Forecasting Tourist Arrivals Based on Fuzzy Approach with Average Length and New Base Mapping

Transcription

1 Forecasing Touris Arrivals Based on Fuzzy Approach wih Average Lengh and New Base Mapping Sii Musleha Ab Mualib Faculy of Compuer & Mahemaical Sciences Universii Teknologi MARA Malaysia Nazirah Ramli Farah Naasha Zainoordin & Zulkifli Ab Ghani Hilmi Faculy of Compuer & Mahemaical Sciences Universii Teknologi MARA Cawangan Pahang Malaysia Absrac Fuzzy ime series model (FTSM) has been developed since years ago o handle daa in linguisic s However mos of he FTSM defined daa in erms of discree fuzzy ses which provide he forecased oupu as a single poin Various inerval lengh mehods have been proposed o obain higher forecasing accuracy bu some of hem canno deal wih daa in large of range This paper proposes an improved fuzzy forecasing model based on rapezoidal fuzzy numbers wih new base mapping The proposed FTSM involves four phases which are daa collecion on ouris arrival esing daa for seasonaliy and rend paern model developmen and verifying he forecasing accuracy The resul shows ha he daa has seasonal and rend paern; and 13 linguisic s in erms of rapezoidal fuzzy numbers were obained Various forecased inervals under differen degree of confidence can be produced compared o a single poin forecased offered by he previous FTSM In addiion a wide range of daa can be caered based on he new base mapping able Keywords Average lengh Base mapping Fuzzy Time series Trapezoidal fuzzy I INTRODUCTION Forecasing is a process of predicing he fuure performance based on he exising hisorical daa Predicing he fuure is imporan in managing and is a key componen of planning and decision making Various radiional forecasing models have been proposed o predic he fuure performance and solving he problem such as Grey Markov model [1] Box Jenkins [] and exponenial smoohing model [3] However he radiional forecasing canno handle he vagueness and incompleeness in daa Moreover he radiional model requires more hisorical daa mus obey normal disribuion and canno be used for forecasing problems peraining linguisic s Thus [4] proposed fuzzy ime series model (FTSM) o deal wih hese issues by using he enrollmen a Universiy of Alabama as a sample se Based on ha mehod [5] concluded ha Song and Chissom s mehod required large amoun of compuaion o derive fuzzy relaions and max min composiion operaions Thus [5] used simple arihmeic operaions for ime series forecasing Thereafer various relaed research works have been made ha follow heir framework such as alering he lengh inerval o improve forecasing accuracy In [4 6] Song and Chissom pariioned he universe of discourse ino seven ISBN BMCRC016 lengh of inervals However he reason o use seven inervals has no been discussed [7] menioned ha he lengh of inerval will affec he forecaser accuracy Thus [7] proposed average based and disribuion based lengh mehod wih smaller error compared o he previous one Since ha many improvemens and alering in defining he lengh of inerval have been proposed In 004 [8] proposed naural pariioning and [9] proposed frequency densiy based mehod [10] used raio based lengh mehod and [11] proposed mean based discreizaion mehod o pariion he universe of discourse However he aforemenioned mehods are complicaed as i needs an exra ask during he calculaion process The average based lengh is significanly used in deermining he universe of discourse [ ] However he base mapping able proposed for his mehod canno caer daa in large of range Therefore his paper proposes forecasing ouris arrivals wih FTS approach wih an average based lengh mehod and consruc he new based mapping able whereby i can caer for daa in large of range The paper is organized as follows Secion inroduces some basic definiion of FTS and TrFNs Secion 3 presens he new base mapping able In Secion 4 we proposed FTS forecasing mehod wih average based lengh and he new base mapping able Secion 5 presens he forecasing of ouris arrivals o illusrae he proposed mehod The discussion is presened in Secion 6 Lasly he paper is concluded in Secion 7 II PRELIMINARIES In his secion some basic conceps of fuzzy ime series and rapezoidal fuzzy numbers (TrFNs) are presened The concep of FTS was firs presened and defined by [4 6] while TrFNs was firs used in [1] The definiions are given as follows: Definiion 1 Le Y 01 be a subse of R and Y be he universe of discourse defined by fuzzy se i i 1 If F consiss of i i 1 hen F is called FTS on Y 01 4

2 Definiion Le F is a FTS F is caused by 1 here exiss a fuzzy relaionship R 1 F F 1 R 1 operaor The relaionship can be denoed as F 1 F Ai Aj if F 1 Ai and F Aj F if such ha where represen as fuzzy Definiion 3 A TrFNs A denoed as A a b c d A or is defined as 0 x a x a a x b b a x 1 b x c (1) d x c x d d c 0 x d III THE NEW BASE MAPPING TABLE I BASE MAPPING TABLE [7] Range Base Table I shows he base mapping able for average lengh mehod proposed by [7] However he base mapping able canno caer for large of daa Therefore a new base mapping able is consruc based on he paern of he exising base mapping able From he paern of he las wo rows in Table I we can consruc he new base mapping able as follows: Assume for n = 1 for range n = for range when n = 1 he range will be o 10 wih base of 1 10 n = he range will be 1 10 o 10 wih base of 10 Therefore based on he paern when n k he range will be 1 10 k 1 o 10 k wih base of 10 k The new base mapping able is simplified in Table II TABLE II NEW BASE MAPPING TABLE n Range Base k 1 10 k 1-10 k 10 k IV PROPOSED FORECASTING MODEL The proposed forecasing model wih he new base mapping involved four phases whereby phase is aken from [15] The procedure of he proposed approach is given as follows Phase 1: Collecing he daa Phase : Invesigaion for seasonaliy and rend paern 3 Line char mehod for deermining seasonaliy and rend paern was used For daa wih seasonal variaion deseasonalize he daa using raio o moving average [15] Phase 3: Model Developmen a Deermine he maximum D max and he minimum D min of he deseasonalize daa D b Define he universe of discourse U D min D1 Dmax D where D 1 and D are proper ineger c Deermine he appropriae lengh of inervals l and pariion he universe of discourse by using he average based lengh mehods as follows: i Calculae he absolue difference beween he nex deseasonalized daa D and deseasonalized daa 1 ( D ) whereby 1 n 1 and calculae he average of he difference beween daa ii Take half of he average Avg new as he lengh iii Deermine he range and base for he lengh in (ii) and rounded he lengh by using he new base mapping able in Table II iv Calculae he number of inerval using formula m D D D D l where l is he lengh max min 1 / afer rounded v Pariion he universe of discourse using he lengh inerval obained in (iv) Assume here are m inervals which are u1 d 1 d u d d3 um1 d m1 dm and um d m dm 1 Remove he inerval ha does no cover he daa d Develop new TrFNs o represen he linguisic erms based on he inerval obained in (v) as follows: A1 d1 d1 d d3 A d1 d d3 d4 : : Am 1 dm dm1 dm dm1 Am dm1 dm dm1 dm1 e Fuzzify he D If he um Am D is locaed in he range u m so f Creae fuzzy logical relaionship (FLR) based on Definiion Then consruc FLR group g Deermine he forecased s O by using he heurisic rules as follows: Rule 1: if he FLR group of A i is empy; Ai hen O Ai Rule : if he FLR group of A i is one o one; A i Aj hen O Aj Rule 3: if he FLR group of A i is one o many; Ai Aj1 Ai Aj Ai Ajp hen Aj1 Aj Ajp O () p h Calculae he final forecased s F based on [15] as follows: 43

3 i If he original hisorical daa only have seasonal paern hen F Si O F Si a Si b Si c Si d Where O a b c d (3) and Si = seasonal index ii If he original hisorical daa only have a rend paern hen F R 1 O F R a R b R c R d (4) Where R = acual iii If he original hisorical daa have boh seasonal and Si rend paerns hen F R 1 Si O Si 1 Si Si 1 Si 1 Si 1 F R Si a R Si b R Si c R 1 Si d Si 1 Si 1 Si 1 Si 1 (5) iv If he original hisorical daa do no have eiher seasonal or rend paern hen F O F a b c d (6) i Defuzzify he forecased cenroid mehod as follows: This mehod is applied o ranslae he TrFNs ino crisp F x x dx (7) x dx Phase 4: Verifying he forecasing accuracy Calculae he roo mean square error (RMSE) o verify he performance of proposed mehod The formulaion is; RMSE = n F R i1 n (8) Where R = acual F = forecased afer defuzzify and n = number of daa V FORECASTING TOURIST ARRIVALS In his secion he proposed FTSM wih average based lengh and new base mapping able are used o predic he upcoming ouris o Malaysia Phase 1: Fig 1 show he FTS daa for monhly ouris arrival o Malaysia from 008 unil 010 The daa is colleced from Johor Tourism Deparmen Phase : The seasonaliy of he daa is deermined based on he graph in Fig 1 I shows ha here is a decreasing paern in he upcoming ouris arrival o Malaysia from January o February and also increasing paern of ouris arrival from May o June Thus he hisorical daa is seasonal since i has similar rend for hree consecuive years Table III shows he deseasonalized obained using he raio o moving average mehod and Fig shows he rend paern by using line char TABLE III DESEASONALIZED VALUE FOR TOURIST ARRIVAL TO MALAYSIA Year Monh Deseasonalized Year Monh Deseasonalized 008 Jan July Feb Aug Mar Sep Apr Oc May Nov June Dec July Jan Aug Feb Sep Mar Oc Apr Nov May Dec June Jan July Feb Aug Mar Sep Apr Oc May Nov June Dec Fig Deseasonalizex daa for rend paern Phase 3: The average based lengh wih new base mapping able is implemened o deermine he appropriae lengh of inerval which is The D min and D max deseasonalized (from Table III) are and respecively By choosing D 1 = and D = 1913 he universe of discourse U is defined as [ ] Thus he number of appropriae inerval is 0 However he inervals which are no covered by he hisorical daa have been removed So ha he new numbers of inervals are 13 which are u u u and u Fig 1 Time series daa of ouris arrivals o Malaysia from January 008 o December

4 The new TrFNs can be defined as follows: A A : : A A Fuzzify he D As an example D for January 008 is Thus he and locaed a range u corresponding fuzzy number is assigned as A 4 Table IV shows some of he corresponding fuzzy number for he ouris arrivals in year 008 TABLE IV CORRESPONDING FUZZY NUMBERS OF THE TOURIST ARRIVALS TO MALAYSIA FOR YEAR 008 Monh D Fuzzy number Monh D Fuzzy number Jan A 4 July A 5 Feb Mar Apr May June A 8 Aug A 3 Sep A 5 Oc A 7 Nov A Dec Afer fuzzifying he D he FLR is creaed and furher he FLR group is produced as shown in Table V TABLE V FUZZY LOGICAL RELATIONSHIP GROUPS Group Fuzzy Logical Relaionship 1 A1 A3 A3 A4 A3 A5 3 A4 A1 A4 A5 A4 A6 A4 A7 A4 A8 4 A5 A4 A5 A7 5 A6 A4 A6 A6 A6 A7 6 A7 A4 A7 A6 A7 A8 A7 A10 7 A8 A3 A8 A7 A8 A8 A8 A9 A8 A1 8 A9 A8 A9 A11 9 A10 A7 A10 A8 A10 A9 A10 A13 10 A11 A10 A11 A1 11 A1 A10 1 A13 A11 The forecased oupu O is calculaed using heurisic rules Then he final forecased F is generaed by using equaion (5) since he D have boh seasonal and rend paern Table VI shows he final forecased of ouris arrivals for year 008 o 010 wih heir defuzzified s A 4 A 1 A 3 A 4 A 6 TABLE VI FORECASTED OUTPUT OF THE TOURIST ARRIVAL TO MALAYSIA IN YEAR 008 TO 010 Monh Forecased oupu Defuzzified Jan 008 Feb 008 ( ) Mar 008 ( ) Apr 008 ( ) May 008 ( ) June 008 ( ) July 008 ( ) Aug 008 ( ) Sep 008 ( ) Oc 008 ( ) Nov 008 ( ) Dec 008 ( ) Jan 009 ( ) Feb 009 ( ) Mar 009 ( ) Apr 009 ( ) May 009 ( ) June 009 ( ) July 009 ( ) Aug 009 ( ) Sep 009 ( ) Oc 009 ( ) Nov 009 ( ) Dec 009 ( ) Jan 010 ( ) Feb 010 ( ) Mar 010 ( ) Apr 010 ( ) May 010 ( ) June 010 ( ) July 010 ( ) Aug 010 ( ) Sep 010 ( ) Oc 010 ( ) Nov 010 ( ) Dec 010 ( ) Jan 011 ( ) VI DISCUSSION In Table VI he forecased ouris arrival for January 011 is ( ) and he possible forecased inervals under differen degree of confidence (DDoC) can be obained The decision analys used TrFNs o find he possible forecased inerval by using he - cu I can be obained under a degree of confidence ( ) from inerval 0 1 Here we ook June and calculae he degree of confidence because of he exisence of a seasonal facor in hese hree consecuive years The forecased inervals are presened as in Table VII VIII and IX below TABLE VII FORECASTED INTERVALS FOR JUNE 008 (FROM = 0 TO 1) Degree of confidence = 0 [ ] = 01 [ ] = 0 [ ] = 03 [ ] = 04 [ ] = 05 [ ] = 06 [ ] = 07 [ ] = 08 [ ] = 09 [ ] = 10 [ ] 45

5 TABLE VIII FORECASTED INTERVALS FOR JUNE 009 (FROM = 0 TO 1) Degree of confidence = 0 [ ] = 01 [ ] = 0 [ ] = 03 [ ] = 04 [ ] = 05 [ ] = 06 [ ] = 07 [ ] = 08 [ ] = 09 [ ] = 10 [ ] TABLE IX FORECASTED INTERVALS FOR JUNE 010 (FROM = 0 TO 1) Degree of confidence = 0 [ ] = 01 [ ] = 0 [ ] = 03 [ ] = 04 [ ] = 05 [ ] = 06 [ ] = 07 [ ] = 08 [ ] = 09 [ ] = 10 [ ] The inerval wih higher of will give smaller inerval of esimaion For example for June 010 when = 0 he inerval is [ ] while = 10 he inerval is [ ] The differences beween inervals are and 471 respecively This indicaes ha he range for higher is more precise in making predicion In addiion his esimaion can be evaluaed by he decision analys Furhermore he previous FTSM produces he forecased s in erms of single poin whereby some informaion will be dissipaed To evaluae he performance of he forecasing model he forecasing accuracy (RMSE) is calculaed The forecased s need o be defuzzified using cenroid mehod and he RMSE obained is 3909 VII CONCLUSION In his sudy he fuzzy forecasing approach using an average based lengh wih new base mapping is proposed The new base mapping able can caer a large of range The propose forecasing model can caer for daa up o 10 k where k Thus his paricular model allows unlimied range of daa compared o he previous base mapping able ha only caers up o10 3 of daa These enhance he forecasing model o be used in more wide applicaion of fuzzy forecasing problem Oher han ha he proposed mehod also produces he forecased oupu in erm of TrFNs which allows forecasers o analyze various forecased inervals under differen degree of confidence insead of single poin offered by he exising FTSM Thus i can provide more informaion on he forecased s ACKNOWLEDGMENT This research is suppored by he Minisry of Educaion Malaysia (MOE) and Universii Teknologi MARA Malaysia (UiTM) under he Research Gran No 600-RMI/RAGS 5/3 (149/014) REFERENCES [1] X Li and W Chen A Grey-Markov predicaion for unemploymen rae of graduaes in China in Grey Sysems and Inelligen Services 009 pp [] K Mahipan N Chuiman and B Kumphon A forecasing model for Thailand s unemploymen rae Mod Appl Sci vol 7 no 7 pp [3] M Brau Economeric models or smoohing exponenial echniques o predic macroeconomic indicaors in Romania Zagreb In Rev Econ Bus vol 15 no pp [4] Q Song and B S Chissom Forecasing enrollmens wih fuzzy ime series Par I Fuzzy Ses Sys vol 54 pp [5] S-M Chen Forecasing enrollmens based on fuzzy ime series Fuzzy ses Sys vol 81 pp [6] Q Song and B S Chissom Forecasing enrollmens wih fuzzy ime series par II Fuzzy Ses Sys vol 6 pp [7] K Huarng Effecive lenghs of inervals o improve forecasing in fuzzy ime series Fuzzy ses Sys vol 13 pp [8] S Li and Y Chen Naural pariioning-based forecasing model for fuzzy ime series in IEEE Inernaional Conference on Fuzzy Sysem 004 pp [9] S Chen and C Hsu A new mehod o forecas enrollmens using fuzzy ime series In J Appl Sci Eng vol pp [10] K Huarng and T H Yu Raio-based lenghs of inervals o improve fuzzy ime series forecasing IEEE Trans Sys Man Cybern Par B vol 36 no pp Apr 006 [11] P Singh and B Borah An efficien ime series forecasing model based on fuzzy ime series Eng Appl Arif Inell vol 6 no 10 pp Nov 013 [1] H Liu An improved fuzzy ime series forecasing mehod using rapezoidal fuzzy numbers Fuzzy Opim Decis Mak vol 6 no 1 pp [13] H T Jasim A G J Salim and K I Ibraheem A Novel Algorihm o Forecas Enrollmen Based on Fuzzy Time Series Appl Appl Mah vol 7 no 1 pp [14] H-T Liu and M-L Wei An improved fuzzy forecasing mehod for seasonal ime series Exper Sys Appl vol 37 no 9 pp Sep 010 [15] H-T Liu An inegraed fuzzy ime series forecasing sysem Exper Sys Appl vol 36 no 6 pp Aug