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In the previous post I highlighted a wavelet based feature extraction model for financial time series. The intention was to use it as a starting point for developing an improved model. To start with let me explain what the model referred to in the previous post is meant to do.

There are two sections to the model. Common to both sections is a DWT smoother which is basically a lowpass filter. The first section analyses the smoothed volatility of the time series to determine the break points which they refer to as “variance change”. The second section performs FFT decomposition on the smoothed time series to obtain the dominant cycle. The trend and turning points are determined from the smoothed time series also.

Here is what I think about the model

• Firstly I don’t support the researchers’ choice of applying the Fourier Transform to determine the dominant cycle period. I explain in this post why FFTs should be avoided when analysing financial time series data. I am in favour of the Hilbert transform mainly for two reasons:
  1. The Hilbert transform works under the assumption that each price point has a phase difference to the previous and subsequant price points. This allows evaluation of the cycle length on a bar-by-bar basis, which unlike the FFT does not impose a constraint on the observation window length.
  2. The Hilbert transform provides a way of visualising interaction of short term cycles with longer term cycles on a phasor plot.
• What I wish to do is to replace the FFT part with the Hilbert Transform and another algorithm to determine the phasor plot of the actual time series - as shown below:

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In the previous post I pointed out a model developed by researchers at the University of Surrey for analysing volatile financial time series. The model is actually quite straight forward to implement and my initial plan was to reproduce it, but haven’t got round to doing so because of other commitments. Nonetheless the plan continues and I shall be posting details of the modifications I wish to make to the model already proposed.

I found a poster presentation drawn up by the same researchers which provides a good summary of the paper they published.

Alternatively you can download a PDF file of the diagram from here.

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This presentation covers almost everything I have been wanting to convey about the usefulness of wavelets.

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The researchers propose a prototype feature extraction system, the block diagram of which is reproduced below:

Questions I ask:

1. It appears that the researchers settled for a fixed wavelet function for doing the decomposition. Are there any benefits in using a hybrid of different wavelet families for decomposing different parts of the time series?
2. Is it possible to create a custom wavelet family with the aim that it works better than the other wavelet families commonly used? What should be the line of approach to this kind of problem? Inductive or Deductive?

I am tempted to write a Matlab model to replicate this system and maybe adapt it a little bit, but there is a paper written by the researchers that needs to be understood first. You can download it from here.

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Here I write about things that may aid those who are new to Signal Processing to grasp a few concepts without necessarily referring to a textbook.

The table below shows some the differences between a Wave and a Wavelet.

Essentially a Wavelet is a small wave with pretty much the opposite characteristics to that of a wave. This is exactly what makes their application appealing to time variant signals such as stock prices. Accompanying wavelets is the Wavelet transform algorithm, of which there are many variants, but all essentially do the same thing using different ways. The table below shows the difference between two contrasting analytical algorithms.

So what are these “domains” we are referring to? Any signal that is a function of time is in the time domain. Stock prices are in the time domain because the each point in time has a price associated with it. Now consider another time domain signal such as a sine wave. Lets assume that this signal has amplitude 2 and takes 5 seconds before it begins to repeat itself i.e. it’s period is 5 seconds. Now the frequency of this signal is the inverse of the period which is 0.2 hertz in our case. Also the amplitude of the signal is the in the frequency domain our sine wave will look like a single spike of height 2 at the point 0.2 hertz. This is because the frequency domain shows the amplitude or power of the signal (y-axis) for different frequencies (x-axis), which is referred to as the spectrum. The spectrum of a stock price takes a more complicated form and is more difficult to determine, but the wavelet transform takes care of that.

Now it is worth pointing out that wavelets are usually combined with other signal processing algorithms to form “systems” that enable the transformation from the time domain into the frequency domain to happen in particular ways, depending on what you want to achieve. For instance instead of determining the spectrum of the stock price one may want to develop a filtering scheme or a trend/cycle estimation scheme. We shall see more of that later.

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For those not too familiar with wavelet theory I have attached some notes here for easy view. The idea of a non-fixed sampling rate is fundamental to the Wavelet Transform and it is very important to understand these concepts before delving into actual applications. The newbie may find some of the material heavy going in terms of the maths involved, but I believe understanding ideas are more important than being able to evaluate intagrals.

The first set of notes presents basic multirate concepts. The notion of upsampling and downsampling are discussed in terms of time and frequency domain characteristics.

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The second set of notes gives a very friendly introduction to the Wavelet Transform and shows it’s advantages over the Fourier Transform.

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I have also attached a very readable paper titled “Wavelets for kids“. You can download it from here

Over the next few posts we shall look at some wavelet decomposition models for financial time series.

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I wish to start discussing wavelets, which is yet another branch of signal processing that has had success in feature extraction and volatility forecasting for time series. I suggest the following three excellent books for the reader who is interested in wavelets:

Additionally there are numerous web resources on wavelets and the like, with about two million hits via google search.

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