MODIFIED COMPOUND SMOOTHER IN MEDIAN ALGORITHM OF SPAN SIZE 42

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1 ODIFIED COPOUND SOOTHER IN EDIAN ALGORITH OF SPAN SIZE *Nurul Nisa Khairol Azmi, ohd Bakri Adam, Norhaslinda Ali Faculy of Comuer and ahemaical Sciences, UniversiiTeknologi ARA Negeri Sembilan Branch, Seremban, Negeri Sembilan, alaysia.,3 Insiue for ahemaical Research, Universii Pura alaysia, 300 UP Serdang, Selangor, alaysia. *Corresonding auhor s nisa@msk.uim.edu.my Submission dae: 30 July 08 Acceed dae: 30 Augus 08 Published dae:30 arch 09 Absrac edian smooher of even window size is obained by averaging he wo middle oins of arranged sequence using arihmeic mean. Some adjusmens were roosed by subsiuing he algorihm of arihmeic mean using quadraic, geomeric, harmonic and conraharmonic mean. The modified running median was aended in 53HT algorihm. This aer is mainly o assess he erformances of roosed adjusmen via simulaion sudy. The resuls show ha modified 53HT using conra harmonic mean erformed he bes in exracing signal from heavy noise. The exraced signal was hen used for forecasing by alying seasonal ARIA algorihm. Forecasing by smoohed values was found o roduce beer forecas han forecasing involving acual values which conained heavy noise. Keywords: comound smooher, running median, non-linear, 53HT, signal, noise.0 INTRODUCTION / BACKGROUND OF THE STUDY Smoohing is a rocess of exracing aern from heavy noise. In ime series, he exisence of heavy noise blurs aern and affecs he reliabiliy and ease of forecasing rocess. Velleman (980) saes, i is imoran o seek for daa smoohers ha are resisan o noise wih occasional "sikes" or long-ailed disribuion. any sudies which have been conduced, rovide romising evidence ha non-linear smooher like running median has a srong abiliy o reduce heavy noise from a series of daa (see Bovik e al., 983; Hird e al., 009 and Sargen e al., 00). Tukey (977) inroduces a non-linear aroach o smoohing, ha is comound smooher. Comound smooher is a combinaion of several algorihms of smoohing, which includes median smooher of various san sizes, weighed moving average, sliing and re-smoohing of he rough. Comound smooher is known as a owerful ool o smooh a daa series wihou excessively disruing he deails of a daa series. Vellemen (98) inroduces a comound smooher ha involves running median of even and odd san size, Hanning and 'wice'. I is called 53HT. The 53HT is only disurbed slighly by long-ailed noise and negligibly by Gaussian whie noise. e Academia Secial Issue GraCe, 08 78

2 Sargen e al. (00) aemed o imrove he comound smooher by combining he smoohing algorihm consising of running median of various san sizes, Hanning and 'wice' o fi he Ausralian fooball layers' erformance. The ouu of smooher was hen used for forecasing using exonenial smoohing mehod. The resul found ha forecas using smoohed daa of comound smooher aroach roduced beer forecas han when using acual daa. Jin and iu (03) rooses a comound smooher ha combines moving average, median smooher, maximum smooher and Hanning in order o reconsruc normalized difference vegeaion index (NDVI) ime series daa, called REH. Even hough, REH has been found o be beer a smoohing he NDVI daa based on secific crieria, Jin and iu (03) acknowledge ha 53HT is a good smooher comared o all. However, imrovemen on he exising comound smooher, 53HT in comarison has ye been exlored..0 ETHODOLOGY. 53HT 53HT is one of he non-linear smoohing echniques ha combines running median, weighed moving average and re-smoohing of he rough. This echnique was firs inroduced by Tukey (977) and exensively described in differen versions by Velleman e al. (98). Le be a doubly-infinie sequence of real daa n,...,,,,..., n. A smooher is defined as an algorihm ha works on o generae a new series ), smoohed values. The algorihm of 53HT is as follows: ( Se : Perform running median of san size wo ( ) median,,, () Se : Re-cener he equaion () ) median ( ), ( ) () Se 3: Nex, equaion () is smoohed again by alying median smooher wih san size hree 3( ) median ), ), ), ), ) (3) Se : Perform running median of san size hree ) median ( ), ( ), ( ) () ( Se 5: Aly weighed moving average or Hanning wih coefficiens 5( ) ( ) ( ) ( ) (5) Se 6: Re-smooh he rough and add he rough o he smoohed values in (5) ) ( ) ( ) (6). odificaion of 53HT 6( The running median of even san size was comued by aking he average of wo subsequen oins in he middle using arihmeic mean. This value is beer han running median of odd san size in he sense ha i reserves he significan sike in a daa series. The smoohed value roduced by running median san size four and re-cenered by running median of san size wo, was a combinaion of Equaion () and (), which can be exressed as follows:, and e Academia Secial Issue GraCe, 08 79

3 ) median,,, median,,, (7) * * ' ' where is he ordered observaion from window in,,, and is he ordered observaion from window in,,,. Some adjusmens were roosed by alying differen yes of means. The yes of means involved were geomeric, quadraic, harmonic and conra harmonic. The modificaions of running median san size are as follows: * Geomeric ean Quadraic ean * * ' ' (8) ( ) * * ' ' ) (9) Harmonic ean ) * * ' ' (0) Conra harmonic ean * * ' ' ( ) () * * ' ' The yes of means ha roduce smaller value han arihmeic mean - geomeric and harmonic, are execed o be more resisan o negaive imulse or block ulse. On he oher hand, quadraic and conra harmonic are execed o be more resonsive o osiive changes in a daa series. Some of he modificaions would no work if he observaions consis of zero or negaive values. Hence, a consan oin should be added o a daa o ensure he smoohed value can be comued. odificaion of 53HT only involved he running median of san size. Uon smoohing rough ar, original algorihm was mainained where he middle oins for running median of san size four and wo were comued via arihmeic mean..3 Simulaion Procedure The evaluaion rocess of smoohing was done hrough he simulaion of signal and noise. The rocess of simulaion was based on rocedure from Conradie e al. (009). Generally, a daa can be decomosed ino he following comonens: Daa = Signal + Noise = () A signal is a combinaion of sinusoidal funcion wih linear curve: ' Signal = = Asin B( C) (3) e Academia Secial Issue GraCe, 08 80

4 wih is he sloe of rend, is he index, A is an amliude, B= is d, and C reresens he dislacemen. Hence, d where d is he eriod and frequency D Asin B( C) D () 0.7 For sine funcion, le, he amliude A =3 and he dislacemen C=. The arameer, A and C were chosen according o Conradie e al. (009). This arameer values will roduce a smooh sine curve. Two hundred values from funcion Asin B( C) were simulaed for beween 0.5 and 9.66 wih he incremens of 0. a moderae frequency of 7 6. This frequency mimics a seasonal flucuaion ha occurs commonly in real racices. Figure shows a sinusoidal of frequency wih linear curve. The noise, {D } was generaed as idenical and indeenden random variables from conaminaed normal disribuion. This is demonsraed as he following: Y Z D Z if if Y, Y 0 wih i.i.d Bernoulli() and indeenden of he. Thus P Y ) PD d PZ d Y P( Y ) PZ d Y 0 Z (5) ( and P( Y 0) so ha P( Y 0) d d (6). wih i.i.d N(0,). To generae noise wih high volailiy, le and =0.75, so ha Var()=(0.75)(5.06) = 3.9. In he simulaion of generaing highly volaile noise, aroximaely 75% of he values came from a N(0,5.06 ) disribuion and aroximaely 5% was from a N(0,) disribuion. Z Figure Sinusoidal funcion of frequency wih linear rend Figure Sinusoidal funcion of frequency lus linear rend wih 75% conaminaed normal noise Figure deics sinusoidal of frequency lus rend wih 75% conaminaed normal noise added. I was hard o caure he general rend and exisence of seasonal oscillaion when 75% conaminaed e Academia Secial Issue GraCe, 08 8

5 normal noise was added. Two hundred signals lus he generaed noise were simulaed and alied o he exising and modified 53HT smooher. The erformances of hese smoohers were evaluaed via esimaed inegraed mean square error (EISE): k n EISE j j. (7) k j n Low EISE indicaes he abiliy for a smooher o recover signal from heavy noise.. Forecasing The exraced signal was hen furher used for forecasing. The mehod of forecasing alied in his aer was seasonal ARIA algorihm which akes ino accoun he rend and seasonaliy a he same ime. Seasonal ARIA can be exressed as follows, according o Box, e al. (05): s ( B ) ( B ) Y ( B) ( B) Z, Z ~ WN (0, ) (8) D d s where Y B B, ( z) z... z, ( z) z... z, ( z) z... z and ( z) z... z. The daa was divided ino wo ars, namely esimaion and evaluaion. In he firs ar of he daa, abou 7 observaions were for he esimaion of arameer and execed values; whereas he res were for he evaluaion of he forecas erformance. The erformance was measured via ean Square Error (SE): h SE ˆ (9) 3.0 RESULT AND DISCUSSION h Table shows he erformance of smoohers measured via EISE. The modified smooher using conra harmonic mean was found o be he bes avenue o exrac sinusoidal signal of frequency from heavy noise. This was vouched by he low value of EISE. Table Performance of exising and modified smooher measured by EISE Tye of modificaion EISE Arihmeic Geomeric.0563 Quadraic Harmonic Conra harmonic The exraced signal from smoohing rocess was subsequenly used for forecasing urose. In his sudy, seasonal ARIA algorihm was alied. The erformances of forecasing wih he inclusion of exraced signal and acual values, were comared. The las observaions were used for evaluaion o deermine wheher forecasing using smoohed values is beer han using acual values. The SE for forecasing using smoohed value was.56 and forecasing wih acual values resuled SE score of The resuls indicaing forecasing using smoohed values roduce beer forecas han forecasing using acual values. Figure 3 shows he forecas using acual value while Figure exhibis forecas involving he alicaion of smoohed values from modified 53HT using conra harmonic mean. e Academia Secial Issue GraCe, 08 8

6 Figure 3 Forecas using acual values Figure Forecas using smoohed values.0 CONCLUSION AND FUTURE WORKS This sudy is mainly o assess he erformance of modified 53HT in cauring sinusoidal lus linear rend signal wih heavy noise added. Noise wih high volailiy was added o he signal and he erformances were measured by recruiing EISE. The resuls show ha modified 53HT using conra harmonic mean erformed he bes in exracing signal from heavy noise. The exraced signal was hen used for forecasing by alying seasonal ARIA algorihm. Forecasing involving smoohed values was found o roduce beer forecas han forecasing wih he inclusion of acual values conaining heavy noise. For fuure works, he erformance of roosed adjusmen o comound smooher will be assessed wih he inclusion of differen yes of signals and noise. Acknowledgemen This research is arially funded from Pura Gran GP/08/ References Boema,. J. (99). Deerminisic roeries of analog median filers. IEEE Transacions on Informaion Theory 37 (6): Bovik, A. C., Huang, T. S. and unson, D. C. (983). A generalizaion of median filering using linear combinaions of order saisics. IEEE Transacions on Acousics, Seech and Signal Processing 3 (6): Box, G. E., Jenkins, G.., Reinsel, G. C., & Ljung, G.. (05). Time series analysis: forecasing and conrol. John Wiley & Sons. New York, USA. Conradie, W., De We, T. and Jankowiz,. D. (009). Performance of nonlinear smoohers in signal recovery. Alied Sochasic odels in Business and Indusry 5 (): 5-. Hird, J. N. and cdermid, G. J. (009). Noise reducion of NDVI ime series: An emirical comarison of seleced echniques. Remoe Sensing of Environmen 3 (): e Academia Secial Issue GraCe, 08 83

7 Jin, Z. and u, B. 03. A novel comound smooher REH o reconsruc ODIS NDVI ime series. IEEE Geoscience and Remoe Sensing Leers 0 (): Sargen, J. and Bedford, A. (00). Imroving Ausralian Fooball League layer erformance forecass using oimized nonlinear smoohing. Inernaional Journal of Forecasing 6 (3): Tukey, J. (977). Exloraory Daa Analysis. Addison-Wesley Publishing Comany, Readings. Velleman, P. F. (980). Definiion and comarison of robus nonlinear daa smoohing algorihms. Journal of he American Saisical Associaion 75 (37): Velleman, P. F. and Hoaglin, D. C. (98). Alicaions, basics, and comuing of exloraory daa analysis. Duxbury Press, Boson, assachuses. Yin, L., Yang, R., Gabbouj,. and Neuvo, Y Weighed median filers : a uorial. IEEE Transacions on Circuis and Sysems II: Analog and Digial Signal Processing 3 (3): e Academia Secial Issue GraCe, 08 8