Research ArticleMulti-Step-Ahead Forecasting Model for Monthly AnchovyCatches Based on Wavelet Analysis

Nibaldo Rodriacuteguez Claudio Cubillos and Joseacute-Miguel Rubio

School of Informatics Engineering Pontificia Universidad Catolica de Valparaıso 2362807 Valparaıso Chile

Correspondence should be addressed to Nibaldo Rodrıguez nibaldorodriguezucvcl

Received 29 July 2013 Revised 29 December 2013 Accepted 12 January 2014 Published 24 February 2014

Academic Editor Hector Pomares

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

This paper presents a 119901-step-ahead forecasting strategy based on two stages to improve pelagic fish-catch time-series modeling byconsidering annual and interannual fluctuations for northern Chile (18∘Sndash24∘S) In the first stage the stationary wavelet transformis used to separate the raw time series into an annual component and an interannual component whereas the periodicities of eachcomponent are obtained using theMorlet wavelet power spectrum In the second stage a linear autoregressivemodel is constructedto predict each component and the unknown 119901-next values are forecasted by the addition of the two predicted components Wedemonstrate the utility of the proposed forecasting model on monthly anchovy-catches time series for periods from January 1963to December 2007 Empirical results obtained for 10-month-ahead forecasting showed the effectiveness of the proposed waveletautoregressive strategy

1 Introduction

During the last decades both the direct method and theiterative method have been generally used in literatureto implement 119901-step-ahead forecasting model [1ndash6] Theiterative method iterates 119901 times the same one-step-aheadforecasting model to obtain the 119901 predicted values whereasthe direct method estimates a set of 119901 forecasting modelseach returning a forecast for the 120591th value 120591 isin 1 2 119901 Inotherwords119901models are calibrated from the time series (onefor each horizon) On the other hand multi-step-ahead fore-casting models play an important role in the management ofmarine resources However the multi-step-ahead forecastingtask is difficult due to factors such as accumulation of errorsreduced accuracy increased uncertainty and nonstationarity[1ndash6]

In recent years linear regression model [7ndash9] and artifi-cial neuronal networks (ANN) [10 11] have been proposed forpelagic fishes forecasting model The disadvantage of modelsbased on linear regressions is the supposition of stationarityand linearity of the time series of pelagic species catchesAlthough ANN allow modelling the nonlinear behaviour oftime series they also have some disadvantages such as the

stagnancy of local minima due to the steepest descent learn-ing method A multilayer perceptron neural network modelfor forecasting the anchovy and sardines catches of northernChile was proposed by [10 11] which reported a coefficient ofdetermination between 82 and 87 Noteworthy is that theanchovy and sardine are important pelagic fisheries resourcesfor economic development of northern Chile

In this paper a 119901-step-ahead forecasting strategy basedon wavelet analysis and linear autoregressive (AR) modelis proposed to improve prediction accuracy of monthlyanchovy catches in the coastal zone of northern Chile Theadvantage of wavelet analysis is its ability to separate lowfrequency components and high frequency components froma nonstationary time series Besides each component is moreregular than the raw time series which can help improvethe forecasting performance [12 13] On the other hand thewavelet analysis was also selected due to its successful usein electricity market [12 13] smoothing methods [14ndash17]financial market [18ndash20] and ecological time series modeling[21 22] Moreover variability analysis at different time scalesbased on the wavelet power spectrumhas shown that climaticoscillations such as the El Nino-Southern Oscillation signifi-cantly affect the abundance of marine species [23 24]

Hindawi Publishing CorporationJournal of Applied MathematicsVolume 2014 Article ID 798464 8 pageshttpdxdoiorg1011552014798464

2 Journal of Applied Mathematics

Finally the proposed 119901-step-ahead forecasting strategyis based on two stages In the first stage the raw timeseries is decomposed using the 3-level stationary wavelettransform (SWT) to separate both the annual component(AC) and the interannual component (IAC) The periodicityof each component at different time scales is detected andquantified using the global Morlet wavelet power spectrumIn the second stage both the AC and IAC are forecastedusing a linear AR model and the unknown 119901-next values areforecasted by the algebraic sum of the two predicted values

The rest of this paper is organized as follows Section 2briefly describes the wavelet analysis The 119901-step-ahead fore-casting strategy is presented in Section 3 The experimentalresults and performance evaluation are presented in Section 4followed by the conclusions in Section 5

2 Wavelet Analysis

In this section we briefly present the continuous wavelettransform and the discrete stationary wavelet transform

21 Continuous Wavelet Transform The continuous wavelettransform (CWT) decomposes a continuous time signal 119909(119905)into a family of ldquodaughter-waveletrdquo functions 120595

119904119887(119905) which

are generated by the dilatation and the translation of amotherwavelet function The CWT coefficients of a real signal 119909(119905)are obtained for the different scales and translations as follows[25 26]

119882(119904 119887) = int

infin

infin

119909 (119905) 120595lowast

119904119887

(119905) 119889119905 (1)

where the lowast denotes the complex conjugate 119904 is a dilatation(scale) factor that controls the width of the wavelet functionand 119887 is a translation factor controlling its locationThe scaledand translated daughter-wavelet function is obtained as

120595119904119887(119905) =

1

radic119904120595(119905 minus 119887

119904) 119904 isin R

+

119887 isin R (2)

where 120595(119905) is the mother wavelet function and the choiceof the wavelet function is not arbitrary There are severalconsiderations whenmaking the choice of a wavelet functionfor example real versus complex wavelets continuous versusdiscrete wavelets and orthogonal versus redundant waveletsHowever all the wavelets share a general feature That is fastoscillations have good time resolution but lower frequencyresolution whereas low oscillations have good frequencybut poor time resolution In this paper the CWT has beenimplemented using the Morlet mother wavelet functiondefined as

120595 (119905) = 120587minus14 exp (119894120596

0) exp(minus119905

2

2) (3)

where 1205960defines the frequency center of the wavelet and

here is set equal to 1205960= 6 as it yields the function to

have zero mean and be localized in both time and frequencyspace as well as providing a good balance between time andfrequency Moreover for the Morlet wavelet the scale 119904 is

inversely related to the Fourier frequency that is 119891 = 1119904which simplifies the interpretation of the wavelet analysis andonemay replace the scale 119904 by the Fourier period 1119891 [25 26]

For discrete time series 119909119899= 119909(119899) 119899 = 0 1 119873 minus 1

with uniform time step 120575119905 its CWT is obtained as a 119869 times 119873matrix whose (119895 119896) elements are given by [27 28]

119882(119895 119896) = radic120575119905

119904

119873minus1

sum

119899=0

119909 (119899) 120595lowast

((119899 minus 119896) 120575119905

119904119895

)

119895 = 0 1 119869 minus 1

(4)

where 119904119895represents a set of scales 119896 = 0 1 119873 minus 1 denotes

the shifting index 119869 determines the largest scale and119873 is thenumber of data points in the time series

The global wavelet power spectrum GWPS of a discretetime series 119909

119899is calculated as [27 28]

GWPS (119904119895) =1

119873

119873minus1

sum

119899=0

1003816100381610038161003816119882 (119895 119896)10038161003816100381610038162

119895 = 0 1 119869 minus 1 (5)

where GWPS represents the cumulate variance contributedby each time scale and |119882(119895 119896)|2 denotes the local variancedistribution of the time series in the time-scale plane

22 Stationary Wavelet Transform The SWT was indepen-dently proposed by several researchers and it is known inliterature under a variety of names such as the nondecimatedwavelet transform invariant wavelet transform and redun-dant wavelet transformThe key feature is that it gives a betterapproximation than the discrete wavelet transform (DWT)since it is redundant linear and shift invariant [14ndash17] TheSWT is similar to the DWT [25 26] in that the high-pass andlow-pass filters are applied to the input signal at each levelbut the output signal is never decimated Instead the filtersare upsampled at each level of decomposition [14]

Now we consider the following discrete time series 1198860=

119909119899with 119873 = 2119869 for some integer 119869 At the first level of SWT

the input signal 1198860is convolved with a low-pass filter ℎ

0of

length 119903 to obtain the approximation coefficient 1198861and with

a high-pass filter 1198920of length 119903 to obtain the detail coefficient

1198891 That is

1198861(119899) = sum

119896

ℎ0(119899 minus 119896) 119886

0(119896) (6a)

1198891(119899) = sum

119896

1198920(119899 minus 119896) 119886

0(119896) (6b)

because no subsampling is performed and 1198861and 119889

1are of

length119873 instead of1198732 as in the DWT case At the next levelof the SWT the approximation coefficient 119886

1is split into two

parts using the same previous scheme but with a new pair offilter ℎ

1and 119892

1 which are obtained by inserting a zero value

between the elements of the filters used in the previous step

Journal of Applied Mathematics 3

InputA time series 119909 = 119909(119899) 119899 = 0 1 119873 minus 1 of119873 samplesOutput Annual and inter-annual component

(1) 119869 larr 3(2)119882119899 larr 1198891198872(3) [119886 119889] larr swt(119909 119869119882119899)(4) Auxlarr zeros(119869119873)(5) 119909119886

larr 119894swt(Aux 119889119882119899)(6) 119909ia larr 119894swt(119886Aux119882119899)(7) return 119909

119886

119909ia

Algorithm 1 Component separation

(ie ℎ0and 119892

0) The general process of the SWT is continued

iteratively for 119895 = 1 119869 minus 1 and is given as

119886119895+1(119899) = sum

119896

ℎ119895(119899 minus 119896) 119886

119895(119896) (7a)

119889119895+1(119899) = sum

119896

119892119895(119899 minus 119896) 119886

119895(119896) (7b)

where ℎ119895and119892119895are obtained by the upsampling operatorThe

upsample operator inserts 2119895minus1 zeros between the elements ofthe filters ℎ

0and 119892

0 respectivelyThe SWT is fully defined by

the choice of a pair of filters (ie ℎ0and 119892

0) and the number

of decomposition step 119869

3 Multi-Step-Ahead Forecasting Strategy

31 Wavelet Preprocessing At this stage the SWT is usedto extract both the AC and IAC by using the Daubechieslow-passhigh-pass filter with four coefficients (Db2) andthree levels of decomposition The periodic behavior of eachcomponent was obtained by using the GWPS with a 95confidence level [27 28] which can be seen in Figures 23 and 4 Algorithm 1 explains the component separationprocess considering the Db2 wavelet filter and 119869-levels ofdecomposition In this algorithm line (3) performs theseparation of components by using 3-level SWT with Db2wavelet filter whereas line (5) and line (6) implement thereconstruction of the AC and the IAC using the inverse 3-level SWT

32Wavelet Autoregressive Forecasting Theproposed119901-step-ahead forecasting strategy is based on direct method Thedirectmethod is to learn119901 single outputmodels each return-ing a direct prediction of the future value 119909(119899+119901) In order topredict the future value119909(119899+119901)firstlywe separate the originaltime series into two components by using Algorithm 1 Thefirst component 119909

119886presents the annual variabilities and is

characterized by fast dynamic while the second component119909ia indicates interannual fluctuation and is characterized byslow dynamics Therefore the proposed forecasting modelwill be the coaddition of two predicted values given by

119909 (119899 + 120591) = 119909119886(119899 + 120591) + 119909ia (119899 + 120591) + 119890 (119899) (8)

where 120591 represents the forecasting horizon and 119890(119899) denotesthe 119899th value of the residual component

The 119909119886(119899+120591) and 119909ia(119899+120591) are estimated by using a linear

autoregressive function with exogenous inputs given by thefollowing two equations respectively

119909119886(119899 + 120591) = 119865

119886120591[119909119886(119899) 119909ia (119899)]

=

119898minus1

sum

119894=0

120572119894119909119886(119899 minus 119894) +

119898minus1

sum

119894=0

120572119898+119894119909ia (119899 minus 119894)

(9)

where [119909119886(119899minus 119894)] represents the endogenous regressor vector

whereas [119909ia(119899 minus 119894)] plays the role of exogenous regressorvector and119898 is the size of a time window

119909ia (119899 + 120591) = 119865ia120591 [119909ia (119899) 119909119886 (119899)]

=

119898minus1

sum

119894=0

120573119894119909ia (119899 minus 119894) +

119898minus1

sum

119894=0

120573119898+119894119909119886(119899 minus 119894)

(10)

where [119909ia(119899 minus 119894)] represents the endogenous regressor vectorand the exogenous regressor vector is denoted by [119909

119894(119899 minus 119894)]

In order to estimate the parameters 120572119894 and 120573

119894 the linear

least square method is used Now suppose a set of 119879 traininginput-output samples then we can perform 119879 equations ofthe following forms

119883119886= 119885120572 (11a)

119883ia = 119885120573 (11b)

where 119885 is the regressor matrix of 119879 rows and 2119898 columnsand 120572 and 120573 are vectors of 2119898 rows and one column Finallythe 120572 and 120573 values are calculated by using the followingequations

120572 = 119885dagger

119883119886 (12a)

120573 = 119885dagger

119883ia (12b)

where (sdot)dagger denotes the Moore-Penrose pseudoinverse [29]

4 Measures of Accuracy Applied inthe Models Performance

In this paper three criteria of accuracy are used to evaluatethe estimation capabilities during the test phase of the evalu-ated models They are root mean square error (RMSE) mean

4 Journal of Applied Mathematics

1963

1965

1967

1969

1971

1973

1975

1977

1979

1981

1983

1985

1987

1989

1991

1993

1995

1997

1999

2001

2003

2005

2007

0

02

04

06

08

1

Time (month)

Nor

mal

ized

obs

erve

d m

onth

ly ca

tche

s

Figure 1 Observed monthly anchovy catches data

absolute percentage error (MAPE) and relative percentageerror (RPE) which are calculated as

RMSE = radic 1119872

119872

sum

119899=1

(119909 (119899) minus 119909 (119899))2

MAPE = 1119872

119872

sum

119899=1

10038161003816100381610038161003816100381610038161003816

119909 (119899) minus 119909 (119899)

119909 (119899)

10038161003816100381610038161003816100381610038161003816times 100

RPE = 119909 (119899) minus 119909 (119899)119909 (119899)

times 100

(13)

where119909(119899) is the actual value atmonth 119899119909(119899) is the predictedvalue at month 119899 and 119872 is the corresponding number oftesting (validation) values

5 Results

The time series data analyzed in this paper corresponds tothe monthly fish catch of anchovy for periods from January1963 to December 2007 (httpwwwsernapescacl) Besidesthe original time series data is smoothed using the followingoperation 119909 = radic119909 The new smoothed original time series isshown in Figure 1Thefirst stepwas to implement the compo-nent separation process presented in Algorithm 1 using threedifferent wavelet filters with three levels of decompositionto select the most suitable wavelet filter The performanceof Algorithm 1 was evaluated using the Daubechies filterSymlets filter and Coiflets filter The first two filters areimplemented using four and six coefficients denoted as Db2Sym2 Db3 and Sym3 respectively whereas the third filteris evaluated using six and twelve coefficients denoted asCoif1 and Coif2 The following step was to detect the mostsignificant periodic fluctuations of each component based onthe GWPS The main goal of the detection phase was to findthe significant peak power that explained the periodicities ofthe time series

The GWPS for the AC and the IAC is presented inFigures 2 3 and 4 respectively In these figures the peakpower indicates the catches high activity The black thick linedesignates the 95 confidence level against red noise and thevalues of the peak power obtained on the black thick line aresignificant [27 28]

Figures 2(a) and 2(b) show the AC time series and theGWPS respectively From Figure 2(b) it can be observedthat there are two peaks of significant power The first peakhas periodicities of 6 months whereas the second peak hasa period of 12 months Therefore the AC time series hasa seasonal pattern On the other hand the AC time seriesobtained by using 4-level SWT with Db2 wavelet filter ispresented in Figure 3(a) whereas Figure 3(b) shows that ACtime series has a cycle of 20months and another of 30monthswhich leads to the conclusion that this new AC time seriesdoes not meet seasonal behavior because it has interannualperiods Therefore only 3-level SWT is used to evaluate theperformance of the wavelet autoregressive (WAR) forecastingmodel proposed

The AIC time series and GWPS are presented in Figures4(a) and 4(b) respectively From Figure 4(b) it can beobserved that there are three peaks of significant powerThe first peak has a period of 84 months and the secondpeak has a periodicity of 31 months while the last peak hasa periodic behavior of 152 months Therefore these threepeaks represent significant interannual fluctuations and canbe associated with the El Nino Southern Oscillation effect[30 31]

Once both the AC and IAC were reconstructed by usingthe 3-level SWT with Db2 wavelet filter each componentwas divided into two parts a training data set (119879 = 360observations 23) and a test data set (119872 = 180 observations13) The training data set was firstly used to estimate theparameters of the WAR forecasting models and the testingdata set was used for computing the performancemeasures ofthemodels and for validation purposesTheWAR forecastingmodel was calibrated with 2119898 = 2 times 30 previous months asinput data due to the interannual variability of the originaltime series In the calibration process overall parameters wereestimated using the linear least squares method

Theperformancemeasures of theWAR forecastingmodelas a function of the forecasting horizon using differentwavelet filters with 3-level SWT are shown in Figure 5 FromFigure 5 it is observed that the Coif2 Db3 and Sym3 gavea poor performance in comparison to Coif1 Sym3 andDb2 On the contrary Db2 and Sym2 achieved the betterperformance Therefore the Db2 wavelet filter was used inall further analysis From Figure 5 it can be seen that theperformance measures significantly increase their values fora time horizon of more than 10-month-ahead forecasting

Once both the AC and IAC have been separately pre-dicted their values must be added in order to obtain the119901-step-ahead anchovy-catches forecasting model In the fol-lowing we present only 10-month-ahead forecasting resultsduring the testing phase whose results are illustrated inFigures 6 7 and 8

The performance evaluation of the 11986511988610[sdot] and 119865

11988611989410[sdot]

models is presented in Figures 6 and 7 As seen in Figures

Journal of Applied Mathematics 5

0020406

Time (month)

Ann

ual m

onth

ly ca

tche

s

1963

1965

1967

1969

1971

1973

1975

1977

1979

1981

1983

1985

1987

1989

1991

1993

1995

1997

1999

2001

2003

2005

2007

minus02

minus04

minus06

(a)

1 3 5 7 9 11 13 15 17 19 21 23 25 270

05

15

2

25

Period (month)

Glo

bal W

PS

(b)

Figure 2 (a) Annual component (119909119886

) from 3-level SWT and (b) GWPS

Time (month)

Ann

ual m

onth

ly ca

tche

s

1963

1965

1967

1969

1971

1973

1975

1977

1979

1981

1983

1985

1987

1989

1991

1993

1995

1997

1999

2001

2003

2005

2007

minus02

minus04

0

02

04

06

(a)

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35Period (month)

Glo

bal W

PS0

05

1

15

2

25

3

35

(b)

Figure 3 (a) Annual component (119909119886

) from 4-level SWT and (b) GWPS

Time (month)

1963

1965

1967

1969

1971

1973

1975

1977

1979

1981

1983

1985

1987

1989

1991

1993

1995

1997

1999

2001

2003

2005

2007

04

06

02

0

Inte

rann

ual m

onth

ly ca

tche

s

(a)

0123456789

10

Period (month)

Glo

bal W

PS

1 9

17

25

33

41

49

57

65

73

81

89

97

105

113

121

129

137

145

153

161

169

(b)

Figure 4 (a) Interannual (119909ia) component from 3-level SWT and (b) GWPS

1 2 3 4 5 6 7 8 9 10 11 12048

1216202428

Forecast time horizon (month)

MA

PE (

)

Coif2Sym3Db3

Coif1Sym2Db2

(a)

1 2 3 4 5 6 7 8 9 10 11 1200001001010020100301004010050100601007010080100901

Forecast time horizon (month)

RMSE

Coif2Sym3Db3

Coif1Sym2Db2

(b)

Figure 5 Forecasting horizon for WAR forecasting

6 Journal of Applied Mathematics

1993

1994

1995

1996

1997

1998

1999

2000

2001

2002

2003

2004

2005

2006

2007

00125

0250375

Actual sample (month)

Ann

ual c

ompo

nent

minus0375

minus025

minus0125

minus05

Estimated catchesActual catches

(a)

050

100150

Rela

tive e

rror

()

1993

1994

1995

1996

1997

1998

1999

2000

2001

2002

2003

2004

2005

2006

2007

Actual sample (month)

minus50

minus100

minus150

(b)

Figure 6 10-month-ahead WAR forecasting for annual component (119909119886

)

00125

0250375

050625

1993

1994

1995

1996

1997

1998

1999

2000

2001

2002

2003

2004

2005

2006

2007

Actual sample (month)Estimated catchesActual catches

Inte

rann

ual c

ompo

nent

(a)

02468

10

Actual sample (month)

Rela

tive e

rror

()

1993

1994

1995

1996

1997

1998

1999

2000

2001

2002

2003

2004

2005

2006

2007

minus8

minus6minus4minus2

minus10

(b)

Figure 7 10-month-ahead WAR forecasting for interannual component (119909ia)

00125

0250375

050625

0750875

Nor

mal

ized

mon

thly

catc

hes

1993

1994

1995

1996

1997

1998

1999

2000

2001

2002

2003

2004

2005

2006

2007

Actual sample (month)Estimated catches a + iaRaw actual catches

X X

X

(a)

010203040

Rela

tive e

rror

()

1993

1994

1995

1996

1997

1998

1999

2000

2001

2002

2003

2004

2005

2006

2007

Actual sample (month)

minus40

minus30

minus20

minus10

(b)

Figure 8 10-month-ahead WAR forecasting for observed anchovy catches versus estimated anchovy catches

6(a) and 7(a) the actual value and predicted value are veryclose for the testing data with low performance measuresThe model 119865

11988610[sdot] achieves a RMSE of 00093 and MAPE of

36 whereas the 119865ia10[sdot] model obtained a RMSE of 00062and MAPE 138 On the other hand from Figure 6(b) itis leveraged that over 80 of the predicted AC catches areacceptable with residual ranging from minus20 to 20 and

Figure 7(b) illustrates that 96 of the predicted IAC catchesvalues are within the range of 4

Figure 8(a) provides the observed monthly anchovy-catches data set versus forecasted anchovy catches whoseforecasting behavior is very accurate for testing data witha RMSE and MAPE of 00093 and 266 respectively Onthe other hand from Figure 8(b) it can be observed that an

Journal of Applied Mathematics 7

important fraction (over 95) of the predicted values areacceptable with residuals ranging from minus10 to 10

6 Conclusions

This paper presented a 119901-month-ahead forecasting strategybased on two stages to improve pelagic fish-catch time-series modeling It is based on wavelet analysis and linearAR modeling In the first stages the two time series areconstructed after 3-level stationary wavelet decompositioncontaining information about the original time series atannual and interannual time scale whereas in the secondstage each time series is modeled by using a linear ARmodel As time series are not independent each AR modelincluded information of the other time series to improve theforecasting accuracyThe 10-month-ahead forecasting resultsshow that the thirty previous months of each time seriescontain valuable information to explain the highest variancelevel of the monthly anchovy catches of northern Chile forperiods from January 1963 to December 2007 Finally thewavelet autoregressive forecasting strategy can be suitable asa very promising methodology for any other marine speciesof the fishing industry

Conflict of Interests

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

Acknowledgments

This work was supported in part by Grant CONICYTFONDECYTRegular 1131105 and by the DI project of thePontificia Universidad Catolica de Valparaıso

References

[1] L Zhang W-D Zhou P-C Chang J-W Yang and F-Z LIldquoIterated time series prediction with multiple support vectorregression modelsrdquo Neurocomputing vol 99 pp 411ndash422 2013

[2] C Hamzacebi D Akay and F Kutay ldquoComparison of directand iterative artificial neural network forecast approaches inmulti-periodic time series forecastingrdquo Expert Systems withApplications vol 36 no 2 pp 3839ndash3844 2009

[3] A Sorjamaa J Hao N Reyhani Y Ji and A LendasseldquoMethodology for long-term prediction of time seriesrdquo Neuro-computing vol 70 no 16ndash18 pp 2861ndash2869 2007

[4] H Cheng P-N Tan J Gao and J Scripps ldquoMultistep-aheadtime series predictionrdquo inAdvances in Knowledge Discovery andData Mining W-K Ng M Kitsuregawa J Li and K ChangEds vol 3918 of Lecture Notes in Computer Science pp 765ndash774 Springer Berlin Germany 2006

[5] D M Kline ldquoMethods for multi-step time series forecastingwith neural networksrdquo in Neural Networks in Business Fore-casting G P Zhang Ed pp 226ndash250 Information ScienceHershey Pa USA 2004

[6] G C Tiao and R S Tsay ldquoSome advances in non-linear andadaptive modelling in time-seriesrdquo Journal of Forecasting vol13 no 2 pp 109ndash131 1994

[7] K I Stergiou E D Christou and G Petrakis ldquoModelling andforecastingmonthly fisheries catches comparison of regressionunivariate and multivariate time series methodsrdquo FisheriesResearch vol 29 no 1 pp 55ndash95 1997

[8] K I Stergiou and E D Christou ldquoModelling and forecastingannual fisheries catches comparison of regression univariateand multivariate time series methodsrdquo Fisheries Research vol25 no 2 pp 105ndash138 1996

[9] K I Stergiou ldquoShort-term fisheries forecasting comparisonof smoothing ARIMA and regression techniquesrdquo Journal ofApplied Ichthyology vol 7 no 4 pp 193ndash204 1991

[10] J C Gutierrez-Estrada C Silva E Yanez N Rodrıguez and IPulido-Calvo ldquoMonthly catch forecasting of anchovy Engraulisringens in the north area of Chile non-linear univariateapproachrdquo Fisheries Research vol 86 no 2-3 pp 188ndash200 2007

[11] E Yanez F Plaza J C Gutierrez-Estrada et al ldquoAnchovy(Engraulis ringens) and sardine (Sardinops sagax) abundanceforecast off northern Chile a multivariate ecosystemic neuralnetwork approachrdquo Progress in Oceanography vol 87 no 1ndash4pp 242ndash250 2010

[12] N A Shrivastava and B K Panigrahi ldquoA hybrid wavelet-ELM based short term price forecasting for electricity marketsrdquoInternational Journal of Electrical Power amp Energy Systems vol55 pp 41ndash50 2014

[13] N Amjady and F Keynia ldquoDay ahead price forecasting ofelectricity markets by a mixed data model and hybrid forecastmethodrdquo International Journal of Electrical Power amp EnergySystems vol 30 no 9 pp 533ndash546 2008

[14] G Nason and B Silverman ldquoThe stationary wavelet transformand some statistical applicationsrdquo inWavelets and Statistics vol103 of Lecture Notes in Statistics pp 281ndash299 Springer NewYork NY USA 1995

[15] R R Coifman and D L Donoho ldquoTranslation-invariant de-noisingrdquo in Wavelets and Statistics vol 103 of Lecture Notes inStatistics pp 125ndash150 Springer New York NY USA 1995

[16] J-C Pesquet H Krim and H Carfantan ldquoTime-invariantorthonormal wavelet representationsrdquo IEEE Transactions onSignal Processing vol 44 no 8 pp 1964ndash1970 1996

[17] D B Percival andA TWaldenWaveletMethods for Time SeriesAnalysis Cambridge University Press Cambridge UK 2000

[18] S Lahmiri ldquoForecasting direction of the S-P500 movementusing wavelet transform and support vector machinesrdquo Inter-national Journal of Strategic Decision Sciences vol 4 no 1 pp78ndash88 2013

[19] T-J Hsieh H-F Hsiao andW-C Yeh ldquoForecasting stock mar-kets using wavelet transforms and recurrent neural networksan integrated system based on artificial bee colony algorithmrdquoApplied Soft Computing vol 11 no 2 pp 2510ndash2525 2011

[20] B-L Zhang R Coggins M A Jabri D Dersch and B FlowerldquoMultiresolution forecasting for futures trading using waveletdecompositionsrdquo IEEETransactions onNeural Networks vol 12no 4 pp 765ndash775 2001

[21] M Hidalgo T Rouyer J C Molinero et al ldquoSynergistic effectsof fishing-induced demographic changes and climate variationon fish population dynamicsrdquo Marine Ecology Progress Seriesvol 426 pp 1ndash12 2011

[22] B Cazelles M Chavez D Berteaux et al ldquoWavelet analysis ofecological time seriesrdquo Oecologia vol 156 no 2 pp 287ndash3042008

[23] T Rouyer J-M Fromentin N C Stenseth and B CazellesldquoAnalysing multiple time series and extending significance

8 Journal of Applied Mathematics

testing in wavelet analysisrdquoMarine Ecology Progress Series vol359 pp 11ndash23 2008

[24] C-H Hsieh C-S Chen T-S Chiu et al ldquoTime series analysesreveal transient relationships between abundance of larvalanchovy and environmental variables in the coastal waterssouthwest of Taiwanrdquo Fisheries Oceanography vol 18 no 2 pp102ndash117 2009

[25] IDaubechiesTenLectures onWavelets SIAMPhiladelphia PaUSA 1992

[26] S Mallat A Wavelet Tour of Signal Processing Academic PressSan Diego Calif USA 1998

[27] C Torrence and G P Compo ldquoA practical guide to waveletanalysisrdquoBulletin of the AmericanMeteorological Society vol 79no 1 pp 61ndash78 1998

[28] A Grinsted J C Moore and S Jevrejeva ldquoApplication of thecross wavelet transform and wavelet coherence to geophysicaltimes seriesrdquo Nonlinear Processes in Geophysics vol 11 no 5-6pp 561ndash566 2004

[29] D SerreMatricesTheory and Applications Springer NewYorkNY USA 2002

[30] A Quetglas F Ordines M Hidalgo et al ldquoSynchronouscombined effects of fishing and climate within a demersalcommunityrdquo ICES Journal of Marine Science vol 70 no 2 pp319ndash328 2013

[31] S E Lluch-Cota A Pares-Sierra V O Magana-Rueda etal ldquoChanging climate in the Gulf of Californiardquo Progress inOceanography vol 87 no 1ndash4 pp 114ndash126 2010

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Journal of Applied Mathematics

Finally the proposed 119901-step-ahead forecasting strategyis based on two stages In the first stage the raw timeseries is decomposed using the 3-level stationary wavelettransform (SWT) to separate both the annual component(AC) and the interannual component (IAC) The periodicityof each component at different time scales is detected andquantified using the global Morlet wavelet power spectrumIn the second stage both the AC and IAC are forecastedusing a linear AR model and the unknown 119901-next values areforecasted by the algebraic sum of the two predicted values

The rest of this paper is organized as follows Section 2briefly describes the wavelet analysis The 119901-step-ahead fore-casting strategy is presented in Section 3 The experimentalresults and performance evaluation are presented in Section 4followed by the conclusions in Section 5

2 Wavelet Analysis

In this section we briefly present the continuous wavelettransform and the discrete stationary wavelet transform

21 Continuous Wavelet Transform The continuous wavelettransform (CWT) decomposes a continuous time signal 119909(119905)into a family of ldquodaughter-waveletrdquo functions 120595

119904119887(119905) which

are generated by the dilatation and the translation of amotherwavelet function The CWT coefficients of a real signal 119909(119905)are obtained for the different scales and translations as follows[25 26]

119882(119904 119887) = int

infin

infin

119909 (119905) 120595lowast

119904119887

(119905) 119889119905 (1)

where the lowast denotes the complex conjugate 119904 is a dilatation(scale) factor that controls the width of the wavelet functionand 119887 is a translation factor controlling its locationThe scaledand translated daughter-wavelet function is obtained as

120595119904119887(119905) =

1

radic119904120595(119905 minus 119887

119904) 119904 isin R

+

119887 isin R (2)

where 120595(119905) is the mother wavelet function and the choiceof the wavelet function is not arbitrary There are severalconsiderations whenmaking the choice of a wavelet functionfor example real versus complex wavelets continuous versusdiscrete wavelets and orthogonal versus redundant waveletsHowever all the wavelets share a general feature That is fastoscillations have good time resolution but lower frequencyresolution whereas low oscillations have good frequencybut poor time resolution In this paper the CWT has beenimplemented using the Morlet mother wavelet functiondefined as

120595 (119905) = 120587minus14 exp (119894120596

0) exp(minus119905

2

2) (3)

where 1205960defines the frequency center of the wavelet and

here is set equal to 1205960= 6 as it yields the function to

have zero mean and be localized in both time and frequencyspace as well as providing a good balance between time andfrequency Moreover for the Morlet wavelet the scale 119904 is

inversely related to the Fourier frequency that is 119891 = 1119904which simplifies the interpretation of the wavelet analysis andonemay replace the scale 119904 by the Fourier period 1119891 [25 26]

For discrete time series 119909119899= 119909(119899) 119899 = 0 1 119873 minus 1

with uniform time step 120575119905 its CWT is obtained as a 119869 times 119873matrix whose (119895 119896) elements are given by [27 28]

119882(119895 119896) = radic120575119905

119904

119873minus1

sum

119899=0

119909 (119899) 120595lowast

((119899 minus 119896) 120575119905

119904119895

)

119895 = 0 1 119869 minus 1

(4)

where 119904119895represents a set of scales 119896 = 0 1 119873 minus 1 denotes

the shifting index 119869 determines the largest scale and119873 is thenumber of data points in the time series

The global wavelet power spectrum GWPS of a discretetime series 119909

119899is calculated as [27 28]

GWPS (119904119895) =1

119873

119873minus1

sum

119899=0

1003816100381610038161003816119882 (119895 119896)10038161003816100381610038162

119895 = 0 1 119869 minus 1 (5)

where GWPS represents the cumulate variance contributedby each time scale and |119882(119895 119896)|2 denotes the local variancedistribution of the time series in the time-scale plane

22 Stationary Wavelet Transform The SWT was indepen-dently proposed by several researchers and it is known inliterature under a variety of names such as the nondecimatedwavelet transform invariant wavelet transform and redun-dant wavelet transformThe key feature is that it gives a betterapproximation than the discrete wavelet transform (DWT)since it is redundant linear and shift invariant [14ndash17] TheSWT is similar to the DWT [25 26] in that the high-pass andlow-pass filters are applied to the input signal at each levelbut the output signal is never decimated Instead the filtersare upsampled at each level of decomposition [14]

Now we consider the following discrete time series 1198860=

119909119899with 119873 = 2119869 for some integer 119869 At the first level of SWT

the input signal 1198860is convolved with a low-pass filter ℎ

0of

length 119903 to obtain the approximation coefficient 1198861and with

a high-pass filter 1198920of length 119903 to obtain the detail coefficient

1198891 That is

1198861(119899) = sum

119896

ℎ0(119899 minus 119896) 119886

0(119896) (6a)

1198891(119899) = sum

119896

1198920(119899 minus 119896) 119886

0(119896) (6b)

because no subsampling is performed and 1198861and 119889

1are of

length119873 instead of1198732 as in the DWT case At the next levelof the SWT the approximation coefficient 119886

1is split into two

parts using the same previous scheme but with a new pair offilter ℎ

1and 119892

1 which are obtained by inserting a zero value

between the elements of the filters used in the previous step

Journal of Applied Mathematics 3

InputA time series 119909 = 119909(119899) 119899 = 0 1 119873 minus 1 of119873 samplesOutput Annual and inter-annual component

(1) 119869 larr 3(2)119882119899 larr 1198891198872(3) [119886 119889] larr swt(119909 119869119882119899)(4) Auxlarr zeros(119869119873)(5) 119909119886

larr 119894swt(Aux 119889119882119899)(6) 119909ia larr 119894swt(119886Aux119882119899)(7) return 119909

119886

119909ia

Algorithm 1 Component separation

(ie ℎ0and 119892

0) The general process of the SWT is continued

iteratively for 119895 = 1 119869 minus 1 and is given as

119886119895+1(119899) = sum

119896

ℎ119895(119899 minus 119896) 119886

119895(119896) (7a)

119889119895+1(119899) = sum

119896

119892119895(119899 minus 119896) 119886

119895(119896) (7b)

where ℎ119895and119892119895are obtained by the upsampling operatorThe

upsample operator inserts 2119895minus1 zeros between the elements ofthe filters ℎ

0and 119892

0 respectivelyThe SWT is fully defined by

the choice of a pair of filters (ie ℎ0and 119892

0) and the number

of decomposition step 119869

3 Multi-Step-Ahead Forecasting Strategy

31 Wavelet Preprocessing At this stage the SWT is usedto extract both the AC and IAC by using the Daubechieslow-passhigh-pass filter with four coefficients (Db2) andthree levels of decomposition The periodic behavior of eachcomponent was obtained by using the GWPS with a 95confidence level [27 28] which can be seen in Figures 23 and 4 Algorithm 1 explains the component separationprocess considering the Db2 wavelet filter and 119869-levels ofdecomposition In this algorithm line (3) performs theseparation of components by using 3-level SWT with Db2wavelet filter whereas line (5) and line (6) implement thereconstruction of the AC and the IAC using the inverse 3-level SWT

32Wavelet Autoregressive Forecasting Theproposed119901-step-ahead forecasting strategy is based on direct method Thedirectmethod is to learn119901 single outputmodels each return-ing a direct prediction of the future value 119909(119899+119901) In order topredict the future value119909(119899+119901)firstlywe separate the originaltime series into two components by using Algorithm 1 Thefirst component 119909

119886presents the annual variabilities and is

characterized by fast dynamic while the second component119909ia indicates interannual fluctuation and is characterized byslow dynamics Therefore the proposed forecasting modelwill be the coaddition of two predicted values given by

119909 (119899 + 120591) = 119909119886(119899 + 120591) + 119909ia (119899 + 120591) + 119890 (119899) (8)

where 120591 represents the forecasting horizon and 119890(119899) denotesthe 119899th value of the residual component

The 119909119886(119899+120591) and 119909ia(119899+120591) are estimated by using a linear

autoregressive function with exogenous inputs given by thefollowing two equations respectively

119909119886(119899 + 120591) = 119865

119886120591[119909119886(119899) 119909ia (119899)]

=

119898minus1

sum

119894=0

120572119894119909119886(119899 minus 119894) +

119898minus1

sum

119894=0

120572119898+119894119909ia (119899 minus 119894)

(9)

where [119909119886(119899minus 119894)] represents the endogenous regressor vector

whereas [119909ia(119899 minus 119894)] plays the role of exogenous regressorvector and119898 is the size of a time window

119909ia (119899 + 120591) = 119865ia120591 [119909ia (119899) 119909119886 (119899)]

=

119898minus1

sum

119894=0

120573119894119909ia (119899 minus 119894) +

119898minus1

sum

119894=0

120573119898+119894119909119886(119899 minus 119894)

(10)

where [119909ia(119899 minus 119894)] represents the endogenous regressor vectorand the exogenous regressor vector is denoted by [119909

119894(119899 minus 119894)]

In order to estimate the parameters 120572119894 and 120573

119894 the linear

least square method is used Now suppose a set of 119879 traininginput-output samples then we can perform 119879 equations ofthe following forms

119883119886= 119885120572 (11a)

119883ia = 119885120573 (11b)

where 119885 is the regressor matrix of 119879 rows and 2119898 columnsand 120572 and 120573 are vectors of 2119898 rows and one column Finallythe 120572 and 120573 values are calculated by using the followingequations

120572 = 119885dagger

119883119886 (12a)

120573 = 119885dagger

119883ia (12b)

where (sdot)dagger denotes the Moore-Penrose pseudoinverse [29]

4 Measures of Accuracy Applied inthe Models Performance

In this paper three criteria of accuracy are used to evaluatethe estimation capabilities during the test phase of the evalu-ated models They are root mean square error (RMSE) mean

4 Journal of Applied Mathematics

1963

1965

1967

1969

1971

1973

1975

1977

1979

1981

1983

1985

1987

1989

1991

1993

1995

1997

1999

2001

2003

2005

2007

0

02

04

06

08

1

Time (month)

Nor

mal

ized

obs

erve

d m

onth

ly ca

tche

s

Figure 1 Observed monthly anchovy catches data

absolute percentage error (MAPE) and relative percentageerror (RPE) which are calculated as

RMSE = radic 1119872

119872

sum

119899=1

(119909 (119899) minus 119909 (119899))2

MAPE = 1119872

119872

sum

119899=1

10038161003816100381610038161003816100381610038161003816

119909 (119899) minus 119909 (119899)

119909 (119899)

10038161003816100381610038161003816100381610038161003816times 100

RPE = 119909 (119899) minus 119909 (119899)119909 (119899)

times 100

(13)

where119909(119899) is the actual value atmonth 119899119909(119899) is the predictedvalue at month 119899 and 119872 is the corresponding number oftesting (validation) values

5 Results

The time series data analyzed in this paper corresponds tothe monthly fish catch of anchovy for periods from January1963 to December 2007 (httpwwwsernapescacl) Besidesthe original time series data is smoothed using the followingoperation 119909 = radic119909 The new smoothed original time series isshown in Figure 1Thefirst stepwas to implement the compo-nent separation process presented in Algorithm 1 using threedifferent wavelet filters with three levels of decompositionto select the most suitable wavelet filter The performanceof Algorithm 1 was evaluated using the Daubechies filterSymlets filter and Coiflets filter The first two filters areimplemented using four and six coefficients denoted as Db2Sym2 Db3 and Sym3 respectively whereas the third filteris evaluated using six and twelve coefficients denoted asCoif1 and Coif2 The following step was to detect the mostsignificant periodic fluctuations of each component based onthe GWPS The main goal of the detection phase was to findthe significant peak power that explained the periodicities ofthe time series

The GWPS for the AC and the IAC is presented inFigures 2 3 and 4 respectively In these figures the peakpower indicates the catches high activity The black thick linedesignates the 95 confidence level against red noise and thevalues of the peak power obtained on the black thick line aresignificant [27 28]

Figures 2(a) and 2(b) show the AC time series and theGWPS respectively From Figure 2(b) it can be observedthat there are two peaks of significant power The first peakhas periodicities of 6 months whereas the second peak hasa period of 12 months Therefore the AC time series hasa seasonal pattern On the other hand the AC time seriesobtained by using 4-level SWT with Db2 wavelet filter ispresented in Figure 3(a) whereas Figure 3(b) shows that ACtime series has a cycle of 20months and another of 30monthswhich leads to the conclusion that this new AC time seriesdoes not meet seasonal behavior because it has interannualperiods Therefore only 3-level SWT is used to evaluate theperformance of the wavelet autoregressive (WAR) forecastingmodel proposed

The AIC time series and GWPS are presented in Figures4(a) and 4(b) respectively From Figure 4(b) it can beobserved that there are three peaks of significant powerThe first peak has a period of 84 months and the secondpeak has a periodicity of 31 months while the last peak hasa periodic behavior of 152 months Therefore these threepeaks represent significant interannual fluctuations and canbe associated with the El Nino Southern Oscillation effect[30 31]

Once both the AC and IAC were reconstructed by usingthe 3-level SWT with Db2 wavelet filter each componentwas divided into two parts a training data set (119879 = 360observations 23) and a test data set (119872 = 180 observations13) The training data set was firstly used to estimate theparameters of the WAR forecasting models and the testingdata set was used for computing the performancemeasures ofthemodels and for validation purposesTheWAR forecastingmodel was calibrated with 2119898 = 2 times 30 previous months asinput data due to the interannual variability of the originaltime series In the calibration process overall parameters wereestimated using the linear least squares method

Theperformancemeasures of theWAR forecastingmodelas a function of the forecasting horizon using differentwavelet filters with 3-level SWT are shown in Figure 5 FromFigure 5 it is observed that the Coif2 Db3 and Sym3 gavea poor performance in comparison to Coif1 Sym3 andDb2 On the contrary Db2 and Sym2 achieved the betterperformance Therefore the Db2 wavelet filter was used inall further analysis From Figure 5 it can be seen that theperformance measures significantly increase their values fora time horizon of more than 10-month-ahead forecasting

Once both the AC and IAC have been separately pre-dicted their values must be added in order to obtain the119901-step-ahead anchovy-catches forecasting model In the fol-lowing we present only 10-month-ahead forecasting resultsduring the testing phase whose results are illustrated inFigures 6 7 and 8

The performance evaluation of the 11986511988610[sdot] and 119865

11988611989410[sdot]

models is presented in Figures 6 and 7 As seen in Figures

Journal of Applied Mathematics 5

0020406

Time (month)

Ann

ual m

onth

ly ca

tche

s

1963

1965

1967

1969

1971

1973

1975

1977

1979

1981

1983

1985

1987

1989

1991

1993

1995

1997

1999

2001

2003

2005

2007

minus02

minus04

minus06

(a)

1 3 5 7 9 11 13 15 17 19 21 23 25 270

05

15

2

25

Period (month)

Glo

bal W

PS

(b)

Figure 2 (a) Annual component (119909119886

) from 3-level SWT and (b) GWPS

Time (month)

Ann

ual m

onth

ly ca

tche

s

1963

1965

1967

1969

1971

1973

1975

1977

1979

1981

1983

1985

1987

1989

1991

1993

1995

1997

1999

2001

2003

2005

2007

minus02

minus04

0

02

04

06

(a)

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35Period (month)

Glo

bal W

PS0

05

1

15

2

25

3

35

(b)

Figure 3 (a) Annual component (119909119886

) from 4-level SWT and (b) GWPS

Time (month)

1963

1965

1967

1969

1971

1973

1975

1977

1979

1981

1983

1985

1987

1989

1991

1993

1995

1997

1999

2001

2003

2005

2007

04

06

02

0

Inte

rann

ual m

onth

ly ca

tche

s

(a)

0123456789

10

Period (month)

Glo

bal W

PS

1 9

17

25

33

41

49

57

65

73

81

89

97

105

113

121

129

137

145

153

161

169

(b)

Figure 4 (a) Interannual (119909ia) component from 3-level SWT and (b) GWPS

1 2 3 4 5 6 7 8 9 10 11 12048

1216202428

Forecast time horizon (month)

MA

PE (

)

Coif2Sym3Db3

Coif1Sym2Db2

(a)

1 2 3 4 5 6 7 8 9 10 11 1200001001010020100301004010050100601007010080100901

Forecast time horizon (month)

RMSE

Coif2Sym3Db3

Coif1Sym2Db2

(b)

Figure 5 Forecasting horizon for WAR forecasting

6 Journal of Applied Mathematics

1993

1994

1995

1996

1997

1998

1999

2000

2001

2002

2003

2004

2005

2006

2007

00125

0250375

Actual sample (month)

Ann

ual c

ompo

nent

minus0375

minus025

minus0125

minus05

Estimated catchesActual catches

(a)

050

100150

Rela

tive e

rror

()

1993

1994

1995

1996

1997

1998

1999

2000

2001

2002

2003

2004

2005

2006

2007

Actual sample (month)

minus50

minus100

minus150

(b)

Figure 6 10-month-ahead WAR forecasting for annual component (119909119886

)

00125

0250375

050625

1993

1994

1995

1996

1997

1998

1999

2000

2001

2002

2003

2004

2005

2006

2007

Actual sample (month)Estimated catchesActual catches

Inte

rann

ual c

ompo

nent

(a)

02468

10

Actual sample (month)

Rela

tive e

rror

()

1993

1994

1995

1996

1997

1998

1999

2000

2001

2002

2003

2004

2005

2006

2007

minus8

minus6minus4minus2

minus10

(b)

Figure 7 10-month-ahead WAR forecasting for interannual component (119909ia)

00125

0250375

050625

0750875

Nor

mal

ized

mon

thly

catc

hes

1993

1994

1995

1996

1997

1998

1999

2000

2001

2002

2003

2004

2005

2006

2007

Actual sample (month)Estimated catches a + iaRaw actual catches

X X

X

(a)

010203040

Rela

tive e

rror

()

1993

1994

1995

1996

1997

1998

1999

2000

2001

2002

2003

2004

2005

2006

2007

Actual sample (month)

minus40

minus30

minus20

minus10

(b)

Figure 8 10-month-ahead WAR forecasting for observed anchovy catches versus estimated anchovy catches

6(a) and 7(a) the actual value and predicted value are veryclose for the testing data with low performance measuresThe model 119865

11988610[sdot] achieves a RMSE of 00093 and MAPE of

36 whereas the 119865ia10[sdot] model obtained a RMSE of 00062and MAPE 138 On the other hand from Figure 6(b) itis leveraged that over 80 of the predicted AC catches areacceptable with residual ranging from minus20 to 20 and

Figure 7(b) illustrates that 96 of the predicted IAC catchesvalues are within the range of 4

Figure 8(a) provides the observed monthly anchovy-catches data set versus forecasted anchovy catches whoseforecasting behavior is very accurate for testing data witha RMSE and MAPE of 00093 and 266 respectively Onthe other hand from Figure 8(b) it can be observed that an

Journal of Applied Mathematics 7

important fraction (over 95) of the predicted values areacceptable with residuals ranging from minus10 to 10

6 Conclusions

This paper presented a 119901-month-ahead forecasting strategybased on two stages to improve pelagic fish-catch time-series modeling It is based on wavelet analysis and linearAR modeling In the first stages the two time series areconstructed after 3-level stationary wavelet decompositioncontaining information about the original time series atannual and interannual time scale whereas in the secondstage each time series is modeled by using a linear ARmodel As time series are not independent each AR modelincluded information of the other time series to improve theforecasting accuracyThe 10-month-ahead forecasting resultsshow that the thirty previous months of each time seriescontain valuable information to explain the highest variancelevel of the monthly anchovy catches of northern Chile forperiods from January 1963 to December 2007 Finally thewavelet autoregressive forecasting strategy can be suitable asa very promising methodology for any other marine speciesof the fishing industry

Conflict of Interests

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

Acknowledgments

This work was supported in part by Grant CONICYTFONDECYTRegular 1131105 and by the DI project of thePontificia Universidad Catolica de Valparaıso

References

[1] L Zhang W-D Zhou P-C Chang J-W Yang and F-Z LIldquoIterated time series prediction with multiple support vectorregression modelsrdquo Neurocomputing vol 99 pp 411ndash422 2013

[2] C Hamzacebi D Akay and F Kutay ldquoComparison of directand iterative artificial neural network forecast approaches inmulti-periodic time series forecastingrdquo Expert Systems withApplications vol 36 no 2 pp 3839ndash3844 2009

[3] A Sorjamaa J Hao N Reyhani Y Ji and A LendasseldquoMethodology for long-term prediction of time seriesrdquo Neuro-computing vol 70 no 16ndash18 pp 2861ndash2869 2007

[4] H Cheng P-N Tan J Gao and J Scripps ldquoMultistep-aheadtime series predictionrdquo inAdvances in Knowledge Discovery andData Mining W-K Ng M Kitsuregawa J Li and K ChangEds vol 3918 of Lecture Notes in Computer Science pp 765ndash774 Springer Berlin Germany 2006

[5] D M Kline ldquoMethods for multi-step time series forecastingwith neural networksrdquo in Neural Networks in Business Fore-casting G P Zhang Ed pp 226ndash250 Information ScienceHershey Pa USA 2004

[6] G C Tiao and R S Tsay ldquoSome advances in non-linear andadaptive modelling in time-seriesrdquo Journal of Forecasting vol13 no 2 pp 109ndash131 1994

[7] K I Stergiou E D Christou and G Petrakis ldquoModelling andforecastingmonthly fisheries catches comparison of regressionunivariate and multivariate time series methodsrdquo FisheriesResearch vol 29 no 1 pp 55ndash95 1997

[8] K I Stergiou and E D Christou ldquoModelling and forecastingannual fisheries catches comparison of regression univariateand multivariate time series methodsrdquo Fisheries Research vol25 no 2 pp 105ndash138 1996

[9] K I Stergiou ldquoShort-term fisheries forecasting comparisonof smoothing ARIMA and regression techniquesrdquo Journal ofApplied Ichthyology vol 7 no 4 pp 193ndash204 1991

[10] J C Gutierrez-Estrada C Silva E Yanez N Rodrıguez and IPulido-Calvo ldquoMonthly catch forecasting of anchovy Engraulisringens in the north area of Chile non-linear univariateapproachrdquo Fisheries Research vol 86 no 2-3 pp 188ndash200 2007

[11] E Yanez F Plaza J C Gutierrez-Estrada et al ldquoAnchovy(Engraulis ringens) and sardine (Sardinops sagax) abundanceforecast off northern Chile a multivariate ecosystemic neuralnetwork approachrdquo Progress in Oceanography vol 87 no 1ndash4pp 242ndash250 2010

[12] N A Shrivastava and B K Panigrahi ldquoA hybrid wavelet-ELM based short term price forecasting for electricity marketsrdquoInternational Journal of Electrical Power amp Energy Systems vol55 pp 41ndash50 2014

[13] N Amjady and F Keynia ldquoDay ahead price forecasting ofelectricity markets by a mixed data model and hybrid forecastmethodrdquo International Journal of Electrical Power amp EnergySystems vol 30 no 9 pp 533ndash546 2008

[14] G Nason and B Silverman ldquoThe stationary wavelet transformand some statistical applicationsrdquo inWavelets and Statistics vol103 of Lecture Notes in Statistics pp 281ndash299 Springer NewYork NY USA 1995

[15] R R Coifman and D L Donoho ldquoTranslation-invariant de-noisingrdquo in Wavelets and Statistics vol 103 of Lecture Notes inStatistics pp 125ndash150 Springer New York NY USA 1995

[16] J-C Pesquet H Krim and H Carfantan ldquoTime-invariantorthonormal wavelet representationsrdquo IEEE Transactions onSignal Processing vol 44 no 8 pp 1964ndash1970 1996

[17] D B Percival andA TWaldenWaveletMethods for Time SeriesAnalysis Cambridge University Press Cambridge UK 2000

[18] S Lahmiri ldquoForecasting direction of the S-P500 movementusing wavelet transform and support vector machinesrdquo Inter-national Journal of Strategic Decision Sciences vol 4 no 1 pp78ndash88 2013

[19] T-J Hsieh H-F Hsiao andW-C Yeh ldquoForecasting stock mar-kets using wavelet transforms and recurrent neural networksan integrated system based on artificial bee colony algorithmrdquoApplied Soft Computing vol 11 no 2 pp 2510ndash2525 2011

[20] B-L Zhang R Coggins M A Jabri D Dersch and B FlowerldquoMultiresolution forecasting for futures trading using waveletdecompositionsrdquo IEEETransactions onNeural Networks vol 12no 4 pp 765ndash775 2001

[21] M Hidalgo T Rouyer J C Molinero et al ldquoSynergistic effectsof fishing-induced demographic changes and climate variationon fish population dynamicsrdquo Marine Ecology Progress Seriesvol 426 pp 1ndash12 2011

[22] B Cazelles M Chavez D Berteaux et al ldquoWavelet analysis ofecological time seriesrdquo Oecologia vol 156 no 2 pp 287ndash3042008

[23] T Rouyer J-M Fromentin N C Stenseth and B CazellesldquoAnalysing multiple time series and extending significance

8 Journal of Applied Mathematics

testing in wavelet analysisrdquoMarine Ecology Progress Series vol359 pp 11ndash23 2008

[24] C-H Hsieh C-S Chen T-S Chiu et al ldquoTime series analysesreveal transient relationships between abundance of larvalanchovy and environmental variables in the coastal waterssouthwest of Taiwanrdquo Fisheries Oceanography vol 18 no 2 pp102ndash117 2009

[25] IDaubechiesTenLectures onWavelets SIAMPhiladelphia PaUSA 1992

[26] S Mallat A Wavelet Tour of Signal Processing Academic PressSan Diego Calif USA 1998

[27] C Torrence and G P Compo ldquoA practical guide to waveletanalysisrdquoBulletin of the AmericanMeteorological Society vol 79no 1 pp 61ndash78 1998

[28] A Grinsted J C Moore and S Jevrejeva ldquoApplication of thecross wavelet transform and wavelet coherence to geophysicaltimes seriesrdquo Nonlinear Processes in Geophysics vol 11 no 5-6pp 561ndash566 2004

[29] D SerreMatricesTheory and Applications Springer NewYorkNY USA 2002

[30] A Quetglas F Ordines M Hidalgo et al ldquoSynchronouscombined effects of fishing and climate within a demersalcommunityrdquo ICES Journal of Marine Science vol 70 no 2 pp319ndash328 2013

[31] S E Lluch-Cota A Pares-Sierra V O Magana-Rueda etal ldquoChanging climate in the Gulf of Californiardquo Progress inOceanography vol 87 no 1ndash4 pp 114ndash126 2010

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Journal of Applied Mathematics 3

InputA time series 119909 = 119909(119899) 119899 = 0 1 119873 minus 1 of119873 samplesOutput Annual and inter-annual component

(1) 119869 larr 3(2)119882119899 larr 1198891198872(3) [119886 119889] larr swt(119909 119869119882119899)(4) Auxlarr zeros(119869119873)(5) 119909119886

larr 119894swt(Aux 119889119882119899)(6) 119909ia larr 119894swt(119886Aux119882119899)(7) return 119909

119886

119909ia

Algorithm 1 Component separation

(ie ℎ0and 119892

0) The general process of the SWT is continued

iteratively for 119895 = 1 119869 minus 1 and is given as

119886119895+1(119899) = sum

119896

ℎ119895(119899 minus 119896) 119886

119895(119896) (7a)

119889119895+1(119899) = sum

119896

119892119895(119899 minus 119896) 119886

119895(119896) (7b)

where ℎ119895and119892119895are obtained by the upsampling operatorThe

upsample operator inserts 2119895minus1 zeros between the elements ofthe filters ℎ

0and 119892

0 respectivelyThe SWT is fully defined by

the choice of a pair of filters (ie ℎ0and 119892

0) and the number

of decomposition step 119869

3 Multi-Step-Ahead Forecasting Strategy

31 Wavelet Preprocessing At this stage the SWT is usedto extract both the AC and IAC by using the Daubechieslow-passhigh-pass filter with four coefficients (Db2) andthree levels of decomposition The periodic behavior of eachcomponent was obtained by using the GWPS with a 95confidence level [27 28] which can be seen in Figures 23 and 4 Algorithm 1 explains the component separationprocess considering the Db2 wavelet filter and 119869-levels ofdecomposition In this algorithm line (3) performs theseparation of components by using 3-level SWT with Db2wavelet filter whereas line (5) and line (6) implement thereconstruction of the AC and the IAC using the inverse 3-level SWT

32Wavelet Autoregressive Forecasting Theproposed119901-step-ahead forecasting strategy is based on direct method Thedirectmethod is to learn119901 single outputmodels each return-ing a direct prediction of the future value 119909(119899+119901) In order topredict the future value119909(119899+119901)firstlywe separate the originaltime series into two components by using Algorithm 1 Thefirst component 119909

119886presents the annual variabilities and is

characterized by fast dynamic while the second component119909ia indicates interannual fluctuation and is characterized byslow dynamics Therefore the proposed forecasting modelwill be the coaddition of two predicted values given by

119909 (119899 + 120591) = 119909119886(119899 + 120591) + 119909ia (119899 + 120591) + 119890 (119899) (8)

where 120591 represents the forecasting horizon and 119890(119899) denotesthe 119899th value of the residual component

The 119909119886(119899+120591) and 119909ia(119899+120591) are estimated by using a linear

autoregressive function with exogenous inputs given by thefollowing two equations respectively

119909119886(119899 + 120591) = 119865

119886120591[119909119886(119899) 119909ia (119899)]

=

119898minus1

sum

119894=0

120572119894119909119886(119899 minus 119894) +

119898minus1

sum

119894=0

120572119898+119894119909ia (119899 minus 119894)

(9)

where [119909119886(119899minus 119894)] represents the endogenous regressor vector

whereas [119909ia(119899 minus 119894)] plays the role of exogenous regressorvector and119898 is the size of a time window

119909ia (119899 + 120591) = 119865ia120591 [119909ia (119899) 119909119886 (119899)]

=

119898minus1

sum

119894=0

120573119894119909ia (119899 minus 119894) +

119898minus1

sum

119894=0

120573119898+119894119909119886(119899 minus 119894)

(10)

where [119909ia(119899 minus 119894)] represents the endogenous regressor vectorand the exogenous regressor vector is denoted by [119909

119894(119899 minus 119894)]

In order to estimate the parameters 120572119894 and 120573

119894 the linear

least square method is used Now suppose a set of 119879 traininginput-output samples then we can perform 119879 equations ofthe following forms

119883119886= 119885120572 (11a)

119883ia = 119885120573 (11b)

where 119885 is the regressor matrix of 119879 rows and 2119898 columnsand 120572 and 120573 are vectors of 2119898 rows and one column Finallythe 120572 and 120573 values are calculated by using the followingequations

120572 = 119885dagger

119883119886 (12a)

120573 = 119885dagger

119883ia (12b)

where (sdot)dagger denotes the Moore-Penrose pseudoinverse [29]

4 Measures of Accuracy Applied inthe Models Performance

In this paper three criteria of accuracy are used to evaluatethe estimation capabilities during the test phase of the evalu-ated models They are root mean square error (RMSE) mean

4 Journal of Applied Mathematics

1963

1965

1967

1969

1971

1973

1975

1977

1979

1981

1983

1985

1987

1989

1991

1993

1995

1997

1999

2001

2003

2005

2007

0

02

04

06

08

1

Time (month)

Nor

mal

ized

obs

erve

d m

onth

ly ca

tche

s

Figure 1 Observed monthly anchovy catches data

absolute percentage error (MAPE) and relative percentageerror (RPE) which are calculated as

RMSE = radic 1119872

119872

sum

119899=1

(119909 (119899) minus 119909 (119899))2

MAPE = 1119872

119872

sum

119899=1

10038161003816100381610038161003816100381610038161003816

119909 (119899) minus 119909 (119899)

119909 (119899)

10038161003816100381610038161003816100381610038161003816times 100

RPE = 119909 (119899) minus 119909 (119899)119909 (119899)

times 100

(13)

where119909(119899) is the actual value atmonth 119899119909(119899) is the predictedvalue at month 119899 and 119872 is the corresponding number oftesting (validation) values

5 Results

The time series data analyzed in this paper corresponds tothe monthly fish catch of anchovy for periods from January1963 to December 2007 (httpwwwsernapescacl) Besidesthe original time series data is smoothed using the followingoperation 119909 = radic119909 The new smoothed original time series isshown in Figure 1Thefirst stepwas to implement the compo-nent separation process presented in Algorithm 1 using threedifferent wavelet filters with three levels of decompositionto select the most suitable wavelet filter The performanceof Algorithm 1 was evaluated using the Daubechies filterSymlets filter and Coiflets filter The first two filters areimplemented using four and six coefficients denoted as Db2Sym2 Db3 and Sym3 respectively whereas the third filteris evaluated using six and twelve coefficients denoted asCoif1 and Coif2 The following step was to detect the mostsignificant periodic fluctuations of each component based onthe GWPS The main goal of the detection phase was to findthe significant peak power that explained the periodicities ofthe time series

The GWPS for the AC and the IAC is presented inFigures 2 3 and 4 respectively In these figures the peakpower indicates the catches high activity The black thick linedesignates the 95 confidence level against red noise and thevalues of the peak power obtained on the black thick line aresignificant [27 28]

Figures 2(a) and 2(b) show the AC time series and theGWPS respectively From Figure 2(b) it can be observedthat there are two peaks of significant power The first peakhas periodicities of 6 months whereas the second peak hasa period of 12 months Therefore the AC time series hasa seasonal pattern On the other hand the AC time seriesobtained by using 4-level SWT with Db2 wavelet filter ispresented in Figure 3(a) whereas Figure 3(b) shows that ACtime series has a cycle of 20months and another of 30monthswhich leads to the conclusion that this new AC time seriesdoes not meet seasonal behavior because it has interannualperiods Therefore only 3-level SWT is used to evaluate theperformance of the wavelet autoregressive (WAR) forecastingmodel proposed

The AIC time series and GWPS are presented in Figures4(a) and 4(b) respectively From Figure 4(b) it can beobserved that there are three peaks of significant powerThe first peak has a period of 84 months and the secondpeak has a periodicity of 31 months while the last peak hasa periodic behavior of 152 months Therefore these threepeaks represent significant interannual fluctuations and canbe associated with the El Nino Southern Oscillation effect[30 31]

Once both the AC and IAC were reconstructed by usingthe 3-level SWT with Db2 wavelet filter each componentwas divided into two parts a training data set (119879 = 360observations 23) and a test data set (119872 = 180 observations13) The training data set was firstly used to estimate theparameters of the WAR forecasting models and the testingdata set was used for computing the performancemeasures ofthemodels and for validation purposesTheWAR forecastingmodel was calibrated with 2119898 = 2 times 30 previous months asinput data due to the interannual variability of the originaltime series In the calibration process overall parameters wereestimated using the linear least squares method

Theperformancemeasures of theWAR forecastingmodelas a function of the forecasting horizon using differentwavelet filters with 3-level SWT are shown in Figure 5 FromFigure 5 it is observed that the Coif2 Db3 and Sym3 gavea poor performance in comparison to Coif1 Sym3 andDb2 On the contrary Db2 and Sym2 achieved the betterperformance Therefore the Db2 wavelet filter was used inall further analysis From Figure 5 it can be seen that theperformance measures significantly increase their values fora time horizon of more than 10-month-ahead forecasting

Once both the AC and IAC have been separately pre-dicted their values must be added in order to obtain the119901-step-ahead anchovy-catches forecasting model In the fol-lowing we present only 10-month-ahead forecasting resultsduring the testing phase whose results are illustrated inFigures 6 7 and 8

The performance evaluation of the 11986511988610[sdot] and 119865

11988611989410[sdot]

models is presented in Figures 6 and 7 As seen in Figures

Journal of Applied Mathematics 5

0020406

Time (month)

Ann

ual m

onth

ly ca

tche

s

1963

1965

1967

1969

1971

1973

1975

1977

1979

1981

1983

1985

1987

1989

1991

1993

1995

1997

1999

2001

2003

2005

2007

minus02

minus04

minus06

(a)

1 3 5 7 9 11 13 15 17 19 21 23 25 270

05

15

2

25

Period (month)

Glo

bal W

PS

(b)

Figure 2 (a) Annual component (119909119886

) from 3-level SWT and (b) GWPS

Time (month)

Ann

ual m

onth

ly ca

tche

s

1963

1965

1967

1969

1971

1973

1975

1977

1979

1981

1983

1985

1987

1989

1991

1993

1995

1997

1999

2001

2003

2005

2007

minus02

minus04

0

02

04

06

(a)

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35Period (month)

Glo

bal W

PS0

05

1

15

2

25

3

35

(b)

Figure 3 (a) Annual component (119909119886

) from 4-level SWT and (b) GWPS

Time (month)

1963

1965

1967

1969

1971

1973

1975

1977

1979

1981

1983

1985

1987

1989

1991

1993

1995

1997

1999

2001

2003

2005

2007

04

06

02

0

Inte

rann

ual m

onth

ly ca

tche

s

(a)

0123456789

10

Period (month)

Glo

bal W

PS

1 9

17

25

33

41

49

57

65

73

81

89

97

105

113

121

129

137

145

153

161

169

(b)

Figure 4 (a) Interannual (119909ia) component from 3-level SWT and (b) GWPS

1 2 3 4 5 6 7 8 9 10 11 12048

1216202428

Forecast time horizon (month)

MA

PE (

)

Coif2Sym3Db3

Coif1Sym2Db2

(a)

1 2 3 4 5 6 7 8 9 10 11 1200001001010020100301004010050100601007010080100901

Forecast time horizon (month)

RMSE

Coif2Sym3Db3

Coif1Sym2Db2

(b)

Figure 5 Forecasting horizon for WAR forecasting

6 Journal of Applied Mathematics

1993

1994

1995

1996

1997

1998

1999

2000

2001

2002

2003

2004

2005

2006

2007

00125

0250375

Actual sample (month)

Ann

ual c

ompo

nent

minus0375

minus025

minus0125

minus05

Estimated catchesActual catches

(a)

050

100150

Rela

tive e

rror

()

1993

1994

1995

1996

1997

1998

1999

2000

2001

2002

2003

2004

2005

2006

2007

Actual sample (month)

minus50

minus100

minus150

(b)

Figure 6 10-month-ahead WAR forecasting for annual component (119909119886

)

00125

0250375

050625

1993

1994

1995

1996

1997

1998

1999

2000

2001

2002

2003

2004

2005

2006

2007

Actual sample (month)Estimated catchesActual catches

Inte

rann

ual c

ompo

nent

(a)

02468

10

Actual sample (month)

Rela

tive e

rror

()

1993

1994

1995

1996

1997

1998

1999

2000

2001

2002

2003

2004

2005

2006

2007

minus8

minus6minus4minus2

minus10

(b)

Figure 7 10-month-ahead WAR forecasting for interannual component (119909ia)

00125

0250375

050625

0750875

Nor

mal

ized

mon

thly

catc

hes

1993

1994

1995

1996

1997

1998

1999

2000

2001

2002

2003

2004

2005

2006

2007

Actual sample (month)Estimated catches a + iaRaw actual catches

X X

X

(a)

010203040

Rela

tive e

rror

()

1993

1994

1995

1996

1997

1998

1999

2000

2001

2002

2003

2004

2005

2006

2007

Actual sample (month)

minus40

minus30

minus20

minus10

(b)

Figure 8 10-month-ahead WAR forecasting for observed anchovy catches versus estimated anchovy catches

6(a) and 7(a) the actual value and predicted value are veryclose for the testing data with low performance measuresThe model 119865

11988610[sdot] achieves a RMSE of 00093 and MAPE of

36 whereas the 119865ia10[sdot] model obtained a RMSE of 00062and MAPE 138 On the other hand from Figure 6(b) itis leveraged that over 80 of the predicted AC catches areacceptable with residual ranging from minus20 to 20 and

Figure 7(b) illustrates that 96 of the predicted IAC catchesvalues are within the range of 4

Figure 8(a) provides the observed monthly anchovy-catches data set versus forecasted anchovy catches whoseforecasting behavior is very accurate for testing data witha RMSE and MAPE of 00093 and 266 respectively Onthe other hand from Figure 8(b) it can be observed that an

Journal of Applied Mathematics 7

important fraction (over 95) of the predicted values areacceptable with residuals ranging from minus10 to 10

6 Conclusions

This paper presented a 119901-month-ahead forecasting strategybased on two stages to improve pelagic fish-catch time-series modeling It is based on wavelet analysis and linearAR modeling In the first stages the two time series areconstructed after 3-level stationary wavelet decompositioncontaining information about the original time series atannual and interannual time scale whereas in the secondstage each time series is modeled by using a linear ARmodel As time series are not independent each AR modelincluded information of the other time series to improve theforecasting accuracyThe 10-month-ahead forecasting resultsshow that the thirty previous months of each time seriescontain valuable information to explain the highest variancelevel of the monthly anchovy catches of northern Chile forperiods from January 1963 to December 2007 Finally thewavelet autoregressive forecasting strategy can be suitable asa very promising methodology for any other marine speciesof the fishing industry

Conflict of Interests

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

Acknowledgments

This work was supported in part by Grant CONICYTFONDECYTRegular 1131105 and by the DI project of thePontificia Universidad Catolica de Valparaıso

References

[1] L Zhang W-D Zhou P-C Chang J-W Yang and F-Z LIldquoIterated time series prediction with multiple support vectorregression modelsrdquo Neurocomputing vol 99 pp 411ndash422 2013

[2] C Hamzacebi D Akay and F Kutay ldquoComparison of directand iterative artificial neural network forecast approaches inmulti-periodic time series forecastingrdquo Expert Systems withApplications vol 36 no 2 pp 3839ndash3844 2009

[3] A Sorjamaa J Hao N Reyhani Y Ji and A LendasseldquoMethodology for long-term prediction of time seriesrdquo Neuro-computing vol 70 no 16ndash18 pp 2861ndash2869 2007

[4] H Cheng P-N Tan J Gao and J Scripps ldquoMultistep-aheadtime series predictionrdquo inAdvances in Knowledge Discovery andData Mining W-K Ng M Kitsuregawa J Li and K ChangEds vol 3918 of Lecture Notes in Computer Science pp 765ndash774 Springer Berlin Germany 2006

[5] D M Kline ldquoMethods for multi-step time series forecastingwith neural networksrdquo in Neural Networks in Business Fore-casting G P Zhang Ed pp 226ndash250 Information ScienceHershey Pa USA 2004

[6] G C Tiao and R S Tsay ldquoSome advances in non-linear andadaptive modelling in time-seriesrdquo Journal of Forecasting vol13 no 2 pp 109ndash131 1994

[7] K I Stergiou E D Christou and G Petrakis ldquoModelling andforecastingmonthly fisheries catches comparison of regressionunivariate and multivariate time series methodsrdquo FisheriesResearch vol 29 no 1 pp 55ndash95 1997

[8] K I Stergiou and E D Christou ldquoModelling and forecastingannual fisheries catches comparison of regression univariateand multivariate time series methodsrdquo Fisheries Research vol25 no 2 pp 105ndash138 1996

[9] K I Stergiou ldquoShort-term fisheries forecasting comparisonof smoothing ARIMA and regression techniquesrdquo Journal ofApplied Ichthyology vol 7 no 4 pp 193ndash204 1991

[10] J C Gutierrez-Estrada C Silva E Yanez N Rodrıguez and IPulido-Calvo ldquoMonthly catch forecasting of anchovy Engraulisringens in the north area of Chile non-linear univariateapproachrdquo Fisheries Research vol 86 no 2-3 pp 188ndash200 2007

[11] E Yanez F Plaza J C Gutierrez-Estrada et al ldquoAnchovy(Engraulis ringens) and sardine (Sardinops sagax) abundanceforecast off northern Chile a multivariate ecosystemic neuralnetwork approachrdquo Progress in Oceanography vol 87 no 1ndash4pp 242ndash250 2010

[12] N A Shrivastava and B K Panigrahi ldquoA hybrid wavelet-ELM based short term price forecasting for electricity marketsrdquoInternational Journal of Electrical Power amp Energy Systems vol55 pp 41ndash50 2014

[13] N Amjady and F Keynia ldquoDay ahead price forecasting ofelectricity markets by a mixed data model and hybrid forecastmethodrdquo International Journal of Electrical Power amp EnergySystems vol 30 no 9 pp 533ndash546 2008

[14] G Nason and B Silverman ldquoThe stationary wavelet transformand some statistical applicationsrdquo inWavelets and Statistics vol103 of Lecture Notes in Statistics pp 281ndash299 Springer NewYork NY USA 1995

[15] R R Coifman and D L Donoho ldquoTranslation-invariant de-noisingrdquo in Wavelets and Statistics vol 103 of Lecture Notes inStatistics pp 125ndash150 Springer New York NY USA 1995

[16] J-C Pesquet H Krim and H Carfantan ldquoTime-invariantorthonormal wavelet representationsrdquo IEEE Transactions onSignal Processing vol 44 no 8 pp 1964ndash1970 1996

[17] D B Percival andA TWaldenWaveletMethods for Time SeriesAnalysis Cambridge University Press Cambridge UK 2000

[18] S Lahmiri ldquoForecasting direction of the S-P500 movementusing wavelet transform and support vector machinesrdquo Inter-national Journal of Strategic Decision Sciences vol 4 no 1 pp78ndash88 2013

[19] T-J Hsieh H-F Hsiao andW-C Yeh ldquoForecasting stock mar-kets using wavelet transforms and recurrent neural networksan integrated system based on artificial bee colony algorithmrdquoApplied Soft Computing vol 11 no 2 pp 2510ndash2525 2011

[20] B-L Zhang R Coggins M A Jabri D Dersch and B FlowerldquoMultiresolution forecasting for futures trading using waveletdecompositionsrdquo IEEETransactions onNeural Networks vol 12no 4 pp 765ndash775 2001

[21] M Hidalgo T Rouyer J C Molinero et al ldquoSynergistic effectsof fishing-induced demographic changes and climate variationon fish population dynamicsrdquo Marine Ecology Progress Seriesvol 426 pp 1ndash12 2011

[22] B Cazelles M Chavez D Berteaux et al ldquoWavelet analysis ofecological time seriesrdquo Oecologia vol 156 no 2 pp 287ndash3042008

[23] T Rouyer J-M Fromentin N C Stenseth and B CazellesldquoAnalysing multiple time series and extending significance

8 Journal of Applied Mathematics

testing in wavelet analysisrdquoMarine Ecology Progress Series vol359 pp 11ndash23 2008

[24] C-H Hsieh C-S Chen T-S Chiu et al ldquoTime series analysesreveal transient relationships between abundance of larvalanchovy and environmental variables in the coastal waterssouthwest of Taiwanrdquo Fisheries Oceanography vol 18 no 2 pp102ndash117 2009

[25] IDaubechiesTenLectures onWavelets SIAMPhiladelphia PaUSA 1992

[26] S Mallat A Wavelet Tour of Signal Processing Academic PressSan Diego Calif USA 1998

[27] C Torrence and G P Compo ldquoA practical guide to waveletanalysisrdquoBulletin of the AmericanMeteorological Society vol 79no 1 pp 61ndash78 1998

[28] A Grinsted J C Moore and S Jevrejeva ldquoApplication of thecross wavelet transform and wavelet coherence to geophysicaltimes seriesrdquo Nonlinear Processes in Geophysics vol 11 no 5-6pp 561ndash566 2004

[29] D SerreMatricesTheory and Applications Springer NewYorkNY USA 2002

[30] A Quetglas F Ordines M Hidalgo et al ldquoSynchronouscombined effects of fishing and climate within a demersalcommunityrdquo ICES Journal of Marine Science vol 70 no 2 pp319ndash328 2013

[31] S E Lluch-Cota A Pares-Sierra V O Magana-Rueda etal ldquoChanging climate in the Gulf of Californiardquo Progress inOceanography vol 87 no 1ndash4 pp 114ndash126 2010

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Journal of Applied Mathematics

1963

1965

1967

1969

1971

1973

1975

1977

1979

1981

1983

1985

1987

1989

1991

1993

1995

1997

1999

2001

2003

2005

2007

0

02

04

06

08

1

Time (month)

Nor

mal

ized

obs

erve

d m

onth

ly ca

tche

s

Figure 1 Observed monthly anchovy catches data

absolute percentage error (MAPE) and relative percentageerror (RPE) which are calculated as

RMSE = radic 1119872

119872

sum

119899=1

(119909 (119899) minus 119909 (119899))2

MAPE = 1119872

119872

sum

119899=1

10038161003816100381610038161003816100381610038161003816

119909 (119899) minus 119909 (119899)

119909 (119899)

10038161003816100381610038161003816100381610038161003816times 100

RPE = 119909 (119899) minus 119909 (119899)119909 (119899)

times 100

(13)

where119909(119899) is the actual value atmonth 119899119909(119899) is the predictedvalue at month 119899 and 119872 is the corresponding number oftesting (validation) values

5 Results

The time series data analyzed in this paper corresponds tothe monthly fish catch of anchovy for periods from January1963 to December 2007 (httpwwwsernapescacl) Besidesthe original time series data is smoothed using the followingoperation 119909 = radic119909 The new smoothed original time series isshown in Figure 1Thefirst stepwas to implement the compo-nent separation process presented in Algorithm 1 using threedifferent wavelet filters with three levels of decompositionto select the most suitable wavelet filter The performanceof Algorithm 1 was evaluated using the Daubechies filterSymlets filter and Coiflets filter The first two filters areimplemented using four and six coefficients denoted as Db2Sym2 Db3 and Sym3 respectively whereas the third filteris evaluated using six and twelve coefficients denoted asCoif1 and Coif2 The following step was to detect the mostsignificant periodic fluctuations of each component based onthe GWPS The main goal of the detection phase was to findthe significant peak power that explained the periodicities ofthe time series

The GWPS for the AC and the IAC is presented inFigures 2 3 and 4 respectively In these figures the peakpower indicates the catches high activity The black thick linedesignates the 95 confidence level against red noise and thevalues of the peak power obtained on the black thick line aresignificant [27 28]

Figures 2(a) and 2(b) show the AC time series and theGWPS respectively From Figure 2(b) it can be observedthat there are two peaks of significant power The first peakhas periodicities of 6 months whereas the second peak hasa period of 12 months Therefore the AC time series hasa seasonal pattern On the other hand the AC time seriesobtained by using 4-level SWT with Db2 wavelet filter ispresented in Figure 3(a) whereas Figure 3(b) shows that ACtime series has a cycle of 20months and another of 30monthswhich leads to the conclusion that this new AC time seriesdoes not meet seasonal behavior because it has interannualperiods Therefore only 3-level SWT is used to evaluate theperformance of the wavelet autoregressive (WAR) forecastingmodel proposed

The AIC time series and GWPS are presented in Figures4(a) and 4(b) respectively From Figure 4(b) it can beobserved that there are three peaks of significant powerThe first peak has a period of 84 months and the secondpeak has a periodicity of 31 months while the last peak hasa periodic behavior of 152 months Therefore these threepeaks represent significant interannual fluctuations and canbe associated with the El Nino Southern Oscillation effect[30 31]

Once both the AC and IAC were reconstructed by usingthe 3-level SWT with Db2 wavelet filter each componentwas divided into two parts a training data set (119879 = 360observations 23) and a test data set (119872 = 180 observations13) The training data set was firstly used to estimate theparameters of the WAR forecasting models and the testingdata set was used for computing the performancemeasures ofthemodels and for validation purposesTheWAR forecastingmodel was calibrated with 2119898 = 2 times 30 previous months asinput data due to the interannual variability of the originaltime series In the calibration process overall parameters wereestimated using the linear least squares method

Theperformancemeasures of theWAR forecastingmodelas a function of the forecasting horizon using differentwavelet filters with 3-level SWT are shown in Figure 5 FromFigure 5 it is observed that the Coif2 Db3 and Sym3 gavea poor performance in comparison to Coif1 Sym3 andDb2 On the contrary Db2 and Sym2 achieved the betterperformance Therefore the Db2 wavelet filter was used inall further analysis From Figure 5 it can be seen that theperformance measures significantly increase their values fora time horizon of more than 10-month-ahead forecasting

Once both the AC and IAC have been separately pre-dicted their values must be added in order to obtain the119901-step-ahead anchovy-catches forecasting model In the fol-lowing we present only 10-month-ahead forecasting resultsduring the testing phase whose results are illustrated inFigures 6 7 and 8

The performance evaluation of the 11986511988610[sdot] and 119865

11988611989410[sdot]

models is presented in Figures 6 and 7 As seen in Figures

Journal of Applied Mathematics 5

0020406

Time (month)

Ann

ual m

onth

ly ca

tche

s

1963

1965

1967

1969

1971

1973

1975

1977

1979

1981

1983

1985

1987

1989

1991

1993

1995

1997

1999

2001

2003

2005

2007

minus02

minus04

minus06

(a)

1 3 5 7 9 11 13 15 17 19 21 23 25 270

05

15

2

25

Period (month)

Glo

bal W

PS

(b)

Figure 2 (a) Annual component (119909119886

) from 3-level SWT and (b) GWPS

Time (month)

Ann

ual m

onth

ly ca

tche

s

1963

1965

1967

1969

1971

1973

1975

1977

1979

1981

1983

1985

1987

1989

1991

1993

1995

1997

1999

2001

2003

2005

2007

minus02

minus04

0

02

04

06

(a)

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35Period (month)

Glo

bal W

PS0

05

1

15

2

25

3

35

(b)

Figure 3 (a) Annual component (119909119886

) from 4-level SWT and (b) GWPS

Time (month)

1963

1965

1967

1969

1971

1973

1975

1977

1979

1981

1983

1985

1987

1989

1991

1993

1995

1997

1999

2001

2003

2005

2007

04

06

02

0

Inte

rann

ual m

onth

ly ca

tche

s

(a)

0123456789

10

Period (month)

Glo

bal W

PS

1 9

17

25

33

41

49

57

65

73

81

89

97

105

113

121

129

137

145

153

161

169

(b)

Figure 4 (a) Interannual (119909ia) component from 3-level SWT and (b) GWPS

1 2 3 4 5 6 7 8 9 10 11 12048

1216202428

Forecast time horizon (month)

MA

PE (

)

Coif2Sym3Db3

Coif1Sym2Db2

(a)

1 2 3 4 5 6 7 8 9 10 11 1200001001010020100301004010050100601007010080100901

Forecast time horizon (month)

RMSE

Coif2Sym3Db3

Coif1Sym2Db2

(b)

Figure 5 Forecasting horizon for WAR forecasting

6 Journal of Applied Mathematics

1993

1994

1995

1996

1997

1998

1999

2000

2001

2002

2003

2004

2005

2006

2007

00125

0250375

Actual sample (month)

Ann

ual c

ompo

nent

minus0375

minus025

minus0125

minus05

Estimated catchesActual catches

(a)

050

100150

Rela

tive e

rror

()

1993

1994

1995

1996

1997

1998

1999

2000

2001

2002

2003

2004

2005

2006

2007

Actual sample (month)

minus50

minus100

minus150

(b)

Figure 6 10-month-ahead WAR forecasting for annual component (119909119886

)

00125

0250375

050625

1993

1994

1995

1996

1997

1998

1999

2000

2001

2002

2003

2004

2005

2006

2007

Actual sample (month)Estimated catchesActual catches

Inte

rann

ual c

ompo

nent

(a)

02468

10

Actual sample (month)

Rela

tive e

rror

()

1993

1994

1995

1996

1997

1998

1999

2000

2001

2002

2003

2004

2005

2006

2007

minus8

minus6minus4minus2

minus10

(b)

Figure 7 10-month-ahead WAR forecasting for interannual component (119909ia)

00125

0250375

050625

0750875

Nor

mal

ized

mon

thly

catc

hes

1993

1994

1995

1996

1997

1998

1999

2000

2001

2002

2003

2004

2005

2006

2007

Actual sample (month)Estimated catches a + iaRaw actual catches

X X

X

(a)

010203040

Rela

tive e

rror

()

1993

1994

1995

1996

1997

1998

1999

2000

2001

2002

2003

2004

2005

2006

2007

Actual sample (month)

minus40

minus30

minus20

minus10

(b)

Figure 8 10-month-ahead WAR forecasting for observed anchovy catches versus estimated anchovy catches

6(a) and 7(a) the actual value and predicted value are veryclose for the testing data with low performance measuresThe model 119865

11988610[sdot] achieves a RMSE of 00093 and MAPE of

36 whereas the 119865ia10[sdot] model obtained a RMSE of 00062and MAPE 138 On the other hand from Figure 6(b) itis leveraged that over 80 of the predicted AC catches areacceptable with residual ranging from minus20 to 20 and

Figure 7(b) illustrates that 96 of the predicted IAC catchesvalues are within the range of 4

Figure 8(a) provides the observed monthly anchovy-catches data set versus forecasted anchovy catches whoseforecasting behavior is very accurate for testing data witha RMSE and MAPE of 00093 and 266 respectively Onthe other hand from Figure 8(b) it can be observed that an

Journal of Applied Mathematics 7

important fraction (over 95) of the predicted values areacceptable with residuals ranging from minus10 to 10

6 Conclusions

This paper presented a 119901-month-ahead forecasting strategybased on two stages to improve pelagic fish-catch time-series modeling It is based on wavelet analysis and linearAR modeling In the first stages the two time series areconstructed after 3-level stationary wavelet decompositioncontaining information about the original time series atannual and interannual time scale whereas in the secondstage each time series is modeled by using a linear ARmodel As time series are not independent each AR modelincluded information of the other time series to improve theforecasting accuracyThe 10-month-ahead forecasting resultsshow that the thirty previous months of each time seriescontain valuable information to explain the highest variancelevel of the monthly anchovy catches of northern Chile forperiods from January 1963 to December 2007 Finally thewavelet autoregressive forecasting strategy can be suitable asa very promising methodology for any other marine speciesof the fishing industry

Conflict of Interests

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

Acknowledgments

This work was supported in part by Grant CONICYTFONDECYTRegular 1131105 and by the DI project of thePontificia Universidad Catolica de Valparaıso

References

[1] L Zhang W-D Zhou P-C Chang J-W Yang and F-Z LIldquoIterated time series prediction with multiple support vectorregression modelsrdquo Neurocomputing vol 99 pp 411ndash422 2013

[2] C Hamzacebi D Akay and F Kutay ldquoComparison of directand iterative artificial neural network forecast approaches inmulti-periodic time series forecastingrdquo Expert Systems withApplications vol 36 no 2 pp 3839ndash3844 2009

[3] A Sorjamaa J Hao N Reyhani Y Ji and A LendasseldquoMethodology for long-term prediction of time seriesrdquo Neuro-computing vol 70 no 16ndash18 pp 2861ndash2869 2007

[4] H Cheng P-N Tan J Gao and J Scripps ldquoMultistep-aheadtime series predictionrdquo inAdvances in Knowledge Discovery andData Mining W-K Ng M Kitsuregawa J Li and K ChangEds vol 3918 of Lecture Notes in Computer Science pp 765ndash774 Springer Berlin Germany 2006

[5] D M Kline ldquoMethods for multi-step time series forecastingwith neural networksrdquo in Neural Networks in Business Fore-casting G P Zhang Ed pp 226ndash250 Information ScienceHershey Pa USA 2004

[6] G C Tiao and R S Tsay ldquoSome advances in non-linear andadaptive modelling in time-seriesrdquo Journal of Forecasting vol13 no 2 pp 109ndash131 1994

[7] K I Stergiou E D Christou and G Petrakis ldquoModelling andforecastingmonthly fisheries catches comparison of regressionunivariate and multivariate time series methodsrdquo FisheriesResearch vol 29 no 1 pp 55ndash95 1997

[8] K I Stergiou and E D Christou ldquoModelling and forecastingannual fisheries catches comparison of regression univariateand multivariate time series methodsrdquo Fisheries Research vol25 no 2 pp 105ndash138 1996

[9] K I Stergiou ldquoShort-term fisheries forecasting comparisonof smoothing ARIMA and regression techniquesrdquo Journal ofApplied Ichthyology vol 7 no 4 pp 193ndash204 1991

[10] J C Gutierrez-Estrada C Silva E Yanez N Rodrıguez and IPulido-Calvo ldquoMonthly catch forecasting of anchovy Engraulisringens in the north area of Chile non-linear univariateapproachrdquo Fisheries Research vol 86 no 2-3 pp 188ndash200 2007

[11] E Yanez F Plaza J C Gutierrez-Estrada et al ldquoAnchovy(Engraulis ringens) and sardine (Sardinops sagax) abundanceforecast off northern Chile a multivariate ecosystemic neuralnetwork approachrdquo Progress in Oceanography vol 87 no 1ndash4pp 242ndash250 2010

[12] N A Shrivastava and B K Panigrahi ldquoA hybrid wavelet-ELM based short term price forecasting for electricity marketsrdquoInternational Journal of Electrical Power amp Energy Systems vol55 pp 41ndash50 2014

[13] N Amjady and F Keynia ldquoDay ahead price forecasting ofelectricity markets by a mixed data model and hybrid forecastmethodrdquo International Journal of Electrical Power amp EnergySystems vol 30 no 9 pp 533ndash546 2008

[14] G Nason and B Silverman ldquoThe stationary wavelet transformand some statistical applicationsrdquo inWavelets and Statistics vol103 of Lecture Notes in Statistics pp 281ndash299 Springer NewYork NY USA 1995

[15] R R Coifman and D L Donoho ldquoTranslation-invariant de-noisingrdquo in Wavelets and Statistics vol 103 of Lecture Notes inStatistics pp 125ndash150 Springer New York NY USA 1995

[16] J-C Pesquet H Krim and H Carfantan ldquoTime-invariantorthonormal wavelet representationsrdquo IEEE Transactions onSignal Processing vol 44 no 8 pp 1964ndash1970 1996

[17] D B Percival andA TWaldenWaveletMethods for Time SeriesAnalysis Cambridge University Press Cambridge UK 2000

[18] S Lahmiri ldquoForecasting direction of the S-P500 movementusing wavelet transform and support vector machinesrdquo Inter-national Journal of Strategic Decision Sciences vol 4 no 1 pp78ndash88 2013

[19] T-J Hsieh H-F Hsiao andW-C Yeh ldquoForecasting stock mar-kets using wavelet transforms and recurrent neural networksan integrated system based on artificial bee colony algorithmrdquoApplied Soft Computing vol 11 no 2 pp 2510ndash2525 2011

[20] B-L Zhang R Coggins M A Jabri D Dersch and B FlowerldquoMultiresolution forecasting for futures trading using waveletdecompositionsrdquo IEEETransactions onNeural Networks vol 12no 4 pp 765ndash775 2001

[21] M Hidalgo T Rouyer J C Molinero et al ldquoSynergistic effectsof fishing-induced demographic changes and climate variationon fish population dynamicsrdquo Marine Ecology Progress Seriesvol 426 pp 1ndash12 2011

[22] B Cazelles M Chavez D Berteaux et al ldquoWavelet analysis ofecological time seriesrdquo Oecologia vol 156 no 2 pp 287ndash3042008

[23] T Rouyer J-M Fromentin N C Stenseth and B CazellesldquoAnalysing multiple time series and extending significance

8 Journal of Applied Mathematics

testing in wavelet analysisrdquoMarine Ecology Progress Series vol359 pp 11ndash23 2008

[24] C-H Hsieh C-S Chen T-S Chiu et al ldquoTime series analysesreveal transient relationships between abundance of larvalanchovy and environmental variables in the coastal waterssouthwest of Taiwanrdquo Fisheries Oceanography vol 18 no 2 pp102ndash117 2009

[25] IDaubechiesTenLectures onWavelets SIAMPhiladelphia PaUSA 1992

[26] S Mallat A Wavelet Tour of Signal Processing Academic PressSan Diego Calif USA 1998

[27] C Torrence and G P Compo ldquoA practical guide to waveletanalysisrdquoBulletin of the AmericanMeteorological Society vol 79no 1 pp 61ndash78 1998

[28] A Grinsted J C Moore and S Jevrejeva ldquoApplication of thecross wavelet transform and wavelet coherence to geophysicaltimes seriesrdquo Nonlinear Processes in Geophysics vol 11 no 5-6pp 561ndash566 2004

[29] D SerreMatricesTheory and Applications Springer NewYorkNY USA 2002

[30] A Quetglas F Ordines M Hidalgo et al ldquoSynchronouscombined effects of fishing and climate within a demersalcommunityrdquo ICES Journal of Marine Science vol 70 no 2 pp319ndash328 2013

[31] S E Lluch-Cota A Pares-Sierra V O Magana-Rueda etal ldquoChanging climate in the Gulf of Californiardquo Progress inOceanography vol 87 no 1ndash4 pp 114ndash126 2010

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Journal of Applied Mathematics 5

0020406

Time (month)

Ann

ual m

onth

ly ca

tche

s

1963

1965

1967

1969

1971

1973

1975

1977

1979

1981

1983

1985

1987

1989

1991

1993

1995

1997

1999

2001

2003

2005

2007

minus02

minus04

minus06

(a)

1 3 5 7 9 11 13 15 17 19 21 23 25 270

05

15

2

25

Period (month)

Glo

bal W

PS

(b)

Figure 2 (a) Annual component (119909119886

) from 3-level SWT and (b) GWPS

Time (month)

Ann

ual m

onth

ly ca

tche

s

1963

1965

1967

1969

1971

1973

1975

1977

1979

1981

1983

1985

1987

1989

1991

1993

1995

1997

1999

2001

2003

2005

2007

minus02

minus04

0

02

04

06

(a)

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35Period (month)

Glo

bal W

PS0

05

1

15

2

25

3

35

(b)

Figure 3 (a) Annual component (119909119886

) from 4-level SWT and (b) GWPS

Time (month)

1963

1965

1967

1969

1971

1973

1975

1977

1979

1981

1983

1985

1987

1989

1991

1993

1995

1997

1999

2001

2003

2005

2007

04

06

02

0

Inte

rann

ual m

onth

ly ca

tche

s

(a)

0123456789

10

Period (month)

Glo

bal W

PS

1 9

17

25

33

41

49

57

65

73

81

89

97

105

113

121

129

137

145

153

161

169

(b)

Figure 4 (a) Interannual (119909ia) component from 3-level SWT and (b) GWPS

1 2 3 4 5 6 7 8 9 10 11 12048

1216202428

Forecast time horizon (month)

MA

PE (

)

Coif2Sym3Db3

Coif1Sym2Db2

(a)

1 2 3 4 5 6 7 8 9 10 11 1200001001010020100301004010050100601007010080100901

Forecast time horizon (month)

RMSE

Coif2Sym3Db3

Coif1Sym2Db2

(b)

Figure 5 Forecasting horizon for WAR forecasting

6 Journal of Applied Mathematics

1993

1994

1995

1996

1997

1998

1999

2000

2001

2002

2003

2004

2005

2006

2007

00125

0250375

Actual sample (month)

Ann

ual c

ompo

nent

minus0375

minus025

minus0125

minus05

Estimated catchesActual catches

(a)

050

100150

Rela

tive e

rror

()

1993

1994

1995

1996

1997

1998

1999

2000

2001

2002

2003

2004

2005

2006

2007

Actual sample (month)

minus50

minus100

minus150

(b)

Figure 6 10-month-ahead WAR forecasting for annual component (119909119886

)

00125

0250375

050625

1993

1994

1995

1996

1997

1998

1999

2000

2001

2002

2003

2004

2005

2006

2007

Actual sample (month)Estimated catchesActual catches

Inte

rann

ual c

ompo

nent

(a)

02468

10

Actual sample (month)

Rela

tive e

rror

()

1993

1994

1995

1996

1997

1998

1999

2000

2001

2002

2003

2004

2005

2006

2007

minus8

minus6minus4minus2

minus10

(b)

Figure 7 10-month-ahead WAR forecasting for interannual component (119909ia)

00125

0250375

050625

0750875

Nor

mal

ized

mon

thly

catc

hes

1993

1994

1995

1996

1997

1998

1999

2000

2001

2002

2003

2004

2005

2006

2007

Actual sample (month)Estimated catches a + iaRaw actual catches

X X

X

(a)

010203040

Rela

tive e

rror

()

1993

1994

1995

1996

1997

1998

1999

2000

2001

2002

2003

2004

2005

2006

2007

Actual sample (month)

minus40

minus30

minus20

minus10

(b)

Figure 8 10-month-ahead WAR forecasting for observed anchovy catches versus estimated anchovy catches

6(a) and 7(a) the actual value and predicted value are veryclose for the testing data with low performance measuresThe model 119865

11988610[sdot] achieves a RMSE of 00093 and MAPE of

36 whereas the 119865ia10[sdot] model obtained a RMSE of 00062and MAPE 138 On the other hand from Figure 6(b) itis leveraged that over 80 of the predicted AC catches areacceptable with residual ranging from minus20 to 20 and

Figure 7(b) illustrates that 96 of the predicted IAC catchesvalues are within the range of 4

Figure 8(a) provides the observed monthly anchovy-catches data set versus forecasted anchovy catches whoseforecasting behavior is very accurate for testing data witha RMSE and MAPE of 00093 and 266 respectively Onthe other hand from Figure 8(b) it can be observed that an

Journal of Applied Mathematics 7

important fraction (over 95) of the predicted values areacceptable with residuals ranging from minus10 to 10

6 Conclusions

This paper presented a 119901-month-ahead forecasting strategybased on two stages to improve pelagic fish-catch time-series modeling It is based on wavelet analysis and linearAR modeling In the first stages the two time series areconstructed after 3-level stationary wavelet decompositioncontaining information about the original time series atannual and interannual time scale whereas in the secondstage each time series is modeled by using a linear ARmodel As time series are not independent each AR modelincluded information of the other time series to improve theforecasting accuracyThe 10-month-ahead forecasting resultsshow that the thirty previous months of each time seriescontain valuable information to explain the highest variancelevel of the monthly anchovy catches of northern Chile forperiods from January 1963 to December 2007 Finally thewavelet autoregressive forecasting strategy can be suitable asa very promising methodology for any other marine speciesof the fishing industry

Conflict of Interests

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

Acknowledgments

This work was supported in part by Grant CONICYTFONDECYTRegular 1131105 and by the DI project of thePontificia Universidad Catolica de Valparaıso

References

[1] L Zhang W-D Zhou P-C Chang J-W Yang and F-Z LIldquoIterated time series prediction with multiple support vectorregression modelsrdquo Neurocomputing vol 99 pp 411ndash422 2013

[2] C Hamzacebi D Akay and F Kutay ldquoComparison of directand iterative artificial neural network forecast approaches inmulti-periodic time series forecastingrdquo Expert Systems withApplications vol 36 no 2 pp 3839ndash3844 2009

[3] A Sorjamaa J Hao N Reyhani Y Ji and A LendasseldquoMethodology for long-term prediction of time seriesrdquo Neuro-computing vol 70 no 16ndash18 pp 2861ndash2869 2007

[4] H Cheng P-N Tan J Gao and J Scripps ldquoMultistep-aheadtime series predictionrdquo inAdvances in Knowledge Discovery andData Mining W-K Ng M Kitsuregawa J Li and K ChangEds vol 3918 of Lecture Notes in Computer Science pp 765ndash774 Springer Berlin Germany 2006

[5] D M Kline ldquoMethods for multi-step time series forecastingwith neural networksrdquo in Neural Networks in Business Fore-casting G P Zhang Ed pp 226ndash250 Information ScienceHershey Pa USA 2004

[6] G C Tiao and R S Tsay ldquoSome advances in non-linear andadaptive modelling in time-seriesrdquo Journal of Forecasting vol13 no 2 pp 109ndash131 1994

[7] K I Stergiou E D Christou and G Petrakis ldquoModelling andforecastingmonthly fisheries catches comparison of regressionunivariate and multivariate time series methodsrdquo FisheriesResearch vol 29 no 1 pp 55ndash95 1997

[8] K I Stergiou and E D Christou ldquoModelling and forecastingannual fisheries catches comparison of regression univariateand multivariate time series methodsrdquo Fisheries Research vol25 no 2 pp 105ndash138 1996

[9] K I Stergiou ldquoShort-term fisheries forecasting comparisonof smoothing ARIMA and regression techniquesrdquo Journal ofApplied Ichthyology vol 7 no 4 pp 193ndash204 1991

[10] J C Gutierrez-Estrada C Silva E Yanez N Rodrıguez and IPulido-Calvo ldquoMonthly catch forecasting of anchovy Engraulisringens in the north area of Chile non-linear univariateapproachrdquo Fisheries Research vol 86 no 2-3 pp 188ndash200 2007

[11] E Yanez F Plaza J C Gutierrez-Estrada et al ldquoAnchovy(Engraulis ringens) and sardine (Sardinops sagax) abundanceforecast off northern Chile a multivariate ecosystemic neuralnetwork approachrdquo Progress in Oceanography vol 87 no 1ndash4pp 242ndash250 2010

[12] N A Shrivastava and B K Panigrahi ldquoA hybrid wavelet-ELM based short term price forecasting for electricity marketsrdquoInternational Journal of Electrical Power amp Energy Systems vol55 pp 41ndash50 2014

[13] N Amjady and F Keynia ldquoDay ahead price forecasting ofelectricity markets by a mixed data model and hybrid forecastmethodrdquo International Journal of Electrical Power amp EnergySystems vol 30 no 9 pp 533ndash546 2008

[14] G Nason and B Silverman ldquoThe stationary wavelet transformand some statistical applicationsrdquo inWavelets and Statistics vol103 of Lecture Notes in Statistics pp 281ndash299 Springer NewYork NY USA 1995

[15] R R Coifman and D L Donoho ldquoTranslation-invariant de-noisingrdquo in Wavelets and Statistics vol 103 of Lecture Notes inStatistics pp 125ndash150 Springer New York NY USA 1995

[16] J-C Pesquet H Krim and H Carfantan ldquoTime-invariantorthonormal wavelet representationsrdquo IEEE Transactions onSignal Processing vol 44 no 8 pp 1964ndash1970 1996

[17] D B Percival andA TWaldenWaveletMethods for Time SeriesAnalysis Cambridge University Press Cambridge UK 2000

[18] S Lahmiri ldquoForecasting direction of the S-P500 movementusing wavelet transform and support vector machinesrdquo Inter-national Journal of Strategic Decision Sciences vol 4 no 1 pp78ndash88 2013

[19] T-J Hsieh H-F Hsiao andW-C Yeh ldquoForecasting stock mar-kets using wavelet transforms and recurrent neural networksan integrated system based on artificial bee colony algorithmrdquoApplied Soft Computing vol 11 no 2 pp 2510ndash2525 2011

[20] B-L Zhang R Coggins M A Jabri D Dersch and B FlowerldquoMultiresolution forecasting for futures trading using waveletdecompositionsrdquo IEEETransactions onNeural Networks vol 12no 4 pp 765ndash775 2001

[21] M Hidalgo T Rouyer J C Molinero et al ldquoSynergistic effectsof fishing-induced demographic changes and climate variationon fish population dynamicsrdquo Marine Ecology Progress Seriesvol 426 pp 1ndash12 2011

[22] B Cazelles M Chavez D Berteaux et al ldquoWavelet analysis ofecological time seriesrdquo Oecologia vol 156 no 2 pp 287ndash3042008

[23] T Rouyer J-M Fromentin N C Stenseth and B CazellesldquoAnalysing multiple time series and extending significance

8 Journal of Applied Mathematics

testing in wavelet analysisrdquoMarine Ecology Progress Series vol359 pp 11ndash23 2008

[24] C-H Hsieh C-S Chen T-S Chiu et al ldquoTime series analysesreveal transient relationships between abundance of larvalanchovy and environmental variables in the coastal waterssouthwest of Taiwanrdquo Fisheries Oceanography vol 18 no 2 pp102ndash117 2009

[25] IDaubechiesTenLectures onWavelets SIAMPhiladelphia PaUSA 1992

[26] S Mallat A Wavelet Tour of Signal Processing Academic PressSan Diego Calif USA 1998

[27] C Torrence and G P Compo ldquoA practical guide to waveletanalysisrdquoBulletin of the AmericanMeteorological Society vol 79no 1 pp 61ndash78 1998

[28] A Grinsted J C Moore and S Jevrejeva ldquoApplication of thecross wavelet transform and wavelet coherence to geophysicaltimes seriesrdquo Nonlinear Processes in Geophysics vol 11 no 5-6pp 561ndash566 2004

[29] D SerreMatricesTheory and Applications Springer NewYorkNY USA 2002

[30] A Quetglas F Ordines M Hidalgo et al ldquoSynchronouscombined effects of fishing and climate within a demersalcommunityrdquo ICES Journal of Marine Science vol 70 no 2 pp319ndash328 2013

[31] S E Lluch-Cota A Pares-Sierra V O Magana-Rueda etal ldquoChanging climate in the Gulf of Californiardquo Progress inOceanography vol 87 no 1ndash4 pp 114ndash126 2010

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Journal of Applied Mathematics

1993

1994

1995

1996

1997

1998

1999

2000

2001

2002

2003

2004

2005

2006

2007

00125

0250375

Actual sample (month)

Ann

ual c

ompo

nent

minus0375

minus025

minus0125

minus05

Estimated catchesActual catches

(a)

050

100150

Rela

tive e

rror

()

1993

1994

1995

1996

1997

1998

1999

2000

2001

2002

2003

2004

2005

2006

2007

Actual sample (month)

minus50

minus100

minus150

(b)

Figure 6 10-month-ahead WAR forecasting for annual component (119909119886

)

00125

0250375

050625

1993

1994

1995

1996

1997

1998

1999

2000

2001

2002

2003

2004

2005

2006

2007

Actual sample (month)Estimated catchesActual catches

Inte

rann

ual c

ompo

nent

(a)

02468

10

Actual sample (month)

Rela

tive e

rror

()

1993

1994

1995

1996

1997

1998

1999

2000

2001

2002

2003

2004

2005

2006

2007

minus8

minus6minus4minus2

minus10

(b)

Figure 7 10-month-ahead WAR forecasting for interannual component (119909ia)

00125

0250375

050625

0750875

Nor

mal

ized

mon

thly

catc

hes

1993

1994

1995

1996

1997

1998

1999

2000

2001

2002

2003

2004

2005

2006

2007

Actual sample (month)Estimated catches a + iaRaw actual catches

X X

X

(a)

010203040

Rela

tive e

rror

()

1993

1994

1995

1996

1997

1998

1999

2000

2001

2002

2003

2004

2005

2006

2007

Actual sample (month)

minus40

minus30

minus20

minus10

(b)

Figure 8 10-month-ahead WAR forecasting for observed anchovy catches versus estimated anchovy catches

6(a) and 7(a) the actual value and predicted value are veryclose for the testing data with low performance measuresThe model 119865

11988610[sdot] achieves a RMSE of 00093 and MAPE of

36 whereas the 119865ia10[sdot] model obtained a RMSE of 00062and MAPE 138 On the other hand from Figure 6(b) itis leveraged that over 80 of the predicted AC catches areacceptable with residual ranging from minus20 to 20 and

Figure 7(b) illustrates that 96 of the predicted IAC catchesvalues are within the range of 4

Figure 8(a) provides the observed monthly anchovy-catches data set versus forecasted anchovy catches whoseforecasting behavior is very accurate for testing data witha RMSE and MAPE of 00093 and 266 respectively Onthe other hand from Figure 8(b) it can be observed that an

Journal of Applied Mathematics 7

important fraction (over 95) of the predicted values areacceptable with residuals ranging from minus10 to 10

6 Conclusions

This paper presented a 119901-month-ahead forecasting strategybased on two stages to improve pelagic fish-catch time-series modeling It is based on wavelet analysis and linearAR modeling In the first stages the two time series areconstructed after 3-level stationary wavelet decompositioncontaining information about the original time series atannual and interannual time scale whereas in the secondstage each time series is modeled by using a linear ARmodel As time series are not independent each AR modelincluded information of the other time series to improve theforecasting accuracyThe 10-month-ahead forecasting resultsshow that the thirty previous months of each time seriescontain valuable information to explain the highest variancelevel of the monthly anchovy catches of northern Chile forperiods from January 1963 to December 2007 Finally thewavelet autoregressive forecasting strategy can be suitable asa very promising methodology for any other marine speciesof the fishing industry

Conflict of Interests

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

Acknowledgments

This work was supported in part by Grant CONICYTFONDECYTRegular 1131105 and by the DI project of thePontificia Universidad Catolica de Valparaıso

References

[1] L Zhang W-D Zhou P-C Chang J-W Yang and F-Z LIldquoIterated time series prediction with multiple support vectorregression modelsrdquo Neurocomputing vol 99 pp 411ndash422 2013

[2] C Hamzacebi D Akay and F Kutay ldquoComparison of directand iterative artificial neural network forecast approaches inmulti-periodic time series forecastingrdquo Expert Systems withApplications vol 36 no 2 pp 3839ndash3844 2009

[3] A Sorjamaa J Hao N Reyhani Y Ji and A LendasseldquoMethodology for long-term prediction of time seriesrdquo Neuro-computing vol 70 no 16ndash18 pp 2861ndash2869 2007

[4] H Cheng P-N Tan J Gao and J Scripps ldquoMultistep-aheadtime series predictionrdquo inAdvances in Knowledge Discovery andData Mining W-K Ng M Kitsuregawa J Li and K ChangEds vol 3918 of Lecture Notes in Computer Science pp 765ndash774 Springer Berlin Germany 2006

[5] D M Kline ldquoMethods for multi-step time series forecastingwith neural networksrdquo in Neural Networks in Business Fore-casting G P Zhang Ed pp 226ndash250 Information ScienceHershey Pa USA 2004

[6] G C Tiao and R S Tsay ldquoSome advances in non-linear andadaptive modelling in time-seriesrdquo Journal of Forecasting vol13 no 2 pp 109ndash131 1994

[7] K I Stergiou E D Christou and G Petrakis ldquoModelling andforecastingmonthly fisheries catches comparison of regressionunivariate and multivariate time series methodsrdquo FisheriesResearch vol 29 no 1 pp 55ndash95 1997

[8] K I Stergiou and E D Christou ldquoModelling and forecastingannual fisheries catches comparison of regression univariateand multivariate time series methodsrdquo Fisheries Research vol25 no 2 pp 105ndash138 1996

[9] K I Stergiou ldquoShort-term fisheries forecasting comparisonof smoothing ARIMA and regression techniquesrdquo Journal ofApplied Ichthyology vol 7 no 4 pp 193ndash204 1991

[10] J C Gutierrez-Estrada C Silva E Yanez N Rodrıguez and IPulido-Calvo ldquoMonthly catch forecasting of anchovy Engraulisringens in the north area of Chile non-linear univariateapproachrdquo Fisheries Research vol 86 no 2-3 pp 188ndash200 2007

[11] E Yanez F Plaza J C Gutierrez-Estrada et al ldquoAnchovy(Engraulis ringens) and sardine (Sardinops sagax) abundanceforecast off northern Chile a multivariate ecosystemic neuralnetwork approachrdquo Progress in Oceanography vol 87 no 1ndash4pp 242ndash250 2010

[12] N A Shrivastava and B K Panigrahi ldquoA hybrid wavelet-ELM based short term price forecasting for electricity marketsrdquoInternational Journal of Electrical Power amp Energy Systems vol55 pp 41ndash50 2014

[13] N Amjady and F Keynia ldquoDay ahead price forecasting ofelectricity markets by a mixed data model and hybrid forecastmethodrdquo International Journal of Electrical Power amp EnergySystems vol 30 no 9 pp 533ndash546 2008

[14] G Nason and B Silverman ldquoThe stationary wavelet transformand some statistical applicationsrdquo inWavelets and Statistics vol103 of Lecture Notes in Statistics pp 281ndash299 Springer NewYork NY USA 1995

[15] R R Coifman and D L Donoho ldquoTranslation-invariant de-noisingrdquo in Wavelets and Statistics vol 103 of Lecture Notes inStatistics pp 125ndash150 Springer New York NY USA 1995

[16] J-C Pesquet H Krim and H Carfantan ldquoTime-invariantorthonormal wavelet representationsrdquo IEEE Transactions onSignal Processing vol 44 no 8 pp 1964ndash1970 1996

[17] D B Percival andA TWaldenWaveletMethods for Time SeriesAnalysis Cambridge University Press Cambridge UK 2000

[18] S Lahmiri ldquoForecasting direction of the S-P500 movementusing wavelet transform and support vector machinesrdquo Inter-national Journal of Strategic Decision Sciences vol 4 no 1 pp78ndash88 2013

[19] T-J Hsieh H-F Hsiao andW-C Yeh ldquoForecasting stock mar-kets using wavelet transforms and recurrent neural networksan integrated system based on artificial bee colony algorithmrdquoApplied Soft Computing vol 11 no 2 pp 2510ndash2525 2011

[20] B-L Zhang R Coggins M A Jabri D Dersch and B FlowerldquoMultiresolution forecasting for futures trading using waveletdecompositionsrdquo IEEETransactions onNeural Networks vol 12no 4 pp 765ndash775 2001

[21] M Hidalgo T Rouyer J C Molinero et al ldquoSynergistic effectsof fishing-induced demographic changes and climate variationon fish population dynamicsrdquo Marine Ecology Progress Seriesvol 426 pp 1ndash12 2011

[22] B Cazelles M Chavez D Berteaux et al ldquoWavelet analysis ofecological time seriesrdquo Oecologia vol 156 no 2 pp 287ndash3042008

[23] T Rouyer J-M Fromentin N C Stenseth and B CazellesldquoAnalysing multiple time series and extending significance

8 Journal of Applied Mathematics

testing in wavelet analysisrdquoMarine Ecology Progress Series vol359 pp 11ndash23 2008

[24] C-H Hsieh C-S Chen T-S Chiu et al ldquoTime series analysesreveal transient relationships between abundance of larvalanchovy and environmental variables in the coastal waterssouthwest of Taiwanrdquo Fisheries Oceanography vol 18 no 2 pp102ndash117 2009

[25] IDaubechiesTenLectures onWavelets SIAMPhiladelphia PaUSA 1992

[26] S Mallat A Wavelet Tour of Signal Processing Academic PressSan Diego Calif USA 1998

[27] C Torrence and G P Compo ldquoA practical guide to waveletanalysisrdquoBulletin of the AmericanMeteorological Society vol 79no 1 pp 61ndash78 1998

[28] A Grinsted J C Moore and S Jevrejeva ldquoApplication of thecross wavelet transform and wavelet coherence to geophysicaltimes seriesrdquo Nonlinear Processes in Geophysics vol 11 no 5-6pp 561ndash566 2004

[29] D SerreMatricesTheory and Applications Springer NewYorkNY USA 2002

[30] A Quetglas F Ordines M Hidalgo et al ldquoSynchronouscombined effects of fishing and climate within a demersalcommunityrdquo ICES Journal of Marine Science vol 70 no 2 pp319ndash328 2013

[31] S E Lluch-Cota A Pares-Sierra V O Magana-Rueda etal ldquoChanging climate in the Gulf of Californiardquo Progress inOceanography vol 87 no 1ndash4 pp 114ndash126 2010

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Journal of Applied Mathematics 7

important fraction (over 95) of the predicted values areacceptable with residuals ranging from minus10 to 10

6 Conclusions

This paper presented a 119901-month-ahead forecasting strategybased on two stages to improve pelagic fish-catch time-series modeling It is based on wavelet analysis and linearAR modeling In the first stages the two time series areconstructed after 3-level stationary wavelet decompositioncontaining information about the original time series atannual and interannual time scale whereas in the secondstage each time series is modeled by using a linear ARmodel As time series are not independent each AR modelincluded information of the other time series to improve theforecasting accuracyThe 10-month-ahead forecasting resultsshow that the thirty previous months of each time seriescontain valuable information to explain the highest variancelevel of the monthly anchovy catches of northern Chile forperiods from January 1963 to December 2007 Finally thewavelet autoregressive forecasting strategy can be suitable asa very promising methodology for any other marine speciesof the fishing industry

Conflict of Interests

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

Acknowledgments

This work was supported in part by Grant CONICYTFONDECYTRegular 1131105 and by the DI project of thePontificia Universidad Catolica de Valparaıso

References

[1] L Zhang W-D Zhou P-C Chang J-W Yang and F-Z LIldquoIterated time series prediction with multiple support vectorregression modelsrdquo Neurocomputing vol 99 pp 411ndash422 2013

[2] C Hamzacebi D Akay and F Kutay ldquoComparison of directand iterative artificial neural network forecast approaches inmulti-periodic time series forecastingrdquo Expert Systems withApplications vol 36 no 2 pp 3839ndash3844 2009

[3] A Sorjamaa J Hao N Reyhani Y Ji and A LendasseldquoMethodology for long-term prediction of time seriesrdquo Neuro-computing vol 70 no 16ndash18 pp 2861ndash2869 2007

[4] H Cheng P-N Tan J Gao and J Scripps ldquoMultistep-aheadtime series predictionrdquo inAdvances in Knowledge Discovery andData Mining W-K Ng M Kitsuregawa J Li and K ChangEds vol 3918 of Lecture Notes in Computer Science pp 765ndash774 Springer Berlin Germany 2006

[5] D M Kline ldquoMethods for multi-step time series forecastingwith neural networksrdquo in Neural Networks in Business Fore-casting G P Zhang Ed pp 226ndash250 Information ScienceHershey Pa USA 2004

[6] G C Tiao and R S Tsay ldquoSome advances in non-linear andadaptive modelling in time-seriesrdquo Journal of Forecasting vol13 no 2 pp 109ndash131 1994

[7] K I Stergiou E D Christou and G Petrakis ldquoModelling andforecastingmonthly fisheries catches comparison of regressionunivariate and multivariate time series methodsrdquo FisheriesResearch vol 29 no 1 pp 55ndash95 1997

[8] K I Stergiou and E D Christou ldquoModelling and forecastingannual fisheries catches comparison of regression univariateand multivariate time series methodsrdquo Fisheries Research vol25 no 2 pp 105ndash138 1996

[9] K I Stergiou ldquoShort-term fisheries forecasting comparisonof smoothing ARIMA and regression techniquesrdquo Journal ofApplied Ichthyology vol 7 no 4 pp 193ndash204 1991

[10] J C Gutierrez-Estrada C Silva E Yanez N Rodrıguez and IPulido-Calvo ldquoMonthly catch forecasting of anchovy Engraulisringens in the north area of Chile non-linear univariateapproachrdquo Fisheries Research vol 86 no 2-3 pp 188ndash200 2007

[11] E Yanez F Plaza J C Gutierrez-Estrada et al ldquoAnchovy(Engraulis ringens) and sardine (Sardinops sagax) abundanceforecast off northern Chile a multivariate ecosystemic neuralnetwork approachrdquo Progress in Oceanography vol 87 no 1ndash4pp 242ndash250 2010

[12] N A Shrivastava and B K Panigrahi ldquoA hybrid wavelet-ELM based short term price forecasting for electricity marketsrdquoInternational Journal of Electrical Power amp Energy Systems vol55 pp 41ndash50 2014

[13] N Amjady and F Keynia ldquoDay ahead price forecasting ofelectricity markets by a mixed data model and hybrid forecastmethodrdquo International Journal of Electrical Power amp EnergySystems vol 30 no 9 pp 533ndash546 2008

[14] G Nason and B Silverman ldquoThe stationary wavelet transformand some statistical applicationsrdquo inWavelets and Statistics vol103 of Lecture Notes in Statistics pp 281ndash299 Springer NewYork NY USA 1995

[15] R R Coifman and D L Donoho ldquoTranslation-invariant de-noisingrdquo in Wavelets and Statistics vol 103 of Lecture Notes inStatistics pp 125ndash150 Springer New York NY USA 1995

[16] J-C Pesquet H Krim and H Carfantan ldquoTime-invariantorthonormal wavelet representationsrdquo IEEE Transactions onSignal Processing vol 44 no 8 pp 1964ndash1970 1996

[17] D B Percival andA TWaldenWaveletMethods for Time SeriesAnalysis Cambridge University Press Cambridge UK 2000

[18] S Lahmiri ldquoForecasting direction of the S-P500 movementusing wavelet transform and support vector machinesrdquo Inter-national Journal of Strategic Decision Sciences vol 4 no 1 pp78ndash88 2013

[19] T-J Hsieh H-F Hsiao andW-C Yeh ldquoForecasting stock mar-kets using wavelet transforms and recurrent neural networksan integrated system based on artificial bee colony algorithmrdquoApplied Soft Computing vol 11 no 2 pp 2510ndash2525 2011

[20] B-L Zhang R Coggins M A Jabri D Dersch and B FlowerldquoMultiresolution forecasting for futures trading using waveletdecompositionsrdquo IEEETransactions onNeural Networks vol 12no 4 pp 765ndash775 2001

[21] M Hidalgo T Rouyer J C Molinero et al ldquoSynergistic effectsof fishing-induced demographic changes and climate variationon fish population dynamicsrdquo Marine Ecology Progress Seriesvol 426 pp 1ndash12 2011

[22] B Cazelles M Chavez D Berteaux et al ldquoWavelet analysis ofecological time seriesrdquo Oecologia vol 156 no 2 pp 287ndash3042008

[23] T Rouyer J-M Fromentin N C Stenseth and B CazellesldquoAnalysing multiple time series and extending significance

8 Journal of Applied Mathematics

testing in wavelet analysisrdquoMarine Ecology Progress Series vol359 pp 11ndash23 2008

[24] C-H Hsieh C-S Chen T-S Chiu et al ldquoTime series analysesreveal transient relationships between abundance of larvalanchovy and environmental variables in the coastal waterssouthwest of Taiwanrdquo Fisheries Oceanography vol 18 no 2 pp102ndash117 2009

[25] IDaubechiesTenLectures onWavelets SIAMPhiladelphia PaUSA 1992

[26] S Mallat A Wavelet Tour of Signal Processing Academic PressSan Diego Calif USA 1998

[27] C Torrence and G P Compo ldquoA practical guide to waveletanalysisrdquoBulletin of the AmericanMeteorological Society vol 79no 1 pp 61ndash78 1998

[28] A Grinsted J C Moore and S Jevrejeva ldquoApplication of thecross wavelet transform and wavelet coherence to geophysicaltimes seriesrdquo Nonlinear Processes in Geophysics vol 11 no 5-6pp 561ndash566 2004

[29] D SerreMatricesTheory and Applications Springer NewYorkNY USA 2002

[30] A Quetglas F Ordines M Hidalgo et al ldquoSynchronouscombined effects of fishing and climate within a demersalcommunityrdquo ICES Journal of Marine Science vol 70 no 2 pp319ndash328 2013

[31] S E Lluch-Cota A Pares-Sierra V O Magana-Rueda etal ldquoChanging climate in the Gulf of Californiardquo Progress inOceanography vol 87 no 1ndash4 pp 114ndash126 2010

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 Journal of Applied Mathematics

testing in wavelet analysisrdquoMarine Ecology Progress Series vol359 pp 11ndash23 2008

[24] C-H Hsieh C-S Chen T-S Chiu et al ldquoTime series analysesreveal transient relationships between abundance of larvalanchovy and environmental variables in the coastal waterssouthwest of Taiwanrdquo Fisheries Oceanography vol 18 no 2 pp102ndash117 2009

[25] IDaubechiesTenLectures onWavelets SIAMPhiladelphia PaUSA 1992

[26] S Mallat A Wavelet Tour of Signal Processing Academic PressSan Diego Calif USA 1998

[27] C Torrence and G P Compo ldquoA practical guide to waveletanalysisrdquoBulletin of the AmericanMeteorological Society vol 79no 1 pp 61ndash78 1998

[28] A Grinsted J C Moore and S Jevrejeva ldquoApplication of thecross wavelet transform and wavelet coherence to geophysicaltimes seriesrdquo Nonlinear Processes in Geophysics vol 11 no 5-6pp 561ndash566 2004

[29] D SerreMatricesTheory and Applications Springer NewYorkNY USA 2002

[30] A Quetglas F Ordines M Hidalgo et al ldquoSynchronouscombined effects of fishing and climate within a demersalcommunityrdquo ICES Journal of Marine Science vol 70 no 2 pp319ndash328 2013

[31] S E Lluch-Cota A Pares-Sierra V O Magana-Rueda etal ldquoChanging climate in the Gulf of Californiardquo Progress inOceanography vol 87 no 1ndash4 pp 114ndash126 2010

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of