A W avelet B ased A pproach O f A thens Stock M arket P redictability

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A W avelet B ased A pproach O f A thens Stock M arket P redictability

A Wavelet Based Approach Of AthensStock Market PredictabilitybyHelen AvaritsiotisCentre For Quantitative FinanceTanaka Business SchoolImperial CollegeLondonSAMOS 12-14 July 2007AFE CONFERENCEA wavelet based approach of Athensstock market predictability1/17


ABSTRACT (1)Development of a process within the framework of my PhD:•Employs the Discrete Wavelet Transform (DWT) using: Haar Wavelet Daubechies-4 Wavelet•Prediction of four ATHEX Indices: Banks Index Telecoms Index Oil and Gas Index Insurance Index• Assessment of Forecasting AccuracySAMOS 12-14 July 2007AFE CONFERENCEA wavelet based approach of Athensstock market predictability2/17


ABSTRACT (2)•Employ the Discrete Wavelet Transform (DWT)•Select number of decomposition levels•Prediction at coefficients level using: Linear Regression - Approximate coefficients Autoregressive Process (AR) - Detail Coefficients•Reconstruction of predicted data using the Inverse DiscreteWavelet Transform (IDWT)•Assess Prediction Results using RMSE and HIT RATESAMOS 12-14 July 2007AFE CONFERENCEA wavelet based approach of Athensstock market predictability3/17


INTRODUCTION•Several models have been developed for time seriesprediction based on historical data (de Gooijer and Hyndman,2006)•Most popular: Auto-Regressive method (AR) Auto-Regressive Moving Average (ARMA)Auto-Regressive Integrated Moving Average (ARIMA)(discussed by Box and Jenkins, 1976 and Diebold, 2004)•Important requirement for aforementioned models isstationarity.•However, financial time series are usually non-stationary.They exhibit complicated patterns, abrupt changes etc. andstationarity transformation can be cumbersome.SAMOS 12-14 July 2007AFE CONFERENCEA wavelet based approach of Athensstock market predictability4/17


BASIC PROPERTIES OF THE DWT•The DWT is a powerful alternative, which has onlyrecently started to be used in relation to financial time seriesanalysis. Non-stationary time series become stationary afterwavelet transformation. (Marsy, 1993)•DWT decomposes time series into different time scales(multiresolution analysis) to reveal underlying structure ofthe series, seasonalities, discontinuities, volatility clustering.•Locality property of the DWT allows wavelets to localiseboth in time and scale (unlike Fourier), but underHeisenburg’s curse.•Correlated signals in the time domain become almostun-correlated in the time-scale domain through the DWT.•DWT inherently ‘smoothes’ data by appropriatethresholding of the wavelet coefficients at all levels.SAMOS 12-14 July 2007AFE CONFERENCEA wavelet based approach of Athensstock market predictability5/17


SHORT LITERATURE REVIEWForecasting techniques have been proposed that use wavelettransforms: Maximal Overlap Discrete Wavelet Transform togetherwith the Haar filter (Fryzlewicz, 2003) “A-trous”-wavelet transfrom (Renaud et al, 2005)In this paper, a new prediction method is proposed, which isessentially a bridge between the wavelet denoisingtechniques and wavelet predictive methods.More specifically we combine the DWT with the AR toproject detail coefficients at each level of the multi-scaledecomposition using the Haar and Daubechies-4 wavelets.SAMOS 12-14 July 2007AFE CONFERENCEA wavelet based approach of Athensstock market predictability6/17


PAPER SYNOPSIS• The DWT is presented, which decomposes time series intowavelet coefficients.• Thresholding methodology of the produced waveletcoefficients for denoising and smoothing of time series.• Prediction methodology using hybrid DWT/AR method• Backtesting experiment results using three different lengthrolling windows using the RMSE and Hit Rate measures forevaluation * .* Using especially developed C code.SAMOS 12-14 July 2007AFE CONFERENCEA wavelet based approach of Athensstock market predictability7/17


THE DISCRETE WAVELET TRANSFORM(DWT) (1)• A dyadic discretised wavelet function is:ψ− m 2 − m, n( t ) = 2 ψ (2 t nm−Where m,n ∈ Ζ control the wavelet dilation and translation respectively.)• For every wavelet function there exists an associatedscaling function:φ− m 2 − m, n( t ) = 2 φ (2 t nm−)The two functions act on the input signal complimentary as ahigh-pass and low-pass filter respectively.SAMOS 12-14 July 2007AFE CONFERENCEA wavelet based approach of Athensstock market predictability8/17


THE DISCRETE WAVELET TRANSFORM(DWT) (2)• By convolution of the wavelet function with a signal thedetail coefficients are produced:1m 1, n k 2 n m , kd+= ∑ g−c2k•Similarly, the approximation coefficients:1m 1, n k 2 n m , kc+= ∑ h−c2kWhere h,g are coefficients that satisfy the equations aboveSAMOS 12-14 July 2007AFE CONFERENCEA wavelet based approach of Athensstock market predictability9/17


HAAR AND DAUBECHIES WAVELETREPRESENATIONS• Haar WaveletNOTE: The left column showsthe ‘father’ wavelets(approximate representation)and the right column the‘mother’ wavelets. (detailrepresentation)• DaubechiesWaveletSAMOS 12-14 July 2007AFE CONFERENCEA wavelet based approach of Athensstock market predictability10/17


WITHIN SCALE DECORRELATION OFDETAIL COEFFICIENTSApplication of the DWT to the historical data produces detail coefficientsfor each level of the decomposition, which show a small within scalecorrelation. As the level of decomposition increases the correlationbetween the detail coefficients of each scale is reduced.Figure 1: Auto-correlation of Detail Coefficients as a Function of DecompositionLevel for the BANKS IndexSAMOS 12-14 July 2007AFE CONFERENCEA wavelet based approach of Athensstock market predictability11/17


WAVELET DENOISING•A time series can be denoised by thresholding of the detailcoefficients at all decomposition levels.λ• Threshold level, :λ =2 σ2 log( N)where N is the length of the decomposed vector, is the variance of thenoise estimated from the variance of the detail coefficients at the firstdecomposition level.• The resulting n thresholded detail coefficients atdecomposition level j:2σsoftdjjj[ n ] = sign ( d [ n ]) ( d [ n ] − λ )SAMOS 12-14 July 2007AFE CONFERENCEA wavelet based approach of Athensstock market predictability12/17


PROPOSED PREDICTION METHODOLOGY We used 512 daily values of the four new-sector indices ofATHEX, from 17/12/2003 to 30/12/2005 as historical data. Applied the DWT varying the rolling window width (64,128,256), employing both the Haar and Daubechies-4wavelets, always using 5 Levels of decomposition. At the last level of decomposition we estimate 1 value forthe approximate coefficients using the least-squaresalgorithm and 1 value for the detail coefficients using theAR model. At each level using the detail coefficients that wereproduced from the decomposition an appropriate number ofdetail coefficients is estimated. The IDWT is applied to produce the forecasts.SAMOS 12-14 July 2007AFE CONFERENCEA wavelet based approach of Athensstock market predictability13/17


PREDICTION ACCURACY MEASURESThe forecasting accuracy has been evaluated using:RMSERMSE =1pp( actual ( t ) predicted ( t ) )∑=−t12HIT RATE⎧ 1 if ∆ Ppred , t≤ 0 and ∆ Pactual , t≤ 0 ⎫⎪ ⎪SameDirection ( ∆ Ppred , t, ∆ Pactual , t) = ⎨ 1 if ∆ Ppred , t≥ 0 and ∆ Pactual , t≥ 0 ⎬⎪ 0 otherwise⎪⎩ ⎭actual t actual t actual tP P P−∆ = −∆ Ppred , t= Ppred , t− Ppred , t − 1, , , 1SAMOS 12-14 July 2007AFE CONFERENCEA wavelet based approach of Athensstock market predictability14/17


PREDICTION RESULTSATHEXIndexHaar Filter Db-4 Filter ARwindow 64 window 128 window 256 window 64 window 128 window 256 window 64RMSE HR RMSE HR RMSE HR RMSE HR RMSE HR RMSE HR RMSE HRBanks 58 0.54 54 0.55 55 0.56 49 0.52 49 0.52 52 0.60 84 0.000Telecoms 57 0.47 52 0.45 54 0.53 49 0.52 48 0.51 52 0.53 79 0.002Oil & Gas 66 0.50 62 0.52 69 0.53 55 0.52 55 0.52 64 0.52 97 0.000Insurance 82 0.52 74 0.54 75 0.51 69 0.53 69 0.58 73 0.54 123 0.002SAMOS 12-14 July 2007AFE CONFERENCEA wavelet based approach of Athensstock market predictability15/17


CONCLUSIONS The forecasting process proposed in combinationwith the Daubechies-4 filter predicts with a relativehigh degree of success all of the four sector-basedindices of ATHEX studied.The AR model used as benchmark showed zeropredictability.The prediction success depends on the rollingwindow length as it was expected.Extension of the proposed process with the use ofIndependent Component Analysis for the betweenscalede-correlation of detail coefficients is expected tofurther improve the forecasting success.SAMOS 12-14 July 2007AFE CONFERENCEA wavelet based approach of Athensstock market predictability16/17


REFERENCESBox, G.P. and G. M. Jenkins (1976). Time Series Analysis: Forecasting and Control,Holden-Day.De Gooijer, J.G. and R. J. Hyndman. (2006). 25 years of time series Forecasting. Int.Journal. of Forecasting, 22, 443-473.Diebold, Francis X. (2004). Elements of Forecasting. 3d Edition, Thomson South-Western.Donoho, D. L. (1995). Denoising via soft thresholding. IEEE transactions on InformationTheory, 41, 613-627.Fryzlewicz, P. (2003). Wavelet Techniques for Time Series and Poison Data. PhDDissertation, University of Bristol, U.K.Gencay, R., F. Selcuk and B. Whitcher (2002). An Introduction to Wavelets and OtherFiltering Methods in Finance and Economics. Academic Press.Mallat, S.G. (1989). A theory for Multiresolution Signal Decomposition: The WaveletRepresentation. IEEE Trans. On Pattern Analysis and Machine Intelligence, 11 (7), 674-692.Marsy, E. (1993).The wavelet transform of stochastic processes with stationary incrementsand its application to fractional Brownian motion. IEEE Transactions on Information Theory,39, 260-264.Ramsey, J. B. (1999). The contribution of wavelets to the analysis of economic andfinancial data. Phil.Trans. R. Soc. Lond. A 357, 2593-2606.Renaud, O., J.-L. Starck, and F. Murtagh. (2005). Wavelet-Based Combined SignalFiltering and Prediction. IEEE Transactions Systems, Man and Cybernetics, part B, 35(6),1241-1251.SAMOS 12-14 July 2007AFE CONFERENCEA wavelet based approach of Athensstock market predictability17/17

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