Exponential Weighted Moving Average (EWMA) Chart Under The Assumption of Moderateness And Its 3 Control Limits

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1 DOI: 0.545/mjis Exponenial Weighed Moving Average (EWMA) Char Under The Assumpion of Moderaeness And Is 3 Conrol Limis KALPESH S TAILOR Assisan Professor, Deparmen of Saisics, M. K. Bhavnagar Universiy, Bhavnagar kalpesh_lr@yahoo.co.in Received: Ocober, 06 Revised: November 9, 06 Acceped: December 5, 06 Published online: March 05, 07 The Auhor(s) 06. This aricle is published wih open access a Absrac Moderae disribuion is a very good alernaive of normal disribuion proposed by Naik V.D and Desai J.M. [4], which has mean deviaion as scale parameer raher han he sandard deviaion. Mean deviaion (δ) is a very good alernaive of sandard deviaion (σ) as mean deviaion is considered o be he mos inuiively and raionally defined measure of dispersion. This fac can be very useful in he field of qualiy conrol o consruc he conrol limis of he conrol chars. On he basis of his fac Naik V.D. and Tailor K.S. [5] have proposed 3δ conrol limis. In 3δ conrol limis, he upper and lower conrol limis are se a 3δ disance from he cenral line where δ is he mean deviaion of sampling disribuion of he saisic being used for consrucing he conrol char. In his paper i has been assumed ha he underlying disribuion of he variable of ineres follows moderae disribuion proposed by Naik V.D and Desai J.M. [4] and 3δ conrol limis of exponenial weighed moving average char are derived. Also an empirical sudy is carried ou o illusrae he use hese chars. Keywords Mean deviaion, Moderae disribuion, Exponenial weighed moving average, 3δ conrol limis.. INTRODUCTION A fundamenal assumpion in he developmen of conrol chars for variables is ha he underlying disribuion of he concerned qualiy characerisic is normal. The normal disribuion is one of he mos imporan disribuions in he saisical inference in which mean ( µ ) and sandard deviaion (σ) are Mahemaical Journal of Inerdisciplinary Sciences Vol-5, No-, March 07 pp. 9

2 Tailor, KS he parameers. Naik V.D and Desai J.M. [4] have suggesed an alernaive of normal disribuion, which is called moderae disribuion. In moderae disribuion mean ( µ ) and mean deviaion (δ) are he pivoal parameers and, hey have properies similar o normal disribuion. Naik V.D. and Tailor K.S. [5] have proposed he concep of 3δ conrol limis on he basis of moderae disribuion. Under his rule, he upper and lower conrol limis are se a 3δ disance from he cenral line where δ is he mean deviaion of sampling disribuion of he saisic being used for consrucing he conrol char. Thus in he proposed conrol chars, under he moderaeness assumpion, he conrol limis for any saisic T should be deermined as follows. Cenral line (CL) Expeced value of T μ Lower Conrol Limi (LCL) Mean of T -3δ T µ -3δ T Upper Conrol Limi (UCL) Mean of T + 3δ T µ + 3δ T Where μ is mean of T and δ T is he mean deviaion of T. I is found ha since δ provides exac average disance from mean and σ provides only an approximae average disance, 3δ limis can be considered o be more raional as compared o 3σ limis. Naik and Tailor have derived 3δ conrol limis of X -char, R-char, s (sample sandard deviaion) char and d (sample mean deviaion) char. Tailor has also suggesed 3δ conrol limis of moving average and moving range chars. Thus, in his paper i is assumed ha he underlying disribuion of he concerned variable follows moderae disribuion and 3δ conrol limis for exponenial weighed moving average is derived. An empirical sudy is also carried ou o illusrae he use of he char.. EXPONENTIAL WEIGHTED MOVING AVERAGE (EWMA) CHART The concep of EWMA char was inroduced by Robers S.W [7]. The exponenially weighed moving average char is a ype of moving mean char in which an exponenially weighed mean is calculaed each ime a new resul becomes available. The EWMA conrol char is a very good alernaive o he Shewhar char, when we are ineresed in deecing small shifs.

3 New weighed mean ( new resul) + ((- ) previous resul ), where is he smoohing consan. I has a value beween 0 and, many saisician use 0., bu choice of has o be lef o he judgmen of he qualiy conrol specialis, he smaller he value of, he greaer he influence of he hisorical daa. The EWMA char is much more effecive han moving average char for deecing small shifs. If i is imporan o recognize small shifs early in he process, hen he value of should be small. If, he EWMA char reduces iself o he usual X -char. This has been used by some organizaions, paricularly in he process indusries, as he basis of new conrol (performance) char sysems. Grea care mus be aken when using hese sysems since hey do no show changes in variabiliy very well and he basis for weighing daa is ofen eiher quesionable or arbirary. Exponenial Weighed Moving Average (EWMA) Char Under The Assumpion of Moderaeness And Is 3 Conrol Limis 3. (3δ) CONTROL LIMITS FOR EXPONENTIAL WEIGHTED MOVING AVERAGE (EWMA) CHART Suppose a measurable qualiy characerisic of he produc is denoed by X. Suppose ha m samples, each of size n, are drawn a more or less regular inerval of ime from he producion processes. These samples are known as subgroups, and for each of hese subgroups he values of exponenial weighed moving mean X are obained. Le he disribuion of he variable X be moderae wih mean µ and mean deviaion δ, hen, as proved by Naik V.D and Desai J.M. [4], he disribuion of X is also moderae wih mean µ and δ mean deviaion. Furher, if he disribuion of X is no moderae, and he n number of unis in each subgroup is 4 or more, hen on he basis of cenral limi heorem for moderae disribuion, i can be said ha X follows moderae disribuion. The EWMA funcion is defined as, Z xi + ( - ) Zi -, where 0< If he individual X are independen random variables wih variance σ n, hen he variance of Z is defined as σ σ α - - ( - ) 3

4 Tailor, KS Therefore σ σ α - - ( - ) σ - - ( - ) α () Since we are assuming moderaeness, he mean error of Z is defined as δ π δ α ( ) () Thus, he 3δ- conrol limis of EWMA char can be deermined as follows Cenral line (C.L) E( X ) Lower conrol limi (L.C.L) E( X )- X (3) 3δ X -3 π δ - n - ( - α) δ X α - - ( - ) (4) Upper conrol limi (U.C.L) E( X )+ 3δ π δ X 3 ( α) n (5) δ X α ( ) Where X and δ are ypically esimaed from preliminary daa as sample mean and sample mean deviaion. As α is small and if increases, he effec of saring value soon dissipaes and he mean error converges o is asympoic value. 4

5 i.e δ π δ - The conrol limis for EWMA char are usually based on he asympoic mean deviaion of he saisic. Hence asympoic 3δ-conrol limis for his char can be derived as following way, Cenral line (C.L) E ( X ) Exponenial Weighed Moving Average (EWMA) Char Under The Assumpion of Moderaeness And Is 3 Conrol Limis Lower conrol limi (L.C.L) E( X )- X (6) 3δ X -3 X π δ - δ - (7) Upper conrol limi (U.C.L) E( X )+ 3δ X + 3 X π δ - δ - (8) 4. AN EMPIRICAL STUDY FOR EWMA CHART: To illusrae he use of EWMA conrol scheme, we use a se of simulaed observaions aken from Lucas J. M and Crosier R.B []. The daa, ogeher wih he corresponding EWMA values, are shown in Table. The arge value is aken o be 0, so he process is in conrol for he firs 0 observaions. The mean level was shifed upward by approximaely one sandard deviaion for he las nine observaions. The parameers of he EWMA are chosen o be α 0.5, δ, n Asympoic 3-conrol limis for EWMA char are obained by 5

6 Tailor, KS Table : Observed value EWMA + (- ) LCL -.00, CL 0 and UCL.00 Similarly, asympoic 3σ-conrol limis for EWMA char are obained by LCL -.34, CL 0 and UCL.34 Now, EWMA char for moderaeness and normaliy assumpions can be consruced as follows. From figure, i can be seen ha under moderaeness assumpion wih 3δ -conrol limis, EWMA char is under he saisical conrol, while under normaliy assumpion wih 3σ-conrol limis, i is ou of conrol as one sample poin falls ouside he UCL. The poin which shows ou of conrol siuaion in EWMA char under normaliy assumpion shows under conrol siuaion in EWMA char under moderaeness assumpion. 6

7 Exponenial Weighed Moving Average (EWMA) Char Under The Assumpion of Moderaeness And Is 3 Conrol Limis Figure : I can also be seen from figure, ha using asympoic conrol limis raher han he ime varying limis, makes he EWMA char much less sensiive o process shifs in he firs few observaions. This could be a significan problem if a large shif occurs early, or if afer an ou of conrol condiion he process is no properly rese. 5. SUMMARY This paper derives he EWMA char under he assumpion of moderaeness. The 3δ - conrol limis are derived for he EWMA char wih ime varying conrol limis. Also he asympoic 3δ - conrol limis are derived for he same char. An empirical sudy is carried ou o illusrae his char. A comparaive sudy is carried ou for he EWMA char under moderaeness assumpion and under normaliy assumpion and i is found ha EWMA char under moderaeness assumpion perform beer han EWMA char under normaliy assumpion. Hence, i is suggesed ha EWMA char under moderaeness assumpion wih 3δ - conrol limis should be used for effecive performance of he char. 6. APPENDIX (a) Moderae Disribuion Suppose he p.d.f. of a disribuion of a random variable X is defined as, 7

8 Tailor, KS X -µ - π δ f( x) e,- < X <, δ > 0 πδ Then, he random variable X may be said o be following moderae disribuion wih parameers μ and δ and may be denoed as X M( µδ, ). I can be proved ha, (i) f ( x ) - (ii) Mean E(x) μ (iii) Mean deviaion E X -π δ (vi) Sandard deviaion π δ () x (v) M.G.F M e π π+ δ 4 (vi) f ( µ - X) f µ + x ( ) (b) 3σ-conrol limis of EWMA char C.L. X U.C.L. L.C.L. X + 3σ - - ( - α) X σ ( α) REFERENCES [] Huner J. S. (986) The Exponenially Weighed Moving Average, Journal of Qualiy Technology, 8, [] Lucas J.M. and Crosier R.B. (98) Fas Iniial Response for CUSUM Qualiy Conrol Schemes, Technomerics, 4, [3] Lucas J.M. and Saccucci M.S. (990) Exponenially Weighed Moving average Schemes, Properies and Enhancemens, Technomerics, 3, 9 [4] Naik V.D and Desai J.M. (05) Moderae Disribuion: A modified normal disribuion which has Mean as locaion parameer and Mean Deviaion as scale parameer, VNSGU Journal of Science and Technology, Vol.4, No

9 [5] Naik V.D and Tailor K.S. (05) On performance of and R-chars under he assumpion of moderaeness raher han normaliy and wih 3 conrol limis raher han 3 conrol limis, VNSGU Journal of Science and Technology, Vol.4, No., [6] Kalpesh S. Tailor (06) Moving average and moving range chars under he assumpion of moderaeness and is 3 conrol limis [7] Robers S.W. (959) Conrol char Tess Based on Geomeric Moving Average Chars. Technomerics, Vol.-, No.-3, pp [8] Tailor K.S. and Naik V.D. (06) Mean deviaion () based conrol limis of SQC chars for sample sandard deviaion(s) and sample mean deviaion (d) and heir performance analysis under 3 conrol limis agains 3 conrol limis, VNSGU Journal of Science and Technology, (Acceped for Publicaion). Exponenial Weighed Moving Average (EWMA) Char Under The Assumpion of Moderaeness And Is 3 Conrol Limis 9