Research Article Application of Empirical Mode...

Research ArticleApplication of Empirical Mode Decomposition with Local LinearQuantile Regression in Financial Time Series Forecasting

Abobaker M Jaber1 Mohd Tahir Ismail1 and Alsaidi M Altaher2

1 School of Mathematical Sciences Universiti Sains Malaysia 11800 Minden Penang Malaysia2 Statistics Department Sebha University Sebha 00218 Libya

Correspondence should be addressed to Abobaker M Jaber jaber3tyahoocouk

Received 15 January 2014 Revised 23 June 2014 Accepted 25 June 2014 Published 22 July 2014

Academic Editor Mohamed Hanafi

Copyright copy 2014 Abobaker M Jaber et al This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

This paper mainly forecasts the daily closing price of stock markets We propose a two-stage technique that combines the empiricalmode decomposition (EMD) with nonparametric methods of local linear quantile (LLQ) We use the proposed technique EMD-LLQ to forecast two stock index time series Detailed experiments are implemented for the proposedmethod in which EMD-LPQEMD andHolt-Wintermethods are comparedThe proposed EMD-LPQmodel is determined to be superior to the EMD andHolt-Winter methods in predicting the stock closing prices

1 Introduction

Recent studies have indicated that financial markets typi-cally follow nonlinear and nonstationary behaviorThereforeforecasting finance using classical techniques is quite diffi-cult The empirical mode decomposition (EMD) exploredby Huang et al [1] is a very powerful tool in modernquantitative finance and has emerged as a powerful statisticalmodeling technique [2 3]The capacity of the EMD to handlenonlinear and nonstationary behaviors has provided bothresearchers and practitioners with an attractive alternativetool The EMD can explain the generation of time series datafrom an alternative perspective by breaking up time seriessignals into smaller numbers of independent and concretelyimplicational intrinsic modes based on scale separationThis distinguishing feature makes the EMD a valuable anddesirable tool for forecasting financial time series signals [4]The current study aims to extract and forecast the trend oftwo stock markets namely the Kuala Lumpur Bursa (KLSE)index and the New Zealand stock market index (NZX50)using the advantages of local linear quantile (LLQ) regressionThe proposedmethod consists of two stages In the first stageLLQ is applied to corrupt and noisy data The remainingseries is subsequently expected to be hidden in the residualsIn the second stage EMD is applied to the residualsThe final

estimate is the summation of the fitting estimates from theLLQ and EMD To extract and forecast the trend usingEMD-LLQ and EMD we summarize the steps as follows(1) A signal is decomposed by the EMD-LLQ and EMD(2) Meaningful intrinsic mode functions (IMFs) (compo-nents) are selected using the fast Fourier transform (FFT)(see [5]) (3) Selected components are added to the residueto obtain the trend (4) The Holt-Winter method is basedon the selected components and provides the forecastingresults

The remainder of this paper is organized as followsSection 2 presents a brief background of the EMD andLLQ Section 3 introduces the proposed method Section 4compares the results of the original EMD algorithm and thenew proposed method by forecasting the daily closing pricesof two stock markets namely the KLSE and the NZX50Section 5 concludes

2 Why EMD-LPQ (Empirical ModeDecomposition Combinedwith Local Linear Quantile Regression)

Due to the edged effects nonparametric techniques such asempirical mode decomposition show a sharp increase in

Hindawi Publishing Corporatione Scientific World JournalVolume 2014 Article ID 708918 5 pageshttpdxdoiorg1011552014708918

2 The Scientific World Journal

variance and bias at points near the boundary The presenceof such problem has dramatic effects on results Varieties ofworks have been reported in literature in order to reducethe effects of boundary problem in traditional EMD Two-stagemethods or couplingmethods nowadays have beenusedextensively for solving such problem for instance [6] appliedneural network to each IMF to restrain the end effect [7]provided an algorithm based on the sigma-pi neural networkwhich is used to extend signals before applying EMD Refer-ence [4] proposed a new two-stage algorithmThe algorithmincludes two steps the extrapolation of the signal throughsupport vector (SV) regression at both endpoints to formthe primary expansion signal and then the primary signalis further expanded through extrema mirror expansion andEMD is performed on the resulting signal to obtain reducedend limitations All previous methods have been shown tohave good solutions to the end point and achieved a higherprecision in an application part as well In this paper we havefollowed [8] The proposed method EMD-LPQ (empiricalmode decomposition combined with local linear quantileregression) is designed to be a robust version of classicalempirical mode decomposition especially in presence of edgeeffect problems

3 Empirical Mode Decomposition

The EMD [1] has proven to be a natural extension of and analternative technique to traditional methods for analyz-ing nonlinear and nonstationary signals such as waveletmethods Fourier methods and empirical orthogonal func-tions [9] In this section we briefly describe the EMDalgorithm The EMD mainly decomposes the data 119910

119905into

smaller signals called IMFs An IMF is a function in whichthe upper and the lower envelopes are symmetric Moreoverthe number of zero-crossings and the numbers of extremesare equal or differ by one at the most [10] The algorithm forextracting IMFs for a given time series 119910

119905is called shifting

and it consists of the following steps

(i) The initial estimates for the residue are set at 1199030(119905) =

119910119905 1198920(119905) = 119903

119896minus1(119905) and 119894 = 1 and the IMF index is set

at 119896 = 1

(ii) The lower minima 119868min119894minus1 and the upper 119868max119894minus1 envel-opes of the signal are constructedusing the cubicspline method

(iii) The mean values 119898119894are computed by averaging the

upper and lower envelopes as 119898119894minus1

= [119868max119894minus1+119868min119894minus1]2

(iv) The mean is subtracted from the original signal thatis 119892119894= 119892119894minus1

minus 119898119894minus1

and 119894 = 119894 + 1 Steps (i) to (iv) arerepeated until119892

119894becomes an IMFHence the 119896th IMF

is given by IMF119870

= 119892119894

(v) The residue is updated via 119903119896(119905) = 119903

119896minus1(119899) minus IMF

119870

This residual component is treatedas new data andsubjected to the previously described process to cal-culate the next IMF

119870+1

(vi) The previous steps are repeated until the final residualcomponent 119903(119909) becomes amonotonic function Thefinal estimation of residue 119903(119909) is subsequently con-sidered

Several methods have been presented to extract trendsfrom a time series Freehand and least squares methods arethe commonly used techniques the former depends on theexperience of users whereas the latter is difficult to use whenthe original series is very irregular [11] The EMD is anothereffective method for extracting trends [6]

4 Local Linear Quantile (LLQ) Regression

The seminal study of Koenker and Bassett [12] introducedparametric quantile regression which is considered an alter-native to the classical regression in both parametric and non-parametric fields Numerous models for the nonparametricapproach have been introduced in statistical literature suchas the locally polynomial quantile regression by Chaudhuri[13] and the kernel methods by Koenker et al [14] In thispaper we adopt the LLQ regression employed byYu and Jones[15]

Let (119909119894 119910119894) 119894 = 1 119899 be bivariate observations To

estimate the 120591th conditional quantile function of response 119910the equation below is defined given 119883 = 119909

119892 (119909) = 119876119910(120591 | 119909) (1)

Let119870 be a positive symmetric unimodal kernel functionand consider the following weighted quantile regressionproblem

min120573isin1198942

119899

sum

119894=1

119908119894(119909) 120588120591(119910119894minus 1205730minus 1205731(119909119894minus 119909)) (2)

where 119908(119909) = 119896((119909119894minus 119909)ℎ)ℎ Once the covariate observa-

tions are centered at point the estimate of 119892(119909) is simply 1205730

which is the first component of the minimizer of (1) anddetermines the estimate of the slope of the function 119892 at point119909

5 Bandwidth Selection

The practical performance of 119876120572(119909) strongly depends on the

selected bandwidth parameter We adopt the strategy of Yuand Jones [15] In sum we employ the automatic bandwidthselection strategy for smoothing conditional quantiles asfollows

(1) We use ready-made and sophisticated methods inselecting ℎmean we employ [16] which explored

The Scientific World Journal 3

a ldquodirect pluginrdquo bandwidth selection procedurewhich relies on asymptotically optimal bandwidth

ℎmean = 1205902119877 (119870) (119887 minus 119886)

11989912058322int 11989810158401015840 (119909)

2

119901 (119909) 119889119909

15

= 1198621(119870)

1205902(119887 minus 119886)

11989912057922

15

(3)

where and for later use we have introduced the array

120579119903119904

= int119898119903(119909)119898119904(119909) 119901 (119909) 119889119909 119903 119904 ge 0 119903 + 119904 even (4)

Again for later use we write down the following estimatorwith a bandwidth 119892 for 120590

2

1205902

119892=

1

]

119899

sum

119894=1

119884119894minus 119892(119883119894)2

(5)

This is simply a normalized residual sum of squares andnormalizing quantity ] sometimes known as the degrees offreedom which is given by ] = 119899 minus 2sum

119894119908119909119894ℎ

(119883119894) +

sum119894ℎ

1199082119909119895ℎ

(119883119894)

Its presented guarantees 119864(1205902

119892| 1198831 119883

119899) = 1205902 where

119898(119909) is either constant or linear [17]

(2) We use ℎ120591

= ℎmean120591(1 minus 120591)120601(Φminus1

(120591))2

15

to obtainall of the other ℎ

120591119904 from ℎmean 120601 and Φ are standard

normal density and distribution functions and ℎmeanis a bandwidth parameter for regressionmean estima-tion with various existing methods This procedureobtains identical bandwidths for the 120591 and (1 minus 120591)

quantiles

6 Proposed Method

The proposed method consists of two stages that automati-cally decrease the boundary effects of EMD [8] At the firststage LLQ which is considered as an excellent boundarytreatment [18] is applied to the corrupted and noisy dataThe remaining series is then expected to be hidden in theresiduals At the second stage EMD is applied to the residualsThe final estimate is the summation of the fitting estimatesfrom LLQ and EMD Compared with EMD this combinationobtains more accurate estimates

This section elaborates the proposedmethod EMD-LLQThe basic idea behind the proposed method is to estimatethe underlying function 119891 with the sum of a set of EMDfunctions 119891EMD and an LLQ function 119891LLQ That is

119891LLQREMD = 119891EMD + 119891LPQR (6)

We estimate the two components 119891EMD and 119891LLQ to ob-tain our proposed estimate 119891EMDLLQ through the followingsteps

(1) The LLQ is applied to the corrupt and noisy data 119910119894

and the trend estimate 119891LLQ is subsequently obtained

(2) The residuals of 119890119894from LLQ that is 119890

119894= 119910119894minus 119891LLQ

are determined

(3) The EMD is applied to 119890119894 given that the remaining

series is expected to be hidden in the residuals Thisstep is accomplished by performing the followingsubsteps this substep is accomplished by performingalgorithms (i) to (vi)

(4) The final estimate is the summation of the fittingestimates from LLQ and EMD and is as follows

119891LLQEMD = 119891EMD + 119903 (119905) (7)

7 Experiment Analysis and Results

In this section we consider the daily closing prices of twostock markets namely KLSE and NZX50 from December3 2007 to December 6 2013 see Figure 1 The last 1030 and 50 days of the KLSE and NZX50 stock indicesare forecasted respectively based on the past sequencesThe selection of these two indices aims to qualitatively andculturally compare the time series of two different marketsThe data used in this study are collected from the websitehttpinfinanceyahoocom We analyze the two indicesbased on the EMD-LLQ and EMD in combination with theFFT and the Holt-Winter methods The approach consists ofseveral steps First we decompose the daily closing pricesof the stock markets into a finite number of componentscalled IMFs and one residue Second we select significantcomponents by applying the FFT to each IMF Third weadd the significant component obtained from step two to theresidue to acquire the trend Finally we employ the Holt-Winter method for forecasting the trend

8 Comparison of Forecasted Data

Two criteria are used to evaluate the forecasting performanceof the different models in empirical studies The forecastingaccuracy measures employed in this study are root meansquare error (RMSE) mean error (MAE) and mean absolutesquare error (MASE) The RMS MA and MASE valuesobtained through the EMD-LLQ EMD and Holt-Wintermethods in each test set for the two index series are sum-marized in Tables 1 and 2 The results demonstrate that theproposed EMD-LLQ method is more successful in all casesin forecasting the stock closing prices than the EMD and theHolt-Winter methods

9 Conclusion and Future Research

Wepropose an EMD-LLQmodel for forecasting future pricesby considering the past sequences of daily stock prices TheEMD-LLQ method is a new two-stage forecasting methodthat combines the EMDandLLQalgorithmThe effectiveness


KLSE closing price indexKL

SE

Index

1800

1600

1400

1200

1000

800

0 500 1000 1500

(a)

Index0 500 1000 1500

5000

4500

4000

3500

3000

2500

NZX50 closing price index

NZX

50

(b)

Figure 1 (a) and (b) KLSE price and NZX50 closing price index respectively

Table 1 Comparison of RMSE MA and MASE values for KLSEusing the Holt-Winter LLQ EMD and EMD-LLQ methods

nhead = 10 RMSE MAE MASEEMD-LLQ

120591(025)8224843 4688303 1330985

EMD-LLQ120591(050)

8222308 4687086 1294356EMD-LLQ

120591(075)8216983 4683845 1351209

EMD-LLQ120591(095)

8223858 4687594 1350173EMD 8243217 4701197 1352341Holt-Winters 8240651 473557 2180672

Table 2 Comparison of RMSE MA and MASE values for NZX50using the Holt-Winter LLQ EMD and EMD-LLQ methods


120591(025)2002658 1146388 141743

EMD-LLQ120591(050)

2003291 1146721 1409419EMD-LLQ

120591(075)2003833 1146989 1460694

EMD-LLQ120591(095)

2002921 1146404 1433273EMD 2005733 1147129 1548274Holt-Winters 2010189 1159261 2257123

of the new model is analyzed by performing experimentson KLSE and NZX50 data The capability of the EMD-LLQfor forecasting the future daily stock closing prices is betterthan that of LLQ EMD and the Holt-Winter methods Theresults demonstrate that EMD-LLQ is an effective methodfor forecasting financial time series Future research shouldimprove the performance of the proposed method Wecan consider several forecasting strategies by analyzing thecomposition of daily stock closing prices Another possi-ble strategy is to apply the Hilbert-Huang transform forselecting meaningful IMFs (components) instead of usingthe FFT

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

The authors would like to thank Universiti Sains Malaysia(USM Short Term Grant 304PMATHS6313045) for finan-cial support

References

[1] N E Huang Z Shen S R Long et al ldquoThe empirical modedecomposition and the Hilbert spectrum for nonlinear andnon-stationary time series analysisrdquo Proceedings of the RoyalSociety of London A Mathematical vol 454 pp 903ndash995 1971

[2] K Drakakis ldquoEmpirical mode decomposition of financial datardquoInternational Mathematical Forum vol 3 no 25 pp 1191ndash12022008

[3] N E Huang M Wu W Qu S R Long and S S P ShenldquoApplications of Hilbert-Huang transform to non-stationaryfinancial time series analysisrdquo Applied Stochastic Models inBusiness and Industry vol 19 no 3 pp 245ndash268 2003

[4] A Lin P Shang G Feng and B Zhong ldquoApplication of empir-ical mode decomposition combined with K-nearest neighborsapproach in financial time series forecastingrdquo Fluctuation andNoise Letters vol 11 no 2 Article ID 1250018 2012

[5] S Rezaei Physiological Synchrony as Manifested in DyadicInteractions University of Toronto 2013

[6] Y Deng W Wang C Qian Z Wang and D Dai ldquoBoundary-processing-technique in EMD method and Hilbert transformrdquoChinese Science Bulletin vol 46 no 11 pp 954ndash960 2001

[7] Z Liu ldquoA novel boundary extension approach for empiricalmode decompositionrdquo in Intelligent Computing pp 299ndash304Springer 2006

[8] A M Jaber M T Ismail and A M Altaher ldquoEmpirical modedecomposition combined with local linear quantile regressionfor automatic boundary correctionrdquoAbstract and Applied Anal-ysis vol 2014 Article ID 731827 8 pages 2014


[9] C D Blakely A Fast Empirical Mode Decomposition Techniquefor Nonstationary Nonlinear Time Series Elsevier Science Ams-terdam The Netherlands 2005

[10] A Amar and Z E A Guennoun ldquoContribution of wavelettransformation and empiricalmodedecomposition tomeasure-ment of US core inflationrdquo Applied Mathematical Sciences vol6 no 135 pp 6739ndash6752 2012

[11] J Fan and R Li ldquoStatistical challenges with high dimensional-ity feature selection in knowledge discoveryrdquo httparxivorgabsmath0602133

[12] R Koenker andG Bassett Jr ldquoRegression quantilesrdquo Economet-rica vol 46 no 1 pp 33ndash50 1978

[13] P Chaudhuri ldquoNonparametric estimates of regression quantilesand their local Bahadur representationrdquoTheAnnals of Statisticsvol 19 no 2 pp 760ndash777 1991

[14] R Koenker P Ng and S Portnoy ldquoQuantile smoothing splinesrdquoBiometrika vol 81 no 4 pp 673ndash680 1994

[15] K Yu andMC Jones ldquoLocal linear quantile regressionrdquo Journalof the American Statistical Association vol 93 no 441 pp 228ndash237 1998

[16] D Ruppert S J Sheather and M P Wand ldquoAn effective band-width selector for local least squares regressionrdquo Journal of theAmerican Statistical Association vol 90 no 432 pp 1257ndash12701995

[17] T C M Lee and V Solo ldquoBandwidth selection for local linearregression a simulation studyrdquo Computational Statistics vol 14no 4 pp 515ndash532 1999

[18] Z Cai and X Xu ldquoNonparametric quantile estimations fordynamic smooth coefficient modelsrdquo Journal of the AmericanStatistical Association vol 103 no 484 pp 1595ndash1608 2008

Submit your manuscripts athttpwwwhindawicom

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MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


variance and bias at points near the boundary The presenceof such problem has dramatic effects on results Varieties ofworks have been reported in literature in order to reducethe effects of boundary problem in traditional EMD Two-stagemethods or couplingmethods nowadays have beenusedextensively for solving such problem for instance [6] appliedneural network to each IMF to restrain the end effect [7]provided an algorithm based on the sigma-pi neural networkwhich is used to extend signals before applying EMD Refer-ence [4] proposed a new two-stage algorithmThe algorithmincludes two steps the extrapolation of the signal throughsupport vector (SV) regression at both endpoints to formthe primary expansion signal and then the primary signalis further expanded through extrema mirror expansion andEMD is performed on the resulting signal to obtain reducedend limitations All previous methods have been shown tohave good solutions to the end point and achieved a higherprecision in an application part as well In this paper we havefollowed [8] The proposed method EMD-LPQ (empiricalmode decomposition combined with local linear quantileregression) is designed to be a robust version of classicalempirical mode decomposition especially in presence of edgeeffect problems

3 Empirical Mode Decomposition

The EMD [1] has proven to be a natural extension of and analternative technique to traditional methods for analyz-ing nonlinear and nonstationary signals such as waveletmethods Fourier methods and empirical orthogonal func-tions [9] In this section we briefly describe the EMDalgorithm The EMD mainly decomposes the data 119910

119905into

smaller signals called IMFs An IMF is a function in whichthe upper and the lower envelopes are symmetric Moreoverthe number of zero-crossings and the numbers of extremesare equal or differ by one at the most [10] The algorithm forextracting IMFs for a given time series 119910

119905is called shifting

and it consists of the following steps

(i) The initial estimates for the residue are set at 1199030(119905) =

119910119905 1198920(119905) = 119903

119896minus1(119905) and 119894 = 1 and the IMF index is set

at 119896 = 1

(ii) The lower minima 119868min119894minus1 and the upper 119868max119894minus1 envel-opes of the signal are constructedusing the cubicspline method

(iii) The mean values 119898119894are computed by averaging the

upper and lower envelopes as 119898119894minus1

= [119868max119894minus1+119868min119894minus1]2

(iv) The mean is subtracted from the original signal thatis 119892119894= 119892119894minus1

minus 119898119894minus1

and 119894 = 119894 + 1 Steps (i) to (iv) arerepeated until119892

119894becomes an IMFHence the 119896th IMF

is given by IMF119870

= 119892119894

(v) The residue is updated via 119903119896(119905) = 119903

119896minus1(119899) minus IMF

119870

This residual component is treatedas new data andsubjected to the previously described process to cal-culate the next IMF

119870+1

(vi) The previous steps are repeated until the final residualcomponent 119903(119909) becomes amonotonic function Thefinal estimation of residue 119903(119909) is subsequently con-sidered

Several methods have been presented to extract trendsfrom a time series Freehand and least squares methods arethe commonly used techniques the former depends on theexperience of users whereas the latter is difficult to use whenthe original series is very irregular [11] The EMD is anothereffective method for extracting trends [6]

4 Local Linear Quantile (LLQ) Regression

The seminal study of Koenker and Bassett [12] introducedparametric quantile regression which is considered an alter-native to the classical regression in both parametric and non-parametric fields Numerous models for the nonparametricapproach have been introduced in statistical literature suchas the locally polynomial quantile regression by Chaudhuri[13] and the kernel methods by Koenker et al [14] In thispaper we adopt the LLQ regression employed byYu and Jones[15]

Let (119909119894 119910119894) 119894 = 1 119899 be bivariate observations To

estimate the 120591th conditional quantile function of response 119910the equation below is defined given 119883 = 119909

119892 (119909) = 119876119910(120591 | 119909) (1)

Let119870 be a positive symmetric unimodal kernel functionand consider the following weighted quantile regressionproblem

min120573isin1198942

119899

sum

119894=1

119908119894(119909) 120588120591(119910119894minus 1205730minus 1205731(119909119894minus 119909)) (2)

where 119908(119909) = 119896((119909119894minus 119909)ℎ)ℎ Once the covariate observa-

tions are centered at point the estimate of 119892(119909) is simply 1205730

which is the first component of the minimizer of (1) anddetermines the estimate of the slope of the function 119892 at point119909

5 Bandwidth Selection

The practical performance of 119876120572(119909) strongly depends on the

selected bandwidth parameter We adopt the strategy of Yuand Jones [15] In sum we employ the automatic bandwidthselection strategy for smoothing conditional quantiles asfollows

(1) We use ready-made and sophisticated methods inselecting ℎmean we employ [16] which explored



ℎmean = 1205902119877 (119870) (119887 minus 119886)

11989912058322int 11989810158401015840 (119909)

2

119901 (119909) 119889119909

15

= 1198621(119870)

1205902(119887 minus 119886)

11989912057922

15

(3)


120579119903119904

= int119898119903(119909)119898119904(119909) 119901 (119909) 119889119909 119903 119904 ge 0 119903 + 119904 even (4)


2

1205902

119892=

1

]

119899

sum

119894=1

119884119894minus 119892(119883119894)2

(5)


119894119908119909119894ℎ

(119883119894) +

sum119894ℎ

1199082119909119895ℎ

(119883119894)


119892| 1198831 119883

119899) = 1205902 where


(2) We use ℎ120591


(120591))2

15




quantiles

6 Proposed Method



119891LLQREMD = 119891EMD + 119891LPQR (6)





119894= 119910119894minus 119891LLQ

are determined




119891LLQEMD = 119891EMD + 119903 (119905) (7)









SE

Index

1800

1600

1400

1200

1000

800

0 500 1000 1500

(a)

Index0 500 1000 1500

5000

4500

4000

3500

3000

2500


NZX

50

(b)




120591(025)8224843 4688303 1330985

EMD-LLQ120591(050)

8222308 4687086 1294356EMD-LLQ

120591(075)8216983 4683845 1351209

EMD-LLQ120591(095)

8223858 4687594 1350173EMD 8243217 4701197 1352341Holt-Winters 8240651 473557 2180672



120591(025)2002658 1146388 141743

EMD-LLQ120591(050)

2003291 1146721 1409419EMD-LLQ

120591(075)2003833 1146989 1460694

EMD-LLQ120591(095)

2002921 1146404 1433273EMD 2005733 1147129 1548274Holt-Winters 2010189 1159261 2257123




Acknowledgment


References



























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











ℎmean = 1205902119877 (119870) (119887 minus 119886)

11989912058322int 11989810158401015840 (119909)

2

119901 (119909) 119889119909

15

= 1198621(119870)

1205902(119887 minus 119886)

11989912057922

15

(3)


120579119903119904

= int119898119903(119909)119898119904(119909) 119901 (119909) 119889119909 119903 119904 ge 0 119903 + 119904 even (4)


2

1205902

119892=

1

]

119899

sum

119894=1

119884119894minus 119892(119883119894)2

(5)


119894119908119909119894ℎ

(119883119894) +

sum119894ℎ

1199082119909119895ℎ

(119883119894)


119892| 1198831 119883

119899) = 1205902 where


(2) We use ℎ120591


(120591))2

15




quantiles

6 Proposed Method



119891LLQREMD = 119891EMD + 119891LPQR (6)





119894= 119910119894minus 119891LLQ

are determined




119891LLQEMD = 119891EMD + 119903 (119905) (7)









SE

Index

1800

1600

1400

1200

1000

800

0 500 1000 1500

(a)

Index0 500 1000 1500

5000

4500

4000

3500

3000

2500


NZX

50

(b)




120591(025)8224843 4688303 1330985

EMD-LLQ120591(050)

8222308 4687086 1294356EMD-LLQ

120591(075)8216983 4683845 1351209

EMD-LLQ120591(095)

8223858 4687594 1350173EMD 8243217 4701197 1352341Holt-Winters 8240651 473557 2180672



120591(025)2002658 1146388 141743

EMD-LLQ120591(050)

2003291 1146721 1409419EMD-LLQ

120591(075)2003833 1146989 1460694

EMD-LLQ120591(095)

2002921 1146404 1433273EMD 2005733 1147129 1548274Holt-Winters 2010189 1159261 2257123




Acknowledgment


References



























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











SE

Index

1800

1600

1400

1200

1000

800

0 500 1000 1500

(a)

Index0 500 1000 1500

5000

4500

4000

3500

3000

2500


NZX

50

(b)




120591(025)8224843 4688303 1330985

EMD-LLQ120591(050)

8222308 4687086 1294356EMD-LLQ

120591(075)8216983 4683845 1351209

EMD-LLQ120591(095)

8223858 4687594 1350173EMD 8243217 4701197 1352341Holt-Winters 8240651 473557 2180672



120591(025)2002658 1146388 141743

EMD-LLQ120591(050)

2003291 1146721 1409419EMD-LLQ

120591(075)2003833 1146989 1460694

EMD-LLQ120591(095)

2002921 1146404 1433273EMD 2005733 1147129 1548274Holt-Winters 2010189 1159261 2257123




Acknowledgment


References



























Volume 2014




Journal of











Journal of


Function Spaces






Algebra



























Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Research Article Application of Empirical Mode...

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