gp forecast programing

8/10/2019 Gp Forecast Programing

1/14

Genetic programming withwavelet-based indicators forfinancial forecastingJin Li1, Zhu Shi2 and Xiaoli Li11

The Centre of Excellence for Research in Computational Intelligence andApplications (CERCIA), School of Computer Science, The University of BirminghamEdgbaston, Birmingham B15 2TT, UK2School of Software Engineering, The University of Science and Technology of ChinaP.R. China

Wavelet analysis, as a promising technique, has been used to approach numerous problems iscience and engineering. Recent years have witnessed its novel application in economic anfinance. This paper is to investigate whether features (or indicators) extracted using the wavelanalysis technique could improve financial forecasting by means of Financial GenetProgramming (FGP), a genetic programming-based forecasting tool. More specifically, predict whether the Dow Jones Industrial Average (DJIA) Index will rise by 2.2% or morwithin the next 21 trading days, we first extract some indicators based on wavelet coefficients othe DJIA time series using a discrete wavelet transform; we then feed FGP with those wavele

based indicators to generate decision trees and make predictions. By comparison with thprediction performance of our previous study, it is suggested that wavelet analysis be capab

of bringing in promising indicators, and improving the forecasting performance of FGP.

Key words: financial forecasting; genetic programming; stock data; wavelet analysis.

1. Introduction

In the past decade, researchers in the fields of applied mathematics and electricaengineering have developed the useful wavelet analysis methods for the multi-scal

Transactions of the Institute of Measurement and Control28, 3 (2006) pp. 285297


2/14

representation and the analysis of complicated signals. Examples of wavelet applications are turbulence analysis, image compression, earthquake prediction, biomedicasignal processing and so forth (eg , Daubechies, 1992; Li and Yao, 2005; Li et al ., 2000Meyer, 1993; Percival and Walden, 2000). More recent years have witnessed its nove

applications in economic and finance (eg , Pan and Wang, 1998; Ramsey and Lampar1998; Ramsey and Zhang, 1997). An overview of wavelets in economic and finance can

be found in the references of Gencay et al . (2002) and Ramsey (2002). Often, thwavelet transform is applied as a decomposition tool to analyse financial time seriesApplications of wavelets are usually focused on studying the dynamics ancorrelation of financial time series, including scaling properties of foreign exchangvolatility (Gencayet al ., 2001), systematic risk in a capital asset pricing model (Gencaet al ., 2003), and the relationship between financial variables and real economic activit(Kim and In, 2003). Apart from these, there are also wavelet applications for financia

forecasting where wavelet coefficients are directly transformed as features, aninput to neural networks for predictions (eg , Arino, 1996; Aussem et al ., 199Murtagh et al ., 2003).

The interest in wavelet analysis in empirical finance is attributed to its advantagesAs opposed to the traditional Fourier techniques, wavelet analysis is able to revealocalized information within the data in the time-scale plane. More specifically, it icapable of decomposing an observed time series into a set of multi-scale or multiresolution constituent time series. This makes it suitable for the analysis of non-lineaand non-stationary financial time series. The time-scale decomposition leads to

number of benefits for financial analysis. Firstly, in theory, one is able to study financial time series at as many more time-scales as possible, rather than at fewtraditional time-scales, like long run and short run. In practice, signals are usualldecomposed into a number of constituent signals by discrete wavelet transformSecondly, through the decomposition, many of anomalies or noises in data can brevealed and therefore can be treated (eg , being removed) separately if necessaryFinally, from the forecasting point of view, it has been made possible to tailor specificomputational forecasting techniques to different constituent time scales and therebgain efficiency of forecast (Ramsey, 2002).

This study applies a discrete wavelet transform to decompose a financial timseries. A number of features are derived based on wavelet coefficients. The difference

between this study and the existing studies in the literature (eg , Arino, 1996; Aussemet al ., 1998; Murtagh et al ., 2003) are as follows. Firstly, the forms of features ardifferent. The features are derivatives from the wavelet coefficients, rather than thvalues of coefficients themselves. The indicators extracted in this study reflect thproperties of the time series in respect of dynamics and statistics. Secondly, oufeatures are generated using coefficients at a certain level, rather than at all levels, witan attempt at removing possible noise from original data. Finally, our approach adopt

the genetic programming technique, rather than neural networks, which is able tgenerate comprehensible decision trees. This makes our method superior to otherssimpl due to the fact that solutions need to be understood b human beings fo

286 Wavelet analysis


3/14

prediction task addressed previously by a genetic programming-based tool, FinanciaGenetic Programming (FGP; Li, 2001). The task is to predict whether an index sharise byr % or more within the nextn periods. Our earlier studies (Li and Tsang, 1999a

b, 2000; Tsang and Li, 2002; Tsang et al. , 1998; Tsang et al. , 2004) made prediction

using some of indicators derived based on some technical analysis rules in textbookIn this study, predictions shall be made using a number of novel indicators extracte

by means of the wavelet analysis technique. It is worth pointing out that sucnovel indicators would probably have more merit to predictability to future pricmovement because they could possibly remove potential noises in original data tsome extent. Provided all experimental settings are same, any improvement in thforecasting performance reported in this study could suggest that wavelet analysis bof value to FGP.

The structure of the paper is as follows. Section 2 describes the FGP system

used. Section 3 introduces wavelet analysis and discusses how the wavelet-baseindicators are generated using the wavelet analysis technique. In section 4, experments and results are reported. The discussions and conclusions are given in thfinal section.

2. FGP for financial forecasting

This section reviews the history of FGP and briefly presents its technical detail fofinancial forecasting. The measures of its prediction performance are also given.

2.1 Overview of FGP

FGP is a major implementation of the Evolutionary Dynamic Data InvestmenEvaluator (EDDIE; Tsang and Li, 2002; Tsang et al. , 1998; Tsang et al. , 2004which is an interactive genetic programming-based financial forecasting tool. It aimto help analysts search the space of interactions and make financial decisionGiven a set of indicators (or features from the point of view of data miningFGP attempts to find interactions among indicators and discover appropriat

corresponding thresholds for indicators. Using genetic programming, FGP generateGenetic Decision Trees (GDTs), which can be understood by human experts. Humaexpertise is channelled into FGP through indicators as the input to the systemIn this way, experts are allowed to experiment with a variety of indicators moreasily. The forecasting performance of FGP crucially depends on the quality of thindicators chosen. This study aims to examine the effectiveness of alternativindicators based on wavelets, instead of some technical analysis indicators used iour previous studies.

FGP system has two versions: namely FGP-1 and FGP-2. FGP-1 is designed to b

able to improve forecasting accuracy by combining experts forecasts from differensources. FGP-2 is designed to be able to improve prediction precision by a constrainh dl Th h dl ll t i k t i t i th t

Liet al. 28


4/14

of financial forecasting problems with demonstrated accuracy (Tsang and Li, 2002In particular, the efficacy of FGP-2 has been examined intensively through a set oprediction tasks: whether an index will rise by r % or more within the nextn perioddenoted byPrn . In this study, FGP-2 is exploited to attack a task, P

2.2

21, on the prices o

the Dow Jones Industrial Average (DJIA). Like our previous study (Li and Tsang2000), we still focus the performance of FGP-2 on the prediction precision (see thdefinition in section 2.2).

FGP-2 generates GDTs to make predictions. An example of a GDT is shown belowwhere a Positive prediction means that the goal can be achieved; Negative meanotherwise.

((IF (MV_50B//18.45) THEN PositiveELSE (IF TRB_5//19.48) AND (Filter_63B/36.24) THEN NegativeELSE Positive

MV_50, TRB_5 and Filter 63 involved in the GDT belong to three types of technicaindicators. They were derived on grounds of three simple technical analysis rules ithe financial literature, eg , Alexander (1964), Fama and Blume (1966) and Brock et a(1992), namely moving average rules, filter rules and trade range break rules. Thesindicators have been argued to have merits to financial forecasting (Brocket al. 1992Sweeney 1988). Our previous study (ie , Li and Tsang, 2000) adopted six indicators afollows to attackPrn on the DJIA.

1) MV_12/Todays price the average price of the last 12 trading days;

2) MV_50/Todays price the average price of the last 50 trading days;3) Filter_5/Todays price the minimum price of the last 5 trading days;4) Filter_63/Todays price the minimum price of the last 63 trading days;5) TRB_5/Todays price the maximum price of the last 5 trading days;6) TRB_50/Todays price the maximum price of the last 50 trading days.

Nevertheless, to find alternative promising indicators is one of the importanmotivations in this study. The hope is that any new derived wavelet-based indicatorwould be better and have more merit to the prediction. As a result, the performance oFGP can be improved in terms of the prediction precision.

For brevity, how FGP works can be explained in pseudo code below. To find mordetails of genetic programming technique, interested readers can refer to Koza (1992

Procedure FGP ( )Begin

Partition whole data into training data and testing data/*While training data is employed to train FGP to find the best-so-far-rule; the testindata is used to determine the performance of predictability of the best-so-far-rule */

Pop1/InitializePopulation(Pop); /*randomly create a population of decision trees. */

Evaluation (Pop); /* calculate fitness of each individual in Pop */Repeat

P R d ti (P )C (P )



5/14

Pop1/Mutation (Pop); /*apply mutation to population */Evaluation (Pop);

Until (TerminationCondition()) /*determine if the termination condition is fired *Apply the best-so-far rule to the testing data;

End

2.2 Performance measures of FGP

The prediction problemPrn can be treated as a binary classification problem. Each dacan be classified as either a positive position or a negative position. A positive positiopredicted by the GDT is sometimes called a buying signal or a recommendation to bu

both of which will be referred to in the following context of this paper. For each GDTwe define the Rate of Correctness (RC), the Rate of Missing Chance (RMC) and th

Rate of Failure (RF) as its prediction performance measures. The Rate of Precision (RPis also given as an important meaningful reference measure for the user, as it measurethe accuracy of buying signals. Formulae for each measure are given through contingency table (Table 1).

As mentioned earlier, FGP-1 generates GDTs, aimed at making prediction aaccurately as possible. Thus, RC on its own is an appropriate fitness function foFGP-1. In contrast, FGP-2 attempts to improve prediction precision, ie , RP, which equivalent to reducing RF. A lower RF means that each positive recommendatiomade by the GDT is more likely to be a good and correct opportunity for the investo

to make a bid. FGP-2 achieves this target by means of a novel constrained fitnesfunction, which is taken as follows.

fw_rcRCw_rmcRMCw_rfRF: Where 05w_rc; w_rmc; and w_rf51

(1

It involves three performance measures, ie , RC, RMC and RF, and three weights, iw_rc, w_rmc andw_rf. The goodness of a GDT is no longer assessed only by its RC, bu

by a synthetical value, which is the weighted sum of its three performance rates. Thuser is allowed to reflect their preferences to any measure by adjusting the weights

Table 1 A contingency table for the binary classification, where a specific prediction ruis invoked

/

RCTP TN

OO

TP TN

NN;

/

RMCFN

O;

/

RF(1RP) FP

N;

#of True Negative Positions[TN]

#of False Positive Positions[FP]

Actual #of negative position(O)/TN/FP

#of False Negative Positions #of True Positive Positions Actual # of positive position

Liet al. 28


6/14

Due to the brittleness of the fitness function (cf., Tsang and Li, 2002), a novel constrainparameter,R/[Pmin,Pmax], is introduced into Function 1, which defines the minimumand maximum percentage of recommendations that is used to enforce FGP to achievthe training data (like most machine learning methods, the assumption is that the tes

data exhibits similar characteristics). The effectiveness of the constraint in the fitnesfunction for achieving more reliable and accurate predictions has been demonstratein our numerous previous studies. In general, FGP-2 allows the user to tune parameter, ie , constraint R , in order to improve RP without affecting the Rsignificantly, though at the price of increasing RMC. Such a scenario recurs in thistudy as well (see section 4).

It is worth emphasizing again that the performance of FGP crucially depends on thquality of indicators chosen by the users. We argue that the higher quality of thindicators used would almost always lead to better performance of FGP. This i

evident in this study.

3. Wavelet-based indicators

In this section, a brief introduction of wavelet analysis is given. We then describe thindicators used in this study, which are derived from wavelet coefficients.

3.1 Discrete wavelet transform

Successful applications in science and engineering demonstrate that the waveletransform is a powerful signal or image processing method. Wavelet transformovercomes the shortcomings of the STFT (short time Fourier transform) by performina multi-resolution analysis of signals (eg , Daubechies, 1992; Meyer, 1993). The waveletransform can be used to describe the content of the different frequency over time of non-stationary time series at a time-scale space. Thus, some of transients that arhidden in the time series can be highlighted. The wavelet transform of a time seriex(t) is defined as

W(a; b) 1ffiffiffia

pgx(t)c

t ba

dt (2

wheret is the time,a/0 andb are scale and translation parameters, respectively; c(is a mother wavelet. W(a ,b) is the coefficients of wavelet transform of x(t). 1/a iproportional to the frequency of the wavelet function. For a small value of a , thwavelet coefficient corresponds roughly to a high-frequency component of a timseries; whereas a big one corresponds to a low-frequency component of the time serieBy adjusting scale parameter a, the wavelet transform can flexibly decompose a tim

series x(t) into multiple resolution constituent time series. Since the wavelecoefficients obtained can indicate local characteristics of a non-stationary time serie

t th ti l t id tif t t t i ti ft t t f t



7/14

series x(n) (n/1, 2,. . ., N), whereNis the number of data points in the time seriecan be defined as follows:

x(n)Xk

CJ

(k)fJ

(2Jnk)XJ

j1X

k

dj(k)c

j(2jnk) (3

wheredj(k) are called wavelet coefficients at the levelj {j/1, 2, . . .,J}, andCJ(k) are thcoefficients at the maximum resolution levelJ. Both values of coefficients are varied bposition, as indicated by the value ofk. The value ofJcan be set up by users from 1 ta maximum integer number, which is sustainable byN(ie , 2JB/N). f(.) is called fathewavelets whereas c(.) is called mother wavelets, both of which are derivable from

basic wavelet (eg , the Haar, the Daublet and the Morlet). The father wavelet providean approximate version of the time series at successive resolutions, whilst the mothe

wavelet captures the detail at each resolution.In summary, given a time series, a basis wavelet and a parameter J, both wavele

coefficients, ie , CJ(k) and dj(k) (j/1, 2, . . ., J) can be calculated by a fast recursivscheme (Meyer, 1993).CJ(k) represents the smooth coefficients that capture the trend othe time series, whereasdj(k), representing increasing finer resolution deviations fromthe smooth trend, can capture higher-frequency oscillations. To what extent tharesultant coefficients CJ(k) smooth the time series is determined by the size ofselected. The largerJis, the more smooth the part of the time series can be captured bCJ(k). The choice ofJis crucial in applications of wavelet analysis to finance and sha

be discussed further in section 4.

3.2 Deriving wavelet-based indicators

In this paper, the energy, entropy and others ofCJ(k), wavelet coefficients at levelJ, arcalculated and they are taken as indicators for FGP-2. The reason for choosing CJ(krather thandj(k), is thatCJ(k) captures major trends of a time series, whereas dj(k) onlcaptures deviations of the time series. Some of the derived indicators describe thfeatures of a financial time series in dynamics whilst others are mere statistics of

financial time series. Given that a financial time series could potentially refledynamics of the movements of financial markets, all indicators could have financiameaning to some extent. The formulae for extracting those features are given below

1) Energy feature . The feature is based on the amplitude with different frequency of a timseries. The energy of wavelet coefficient at each resolutions level j/1, 2, . . ., Jwith sliding window (l is the window size) with index i is written as:

(BEj)i

Xk

C2j (l)

i(4

) f Th f h f h l ff

Liet al. 29


8/14

(Hj)i

X

l

pl;j ln(pl;j)

i

(5

3) Curve length . The feature is to compute the trajectory of a wavelet coefficient. If a curvlength is long, the change of system is severe; slight otherwise. The formula ocomputation of the curve length is below:

CL[j]Xli1

jCj(i1)Cj(i)j (6

4) Non-linear energy. The feature is to describe the local change of energy informationwhich can be used to extract the spikes in the wavelet coefficients. The equation o

non-linear energy is below:

NE[j]Xli1

jCj(i)Cj(i)Cj(i1)Cj(i1)j (7

5) Statistic features . Some basic statistics can also be applied to extract some of featurefrom the wavelet coefficients. They are listed as follows:

Mean: Mean[j]1l

Xli1

Cj(i)

Maximum: f max(j)max(Cj(1); :::; Cj(j):::; Cj(l))

Minimum: f min(j)min(Cj(1); :::; Cj(j):::; Cj(l))

Median: f med(j)median(Cj(1); :::; Cj(j):::; Cj(l))

Standard deviation: STD(j)1

l Xl

i

Cj(i)mean(Cj(i)

2(8

All nine different indicators are adopted by FGP. Note that any feature above at a timindex,i, is calculated using a fixed sliding window that covers preceding l coefficienvalues (ie , widow size/l), because only previous coefficients are available at timindex i . We did not conduct any feature selection process in this study, as genetiprogramming itself has the capability of selecting more promising indicatoradaptively via its genetic operators such as reproduction, crossover and mutationwhile evolving decision trees.

4 E im nt nd lt



9/14

to know any performance improvement of FGP-2 in reducing RF, or increasinRP equivalently.

For a fair comparison, we follow our earlier study of Li and Tsang (2000). Thmajor parameter settings for FGP, such as population size, crossover rate, mutatio

rate, selection strategy and termination criteria, etc., are the same and listed iTable 2. The termination criteria are similar as well, which are either that FGreaches the maximum generation number (ie , 30), or that it runs out of th30 min time that we set (as opposed to the maximum running time, 2 h in ouprevious study), whichever is achieved first. Except for the nine novel indicatorused here, which replace six technical analysis indicators used previously, therare no other differences. Indeed, the dataset used is the same, which are thDJIA closing index data from 07/04/1969 to 09/04/1981 (a total of 3035 tradindays). We split up the whole dataset into the training dataset from 07/04

1969 to 11/10/1976 (1900 trading days) and the test dataset from 12/10/1976 t09/04/1981 (1135 trading days). Both the training period and the test periocontain roughly 50% of positive positions. GDTs, generated by FGP-2 on the trainindataset, are tested on the test dataset. We report performance results of the GDTon the test dataset.

A basis wavelet of Daubechies 4 is used in this study. To calculate those ninindicators, we setJ/2 and window size/64 (ie , l/64), which deliver better resultthan those using other levels and/or other window sizes experimentally. Ouextensive experiments, which are not reported in this paper, suggest a rule of thumfor selecting the level Jand window size l . In general, Jis preferable to be a smallefigure more than 1, eg , 2 or 3, since J/1 usually results in indicators with lowequality because of possible noisy information in data not being filtered; whilst a highe

Table 2 Tableau for the parameters of FGP-2 experiments

Input terminals (9 wavelets-basedindicators)

BE3, H3, CL[3], NE[3], Mean[3],fmax(3),fmax(3), fmax(3), STD(3), and Real values

Prediction terminals {0, 1}: 1 means Positive; 0 means Negative

Non-terminals If-then-else, And, Or, Not,/

,]/

,B/

,5/

,/

Crossover rate 0.9Mutation rate 0.01Population size 1200Maximum no. of generations 30Termination criterion The maximum number of generations has bee

run or FGP-2 has run for more than 0.5 hSelection strategy Tournament selection, size/4Max depth of individual

programs17

Max depth of initial individualprograms 4Run times 00.5 h

Liet al. 29


10/14

J (eg , 4 or 5) usually leads to ineffective indicators, because of useful data informationprobably being eliminated. Window size l is preferable to be three times longer as thlength of future days that the prediction covers. Similarly, any bigger l or smallerwould potentially worsen the indicators derived because of possibly impropeinformation being taken into account.

Similarly, we take four non-overlapped R/

[Pmin

, Pmax

] values (ie , R1/

[35, 50R2/[20, 35]; R3/[15, 20]; R4/[10, 15]) for the constrained fitness functionrespectively, and run FGP-2 in turn. For each R , 10 independent runs were completeand total 10 GDTs were produced consequently. Table 3 lists the performances oall 10 GDTs and their mean and standard deviation for R1/[35, 50]. The RP, RCand RMC are 0.7270, 0.6281 and 0.5383 (which increase 12.76% and 9.09% in Rand RC, respectively and reduce 11.91% in RMC), compared with 0.5994, 0.537and 0.6574, respectively, obtained in the earlier study (see the last row in Table 5A statistical two-tailed unpaired t-test has been applied to determine whethethe results differences between two groups are statistically significant. Showin Table 4 are t-values and their corresponding p-values under each measureindicating that the generated GDTs statistically exhibit much better performancunder all of three criteria at a significant level of aB/1.0/107. The beauty is thathe performance improvement does not scarifice the overall number of buyinrecommendations (ie , TP/FP), as 377 here vs 339 previously is observed usindifferent indicators.

Table 4 t-statistics for comparing mean performances of two groups (the results using the

wavelet-based indicators versus the results using the technical indicators forR/[35%, 50%

For RP For RC For RMC

Table 3 The results of 10 GDTs generated by FGP-2 using R1/[35, 50]

GDTs Precision RMC RC TP FP TN FN

GDT 1 0.7358 0.5389 0.6326 273 98 445 319

GDT 2 0.7124 0.5355 0.6229 275 111 432 317GDT 3 0.7128 0.5304 0.6247 278 112 431 314GDT 4 0.7437 0.5000 0.6493 296 102 441 296GDT 5 0.7557 0.5557 0.6352 263 85 458 329GDT 6 0.6659 0.5118 0.6053 289 145 398 303GDT 7 0.7564 0.5541 0.6361 264 85 458 328GDT 8 0.7304 0.5743 0.6185 252 93 450 340GDT 9 0.7169 0.5338 0.6256 276 109 434 316GDT 10 0.7396 0.5490 0.6308 267 94 449 325MEAN 0.7270 0.5383 0.6281 273.3 103.4 439.6 318.7

STD 0.0254 0.0206 0.0112 12.2 16.7 16.7 12.2



11/14

5

Th

eresultsofmeansandstandarddeviationsfor4

Rsintermsoftwodistin

cttypesofindicatorsusedinFGP-2

M

eanperformanceusingwavelets-basedindicators

R

Precision

RMC

RC

TP

FP

TN

FN

5]

Mean

0.8

539

0.9

919

0

.4813

4.8

1.5

541.5

587.2

STD

0.1

206

0.0

160

0

.0051

9.5

3.9

3.9

9.5

0]

Mean

0.7

680

0.9

331

0

.5012

39.6

13.7

529.3

552.4

STD

0.0

892

0.0

366

0

.0131

21.7

7.7

7.7

21.7

5]

Mean

0.7

646

0.7

620

0

.5665

140.9

40.9

502.1

451.1

STD

0.0

407

0.1

231

0

.0469

72.9

20.6

20.6

72.9

0]

Mean

0.7

270

0.5

383

0

.6281

273.3

103.4

439.6

318.7

STD

0.0

254

0.0

206

0

.0112

12.2

16.7

16.7

12.2

M

eanperformanceusingte

chincalanalysisindicator

s

R

RF

RMC

RC

TP

FP

TN

FN

5]

Mean

0.7

140

0.9

405

0

.4970

35.2

14.1

528.9

556.8

STD

0.0

622

0.0

165

0

.0076

9.8

5.2

5.2

9.8

0]

Mean

0.6

898

0.8

569

0

.5174

84.7

40.4

502.6

507.3

STD

0.0

521

0.0

641

0

.0167

38.0

26.5

26.5

38.0

5]

Mean

0.6

400

0.7

525

0

.5341

146.5

83.3

459.7

445.5

STD

0.0

259

0.0

550

0

.0119

32.6

23.4

23.4

32.6

0]

Mean

0.5

994

0.6

574

0

.5372

202.8

136.1

406.9

389.2

STD

0.0

141

0.0

299

0

.0048

17.7

17.5

17.5

17.7

Liet al. 29


12/14

For brevity, for other three Rs, we only list their means and standard deviationin Table 5. The results suggest that the performance of RP have been improved foall of four Rs, though the performances of RMC and RC have not been improvefor all four Rs. The experimental here is encouraging because the improvement o

RP is what FGP-2 aims to achieve. Therefore, in terms of the prediction precisio(ie , RP), the wavelet-based indicators are superior to the technical analysindicators in this study.

5. Discussions and conclusions

Motivated by potential benefits that wavelet analysis could bring into financiaforecasting, in this paper, we have applied the wavelet analysis technique to extracsome indicators from wavelet coefficients at a certain level in respect of dynamic

and statistics of the time series concerned. In particular, we have examined whethewavelet analysis could be used to improve the forecasting performance of ougenetic programming-based tool, FGP, through those indicators extracted. Theffectiveness of those novel indicators has been demonstrated by tackling a specifiprediction task with FGP-2. By comparison with the results in earlier work, ouexperimental results on DJIA index data have suggested that the novel indicators bcapable of producing better forecasting performance in terms of the predictioprecision. We may argue that the wavelet-based indicators may have much mormerit to the prediction problem studied here in comparison with the technicaanalysis indicators.

We are still far from understanding the role that wavelet analysis could play ifinancial forecasting. Further research is worth being conducted through answerinthe following questions. Firstly, should we derive more indicators at differenlevels, rather than one level in this study, and then make predictions using aindicators at different level; or should we derive indicators, then make a predictioat different levels, and finally make ultimate predictions by combining thosindividual forecasts? If we do, could we achieve better forecasting resultsSecondly, are we better equipped with the wavelet analysis techniques to handlthe prediction task here? For example, are our forecasting results sensitive to th

choices of wavelet types? Finally, what financial and economic problems are morsuitable for wavelet analysis to tackle? If all the above questions could banswered, we believe that we certainly would be in a better position or have

better chance to make wavelet analysis more successful in finance and economicGiven the successes of wavelets in other diverse fields and in finance aneconomics so far, we would expect positive return in financial applications witwavelets in the future.



13/14

References

Alexander, S.S. 1964: Price movement inspeculative markets: trend or random walks,

No. 2. in: P. Cootner, editor, The random

character of stock market prices . MIT Press,33872.

Arino, M.A. 1996: Forecasting time series viadiscrete wavelet transform. Working paper,University de Navarra, Barcelona.

Aussem, A., Campbell, J. and Murtagh, F.1998: Wavelet-based feature extraction anddecomposition strategies for financial fore-casting.Journal of Computational Intelligence inFinance , March/April, 512.

Brock, W., Lakonishok, J. and LeBaron, B.1992: Simple technical trading rules and thestochastic properties of stock returns. Journalof Finance 47, 173164.

Daubechies, I. 1992: Ten lectures on wave-lets. Conference Series in Applied Math.SIAM.

Fama, E.F. and Blume, M.E. 1966: Filter rulesand stock-market trading. Journal of Business39, 22641.

Gencay, R., Selcuk, F. and Whitcher, B. 2001:Scaling properties of foreign exchange vola-tility. Physica 289, 24966.

Gencay, R., Selcuk, F. and Whitcher, B. 2002:An introduction to wavelets and other filteringmethods in finance and economics . AcademicPress.

Gencay, R., Selcuk, F. and Whitcher, B. 2003:Systematic risk and time scales. QuantitativeFinance 3, 10816.

Kim, S. and In, F. 2003: The relationshipbetween financial variables and real eco-nomic activity: evidence from spectral andwavelet analyses. Studies in Nonlinear Dy-namics and Econometrics 7, 1183201.

Koza, J.R. 1992: Genetic programming: on theprogramming of computers by means ofnatural selection. MIT Press.

Li, J. and Tsang, E.P.K. 1999a: Improvingtechnical analysis predictions: an applicationof genetic programming. Proceedings of the12th International Florida AI Research SocietyConference , Orlando, FL, 10812.

Li, J. and Tsang, E.P.K. 1999b: Investmentdecision making using FGP: a case study.Proceedings of the 1999 Congress on Evolution-

C t ti IEEE P 1253 59

Li, J.and Tsang, E.P.K.2000: Reducing failurein investment recommendations using g

netic programming. Proceedings of the Six

International Conference on Computing in Ecnomics and Finance . Society for Computtional Economics, Barcelona.

Li, X. and Yao, X. 2005: Multi-scale statisticprocess monitoring in machining.IEEE TranIndustrial Electronics 52, 92224.

Li, X., Tso, S. K. and Wang, J. 2000: Real timtool condition monitoring using waveltransforms and fuzzy techniques.IEEE Tranactions on Systems, Man, and Cybernetics PaC: Applications and Reviews 31, 35257.

Murtagh, F., Starck, J.L. and Renaud, O. 200On neuro-wavelet modeling. The Journal Decision Supprort System 37, 47584.

Percival, D.B. and Walden, A.T. 2000: Wavelmethods for time series analysis. CambridgPress.

Meyer, Y. 1993: Wavelets, applications and algrithms . SIAM.

Pan, Z. and Wang, X. 1998: A stochastic nonlinear regression estimator using waveletComputational Economics 11, 89102.

Ramsey, J. B.and Zhang, Z.1997: The analysof foreign exchange data using waveformdictionaries. Journal of Empirical Finance 34172.

Ramsey, J.B. and Lampart, C. 1998: Thdecomposition of economic relationships btime scale using wavelets: expenditure anincome. Studies in Nonlinear Dynamics anEconometrics 3, 2342.

Ramsey, J.B. 2002: Wavelets in economics anfinance: past and future. Studies in NonlineaDynamics and Econometrics 6, 127.

Sweeney, R.J. 1988: Some new filter rule testmethods and results. Journal of Financial anQuantitative Analysis 23, 285300.

Tsang, E.P.K. and Li, J. 2002: EDDIE fofinancial forecasting, in S.-H. Chen editoGenetic Algorithms and Programming in Com

putational Finance. Kluwer Series in Computational Finance, Chapter 7, 16174.

Tsang, E.P.K., Li, J. and Butler, J.M. 199EDDIE beats the bookies.International Journof Software, Practice and Experience 28, 103343.

T E P K Y P d Li J 2004 EDDIE

Liet al. 29


14/14

gp forecast programing

Documents