Forecasting the Movement of Share Market Price using Fuzzy Time Series

Transcription

1 Iteratoal Joural of Fuzzy Mathematcs ad Systems. Volume 1, Number 1 (2011), pp Research Ida Publcatos Forecastg the Movemet of Share Market Prce usg Fuzzy Tme Seres B.P. Josh ad Sajay Kumar Ruhela Departmet of Mathematcs, Statstcs & Computer Scece G. B. Pat Uversty of grculture & Techology, Patagar (Uttarakhad) Ida E-mal: bpjosh.13march@gmal.com bstract There are varous methods establshed o tme seres data havg lgustc values for forecastg the future values wth the help of fuzzy tme seres forecastg. However, the major problem fuzzy tme seres forecastg s the accuracy the forecasted values. The preset paper proposes a ew method of forecastg usg fuzzy tme seres. The proposed method s based o hghest membershp grade ad s a smplfed computatoal approach for the forecastg. The proposed method s mplemeted to forecast the movemet of share market prce of State Bak of Ida (SBI) at Bombay Stock Exchage (BSE). The forecasted data have bee compared wth the results obtaed by the method gve by Sgh [5] to show the superorty of proposed method. Keywords: Fuzzy tme seres, Fuzzy membershp grade, Fuzzy logcal relatos, movemet of market prce. Itroducto Fuzzy set theory ad the cocept of lgustc varables ad ts applcato to approxmate reasog developed by Zadeh [9] has bee successfully employed by Sog ad Chssom [6-8] fuzzy tme seres forecastg. Sog ad Chssom [6-8] developed the fuzzy tme seres models ad mplemeted the developed models o the hstorcal erollmet data of Uversty of labama. Che [1-3] preseted a smplfed method for tme seres forecastg usg the arthmetc operatos rather tha complcated max-m composto operatos, there are may researchers used the cocept of fuzzy tme seres forecastg. Huarg [4] preseted a heurstc model for tme seres forecastg usg heurstc creasg ad

2 74 B.P. Josh ad Sajay Kumar Ruhela decreasg relatos to mprove the forecast of erollmets ad also mplemeted t for Tawa Futures Exchage (TIFEX) forecastg. I ths paper, we preset a ew method for forecastg the movemets of market share prce wth fuzzy tme seres. It provdes smple computatoal ad mmzes the tme of geeratg relatoal equatos by usg complex m-max composto operatos. It overcomes the dffculty of searchg a sutable defuzzfcato procedure provdg crsp output of better accuracy. Further, the proposed method s mplemeted to forecast the market prce of a State Bak of Ida s share at Bombay Stock Exchage (BSE). Some basc cocepts of fuzzy tme seres The varous deftos ad propertes of fuzzy tme seres forecastg foud, summarzed ad are preseted as: Defto 1. fuzzy set s a class of objects wth a cotuum of grade of membershp. Let U be the Uverse of dscourse wth U = {u 1, u 2, u 3,..., u,}, where u are possble lgustc values of U, the a fuzzy set of lgustc varables of U s defed by = μ + ( u1) / u1 + μ ( u2 ) / u 2 + μ ( u3 ) / u μ ( u ) / u Where μ s the membershp fucto of the fuzzy set, such that μ : U [ 0,1]. Defto 2. Let Y(t) (t =...,0,1, 2, 3,...), s a subset of R, be the Uverse of dscourse o whch fuzzy sets f (t), ( = 1, 2, 3,...) are defed ad F(t) s the collecto of f, the F(t) s defed as fuzzy tme seres o Y(t). Defto 3. Suppose F(t) s caused oly by F(t-1) ad s deoted by F ( t 1) F ( t ); the there s a fuzzy relatoshp betwee F(t) ad F(t-1) ad ca be expressed as the fuzzy relatoal equato: F(t) = F(t-1) o R(t, t-1) here, o s Max M composto operator. The relato R s called frst-order model of F(t). Further f fuzzy relato R(t, t-1) of F(t) s depedet of tme t, that s to say for dfferet tmes t 1 ad t 2, R(t 1, t 1-1) = R(t 2, t 2-1), the F(t) s called a tme varat fuzzy tme seres. Defto 4. If F(t) s caused by more fuzzy sets, F(t-), F(t-+1),...,F(t-1), the fuzzy relatoshp s represeted by...,,, j

3 Forecastg the Movemet of Share Market Prce 75 here, F t ) =, F ( t + 1) =,......, F ( t 1) =. ( 1 2 Ths relatoshp s called th -order fuzzy tme seres model. Methodology to mplemet proposed method I ths secto, we preset the stepwse procedure of the proposed method of forecastg by fuzzy tme seres. Step 1. Defe the Uverse of dscourse, U based o the rage of avalable tme seres data, by rule U = [D m -D 1, D max -D 2 ] where D 1 ad D 2 are two proper postve umbers. Step 2. Partto the Uverse of dscourse to equal legth of tervals: u 1, u 2,..., u m. The umber of tervals wll be accordace wth the umber of lgustc varables (tragular fuzzy sets) 1, 2,..., m, to be cosdered. Step 3. Costruct the fuzzy sets accordace wth the tervals Step 2 ad apply the tragular membershp rule to each terval each tragular fuzzy set so costructed. Step 4. Fuzzfy the data ad establsh the fuzzy logcal relatoshps by the rule: If s the fuzzy producto of moth ad j s the fuzzfy producto of moth + 1, the the fuzzy logcal relato s deoted as j. Here s called curret state ad j s ext state. Step 5. Utlzg the fuzzy logcal relatoshps, obta the fuzzfed output. fter that defuzzfed ad get crsp output. Step 6. I tme seres forecastg, the forecastg accuracy of a model s commoly measured terms of mea square error (MSE) or terms of average error. Lower the MSE or average error, better the forecastg method. The MSE ad forecastg error are defed as Mea Square error = = 1 ( actual value forecasted value forecasted actual value Forecast g error ( percetage) = 100. actual value sum of forecastg error verage forecast gerror ( percetage) = 100. umbersof errors ) 2 (1) (2) (3)

4 76 B.P. Josh ad Sajay Kumar Ruhela Forecastg market prce of SBI s share prce wth proposed model I ths secto, the proposed method s mplemeted to forecast market prce data of SBI share at BSE. Step 1: ual report of SBI [10] for the year s used to defe D m ad D max. The for ths we defe uverse of dscourse U = [1350, 2550]. Step 2: The Uverse of dscourse s parttoed to sx tervals of lgustc values: u 1 = [1350, 1550], u 2 = [1550, 1750], u 3 = [1750, 1950], u 4 = [1950, 2150], u 5 = [2150, 2350], u 6 = [2350, 2550]. Step 3: Sx fuzzy sets 1, 2,..., 6, are defed o the uverse of dscourse U ad gve Fg. 1. Fgure 1: Fuzzy sets 1, 2,..., 6, Table 1: ctual market prce of SBI share at BSE. Moths SBI s Share Prce at BSE (Rs.) prl May Jue July ugust September October November December Jauary February March

5 Forecastg the Movemet of Share Market Prce 77 Step 4: The tme seres data s fuzzfed wth tragular membershp fucto ad are placed Table 2. Table 2: Fuzzfcato of actual market prce of SBI s share. Moths Fuzzfed value prl May Jue July ugust September October November December Jauary February March Step 5: Fuzzy logcal relatoshps ad fuzzy logcal relatoshp groups of market prce obtaed ad are as (Table 3 ad 4). Table 3: Fuzzy logcal relatoshps of market prce Table 4: Fuzzy logcal relatoshp groups Step 5: The followg table shows the forecasted market prce of SBI share at BSE obtaed by proposed model. Results obtaed by the method gve by Sgh [5] are also cluded for the comparso.

6 78 B.P. Josh ad Sajay Kumar Ruhela Table 5: Forecasted market prce of SBI share. Moths Market prce of SBI s Share at BSE (Rs.) ctual Forecasted Sgh s method prl May Jue July ugust September October November December Jauary February March Step 6: To have a comparso of accuracy forecasted values of our proposed model wth Sgh model, the mea square error (MSE) ad average error of forecast have bee computed ad have bee compared wth Sgh method Table 6. The MSE or average error of forecastg of a method s measure of accuracy of that forecastg method. Lower the MSE or average error, better the forecastg method. Table 6: Comparso of MSE ad average error. Model Proposed Model Sgh Model Mea Square Error(MSE) verage Forecastg Error Fgure 2: ctual versus Proposed model ad Sgh Model of SBI s at BSE.

7 Forecastg the Movemet of Share Market Prce 79 Coclusos I ths paper we have proposed dfferet method of fuzzfcato, whch gves dfferet fuzzy logc relatos. The algorthm of the proposed method s smple. It mmzes the complcated computatos of fuzzy relatoal matrces ad search for a sutable defuzzfcato process ad provdes the forecasted values of better accuracy. Further the method has also bee mplemeted to forecast the market prce of State Bak of Ida s share at Bombay Stock Exchage (BSE). The proposed method s also compared wth the method gve by Sgh [5]. The comparso of MSE ad average forecasted error gve Table 6 shows the superorty of the proposed model over the Sgh s model as t provdes forecast of hgher accuracy. Refereces [1] Che, S.M., Hsu, C.C.: ew Method to Forecast Erollmets usg Fuzzy Tme Seres, Iteratoal Joural of ppled Sceces ad Egeerg. 2 (3), (2004) [2] Che, S.M.: Forecastg Erollmets based o Fuzzy Tme Seres, Fuzzy Sets ad Systems. 81, (1996) [3] Che, S.M.: Forecastg Erollmets based o Hgh-Order Fuzzy Tme Seres, Cyberetcs ad Systems: Iteratoal Joural. 33, 1 16(2002) [4] Huarg, K.: Heurstc Models of Fuzzy Tme Seres for Forecastg, Fuzzy Sets ad Systems. 123, (2001) [5] Sgh, S.R.: smple Method of Forecastg based o Fuzzy Tme Seres, ppled Mathematcs ad Computato. 186, (2007) [6] Sog, Q., Chssom, B.: Fuzzy Tme Seres ad Its Models, Fuzzy Sets ad Systems. 54, (1993) [7] Sog, Q., Chssom, B.S.: Forecastg Erollmets wth Fuzzy Tme Seres Part I, Fuzzy Sets ad Systems. 54, 1 9(1993) [8] Sog, Q., Lelad, R.P.: Learg Defuzzfcato Techques ad pplcatos, Fuzzy Sets ad Systems. 81 (3), (1996) [9] Zadeh, L..: Fuzzy set, Iformato ad Cotrol. 8, (1965) [10] ual Report of State Bak of Ida, ( )