, MAE and MSE can monior he forecas errors and discover he change in paern as quickly as possible before forecass become very inaccurae. There is also oher accuracy measures have been used o evaluae he performance of forecasing mehods. Mahmoud has been lised some of hem (Mahmoud, 1984). Makridakis used MAPE, MSE, AR, MdAPE and PB (Makridakis e al., 1982). Chafield (Chafield, 1988) and Armsrong (Armsrong & Collopy, 1992) poined ou ha he MSE is no appropriae for comparisons beween series as i is scale dependen. Makridakis (Makridakis, Wheelwrigh & Hydman, 1998) noed ha MAPE also has problems when he series has values close o zero. Armsrong and Collopy recommended he use of relaive absolue errors GMRAE and MdRAE alhough relaive errors have infinie variance and undefined mean (Armsrong & Collopy, 1992). In a sudy MAPE, MdAPE, PB, AR, GMRAE and MdRAE is used (Fildes, Hibon, Makridakis & Meade, 1998). The M3-compeiion use hree differen measures: MdRAE, smape and SMdAPE (Makridakis & Hibon, 2000). The symmeric measures were proposed by Makridakis (Makridakis, 1993). Table 1.3 displays commonly used forecas accuracy measures.

25 16 Table 1.3 Commonly used forecas accuracy measures (Gooijer and Hyndman, 2006) MSE Mean squared error =mean( e ) RMSE Roo mean squared error = MSE MAE Mean Absolue error =mean( e ) MdAE Median absolue error =median( e ) MAPE Mean absolue percenage error =mean( p ) MdAPE Median absolue percenage error =median( p ) smape Symmeric mean absolue percenage error =mean( 2 Y ˆ / ˆ Y Y Y ) smdape Symmeric median absolue percenage =median( 2 Y ˆ / ˆ Y Y Y error MRAE Mean relaive absolue error =mean( r ) MdRAE Median relaive absolue error =median( r ) GMRAE Geomeric mean relaive absolue error =gmean( r ) RelMAE Relaive mean absolue error =MAE/MAE b RelRMS E Relaive roo mean squared error =RMSE/RMSE b LMR Log mean squared error =log(relmse) PB Percenage beer =100 mean(i{ r <1}) 2 ) PB(MAE ) Percenage beer (MAE) =100 mean(i{mae<mae b }) PB(MSE) Percenage beer (MSE) =100 mean(i{mse<mse b })

26 CHAPTER TWO SMOOTHING There are several smoohing echniques for esimaing he numerical values of he coefficiens from noisy observaions of he underlying process. Smoohing is a process like curve fiing, bu here is a disincion. In a curve-fiing problem, one has a se of daa o which some appropriae curve is o be fied. The compuaions are done once, and he curve should fi equally well o he enire se of daa. A smoohing problem sars he same way, wih good clean daa and a reasonable model o represen he process being forecas. The model is fied o he daa; ha is, he coefficiens in he model are esimaed from he daa available o dae. So far, he problem is a simple curve-fiing problem. There are wo differences. Firs, he model should fi curren daa very well, bu i is no imporan ha daa obained a long ime ago fi so well. Second, he compuaions are repeaed wih each new observaion. The process is essenially ieraive, so ha is i imporan ha he compuaional procedures be fas and simple. 2.1 Moving Average In moving average echnique, model is assumed o be a consan model. Therefore, model for he underlying process is a (2.1) and he observaions include random noise X a (2.2) where he noise samples { } have an average value of zero. I is quie possible ha in differen pars of he sequence of observaions, widely separaed from each oher, he value of he single coefficien a will change. Bu in any local segmen, a single value gives a reasonably good model of he process (Brown, 1964). 17

27 18 The curren value of a can be esimaed by some sor of an average. Since he value can change gradually wih ime, he average compued a any ime should place more weigh on curren observaions han on hose obained a long ime ago. The moving average is in common use for ha reason. Now, M X X X N 1 N 1 (2.3) is he acual average of he N mos recen observaions, compued a ime. Is value is useful as an esimae of he coefficien â. The process of compuing moving average is quie simple and sraighforward. I is accurae: he average minimizes he sum of squares of he differences beween he mos recen N observaions and he esimae of he coefficien in he model. The rae of response is conrolled by he choice of he number N of observaions o be averaged. If N is large, he esimaes will be very sable. If he observaions come from a consan process, where a has a rue value, and where he noise samples { } are random samples from normal disribuion wih zero mean and variance 2, hen he average is an unbiased esimae of he 2 coefficien a, and he variance of he successive esimaes is 2 M / N. X1 X 2 X N E( M ) E N E( X1) E( X 2 ) E( X n ) N E( a 1) E( a 2 ) E( a N ) N a N N (2.4) a

28 19 X1 X 2 X N V ( M ) V N V ( X1) V ( X 2) V ( X n ) 2 N 2 N N 2 (2.5) N 2 The average age of he daa used in moving average is k N 1 N 1 N 2 (2.6) Following able summarizes he process. If we choose N=3 hen here will be no prediced values for he firs wo values of he ime series. Beginning from observaion hree, prediced values will be calculaed as he average of he las 3 observaions. Table 2.1 Moving average example Acual Values Moving Average Absolue Error Squared Error For example; M 3 x3 x2 x1 / 3 (9 8 9) / or M 9 x9 x8 x7 / 3 ( ) / 3 13

29 20 Now, MAE or MSE can be calculaed from he simulaed predicions and acual values. n e MAE = n 10 MSE = n 1 e n For moving average m-periods-ahead forecas for any fuure observaion a ime is equal o moving average calculaed a ime is ˆ m X M (2.7) and herefore one-period-ahead forecas is given by ˆ 1 X M (2.8) 2.2 Exponenial Smoohing Exponenial smoohing is probably he mos widely used class of procedures for smoohing discree ime series in order o forecas he fuure. I weighs pas observaions using exponenially decreasing weighs. In oher words, recen observaions are given relaively more weigh in forecasing han he older observaions. In exponenial smoohing, here are one or more smoohing parameers o be deermined and hese choices deermine he weighs, which are exponenially decreasing weighs as he observaions geing older, assigned o he observaions. This is a desired siuaion because fuure evens usually depend more on recen daa han on daa from a long ime ago. This gives he power of adjusing an early forecas wih he laes observaion. In he case of moving averages, which is anoher echnique of smoohing, he weighs assigned o he observaions are he same and equal o 1/ N so newes and oldes daa have he same weighs for forecasing.

30 21 There are also oher differen ypes of forecasing procedures bu exponenial smoohing mehods are widely used in indusry. Their populariy is due o several pracical consideraions in shor-range forecasing (Gardner, 1985); model formulaions are relaively simple model componens and parameers have some inuiive meaning only limied daa sorage and compuaional effor is needed racking signal ess for forecas conrol are easy o apply accuracy can be obained wih minimal effor in model idenificaion 2.3 Hisory of Exponenial Smoohing Exponenial smoohing mehods originaed by he works of Brown (Brown, 1959), (Brown, 1964), Hol (Hol, 1957) and Winers (Winers, 1960). The mehod was independenly developed by Brown and Hol. Robers G. Brown originaed he exponenial smoohing while he was working for he US Navy during World War II (Gass & Harris, 2000). Brown was assigned o design a racking sysem for fireconrol informaion o compue he locaion of submarines. Brown s racking model was essenially simple exponenial smoohing of coninues daa. During he early 1950s, Brown exended simple exponenial o discree daa and developed mehods for rends and seasonaliy. In 1956, Brown presened his work on exponenial smoohing a a conference and his formed he basis of Brown s firs book (Brown, 1959). By he way, Charles C. Hol, wih he suppor of he Office of Naval Research, worked independenly of Brown o develop a similar mehod for exponenial smoohing of addiive rends and enirely differen mehod for smoohing seasonal daa. Hol s original work was documened in an ONR memorandum (Hol, 1957) and wen unpublished unil recenly (Hol 2004a, 2004b).

31 22 A simple classificaion of he rend and seasonal paerns provided by Pegels (Pegels, 1969). Box and Jenkins (Box & Jenkins, 1970), Robers (Robers, 1982), and Abraham and Ledoler (Abraham & Ledoler, 1983) showed ha some linear exponenial smoohing forecass originae from special cases of ARIMA models. Gardner published his firs paper providing a deailed review of exponenial smoohing (Gardner, 1985). Up o his paper, many believed ha exponenial smoohing should be disregarded since i was a special case of ARIMA (Gardner, 2006). Since 1985, many works showed ha exponenial smoohing mehods are opimal for every general class of models ha is in fac broader han he ARIMA. Since 1980, he empirical properies of he mehods sudied by Barolomei (Barolomei & Swee, 1989) and Makridakis (Makridakis & Hibon, 1991), new proposals of esimaion or iniializaion are inroduced by Ledoler, (Ledoler & Abraham, 1984), forecass are evaluaed by McClain (McClain, 1988) and Swee (Swee & Wilson, 1988), and saisical models are concerned by McKenzie (McKenzie, 1984). Numerous variaions on he original mehods have been proposed (Carreno & Madinaveiia, 1990), (Williams & Miller, 1999), (Rosas & Guerrero, 1994), (Lawon, 1998), (Robers, 1982), (McKenzie, 1986). Good forecasing performance of exponenial smoohing mehods has been showed by several auhors (Sachell & Timmermann, 1995), (Chafield e al., 2001), (Hyndman, 2001). Many conribuions were made by researchers o exend he original work of Brown and Hol. These conribuions were made for differen forecas profiles. These profiles are given in Figure 2.1.

32 23 Figure 2.1 Forecas profiles from exponenial smoohing (Gardner, 1985) There are a lo of mehods for he forecas profiles above. Table 2.2 conains equaions for he sandard mehods of exponenial smoohing, all of which are exensions of he work of Brown (1959, 1964), Hol (1957) and Winers (1960). For each ype of rend, here are wo secions of equaions: he firs give recurrence forms and he second gives equivalen error-correcion forms. Recurrence forms were used in he original work by Brown and Hol and are sill widely used in pracice, bu error-correcion forms are simpler.

33 Table 2.2 Equaions for he sandard mehods of exponenial smoohing (Gardner, 2006) 24

34 25 Table 2.3 Noaion for exponenial smoohing (Gardner, 2006) Symbol Definiion Smoohing parameer for he level of he series Smoohing parameer for he rend Smoohing parameer for seasonal indices Auoregressive or damping parameer Discoun facor, 0 1 S Smoohed level of he series, compued afer X is observed T Smoohed addiive rend a he end of period R I X m p Smoohed muliplicaive rend a he end of period Smoohed seasonal index a he end of period Observed value of he ime series in period Number of periods in he forecas lead-ime Number of periods in he seasonal cycle Xˆ ( m ) Forecas for m periods ahead from origin e One-sep-ahead forecas error, e X Xˆ (1) C V D Q Z P Y Cumulaive renormalizaion facor for seasonal indices Transiion variable in smooh ransiion exponenial smoohing Observed value of nonzero demand in he Croson mehod Observed iner-arrival ime of ransacions in he Croson mehod Smoohed nonzero deman in he Croson mehod Smoohed iner-arrival ime in he Croson mehod Esimaed demand per uni ime in he Croson mehod 2.4 Simple Exponenial Smoohing In simple exponenial smoohing mehod model for he underlying process is assumed o be a consan model like moving average and he ime series is represened by X a (2.9) where is random componen wih mean zero and variance 2. The value of a is assumed o be consan in any local segmen of he series bu may change slowly over ime. This is he model wih no rend and no seasonaliy in Table 2.2 and he smoohing equaion for simple exponenial smoohing in recurrence form is given by

35 26 1 ) 1 ( S X S (2.10) where S is he smoohing saisic (or smoohed value) and is he smoohing consan. I can be seen ha he new smoohed value is he weighed sum of he curren observaion and he previous smoohed value. The weigh of he mos recen observaion is and he weigh of he mos recen smoohed value is (1- ). Then, 1 S can be wrien as ) 1 ( S X S (2.11) subsiuing 1 S in equaion 2.10 wih is componen (equaion 2.11) we can wrie S as ) (1 ) (1 ) (1 ) (1 S X X S X X S (2.12) and replacing 2 S in equaion 2.12 wih is componen we have ) (1 ) (1 ) (1 ) (1 ) (1 ) (1 S X X X S X X X S (2.13) repeaing he subsiuion for 4 3, S S and so on up o 0 S finally we have ) (1 ) (1 ) (1 ) (1 ) (1 ) (1 S X X X X X X S (2.14) where 0 S is saring value and i is ofen called as iniial value. Equaion 2.14 can also be wrien like his ) (1 ) (1 k k k S X S (2.15) As i seen from Equaion 2.14 or 2.15, S is he weighed average of all pas observaions and he saring value 0 S. The weighs are decrease exponenially depending on he value of parameer (smoohing consan).

36 27 For example, if smoohing consan is equal o 0.3 hen he weigh associaed wih he las observaion is equal o 0.3 and he weighs assigned o previous observaions are 0.210, 7, 3, 2, and so on. Figure 2.2 shows he weighs given o observaions when value is 0.3. These weighs appear o decline exponenially when conneced by a smooh curve. This is why i is called exponenial smoohing. More weighs given o mos recen observaions and weighs decrease geomerically wih age Weighs Assigned o Observaions Weigh Age Figure 2.2 Weighs assigned o observaions when is 0.3 Weighs assigned by simple exponenial smoohing are non-negaive and sum o uniy, since 1 1 (1 ) (1 ) (1 ) (1 ) 1 (1 ) k 0 k 1 (1 ) (1 ) 1 (1 ) (1 ) 1 (2.16)

37 28 The value of he parameers and S 0 mus be given o calculae he smoohed values. Depending on he chosen value of hese parameers, accuracy of simple exponenial smoohing may vary. There a number of mehods and suggesions o choose he smoohing consan and iniial value which we discuss in deail in secion Now, i is possible o calculae he expeced value and variance of he smoohing saisic S. For sufficienly large, he expeced value of S is so k E( S ) E (1 ) X k (1 ) S0 (1 ) S0 0when k0 0 k (1 ) E( X ) k k i 1 E( X ) (1 ) Noe ha ( r) k 0 i 0 1 r 1 a 1 (1 ) a a (2.17) S is unbiased esimaor of he consan a when. Therefore, for fuure forecass. The variance of S is S can be used k V ( S ) V (1 ) X k (1 ) S0 k0 2 0 V x 2 k ( ) (1 ) k k (1 ) V ( X ) (1 ) k 2 (2.18)

38 29 The weigh of he observaion X k is given as k w (1 ) k 0,1,2,, 1 (2.19) X k and weigh of he saring value is w S0 (1 ) (2.20) For simple exponenial smoohing m-periods-ahead forecasing is given by Xˆ m S m 1, 2,3, (2. 21) herefore one-period-ahead forecasing is given by ˆ 1 X S (2.22) The average age is he age of each piece of daa used in he average. In he exponenial smoohing process, he weigh given o daa k periods ago is (1 ) k so ha he average age of daa is k k 1 1 ka k k (1 ) 2 2 k 0 (1 (1 )) (1 a) (2.23) Since i is possible o derive a smoohing parameer which gives approximaely he same forecass as an unweighed moving average of any given number of periods, some researchers has concluded ha simple smoohing has no imporan accuracy (Adam, 1973), (McLeavey, Lee & Evere, 1981), (Armsrong, 1978), (Elon & Gruber, 1972), (Kirby, 1966). However Makridakis has found ha simple smoohing was significanly more accurae han unweighed moving average in a sample of 1001 ime seris (Makridakis, e all, 1982). Muh was he firs of many o prove ha simple smoohing is opimal for he ARIMA(0, 1, 1) (Muh, 1960). Harrison (1967), Nerlove and Wage (1964), and Theil and Wage (1964) showed ha simple smoohing is opimal wih α deermined by he raio of he variances of he noise processes. Robusness of simple smoohing has also been prediced by oher researches. Cogger (1973), Cohen (1963), Cox (1967) and Pandi and Wu (1974) argued ha a

39 30 more complex models may no yield significanly smaller errors. Robusness was suppored by Makridakis (Makridakis, e all, 1982). Simple smoohing was he bes overall choice for one-period-ahead forecasing. More evidence of robusness is given by he simulaion sudy of Gross and Craig (Gross & Craig, 1974) Smoohing Consan and Saring Value Parameer selecion is an imporan problem of simple exponenial smoohing. The value of smoohing consan and saring value mus be iniialized o sar he recurrence formula of S. There are differen mehods for choosing boh smoohing consan and saring value bu here is no any proven evidence favoring any paricular mehod. The firs problem is choosing he smoohing consan. I is cerain ha value should fall ino he inerval beween 0 and 1. There are wo exreme cases when is zero or one. If is equal o zero hen observaions are ignored enirely and he smoohed value consiss enirely of he saring value S 0. If is equal o one hen he previous observaions are ignored and he value of he smoohed value will equal o curren observaion. Values of in-beween 0 and 1 will produce inermediae resuls. However, i is obvious ha when is close o 1 more weighs pu on he recen observaions and when i close o 0 more weigh pu on he earlier observaions. So i is crucial o choose a proper value. The effec of smoohing consan is shown in Figure 2.3. Daa poins wih marker diamond is he acual daa, marker square represens forecass when 0.1 and marker riangular represens forecass when 0.9. I can be seen ha when 0.9 simple exponenial smoohing responses more rapidly o flucuaions. When a 0.1 response o flucuaions are very slow.

40 31 Figure 2.3 Effec of smoohing consan However, a big smoohing consan does no mean a beer forecas. Figure 2.4 shows he forecass for a 0.1 and a 0.9. As i seen from he graph using a big value for smoohing consan may cause o large forecas errors bu a small value may also cause o no respond o a rend quickly. So, i is very imporan o decide he value of he smoohing consan. Figure 2.4 Forecass

41 32 There are many heoreical and empirical argumens for selecing an appropriae smoohing value (Gardner, 1985). Gardner repors ha an smaller han 0.30 is usually recommended (Gardner, 1985). However, some sudies recommend values above 0.30 since frequenly yielded he bes forecass (Mongomery & Johson, 1976, Makridakis e al., 1982). I was also concluded ha i is bes o deermine an opimum from he daa raher han guessing i (Fildes e al., 1998). In pracice, he smoohing parameer is ofen chosen by a grid search of he parameer space; ha is, differen soluions for are ried saring, for example, from = 0.1 o = 0.9, wih incremens of 0.1. Then he value which produces he smalles sum of squares (or mean squares) for he residuals is chosen as he smoohing consan. In addiion, besides he ex pos MSE crierion, here are oher saisical measures (for example mean absolue error, or mean absolue percenage) error ha can be used o deermine he opimum value. The second problem is choosing he saring value and i is known as iniializaion problem. The weigh of S0 may be quie large when a small is chosen and he ime series relaively shor. Then he choice of he saring value becomes more imporan. Depending on he chosen value of, saring value can effec he qualiy of forecass for many observaions. Table 2.4 shows an example of he weighs given o he saring value and observaions when 0.1 and 0.9 for nine observaions. The weigh given o saring value is which is bigger han all weighs given o oher values when 0.1. Even if he weigh of las observaion is much smaller han he weigh given o saring value. When 0.9 he weigh given o saring value is oo small as expeced.

42 33 Table 2.4 Weighs given o he saring value and observaions Weigh when 0.1 Weigh when 0.9 Saring Value Observaion Figure 2.5 shows a line graphic of he weighs given in Table 2.4. I easy o see he effec of chosen value of o he saring value. A small value is ofen used when more weigh waned o given o he previous observaions bu i hen causes o give more weigh o saring value.

43 34 Figure 2.5 Weighs given o observaions and S0 when 0.1 and 0.9 Mehods for compuing S0 have been developed by a number of researchers. Brown s original suggesion is simply using he mean of he daa for S 0. Ledoler and Abraham (Ledoler & Abraham, 1984) recommended backcasing o obain S 0. When only a few daa poins are available, i can be difficul o choose a saring value. Gilchris proposed using an exac DLS formulaion for S (Gilchris, 1976). Similar ideas for eliminaing he need o esimae saring values are discussed by Cogger (Cogger, 1973), McClain (McClain, 1981), Taylor (Taylor, 1981) and Wade (Wade, 1967). Anoher alernaive wih a limied number of daa poins is o use Bayesian mehods o combine a prior esimae of he level wih an average of he available daa (Cohen, 1966), (Jonhson & Mongomery, 1974) and (Taylor, 1981). Table 2.5 includes an example. There are nine observaions and firs observaion is also used as saring value. Smoohed values and weighs are given in he able and Figure 2.6 displays he acual daa and smoohed values ogeher.

44 35 Table 2.5 Smoohed values for 0.1 and S0 x1 Acual Smoohed Weigh Saring Value Observaion Figure 2.6 Smoohed values when 0.1 and S0 x1 Figure 2.7 shows he effec of using differen mehods for saring value when 0.1. When a small a value is chosen, deciding which saring value will be used becomes more imporan. The chosen mehod affecs he smoohed values for many observaions.

45 36 Figure 2.7 Using differen mehods for saring value when 0.1 Figure 2.8 shows he effec of using differen mehods for saring value when 0.9. I is shown ha chosen mehod for saring value does no maer for big values of. Figure 2.8 Using differen mehods for saring value when 0.9

46 Double Exponenial Smoohing The ideas of exponenial smoohing can be exended o provide esimaes of he wo coefficiens in a linear model. When he process mean changes linearly wih ime hen he model will be X a b (2.24) In his model he average level of he ime series changes in a linear fashion over ime. The slope of he model is b while he inercep a ime 0 is a. The ime series can be described by he rend implied by his sraigh line combined wih random flucuaions. If he slope b of he rend line is greaer han 0, his implies ha he average level of ime series increases as ime advance, whereas if he slope b is less han 0, his implies ha he average level of he ime series decreases as ime advances. The simple smoohing model will lag his process for infinie ime. If simple exponenial smoohing were applied o he observaions from he linear process above, we would obain a he end of period S X (1 ) S 1 (2.25) and hen 1 k S X S where (1 ) k 0 k 0 (2.26) By aking expeced value 1 k E( S ) E( X ) S k 0 1 k0 k k E a b( k) S 0 0 (2.27) as, 0, we obain

47 38 k k E( S ) ( a bt ) b k k 0 k 0 (1 ) a b b (2.28) Since E( X ) a b, we have (1 ) E( S ) E( X ) b (2.29) This shows ha, for a linear model, he firs-order exponenially smoohed saisic, S, will end o lag by an amoun equal o (( 1 ) / ) b. There are several ways o adjus for he lag in simple smoohing. Hol (Hol e al., 1960) and Winers (Winers, 1960), uses separae parameers o smooh he level and rend of he series. The Brown models use a single parameer o smooh boh componens. The approach ofen used o deermine updaed esimaes of a and b is known as double exponenial smoohing. The specific formula for double exponenial smoohing for wo parameer is given by S X (1 )( S b ) 1 1 b ( S S ) (1 ) b 1 1 (2.30) The firs smoohing equaion adjus S direcly for he rend of he previous period,b -1, by adding i o he las smoohed value of S 1. This helps o eliminae he lag and brings S o he appropriae base of he curren value. As in he case for simple exponenial smoohing, here are a variey of schemes o se iniial values for S0 and b 0 in double exponenial smoohing. S0 x 1. Here are hree suggesions for b0 in general se o b x x / 1 b x x x x x x b x x n 0 n 1 / 3 (2.31)

48 39 For parameers and, values generally less han 0.3 have been recommended for he Hol-Winers models. For he Brown models, α values of 0.2 or less are generally acceped (Brown, 1964), (Couie, e al., 1964), (Harrison, 1967), and (Mongomery and Johson, 1976). In oher applicaions, a wider range of parameers should be considered. Makridakis (Makridakis, e all, 1982) and Chafield (Chafield, 1978) found ha he mos accurae parameers were frequenly in he range Even parameers above 1.0 should be considered. McClain and Thomas showed ha he non-seasonal model is sable over he range 0 < α < 2 and 0 < γ < (4-2 α)/ α. The Brown non-seasonal linear rend model is sable for 0 < α < 2. Using compuaional mehods, Swee (Swee, 1985) reached he following conclusions on he parameers for seasonal models: if he lengh of he seasonal cycle is four periods, he model is always sable for parameers beween 0 and 1. If he cycle is 12 periods, he model is no necessarily sable for parameers in his range. A grid search can be used o find he parameer se which minimizes he fied MSE. The m-periods-ahead forecas is given by Xˆ m S b m (2.32) The one-period-ahead forecas is given by ˆ 1 X S b (2.33) Table 2.6 includes and example o compare he smoohed values obained from simple exponenial smoohing and double exponenial smoohing. For simple exponenial smoohing 0.1 and S0 x1 is used and for double exponenial smoohing 0.1, 0.1, S0 x1 and b 0 ( x 4 x 1 ) / 3 is used for iniial values. Double exponenial smoohing performs beer han simple exponenial smoohing since here is an adjusmen for he rend in double exponenial smoohing. I is easy o see he performance of he double exponenial smoohing in Figure 2.9. Table 2.6 Double exponenial smoohing vs Single exponenial smoohing Daa Double Exponenial Smoohing Simple Exponenial Smoohing

49 Daa Simple 1 Double Figure 2.9 Double exponenial smoohing vs simple exponenial smoohing When 0.9 is used and oher parameers lef as he same, double exponenial smoohing gives beer response o rend componen owards he end of he series (Figure 2.10). Figure 2.11 and Figure 2.12 displays oher examples. For small values rend componen addiion o he double exponenial smoohing may cause beer forecass for double exponenial smoohing. Bu when value is close o 1, rend conribuion seems o geing loose is imporance for double exponenial smoohing.

50 Daa Simple Double Figure 2.10 DES for 0.1 and 0.9 vs SES for Daa Simple Double Figure 2.11 DES for 0.3 and 0.9 vs SES for 0.3

51 Daa Simple Double Figure 2.12 DES for 0.8 and 0.9 vs SES for Triple Exponenial Smoohing Triple exponenial smoohing is used if here is a rend and seasonaliy in he daa. A hird equaion is inroduced o care of he seasonaliy. The equaions for riple exponenial smoohing are given by X S (1 )( S b ) 1 1 IL b ( S S ) (1 ) b I 1 1 X (1 ) I S L (2.34) where I is he seasonal index and is a consan beween 0 and 1. The general formula o esimae iniial rend is given by b 0 1 X L1 X1 X L2 X 2 X LL X L L L L L (2.35) Iniial values for he seasonal indices are calculaed in hree seps. Le consider a daa consis of 4 years wih 3 periods. Then

52 43 Sep 1. Compue he average of each of he 4 years A p 3 i X i 1 p 1,2,,4 3 Sep 2. Divide he observaions by he appropriae yearly mean X1 / A1 4 / 2 X 2 / A1 5 / 2 X 3 / A1 6 / 2 X A X 7 / A3 X10 / A4 X A X 8 / A3 X11 / A4 X A X 9 / A3 X12 / A4 Sep 3. Calculae seasonal indices as he average of each row I ( X / A X / A X / A X / A ) / I ( X / A X / A X / A X / A ) / I ( X / A X / A X / A X / A ) / The m-periods-ahead forecas is given by Xˆ ( S b m) I (2.36) m Lm The one-period-ahead forecas is given by Xˆ ( S b ) I (2.37) 1 L1

53 CHAPTER THREE MODIFIED EXPONENTIAL SMOOTHING 3.1 Modified Simple Exponenial Smoohing If m is any ineger (1 m ) and leing ( m / ) hen simple exponenial smoohing equaion can be wrien as m m S X S 1 (3.1) hen S is weighed average of all pas observaions. We firs demonsrae S can be wrien as a linear combinaion of pas daa and hen observe ha he weighs given o pas observaions are nonnegaive and sum o uniy, hus making i possible o inerpre S as a weighed average of observaions. Now S 1 can be wrien as m m1 S X S (3.2) subsiuing for S 1 wih is componen in equaion (3.1) we obain m m m m m 1 S X X S (3.3) Now S 2 can be wrien as m m2 S X S (3.4) and subsiuing S 2 we obain m m m m m m1 S X X 1 X m m1 m2 S (3.5) 44

54 45 and repeaing for S 3 and so on we may finally obain he general form of S such ha m m m m m m1 S X X 1 X m m m1 m2 X (3.6) m m m X 1 2 m2 m1 m m Sm 1 m 2 m1 m1 or ( m1) m ( m)!( k 1)! ( m)! m! k m k 0 ( m k)!( 1)!! S X S (3.7) or k 1 ( m1) m 1 1 S X S m m k m k 0 (3.8) where Sm is a simple average of firs m observaions i.e. S m X1 X 2.. X m m (3.9) We observed ha he curren smoohed esimae S is he weighed average of he observaions for all earlier periods and he saring value S m. Noe ha he weigh for observaion X k is m ( m)!( k 1)! wx k k 0,1,, ( m 1) (3.10) ( m k)!( 1)!

55 46 and i is beween 0 and 1 for all k and any given value of m. So he saisic S is a weighed average of all pas observaions. Now we mus prove ha hese weighs sum o uniy. Suppose =3 and m=2 hen w x k 2 (1)!(2 k)! 3 (1 k)!(2)! k 0 (3.11) so w x 3 2 (1)!(2)! 2 3 (1)!(2)! 3 (3.12) and w S 2 2 (1)!(1)! 1 3 (0)!(2)! 3 (3.13) and sum of he weighs is one. And suppose =5 and m=2 hen w x k 2 (3)!(4 k)! 5 (3 k)!(4)! k 0,1, 2 (3.14) corresponding weighs are 4/10, 3/10, 2/10 and he weigh of S2 is 1/10 which are sum o uniy again. For a general form, using probabilisic argumens we can prove ha sum of he weighs (>m) for X, m m m m m m m 2 m 1 (3.15) and weigh of S m m m m 2 m 1 (3.16) is equal o one. Consider an urn ha conains balls such ha a number of m minoriy balls are of one color (whie, say) and (-m) balls are of anoher majoriy color (black, say). The disribuion of he number of draws y unil a specified number

56 47 c of whie ball is obained as he negaive hyper geomeric disribuion (Johnson & Kos, 1969), given by y 1 y c 1 m c f ( y; m; ; c) P( Y y) y c, c 1,, c m m (3.17) wih parameers saisfying 1 c m. Assume ha y is he number of draws unil firs whie ball is obained, i.e., c=1. This yield us y m 1 f ( y; m; ) P( Y y) y 1,2,, m 1 m and expeced value and variance of random variable Y are (3.18) 1 E ( Y ), m 1 ( 1)( m) 1 ( Y ) 1 ( m 1)( m 2) m 1 Var (3.19) Le X Y 1 hen we obain weigh of observaions for modified simple exponenial smoohing process wih E( X ) ( m) / ( m 1). Now i is clear ha he weighs for he observaions and iniial value are nohing more han probabiliies of negaive hyper geomeric for given m and from y=1, 2,, -m+1. Therefore i is clear ha sum of he weighs X, X 1,, X m1 and weigh of Sm is equal o 1.

57 Table 3.1 Weighs given by Modified Simple Exponenial Smoohing and Simple Exponenial Smoohing k 0 1 -(m+1) -m -2-1 X X 1 X m 1 X m X 2 X1 m m m m m 2 MSES 1 1 m SES (1 ) (1 ) m 1 (1 ) m 2 (1 ) (1 ) 1 Saring Value MSES m m m 1 SES (1 ) Table 3.2 Age of daa MSES SES Age of daa m m

58 If we use corresponding level for a seleced m values and for any, hen he weighs given by MSES and SES can be compared by using Table 3.3. Table 3.3 Weighs given by Modified Simple Exponenial Smoohing and Simple Exponenial Smoohing MSES SES k 0 1 -(m+1) -m -2-1 X X 1 X m 1 X m X 2 X1 m m m m m 1 1 m m m m1 2 m m m m m m m m m m 1 Saring Value MSES SES m m m 1 m 49

59 50 There are wo exreme cases in he definiion of S according o seleced m value. When m is equal o 1 hen S X X... X S where S 1 X 1 (3.20) and he weigh of curren and all pas observaions are equal o equal o hen all he weighs are given o he las observaion 1. When m is S X (3.21). When 1<m< hen he weighs decrease geomerically wih he age of he observaions. For he sake of simpliciy and demonsraion, choose m=3 hen 3 3 S X S 1 (3.22) and if he number of observaions is equal o 10 hen S10 X10 X9 X8 X7 X X5 X S S 0.30 X X X X X X X S is obained where S3 is a simple average of firs hree observaion. The weigh associaed wih he curren observaion is 0.30 and he weighs assigned o previous observaions are,,, and so on respecively, as i seen he weighs of older observaions decrease geomerically wih he age of he observaions (Figure 3.1). Noe ha, he weigh of he saring value S3 is smaller han he weigh of X 4.

60 Weighs when m=3 and = x10 x9 x8 x7 x6 x5 x4 s3 Figure 3.1 Weighs given by modified simple exponenial smoohing when m=3 and =10 Figure 3.2 shows he weighs assigned by modified simple exponenial smoohing when m=4. They are sill decreasing wih he age of observaions and he weigh of he saring value S4 is smaller han he weigh of X Weighs when m=4 and = m= x10 x9 x8 x7 x6 x5 s4 Figure 3.2 Weighs given by modified simple exponenial smoohing when m=4 and =10

61 52 1 Weighs for m=2, 3, 4, 5, 6, 7, 8, 9 when =10 Weigh m=2 m=3 m=4 m=5 m=6 m=7 m=8 m= Age Figure 3.3 Weigh of X k for k=10, 9,, m and m=2, 3, 4, 5, 6, 7, 8 when =10 A comparison of he weighs assigned by modified simple exponenial smoohing for =10 and differen value of m=2, 3, 4, 5, 6, 7, 8, 9 are shown in Figure 3.3. Because hese weighs appear o decline exponenially when conneced by a smooh curve, he name exponenial smoohing can be applied o his procedure. Anoher resul is ha, he weigh assigned o saring value is always smaller han he weigh of previous observaion for all m values. Wheher he value of m is small or no, i does no affec he conribuion of he saring value o he smoohed value (and also o forecas). However, for simple exponenial smoohing i is seen ha small value migh cause o give more weigh o saring value han he weighs given o many previous observaions and his may effec he forecass. We can compare he effec of saring value when a small smoohing consan is chosen for boh mehods simple exponenial smoohing and modified simple exponenial smoohing. When we add smoohed values obained from modified simple exponenial o he Figure 2.6, i can be see ha saring value does no effecs he smoohed values oo much for modified simple exponenial smoohing.

62 53 90 SES vs MSES for small smoohing consan Daa 75 Acual α=0.1 m= Age Figure 3.4 Effec of saring value for small smoohing consan The weigh given o he observaions and iniial value of simple exponenial smoohing for =0.3 and modified simple exponenial smoohing for m=3 for =10 is given in Figure 3.5 As i seen, he weigh of las five observaions for modified simple exponenial smoohing is greaer han weigh of simple exponenial smoohing. Boh mehods assign equal weigh for fourh observaion bu before his observaion weigh of modified simple exponenial smoohing for oher observaions decline faser han simple exponenial smoohing. In conras o simple exponenial smoohing, weigh of iniial value S3smaller han las weigh x4 for modified simple exponenial smoohing.

63 SES MSES x10 x9 x8 x7 x6 x5 x4 x3/s3 x2 x1 s0 SES MSES 0.30 Figure 3.5 Comparison of weighs for simple exponenial smoohing (=0.3) and modified simple exponenial smoohing (T=10, m=3) The weigh given o he observaions and iniial value for simple exponenial smoohing for =0.2 and for modified simple exponenial smoohing for m=3 and =10 is given in Figure 3.6. Modified simple exponenial smoohing assigns more weigh o recen observaions up o sevenh observaion. The weighs assigned by boh mehods are equal for sevenh observaion and from sevenh observaion o firs observaion more weigh is given by simple exponenial smoohing. Noice ha, weigh of saring value S0 is greaer han he weighs assigned o observaions up o sevenh observaion for simple exponenial smoohing. However, weigh of saring value S3 is smaller han he weigh of all observaions for modified simple exponenial smoohing.

64 SES MSES 0 0 x15 x14 x13 x12 x11 x10 x9 x8 x7 x6 x5 x4 x3/s3 x2 x1 s0 SES MSES Figure 3.6 Comparison of weighs for simple exponenial smoohing (=0.2) and modified simple exponenial smoohing (=15, m=3) The assigned weigh for more recen daa by modified simple exponenial smoohing for any given and m is always greaer han simple exponenial smoohing wih corresponding value. The difference beween weighs for wo mehods is sricly increasing up o cerain ime, afer reaching maximum poin his difference sars o decrease unil boh mehod equal o each oher. Afer equaliy, simple exponenial smoohing gives more weigh hen modified simple exponenial smoohing for middle par of he observaions. Le compare he weigh differences of MSES and SES for differen values of m, and corresponding value. When m is equal o 1 smoohing process for MSES can be wrien as S X X X S where S1 X1 and he weigh of curren and all pas observaions are equal o 1/ and ha can no be achieved in he SES. However if we choose α=1/ in he SES hen

65 56 S X (1 ) X (1 ) X (1 ) X (1 ) S X X X X S is obained. Therefore when m is equal o 1 weigh of X k in MSES will always be greaer han ha in SES for k=0, 1, 2,, -1 for any value of. Figure 2 shows he graphs of differences when m=1 and =10, 30 and 50. All differences are posiive since weighs of MSES are greaer han weighs of SES. 00 Wmses-Wses Weigh k =10 =30 =50 Figure 3.7 Weigh differences when m=1 and =10, 30 and 50 When m is equal o wo, MSES gives more weighs o recen observaions up o a cerain poin and afer ha poin SES weighs become bigger for older observaions. Figure 3 shows he weigh differences for =10, =30 and =50. As i seen from he graph, he weigh differences for recen observaions are geing large when value is geing smaller.

66 57 50 Wmses-Wses Weigh k =10 =30 =50 Figure 3.8 Weigh differences when m=2 and =10, 30 and 50 The same resul is also obained for all m values beween 2 and -1. Weighs given by MSES are bigger up o a cerain poin and afer ha poin weighs of SES are bigger. Noice ha MSES gives more weighs o recen observaions. Following figures show he weigh differences for differen m (m>2) and values.

67 58 00 Wmses-Wses Weigh k =10 =30 =50 Figure 3.9 Weigh differences when m=3 and =10, 30 and Wmses-Wses Weigh k =10 =30 =50 Figure 3.10 Weigh differences when m=5 and =10, 30 and 50.

68 59 Up o now, we showed ha modified simple exponenial smoohing assigns exponenially decreasing weighs o observaions, weighs are non-negaive and sum o uniy. Smoohing saisic S can be wrien as a linear combinaion of he observaions and saring value. In addiion, he chosen value of smoohing consan does no influence he conribuion of he saring consan o laer smoohed values. And modified simple exponenial smoohing assigns even more weigh han simple exponenial smoohing o mos recen observaions and less weigh o earlier observaions. Now we can find he expeced value of S exponenial smoohing o prove ha parameer a. S for modified simple is also an unbiased esimaor of unknown k 1 ( m1) m 1 1 E( S ) E x S m m k m k 0 k 1 ( m1) m 1 1 E( X ) k0 m m E( X ) a (3.23) Therefore S is unbiased esimaor of he unknown parameer a.

69 60 Variance of S for modified simple exponenial smoohing is k m m 1 ( m)! m! V ( S ) V X S m 1k m k1! k m m 1 ( m)! m! V X k 1! m V X m k 1 k m 1 m V ( S ) m (3.24) Average age of daa for MSES is k m m 1 (3.25) since his sum is expeced value of sum for negaive hyper geomeric random variable if Y sars from 0. Average age of MSES is smaller han average age of SES since MSES gives more weigh o recen observaions m-periods-ahead forecas made by modified simple exponenial smoohing is ˆ m X S (3.26) And one-period-ahead forecas made by modified simple exponenial smoohing is ˆ 1 X S (3.27) 3.2 Modified Double Exponenial Smoohing The same idea can be applied o double exponenial smoohing. If m is any ineger (1 m ) and leing ( m / ) hen double exponenial smoohing equaions can be wrien as

70 61 m m S X ( S 1 b 1) b ( S S ) (1 ) b 1 1 (3.28) where Sm is again he average of firs m observaions. b is he rend facor and i lef as i was. The parameer used in he second equaion gives exponenially decreasing weighs o he differences of successive smoohed values. The value of is beween 0 and 1. Iniial value for second equaion is b0 and here are differen suggesions o se i. Taking he firs value in he daa, or average of he differences of firs hree couple daa, or fiing a linear regression model are some of he suggesed mehods o se he iniial value of b 0. The smoohing equaion of modified double exponenial smoohing is similar o he smoohing equaion of modified simple exponenial smoohing. The only difference is he erm b 1 added o S 1. So, i is obvious ha weighs are decreasing exponenially and sum o uniy. Modified double exponenial smoohing gives also more weighs han double exponenial smoohing o recen observaions like modified simple exponenial smoohing. m-periods-ahead forecas made by modified double exponenial smoohing is Xˆ m S b m (3.29) And one-period-ahead forecas made by modified double exponenial smoohing is ˆ 1 X S b (3.30) 3.3 Modified Triple Exponenial Smoohing The same idea can also be applied o riple exponenial smoohing. If m is any ineger (1 m ) and leing ( m / ) hen riple exponenial smoohing equaions can be rewrien as

71 62 m X m S ( S b ) 1 1 IL b ( S S ) (1 ) b 1 1 X Iı (1 ) I S L (3.31) where Sm is he average of firs m observaions. b is he rend facor and I is he seasonal index. The recurring equaions b and I lef as i was. The value of parameers and are beween 0 and 1. Iniial value for second equaion is b0 and he general formula o esimae he iniial rend is given by b 0 1 X L1 X1 X L2 X 2 X LL xl L L L L (3.32) The smoohing equaion of modified riple exponenial smoohing is similar o he smoohing equaion of modified simple exponenial smoohing. The only difference is he erm b 1 added o S 1 and erm I L dividing X. So, i is obvious ha weighs are decreasing exponenially and sum o uniy. Modified riple exponenial smoohing gives also more weighs han double exponenial smoohing o recen observaions like modified simple exponenial smoohing. The m-periods-ahead forecas is given by Xˆ ( S b m) I (3.33) m Lm and one-period-ahead forecas is given by Xˆ ( S b ) I (3.34) 1 L1

72 CHAPTER FOUR APPLICATION A compuer program is wrien o compare exponenial smoohing mehods and modified exponenial smoohing mehods. Php programming language is used for programming. Daa is sored in a daabase using Mysql. A web server is used o consruc user inerface (Figure 4.1). Figure 4.1 User Inerface Compuer program consiss of wo pars; Daa operaions and Analysis operaions. Daa is sored in a daabase and in he analysis pages i is possible o choose a daa se from he daabase. Since i is serving as a web sie any one who have an inerne connecion may access o i. Using a cenralized daabase will conribue o consruc a daa pool. 63

73 Daa Operaions Daa operaions provide us o sore daa in a daabase o be used laer in he analysis. Creaing or imporing a new daa se, updaing and deleing an exising daa se are he basic jobs of he daa operaions (Figure 4.2). Figure 4.2 Daa Operaions Afer clicking he buon Creae New Daa Se a page will load o creae a new daa se (Figure 4.3). Daa and some oher info o describe he daa will be enered here. A new inpu box will be added o he form every ime afer a new daa is enered o he las inpu box alhough here seems o be only one inpu box o ener daa. New daa se can be saved by clicking he buon SAVE afer enering he required info and daa. Daa ses are sored in mysql daabase and in he analysis pages a desired daa se can be chosen easily.

74 65 Figure 4.3 Creaing a New Daa Se I is possible o display some descripive info of any daase. Daa iself, ime char plo and some descripive saisics can be displayed a his page (Figure 4.4). Figure 4.4 Describing a Daa Se

75 Analysis Operaions The second par of he program is he analysis par. Here, smoohed values, fis, errors for he exising mehods and modified mehods can be calculaed, graphics for smoohed values, fis or errors can be displayed and differen accuracy measures can be calculaed o compare he mehods. There are hree buons a he righ side of he main page o run analyses (Figure 4.1). In fac here is no difference abou he calculaions made by hese hree ypes of run, only difference is he reporing forma and he number of he daa se ha will be used in he analysis Run Mehods When Run Mehods seleced hen only one seleced daa se will be used and all he calculaions and graphics will be displayed in deail while reporing. Afer clicking Run Mehods buon a new page will load o allow user o selec a daa se from he daabase and exponenial smoohing ype (ie, single, double or riple) (Figure 4.5). Figure 4.5 Run Mehods Screen

76 67 A new screen will be displayed o allow user o ener he necessary values for he parameers ha will be used by he mehods afer selecing a daa se from he selec box and clicking he buon of he desired smoohing ype. For example, if Simple Exponenial buon is clicked hen simple exponenial smoohing and modified simple exponenial smoohing will be compared. Value of for simple exponenial smoohing and value of m for modified simple exponenial smoohing mus be enered in his page (Figure 4.6). A value for m will be suggesed afer enering a value for. User can choose suggesed value or ener anoher value. Also a mehod o calculae iniial value for simple exponenial smoohing mus be seleced from he selec box ha includes he mos frequenly menioned mehods in he lieraure. Figure 4.6 Parameer Screen There are various ables and graphics in he reporing page o compare he mehods. Several abs are used for simpliciy. The firs ab includes a able ha displays he original daa and smoohed values for simple exponenial smoohing and modified simple exponenial smoohing. A ime char plo of original daa and smoohed values is also displayed in he firs ab (Figure 4.7).

77 68 I is possible o see he relaed calculaion of a smoohed value moving mouse over a cell of he able. The values used in he calculaion of smoohed value will be colorized and he formula will be displayed in a pop up message balloon. Figure 4.7 Smoohed values A he op of he page, analyzed daa se is lised in a selec box and used parameers are also displayed as a web form. I is possible o ener a new or m value o rerun he analysis for differen values of and m. I is also possible o change he daa se from he lis box o run same analysis for a differen ime series daa. Second ab in he reporing page displays he forecased values (fis) and graph of hose values (Figure 4.8). Third and fourh abs are display he errors and percen errors respecively (Figure 4.9, Figure 4.10).

78 69 Figure 4.8 Fis Figure 4.9 Errors

79 70 Figure 4.10 Percen Error The fifh ab displays he accuracy measures o compare he mehods. Seven error measures are calculaed here; Mean Absolue Error (MAE), Mean Squared Error (MSE), roo Mean Squared Error (rmse), Mean Absolue Percen Error (MAPE), Symmeric Mean Absolue Percen Error (smape), Relaive Average Ranking of Absolue Percen Error (rarsape) and Percen Beer (pbeer) (Figure 4.11). n x xˆ 1 MAE = n n MSE = x xˆ 1 2 rmse= MSE n n ( x xˆ ) / x MAPE= 1 *100 n n ( x xˆ ) / (( x xˆ ) / 2) smape= 1 n *100

80 71 Figure 4.11 Accuracy Measures And he las ab displays he weighs assigned by he mehods (Figure 4.12). Figure 4.12 Weighs

81 Run Mehods on Daases The second ype of analyses is o make same analysis for a group of daa ses. Some of daa ses can be grouped when creaing hem, for example 111 ime series daa from Makridakis compeiion can be combined in a group. All he ime series in his group will auomaically be analyzed when his group is seleced. A new page will be loaded and here will be a selec box o selec a daa group and four buons o describe seleced daa group or run an analysis on a desired mehod afer clicking Run Mehods on Daases in main menu (Figure 4.13). Figure 4.13 Run Mehods on Daases Afer selecing a daa group from selec box, size, mean and variance of each daase in he group can be lised by clicking Describe Daa buon (Figure 4.14). This will give some inroducory informaion o user abou he size, mean, variance, median, min and max value of each he daa se in ha group. The se number lised in he repor includes a link o change display o deailed daa descripion page. Deailed informaion abou each daa se can be displayed by clicking he corresponding se number or graph icon in he lis.

82 73 Figure 4.14 Describing a group of daa ses One of he hree buons named as Simple Smoohing, Double Smoohing and Triple Smoohing mus be clicked afer selecing a group from he selec box o make he analysis on a desired smoohing mehod (Figure 4.13). Firs, a page will be displayed o allow user o ener required values for parameers ha used in he calculaions of mehods. For example, value and iniial value mehod for simple exponenial smoohing and m value for modified simple exponenial smoohing will be asked for simple smoohing. The values enered here will be used for all daa ses included in he seleced daa group. Since he sizes of daa ses are differen, corresponding m value mus also be differen. Regardless of he daase size he enered m value will be used since here is only one inpu box o ener m value. However, leaving m value as empy will cause o selecing corresponding m value auomaically by he program for each daa se in he daa group for a given value (Figure 4.15).

83 74 Figure 4.15 Simple Smoohing Reporing page includes seven abs (Figure 4.16). Each ab displays he resuls for a differen accuracy measure. For example, resuls for MAE is included in he firs ab. Daa se number, size of he daa se, value, accuracy measures calculaed for classical and modified mehod, and corresponding m value are lised in a row for each daa se in he seleced daa group. Performance of he classical mehod and modified mehod can be compared for each daa se using he seven differen accuracy measures included here. There are also wo icons in each row in he lis. One may open he deailed analysis page for any daa se by clicking he firs icon and daa describe page can be opened for any daa se by clicking he second icon.

84 75 Figure 4.16 Repor page Run Mehods Auomaically The hird buon o make analysis in he main menu (Figure 4.1) is Run Mehods Auomaically. I is imporan o ry differen values since we don know he opimum value for a daa se. Accuracy measures are calculaed for differen values of and corresponding m values for a seleced daa se or a daa group when his ype of analysis seleced. Differen values, saring from 0.1 and incremening by 0.1 up o 0.9, and corresponding m values are used and accuracy measures are calculaed o compare he mehods for differen levels of. A new page will be loaded afer clicking he buon Run Mehods Auomaically (Figure 4.17). There are wo secions in his page. Lef side is used o run only one daa se for differen levels and corresponding m values, righ side is used o run a daa group again for differen levels and corresponding m values.

85 76 Figure 4.17 Run Mehods Auomaically When a daa se is seleced and a buon is clicked from lef side relaed smoohing ype will be analyzed. Differen values, saring from 0.1 up o 0.9 incremening by 0.1, and corresponding m values will be used and accuracy measures will be calculaed and displayed for each level of (Figure 4.18).

86 77 Figure 4.18 Analysis resuls for only one daa se When a daa group is seleced from righ side and clicked a buon below relaed smoohing ype will be analyzed for each daa se in ha group. Saring from 0.1 up o 0.9 incremening by 0.1, differen values and corresponding m values will be used and accuracy measures will be calculaed for each daa se in he seleced daa group. Daa se averages will be calculaed for each accuracy measure and displayed in a abulaed lis (Figure 4.19). Probably, classical mehod will perform beer for some daa ses in he daa group and modified mehod will perform beer for ohers. So, number of imes ha each mehod performed beer han oher is calculaed and percenages are displayed a he end of he lis for each accuracy measures (Figure 4.20).

87 78 Figure 4.19 Analysis resuls for a daa group Figure 4.20 Percenages for each accuracy measures

88 CHAPTER FIVE EMPIRICAL COMPARISONS Empirical sudies play an imporan role o undersand various behaviors of forecasing mehods herefore an empirical sudy is developed o compare he performance of he modified and classical exponenial smoohing mehods. Firs of all, real ime series from Makridakis compeiions ha are well known and mosly used by he researchers are used in he sudy. There are 111 ime series in Makridakis compeiion (Makridakis, 1979) and 1001 ime series in M-compeiion (Makridakis, 1982). Secondly, here are many accuracy measures have been used o evaluae he performance of forecasing. Mos commonly used accuracy measures are included in he sudy which are Mean Absolue Error (MAE), Mean Squared Error (MSE), roo Mean Squared Error (rmse), Mean Absolue Percen Error (MAPE), Symmeric Mean Absolue Percen Error (smape), Relaive Average Ranking of Absolue Percen Error (rarsape) and percen Beer (pbeer). Finally, boh in-sample performance and ou-of-sample performance is compared. The in-sample performance comparison is based on one-sep-ahead forecass and mosly used comparison ype for he earlier sudies. Laer, he ou-of-sample performance have been sared o be use which is based on forecasing he daa in he hold-ou period. 5.1 Modified Simple Exponenial Smoohing vs. Simple Exponenial Smoohing In-sample Performance There are 111 ime series in he Makridakis compeiion (Makridakis, 1979) and 1001 ime series in M-Compeiion (Makridakis, 1982). All ime series are used o compare in-sample performance of he mehods modified simple exponenial 79

89 80 smoohing and simple exponenial smoohing and summary of he comparison resuls are given in a brief able a he end of his secion. Le firs look a some deailed comparison resuls for a few ime series from hese oal 1112 ime series. Le sar wih he firs ime series from Makridakis Compeiion. There are 472 daa in his series. Mean of he daa is and variance is The ime series plo is given in Figure 5.1. Figure 5.1 Time series plo of firs sample To see he deailed comparison beween he mehods modified simple exponenial smoohing and simple exponenial smoohing Run Mehods mus be clicked from he home page (Figure 5.2).

90 81 Figure 5.2 Home page A new page will be displayed o selec a ime series from he daabase and he smoohing ype. Choose Sample 1.1 from he selec box and click Simple Smoohing buon o compare modified simple exponenial smoohing and simple exponenial smoohing (Figure 5.3).

91 82 Figure 5.3 Selecing a ime series and smoohing ype Nex sep is enering he values required o run analysis. Corresponding value for m will be recommended afer enering a value for. User may choose o ener recommended value or ener anoher value. Le ener 0.1 for hen he corresponding value for m is recommended as 47. Le use recommended m value so ener 47 for m. Afer selecing a mehod for calculaing iniial value for simple exponenial smoohing click SEND buon o sar analysis (Figure 5.4).

92 83 Figure 5.4 Enering required values o sar he analysis Repor page includes some descripion abou he seleced ime series and enered values for he parameers and 7 abs o display resul for he analysis. The firs ab displays he smoohed values and a graph of original values and he smoohed values obained from each mehod. I is possible o see he calculaion of a smoohed value moving mouse over a relaed able cell (Figure 5.5).

93 84 Figure 5.5 Original daa and smoohed values obained from each mehod In he graph, blue color is used for original daa, red color is used classical mehod and green color is used for modified mehod. I is easy o see ha smoohed values for each mehod are far away from he original daa. This may be a visual clue o user o increase he values of and m. The second ab displays he forecass. Remember ha he forecas value a ime was he previous smoohed value for simple exponenial smoohing. This ab displays he original daa and forecass obained from each mehod and includes a graph of hem (Figure 5.6). When we look a o he graph i can be said ha forecased values are no fi good enough o original daa since smoohed values were also far away from he original daa. By he way selecing anoher and m values may produce beer resuls. I may be beer o rerun he analysis for a differen value of and corresponding m value afer finishing he examining all abs.

94 85 Figure 5.6 One-sep-ahead forecass The hird ab is he Errors ab. Errors calculaed for each mehod lised in his ab and also a graph of he errors is displayed (Figure 5.7). Here red color is used for simple exponenial smoohing and green color is used for modified simple exponenial smoohing. I is hard o say somehing abou he errors when we look a o he graph since here are a lo of daa in he graph. However, i seems o be ha errors obained from modified simple exponenial smoohing are mosly smaller han he errors obained from simple exponenial smoohing.

95 86 Figure 5.7 Errors The nex ab is percen Errors ab which displays he percen errors (Figure 5.8). Percen errors for modified simple exponenial smoohing are mosly smaller han simple exponenial smoohing when we look a o he graphic in Figure 5.8.

96 87 Figure 5.8 Percen errors The fifh ab is he ab ha includes he resuls for he accuracy measures. This is probably he mos imporan ab of he repor page. According o he values of accuracy measures lised in his ab he accuracy of he modified simple exponenial smoohing and simple exponenial smoohing can be compared. The oher abs have some inuiive meanings o user bu his ab helps o see he accuracy of each mehods. When value is 0.1 and he corresponding m value is 47, for he ime series Sample 1.1 from Makridakis Compeiion, MAE value is for simple exponenial smoohing and i is for modified simple exponenial smoohing (Figure 5.9). I can be said ha modified simple exponenial smoohing is performed beer hen simple exponenial smoohing according o MAE.

97 88 Figure 5.9 Accuracy measures MSE value for modified simple exponenial smoohing (MSES), ha is , is smaller han MSE value for simple exponenial smoohing (SES) ha is I can be said ha MSES performed beer han SES according o he MSE. Of course rmse value for MSES is smaller han rmse value for SES since rmse is squared roo of MSE. Mean Absolue Percen Error (MAPE) value for MSES is 37 which is smaller han MAPE value for SES ha is 796. smape value for MSES is smaller han smape value for SES and rarsape value for MSES is smaller han rarsape value for SES. MSES sill performs beer han SES according o hese hree accuracy measures. The bigger value is beer for he accuracy measure pbeer. pbeer value for MSES is 25 and pbeer value for SES is MSES again performs beer according o he accuracy measure pbeer since his ime bigger value is beer. All accuracy measures used in he empirical sudy showed ha modified simple exponenial smoohing performs beer han simple exponenial smoohing when

98 89 is 0.1 and he corresponding m value is 47. However, his does no mean ha hese are he bes resuls for his ime series. Analysis mus be repeaed for differen and corresponding m value. Le ake as 0.2 and hen corresponding m value as 95 and repea he analysis. Alhough i is possible o invesigae each ab from firs o fourh ab le look a o he accuracy measures ab direcly (Figure 5.10). Figure 5.10 Accuracy measures when =0.2 and m=95 Values of all accuracy measures, of course excep pbeer, for modified simple exponenial smoohing is smaller han hose for simple exponenial smoohing and pbeer value for modified simple exponenial smoohing is bigger han simple exponenial smoohing when =0.2 and he corresponding m=95. I is possible o say ha modified simple exponenial smoohing again performs beer han simple exponenial smoohing according o all accuracy measures when =0.2 and m=95. Now we can incremen value by 0.1 and rerun he analysis for =0.3 and corresponding m=142 hen again incremening by 0.1 rerun again for =0.4 and corresponding m=189 and so on. However i is possible o do analysis auomaically for differen values of saring from 0.1 incremened by 0.1 up o 0.9 and he corresponding m values for he relaed value.

99 90 Afer clicking Run Mehods Auomaically buon from home page (Figure 5.2), selec Sample 1.1 for Daa Se from he selec box and click he simple smoohing buon on he lef side (Figure 5.11). Figure 5.11 Running simple smoohing auomaically for Sample 1.1 Saring from 0.1 and incremening by 0.1 up o 0.9 differen values and corresponding m values will be used and accuracy measures will be calculaed for each level auomaically. Only calculaed values of accuracy measures will be displayed in a able excep MSE and MAPE since rmse is already squared roo of MSE and here is no remarkable difference beween MAPE and smape (Figure 5.12).

100 91 Figure 5.12 Resuls for Sample 1.1 As i seen from he able modified simple exponenial smoohing performs beer han simple exponenial smoohing for all accuracy measures for values saring 0.1 up o 0.8 excep pbeer value for =0.7. And simple exponenial smoohing performs beer han modified simple exponenial smoohing for all accuracy measures for values 0.8 and 0.9 excep rarsape value when =0.8. I can be said ha modified simple exponenial smoohing mosly performed beer han simple exponenial smoohing according o he accuracy measures for differen levels. There are 5 accuracy measures and 9 differen levels so here are oal 45 values for accuracy measures. Simple exponenial smoohing performed beer 10 imes and modified simple exponenial smoohing performed beer 35 imes for his ime series. As a resul modified simple exponenial smoohing 78% performed beer han simple exponenial smoohing according o he accuracy measures for his ime series. The las row in he able includes he average values of each column. Modified simple exponenial smoohing performed beer han simple exponenial smoohing according o all accuracy measures on he average.

101 92 Now we can move o nex ime series in he daabase. There is a selec box ha lising all ime series and navigaion buons on he op-righ of he repor page. I is possible o move previous, nex or a desired ime series direcly using selec box or navigaion buons. Le selec Sample 1.2 from he selec box. A new analysis will be done auomaically for he ime series seleced from he lis and resuls for accuracy measures will be lised again (Figure 5.13). Figure 5.13 Accuracy measures for Sample 1.2 We see ha all MAE, rmse, smape, rarsape values for modified simple exponenial smoohing are smaller han hose values for simple exponenial smoohing and all pbeer values for modified simple exponenial smoohing are bigger han hose values for simple exponenial smoohing for all levels for Sample 1.2. Therefore i is possible o say ha modified simple exponenial smoohing performs beer han simple exponenial smoohing according o he all accuracy measures for all levels. Modified simple exponenial smoohing performs 100% beer han simple exponenial smoohing for his ime series. Nex ime series is Sample 1.3 (Figure 5.14)

102 93 Figure 5.14 Resuls for sample 1.3 This ime simple exponenial smoohing performs beer han modified simple exponenial smoohing for small values and modified simple exponenial smoohing performs beer han simple exponenial smoohing for big values. There is no any value for accuracy measures for values 0.8 and 0.9 because he sample size of his ime series is 11 and here is no enough smoohed value for higher value of for his ime series. I will no be appropriae o calculae accuracy measures so he aserisks are displayed for levels 0.8 and 0.9 Simple exponenial smoohing 14 imes performed beer and modified simple exponenial smoohing 21 imes performed beer. So modified simple exponenial smoohing 67% performs beer han simple exponenial smoohing for his ime series. When we look a o he average values, modified simple exponenial smoohing performs beer for 4 accuracy measures and simple exponenial smoohing performs beer for one accuracy measure. All ime series in he Makridakis compeiion can be analyzed by his way bu here is also anoher way of considering he performance of all ime series. All ime series in a predefined ime series group can be auomaically analyzed by using he

103 94 righ side of he Run Mehods Auomaically page (Figure 5.15). Daa Se Group selec box will display he daa group names if creaed while enering new ime series o daabase. There are wo daa group in our daabase, one for 111 ime series for Makridakis Compeiion and one for 1001 ime series for M-Compeiion. When a daa se group is analyzed, all he ime series in ha group will ake place in he analysis. Accuracy measures will be calculaed as usual way for differen values saring from 0.1 up o 0.9 incremening by 0.1 and corresponding m values for each ime series in he group. Average values will be calculaed for each accuracy measure for all ime series and hese averages will be abulaed in he repor page. Figure 5.15 Running analysis for a daa group Le selec 111Daa from selec box lis and click o Simple Smoohing buon. The average values of all accuracy measures will be displayed for values saring from 0.1 up o 0.9 incremening by 0.1 and corresponding m values for 111 imes series included in he Makridakis compeiion (Figure 5.16).

104 95 Figure 5.16 Accuracy measures for all ime series in Makridakis Compeiion The firs row includes he resuls for firs ime series ha is Sample 1.1. If we check he values wih he average row of he Figure 5.12, we can see ha he resuls of he firs row and he average row in Figure 5.12 are idenical as expeced. I is possible o say ha modified simple exponenial smoohing performs beer han simple exponenial smoohing according o all accuracy measures for Sample 1.1. The second row includes he resuls for Sample 1.2. Modified simple exponenial smoohing again performs beer han simple exponenial smoohing for all accuracy measures for Sample 1.2. Sample 2.1, Sample 2.4, Sample 2.5, Sample 2.6, Sample 2.8, Sample 2.9 and Sample 2.10 are he ohers ime series ha modified simple exponenial smoohing performs beer han simple exponenial smoohing according o all accuracy measures. For Sample 1.4 and Sample 2.2, modified simple exponenial smoohing performs beer han simple exponenial smoohing according o mos of he accuracy measures, simple exponenial smoohing performs beer for 3 of hem. Simple exponenial smoohing performs beer han modified simple exponenial for Sample 2.3 and Sample 2.7 according o all accuracy measures. As resul i is

105 96 possible o say ha modified simple exponenial smoohing performs beer for 8 ime series and simple exponenial smoohing performs beer for 2 ime series for firs 10 ime series in he Makridakis compeiion. Looking ime plo of he series Sample 2.3 and Sample 2.7 may helpful o undersand why modified simple exponenial smoohing does no perform beer for hese wo series (Figure 5.17, Figure 5.18). Figure 5.17 Time series plo for Sample 2.3

106 97 Figure 5.18 Time series plo for Sample 2.7 The firs salien hing when we look a o he graphs is he irregular componen (refer o Figure 1.5). There seems o be irregular movemens ha follow no recognizable or regular paern for boh ime series. These flucuaions may be caused by unusual evens or may conain noisy or random componen of he daa. Producing big errors for his ype of series may be an advanage. So, we can say ha modified simple exponenial smoohing reac beer han simple exponenial smoohing for irregular ime series insead of saying modified simple exponenial did no performed well. Resuls for 111 ime series given in Table 5.1. From he able i can bee seen ha modified simple exponenial smoohing performs beer for some series and simple exponenial smoohing performs beer for ohers according o accuracy measures. So i is imporan o know how many imes each mehod performed beer han he oher. The las row of he able displays he percenage of success for each mehod. Modified simple exponenial smoohing performed 70% beer according o MAE, 66% beer according o rmse, 74% beer according o smape, 77% beer according o rarsape and 70% beer according o pbeer.

107 98 Table 5.1 Accuracy measures for 111 ime series Simple Exponenial Smoohing Time Series MAE rmse MAE rmse Sample Sample Sample Sample Sample Sample Sample Sample Sample Sample 2.6 smape rarsape pbeer Modified Simple Exponenial Smoohing smape rarsape pbeer 5 Sample Sample Sample Sample 2.10 Sample 2.11 Sample Sample Sample Sample Sample Sample Sample Sample Sample Sample Sample Sample Sample Sample Sample Sample Sample Sample

108 99 Sample Sample Sample Sample Sample Sample 5.8 Sample Sample Sample Sample Sample Sample Sample Sample Sample Sample Sample Sample Sample Sample Sample Sample Sample Sample Sample Sample Sample Sample Sample Sample Sample Sample Sample Sample Sample 5.10 Sample 8.4

109 100 Sample Sample Sample Sample Sample Sample Sample Sample Sample Sample Sample Sample Sample Sample Sample Sample Sample Sample Sample Sample Sample Sample Sample Sample Sample Sample Sample Sample Sample Sample Sample Sample Sample Sample Sample Sample

110 101 Sample Sample Sample Sample Sample Sample pbeer Time series plos of he series for which modified simple exponenial smoohing seems o be no performed beer given in Appendix A. The common characerisic of hose series seems o be he irregular componen. They appear o have random flucuaions which causes modified simple exponenial smoohing o produce forecass error bigger hen simple exponenial smoohing. The oher se used o compare modified simple exponenial smoohing and simple exponenial smoohing is he ime series used in M-Compeiion (Makridakis e al,. 1982). There are 1001 ime series differen sources and hese series also used by a lo of researchers. The same analysis is repeaed for 1001 ime series in M-Compeiion and resul able given in Appendix B. The firs and he las screen is given in Figure 5.19, Figure 5.20 respecively. When we look a o he las row of he able o see he percenages of success for boh mehods, modified simple exponenial smoohing again performs beer han simple exponenial smoohing. Modified simple exponenial smoohing performed 71%, 67%, 70%, 79% and 70% beer han simple exponenial smoohing according o accuracy measures MAE, rmse, smape, rarsape and pbeer respecively.

111 102 Figure 5.19 Firs resul screen of 1001 ime series Figure 5.20 Las resul screen of 1001 ime series I is also possible o run auomaic analysis based on he grid search for m value. This ime all m values and corresponding values will be used in comparison. Grid

112 103 search on m mus be seleced o make analysis based on he differen values of m and corresponding values (Figure 5.21). Figure 5.21 Grid search on m Figure 5.22 Resuls for Sample 1.2

113 104 Resuls for Sample 1.2 are given in Figure Saring from 1, all values for m and corresponding values are used o calculae he accuracy measures. Table 5.2 shows he resul. Modified simple exponenial smoohing performed 74%, 70%, 78%, 76% and 72% beer han simple exponenial smoohing according o accuracy measures which are very similar o he values in Table 5.1. Table 5.2 Accuracy measures of 111 ime series for grid search on m value Simple Exponenial Smoohing Time Series MAE rmse Sample Sample Sample Sample 1.4 Modified Simple Exponenial Smoohing smape rarsape pbeer MAE rmse Sample Sample Sample Sample Sample Sample Sample Sample Sample Sample Sample Sample Sample Sample Sample Sample Sample Sample Sample Sample Sample Sample Sample smape rarsape pbeer

114 105 Sample Sample Sample Sample Sample Sample Sample Sample Sample Sample Sample Sample Sample Sample Sample Sample Sample Sample Sample 5.10 Sample Sample Sample Sample Sample Sample Sample Sample Sample Sample Sample Sample Sample Sample Sample Sample Sample

115 106 Sample Sample Sample Sample Sample Sample Sample Sample Sample Sample Sample Sample Sample Sample Sample Sample Sample Sample Sample Sample Sample Sample Sample Sample Sample Sample Sample Sample Sample Sample Sample Sample Sample Sample Sample Sample

116 107 Sample Sample Sample Sample Sample Sample Sample Sample Sample Sample Sample Sample 12.9 pbeer Ou-of-sample Performance The firs compeiion by Makridakis (1979) was designed as an in-sample performance comparison so only 1001 series in M-Compeiion (1982) is used o ge he ou-of-sample performance of he mehods. There is differen number of observaions lef as pos forecas values for each series and hese observaions are used o compare forecas accuracy of he mehods. The code of he firs series is YAF2. There is 22 observaions for calculae he smoohed values and 6 observaions o use pos forecasing. Figure 5.23 displays he smoohed values and observaions. And, Figure 5.24 displays he forecass for 6 observaions. value is chosen as 0.1 and corresponding m value is used as 2. When look a o he figures, i is easy o see ha modified simple exponenial smoohing is perform beer han simple exponenial smoohing for boh smoohing and forecasing when 0.1 and m 2.

117 108 Figure 5.23 Smoohed values Figure 5.24 Forecass

118 109 Figure 5.25 Accuracy measures for YAF2 when 0.1 and m 2 I can be seen ha modified simple exponenial smoohing performs beer han simple exponenial smoohing according o all accuracy measures for YAF2 when 0.1 and m 2 (Figure 5.25). For differen values and corresponding m values auomaed analysis can be run. The resuls are given in Figure I can be said ha modified simple exponenial smoohing performs beer han simple exponenial smoohing according o all accuracy measures for all values from 0.1 o 0.9 incremened by 0.1 and corresponding m values. Appendix C includes he ou-of-sample performance for all 1001 series. Modified simple exponenial smoohing performs 71%, 67%, 70%, 79% and 70% beer han simple exponenial smoohing according o MAE, rmse, smape, rarsape and pbeer respecively.

119 110 Figure 5.26 Accuracy measures for YAF2 for differen incremened by 0.1 and corresponding m values 5.2 levels from 0.1 o 0.9 Modified Double Exponenial Smoohing vs Double Exponenial Smoohing In-sample Performance Same operaions can be run for double exponenial smoohing. Click Run Mehods buon a home page (Figure 5.2) o run he analysis manually and hen click Double Smoohing buon afer selecing a daa se from selec box (Figure 5.3). A new page will be loaded o ener he required parameers for double exponenial smoohing and modified double exponenial smoohing

120 111 Figure 5.27 Parameer inpu screen for double exponenial smoohing Generally, values less han 0.3 have been recommended for parameers and for he Hol-Winers models (Gardner, 1985). Le ener 0.3 for boh and, and recommended value 142 for m. And click SEND buon. A similar page o simple exponenial smoohing will load bu his ime he resuls for double exponenial smoohing will be displayed. Figure 5.28, Figure 5.29, Figure 5.30, Figure 5.31 and Figure 5.32 displays smoohed values, fis, errors, percen errors and accuracy measures respecively for modified double exponenial smoohing and double exponenial smoohing for Sample 1.2. When we look a o accuracy measures in Figure 5.32, i can be concluded ha modified double exponenial smoohing performs beer han double exponenial smoohing. Figure 5.33 and Figure 5.34 displays he accuracy measures when is equal o 0.5 and 0.7 respecively while and m remain same. Modified double exponenial smoohing performs beer han double exponenial smoohing wih hese seings. Differen values for, and m can be ried bu i is also possible o run analysis auomaically by clicking Run Mehods Auomaically from home page (Figure 5.2). Figure 5.35 displays he resul of auomaed analyzing.

121 112 Figure 5.28 Smoohed values Figure 5.29 Fis

122 113 Figure 5.30 Errors Figure 5.31 Percen errors

123 114 Figure 5.32 Accuracy measures Figure 5.33 Accuracy measures when 0.5, 0.3 and m=30

124 115 Figure 5.34 Accuracy measures when 0.7, 0.3 and m=30 Figure 5.35 Analyzing auomaically for differen, and m values The 64 ime series ou of 111 ime series seems o be a candidae for double exponenial smoohing. Table 5.3 displays he resuls for hem

125 116 Table 5.3 Resuls for 64 ime series ou of 111 Double Exponenial Smoohing Time Series MAE Sample 1.1 Sample 1.2 Sample 1.4 Sample 2.1 Sample 2.5 Sample 2.6 Sample 2.12 Sample 2.13 Sample 3.2 Sample 3.3 Sample 3.7 Sample 4.2 Sample 4.4 Sample 4.6 Sample 4.7 Sample 5.1 Sample 5.3 Sample 5.4 Sample 5.5 Sample 5.7 Sample 5.10 Sample 6.1 Sample Sample Sample 7.2 Sample 7.3 Sample 7.4 Sample 7.6 Sample 7.7 Sample 7.9 Sample 7.11 Sample 7.12 Sample Sample Sample Sample Sample Sample Sample Sample 8.6 Sample 8.7 Sample 8.9 Sample 8.10 Sample 9.1 Sample 9.3 Sample 9.4 Sample 9.6 Sample 9.7 Sample 9.8 Sample 9.9 Sample 9.10 Sample 9.12 Sample 10.1 Sample 10.2 Sample 11.1 Sample 12.1 Sample 12.2 Sample 12.9 Sample Sample Sample Sample pbeer rmse Modified Double Exponenial Smoohing SMAPE rarsape pbeer MAE rmse smape rarsape pbeer

126 117 I can be said ha modified double exponenial smoohing performs 73%, 71%, 68%, 74% and 77 beer han double exponenial smoohing according o accuracy measures MAE, MSE, rmse, rarsape and pbeer respecively for 64 ime series appear o be have a rend and proper for double smoohing. Appendix D includes he ime series plo of he series ha modified double exponenial smoohing seems o be no performed well. The common characerisic of he series is he irregular componen hey have alhough here seems o be a long erm rend Ou-of-sample Performance 1001 ime series from M-Compeiion is used o compare he ou-of-sample performance of he mehods. Appedix E includes he resuls. I can be said ha modified double exponenial smoohing performed 74%, 71%, 74%, 75% and 72% beer han double exponenial smoohing according o accuracy measures MAE, rmse, smape, rarsape and pbeer respecively.

127 CHAPTER SIX CONCLUSION A new mehod named modified exponenial smoohing is inroduced in his hesis. A new smoohing consan and saring value is developed. The aim is o give even more weighs han given by classical mehods o mos recen observaions by a new smoohing consan and inroduce a new saring value whose conribuion o he forecass is no influenced by he value of smoohing consan. Firs, i is proved ha modified mehod provides he basic properies of he classical one. Such as, weighs given by modified mehod are exponenially decreasing giving higher weighs o mos recen observaions and weighs sum o uniy. So, he smoohed saisic S can be wrien as a linear combinaion of previous observaion. For example, S is given below for simple exponenial smoohing and modified simple exponenial smoohing respecively S X (1 ) S 1 m m S X S 1 I is also proved ha S is unbiased esimaor for boh simple exponenial smoohing and modified simple exponenial smoohing. The general form of S for simple exponenial smoohing and modified simple exponenial smoohing is given by 1 S (1 ) k X k (1 ) S 0 k 0 and 118

128 119 S ( m 1) k 0 k 1 m 1 x 1 S k m m m Weigh of observaion X k for simple exponenial smoohing is given as wx k (1 ) k k 0,1,2,, 1 and weigh of he saring for simple exponenial smoohing value is ws0 (1 ) Weigh of observaion X k for modified simple exponenial smoohing is given as m ( m)!( k 1)! wx k ( m k )!( 1)! k 0,1,, (m 1) and weigh of he saring value for modified simple exponenial smoohing is m m 1 1 wsm 1 m 1 I is also proved ha modified mehod give higher weighs o mos recen observaions and saring value is no affeced by he value of smoohing consan. An empirical sudy is developed o compare mehods. 111 ime series of Makridakis Compeiion and 1001 ime series of M-Compeiion are included in he empirical sudy. Seven accuracy measures are used o compare in-sample performance and ou-of-sample performance of. Empirical resuls showed us ha modified mehods performed beer han classical mehods for mos of he ime.

129 REFERENCES Abraham, B., & Ledoler, J. (1983). Saisical mehods for forecasing. New York, John Wiley and Sons. Adam, E. E. (1973). Individual Iem Forecasing Model Evaluaion, Decision Sciences, 4: p Armsrong, J. S. (1978). Long-Range Forecasing, New York: Wiley Chaper 7. Armsrong, J. S., Collopy, F., (1992). Error Measures for Generalizing abou Forecasing Mehods: Empirical Comparisons. Inernaional Journal of Forecasing. 8, p Barolomei, S. M., & Swee, A. L. (1989). A noe on a comparison of exponenial smoohing mehods for forecasing seasonal series. Inernaional Journal of Forecasing. 5, p Box, G. E. P., & Jenkins, G. M. (1970). Time series analysis: Forecasing and conrol. San Francisco, Holden Day. Bowerman B. L., & O Connell R.T., (1987). Time Series Forecasing Unified Conceps and Compuer Implemenaion (2 nd ed.). Boson: Duxbury Press. Brown, R.G, (1959). Smoohing: forecasing for invenory conrol. New York: McGraw-Hill. Brown, R.G. (1964). Smoohing, Forecasing and Predicion of Discree Time Series. NJ: Prencice-Hall. Bruce, L.B., & Richard, T.O., (1979). Forecasing & Time Series. California. Carreno, J., & Madinaveiia, J. (1990). A modificaion of ime series forecasing mehods for handling announced price increases. Inernaional Journal of Forecasing. 6,

130 121 Chafield, C., (1988). Wha is he bes mehod of forecasing? Journal of Applied Saisics. 15, p Cogger, K. O., (1973). Exensions of he Fundemenal Theorem of Exponenial Smoohing, Managemen Science, 19, p Cohen, G. D., (1963). A noe on exponenional smoohing and auocorrelaed inpus, Operaion Research, 11, p Cohen, G. D., (1966). Bayesian adjusmen of sales forecass in muli-iem invenory conrol sysem. Journal of Indusrial Engineering. 17, p Couie, G. A., e al. (1964). Shor-erm Forecasing. Mahemaical and saisical echniques for indusry. Monograph ; no.2. Cox, D. R., (1967). Predicion by exponenially weighed moving averages and relaed mehods, Journal of he Royal Saisical Sociey, 23, p Elon, E. J., & Gruber, H. J. (1972), Earning esimaes and he accuracy of expecaional daa, Managemen Science, 18, p Fildes, R., Hibon, M., Makridakis, S., & Maeda, N., (1998). Generalising abou univariae forecasing mehods: Furher empirical evidence. Inernaional Journal of Forecasing. 14, p Gardner, Jr. E. S., (1985). Exponenial smoohing: The sae of he ar. Journal of Forecasing, 4, p1-28. Gardner, Jr. E. S., (2006). Exponenial Smoohing: The sae of he ar Par II. Inernaional Journal of Forecas. 22, p Gass, S.I., & Harris, C.M., (2000). Encyclopedia of Operaions Research and Managemen Science (Cenennial ediion). Dordrech, The Neherlands. George, E.P.B., & Gwilym, M.J., & Gregory, C.R., (1994). Time Series Analysis Forecasing and Conrol. New Jersey: Prenice-Hall.

131 122 Gilchris, W. W., (1976). Saisical forecasing. London, Wiley. Gooijer, J. G. & Hyndman, R. J., (2006). 25 years of ime series forecasing. Inernaional Journal of Forecasing. 22, p Gross, D, & Craig, R. J., (1974). A comparison of maximum likelihood, exponenial smoohing and Bayes forecasing procedures in inverory modeling, Inernaional Journal of Producion Research, 12, p Harrison, P. J., (1967). Exponenial Smoohing and Shor erm Sales for Forecasing, Managemen Sciences, 13, p Hol, C.C. (1957). Forecasing seasonals and rends by exponenially weighed moving averages, ONR Memorandum, Vol52, Pissburgh, PA: Carnegie Insiue of Technology. Hol, C.C., Modigliani, F., Muh, J.F., Simon, H.A., (1960). Planning Producion, Invenories and Work Force. Englewood Cliffs, N.J, Prenice-Hall. Hol, C.C. (2004a). Forecasing seasonals and rends by exponenially weighed moving averages, Inernaional Journal of Forecasing, 20, p5-10. Hol, C.C. (2004b). Auhor s rerospecive on Forecasing seasonal and rends by exponenially weighed moving averages, Inernaional Journal of Forecasing, 20, p Hyndman, R. J. (2001). I s ime o move from wha o why. Inernaional Journal of Forecasing, 17, p Johnson, L. A., & Mongomery, D. C., (1974). Operaions Research in Producion Planning, Scheduling, and Invenory Conrol. New York, Wiley. Johnson, N.L. and Kos, S., (1969). Disribuions in Saisics: Discree Disribuions. John Wiley, New York Kirby, R. M., (1966). A Comparison of Shor and Medium range Forecasing Mehods. Managemen Sciences, 13, p

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135 APPENDIX A Figure 6.1 Time series plo of Sample 2.3, Sample 2.7, Sample 3.2, Sample 4.1 Figure 6.2 Time series plo of Sample 4.7, Sample 5.2, Sample 5.6, Sample

136 127 Figure 6.3 Time series plo of Sample 6.3, Sample 7.1.2, Sample 7.2, Sample 7.5 Figure 6.4 Time series plo of Sample 7.6, Sample 7.8, Sample 8.1.2, Sample 8.7

137 128 Figure 6.5 Time series plo of Sample 8.8.1, Sample 8.10, Sample 9.2, Sample 11.1 Figure 6.6 Time series plo of Sample , Sample 12.4, Sample 12.5, Sample

138 129 Figure 6.7 Time series plo of Sample 12.7, Sample , Sample