PROC NLP Approach for Optimal Exponential Smoothing Srihari Jaganathan, Cognizant Technology Solutions, Newbury Park, CA.

Transcription

1 PROC NLP Approach for Opimal Exponenial Smoohing Srihari Jaganahan, Cognizan Technology Soluions, Newbury Park, CA. ABSTRACT Esimaion of smoohing parameers and iniial values are some of he basic requiremens of forecasing ime series using exponenial smoohing. Recenly several researchers have proposed a nonlinear opimizaion based framework o esimae he smoohing and iniial values. The gis of his paper is o propose a PROC NLP (SAS/OR ) based approach o solve he nonlinear opimizaion problem. A mehodology o implemen nonlinear opimizaion problem wih PROC NLP is developed. Deailed synax and he descripion of he proposed mehodology is presened. Two empirical case sudies are presened in order o es he efficacy of he proposed mehodology. A comparaive sudy of he proposed mehodology wih ime series forecasing sysem (TSFS) in SAS/ETS is presened. Numerical experience based on he comparaive sudy indicaes ha he proposed mehodology performs beer han he TSFS. Keywords: Exponenial smoohing, opimizaion, PROC NLP, SAS/OR, SAS/ETS, Time series, forecasing. INTRODUCTION Exponenial smoohing is a popular ime series based forecasing echnique. Exponenial smoohing uilizes pas values o predic he fuure oucomes. Smoohing parameer or weighs are used o exrapolae a ime series daa by giving imporance o he mos recen daa. In relaive sense he prediced value is influenced by he mos recen ime series daa poin raher han an old ime series daa poin. Exponenial smoohing can accommodae rend and seasonaliy inheren in a ime series daa. There are four major seps involved in forecasing using exponenial smoohing mehod.. Esimaing he opimal smoohing parameers.. Esimaing he iniial values for rend. 3. Esimaing he iniial values for seasonaliy. 4. Normalizing he iniial seasonal values. There are several heurisic approaches o esimae he iniial and smoohing parameers. Recenly several researchers (Rasmussen, 004 and Bermudez e al., 007) have applied opimizaion based approach o esimae he iniial and smoohing values. The opimizaion problem is formulaed as a nonlinear programming problem and solved using an appropriae numerical mehod. PROC NLP is a par of SAS/OR module and can be used o solve general nonlinear opimizaion problem. The main purpose of his paper is o propose a PROC NLP based approach o solve he nonlinear programming problem o esimae he smoohing parameers and iniial values. In addiion, he normalizing of seasonal values is modeled as consrain and solved as a consrained nonlinear opimizaion problem. This paper explois he algebraic modeling capabiliies of SAS and demonsraes he usage of PROC NLP o solve he opimizaion problem o esimae he iniial values, smoohing parameers and normalize he seasonal parameers. Two empirical case sudies are presened o compare he proposed mehodology is compared wih he SAS/ETS ime series forecasing sysem (abbreviaed as TSFS). Based on he case sudy his paper concludes ha he proposed mehodology ouperforms he TSFS. The organizaion of his paper as follows: In he following secion an overview of exponenial smoohing is presened wih he underlying equaions. Nex, modeling of he nonlinear opimizaion problem o esimae iniial values and smoohing weighs is presened. The proposed mehodology is developed wih he synax and relevan descripion of PROC NLP. Two empirical case sudies are presened and compared wih SAS TSFS. Final secion concludes wih he discussion abou he proposed mehodology. EXPONENTIAL SMOOTHING AN OVERVIEW Exponenial smoohing is based on combinaion of hree parameers Level, Trend and Seasonaliy. There are differen forms of exponenial smoohing, and are classified mainly based on rend and seasonaliy componen. Table gives he exponenial smoohing mehods based on Pegel s Classificaion (Makridakris e al., 998). Trend Seasonaliy N (None) A (Addiive) M () N (None) N N N A N M A (Addiive) A N A A A M M () M N M A M M Table. Exponenial Smoohing Mehod Pegel s Classificaion

2 Of all he mehods in he Table, he Addiive Trend Addiive Seasonaliy (A A) and he Addiive Trend Seasonaliy (A M) are he mos popular and commonly used Exponenial smoohing Mehods originally developed by Winers. These mehods are also called as Hols Winers exponenial smoohing mehod. The Hols Winer mehod wih addiive seasonaliy can be expressed using equaion,, 3, 4 and 5 (Bermudez e al., 007). Level: L = α( Y S k) + ( α)( L + b ) () Trend: b = β( L L ) + ( β) b () Seasonal: S = γ( Y L) + ( γ) S k (3) Prediced: P = L + b + S k (4) Forecas: F+ m= L+ bm + S k+ m (5) The Hols Winer mehod wih muliplicaive seasonaliy can be expressed using equaion 6, 7, 8, 9 and 0 (Bermudez e al., 007). The noaions for all he equaions are presened in Table. Y Level: L = α( ) + ( α)( L + b ) (6) S k Trend: b = β( L L ) + ( β) b (7) Y Seasonal: S = γ( ) + ( γ) S k (8) L Prediced: P = ( L + b) S k (9) Forecas: F = ( L + bm) S (0) Symbol + m k+ m Definiion Time period,,.,n Y Observed value for ime period. L Level of he series a ime period. b Trend componen of he series a ime period. S Seasonal componen of he series a ime period. m Periods ahead. P Prediced values for period. F + m Forecas for m periods ahead. k α β γ Lengh of he seasonaliy (Eg. Monhs, Quarers in a year.). Smoohing consan for level. Smoohing consan for rend componen. Smoohing consan for seasonal componen. Table. Noaions Used.

3 OPTIMIZATION MODELING There are differen error measures ha can be used o esimae he accuracy of a forecasing mehod. Roo mean squared error (RMSE), mean absolue percenage error (MAPE), and mean absolue error (MAE) are some of he commonly used merics (Makridakris e al., 998). RMSE is he square roo of he sum of squared difference beween he acual and prediced value. An ideal forecasing mehod should have a RMSE of 0 or minimum, bu in pracice i is difficul o obain such a perfec fi, hence i is appropriae o model he problem wih he leas RMSE. This naurally leads o an opimizaion problem, wherein he objecive is o minimize he RMSE. This paper uilizes he RMSE as an error minimizaion echnique in he opimizaion model, however oher error measure such as MAE, MAPE, ec., can also be used as an error minimizaion echniques. ESTIMATION OF SMOOTHING PARAMETERS (α, β and γ) The smoohing parameers α, β and γ can be obained by minimizing he RMSE error meric and consraining he parameers o (0, ). The opimizaion model is presened in (). The value of P is obained from Equaion 4 or Equaion 9. n Min Y P αβγ,, ( n ) ( ) αβγ,, 0, = ( ) () ESTIMATION OF INITIAL VALUES (L 0, B 0 AND S 0, S -,, S -K ) Esimaion of iniial values is an imporan par of any exponenial smoohing based forecasing. There are several differen approaches in esimaing he iniial values. An excellen caalog of differen mehods is presened in (Bermudez e al., 007). Recenly several researchers have considered he iniial values as a decision variable and solve he opimizaion model in (). The opimizaion model in () can be modified o accommodae he iniial values δ = (L 0, b 0 and S 0, S -,, S -k ) and is represened in (). n Min Y P αβγδ,,, ( ) ( n ) = ( ) ( L, b, S,..., S ) αβγ,, 0, δ = 0 0 k 0 () NORMALIZATION OF SEASONAL PARAMETERS: According o (Rasmussen, 004) normalizing he iniial seasonal parameers is an imporan requiremen in Exponenial smoohing. In his paper, he iniial values for boh addiive and muliplicaive models are normalized. For addiive model he sum of iniial seasonal values should equal o 0 and in he case of muliplicaive model he sum of iniial seasonal values should equal o (Bermudez e al., 007). Opimizaion model in () can be modified o accommodae he normalizing consrains and is shown in (3). n Min Y P αβγδ,,, Subjec o : k n= k n= S S n n ( n ) = ( ) ( L, b, S,..., S ) αβγ,, 0, δ = 0 0 k 0 ( ) = K if muliplicaive version. = 0 if addiive version. (3) 3

4 PROPOSED METHODOLOGY WITH PROC NLP Rasmussen (Rasmussen, 004) proposed a NLP and spreadshee based mehodology o opimally obain smoohing parameers and iniial values. SAS and SAS/OR module are widely used in business analyics in various indusries and organizaions o solve opimizaion problems. In his paper we uilize Proc NLP and propose a framework o obain he iniial values and smoohing weighs. As saed in he previous secion, he opimizaion model o obain smoohing parameers and iniial values is a consrained nonlinear problem. An appropriae numerical mehod is required o solve his problem. Proc NLP is a SAS procedure ha is a par of SAS/OR module. Proc NLP is capable of solving boh consrained and unconsrained nonlinear opimizaion problems. Proc NLP uilizes several well esablished numerical mehods o opimize an objecive funcion, i.e., o eiher maximize or minimize he objecive funcion. Since he opimizaion problem presened in (3) is a nonlinear consrained opimizaion problem Proc NLP is uilized o minimize he objecive funcion RMSE. The synax for he proposed mehodology is as follows: Proc NLP; run; array y[55] ; array l[55] l - l55; array b[55] b - b55; array s[55] s - s55; array p[55] p - p55; array x[7 ] x - x7; /* Decision Variables */ array z[55] z - z55; min q; parms x = 0, x = 5, x3 - x4 =.5, x5 - x7 = 0.3 ; bounds 0.0 <= x <= 30,0.0<= x3 - x4 <=0,0<= x5 - x7<= ; nlincon c = ; [OR] nlincon c = 0; c = sum(of x3 - x4); l[]=x[]; /* Iniial Level Componen */ b[]=x[]; /* Iniial Trend Componen */ do j = o ; s[j] = x[j+]; /* Iniial Seasonal Componen */ a = x[5]; /*Smoohing Parameer α */ v = x[6]; /*Smoohing Parameer β */ g = x[7]; /*Smoohing Parameer γ */ do i = 3 o 55; l[i] = a*(y[i]/s[i-k])+(-a)*(l[i-]+b[i-]); /* Eq. ()*/ b[i] = v*(l[i]-l[i-])+(-v)*b[i-]; /* Eq. ()*/ s[i] = g*(y[i]/l[i])+(-g)*(s[i-k]); /* Eq. (3)*/ p[i] = (l[i-]+b[i-])*s[i-k]; /* Eq. (4)*/ z[i] = (y[i] - p[i])**; [OR] do i = 3 o 55; l[i] = a*(y[i]-s[i-k])+(-a)*(l[i-]+b[i-]); /* Eq. (6)*/ b[i] = v*(l[i]-l[i-])+(-v)*b[i-]; /* Eq. (7)*/ s[i] = g*(y[i]-l[i])+(-g)*(s[i-k]); /* Eq. (8)*/ p[i] = l[i-]+b[i-]+s[i-k]; /* Eq. (9)*/ z[i] = (y[i] - p[i])**; q = ((sum (of z3-z55))/ (44-))**0.5; 4

5 The descripion of he Synax is as follows: Declare array for he decision variable as well he rend, level, seasonal and decision variables. The dimension of he array is deermined based on he lengh of seasonaliy K and he number of observed values n. Dimension of array is equal o he sum of K and n. For example if here are 43 observed values and lengh of seasonaliy is hen he array dimension will be 55. The iniial values hru in array Y[55] Is filled wih 0 in order o faciliae array processing. A deailed explanaion is provided in he nex secion. Minimize he objecive funcion RMSE in (3). Declare he iniial values for smoohing parameers, level, rend and seasonal parameers. Specify bounds for he smoohing parameers, level, rend and seasonal parameers. Alhough no required for he seasonal, level and rend iniial values, i is always good o menion bounds in order o narrow he search space and obain a global soluion. Specify he consrain in order o normalize he seasonal parameers. Equae o K for muliplicaive model and 0 for addiive model. Assign he iniial and smoohing parameers wih appropriae array. This is for Addiive model. Equaions hru Equaion 4 are represened and in addiion he squared difference beween he observed and he prediced value. This is for model. Equaions 6 hru Equaion 9 are represened and in addiion he squared difference beween he observed and he prediced value. RMSE formulae for he objecive funcion is presened, Proposed mehodology can be summarized using following seps:. Formulae he nonlinear opimizaion problem as in (3) using Proc NLP.. Declare arrays for he level, rend, seasonal and smoohing parameers. 3. Specify he iniial values, bounds and consrains. 4. Depending on he addiive or muliplicaive model use equaions hru 4 or Equaion 6 hru Solve he opimizaion problem. 6. Look for convergence in he log. If Proc NLP has converged go o sep 6, else go o sep 3 and revise he iniial values and loosen he bounds (if specified). 7. End. Imporan caveas o be considered for he proposed mehodology are:. Proc NLP is a local opimizaion echnique, so i canno guaranee a global soluion. However, muliple sar of he Proc NLP procedure can lead closer o global soluion.. Proc NLP is heavily influenced by he saring parameers, differen saring parameers can be experimened. EMPIRICAL CASE STUDIES In his secion we presen wo case sudies drawn from (Cohen, 996). The firs is he elecriciy produc rae case sudy and he second case sudy is he Dow Jones Index (DJI). Boh he case sudies are compared wih SAS TSFS which is par of SAS/ETS. Boh he case sudies are solved using Addiive as well as models. The resuls of he for he elecriciy producion rae case sudy are presened in Table 3, 4 and Figure. I is eviden from Table 3 and 4 ha he proposed mehod ouperforms he TSFS of SAS/ETS in all he error measure considered. The source code for he muliplicaive model is given as follows. proc nlp; array y[55]

6 ; array l[55] l - l55; array b[55] b - b55; array s[55] s - s55; array p[55] p - p55; array x[7] x - x7; array z[55] z - z55; min q; parms x = 9000, x = 5, x3 - x4 =.5, x5 - x7 = 0.3 ; bounds 0.0 <= x <= 30, 0.0<= x3 - x4 <=0, 0<= x5 - x7<= ; nlincon c = ; c = sum(of x3 - x4); l[]=x[]; b[]=x[]; do j = o ; s[j] = x[j+]; a = x[5]; v = x[6]; g = x[7]; k = ; do i = 3 o 55; l[i] = a*(y[i]/s[i-k])+(-a)*(l[i-]+b[i-]); b[i] = v*(l[i]-l[i-])+(-v)*b[i-]; s[i] = g*(y[i]/l[i])+(-g)*(s[i-k]); p[i] = (l[i-]+b[i-])*s[i-k]; z[i] = (y[i] - p[i])**; q = ((sum(of z3-z55))/43)**0.5; run; The resuls for he DJI case sudy are presened in Table 4, Table 5 and Figure. In his case sudy also, he proposed mehod also ouperforms he TSFS of SAS/ETS. The proposed mehod has a RMSE of compared wih and obained from SAS/ETS. Error Measure Proc NLP - Addiive TSFS Mean-Squared Error (MSE) Roo Mean-Squared Error (RMSE) * Mean Absolue Error (MAE) Mean Abs. Percenage Error (MAPE) Table 3. Elecriciy Producion Rae case sudy Proposed Mehod vs. TSFS 6

7 .8 x 05.6 Acual Prediced Elecricy Producion Daa.4 Millions of KW HR Time Period Figure. Elecriciy Producion Rae Acual vs. Prediced Consans Proc NLP - Addiive Level Smoohing Weigh (α) Trend Smoohing Weigh (β) Seasonal Smoohing Weigh (γ) Iniial Smoohed Level (L 0 ) Iniial Smoohed Trend (T 0 ) Smoohed Seasonal Facor (S- ) Smoohed Seasonal Facor (S- 0 ) Smoohed Seasonal Facor 3 (S- 9 ) Smoohed Seasonal Facor 4 (S- 8 ) Smoohed Seasonal Facor 5 (S- 7 ) Smoohed Seasonal Facor 6 (S- 6 ) Smoohed Seasonal Facor 7 (S- 5 ) Smoohed Seasonal Facor 8 (S- 4 ) Smoohed Seasonal Facor 9 (S- 3 ) Smoohed Seasonal Facor 0 (S- ) Smoohed Seasonal Facor (S- ) Smoohed Seasonal Facor (S 0 ) Table 4. Consans for Elecriciy Case Sudy Error Measure Proc NLP - Addiive TSFS Mean-Squared Error (MSE) Roo Mean-Squared Error (RMSE) * Mean Absolue Error (MAE) Mean Abs. Percenage Error (MAPE) Table 5. DJI case sudy Proposed Mehod vs. TSFS 7

8 Consans Proc NLP - Addiive TSFS Level Smoohing Weigh (α) Trend Smoohing Weigh (β) Seasonal Smoohing Weigh (γ) Iniial Smoohed Level (L 0 ) Iniial Smoohed Trend (T 0 ) Smoohed Seasonal Facor (S- ) Smoohed Seasonal Facor (S- 0 ) Smoohed Seasonal Facor 3 (S- 9 ) Smoohed Seasonal Facor 4 (S- 8 ) Smoohed Seasonal Facor 5 (S- 7 ) Smoohed Seasonal Facor 6 (S- 6 ) Smoohed Seasonal Facor 7 (S- 5 ) Smoohed Seasonal Facor 8 (S- 4 ) Smoohed Seasonal Facor 9 (S- 3 ) Smoohed Seasonal Facor 0 (S- ) Smoohed Seasonal Facor (S- ) Smoohed Seasonal Facor (S 0 ) Table 6. Consans for DJI Case Sudy 4000 Dow Jones Index Daa 3500 Acual Prediced 3000 DJI Time Period Figure. Dow Jones Index. Acual vs. Prediced. CONCLUSION This paper proposed a PROC NLP based approach for opimal exponenial smoohing based forecasing. This paper demonsraed he inuiive usage of PROC NLP procedure o opimize and esimae he smoohing weighs and iniial values. Synax of he PROC NLP in relaion o he proposed mehodology was presened wih a deailed descripion. Two empirical case sudies were presened using he proposed mehodology. The proposed mehodology performed beer when compared agains TSFS of SAS/ETS. Based on he case sudy, wih superior performance, and also due o he ease of modeling wih PROC NLP, he proposed mehodology can be an aracive alernaive o model exponenial smoohing. REFERENCE Rasmussen, R. (004) On ime series daa and opimal parameers, Omega. 3: 0. Bermudez, J.D., J.V. Segura and E. Vercher (006) Improving demand forecasing accuracy using nonlinear programming sofware, Journal of he operaional research sociey, 57:

9 Makridakris, S., S.C. Wheelwrigh and R.J. Hyndman (998) Forecasing mehods and applicaions, 3 rd ed., New York: Wiley. Cohen, B.L. (996) Forecasing example for business and economics using SAS sysem. Cary, NC: SAS Insiue. CONTACT INFORMATION Your commens and quesions are valued and encouraged. Conac he auhor a: Srihari Jaganahan, Cognizan Technology Soluions. Newbury Park, CA. sriharin@gmail.com SAS and all oher SAS Insiue Inc. produc or service names are regisered rademarks or rademarks of SAS Insiue Inc. in he USA and oher counries. indicaes USA regisraion. Oher brand and produc names are rademarks of heir respecive companies. 9