MODWTMaximal Overlap Discrete Wavelet Transform
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The paper presents an approach depicting two models, namely, hybrid MODVVT-ANN and hybrid MODWT-SVR models to predict 1-step ahead forecasts for weekly National Stock Exchange Fifty price index, where the time series is first decomposed to different sub-series using MODWT. Then, these sub-series are predicted independently using two machine learning models and are aggregated to obtain the final forecasts.
To overcome the above two limitations, a modification of DWT called Maximal Overlap Discrete Wavelet Transform (MODWT) is used.
The proposed hybrid models (MODWT-ANN and MODWT-SVR) for prediction of stock price integrate the advantages of a decomposition model (namely, MODWT) and machine learning models (namely, ANN and SVR).
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] = from MODWT wavelet details reconstructed returns of the stock index A at scale [[tau].sub.j],
First we transform the indices' return series by MODWT. We obtain wavelet and scaling coefficients, which we use to study the energy decomposition of the return series at different time scales (j).
1.1 Description of the Maximal Overlap Discrete Wavelet Transform (MODWT)
Since the wavelet correlation estimator is usually constructed by the MODWT and is simply made up of the wavelet covariance and the wavelet variance for [X.sub.t] and [Y.sub.t], The MODWT estimator of the wavelet correlation can be expressed as
In this novel methodology, for any multivariate stochastic process [X.sub.t] = ([x.sub.1t], [x.sub.2t], ..., [x.sub.nt]), the wavelet coefficients [W.sub.jt] = ([w.sub.1j]t, [w.sub.2jt], ..., [w.sub.njt]) are obtained by applying the MODWT to each [] process at scale [[lambda].sub.j].
For any positive integer, J0, the level J0 MODWT of X is a transform consisting of the [J.sub.0] + 1 vectors [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] all of which have dimension N.
for t = 0,..., N - 1, where [[??].sub.j,l] and [[??].sub.j,l] are the jth MODWT wavelet and scaling filters.
Each wavelet packet has an associated filter, which when convolved with the data produces N wavelet packet coefficients, rather like the MODWT filter for a given value of j.
Like the DWT and MODWT described above, the MODWPT coefficients can be used in a multi-resolution analysis to estimate the partition of variance between frequency intervals (the wavelet packet variance) and to test for significant changes in variance across the space for a given packet.