Sooner or later every neural network practitioner learns that the choice and preprocessing of input data is the most critical task when noisy data is used. Neural networks tend to perform better when filtered data used as inputs 'under-reacts'; filters should yield values that are smooth, and consistent when faced with similar patterns. Reliability of filter response is even more important when the training data set size is relatively small since the neural network easily can memorize the historical patterns without retaining an ability to recognize them in real time when the price action will be different than the historical price action. Wavelet filtered variables provide an excellent form of preprocessing for a neural network model in addition to otherwise proper normalization.

Noise Elimination and Feature Detection
Noise is the bane of most trading systems used by off-the-floor traders. One person's noise is likely to be another person's information, so it is important to avoid generalizing about precisely what constitutes noise. However, once noise has been defined in the context of an application or trading system, it is relatively easy to design a filter to eliminate it: We simply define a filter that captures the noise, then subtract the captured noise from the original series. What remains is (presumably) the important information. 

The complement to noise elimination is feature detection. For information varying at an approximately known rate it is appropriate to apply a filter that is concentrated at the expected rate. The output of this filter will be the information in which we are most interested, with extraneous variation removed. 

Feature detection is closely related to noise reduction and the difference is largely a question of perspective. Perhaps we believe that very high frequency information in a price series is largely due to the activity of floor traders or otherwise represents untradeable information in the context of our system. In such a case we will want to remove the filtered information. On the other hand, perhaps when the larger commercial account traders enter a market their presence causes some short-term price fluctuations that our model recognizes as indicative of future price changes. In this case we will want to keep the short-term filtered information. 

Sometimes we have no idea where the important information may lie. Worse still, the various periodic features are all jumbled together in an incomprehensible mess in the raw series. Vital information may be hidden under worthless clutter, invisible to the naked eye as well as to most prediction models. This situation is best remedied by using multiple filters to separate the original series into two or more components, each of which can be examined separately without interference from other components. When we are wallowing in ignorance, filtering for information separation can be most useful.

Why Use Wavelets?
Most people would agree that extended repetition in financial series is the exception rather than the rule. Almost any time extended repetition becomes apparent, alert observers capitalize on it and thereby eliminate it. Hence there is a need for filters that extend across relatively short stretches of time. 

Probably the best known family of short-term filters is wavelets. The term wavelet comes from the fact that it is a little wave. There is an infinite number of wavelets that can be used for analyzing a series, and nearly this many are in use today. Wavelets have many valuable properties. Some of these properties are the following:

·    All of the individual wavelets in a family are derived from the same mother wavelet. Once we know the properties of the mother wavelet, we know the properties of all members of the entire family.

·    Wavelets easily lend themselves to shifting along time. If we do things correctly, a feature that appears at some particular time will reveal itself in a wavelet analysis in a consistent way. Its wavelet characteristic will be the same, no matter when the feature occurs. This lets us use history to predict the future.

·    Wavelets are inherently based on a meaningful time scale. For skilled users, this is no great advantage because it is not terribly difficult to learn to think in terms of frequencies. But for beginners and casual users, it is much simpler to think in terms of phenomena whose cycle length is a specified number of samples.

For these and other more technical reasons, wavelets make an excellent choice for analyzing financial series.

Why are Morlet Wavelets Valuable?
There is a wide variety of wavelets from which one can choose. For many reasons, the Morlet wavelet is probably the best choice for feature detection in financial series. These reasons include the following:

· Morlet wavelets are naturally robust against shifting a feature in time. Little or no special precautions are needed to ensure that a feature will make itself known in the same way no matter when it occurs. Daubechies wavelets, and in fact all orthogonal wavelets so fond to the signal processing community, present great challenges in ensuring consistency across time.

· A famous mathematical formula called the Heisenberg Uncertainty Principle decrees (roughly) that no filter can, with arbitrary accuracy, simultaneously locate a feature in terms of both its period and its time of appearance. In order to gain more precision in one, the other must be sacrificed. The laws of physics are quite firm here. This principle imposes a bound on how well a wavelet can detect a feature. The Morlet wavelet, for all practical purposes, achieves this bound. In other words, other filters can do no better at simultaneously locating a feature in terms of its period and when it appears. Most other wavelets do worse, and many wavelets and other filters do considerably worse. This is a valuable property.

· Morlet wavelets have a very intuitive nature and definition.

Why Use Gabor Filters?
Wavelets are very narrow in their capabilities. They act as bandpass filters only, a given wavelet responds only to periodic variation in the vicinity of its center frequency. Sometimes we want to keep not only the variation around the center frequency, but all variation above or below as well. Lowpass filters pass all information at or below a given cutoff frequency, and highpass filters pass all information at or above a given cutoff frequency. These types of behavior are not available from a strictly implemented wavelet. A good way to generalize wavelet filters to include lowpass and highpass operations is the Morlet's close relative, Gabor filters.

Gabor filters exactly meet the Heisenberg Uncertainty Limit; there is no other filter that offers a better combination of reliability and the ability to hone in on events in both time and frequency. 

For more information about WAVEFIN contact:

John Bonn / Steve Helme
Cornice Research LLC
3496 Moore Circle
Flagstaff, AZ 86001
nstinfo@corniceresearch.com
www.corniceresearch.com