Research Article A Unified Weight Formula for Calculating the Sample Variance from Weighted Successive Differences
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1 Discrete Dyamics i Nature ad Society Article ID pages Research Article A Uified Weight Formula for Calculatig the Sample Variace from Weighted Successive Differeces Kuo-Hug Lo 12 Tie-Lug Su 1 ad Juei-Chao Che 3 1 Departmet of Idustrial Egieerig ad Maagemet Yua Ze Uiversity Zhogli Taiwa 2 Departmet of Marketig & Logistics Maagemet Yu Da Uiversity of Sciece ad Techology Miaoli City Taiwa 3 Departmet of Statistics ad Iformatio Sciece Fu Je Catholic Uiversity New Taipei City Taiwa Correspodece should be addressed to Juei-Chao Che; @mailfjuedutw Received 1 July 2014; Accepted 6 August 2014; Published 27 August 2014 Academic Editor: Chi-Chia Wu Copyright 2014 Kuo-Hug Lo et al This is a ope access article distributed uder the Creative Commos Attributio Licese which permits urestricted use distributio ad reproductio i ay medium provided the origial work is properly cited The basic formula to calculate sample variace is based o the sum of squared differeces from mea From computatioal perspective mea calculatio is odesired as it ca itroduce computig errors Previous researches have proposed to use weighted formula of the successive differeces to calculate sample variace to avoid mea calculatio But their weighted formula is ot i a uified format i the sese that it has to be represeted as two formulas This paper proposes a uified weight formula for sample variace calculatio from weighted successive differeces A proof is provided to show that sample variace calculated usig the proposed uified weighted formula is mathematically equivalet to the basic defiitio 1 Itroductio Sample variace calculatio is a fudametal task i may data aalysis applicatios The basic formula for calculatig a sample variace is based o the sum of squared differeces from mea Give that a set of data is x 1 x 2 x the sample variace deoted as S 2 is calculated as follows: S 2 SS 2 (1) ( 1) SS (x i x ) 2 isthesumofsquareddiffereces from the mea ad x (1/) x i is the sample mea Vo Neuma et al 1 poitedoutthat(1) doesottake ito accout the order of the observatios They proposed to istead use successive differeces of data so that the order ca be cosidered Specifically they used D ( 1) i2 d 2 i (2) the subscript i refers to temporal order of the data ad d i x i x i 1 Defie{x 0 d 1 d } as the successive differeces of the iput data From computatioal perspective vo Neuma s formula is also advatageous as it avoids a mea calculatio that may itroduce roudig errors The problem with D 2 is that it is ot mathematically equivalet to the basic defiitio This problem was idepedetlysolvedbyeiloadchowdhury2 ad Joarder 3 weighted successive differeces were used to derive a formula that is mathematically equivalet to the basic defiitio Eilo ad Chowdhury 2 cosidered a job schedulig problem they wated to miimize the variace of the job s waitig time Let y i be the waitig time of the ith job By defiitio y 1 0asthefirstjobdoesothavewaitig time ad y i i 1 j1 p jfori 12 is the umber of jobs ad p j is the processig time of job j The objective is to miimize the variace of the waitig time or equivaletly SS (y i y ) 2 For this purpose there is a eed to quickly update SS whe job i ad j are swapped Notice that whe job i ad j areswappedmostofthejobs waitig time will chage accordigly ad y ad SS have to be recalculated To avoid recalculatig y whe updatig SS Eilo ad Chowdhury derived a formula to calculate SS from successive differeces By defiitio the successive
2 2 Discrete Dyamics i Nature ad Society differeces of the waitig time are the processig time; that is p i y i+1 y i 2 1So (y i y ) ( ( i) ip 2 i j1 c ij p ip j 1 ij1ij ( i) jp i p j ) c j i j ij {( i) ij12 1 (4) ( j) i i < j Equatio (3) isotageeralformulaforcalculatigss as y 1 is zero Vai ad Raghavachari 4 gave a more geeral formula by cosiderig the job s completio time rather tha waitig time Let z i i j1 p j bethecompletiotimeofjob itheyrewrote(3) as follows: (z i z ) ( i+1)(i 1) p 2 i (i 1) p i 1 c ij p i p j j1 ji+1 ( j+1)p j ( i+1)(j 1) i j c ij { (6) ( j+1)(i 1) i < j for ij12 I a idepedet work Joarder 3also derived a formulasimilar to (5) He the coverted its double sum structure ito a quadratic form i (3) (5) (x i x ) 2 1 dt 1 c d 1 (7) d T 1 d 1 d 2 d 1 isavectorofthesuccessive differeces ad c c ij is a weight matrix with c ij as defied i (6) Oe problem with the weight formula i (6)isthatitisot i a uified format but has to be represeted as two formulas This deficiecy prohibits a compact represetatio that would facilitate further derivatios To solve this problem we derive a uified weight formula for sample variace calculatios from weighted successive differeces Joarder 3 derived a updatig formula to calculate a variace from weighted successive differeces But his formula cotais a dyamically icreasedumberofupdatigitemsusigtheuifiedweight formula we show 5 that we ca improve Joarder s formula by reducig the updatig items to a fixed umber of oly two items 2 Mai Results Theorem 1 Give that a temporally order of the observatios x 1 x 2 x the sum of squared differeces about the mea ca be represeted as SS (x i x ) 2 1 dt 1 w d 1 (8) d T 1 d 1 d 2 d 1 {d i x i x i 1 }forx 0 0 i 12adw w ij are the symmetric matrix with w ij (+1) (i+j 1) ij 2 (i + j + i j ) Proof First write ij12 (9) (x i x ) 2 (x 1 x 1 ) T (x 1 x 1 ) (10) for x T x 1 x 2 x 1 Nowx cabepresetedas for x T d 1 2 d i P d d P d d i that is the row i colum j elemet of P is p ij { 1 j i 0 j > i for ij12 Next the mea of x cabewritteas 1 (11) (12) (13) x 1 i x 1 ( i+1) d i (14) I vector form this is x 1 Q d 1 (15) 1 1 Q ; (16)
3 Discrete Dyamics i Nature ad Society 3 the row i colum j elemet of Q is Now observe that q ij 1 ( j + 1) ij12 (17) (x 1 x 1 ) T (x 1 x 1 ) (P Q ) d 1 T (P Q ) d 1 d T 1 (P Q ) T (P Q ) d 1 d T 1 (PT P P T Q Q T P + Q T Q ) d 1 (18) Thus we eed to obtai expressios for calculatig P T P P T Q Q T P adq T Q First The L P T P d p im p mj d l ij (19) l ij { j+1 i j i+1 i>j (20) 1 2 (i + j + i j 2) ij12 B P T Q d p im q mj d 1 1 b ij b ij 1 ( j+1)( i+1) b ij (b ij ) T 1 ( i+1)( j+1) ij12 ij12 (21) (22) That is Fially P T Q (P T Q ) T Q T P (23) R Q T Q d d 1 1 q im q mj r ij (24) r ij 1 ( i+1)( j+1) ij12 (25) We ow ca see that P T Q Q T P Q T Q ad hece P T P P T Q Q T P + Q T Q (26) P T P Q T Q A direct calculatio produces as follows: l ij r ij 1 2 (i + j + i j 2) 1 ( j+1)( i+1) 1 (2 2 (i + j + i j )+ ( i+1)( j + 1)) 1 ((+1)(i+j 1) ij 2 (i + j + i j )) 1 w ij (27) Thus P T P Q T Q w ad the proof is complete 3 Numerical Example Thissectiogivesaumericalexampletoillustratesample variace calculatio usig the ouified weight formula c ij givei(6)adtheuifiedformulaw ij givei(9) We take a sample data set x 1 5 x 2 14 x 3 9adx 4 6from Ross 6 Page 145 the data are used to illustrate the variace updatig process usig the oe-pass algorithm proposed i va Reeke 7 The successive differeces for this data set are d 1 5 d (28) d d
4 4 Discrete Dyamics i Nature ad Society ad the successive differeces vector is d T Usig the ouified weight formula the weight matrix c 4 4 c ij 4 4 is costructed usig two formulas:oeforthelowertriagularmatrixadoeforthe strictly upper triagular matrix For the lower triagular matrix with i j c ij ( i + 1)(j 1) For example c 11 (4 1+1)(1 1) 0 c 21 ( )(1 1) 0 c 31 (4 3+1)(1 1)0adc 32 ( )(2 1) 2 For the strictly upper triagular matrix with i < jthe weight formula is c ij ( j+1)(i 1) For example c 12 (4 2+1)(1 1)0 c 13 (4 3+1)(1 1)0ad c 23 (4 3+1)(2 1)2 Combiig the lower triagular matrix ad the strictly upper triagular matrix we ca get The variace is the calculated as S dt 1 4 c 4 4d 4 1 c (29) (30) Now with our approach the weight matrix W 4 4 w ij 4 4 is costructed usig the uified weight formula give i (9) For example w 12 (4 + 1)( ) 1 2 (4/2)( ) 0 ad w 21 (4+1)(2+1 1) 2 1 (4/2)( ) 0Similarlyw 23 ad w 32 are calculated as w 23 (4+1)(2+3 1) 2 3 (4/2)( ) 2 ad w 32 (4 + 1)( ) 3 2 (4/2)( ) 2The otherweightsarecalculatediasimilarmaertoproduce prohibits compact represetatio of further derivatios This paper derives a uified weight formula for calculatig a sample variace from weighted successive differeces We have employed this compute formula to improve variace updatig formula i Vai ad Raghavachari 4 or Joarder 3 Coflict of Iterests The authors declare that there is o coflict of iterests regardig the publicatio of this paper Refereces 1 JvoNeumaRHKetHRBellisoadBIHart The mea square successive differece The Aals of Mathematical Statisticsvol12pp S Eilo ad I G Chowdhury Miimisig waitig time variace i the sigle machie problem Maagemet Sciecevol 23 o 6 pp A H Joarder Sample variace ad first-order differeces of observatios The Mathematical Scietistvol28o2pp V Vai ad M Raghavachari Determiistic ad radom sigle machie sequecig with variace miimizatio Operatios Researchvol35o1pp TLSuKHLoadJCChe Aupdatigformulato calculate sample variace from weighted successive differeces mauscript 6 SMRossItroductio to Probability ad Statistics for Egieers ad Scietists Wiley New York NY USA A J va Reeke Dealig with eelys algorithms Commuicatios of the ACMvol11o3pp The variace is the calculated as S dt 1 4 w 4 4d 4 1 W (31) (32) Coclusios Sample variace calculatio usig weighted successive differeces is advatageous from a computatioal perspective as it avoids a mea calculatio which may itroduce roudig errors However the weight formula that has bee proposed i previous research is ot i a uified format Istead it has to be represeted as two formulas This deficiecy
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