research article local prediction of chaotic time...

Research ArticleLocal Prediction of Chaotic Time Series Based on PolynomialCoefficient Autoregressive Model

Liyun Su and Chenlong Li

School of Mathematics and Statistics Chongqing University of Technology Chongqing 400054 China

Correspondence should be addressed to Liyun Su cloudhopping163com

Received 13 July 2014 Accepted 29 September 2014

Academic Editor Erol Egrioglu

Copyright copy 2015 L Su and C LiThis is an open access article distributed under theCreativeCommonsAttribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

We apply the polynomial function to approximate the functional coefficients of the state-dependent autoregressivemodel for chaotictime series predictionWe present a novel local nonlinearmodel called local polynomial coefficient autoregressive prediction (LPP)model based on the phase space reconstruction The LPP model can effectively fit nonlinear characteristics of chaotic time serieswith simple structure and have excellent one-step forecasting performance We have also proposed a kernel LPP (KLPP) modelwhich applies the kernel technique for the LPPmodel to obtain better multistep forecasting performanceThe proposedmodels areflexible to analyze complex and multivariate nonlinear structures Both simulated and real data examples are used for illustration

1 Introduction

Chaos is widely encountered in nature and many scientificareas since Lorenz found chaotic motion in his research onmeteorology [1] Over the latest decades researchers in manyfields have paidmuch attention to chaos [2ndash13] andmodelingand prediction of chaotic time series have become popular inthe areas of meteorology [1 2] medicine [3] economics [4]signal processing [5] traffic flow [6] power load [7] Sunspotprediction [8ndash11] and many others [12 13]

Although prediction of chaotic time series is very diffi-cult the chaos theory [14] provides a useful tool to predictchaotic time series With the development of chaos theoryand research on its application technique the global [15]and local [16] methods are proposed as two main categoriesThe global method makes an attempt to approximate thewhole time series on all attractors and seeks a functionwhich is valid at every point Then we can get the futurevalues from knowledge of all the previous elements of thetime series However the parameters may be changed whennew information is added into the model The local methodonly use part of the past information to approximate thelocal attractor The future values can be inferred by using

the neighborhoods of the current points Many predictionmethods have gained popularity in practice during the lastdecades based on two main categories Such as adaptiveprediction [17ndash19] the support vector machine (SVM) [20ndash27] polynomial estimation [28ndash30] and neural network [8ndash11 31ndash36]

Recently some researchers studies have shown that localmethods can obtain generally better results than thoseobtained with global methods [18] And some researchersfound that the forecast accuracy can be improved by usingsome combining techniques both in the global and localmethod By using those combining techniques the param-eters can be obtained faster and the analysis of residualcan be underestimated if residuals are not randomnessSuch as SVM combined with neurofuzzy model [21] SVMcombined with PCA [26] ARMA combined with RESN [8]and neural network combined with neurofuzzy model [34]Those methods can obtain generally better results than thoseobtained with single model but they are complex affected bypersonal experience and easy to be overfitted

To overcome these disadvantages we propose to approx-imate the local attractor by using local polynomial coeffi-cient autoregressive prediction (LPP) model The functional

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2015 Article ID 901807 14 pageshttpdxdoiorg1011552015901807

2 Mathematical Problems in Engineering

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Real valueLPP value

KLPP value

x(t)

(a)

Time0 50 100 150 200 250 300 350 400 450 500

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minus15

minus1

minus05

times10minus4

(b)

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00102030405060708

Real valueLPP value

KLPP value

x(t)

(c)

Time0 50 100 150 200 250 300

0

02

Erro

r

LPP errorKLPP error

minus08

minus06

minus04

minus02

(d)

Figure 1 Results of the Lorenz time series (a) and (b) are one-step prediction value and error respectively (c) and (d) aremultistep predictionvalue and error respectively

coefficients of the state-dependent autoregressive model caneffectively fit nonlinear characteristics of nonlinear timeseries By using the polynomial function to approximate thefunctional coefficients we can use LPPmodel to approximatethe local attractor The local attractor can be obtained byutilizing Takens embedding theorem [10] The kernel predic-tion can improve the forecast accuracy [37 38] We combinethe LPP with the kernel technique to develop a kernel LPP(KLPP) model to improve the multistep forecast accuracyThe KLPP model can be applied to choose proper radiusof the nearest neighbors by using the kernel function Inthis work we compare our models with those reported inthe literature to illustrate the effectiveness of our models inpredicting chaotic time series

The paper is organized as follows In Section 2 theconcept of the state-dependent autoregressivemodel the LPPmodel and the KLPP model are proposed and the optimalparameter set is established by using GDF which is a tool toevaluate the goodness of the model with the chosen numberof neighbors [12] In Section 3 the prediction errors of thegenerated time series are computed to analyze the predic-tion effectiveness Section 3 uses three simulated chaoticsystems and one real life time series as examples to evaluate

the proposed modelsThe conclusion of this paper is given inSection 4

2 The LPP Model and KLPP Model

21 Data Structure in Phase Space Reconstruction Supposethat we have a scalar chaotic time series 119909

1 1199092 119909

119899 The

first step of a local prediction is the phase space reconstruc-tion According to Takens embedding theory the phase spacecan be reconstructed and the construct phase points 119883

119905=

(119909119905 119909119905+120591 119909

119905+(119898minus1)120591)119879 where 119905 = 1 2 119872 and 119872 =

119899minus(119898minus1)120591The embedding dimension119898 and the time delay120591 can be obtained by the Cao method [39]Then a continuedvector mapping 119865 R119898 rarr Rm or 119891 R119898 rarr R can be usedto describe the unknown evolution from 119883

119905to 119883119905+1

That is119883119905+1= 119865(119883

119905) and 119909

119905+1= 119891(119909

119905)

22 The LPP Model The state-dependent autoregressivemodel can be described as follows

119909119905= 1198920(119883119905minus1) +

119898

sum

119895=1

119892119895(119883119905minus1) 119909119905minus119895 (1)

Mathematical Problems in Engineering 3

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(a)

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KLPPLPP

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Time0 50 100 150 200 250 300 350 400 450 500

0005

01015

02025

03035

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KLPP

(d)

Figure 2 Results of the Lorenz time series one-step prediction (a) is the polynomial order (b) is the delay of the lag variable (c) is thenumber of nearest neighbor points (d) is the radius of the neighbors

where119883119905minus1= (119909119905minus1 119909119905minus2 119909

119905minus119898) and functional coefficient

119892119895(sdot) 119895 = 0 1 119898 is a continued vector mapping 119892

119895

R119898 rarr ROn main purpose of this work is to estimate the mapping

119891 by using this model The state-dependent autoregressivemodel in an 119898-dimensional reconstructed phase space canbe given as follows

119909119905+1= 1198920(119883119905) +

119898

sum

119895=1

119892119895(119883119905) 119909119905minus(119895minus1)120591

(2)

where 119883119905= (119909

119905 119909119905minus120591 119909

119905minus(119898minus1)120591) 119898 is the embedding

dimension and 120591 is the time delayWe approximate 119892

119895(119883119905) by a polynomial function

119892119895(119883119905) = 119886119895(119906) = sum

119903

119894=0119886119895119894119906119894 where the 119906 as the lag 119889(isin (120591 2120591

(119898minus1)120591)) variable 119909119905minus119889

and 119895 = 0 1 119898Then the LPPmodel in an 119898-dimensional phase space can be described asfollows

119909119905+1= 1198860(119906) +

119898

sum

119895=1

119886119895(119906) 119909119905minus(119895minus1)120591

(3)

where 119886119895(119906) = sum

119903

119894=0119886119895119894119906119894 the 119906 is the lag 119889(isin (120591 2120591 (119898 minus

1)120591)) variable 119909119905minus119889

and 119903 is the length of the polynomialfunction Then we have

119909119905+1=

119903

sum

119894=0

1198860119894(119909119905minus119889)119894

+

119898

sum

119895=1

119903

sum

119894=0

119886119895119894(119909119905minus119889)119894

119909119905minus(119895minus1)119903

(4)

The classical state-dependent autoregressivemodel has short-comings such as low accuracy and low processing speedand it is affected by personal experience The proposedmethod use the polynomial function to approximate thefunctional coefficients of the state-dependent autoregressivemodel which makes it characterized by parameter model toreduce personal experience and improves processing speedAlso the proposedmethod is combinedwith the chaos theorywhich makes it obtain high prediction accuracy

221 The LPP Estimation and Determination of the OptimalParameters (119902119903) In order to obtain LPP model as theapproximation of themapping119891 at the current state point119883

119872

in the reconstructed phase space we select 119902 nearest neighborpoints 119883

119872119895(119895 = 1 2 119902) by using the Euclidean distances

119889(119894) = 119883119894minus1198831198722(119894 = 1 2 119872minus1) and fit a LPPmodel to


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(d)

Figure 3 Results of the Lorenz time series multistep prediction (a) is the polynomial order (b) is the delay of the lag variable (c) is thenumber of nearest neighbor points (d) is the radius of the neighbors

predict the future point 119909119872119895+(119898minus1)120591+1

of 119883119872119895(119895 = 1 2 119902)

We can estimate the parameters 119886119895119894through the least square

equation

min119886119895119894isinR

119902

sum

119905=1

[

[

119909119872119905+(119898minus1)120591+1

minus

119898

sum

119895=0

119903

sum

119894=0

119886119895119894119909119894

119872119905+(119898minus1)120591minus119889119909119872119905+(119895minus1)120591

]

]

2

(5)

where 119895 = 0 1 119898 and 119894 = 0 1 119903Define 120579 equiv (119886

00 119886

0119903 119886

1198980 119886

119898119903)119879

(119898+1)(119903+1)times1and

set

120594 =

[

[

[

[

[

11988011199091198721minus120591

11988011199091198721

sdot sdot sdot 11988011199091198721+(119898minus1)120591

11988021199091198722minus120591

11988021199091198722


d

119880119902119909119872119902minus120591

119880119902119909119872119902


]

]

]

]

]119902times(119898+1)(119903+1)

119884 = (1199091198721+(119898minus1)120591+1

119909119872119902+(119898minus1)120591+1

)

119879

119880119905= (1 119909

119872119905+(119898minus1)120591minus119889 119909

119903

119872119905+(119898minus1)120591minus119889)

119909119872119905minus120591

equiv 1 (119905 = 1 2 119902)

(6)

It follows from least squares theory that

120579 = 120594

119879120594

minus1

120594119879119884 (7)

By using (4) the prediction value 119909119872+(119898minus1)120591+1

can be calcu-lated We add 119909

119872+(119898minus1)120591+1to the training set and utilize the

prediction point 119883119872+1

= (119909119872+1

119909119872+1+120591

119909119872+1+(119898minus1)120591

)119879

as the last phase point 119883119872+1

Then we can compute themultistep prediction 119909

119872+(119898minus1)120591+2by the same scheme in the

forecast of 119909119872+(119898minus1)120591+1

By using a concept of generalized degrees of freedom

(GDF) [12 13] the error variance can be measured And theoptimal parameters of the prediction model can be obtainedby comparing the calculated GDF with different parameters


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minus3

minus2

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times10minus4

(b)

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KLPP value

x(t)

(c)

Time0 50 100 150 200 250 300 350 400 450 500

0

0005

001

0015

Erro

r

LPP errorKLPP error

minus001

minus0005

(d)

Figure 4 Results of Mackey-Glass time series (a) and (b) are one-step prediction value and error respectively (c) and (d) are multistepprediction value and error respectively

The unbiased estimators 2GDF can be obtained in every stepof the prediction with different parameters as follows

2

GDF =(119884 minus ) (119884 minus )

119879

119902 minus 119863

(8)

where = 120594120579 = 120594120594

119879120594minus1120594119879119884 119863 = tr119867 =

tr120594120594119879120594minus1120594119879 and 119902 is the number of nearest neighborpoints

23 The KLPP Model We combine the LPP with the kerneltechnique to develop a kernel LPP (KLPP) model The KLPPmodel can be applied to choose proper radius of the nearestneighbors by using the kernel function We suppose that thespatial correlation between phase points could be measuredby the kernel function For the choice of the kernel functionwe adopt the Epanechnikov kernel It is a truncated functionand is defined as follows

119870(119911119905) =

3

4

(1 minus 1199112

119905)+ (9)

where 119911119905equiv 119889(119905) minus min119889(1) 119889(2) 119889(119902) (or 119911

119905equiv 119889(119905))

Epanechnikov kernel is the optimal kernel for the kernel

density estimation [40] and the local polynomial estimation[41] which minimizes the mean square error Also it is atruncated function which will help choose the neighborhoodradius So we adopt the Epanechnikov kernel for predictionalthough there are other kernel functions such as normalfunction

The estimators 119886119895119894is theminimizer of the sumofweighted

squares

119902

sum

119905=1

[

[

119909119872119905+(119898minus1)120591+1

minus

119898

sum

119895=0

119903

sum

119894=0

119886119895119894119909119894


]

]

2

119870ℎ(119911119905)

(10)

where 119895 = 0 1 119898 119894 = 0 1 119903 and 119870ℎ(119911119905) equiv

(1ℎ)119870(119911119905ℎ) The parameter ℎ is used to control the bound-

ary of the nonnegative kernel function which is introducedto choose radius of the nearest neighbors and emphasizeneighboring observations around119883

119872when estimating 119886

119895119894 It

is clear to see that the prediction accuracy of KLPP modelis sensitive to the radius of the neighbors In this study


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LPPKLPP

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0 50 100 150 200 250 300 350 400 450 500005

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KLPP

(d)

Figure 5 Results of Mackey-Glass time series one-step prediction (a) is the polynomial order (b) is the delay of the lag variable (c) is thenumber of nearest neighbor points (d) is the radius of the neighbors

we choose the parameter ℎ in terms of GDF and the optimalvalue of ℎ is the minimizer of the estimators 2GDF

Thus the estimators 119886119895119894can be obtained from least squares

theory that

120579 = 120594

119879119882120594

minus1

120594119879119882119884 (11)

where 119884 = (1199091198721+(119898minus1)120591+1

119909119872119902+(119898minus1)120591+1

)119879 and 119882 is a

119902 times 119902 diagonal matrix with 119870ℎ(119911119894) equiv (1ℎ)119870(119911

119894ℎ) as its

119894th diagonal element 120594 is a 119902 times [(119898 + 1)(119903 + 1)] matrixwith [119880

119894119909119872119894minus120591

119880119894119909119872119894+(119898minus1)120591

] as its 119894th row and 119880119894=

[1 119909119872119894+(119898minus1)120591minus119889

119909119903

119872119894+(119898minus1)120591minus119889] And the GDF method

chooses ℎ to minimize

2

GDF (ℎ) =(119884 minus ) (119884 minus )

119879

119902 minus 119863

(12)

where = 120594120579 = 120594120594

119879119882120594minus1120594119879119882119884 119863 = tr119867 =

tr120594120594119879119882120594minus1120594119879119882 and 119902 is the number of nearest neighborpoints with the Euclidean distance 119889(119894) = 119883

119894minus 119883119872 (119894 =

1 2 119872 minus 1) less than ℎNow we outline the algorithm based on the basic idea of

our models

Step 1 For a scalar time series 1199091 1199092 119909

119899 the phase space

reconstruct with the embedding dimension 119898 and the timedelay 120591 by the autocorrelation function method and the Caomethod that is119883

119905= (119909119905 119909119905+120591 119909

119905+(119898minus1)120591)119879

Step 2 Compute the Euclidian distances 119889(119894) = 119883119894minus

1198831198722(119894 = 1 2 119872minus1) and select the Epanechnikov kernel

as the spatial correlation between phase points Then thekernel coefficient can be calculated by (9)

Step 3 Identify the parameters 119886119895119894from (11)

Step 4 Calculate the 2GDF in every step of the predictionwith different parameters ℎ 119903 and 119886

119895119894and select the optimal

parameters that make 2GDF get minimum

Step 5 Fit the KLPP with the optimal parameters andcalculate the prediction value 119909

119872+(119898minus1)120591+1

Some additional remarks are now in order(i)With the Epanechnikov kernel let 119896 be from 1 to 15 and

ℎ119896= ℎmin+((ℎmaxminusℎmin)14)times(119896minus1) in Step 4 where ℎmin is

the Euclidian distance between the 2(119898 + 1)(119903 + 1)th nearestneighbor and the reference point and ℎmax is the Euclidian


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Figure 6 Results of Mackey-Glass time series multistep prediction (a) is the polynomial order (b) is the delay of the lag variable (c) is thenumber of nearest neighbor points (d) is the radius of the neighbors

distance between the [3(119898+1)(119903+1)+10]th nearest neighborand the reference point

(ii) We acquire one-step and multistep prediction valuesby using real value and prediction value to the training set asthe prediction steps developed respectively

(iii) To speed up the computation and avoid overfittingwe may let 119903 le 5

(iv) For the algorithm of LPP model we calculate theparameters 119886

119895119894from (7) in Step 3

3 Numerical Experiments andPerformance Evaluation

Here we consider three simulated chaotic systems LorenzMackey-Glass and Henon and one real life time seriesSunspot time series as examples to evaluate the proposedmodels Later the results of proposed models are comparedwith the results reported in the literature for the above exam-ples For themultistep prediction of chaotic time series it willproduce data overflow when order of polynomial function islarger and normalization can avoid this phenomenon Thus

those chaotic time series are selected and scaled between[0 1] as follows

119909minusnew (119905) = 119909 (119905) minusmin (119909)

max (119909) minusmin (119909) (13)

The prediction errors of the generated time series arecomputed to analyze the prediction effectiveness and com-pare the presented models with the results in the literaturewhich are maximum absolute error (MAE) root meansquared error (RMSE) and normalized mean squared error(NMSE) Namely

MAE = max 1003816100381610038161003816119910 (119905) minus 119910 (119905)

1003816100381610038161003816

RMSE = radic 1119899

119899

sum

119905=1

(119910 (119905) minus 119910 (119905))2

NMSE =sum119899

119905=1(119910 (119905) minus 119910 (119905))

2

sum119899

119905=1(119910 (119905) minus 119910)

2

(14)


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minus1

minus15

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KLPP value

(a)

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times10minus5

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(b)

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KLPP value

(c)

Time0 5 10 15 20 25 30 35 40 45 50

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minus2

minus1

LPP errorKLPP error

(d)

Figure 7 Results of Henon time series (a) and (b) are one-step prediction value and error respectively (c) and (d) are multistep predictionvalue and error respectively

31 Performance Analysis in Lorenz Equations The Lorenztime series can be produced as follows [1]

= 120590 (119910 minus 119909)

119910 = (119903 minus 119911) 119909 minus 119910

= 119909119910 minus 119887119911

(15)

where 120590 119903 and 119887 are dimensionless parameters and mostcommonly selected to be 120590 = 10 119903 = 28 and 119887 = 83The standard fourth-order Runge-Kutta method is used toget the Lorenz time series and the 119909-coordinate is used forprediction A time series with a length of 2000 is randomlygenerated 1800 samples are used for training and the restare treated as testing The results of one-step and multisteppredictions for Lorenz time series are shown in Figure 1 inwhich we use LPP model and KLPP model

From Figure 1 it can be seen that for the one-stepprediction both the prediction values of LPP and KLPPmodels have small error values For the multistep predictionwe find out that KLPP model has smaller error values

than LPP model and the values of prediction are in goodagreement with the real time series

The results of prediction are shown in Figures 2 and 3From Figure 2 we can see that the LPP model parametersare in good agreement with the KLPPmodel parametersThepolynomial ordersmainly take one or two which avoids over-fitting The lag variables are selected from 119909

119905minus120591to 119909119905minus(119898minus1)120591

atdifferent phase pointsThenumber of nearest neighbor pointsof both LPP model and KLPP model is chosen in the vicinityof 50 respectively The radius of the neighbors changes from005 to 03 From Figures 1 and 2 for one-step prediction wecan see that both LPP model and KLPP model have similarprediction performance For the multistep prediction theprediction values of LPP start diverging significantly from the120th time step and the error values are larger than KLPPmodel In Figure 3 it can be seen that the parameters almostdo not change in account with the parameters in Figure 2This implies that the kernel can improve the performance ofthe multistep prediction of chaotic time series

32 Performance Analysis in Mackey-Glass Equations TheMackey-Glass model has been used in literature as abenchmark model due to its chaotic characteristics [42]


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Figure 8 Results of Henon time series one-step prediction (a) is the polynomial order (b) is the delay of the lag variable (c) is the numberof nearest neighbor points (d) is the radius of the neighbors

Mackey-Glass time series is generated by the following dis-crete form

119909 (119905 + 1) = (1 minus 119887) 119909 (119905) +

119886119909 (119905 minus 120591)

1 + 119909 (119905 minus 120591)10 (16)

where 119886 = 02 119887 = 01 and 120591 = 17 and initial conditions119909(119905)|119905le0

= 12 Thus we can obtain a scalar chaotic timeseries samples set with length of 2300 from (16) A segment of1800 samples is used for training while the remaining part istreated as testing data Figure 3 compares the real value withprediction value for the rest samples for testing

From Figure 4 we can see that both the one-step predic-tion and the multistep prediction values of LPP and KLPPmodels have small error values And the multistep predictionvalues come up with the real time series with a length of 500The multistep prediction values are of more accuracy thanLorenz time series which means the proposed models havedifferent performances in different chaotic time series

From Figures 5 and 6 we can see that the parameters arealmost the same The polynomial orders mainly take one ortwo The number of nearest neighbor points is chosen in thevicinity of 50 and the radius of the neighbors changes from005 to 03

33 Performance Analysis in Henon Equations Henon map-ping is an evident dynamic system [43] Its equations arewritten by

119909 (119905 + 1) = 1 minus 119886119909 (119905)2+ 119910 (119905)

119910 (119905 + 1) = 119887119909 (119905)

(17)

where 119886 = 14 and 119887 = 03 The 119909 values with a length of1800 are generated to reconstruct the state space as trainingdata and 500 samples are used for testing Results are shownin Figures 7 8 and 9

From Figure 7 we can see that the one-step predictionvalues of LPP and KLPP models have small error valuesThe multistep prediction values of LPP and KLPP modelsstart diverging significantly from the 25th time step and 35thstep respectively From Figures 8 and 9 we can see that thepolynomial orders mainly take 3 4 and 5 the number ofnearest neighbor points is chosen from 50 to 100This impliesthat the proposed models are overfitting

34 Performance Analysis in Sunspot Time Series TheSunspot time series is a good indication of solar activity forsolar cycles The monthly smoothed Sunspot time series has


0 5 10 15 20 25 30 35 40 45 500

1

2

3

4

5

Time

The p

olyn

omia

l ord

er

LPPKLPP

(a)

Time0 5 10 15 20 25 30 35 40 45 50

1

2

3

4

5

6

Dela

y of

the l

ag v

aria

ble

LPPKLPP

(b)

Time0 5 10 15 20 25 30 35 40 45 50

020406080

100120140

Num

ber o

f nea

rest

neig

hbor

poi

nts

LPPKLPP

(c)

Time0 5 10 15 20 25 30 35 40 45 50

02

04

06

08

1

12

Radi

us o

f the

nei

ghbo

rs

KLPP

(d)

Figure 9 Results of Henon time series multistep prediction (a) is the polynomial order (b) is the delay of the lag variable (c) is the numberof nearest neighbor points (d) is the radius of the neighbors

been obtained from the SIDC (World Data Center for theSunspot Index) To compare the results with different modelsin the literature data are selected in the same conditionsreported by [9 10] Sunspot series from November 1834 toJune 2001 (2000 points) are selected and scaled between [0 1]The first 1000 samples of time series are selected to train theprediction models and the remainder 1000 samples are keptto test the prediction models

From Figure 10 it can be seen that for the one-stepprediction both prediction values of LPP and KLPP modelshave accuracy prediction For the multistep prediction wefind out that KLPP model has smaller error values than LPPmodel and the values of prediction are in good agreementwith the real time series The prediction values of LPP andKLPP models start diverging significantly from the 10th and320th time step respectively This implies that the kernelcan improve the performance of the multistep prediction ofchaotic time series

From Figures 11 and 12 we can see that the polynomialorders mainly take one or two The number of nearestneighbor points of both LPP and KLPP models is chosen inthe vicinity of 20 respectively The radius of the neighborschanges from 005 to 03 except for a few phase points Forone-step prediction the LPP and KLPP models have better

Table 1 The comparative results of different models of Lorenz timeseries

Prediction model RMSE MAE NMSERBF-OLS (2006) [34] 141119864 minus 09

ERNN (2007) [9] 879119864 minus 06 990119864 minus 10

PSO-LSSVM (2009) [25] 180119864 minus 04

FBMNN (2009) [10] 205119864 minus 05

HE-NARXNN (2010) [11] 108119864 minus 04 198119864 minus 10

IEOP (2010) [33] 105119864 minus 05

CSAA-SVR (2012) [23] 876119864 minus 04

COA-SVR (2012) [24] 303119864 minus 03

ESN (2012) [35] 250119864 minus 03

HR (2012) [35] 150119864 minus 03

WESN (2012) [31] 958119864 minus 04

ARMA-RESN (2013) [8] 303119864 minus 06

RBFNN (2013) [10] 500119864 minus 03

LNF (2013) [21] 640119864 minus 06

IEC-LSSVM (2014) [20] 118119864 minus 07 441119864 minus 07

The proposed LPP 203119864 minus 05 104119864 minus 04 105119864 minus 08

The proposed KLPP 203119864 minus 05 117119864 minus 04 104119864 minus 08


0 100 200 300 400 500 600 700 800 900 10000

05

1

15

2

Time

x(t)

Real valueLPP value

KLPP value

(a)

Time0 100 200 300 400 500 600 700 800 900 1000

0020406

Erro

r

minus02

minus04

minus06

minus08

LPP errorKLPP error

(b)

Time0 50 100 150 200 250 300 350

002040608

1

minus02

minus04

x(t)

Real valueLPP value

KLPP value

(c)

Time0 50 100 150 200 250 300 350

0

05

1

15

Erro

r

minus05

LPP errorKLPP error

(d)

Figure 10 Results of Sunspot time series (a) and (b) are one-step prediction value and error respectively (c) and (d) aremultistep predictionvalue and error respectively

Table 2 The comparative results of different models of Mackey-Glass time series


ERNN (2007) [9] 420119864 minus 05 315119864 minus 08

PSO-LSSVM (2009) [25] 280119864 minus 03

FBMNN (2009) [10] 302119864 minus 04


IEOP (2010) [33] 194119864 minus 05

ODCF-SVM (2012) [22] 160119864 minus 02

LNF (2013) [21] 790119864 minus 04

LSSVR (2013) [27] 570119864 minus 03




prediction performance for Sunspot time series And for themultistep prediction of Sunspot time series the KLPP modelhas more accuracy prediction than LPP model

35 Results and Discussion We compare the proposed mod-els with some of the models reported in the literature

Table 3The comparative results of different models of Henon timeseries

Prediction model RMSE MAE NMSELSSVM (2005) [10] 440119864 minus 03

RBF-OLS (2009) [10] 413119864 minus 02

FBMNN (2009) [10] 270119864 minus 05

Incremental Algorithm(2012) [36] 785119864 minus 02

K2 Algorithm (2012)[36] 114119864 minus 02



and the results are shown in Tables 1ndash4 In Table 1 we can seethat the proposed models in this paper are better than someof the existing methods But the best results are of ERNN[9] ARMA-RESN [8] and IEC-LSSVM [20] This is becausethese methods have optimized the values for the embeddingdimensions for the phase space reconstruction They alsohave the advantage of the architectural properties of differ-ent models in residual analysis or some models combinedwith different models which further improve the results


0 100 200 300 400 500 600 700 800 900 10000

1

2

3

4

5

Time

The p

olyn

omia

l ord

er

LPPKLPP

(a)

Time0 100 200 300 400 500 600 700 800 900 1000

115

225

335

445

5

Dela

y of

the l

ag v

aria

ble

LPPKLPP

(b)

Time100 200 300 400 500 600 700 800 900 1000

020406080

100120140

Num

ber o

f nea

rest

neig

hbor

poi

nts

LPPKLPP

(c)

Time0 100 200 300 400 500 600 700 800 900 1000

002040608

11214

Radi

us o

f the

nei

ghbo

rs

KLPP

(d)

Figure 11 Results of Sunspot time series one-step prediction (a) is the polynomial order (b) is the delay of the lag variable (c) is the numberof nearest neighbor points (d) is the radius of the neighbors

Table 4The comparative results of differentmodels of Sunspot timeseries

Prediction model RMSE MAE NMSEERNN (2007) [9] 129119864 minus 02 280119864 minus 03

RBF-OLS (2009) [10] 460119864 minus 02


ARMA-RESN (2013) [8] 157937

The proposed LPP 270119864 minus 02(393) 048 640119864 minus 03

The proposed KLPP 330119864 minus 02(480) 052 950119864 minus 03

These are also seen for the rest of time series in Tables 2ndash4 And the data cannot be selected in the same conditionsreported in the literature thus the conclusions from Tables1ndash4 have a few mistakes

Table 2 presents that the proposed models are better thanmost of the existingmethods except for the IEC-LSSVM [20]In Table 3 the proposed models have the best results For thereal-world time series in Table 4 we can see that the proposedmodels are better than some of the existing methods whichis similar to the best results reported in the literature

4 Conclusions

In this paper we apply the polynomial function to approx-imate the functional coefficients of the state-dependentautoregressive model Based on the phase space reconstruc-tion we present a novel local nonlinear model called LPPmodel for chaotic time series prediction The LPP model caneffectively fit nonlinear characteristics of chaotic time serieswith simple structure and have better one-step forecastingperformance But we find out that the LPP model havebad multistep forecasting performance Then we propose akernel LPPmodel which applies the kernel technique for theLPP model so that it may have better multistep forecastingperformance than LPP model

The LPP and KLPP models are simple and are notaffected by personal experience Simulated and real dataexamples illustrate that the proposed models are flexibleto analyze complex and multivariate nonlinear structuresThe numerical experiments show the KLPP model has moreaccuracy prediction than LPPmodel formultistep predictionWe compare LPP andKLPPmodelswith differentmodels andthe results show that our models are feasible


0 50 100 1500

1

2

3

4

5

Time

The p

olyn

omia

l ord

er

LPPKLPP

(a)

Time0 50 100 150

115

225

335

445

5

Dela

y of

the l

ag v

aria

ble

LPPKLPP

(b)

Time20 40 60 80 100 120 140

020406080

100120140

Num

ber o

f nea

rest

neig

hbor

poi

nts

LPPKLPP

(c)

Time0 50 100 150 200 250 300 350

001020304050607

Radi

us o

f the

nei

ghbo

rs

KLPP

(d)

Figure 12 Results of Sunspot time series multistep prediction (a) is the polynomial order (b) is the delay of the lag variable (c) is the numberof nearest neighbor points (d) is the radius of the neighbors

Conflict of Interests

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

Acknowledgments

The project was supported by Natural Science Founda-tion Project of China (Grant no 11471060) Fundamen-tal and Advanced Research Project of CQ CSTC ofChina (Grant no cstc2014jcyjA40003) and Natural ScienceFoundation Project of CQ CSTC of China (Grant noCSTC2012jjA00037)

References

[1] E N Lorenz ldquoDeterministic non-periodic flowrdquo Journal of At-mospheric Sciences vol 20 no 2 pp 130ndash141 1963

[2] E N Lorenz The Essence of Chaos University of WashingtonPress Seattle Wash USA 1993

[3] H Adeli S Ghosh-Dastidar and N Dadmehr ldquoA spatio-temporal wavelet-chaos methodology for EEG-based diagnosisof Alzheimerrsquos diseaserdquoNeuroscience Letters vol 444 no 2 pp190ndash194 2008

[4] A Das and P Das ldquoChaotic analysis of the foreign exchangeratesrdquoAppliedMathematics and Computation vol 185 no 1 pp388ndash396 2007

[5] M B Kennel and S Isabelle ldquoMethod to distinguish possiblechaos from colored noise and to determine embedding param-etersrdquo Physical Review A vol 46 no 6 pp 3111ndash3118 1992

[6] Y-M Zhang X-J Wu and S-L Bai ldquoChaotic characteristicanalysis for traffic flow series and DFPSOVF predictionmodelrdquoActa Physica Sinica vol 62 no 19 Article ID 190509 2013

[7] G Y Zhang Y GWu Y Zhang et al ldquoA simple model for probabilistic inter val forecasts of wind power chaotic time seriesrdquoActa Physica Sinica in China vol 63 no 13 Article ID 1388012014

[8] M Han andM-L Xu ldquoA hybrid predictionmodel of multivari-ate chaotic time series based on error correctionrdquo Acta PhysicaSinica vol 62 no 12 Article ID 120510 7 pages 2013

[9] Q-L Ma Q-L Zheng H Peng T-W Zhong and L-Q XuldquoChaotic time series prediction based on evolving recurrentneural networksrdquo in Proceedings of the 6th International Con-ference on Machine Learning and Cybernetics (ICMLC rsquo07) vol6 pp 3496ndash3500 IEEE August 2007

[10] Q-L Ma Q-L Zheng H Peng and J-W Qin ldquoChaotic timeseries prediction based on fuzzy boundary modular neuralnetworksrdquoActa Physica Sinica vol 58 no 3 pp 1410ndash1419 2009


[11] M Ardalani-Farsa and S Zolfaghari ldquoChaotic time seriesprediction with residual analysis method using hybrid Elman-NARX neural networksrdquoNeurocomputing vol 73 no 13ndash15 pp2540ndash2553 2010

[12] A W Jayawardena W K Li and P Xu ldquoNeighbourhoodselection for local modelling and prediction of hydrologicaltime seriesrdquo Journal of Hydrology vol 258 no 1ndash4 pp 40ndash572002

[13] D She and X Yang ldquoA new adaptive local linear predictionmethod and its application in hydrological time seriesrdquo Math-ematical Problems in Engineering vol 2010 Article ID 20543815 pages 2010

[14] F Takens ldquoDetecting strange attractors in turbulencerdquo inDynamical Systems and Turbulence D A Rand and L S YoungEds vol 898 of Lecture Notes in Mathematics 366C381 1981

[15] L Cao Y Hong H Fang and G He ldquoPredicting chaotic timeseries withwavelet networksrdquo PhysicaD Nonlinear Phenomenavol 85 no 1-2 pp 225ndash238 1995

[16] J D Farmer and J J Sidorowich ldquoPredicting chaotic timeseriesrdquo Physical Review Letters vol 59 no 8 pp 845ndash848 1987

[17] J Du Y-J Cao Z-J Liu et al ldquoLocal higher-orderVolterra filtermulti-step predictionmodel of chaotic time seriesrdquoActa PhysicaSinica vol 58 no 9 pp 5997ndash6005 2009

[18] H-C Li J-S Zhang and X-C Xiao ldquoNeural Volterra filter forchaotic time series predictionrdquo Chinese Physics B vol 14 no 11pp 2181ndash2188 2005

[19] Y-P Zhao L-Y Zhang D-C Li L-F Wang and H-Z JiangldquoChaotic time series prediction using filtering window basedleast squares support vector regressionrdquoActa Physica Sinica vol62 no 12 Article ID 120511 2013

[20] Z J Tang F Ren T Peng et al ldquoA least square support vectormachine prediction algorithm for chaot ic time series based onthe it erative error correctionrdquoActa Physica Sinica in China vol63 no 5 Article ID 050505 10 pages 2014

[21] A Miranian and M Abdollahzade ldquoDeveloping a local least-squares support vector machines-based neuro-fuzzy model fornonlinear and chaotic time series predictionrdquo IEEE Transac-tions on Neural Networks and Learning Systems vol 24 no 2pp 207ndash218 2013

[22] Y-H Yu and J-D Song ldquoRedundancy-test-based hyper-parameters selection approach for support vector machines topredict time seriesrdquo Acta Physica Sinica vol 61 no 17 ArticleID 170516 14 pages 2012

[23] Y X Hu and H T Zhang ldquoPrediction of the chaotic timeseries based on chaotic simulated annealing and support vectormachinerdquo Physics Procedia vol 25 pp 506ndash512 2012

[24] Y X Hu andH T Zhang ldquoChaos optimizationmethod of SVMparameters selection for chaotic time series forecastingrdquo PhysicsProcedia vol 25 pp 588ndash594 2012

[25] P Liu and J Yao ldquoApplication of least square support vectormachine based on particle swarm optimization to chaotic timeseries predictionrdquo in Proceedings of the IEEE InternationalConference on Intelligent Computing and Intelligent Systems(ICIS 09) vol 4 pp 458ndash462 Shanghai China November2009

[26] S J Dong D H Sun and B P Tang ldquoA fault diagnosis methodfor rotatingmachinery based on PCA andMorlet Kernel SVMrdquoMathematical Problems in Engineering vol 2014 Article ID293878 8 pages 2014

[27] J Ma Y Liu Y Guo and X Zhao ldquoMulti-parameter predictionfor chaotic time series based on least squa res support vector

regressionrdquo in Proceedings of the 32nd Chinese Control Confer-ence (CCC rsquo13) pp 6139ndash6142 July 2013

[28] Y D Zhou H Ma W Y Lu and H Q Wang ldquoPredictionof chaotic time series using multivariate local polynomialregressionrdquo Acta Physica Sinica vol 56 no 12 pp 6809ndash68142007

[29] L-Y Su ldquoPrediction of multivariate chaotic time series withlocal polynomial fittingrdquo Computers ampMathematics with Appli-cations vol 59 no 2 pp 737ndash744 2010

[30] L-Y Su Y-J Ma and J-J Li ldquoApplication of local polynomialestimation in suppressing strong chaotic noiserdquo Chinese PhysicsB vol 21 no 2 Article ID 020508 2012

[31] T Song and H Li ldquoChaotic time series prediction based onwavelet echo state networkrdquo Wuli XuebaoActa Physica Sinicavol 61 no 8 Article ID 080506 2012

[32] L Zhang F Tian S Liu L Dang X Peng and X Yin ldquoChaotictime series prediction of E-nose sensor drift in embedded phasespacerdquo Sensors and Actuators B Chemical vol 182 pp 71ndash792013

[33] C-T Zhang Q-L Ma and H Peng ldquoChaotic time seriesprediction based on information entropy optimized parametersof phase space reconstructionrdquo Acta Physica Sinica vol 59 no11 pp 7623ndash7629 2010

[34] A Gholipour B N Araabi and C Lucas ldquoPredicting Chaotictime series using neural and neurofuzzy models a comparativestudyrdquoNeural Processing Letters vol 24 no 3 pp 217ndash239 2006

[35] X Wang and M Han ldquoMultivariate chaotic time series pre-diction based on Hierarchic Reservoirsrdquo in Proceedings of theIEEE International Conference on SystemsMan andCybernetics(SMC rsquo12) pp 384ndash388 October 2012

[36] C-Y Li Y-L Yang and H-W Zhang ldquoPrediction of chaotictime series based on incremental method for Bayesian networklearningrdquo in Proceedings of the 31st Chinese Control Conference(CCC rsquo12) pp 4245ndash4249 July 2012

[37] J L Qu X F Wang Y C Qiao G Feng and Y Z Di ldquoAnimproved local weighted Lin ear prediction model for chaotictime seriesrdquo Chinese Physics Letters vol 31 no 2 Article ID020503 2014

[38] H Qu W T Ma and J H Zhao ldquoKernel least mean kurtosisbased online chaotic time series predictionrdquo Chinese PhysicsLetters vol 30 no 11 Article ID 110505 2013

[39] L Cao ldquoPractical method for determining the minimumembedding dimension of a scalar time seriesrdquo Physica D vol110 no 1-2 pp 43ndash50 1997

[40] V A Epanechnikov ldquoNonparametric estimation of a multi-dimensional probability densityrdquo Theory of Probability and ItsApplications vol 13 pp 153ndash158 1969

[41] J Fan T Gasser I Gijbels M Brockmann and J Engel ldquoLocalpolynomial regression optimal kernels and asymptotic mini-max efficiencyrdquoAnnals of the Institute of StatisticalMathematicsvol 49 no 1 pp 79ndash99 1997

[42] M C Mackey and L Glass ldquoOscillation and chaos in physio-logical control systemsrdquo Science vol 197 no 4300 pp 287ndash2891977

[43] MHenon and Y Pomeau ldquoTwo strange attractors with a simplestructurerdquo in Turbulence and Navier Stokes Equations vol 565of Lecture Notes in Mathematics pp 29ndash68 Springer BerlinGermany 1976

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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


0 50 100 150 200 250 300 350 400 450 5000

02

04

06

08

1

Time

Real valueLPP value

KLPP value

x(t)

(a)

Time0 50 100 150 200 250 300 350 400 450 500

0

05

1

15

Erro

r

LPP errorKLPP error

minus15

minus1

minus05

times10minus4

(b)

Time0 50 100 150 200 250 300

00102030405060708

Real valueLPP value

KLPP value

x(t)

(c)

Time0 50 100 150 200 250 300

0

02

Erro

r

LPP errorKLPP error

minus08

minus06

minus04

minus02

(d)

Figure 1 Results of the Lorenz time series (a) and (b) are one-step prediction value and error respectively (c) and (d) aremultistep predictionvalue and error respectively

coefficients of the state-dependent autoregressive model caneffectively fit nonlinear characteristics of nonlinear timeseries By using the polynomial function to approximate thefunctional coefficients we can use LPPmodel to approximatethe local attractor The local attractor can be obtained byutilizing Takens embedding theorem [10] The kernel predic-tion can improve the forecast accuracy [37 38] We combinethe LPP with the kernel technique to develop a kernel LPP(KLPP) model to improve the multistep forecast accuracyThe KLPP model can be applied to choose proper radiusof the nearest neighbors by using the kernel function Inthis work we compare our models with those reported inthe literature to illustrate the effectiveness of our models inpredicting chaotic time series

The paper is organized as follows In Section 2 theconcept of the state-dependent autoregressivemodel the LPPmodel and the KLPP model are proposed and the optimalparameter set is established by using GDF which is a tool toevaluate the goodness of the model with the chosen numberof neighbors [12] In Section 3 the prediction errors of thegenerated time series are computed to analyze the predic-tion effectiveness Section 3 uses three simulated chaoticsystems and one real life time series as examples to evaluate

the proposed modelsThe conclusion of this paper is given inSection 4

2 The LPP Model and KLPP Model

21 Data Structure in Phase Space Reconstruction Supposethat we have a scalar chaotic time series 119909

1 1199092 119909

119899 The

first step of a local prediction is the phase space reconstruc-tion According to Takens embedding theory the phase spacecan be reconstructed and the construct phase points 119883

119905=

(119909119905 119909119905+120591 119909

119905+(119898minus1)120591)119879 where 119905 = 1 2 119872 and 119872 =

119899minus(119898minus1)120591The embedding dimension119898 and the time delay120591 can be obtained by the Cao method [39]Then a continuedvector mapping 119865 R119898 rarr Rm or 119891 R119898 rarr R can be usedto describe the unknown evolution from 119883

119905to 119883119905+1

That is119883119905+1= 119865(119883

119905) and 119909

119905+1= 119891(119909

119905)

22 The LPP Model The state-dependent autoregressivemodel can be described as follows

119909119905= 1198920(119883119905minus1) +

119898

sum

119895=1

119892119895(119883119905minus1) 119909119905minus119895 (1)


50 100 150 200 250 300 350 400 450 5000

1

2

3

4

5

Time

The p

olyn

omia

l ord

er

LPPKLPP

(a)

Time50 100 150 200 250 300 350 400 450 500

1

2

3

4

5

6

Dela

y of

the l

ag v

aria

ble

KLPPLPP

(b)

Time50 100 150 200 250 300 350 400 450 500

0

50

100

150

Num

ber o

f nea

rest

neig

hbor

poi

nts

KLPPLPP

(c)

Time0 50 100 150 200 250 300 350 400 450 500

0005

01015

02025

03035

Radi

us o

f the

nei

ghbo

rs

KLPP

(d)





119895



119909119905+1= 1198920(119883119905) +

119898

sum

119895=1

119892119895(119883119905) 119909119905minus(119895minus1)120591

(2)

where 119883119905= (119909

119905 119909119905minus120591 119909




119892119895(119883119905) = 119886119895(119906) = sum

119903




119909119905+1= 1198860(119906) +

119898

sum

119895=1

119886119895(119906) 119909119905minus(119895minus1)120591

(3)

where 119886119895(119906) = sum

119903


1)120591)) variable 119909119905minus119889


119909119905+1=

119903

sum

119894=0

1198860119894(119909119905minus119889)119894

+

119898

sum

119895=1

119903

sum

119894=0

119886119895119894(119909119905minus119889)119894

119909119905minus(119895minus1)119903

(4)



119872





0 20 40 60 80 100 1200

051

152

253

354

Time

The p

olyn

omia

l ord

er

LPPKLPP

(a)

Time0 20 40 60 80 100 120

1

2

3

4

5

6

Dela

y of

the l

ag v

aria

ble

LPPKLPP

(b)

Time10 20 30 40 50 60 70 80 90 100 110 120

1020304050607080

Num

ber o

f nea

rest

neig

hbor

poi

nts

LPPKLPP

(c)

Time0 50 100 150 200 250 300

0

005

01

015

02

Radi

us o

f the

nei

ghbo

rs

KLPP

(d)



of 119883119872119895(119895 = 1 2 119902)


equation

min119886119895119894isinR

119902

sum

119905=1

[

[

119909119872119905+(119898minus1)120591+1

minus

119898

sum

119895=0

119903

sum

119894=0

119886119895119894119909119894


]

]

2

(5)


00 119886

0119903 119886

1198980 119886

119898119903)119879

(119898+1)(119903+1)times1and

set

120594 =

[

[

[

[

[

11988011199091198721minus120591

11988011199091198721


11988021199091198722minus120591

11988021199091198722


d

119880119902119909119872119902minus120591

119880119902119909119872119902


]

]

]

]

]119902times(119898+1)(119903+1)

119884 = (1199091198721+(119898minus1)120591+1

119909119872119902+(119898minus1)120591+1

)

119879

119880119905= (1 119909

119872119905+(119898minus1)120591minus119889 119909

119903

119872119905+(119898minus1)120591minus119889)

119909119872119905minus120591

equiv 1 (119905 = 1 2 119902)

(6)


120579 = 120594

119879120594

minus1

120594119879119884 (7)





= (119909119872+1

119909119872+1+120591

119909119872+1+(119898minus1)120591

)119879








0 50 100 150 200 250 300 350 400 450 5000

02

04

06

08

1

Time

Real valueLPP value

KLPP value

x(t)

(a)

Time0 50 100 150 200 250 300 350 400 450 500

0

1

2

3

Erro

r

LPP errorKLPP error

minus3

minus2

minus1

times10minus4

(b)

Time0 50 100 150 200 250 300 350 400 450 500

0

02

04

06

08

1

Real valueLPP value

KLPP value

x(t)

(c)

Time0 50 100 150 200 250 300 350 400 450 500

0

0005

001

0015

Erro

r

LPP errorKLPP error

minus001

minus0005

(d)



2

GDF =(119884 minus ) (119884 minus )

119879

119902 minus 119863

(8)

where = 120594120579 = 120594120594

119879120594minus1120594119879119884 119863 = tr119867 =



119870(119911119905) =

3

4

(1 minus 1199112

119905)+ (9)


119905equiv 119889(119905))




squares

119902

sum

119905=1

[

[

119909119872119905+(119898minus1)120591+1

minus

119898

sum

119895=0

119903

sum

119894=0

119886119895119894119909119894


]

]

2

119870ℎ(119911119905)

(10)

where 119895 = 0 1 119898 119894 = 0 1 119903 and 119870ℎ(119911119905) equiv




119895119894 It



0 50 100 150 200 250 300 350 400 450 5000

1

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The p

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(a)

0 50 100 150 200 250 300 350 400 450 5001

2

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Dela

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LPPKLPP

(b)

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LPPKLPP

(c)

0 50 100 150 200 250 300 350 400 450 500005

01

015

02

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rs

Time

KLPP

(d)




theory that

120579 = 120594

119879119882120594

minus1

120594119879119882119884 (11)

where 119884 = (1199091198721+(119898minus1)120591+1

119909119872119902+(119898minus1)120591+1

)119879 and 119882 is a


119894ℎ) as its


119894119909119872119894minus120591

119880119894119909119872119894+(119898minus1)120591

] as its 119894th row and 119880119894=

[1 119909119872119894+(119898minus1)120591minus119889

119909119903



2

GDF (ℎ) =(119884 minus ) (119884 minus )

119879

119902 minus 119863

(12)

where = 120594120579 = 120594120594

119879119882120594minus1120594119879119882119884 119863 = tr119867 =


119894minus 119883119872 (119894 =


our models




119905= (119909119905 119909119905+120591 119909

119905+(119898minus1)120591)119879









119872+(119898minus1)120591+1





0 50 100 150 200 250 300 350 400 450 5000

1

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The p

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LPPKLPP

(a)

0 50 100 150 200 250 300 350 400 450 5001

2

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Dela

y of

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Time

LPPKLPP

(b)

50 100 150 200 250 300 350 400 450 5000

20

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(c)

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Time

KLPP

(d)






119895119894from (7) in Step 3





max (119909) minusmin (119909) (13)


MAE = max 1003816100381610038161003816119910 (119905) minus 119910 (119905)

1003816100381610038161003816


119899

sum

119905=1

(119910 (119905) minus 119910 (119905))2

NMSE =sum119899

119905=1(119910 (119905) minus 119910 (119905))

2

sum119899

119905=1(119910 (119905) minus 119910)

2

(14)


0 50 100 150 200 250 300 350 400 450 500

0

05

1

15

Time

minus05

minus1

minus15

x(t)

Real valueLPP value

KLPP value

(a)

Time0 50 100 150 200 250 300 350 400 450 500

0

05

1

Erro

r

minus05

minus1

minus15

times10minus5

LPP errorKLPP error

(b)

Time0 5 10 15 20 25 30 35 40 45 50

0

05

1

15

minus05

minus1

minus15

x(t)

Real valueLPP value

KLPP value

(c)

Time0 5 10 15 20 25 30 35 40 45 50

0

1

2

3

Erro

r

minus3

minus2

minus1

LPP errorKLPP error

(d)



= 120590 (119910 minus 119909)

119910 = (119903 minus 119911) 119909 minus 119910

= 119909119910 minus 119887119911

(15)









0 50 100 150 200 250 300 350 400 450 5000

1

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Time

The p

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LPPKLPP

(a)

Time0 50 100 150 200 250 300 350 400 450 500

1

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Dela

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LPPKLPP

(b)

Time0 50 100 150 200 250 300 350 400 450 500

0

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LPPKLPP

(c)

Time0 50 100 150 200 250 300 350 400 450 500

0

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KLPP

(d)



119909 (119905 + 1) = (1 minus 119887) 119909 (119905) +

119886119909 (119905 minus 120591)

1 + 119909 (119905 minus 120591)10 (16)






119909 (119905 + 1) = 1 minus 119886119909 (119905)2+ 119910 (119905)

119910 (119905 + 1) = 119887119909 (119905)

(17)





0 5 10 15 20 25 30 35 40 45 500

1

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The p

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er

LPPKLPP

(a)

Time0 5 10 15 20 25 30 35 40 45 50

1

2

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Dela

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the l

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LPPKLPP

(b)

Time0 5 10 15 20 25 30 35 40 45 50

020406080

100120140

Num

ber o

f nea

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poi

nts

LPPKLPP

(c)

Time0 5 10 15 20 25 30 35 40 45 50

02

04

06

08

1

12

Radi

us o

f the

nei

ghbo

rs

KLPP

(d)







ERNN (2007) [9] 879119864 minus 06 990119864 minus 10

PSO-LSSVM (2009) [25] 180119864 minus 04

FBMNN (2009) [10] 205119864 minus 05


IEOP (2010) [33] 105119864 minus 05

CSAA-SVR (2012) [23] 876119864 minus 04

COA-SVR (2012) [24] 303119864 minus 03

ESN (2012) [35] 250119864 minus 03

HR (2012) [35] 150119864 minus 03

WESN (2012) [31] 958119864 minus 04

ARMA-RESN (2013) [8] 303119864 minus 06

RBFNN (2013) [10] 500119864 minus 03

LNF (2013) [21] 640119864 minus 06





0 100 200 300 400 500 600 700 800 900 10000

05

1

15

2

Time

x(t)

Real valueLPP value

KLPP value

(a)

Time0 100 200 300 400 500 600 700 800 900 1000

0020406

Erro

r

minus02

minus04

minus06

minus08

LPP errorKLPP error

(b)

Time0 50 100 150 200 250 300 350

002040608

1

minus02

minus04

x(t)

Real valueLPP value

KLPP value

(c)

Time0 50 100 150 200 250 300 350

0

05

1

15

Erro

r

minus05

LPP errorKLPP error

(d)




ERNN (2007) [9] 420119864 minus 05 315119864 minus 08

PSO-LSSVM (2009) [25] 280119864 minus 03

FBMNN (2009) [10] 302119864 minus 04


IEOP (2010) [33] 194119864 minus 05

ODCF-SVM (2012) [22] 160119864 minus 02

LNF (2013) [21] 790119864 minus 04

LSSVR (2013) [27] 570119864 minus 03








RBF-OLS (2009) [10] 413119864 minus 02

FBMNN (2009) [10] 270119864 minus 05







0 100 200 300 400 500 600 700 800 900 10000

1

2

3

4

5

Time

The p

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l ord

er

LPPKLPP

(a)

Time0 100 200 300 400 500 600 700 800 900 1000

115

225

335

445

5

Dela

y of

the l

ag v

aria

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LPPKLPP

(b)

Time100 200 300 400 500 600 700 800 900 1000

020406080

100120140

Num

ber o

f nea

rest

neig

hbor

poi

nts

LPPKLPP

(c)

Time0 100 200 300 400 500 600 700 800 900 1000

002040608

11214

Radi

us o

f the

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rs

KLPP

(d)




RBF-OLS (2009) [10] 460119864 minus 02


ARMA-RESN (2013) [8] 157937





4 Conclusions




0 50 100 1500

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(a)

Time0 50 100 150

115

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5

Dela

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LPPKLPP

(b)

Time20 40 60 80 100 120 140

020406080

100120140

Num

ber o

f nea

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neig

hbor

poi

nts

LPPKLPP

(c)

Time0 50 100 150 200 250 300 350

001020304050607

Radi

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f the

nei

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rs

KLPP

(d)




Acknowledgments


References





















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










50 100 150 200 250 300 350 400 450 5000

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(a)

Time50 100 150 200 250 300 350 400 450 500

1

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KLPPLPP

(b)

Time50 100 150 200 250 300 350 400 450 500

0

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Num

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neig

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KLPPLPP

(c)

Time0 50 100 150 200 250 300 350 400 450 500

0005

01015

02025

03035

Radi

us o

f the

nei

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rs

KLPP

(d)





119895



119909119905+1= 1198920(119883119905) +

119898

sum

119895=1

119892119895(119883119905) 119909119905minus(119895minus1)120591

(2)

where 119883119905= (119909

119905 119909119905minus120591 119909




119892119895(119883119905) = 119886119895(119906) = sum

119903




119909119905+1= 1198860(119906) +

119898

sum

119895=1

119886119895(119906) 119909119905minus(119895minus1)120591

(3)

where 119886119895(119906) = sum

119903


1)120591)) variable 119909119905minus119889


119909119905+1=

119903

sum

119894=0

1198860119894(119909119905minus119889)119894

+

119898

sum

119895=1

119903

sum

119894=0

119886119895119894(119909119905minus119889)119894

119909119905minus(119895minus1)119903

(4)



119872





0 20 40 60 80 100 1200

051

152

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354

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LPPKLPP

(a)

Time0 20 40 60 80 100 120

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LPPKLPP

(b)

Time10 20 30 40 50 60 70 80 90 100 110 120

1020304050607080

Num

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LPPKLPP

(c)

Time0 50 100 150 200 250 300

0

005

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rs

KLPP

(d)



of 119883119872119895(119895 = 1 2 119902)


equation

min119886119895119894isinR

119902

sum

119905=1

[

[

119909119872119905+(119898minus1)120591+1

minus

119898

sum

119895=0

119903

sum

119894=0

119886119895119894119909119894


]

]

2

(5)


00 119886

0119903 119886

1198980 119886

119898119903)119879

(119898+1)(119903+1)times1and

set

120594 =

[

[

[

[

[

11988011199091198721minus120591

11988011199091198721


11988021199091198722minus120591

11988021199091198722


d

119880119902119909119872119902minus120591

119880119902119909119872119902


]

]

]

]

]119902times(119898+1)(119903+1)

119884 = (1199091198721+(119898minus1)120591+1

119909119872119902+(119898minus1)120591+1

)

119879

119880119905= (1 119909

119872119905+(119898minus1)120591minus119889 119909

119903

119872119905+(119898minus1)120591minus119889)

119909119872119905minus120591

equiv 1 (119905 = 1 2 119902)

(6)


120579 = 120594

119879120594

minus1

120594119879119884 (7)





= (119909119872+1

119909119872+1+120591

119909119872+1+(119898minus1)120591

)119879








0 50 100 150 200 250 300 350 400 450 5000

02

04

06

08

1

Time

Real valueLPP value

KLPP value

x(t)

(a)

Time0 50 100 150 200 250 300 350 400 450 500

0

1

2

3

Erro

r

LPP errorKLPP error

minus3

minus2

minus1

times10minus4

(b)

Time0 50 100 150 200 250 300 350 400 450 500

0

02

04

06

08

1

Real valueLPP value

KLPP value

x(t)

(c)

Time0 50 100 150 200 250 300 350 400 450 500

0

0005

001

0015

Erro

r

LPP errorKLPP error

minus001

minus0005

(d)



2

GDF =(119884 minus ) (119884 minus )

119879

119902 minus 119863

(8)

where = 120594120579 = 120594120594

119879120594minus1120594119879119884 119863 = tr119867 =



119870(119911119905) =

3

4

(1 minus 1199112

119905)+ (9)


119905equiv 119889(119905))




squares

119902

sum

119905=1

[

[

119909119872119905+(119898minus1)120591+1

minus

119898

sum

119895=0

119903

sum

119894=0

119886119895119894119909119894


]

]

2

119870ℎ(119911119905)

(10)

where 119895 = 0 1 119898 119894 = 0 1 119903 and 119870ℎ(119911119905) equiv




119895119894 It



0 50 100 150 200 250 300 350 400 450 5000

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The p

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LPPKLPP

(a)

0 50 100 150 200 250 300 350 400 450 5001

2

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Dela

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the l

ag v

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LPPKLPP

(b)

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(c)

0 50 100 150 200 250 300 350 400 450 500005

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Time

KLPP

(d)




theory that

120579 = 120594

119879119882120594

minus1

120594119879119882119884 (11)

where 119884 = (1199091198721+(119898minus1)120591+1

119909119872119902+(119898minus1)120591+1

)119879 and 119882 is a


119894ℎ) as its


119894119909119872119894minus120591

119880119894119909119872119894+(119898minus1)120591

] as its 119894th row and 119880119894=

[1 119909119872119894+(119898minus1)120591minus119889

119909119903



2

GDF (ℎ) =(119884 minus ) (119884 minus )

119879

119902 minus 119863

(12)

where = 120594120579 = 120594120594

119879119882120594minus1120594119879119882119884 119863 = tr119867 =


119894minus 119883119872 (119894 =


our models




119905= (119909119905 119909119905+120591 119909

119905+(119898minus1)120591)119879









119872+(119898minus1)120591+1





0 50 100 150 200 250 300 350 400 450 5000

1

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The p

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er

LPPKLPP

(a)

0 50 100 150 200 250 300 350 400 450 5001

2

3

4

5

6

Dela

y of

the l

ag v

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ble

Time

LPPKLPP

(b)

50 100 150 200 250 300 350 400 450 5000

20

40

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Num

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Time

LPPKLPP

(c)

0 50 100 150 200 250 300 350 400 450 500005

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Time

KLPP

(d)






119895119894from (7) in Step 3





max (119909) minusmin (119909) (13)


MAE = max 1003816100381610038161003816119910 (119905) minus 119910 (119905)

1003816100381610038161003816


119899

sum

119905=1

(119910 (119905) minus 119910 (119905))2

NMSE =sum119899

119905=1(119910 (119905) minus 119910 (119905))

2

sum119899

119905=1(119910 (119905) minus 119910)

2

(14)


0 50 100 150 200 250 300 350 400 450 500

0

05

1

15

Time

minus05

minus1

minus15

x(t)

Real valueLPP value

KLPP value

(a)

Time0 50 100 150 200 250 300 350 400 450 500

0

05

1

Erro

r

minus05

minus1

minus15

times10minus5

LPP errorKLPP error

(b)

Time0 5 10 15 20 25 30 35 40 45 50

0

05

1

15

minus05

minus1

minus15

x(t)

Real valueLPP value

KLPP value

(c)

Time0 5 10 15 20 25 30 35 40 45 50

0

1

2

3

Erro

r

minus3

minus2

minus1

LPP errorKLPP error

(d)



= 120590 (119910 minus 119909)

119910 = (119903 minus 119911) 119909 minus 119910

= 119909119910 minus 119887119911

(15)









0 50 100 150 200 250 300 350 400 450 5000

1

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Time

The p

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l ord

er

LPPKLPP

(a)

Time0 50 100 150 200 250 300 350 400 450 500

1

2

3

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5

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Dela

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the l

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LPPKLPP

(b)

Time0 50 100 150 200 250 300 350 400 450 500

0

50

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LPPKLPP

(c)

Time0 50 100 150 200 250 300 350 400 450 500

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KLPP

(d)



119909 (119905 + 1) = (1 minus 119887) 119909 (119905) +

119886119909 (119905 minus 120591)

1 + 119909 (119905 minus 120591)10 (16)






119909 (119905 + 1) = 1 minus 119886119909 (119905)2+ 119910 (119905)

119910 (119905 + 1) = 119887119909 (119905)

(17)





0 5 10 15 20 25 30 35 40 45 500

1

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The p

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er

LPPKLPP

(a)

Time0 5 10 15 20 25 30 35 40 45 50

1

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5

6

Dela

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the l

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ble

LPPKLPP

(b)

Time0 5 10 15 20 25 30 35 40 45 50

020406080

100120140

Num

ber o

f nea

rest

neig

hbor

poi

nts

LPPKLPP

(c)

Time0 5 10 15 20 25 30 35 40 45 50

02

04

06

08

1

12

Radi

us o

f the

nei

ghbo

rs

KLPP

(d)







ERNN (2007) [9] 879119864 minus 06 990119864 minus 10

PSO-LSSVM (2009) [25] 180119864 minus 04

FBMNN (2009) [10] 205119864 minus 05


IEOP (2010) [33] 105119864 minus 05

CSAA-SVR (2012) [23] 876119864 minus 04

COA-SVR (2012) [24] 303119864 minus 03

ESN (2012) [35] 250119864 minus 03

HR (2012) [35] 150119864 minus 03

WESN (2012) [31] 958119864 minus 04

ARMA-RESN (2013) [8] 303119864 minus 06

RBFNN (2013) [10] 500119864 minus 03

LNF (2013) [21] 640119864 minus 06





0 100 200 300 400 500 600 700 800 900 10000

05

1

15

2

Time

x(t)

Real valueLPP value

KLPP value

(a)

Time0 100 200 300 400 500 600 700 800 900 1000

0020406

Erro

r

minus02

minus04

minus06

minus08

LPP errorKLPP error

(b)

Time0 50 100 150 200 250 300 350

002040608

1

minus02

minus04

x(t)

Real valueLPP value

KLPP value

(c)

Time0 50 100 150 200 250 300 350

0

05

1

15

Erro

r

minus05

LPP errorKLPP error

(d)




ERNN (2007) [9] 420119864 minus 05 315119864 minus 08

PSO-LSSVM (2009) [25] 280119864 minus 03

FBMNN (2009) [10] 302119864 minus 04


IEOP (2010) [33] 194119864 minus 05

ODCF-SVM (2012) [22] 160119864 minus 02

LNF (2013) [21] 790119864 minus 04

LSSVR (2013) [27] 570119864 minus 03








RBF-OLS (2009) [10] 413119864 minus 02

FBMNN (2009) [10] 270119864 minus 05







0 100 200 300 400 500 600 700 800 900 10000

1

2

3

4

5

Time

The p

olyn

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l ord

er

LPPKLPP

(a)

Time0 100 200 300 400 500 600 700 800 900 1000

115

225

335

445

5

Dela

y of

the l

ag v

aria

ble

LPPKLPP

(b)

Time100 200 300 400 500 600 700 800 900 1000

020406080

100120140

Num

ber o

f nea

rest

neig

hbor

poi

nts

LPPKLPP

(c)

Time0 100 200 300 400 500 600 700 800 900 1000

002040608

11214

Radi

us o

f the

nei

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rs

KLPP

(d)




RBF-OLS (2009) [10] 460119864 minus 02


ARMA-RESN (2013) [8] 157937





4 Conclusions




0 50 100 1500

1

2

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Time

The p

olyn

omia

l ord

er

LPPKLPP

(a)

Time0 50 100 150

115

225

335

445

5

Dela

y of

the l

ag v

aria

ble

LPPKLPP

(b)

Time20 40 60 80 100 120 140

020406080

100120140

Num

ber o

f nea

rest

neig

hbor

poi

nts

LPPKLPP

(c)

Time0 50 100 150 200 250 300 350

001020304050607

Radi

us o

f the

nei

ghbo

rs

KLPP

(d)




Acknowledgments


References





















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 20 40 60 80 100 1200

051

152

253

354

Time

The p

olyn

omia

l ord

er

LPPKLPP

(a)

Time0 20 40 60 80 100 120

1

2

3

4

5

6

Dela

y of

the l

ag v

aria

ble

LPPKLPP

(b)

Time10 20 30 40 50 60 70 80 90 100 110 120

1020304050607080

Num

ber o

f nea

rest

neig

hbor

poi

nts

LPPKLPP

(c)

Time0 50 100 150 200 250 300

0

005

01

015

02

Radi

us o

f the

nei

ghbo

rs

KLPP

(d)



of 119883119872119895(119895 = 1 2 119902)


equation

min119886119895119894isinR

119902

sum

119905=1

[

[

119909119872119905+(119898minus1)120591+1

minus

119898

sum

119895=0

119903

sum

119894=0

119886119895119894119909119894


]

]

2

(5)


00 119886

0119903 119886

1198980 119886

119898119903)119879

(119898+1)(119903+1)times1and

set

120594 =

[

[

[

[

[

11988011199091198721minus120591

11988011199091198721


11988021199091198722minus120591

11988021199091198722


d

119880119902119909119872119902minus120591

119880119902119909119872119902


]

]

]

]

]119902times(119898+1)(119903+1)

119884 = (1199091198721+(119898minus1)120591+1

119909119872119902+(119898minus1)120591+1

)

119879

119880119905= (1 119909

119872119905+(119898minus1)120591minus119889 119909

119903

119872119905+(119898minus1)120591minus119889)

119909119872119905minus120591

equiv 1 (119905 = 1 2 119902)

(6)


120579 = 120594

119879120594

minus1

120594119879119884 (7)





= (119909119872+1

119909119872+1+120591

119909119872+1+(119898minus1)120591

)119879








0 50 100 150 200 250 300 350 400 450 5000

02

04

06

08

1

Time

Real valueLPP value

KLPP value

x(t)

(a)

Time0 50 100 150 200 250 300 350 400 450 500

0

1

2

3

Erro

r

LPP errorKLPP error

minus3

minus2

minus1

times10minus4

(b)

Time0 50 100 150 200 250 300 350 400 450 500

0

02

04

06

08

1

Real valueLPP value

KLPP value

x(t)

(c)

Time0 50 100 150 200 250 300 350 400 450 500

0

0005

001

0015

Erro

r

LPP errorKLPP error

minus001

minus0005

(d)



2

GDF =(119884 minus ) (119884 minus )

119879

119902 minus 119863

(8)

where = 120594120579 = 120594120594

119879120594minus1120594119879119884 119863 = tr119867 =



119870(119911119905) =

3

4

(1 minus 1199112

119905)+ (9)


119905equiv 119889(119905))




squares

119902

sum

119905=1

[

[

119909119872119905+(119898minus1)120591+1

minus

119898

sum

119895=0

119903

sum

119894=0

119886119895119894119909119894


]

]

2

119870ℎ(119911119905)

(10)

where 119895 = 0 1 119898 119894 = 0 1 119903 and 119870ℎ(119911119905) equiv




119895119894 It



0 50 100 150 200 250 300 350 400 450 5000

1

2

3

4

5

Time

The p

olyn

omia

l ord

er

LPPKLPP

(a)

0 50 100 150 200 250 300 350 400 450 5001

2

3

4

5

6

Time

Dela

y of

the l

ag v

aria

ble

LPPKLPP

(b)

50 100 150 200 250 300 350 400 450 5000

20

40

60

80

100

120

Num

ber o

f nea

rest

neig

hbor

poi

nts

Time

LPPKLPP

(c)

0 50 100 150 200 250 300 350 400 450 500005

01

015

02

025

03

Radi

us o

f the

nei

ghbo

rs

Time

KLPP

(d)




theory that

120579 = 120594

119879119882120594

minus1

120594119879119882119884 (11)

where 119884 = (1199091198721+(119898minus1)120591+1

119909119872119902+(119898minus1)120591+1

)119879 and 119882 is a


119894ℎ) as its


119894119909119872119894minus120591

119880119894119909119872119894+(119898minus1)120591

] as its 119894th row and 119880119894=

[1 119909119872119894+(119898minus1)120591minus119889

119909119903



2

GDF (ℎ) =(119884 minus ) (119884 minus )

119879

119902 minus 119863

(12)

where = 120594120579 = 120594120594

119879119882120594minus1120594119879119882119884 119863 = tr119867 =


119894minus 119883119872 (119894 =


our models




119905= (119909119905 119909119905+120591 119909

119905+(119898minus1)120591)119879









119872+(119898minus1)120591+1





0 50 100 150 200 250 300 350 400 450 5000

1

2

3

4

5

Time

The p

olyn

omia

l ord

er

LPPKLPP

(a)

0 50 100 150 200 250 300 350 400 450 5001

2

3

4

5

6

Dela

y of

the l

ag v

aria

ble

Time

LPPKLPP

(b)

50 100 150 200 250 300 350 400 450 5000

20

40

60

80

100

120

Num

ber o

f nea

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neig

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poi

nts

Time

LPPKLPP

(c)

0 50 100 150 200 250 300 350 400 450 500005

01

015

02

025

03

Radi

us o

f the

nei

ghbo

rs

Time

KLPP

(d)






119895119894from (7) in Step 3





max (119909) minusmin (119909) (13)


MAE = max 1003816100381610038161003816119910 (119905) minus 119910 (119905)

1003816100381610038161003816


119899

sum

119905=1

(119910 (119905) minus 119910 (119905))2

NMSE =sum119899

119905=1(119910 (119905) minus 119910 (119905))

2

sum119899

119905=1(119910 (119905) minus 119910)

2

(14)


0 50 100 150 200 250 300 350 400 450 500

0

05

1

15

Time

minus05

minus1

minus15

x(t)

Real valueLPP value

KLPP value

(a)

Time0 50 100 150 200 250 300 350 400 450 500

0

05

1

Erro

r

minus05

minus1

minus15

times10minus5

LPP errorKLPP error

(b)

Time0 5 10 15 20 25 30 35 40 45 50

0

05

1

15

minus05

minus1

minus15

x(t)

Real valueLPP value

KLPP value

(c)

Time0 5 10 15 20 25 30 35 40 45 50

0

1

2

3

Erro

r

minus3

minus2

minus1

LPP errorKLPP error

(d)



= 120590 (119910 minus 119909)

119910 = (119903 minus 119911) 119909 minus 119910

= 119909119910 minus 119887119911

(15)









0 50 100 150 200 250 300 350 400 450 5000

1

2

3

4

5

Time

The p

olyn

omia

l ord

er

LPPKLPP

(a)

Time0 50 100 150 200 250 300 350 400 450 500

1

2

3

4

5

6

Dela

y of

the l

ag v

aria

ble

LPPKLPP

(b)

Time0 50 100 150 200 250 300 350 400 450 500

0

50

100

150

Num

ber o

f nea

rest

neig

hbor

poi

nts

LPPKLPP

(c)

Time0 50 100 150 200 250 300 350 400 450 500

0

05

1

15

Radi

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f the

nei

ghbo

rs

KLPP

(d)



119909 (119905 + 1) = (1 minus 119887) 119909 (119905) +

119886119909 (119905 minus 120591)

1 + 119909 (119905 minus 120591)10 (16)






119909 (119905 + 1) = 1 minus 119886119909 (119905)2+ 119910 (119905)

119910 (119905 + 1) = 119887119909 (119905)

(17)





0 5 10 15 20 25 30 35 40 45 500

1

2

3

4

5

Time

The p

olyn

omia

l ord

er

LPPKLPP

(a)

Time0 5 10 15 20 25 30 35 40 45 50

1

2

3

4

5

6

Dela

y of

the l

ag v

aria

ble

LPPKLPP

(b)

Time0 5 10 15 20 25 30 35 40 45 50

020406080

100120140

Num

ber o

f nea

rest

neig

hbor

poi

nts

LPPKLPP

(c)

Time0 5 10 15 20 25 30 35 40 45 50

02

04

06

08

1

12

Radi

us o

f the

nei

ghbo

rs

KLPP

(d)







ERNN (2007) [9] 879119864 minus 06 990119864 minus 10

PSO-LSSVM (2009) [25] 180119864 minus 04

FBMNN (2009) [10] 205119864 minus 05


IEOP (2010) [33] 105119864 minus 05

CSAA-SVR (2012) [23] 876119864 minus 04

COA-SVR (2012) [24] 303119864 minus 03

ESN (2012) [35] 250119864 minus 03

HR (2012) [35] 150119864 minus 03

WESN (2012) [31] 958119864 minus 04

ARMA-RESN (2013) [8] 303119864 minus 06

RBFNN (2013) [10] 500119864 minus 03

LNF (2013) [21] 640119864 minus 06





0 100 200 300 400 500 600 700 800 900 10000

05

1

15

2

Time

x(t)

Real valueLPP value

KLPP value

(a)

Time0 100 200 300 400 500 600 700 800 900 1000

0020406

Erro

r

minus02

minus04

minus06

minus08

LPP errorKLPP error

(b)

Time0 50 100 150 200 250 300 350

002040608

1

minus02

minus04

x(t)

Real valueLPP value

KLPP value

(c)

Time0 50 100 150 200 250 300 350

0

05

1

15

Erro

r

minus05

LPP errorKLPP error

(d)




ERNN (2007) [9] 420119864 minus 05 315119864 minus 08

PSO-LSSVM (2009) [25] 280119864 minus 03

FBMNN (2009) [10] 302119864 minus 04


IEOP (2010) [33] 194119864 minus 05

ODCF-SVM (2012) [22] 160119864 minus 02

LNF (2013) [21] 790119864 minus 04

LSSVR (2013) [27] 570119864 minus 03








RBF-OLS (2009) [10] 413119864 minus 02

FBMNN (2009) [10] 270119864 minus 05







0 100 200 300 400 500 600 700 800 900 10000

1

2

3

4

5

Time

The p

olyn

omia

l ord

er

LPPKLPP

(a)

Time0 100 200 300 400 500 600 700 800 900 1000

115

225

335

445

5

Dela

y of

the l

ag v

aria

ble

LPPKLPP

(b)

Time100 200 300 400 500 600 700 800 900 1000

020406080

100120140

Num

ber o

f nea

rest

neig

hbor

poi

nts

LPPKLPP

(c)

Time0 100 200 300 400 500 600 700 800 900 1000

002040608

11214

Radi

us o

f the

nei

ghbo

rs

KLPP

(d)




RBF-OLS (2009) [10] 460119864 minus 02


ARMA-RESN (2013) [8] 157937





4 Conclusions




0 50 100 1500

1

2

3

4

5

Time

The p

olyn

omia

l ord

er

LPPKLPP

(a)

Time0 50 100 150

115

225

335

445

5

Dela

y of

the l

ag v

aria

ble

LPPKLPP

(b)

Time20 40 60 80 100 120 140

020406080

100120140

Num

ber o

f nea

rest

neig

hbor

poi

nts

LPPKLPP

(c)

Time0 50 100 150 200 250 300 350

001020304050607

Radi

us o

f the

nei

ghbo

rs

KLPP

(d)




Acknowledgments


References





















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 50 100 150 200 250 300 350 400 450 5000

02

04

06

08

1

Time

Real valueLPP value

KLPP value

x(t)

(a)

Time0 50 100 150 200 250 300 350 400 450 500

0

1

2

3

Erro

r

LPP errorKLPP error

minus3

minus2

minus1

times10minus4

(b)

Time0 50 100 150 200 250 300 350 400 450 500

0

02

04

06

08

1

Real valueLPP value

KLPP value

x(t)

(c)

Time0 50 100 150 200 250 300 350 400 450 500

0

0005

001

0015

Erro

r

LPP errorKLPP error

minus001

minus0005

(d)



2

GDF =(119884 minus ) (119884 minus )

119879

119902 minus 119863

(8)

where = 120594120579 = 120594120594

119879120594minus1120594119879119884 119863 = tr119867 =



119870(119911119905) =

3

4

(1 minus 1199112

119905)+ (9)


119905equiv 119889(119905))




squares

119902

sum

119905=1

[

[

119909119872119905+(119898minus1)120591+1

minus

119898

sum

119895=0

119903

sum

119894=0

119886119895119894119909119894


]

]

2

119870ℎ(119911119905)

(10)

where 119895 = 0 1 119898 119894 = 0 1 119903 and 119870ℎ(119911119905) equiv




119895119894 It



0 50 100 150 200 250 300 350 400 450 5000

1

2

3

4

5

Time

The p

olyn

omia

l ord

er

LPPKLPP

(a)

0 50 100 150 200 250 300 350 400 450 5001

2

3

4

5

6

Time

Dela

y of

the l

ag v

aria

ble

LPPKLPP

(b)

50 100 150 200 250 300 350 400 450 5000

20

40

60

80

100

120

Num

ber o

f nea

rest

neig

hbor

poi

nts

Time

LPPKLPP

(c)

0 50 100 150 200 250 300 350 400 450 500005

01

015

02

025

03

Radi

us o

f the

nei

ghbo

rs

Time

KLPP

(d)




theory that

120579 = 120594

119879119882120594

minus1

120594119879119882119884 (11)

where 119884 = (1199091198721+(119898minus1)120591+1

119909119872119902+(119898minus1)120591+1

)119879 and 119882 is a


119894ℎ) as its


119894119909119872119894minus120591

119880119894119909119872119894+(119898minus1)120591

] as its 119894th row and 119880119894=

[1 119909119872119894+(119898minus1)120591minus119889

119909119903



2

GDF (ℎ) =(119884 minus ) (119884 minus )

119879

119902 minus 119863

(12)

where = 120594120579 = 120594120594

119879119882120594minus1120594119879119882119884 119863 = tr119867 =


119894minus 119883119872 (119894 =


our models




119905= (119909119905 119909119905+120591 119909

119905+(119898minus1)120591)119879









119872+(119898minus1)120591+1





0 50 100 150 200 250 300 350 400 450 5000

1

2

3

4

5

Time

The p

olyn

omia

l ord

er

LPPKLPP

(a)

0 50 100 150 200 250 300 350 400 450 5001

2

3

4

5

6

Dela

y of

the l

ag v

aria

ble

Time

LPPKLPP

(b)

50 100 150 200 250 300 350 400 450 5000

20

40

60

80

100

120

Num

ber o

f nea

rest

neig

hbor

poi

nts

Time

LPPKLPP

(c)

0 50 100 150 200 250 300 350 400 450 500005

01

015

02

025

03

Radi

us o

f the

nei

ghbo

rs

Time

KLPP

(d)






119895119894from (7) in Step 3





max (119909) minusmin (119909) (13)


MAE = max 1003816100381610038161003816119910 (119905) minus 119910 (119905)

1003816100381610038161003816


119899

sum

119905=1

(119910 (119905) minus 119910 (119905))2

NMSE =sum119899

119905=1(119910 (119905) minus 119910 (119905))

2

sum119899

119905=1(119910 (119905) minus 119910)

2

(14)


0 50 100 150 200 250 300 350 400 450 500

0

05

1

15

Time

minus05

minus1

minus15

x(t)

Real valueLPP value

KLPP value

(a)

Time0 50 100 150 200 250 300 350 400 450 500

0

05

1

Erro

r

minus05

minus1

minus15

times10minus5

LPP errorKLPP error

(b)

Time0 5 10 15 20 25 30 35 40 45 50

0

05

1

15

minus05

minus1

minus15

x(t)

Real valueLPP value

KLPP value

(c)

Time0 5 10 15 20 25 30 35 40 45 50

0

1

2

3

Erro

r

minus3

minus2

minus1

LPP errorKLPP error

(d)



= 120590 (119910 minus 119909)

119910 = (119903 minus 119911) 119909 minus 119910

= 119909119910 minus 119887119911

(15)









0 50 100 150 200 250 300 350 400 450 5000

1

2

3

4

5

Time

The p

olyn

omia

l ord

er

LPPKLPP

(a)

Time0 50 100 150 200 250 300 350 400 450 500

1

2

3

4

5

6

Dela

y of

the l

ag v

aria

ble

LPPKLPP

(b)

Time0 50 100 150 200 250 300 350 400 450 500

0

50

100

150

Num

ber o

f nea

rest

neig

hbor

poi

nts

LPPKLPP

(c)

Time0 50 100 150 200 250 300 350 400 450 500

0

05

1

15

Radi

us o

f the

nei

ghbo

rs

KLPP

(d)



119909 (119905 + 1) = (1 minus 119887) 119909 (119905) +

119886119909 (119905 minus 120591)

1 + 119909 (119905 minus 120591)10 (16)






119909 (119905 + 1) = 1 minus 119886119909 (119905)2+ 119910 (119905)

119910 (119905 + 1) = 119887119909 (119905)

(17)





0 5 10 15 20 25 30 35 40 45 500

1

2

3

4

5

Time

The p

olyn

omia

l ord

er

LPPKLPP

(a)

Time0 5 10 15 20 25 30 35 40 45 50

1

2

3

4

5

6

Dela

y of

the l

ag v

aria

ble

LPPKLPP

(b)

Time0 5 10 15 20 25 30 35 40 45 50

020406080

100120140

Num

ber o

f nea

rest

neig

hbor

poi

nts

LPPKLPP

(c)

Time0 5 10 15 20 25 30 35 40 45 50

02

04

06

08

1

12

Radi

us o

f the

nei

ghbo

rs

KLPP

(d)







ERNN (2007) [9] 879119864 minus 06 990119864 minus 10

PSO-LSSVM (2009) [25] 180119864 minus 04

FBMNN (2009) [10] 205119864 minus 05


IEOP (2010) [33] 105119864 minus 05

CSAA-SVR (2012) [23] 876119864 minus 04

COA-SVR (2012) [24] 303119864 minus 03

ESN (2012) [35] 250119864 minus 03

HR (2012) [35] 150119864 minus 03

WESN (2012) [31] 958119864 minus 04

ARMA-RESN (2013) [8] 303119864 minus 06

RBFNN (2013) [10] 500119864 minus 03

LNF (2013) [21] 640119864 minus 06





0 100 200 300 400 500 600 700 800 900 10000

05

1

15

2

Time

x(t)

Real valueLPP value

KLPP value

(a)

Time0 100 200 300 400 500 600 700 800 900 1000

0020406

Erro

r

minus02

minus04

minus06

minus08

LPP errorKLPP error

(b)

Time0 50 100 150 200 250 300 350

002040608

1

minus02

minus04

x(t)

Real valueLPP value

KLPP value

(c)

Time0 50 100 150 200 250 300 350

0

05

1

15

Erro

r

minus05

LPP errorKLPP error

(d)




ERNN (2007) [9] 420119864 minus 05 315119864 minus 08

PSO-LSSVM (2009) [25] 280119864 minus 03

FBMNN (2009) [10] 302119864 minus 04


IEOP (2010) [33] 194119864 minus 05

ODCF-SVM (2012) [22] 160119864 minus 02

LNF (2013) [21] 790119864 minus 04

LSSVR (2013) [27] 570119864 minus 03








RBF-OLS (2009) [10] 413119864 minus 02

FBMNN (2009) [10] 270119864 minus 05







0 100 200 300 400 500 600 700 800 900 10000

1

2

3

4

5

Time

The p

olyn

omia

l ord

er

LPPKLPP

(a)

Time0 100 200 300 400 500 600 700 800 900 1000

115

225

335

445

5

Dela

y of

the l

ag v

aria

ble

LPPKLPP

(b)

Time100 200 300 400 500 600 700 800 900 1000

020406080

100120140

Num

ber o

f nea

rest

neig

hbor

poi

nts

LPPKLPP

(c)

Time0 100 200 300 400 500 600 700 800 900 1000

002040608

11214

Radi

us o

f the

nei

ghbo

rs

KLPP

(d)




RBF-OLS (2009) [10] 460119864 minus 02


ARMA-RESN (2013) [8] 157937





4 Conclusions




0 50 100 1500

1

2

3

4

5

Time

The p

olyn

omia

l ord

er

LPPKLPP

(a)

Time0 50 100 150

115

225

335

445

5

Dela

y of

the l

ag v

aria

ble

LPPKLPP

(b)

Time20 40 60 80 100 120 140

020406080

100120140

Num

ber o

f nea

rest

neig

hbor

poi

nts

LPPKLPP

(c)

Time0 50 100 150 200 250 300 350

001020304050607

Radi

us o

f the

nei

ghbo

rs

KLPP

(d)




Acknowledgments


References





















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 50 100 150 200 250 300 350 400 450 5000

1

2

3

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The p

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l ord

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LPPKLPP

(a)

0 50 100 150 200 250 300 350 400 450 5001

2

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5

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Dela

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ag v

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LPPKLPP

(b)

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20

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(c)

0 50 100 150 200 250 300 350 400 450 500005

01

015

02

025

03

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f the

nei

ghbo

rs

Time

KLPP

(d)




theory that

120579 = 120594

119879119882120594

minus1

120594119879119882119884 (11)

where 119884 = (1199091198721+(119898minus1)120591+1

119909119872119902+(119898minus1)120591+1

)119879 and 119882 is a


119894ℎ) as its


119894119909119872119894minus120591

119880119894119909119872119894+(119898minus1)120591

] as its 119894th row and 119880119894=

[1 119909119872119894+(119898minus1)120591minus119889

119909119903



2

GDF (ℎ) =(119884 minus ) (119884 minus )

119879

119902 minus 119863

(12)

where = 120594120579 = 120594120594

119879119882120594minus1120594119879119882119884 119863 = tr119867 =


119894minus 119883119872 (119894 =


our models




119905= (119909119905 119909119905+120591 119909

119905+(119898minus1)120591)119879









119872+(119898minus1)120591+1





0 50 100 150 200 250 300 350 400 450 5000

1

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(a)

0 50 100 150 200 250 300 350 400 450 5001

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Time

LPPKLPP

(b)

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(c)

0 50 100 150 200 250 300 350 400 450 500005

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Time

KLPP

(d)






119895119894from (7) in Step 3





max (119909) minusmin (119909) (13)


MAE = max 1003816100381610038161003816119910 (119905) minus 119910 (119905)

1003816100381610038161003816


119899

sum

119905=1

(119910 (119905) minus 119910 (119905))2

NMSE =sum119899

119905=1(119910 (119905) minus 119910 (119905))

2

sum119899

119905=1(119910 (119905) minus 119910)

2

(14)


0 50 100 150 200 250 300 350 400 450 500

0

05

1

15

Time

minus05

minus1

minus15

x(t)

Real valueLPP value

KLPP value

(a)

Time0 50 100 150 200 250 300 350 400 450 500

0

05

1

Erro

r

minus05

minus1

minus15

times10minus5

LPP errorKLPP error

(b)

Time0 5 10 15 20 25 30 35 40 45 50

0

05

1

15

minus05

minus1

minus15

x(t)

Real valueLPP value

KLPP value

(c)

Time0 5 10 15 20 25 30 35 40 45 50

0

1

2

3

Erro

r

minus3

minus2

minus1

LPP errorKLPP error

(d)



= 120590 (119910 minus 119909)

119910 = (119903 minus 119911) 119909 minus 119910

= 119909119910 minus 119887119911

(15)









0 50 100 150 200 250 300 350 400 450 5000

1

2

3

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l ord

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LPPKLPP

(a)

Time0 50 100 150 200 250 300 350 400 450 500

1

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LPPKLPP

(b)

Time0 50 100 150 200 250 300 350 400 450 500

0

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(c)

Time0 50 100 150 200 250 300 350 400 450 500

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f the

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rs

KLPP

(d)



119909 (119905 + 1) = (1 minus 119887) 119909 (119905) +

119886119909 (119905 minus 120591)

1 + 119909 (119905 minus 120591)10 (16)






119909 (119905 + 1) = 1 minus 119886119909 (119905)2+ 119910 (119905)

119910 (119905 + 1) = 119887119909 (119905)

(17)





0 5 10 15 20 25 30 35 40 45 500

1

2

3

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The p

olyn

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l ord

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LPPKLPP

(a)

Time0 5 10 15 20 25 30 35 40 45 50

1

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Dela

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the l

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LPPKLPP

(b)

Time0 5 10 15 20 25 30 35 40 45 50

020406080

100120140

Num

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f nea

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hbor

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LPPKLPP

(c)

Time0 5 10 15 20 25 30 35 40 45 50

02

04

06

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1

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Radi

us o

f the

nei

ghbo

rs

KLPP

(d)







ERNN (2007) [9] 879119864 minus 06 990119864 minus 10

PSO-LSSVM (2009) [25] 180119864 minus 04

FBMNN (2009) [10] 205119864 minus 05


IEOP (2010) [33] 105119864 minus 05

CSAA-SVR (2012) [23] 876119864 minus 04

COA-SVR (2012) [24] 303119864 minus 03

ESN (2012) [35] 250119864 minus 03

HR (2012) [35] 150119864 minus 03

WESN (2012) [31] 958119864 minus 04

ARMA-RESN (2013) [8] 303119864 minus 06

RBFNN (2013) [10] 500119864 minus 03

LNF (2013) [21] 640119864 minus 06





0 100 200 300 400 500 600 700 800 900 10000

05

1

15

2

Time

x(t)

Real valueLPP value

KLPP value

(a)

Time0 100 200 300 400 500 600 700 800 900 1000

0020406

Erro

r

minus02

minus04

minus06

minus08

LPP errorKLPP error

(b)

Time0 50 100 150 200 250 300 350

002040608

1

minus02

minus04

x(t)

Real valueLPP value

KLPP value

(c)

Time0 50 100 150 200 250 300 350

0

05

1

15

Erro

r

minus05

LPP errorKLPP error

(d)




ERNN (2007) [9] 420119864 minus 05 315119864 minus 08

PSO-LSSVM (2009) [25] 280119864 minus 03

FBMNN (2009) [10] 302119864 minus 04


IEOP (2010) [33] 194119864 minus 05

ODCF-SVM (2012) [22] 160119864 minus 02

LNF (2013) [21] 790119864 minus 04

LSSVR (2013) [27] 570119864 minus 03








RBF-OLS (2009) [10] 413119864 minus 02

FBMNN (2009) [10] 270119864 minus 05







0 100 200 300 400 500 600 700 800 900 10000

1

2

3

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The p

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l ord

er

LPPKLPP

(a)

Time0 100 200 300 400 500 600 700 800 900 1000

115

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5

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y of

the l

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LPPKLPP

(b)

Time100 200 300 400 500 600 700 800 900 1000

020406080

100120140

Num

ber o

f nea

rest

neig

hbor

poi

nts

LPPKLPP

(c)

Time0 100 200 300 400 500 600 700 800 900 1000

002040608

11214

Radi

us o

f the

nei

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rs

KLPP

(d)




RBF-OLS (2009) [10] 460119864 minus 02


ARMA-RESN (2013) [8] 157937





4 Conclusions




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(a)

Time0 50 100 150

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(b)

Time20 40 60 80 100 120 140

020406080

100120140

Num

ber o

f nea

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neig

hbor

poi

nts

LPPKLPP

(c)

Time0 50 100 150 200 250 300 350

001020304050607

Radi

us o

f the

nei

ghbo

rs

KLPP

(d)




Acknowledgments


References





















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 50 100 150 200 250 300 350 400 450 5000

1

2

3

4

5

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The p

olyn

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l ord

er

LPPKLPP

(a)

0 50 100 150 200 250 300 350 400 450 5001

2

3

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5

6

Dela

y of

the l

ag v

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ble

Time

LPPKLPP

(b)

50 100 150 200 250 300 350 400 450 5000

20

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Num

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(c)

0 50 100 150 200 250 300 350 400 450 500005

01

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f the

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rs

Time

KLPP

(d)






119895119894from (7) in Step 3





max (119909) minusmin (119909) (13)


MAE = max 1003816100381610038161003816119910 (119905) minus 119910 (119905)

1003816100381610038161003816


119899

sum

119905=1

(119910 (119905) minus 119910 (119905))2

NMSE =sum119899

119905=1(119910 (119905) minus 119910 (119905))

2

sum119899

119905=1(119910 (119905) minus 119910)

2

(14)


0 50 100 150 200 250 300 350 400 450 500

0

05

1

15

Time

minus05

minus1

minus15

x(t)

Real valueLPP value

KLPP value

(a)

Time0 50 100 150 200 250 300 350 400 450 500

0

05

1

Erro

r

minus05

minus1

minus15

times10minus5

LPP errorKLPP error

(b)

Time0 5 10 15 20 25 30 35 40 45 50

0

05

1

15

minus05

minus1

minus15

x(t)

Real valueLPP value

KLPP value

(c)

Time0 5 10 15 20 25 30 35 40 45 50

0

1

2

3

Erro

r

minus3

minus2

minus1

LPP errorKLPP error

(d)



= 120590 (119910 minus 119909)

119910 = (119903 minus 119911) 119909 minus 119910

= 119909119910 minus 119887119911

(15)









0 50 100 150 200 250 300 350 400 450 5000

1

2

3

4

5

Time

The p

olyn

omia

l ord

er

LPPKLPP

(a)

Time0 50 100 150 200 250 300 350 400 450 500

1

2

3

4

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6

Dela

y of

the l

ag v

aria

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LPPKLPP

(b)

Time0 50 100 150 200 250 300 350 400 450 500

0

50

100

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ber o

f nea

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LPPKLPP

(c)

Time0 50 100 150 200 250 300 350 400 450 500

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Radi

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f the

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rs

KLPP

(d)



119909 (119905 + 1) = (1 minus 119887) 119909 (119905) +

119886119909 (119905 minus 120591)

1 + 119909 (119905 minus 120591)10 (16)






119909 (119905 + 1) = 1 minus 119886119909 (119905)2+ 119910 (119905)

119910 (119905 + 1) = 119887119909 (119905)

(17)





0 5 10 15 20 25 30 35 40 45 500

1

2

3

4

5

Time

The p

olyn

omia

l ord

er

LPPKLPP

(a)

Time0 5 10 15 20 25 30 35 40 45 50

1

2

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Dela

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the l

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LPPKLPP

(b)

Time0 5 10 15 20 25 30 35 40 45 50

020406080

100120140

Num

ber o

f nea

rest

neig

hbor

poi

nts

LPPKLPP

(c)

Time0 5 10 15 20 25 30 35 40 45 50

02

04

06

08

1

12

Radi

us o

f the

nei

ghbo

rs

KLPP

(d)







ERNN (2007) [9] 879119864 minus 06 990119864 minus 10

PSO-LSSVM (2009) [25] 180119864 minus 04

FBMNN (2009) [10] 205119864 minus 05


IEOP (2010) [33] 105119864 minus 05

CSAA-SVR (2012) [23] 876119864 minus 04

COA-SVR (2012) [24] 303119864 minus 03

ESN (2012) [35] 250119864 minus 03

HR (2012) [35] 150119864 minus 03

WESN (2012) [31] 958119864 minus 04

ARMA-RESN (2013) [8] 303119864 minus 06

RBFNN (2013) [10] 500119864 minus 03

LNF (2013) [21] 640119864 minus 06





0 100 200 300 400 500 600 700 800 900 10000

05

1

15

2

Time

x(t)

Real valueLPP value

KLPP value

(a)

Time0 100 200 300 400 500 600 700 800 900 1000

0020406

Erro

r

minus02

minus04

minus06

minus08

LPP errorKLPP error

(b)

Time0 50 100 150 200 250 300 350

002040608

1

minus02

minus04

x(t)

Real valueLPP value

KLPP value

(c)

Time0 50 100 150 200 250 300 350

0

05

1

15

Erro

r

minus05

LPP errorKLPP error

(d)




ERNN (2007) [9] 420119864 minus 05 315119864 minus 08

PSO-LSSVM (2009) [25] 280119864 minus 03

FBMNN (2009) [10] 302119864 minus 04


IEOP (2010) [33] 194119864 minus 05

ODCF-SVM (2012) [22] 160119864 minus 02

LNF (2013) [21] 790119864 minus 04

LSSVR (2013) [27] 570119864 minus 03








RBF-OLS (2009) [10] 413119864 minus 02

FBMNN (2009) [10] 270119864 minus 05







0 100 200 300 400 500 600 700 800 900 10000

1

2

3

4

5

Time

The p

olyn

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l ord

er

LPPKLPP

(a)

Time0 100 200 300 400 500 600 700 800 900 1000

115

225

335

445

5

Dela

y of

the l

ag v

aria

ble

LPPKLPP

(b)

Time100 200 300 400 500 600 700 800 900 1000

020406080

100120140

Num

ber o

f nea

rest

neig

hbor

poi

nts

LPPKLPP

(c)

Time0 100 200 300 400 500 600 700 800 900 1000

002040608

11214

Radi

us o

f the

nei

ghbo

rs

KLPP

(d)




RBF-OLS (2009) [10] 460119864 minus 02


ARMA-RESN (2013) [8] 157937





4 Conclusions




0 50 100 1500

1

2

3

4

5

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The p

olyn

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l ord

er

LPPKLPP

(a)

Time0 50 100 150

115

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5

Dela

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the l

ag v

aria

ble

LPPKLPP

(b)

Time20 40 60 80 100 120 140

020406080

100120140

Num

ber o

f nea

rest

neig

hbor

poi

nts

LPPKLPP

(c)

Time0 50 100 150 200 250 300 350

001020304050607

Radi

us o

f the

nei

ghbo

rs

KLPP

(d)




Acknowledgments


References





















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 50 100 150 200 250 300 350 400 450 500

0

05

1

15

Time

minus05

minus1

minus15

x(t)

Real valueLPP value

KLPP value

(a)

Time0 50 100 150 200 250 300 350 400 450 500

0

05

1

Erro

r

minus05

minus1

minus15

times10minus5

LPP errorKLPP error

(b)

Time0 5 10 15 20 25 30 35 40 45 50

0

05

1

15

minus05

minus1

minus15

x(t)

Real valueLPP value

KLPP value

(c)

Time0 5 10 15 20 25 30 35 40 45 50

0

1

2

3

Erro

r

minus3

minus2

minus1

LPP errorKLPP error

(d)



= 120590 (119910 minus 119909)

119910 = (119903 minus 119911) 119909 minus 119910

= 119909119910 minus 119887119911

(15)









0 50 100 150 200 250 300 350 400 450 5000

1

2

3

4

5

Time

The p

olyn

omia

l ord

er

LPPKLPP

(a)

Time0 50 100 150 200 250 300 350 400 450 500

1

2

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5

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Dela

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the l

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LPPKLPP

(b)

Time0 50 100 150 200 250 300 350 400 450 500

0

50

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Num

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LPPKLPP

(c)

Time0 50 100 150 200 250 300 350 400 450 500

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f the

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rs

KLPP

(d)



119909 (119905 + 1) = (1 minus 119887) 119909 (119905) +

119886119909 (119905 minus 120591)

1 + 119909 (119905 minus 120591)10 (16)






119909 (119905 + 1) = 1 minus 119886119909 (119905)2+ 119910 (119905)

119910 (119905 + 1) = 119887119909 (119905)

(17)





0 5 10 15 20 25 30 35 40 45 500

1

2

3

4

5

Time

The p

olyn

omia

l ord

er

LPPKLPP

(a)

Time0 5 10 15 20 25 30 35 40 45 50

1

2

3

4

5

6

Dela

y of

the l

ag v

aria

ble

LPPKLPP

(b)

Time0 5 10 15 20 25 30 35 40 45 50

020406080

100120140

Num

ber o

f nea

rest

neig

hbor

poi

nts

LPPKLPP

(c)

Time0 5 10 15 20 25 30 35 40 45 50

02

04

06

08

1

12

Radi

us o

f the

nei

ghbo

rs

KLPP

(d)







ERNN (2007) [9] 879119864 minus 06 990119864 minus 10

PSO-LSSVM (2009) [25] 180119864 minus 04

FBMNN (2009) [10] 205119864 minus 05


IEOP (2010) [33] 105119864 minus 05

CSAA-SVR (2012) [23] 876119864 minus 04

COA-SVR (2012) [24] 303119864 minus 03

ESN (2012) [35] 250119864 minus 03

HR (2012) [35] 150119864 minus 03

WESN (2012) [31] 958119864 minus 04

ARMA-RESN (2013) [8] 303119864 minus 06

RBFNN (2013) [10] 500119864 minus 03

LNF (2013) [21] 640119864 minus 06





0 100 200 300 400 500 600 700 800 900 10000

05

1

15

2

Time

x(t)

Real valueLPP value

KLPP value

(a)

Time0 100 200 300 400 500 600 700 800 900 1000

0020406

Erro

r

minus02

minus04

minus06

minus08

LPP errorKLPP error

(b)

Time0 50 100 150 200 250 300 350

002040608

1

minus02

minus04

x(t)

Real valueLPP value

KLPP value

(c)

Time0 50 100 150 200 250 300 350

0

05

1

15

Erro

r

minus05

LPP errorKLPP error

(d)




ERNN (2007) [9] 420119864 minus 05 315119864 minus 08

PSO-LSSVM (2009) [25] 280119864 minus 03

FBMNN (2009) [10] 302119864 minus 04


IEOP (2010) [33] 194119864 minus 05

ODCF-SVM (2012) [22] 160119864 minus 02

LNF (2013) [21] 790119864 minus 04

LSSVR (2013) [27] 570119864 minus 03








RBF-OLS (2009) [10] 413119864 minus 02

FBMNN (2009) [10] 270119864 minus 05







0 100 200 300 400 500 600 700 800 900 10000

1

2

3

4

5

Time

The p

olyn

omia

l ord

er

LPPKLPP

(a)

Time0 100 200 300 400 500 600 700 800 900 1000

115

225

335

445

5

Dela

y of

the l

ag v

aria

ble

LPPKLPP

(b)

Time100 200 300 400 500 600 700 800 900 1000

020406080

100120140

Num

ber o

f nea

rest

neig

hbor

poi

nts

LPPKLPP

(c)

Time0 100 200 300 400 500 600 700 800 900 1000

002040608

11214

Radi

us o

f the

nei

ghbo

rs

KLPP

(d)




RBF-OLS (2009) [10] 460119864 minus 02


ARMA-RESN (2013) [8] 157937





4 Conclusions




0 50 100 1500

1

2

3

4

5

Time

The p

olyn

omia

l ord

er

LPPKLPP

(a)

Time0 50 100 150

115

225

335

445

5

Dela

y of

the l

ag v

aria

ble

LPPKLPP

(b)

Time20 40 60 80 100 120 140

020406080

100120140

Num

ber o

f nea

rest

neig

hbor

poi

nts

LPPKLPP

(c)

Time0 50 100 150 200 250 300 350

001020304050607

Radi

us o

f the

nei

ghbo

rs

KLPP

(d)




Acknowledgments


References





















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 50 100 150 200 250 300 350 400 450 5000

1

2

3

4

5

Time

The p

olyn

omia

l ord

er

LPPKLPP

(a)

Time0 50 100 150 200 250 300 350 400 450 500

1

2

3

4

5

6

Dela

y of

the l

ag v

aria

ble

LPPKLPP

(b)

Time0 50 100 150 200 250 300 350 400 450 500

0

50

100

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Num

ber o

f nea

rest

neig

hbor

poi

nts

LPPKLPP

(c)

Time0 50 100 150 200 250 300 350 400 450 500

0

05

1

15

Radi

us o

f the

nei

ghbo

rs

KLPP

(d)



119909 (119905 + 1) = (1 minus 119887) 119909 (119905) +

119886119909 (119905 minus 120591)

1 + 119909 (119905 minus 120591)10 (16)






119909 (119905 + 1) = 1 minus 119886119909 (119905)2+ 119910 (119905)

119910 (119905 + 1) = 119887119909 (119905)

(17)





0 5 10 15 20 25 30 35 40 45 500

1

2

3

4

5

Time

The p

olyn

omia

l ord

er

LPPKLPP

(a)

Time0 5 10 15 20 25 30 35 40 45 50

1

2

3

4

5

6

Dela

y of

the l

ag v

aria

ble

LPPKLPP

(b)

Time0 5 10 15 20 25 30 35 40 45 50

020406080

100120140

Num

ber o

f nea

rest

neig

hbor

poi

nts

LPPKLPP

(c)

Time0 5 10 15 20 25 30 35 40 45 50

02

04

06

08

1

12

Radi

us o

f the

nei

ghbo

rs

KLPP

(d)







ERNN (2007) [9] 879119864 minus 06 990119864 minus 10

PSO-LSSVM (2009) [25] 180119864 minus 04

FBMNN (2009) [10] 205119864 minus 05


IEOP (2010) [33] 105119864 minus 05

CSAA-SVR (2012) [23] 876119864 minus 04

COA-SVR (2012) [24] 303119864 minus 03

ESN (2012) [35] 250119864 minus 03

HR (2012) [35] 150119864 minus 03

WESN (2012) [31] 958119864 minus 04

ARMA-RESN (2013) [8] 303119864 minus 06

RBFNN (2013) [10] 500119864 minus 03

LNF (2013) [21] 640119864 minus 06





0 100 200 300 400 500 600 700 800 900 10000

05

1

15

2

Time

x(t)

Real valueLPP value

KLPP value

(a)

Time0 100 200 300 400 500 600 700 800 900 1000

0020406

Erro

r

minus02

minus04

minus06

minus08

LPP errorKLPP error

(b)

Time0 50 100 150 200 250 300 350

002040608

1

minus02

minus04

x(t)

Real valueLPP value

KLPP value

(c)

Time0 50 100 150 200 250 300 350

0

05

1

15

Erro

r

minus05

LPP errorKLPP error

(d)




ERNN (2007) [9] 420119864 minus 05 315119864 minus 08

PSO-LSSVM (2009) [25] 280119864 minus 03

FBMNN (2009) [10] 302119864 minus 04


IEOP (2010) [33] 194119864 minus 05

ODCF-SVM (2012) [22] 160119864 minus 02

LNF (2013) [21] 790119864 minus 04

LSSVR (2013) [27] 570119864 minus 03








RBF-OLS (2009) [10] 413119864 minus 02

FBMNN (2009) [10] 270119864 minus 05







0 100 200 300 400 500 600 700 800 900 10000

1

2

3

4

5

Time

The p

olyn

omia

l ord

er

LPPKLPP

(a)

Time0 100 200 300 400 500 600 700 800 900 1000

115

225

335

445

5

Dela

y of

the l

ag v

aria

ble

LPPKLPP

(b)

Time100 200 300 400 500 600 700 800 900 1000

020406080

100120140

Num

ber o

f nea

rest

neig

hbor

poi

nts

LPPKLPP

(c)

Time0 100 200 300 400 500 600 700 800 900 1000

002040608

11214

Radi

us o

f the

nei

ghbo

rs

KLPP

(d)




RBF-OLS (2009) [10] 460119864 minus 02


ARMA-RESN (2013) [8] 157937





4 Conclusions




0 50 100 1500

1

2

3

4

5

Time

The p

olyn

omia

l ord

er

LPPKLPP

(a)

Time0 50 100 150

115

225

335

445

5

Dela

y of

the l

ag v

aria

ble

LPPKLPP

(b)

Time20 40 60 80 100 120 140

020406080

100120140

Num

ber o

f nea

rest

neig

hbor

poi

nts

LPPKLPP

(c)

Time0 50 100 150 200 250 300 350

001020304050607

Radi

us o

f the

nei

ghbo

rs

KLPP

(d)




Acknowledgments


References





















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 5 10 15 20 25 30 35 40 45 500

1

2

3

4

5

Time

The p

olyn

omia

l ord

er

LPPKLPP

(a)

Time0 5 10 15 20 25 30 35 40 45 50

1

2

3

4

5

6

Dela

y of

the l

ag v

aria

ble

LPPKLPP

(b)

Time0 5 10 15 20 25 30 35 40 45 50

020406080

100120140

Num

ber o

f nea

rest

neig

hbor

poi

nts

LPPKLPP

(c)

Time0 5 10 15 20 25 30 35 40 45 50

02

04

06

08

1

12

Radi

us o

f the

nei

ghbo

rs

KLPP

(d)







ERNN (2007) [9] 879119864 minus 06 990119864 minus 10

PSO-LSSVM (2009) [25] 180119864 minus 04

FBMNN (2009) [10] 205119864 minus 05


IEOP (2010) [33] 105119864 minus 05

CSAA-SVR (2012) [23] 876119864 minus 04

COA-SVR (2012) [24] 303119864 minus 03

ESN (2012) [35] 250119864 minus 03

HR (2012) [35] 150119864 minus 03

WESN (2012) [31] 958119864 minus 04

ARMA-RESN (2013) [8] 303119864 minus 06

RBFNN (2013) [10] 500119864 minus 03

LNF (2013) [21] 640119864 minus 06





0 100 200 300 400 500 600 700 800 900 10000

05

1

15

2

Time

x(t)

Real valueLPP value

KLPP value

(a)

Time0 100 200 300 400 500 600 700 800 900 1000

0020406

Erro

r

minus02

minus04

minus06

minus08

LPP errorKLPP error

(b)

Time0 50 100 150 200 250 300 350

002040608

1

minus02

minus04

x(t)

Real valueLPP value

KLPP value

(c)

Time0 50 100 150 200 250 300 350

0

05

1

15

Erro

r

minus05

LPP errorKLPP error

(d)




ERNN (2007) [9] 420119864 minus 05 315119864 minus 08

PSO-LSSVM (2009) [25] 280119864 minus 03

FBMNN (2009) [10] 302119864 minus 04


IEOP (2010) [33] 194119864 minus 05

ODCF-SVM (2012) [22] 160119864 minus 02

LNF (2013) [21] 790119864 minus 04

LSSVR (2013) [27] 570119864 minus 03








RBF-OLS (2009) [10] 413119864 minus 02

FBMNN (2009) [10] 270119864 minus 05







0 100 200 300 400 500 600 700 800 900 10000

1

2

3

4

5

Time

The p

olyn

omia

l ord

er

LPPKLPP

(a)

Time0 100 200 300 400 500 600 700 800 900 1000

115

225

335

445

5

Dela

y of

the l

ag v

aria

ble

LPPKLPP

(b)

Time100 200 300 400 500 600 700 800 900 1000

020406080

100120140

Num

ber o

f nea

rest

neig

hbor

poi

nts

LPPKLPP

(c)

Time0 100 200 300 400 500 600 700 800 900 1000

002040608

11214

Radi

us o

f the

nei

ghbo

rs

KLPP

(d)




RBF-OLS (2009) [10] 460119864 minus 02


ARMA-RESN (2013) [8] 157937





4 Conclusions




0 50 100 1500

1

2

3

4

5

Time

The p

olyn

omia

l ord

er

LPPKLPP

(a)

Time0 50 100 150

115

225

335

445

5

Dela

y of

the l

ag v

aria

ble

LPPKLPP

(b)

Time20 40 60 80 100 120 140

020406080

100120140

Num

ber o

f nea

rest

neig

hbor

poi

nts

LPPKLPP

(c)

Time0 50 100 150 200 250 300 350

001020304050607

Radi

us o

f the

nei

ghbo

rs

KLPP

(d)




Acknowledgments


References





















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 100 200 300 400 500 600 700 800 900 10000

05

1

15

2

Time

x(t)

Real valueLPP value

KLPP value

(a)

Time0 100 200 300 400 500 600 700 800 900 1000

0020406

Erro

r

minus02

minus04

minus06

minus08

LPP errorKLPP error

(b)

Time0 50 100 150 200 250 300 350

002040608

1

minus02

minus04

x(t)

Real valueLPP value

KLPP value

(c)

Time0 50 100 150 200 250 300 350

0

05

1

15

Erro

r

minus05

LPP errorKLPP error

(d)




ERNN (2007) [9] 420119864 minus 05 315119864 minus 08

PSO-LSSVM (2009) [25] 280119864 minus 03

FBMNN (2009) [10] 302119864 minus 04


IEOP (2010) [33] 194119864 minus 05

ODCF-SVM (2012) [22] 160119864 minus 02

LNF (2013) [21] 790119864 minus 04

LSSVR (2013) [27] 570119864 minus 03








RBF-OLS (2009) [10] 413119864 minus 02

FBMNN (2009) [10] 270119864 minus 05







0 100 200 300 400 500 600 700 800 900 10000

1

2

3

4

5

Time

The p

olyn

omia

l ord

er

LPPKLPP

(a)

Time0 100 200 300 400 500 600 700 800 900 1000

115

225

335

445

5

Dela

y of

the l

ag v

aria

ble

LPPKLPP

(b)

Time100 200 300 400 500 600 700 800 900 1000

020406080

100120140

Num

ber o

f nea

rest

neig

hbor

poi

nts

LPPKLPP

(c)

Time0 100 200 300 400 500 600 700 800 900 1000

002040608

11214

Radi

us o

f the

nei

ghbo

rs

KLPP

(d)




RBF-OLS (2009) [10] 460119864 minus 02


ARMA-RESN (2013) [8] 157937





4 Conclusions




0 50 100 1500

1

2

3

4

5

Time

The p

olyn

omia

l ord

er

LPPKLPP

(a)

Time0 50 100 150

115

225

335

445

5

Dela

y of

the l

ag v

aria

ble

LPPKLPP

(b)

Time20 40 60 80 100 120 140

020406080

100120140

Num

ber o

f nea

rest

neig

hbor

poi

nts

LPPKLPP

(c)

Time0 50 100 150 200 250 300 350

001020304050607

Radi

us o

f the

nei

ghbo

rs

KLPP

(d)




Acknowledgments


References





















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 100 200 300 400 500 600 700 800 900 10000

1

2

3

4

5

Time

The p

olyn

omia

l ord

er

LPPKLPP

(a)

Time0 100 200 300 400 500 600 700 800 900 1000

115

225

335

445

5

Dela

y of

the l

ag v

aria

ble

LPPKLPP

(b)

Time100 200 300 400 500 600 700 800 900 1000

020406080

100120140

Num

ber o

f nea

rest

neig

hbor

poi

nts

LPPKLPP

(c)

Time0 100 200 300 400 500 600 700 800 900 1000

002040608

11214

Radi

us o

f the

nei

ghbo

rs

KLPP

(d)




RBF-OLS (2009) [10] 460119864 minus 02


ARMA-RESN (2013) [8] 157937





4 Conclusions




0 50 100 1500

1

2

3

4

5

Time

The p

olyn

omia

l ord

er

LPPKLPP

(a)

Time0 50 100 150

115

225

335

445

5

Dela

y of

the l

ag v

aria

ble

LPPKLPP

(b)

Time20 40 60 80 100 120 140

020406080

100120140

Num

ber o

f nea

rest

neig

hbor

poi

nts

LPPKLPP

(c)

Time0 50 100 150 200 250 300 350

001020304050607

Radi

us o

f the

nei

ghbo

rs

KLPP

(d)




Acknowledgments


References





















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 50 100 1500

1

2

3

4

5

Time

The p

olyn

omia

l ord

er

LPPKLPP

(a)

Time0 50 100 150

115

225

335

445

5

Dela

y of

the l

ag v

aria

ble

LPPKLPP

(b)

Time20 40 60 80 100 120 140

020406080

100120140

Num

ber o

f nea

rest

neig

hbor

poi

nts

LPPKLPP

(c)

Time0 50 100 150 200 250 300 350

001020304050607

Radi

us o

f the

nei

ghbo

rs

KLPP

(d)




Acknowledgments


References





















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra



















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra









research article local prediction of chaotic time...

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