2. A Daubechies 4 Wavelet; 5 Level decomposition of the Hang Seng Index.
3. Using a Wavelet to De-Noise the Hang Seng Index:more efficient than a Moving Average, with no lag. (click on image to see how the Wavelet 'hugs' the time series closely, yet gets rid of daily noise) 4. A 1-D complex continuous wavelet of the Straits Times Index in Jet Color Mode. Self-similar patterns are indications of fractals i.e. the Index is not totally random but has its own 'memory'.
5. A 1-D Continuous Wavelet of the Straits Times Index in Prism Color Mode. Again we see patterns of sorts. But it takes a skilled Wavelets practioner to interpret, Unfortunately, I'm just beginning to understand Wavelets.
Financial markets historical data are essentially the same as digital signals. And therefore they can be processed and analyzed with the tools used in digital signal processing. Fourier transforms are not suitable for the decomposition of financial data which are non-stationary, unlike digital signals from electronic devices which tend to be more stationary i.e. having a constant Mean and Variance. Fourier transforms capture the frequencies of the signals and their amplitude, but not the time. Wavelets can capture both time and frequency aspects of a signal. Wavelets are much better at de-noising and compressing signals than traditional DSP tools.New image compression standards like jpeg 4 use wavelets. And for getting rid of noise, but still retaining the essence of data, wavelets are unbeatable. That is why they are used in practically all modern fingerprint recognition systems which have huge database like those used by Immigration Authorities and Crime Prevention bodies.
For financial data, wavelets are suitable for analysis because they are able to spot subtle discontinuties in the data pattern which can be a sign of big changes coming.
Unfortunately, Wavelets are difficult to apply. It still takes experience and expertise to choose the appropriate class of Wavelets [there are about 10 types of Wavelets], and to stretch them to the right length to fit the data. It is still as much an art as a science, although new techniques such as Wavelet Packets and the Lifting algorithm do help a bit. But Wavelets in the hands of a skilled practitioner can definitely be a big help in the analysis of highly speculative stocks for anticipation of their sudden huge moves.
Self-Organizing Maps are a class of Artificial Intelligence. Artificial Intelligence is basically divided into two classes-Supervised and Unsupervised. Supervised AI like Back Propagation Neural Networks have to be taught by showing some of the data, and then are required to apply what they learn on unseen data i.e. be able to generalize. Self-Organizing Maps do not have to be taught. They are able to classify multi-dimensional data without supervision putting them into homogenous clusters i.e. each cluster holds data that are similar in characteristics to each other. Top image shows how a Self-Organizing Map organized the data of 2800 U.S. stocks into clusters of similarity taking into account their P/E, M/B, Volatility, Market Cap, Volume, Earnings Estimate and two dozen other fundamental variables- all at once.
The images above are a by-product of using these new technologies for the analysis of financial data. They show that beauty is everywhere- even in the realm of man-made digital technology.