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cycle is a rhythmic oscillation that has an identifiable frequency (e.g., 0.1 cycle per day) or, equivalently, periodicity (e.g., 10 days per cycle). In the previous two chapters, phenomena that are cyclic in nature were discussed. Those cycles were exogenous in origin and of a known, if not fixed, periodicity. Seasonality, one such form of cyclic phenomena, is induced by the periodicity and recurrence of the seasons and, therefore, is tied into an external driving force. However, while all seasonality is cyclic, not all cycles are seasonal. In this chapter, cycles that can be detected in price data alone, and that do not necessarily have any external driving source, are considered. Some of these cycles may be due to as yet unidentified influences, Others may result only from resonances in the markets. Whatever their source, these are the kinds of cycles that almost every trader has seen when examining charts. In the old days, a trader would take a comb-like instrument, place it on a chart, and look for bottoms and tops occurring with regular intervals between them. The older techniques have now been made part of modem, computerized charting programs, making it easy to visually analyze cycles. When it comes to the mechanical detection and analysis of cycles, ma*imum entropy spectral analysis (MESA) has become the preeminent technique.
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CYCLE DETECTION USING MESA Currently, there are at least three major software products for traders that employ the maximum entropy method for the analysis of market cycles: Cycle Trader (Bressert), MESA (Ehlers, 800.633.6372), and TradeCycles (Scientific Consultant Services, 516-696-3333, and Ruggiero Associates, 800-21 l-9785). This kind of analysis has
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been found useful by many market technicians. For example, Ruggiero (October 1996) contends that adaptive breakout systems that make use of the maximum entropy method (MEM) of cycle analysis perform better than those that do not. Maximum entropy is an elegant and efficient way to determine cyclic activity in a time series, Its particular strength is its ability to detect sharp spectral features with small amounts of data, a desirable characteristic when it comes to analyzing market cycles. The technique has been extensively studied, and implementations using maximum entropy have become polished relative to appropriate preprocessing and postprocessing of the data, as required when using that algorithm. A number of problems, however, exist with the maximum entropy method, as well as with many other mathematical methods for determining cycles. MEM, for example, is somewhat finicky. It can be extremely sensitive to small changes in the data or in such parameters as the number of poles and the look-back period. In addition, the price data must not only be de-trended or differenced, but also be passed through a low-pass filter for smoothing before the data can be handed to the maximum entropy algorithm; the algorithm does not work very well on noisy, raw data. The problem with passing the data through a filter, prior to the maximum entropy cycle extraction, is that lag and phase shifts are induced. Consequently, extrapolations of the cycles detected can be incorrect in terms of phase and timing unless additional analyses are employed. DETECTING CYCLES USING FILTER BANKS For a long time, we have been seeking a method other than maximum entropy to detect and extract useful information about cycles. Besides avoiding some of the problems associated with maximum entropy, the use of a novel approach was also a goal: When dealing with the markets, techniques that are novel sometimes work better simply because they are different from methods used by other traders. One such approach to detecting cycles uses banks of specially designed band-pass filters. This is a method encountered in electronics engineering, where filter banks are often used for spectral analysis. The use of a filter bank approach allows the bandwidth, and other filter characteristics, to be tailored, along with the overlap between successive filters in the bank. This technique helps yield an effective, adaptive response to the markets. A study we conducted using filter banks to explore market cycles produced profitable results (Katz and McCormick, May 1997). Between January 3, 1990, and November 1, 1996, a filter bank trading model, designed to buy and sell on cycle tops and bottoms, pulled $114,950 out of the S&P 500. There were 204 trades, of which 50% were profitable. The return-on-account was 651% (not annualized). Both the long and short sides had roughly equal profitability and a similar percentage of winning trades. Various parameters of the model had been optimized, but almost all parameter values tested yielded profitable results. The para-
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meters determined on the S&P 500 were used in a test of the model on the T-Bonds without any changes in the parameters, this market also traded profitably, retuming a 254% profit. Given the relative crudeness of the filters used, these results were very encouraging. In that study, the goal was simply to design a zero-lag filtering system. The filters were analogous to resonators or tuned circuits that allow signals of certain frequencies (those in the pass-band) to pass through unimpeded, while stopping signals of other frequencies (those in the stop-band). To understand the concept of using@ers, think of the stream of prices from the market as analogous to the electrical voltage fluctuations streaming down the cable from an aerial on their way to a radio receiver. The stream of activity contains noise (background noise or hiss and static), as well as strong signals (modulated cycles) produced by local radio stations. When the receiver is tuned across the band, a filter s resonant or center frequency is adjusted. In many spots on the band, no signals are heard, only static or noise. This means that there are no strong signals, of the frequency to which the receiver is tuned, to excite the resonator. Other regions on the band have weak signals present, and when still other frequencies are tuned in, strong, clear broadcasts are heard; i.e., the filter s center (or resonant frequency) corresponds to the cyclic electrical activity generated by a strong station. What is heard at any spot on the dial depends on whether the circuits in the radio are resonating with anything, i.e., any signals coming in through the aerial that have the same frequency as that to which the radio is tuned. If there are no signals at that frequency, then the circuits are only randomly stimulated by noise. If the radio is tuned to a certain frequency and a strong signal comes in, the circuits resonate with a coherent excitation. In this way, the radio serves as a resonating filter that may be tuned to different frequencies by moving the dial across the band. When the filter receives a signal that is approximately the same frequency as its resonant or center frequency, it responds by producing sound (after demodulation). Traders try to look for strong signals in the market, as others might look for strong signals using a radio receiver, dialing through the different frequencies until a currently broadcasting station-strong market cycle--comes in clearly. To further explore the idea of resonance, consider a tuning fork, one with a resonant frequency of 440 hertz (i.e., 440 cycles per second), that is in the same room as an audio signal generator connected to a loud speaker. As the audio generator s frequency is slowly increased from 401 hertz to 402 to 403, etc., the resonant frequency of the tuning fork is gradually approached. The nearer the audio generator s frequency gets to that of the tuning fork, the more the tuning fork picks up the vibration from the speaker and begins to emit its tone, that is, to resonate with the audio generator s output. When the exact center point of the fork s tuning (440 hertz) is reached, the fork oscillates in exact unison with the speaker s cone; i.e., the correlation between the tuning fork and the speaker cone is perfect. As the frequency of the sound emitted from the speaker goes above (or
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below) that of the tuning fork, the tuning fork still resonates, but it is slightly out of synch with the speaker (phase shif occurs) and the resonance is weaker. As driving frequencies go further away from the resonant frequency of the tuning fork, less and less signal is picked up and responded to by the fork. If a large number of tuning forks (resonators or filters) are each tuned to a slightly different frequency, then there is the potential to pick up a multitude of frequencies or signals, or, in the case of the markets, cycles. A particular filter will resonate very strongly to tire cycle it is tuned to, while the other filters will not respond because they are not tuned to the frequency of that cycle. Cycles in the market may be construed in the same manner as described above-as if they were audio tones that vary over time, sometimes stronger, sometimes weaker. Detecting market cycles may be attempted using a bank of filters that overlap, but that are separate enough to enable one to be found that strongly resonates with the cyclic activity dominant in the market at any given time. Some of the filters will resonate with the current cyclic movement of the market, while others will not because they are not tuned to the frequency/periodicity of current market activity. When a filter passes a signal that is approximately the same frequency as the one to which it is tuned, the filter will behave like the tuning fork, i.e., have zero lag (no phase shift); its output will be in synchrony with the cycle in the market. In addition, the filter output will be fairly close to a perfect sine wave and, therefore, easy to use in making trading decisions. The filter bank used in our earlier study contained Butterworth band-pass filters, the code of which was rather complex, but was fully disclosed in the Easy Language of TradeStation.
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