Understanding Market Cycles

by John Ehlers

The use of cycles is perhaps the most widely misunderstood aspect of technical analysis of the markets. This is due, in part, to a wide variety of disparate approaches ranging from astrology to wavelets being lumped into a cycles category. The purpose of this tutorial is to present a logical and consistent perspective on what cycles are and how they can be used to enhance technical analysis.

I was originally attracted to the use of cycles because it is one parameter on the charts that can be scientifically measured. These measurements can be used to dynamically modify conventional indicators, such as RSI, Stochastics and Moving Averages. Better yet, our research has provided superior indicators derived directly from cycle theory. The successful application of cycles to technical analysis is proven by mechanical trading systems.

The following sections are more or less independent but weave together to establish a basis for a scientific approach to trading. Some sections should be an easy read. Other sections might become too technical for many traders. If you feel uncomfortable in a section, just skip it for the time being and plan to return to it later. The punch line of this tutorial is in the final section, where we show how to correlate the indicators for a consistent analytical approach.

HISTORICAL PERSPECTIVE

Humans have observed cyclic recurring processes in nature since the earliest times and, over time, developed the basic concepts used in modern spectral estimation. Ancient civilizations were able to design calendars and time measures from their observations of the periodicities in the length of the day, the length of the year, the seasonal changes, the phases of the moon and the motion of the planets and stars. In the sixth century BC, Pythagoras developed a relationship between the periodicity of musical notes produced by a fixed tension string and a number representing the length of the string. He believed that the essence of harmony was inherent in the numbers. Pythagoras extended the relationship to describe the harmonic motion of heavenly bodies, describing the motion as the "music of the spheres."

Sir Isaac Newton provided the mathematical basis for modern spectral analysis. In the 17th century, he discovered that sunlight passing through a glass prism expanded into a band of many colors. He determined that each color represented a particular wavelength of light and that the white light of the sun contained all wavelengths. He invented the word spectrum as a scientific term to describe the band of light colors.

Daniel Bournoulli developed the solution to the wave equation for the vibrating musical string in 1738. Later, in 1822, the French engineer Jean Baptiste Joseph Fourier extended the wave equation results by asserting that any function could be represented as an infinite summation of sine and cosine terms. The mathematics of such representations has become known as harmonic analysis due to the harmonic relationship between the sine and cosine terms. Fourier Transforms, the frequency description of time domain events (and vice versa), have been named in his honor.

Norbert Wiener provided the major turning point for the theory of spectral analysis in 1930 when he published his classic paper "Generalized Harmonic Analysis." Among his contributions were precise statistical definitions of autocorrelation and power spectral density for stationary random processes. The use of Fourier Transforms, rather than the Fourier Series of traditional harmonic analysis, enabled Wiener to define spectra in terms of a continuum of frequencies rather than as discrete harmonic frequencies.

John Tukey is the pioneer of modern empirical spectral analysis. In 1949, he provided the foundation for spectral estimation using correlation estimates produced from finite time sequences. Many of the terms of modern spectral estimation (such as aliasing, windowing, prewhitening, tapering, smoothing and decimation) are attributed to Tukey. In 1965, he collaborated with Jim Cooley to describe an efficient algorithm for digital computation of the Fourier Transform. Unfortunately, this Fast Fourier Transform (FFT) is not suitable for analysis of market data.

The work of John Burg was the prime impetus for the current interest in high-resolution spectral estimation from limited time sequences. He described his high-resolution spectral estimate in terms of a maximum entropy formalism in his 1975 doctoral thesis and has been instrumental in the development of modeling approaches to high-resolution spectral estimation. Burg's approach was initially applied to the geophysical exploration for oil and gas through the analysis of seismic waves.

The Burg approach is also applicable to technical market analysis because it produces high-resolution spectral estimates using minimal data. This is important because the short-term market cycles are always shifting. Another benefit of the approach is that it is maximally responsive to the selected data length and is not subject to distortions due to end effects at the ends of the data sample. The trading program, MESA, is an acronym for Maximum Entropy Spectral Analysis.

PHILOSOPHICAL FOUNDATION FOR MARKET CYCLES

It has been written that the market is truly efficient and follows the random walk principle. The fact that Paul Tudor Jones, Larry Williams and a host of other notable traders consistently pull money from the market disproves the categorical assertion. However, a more detailed analysis of the random walk theory could yield some interesting results. 

Brownian motion is a random walk, which, for example, describes the path of a molecule of oxygen in a cubic foot of air. That molecule is free to move in three-dimensional space. The market is more constrained. Prices can only move up and down.  Time can only go forward. There is a more constrained version of the random walk, called the Drunkard's Walk. In this version, the "Drunk" staggers from point A to point B.  We want to examine two formulations of the problem. 

In the first formulation, the "Drunk" flips a coin, and, depending on whether the coin turns up heads or tails, takes a step to the right or left with each step forward. That is, the random variable is direction. The solution to this formulation is a rather famous differential equation called the Diffusion Equation. The Diffusion Equation describes many kinds of physical phenomena, such as the heat traveling up the shaft of a silver spoon when it is placed in a hot cup of coffee or the path of smoke particles leaving a smokestack.

In the second formulation, the "Drunk" again flips the coin. This time, however, he asks himself whether he should take a step in the same direction as the last one or in the opposite direction, depending on the outcome of the coin flip. The solution to this formulation is an equally famous (among mathematicians) differential equation called the Telegrapher's Equation. As the name implies, the Telegrapher's equation describes the way waves travel on a telegraph line. Lo and behold, we have a potentially cyclic solution to what started out to be a random walk problem!

A physical phenomenom embodying both these formulations of the Drunkard's Walk is the meandering of a river. Looking at the aerial photograph of any river in the world, you can see that there are places where the river path is more or less random and other places where the meanderings have a distinctive wavelike pattern. The explanation for these patterns is that the river is attempting to maintain a constant slope on its path to the sea, following the path of least resistance for the conservation of energy. The river attempts to maintain the constant slope by weaving to and fro in a manner similar to a skier maintaining a constant speed as he comes down the mountain. Taken in aggregate, the meanderings are not related to each other and are, therefore, random. However, if you are in a boat on any given meandering, it appears to be coherent and you can pretty well predict where the river is headed for a short distance.

How the Philosophy Translates to Market Patterns

So, here is the leap of assumptions for application of theory to the market. The market charts are similar to the aerial photograph of a river. There are places where the chart movement appears random and other places where distinctive cyclic patterns can be observed. There are plenty of forces on the market, such as greed, fear, etc., which in aggregate force the market to follow the path of least resistance. In this sense, the market is satisfying the conservation of energy. If this is true, then we can apply the Drunkard's Walk analysis to the market. There are times when the market is in a Trend Mode. In this case, the market path is similar to smoke coming from a smokestack being bent in a general direction by the breeze. And, in this case, the best predictor of the random variable is the (moving) average. There are other times when the market is in a Cycle Mode. In this case, the best predictor of a cyclic turning point is an "oscillator" that senses the change in momentum.

Think of it this way. Ask yourself IF the composite group of traders ask:

Will the direction of the market change?

OR

Will the trend continue?

The significant point for our technical analysis is that the market can be divided into two different modes:  the Trend Mode and the Cycle Mode. These two modes are traded in distinctly different, and often opposite, ways. Regardless, the market in the larger perspective is behaving randomly. Our goal as technical analysts is to exploit the short- term behavior.

TYPES OF CYCLE MEASUREMENTS

There are three methods commonly used for measuring market cycles. These are:

            1.  Cycle Finders

            2.  Fast Fourier Transforms (FFTs)

            3.  Maximum Entropy Spectral Analysis (MESA)

Cycle Finders are found in every toolbox software. These cycle finders basically enable you to measure the distance between successive major bottoms or successive major tops. The resulting cycle length is just the number of bars between these maxima or minima. Cycle finders are perhaps the second best way to measure market cycles. They have immediate application to the current cycle. One disadvantage is that the measurement can only be made at discrete intervals and is not continuous. A larger disadvantage is that there is a temptation to correlate a number of successive cycles. From our Drunkard's Walk discussion, we concluded that cycles can come and go in the market, and it is not necessarily true that we can correlate a string of them.

The Fast Fourier Transform (FFT) and Why It's Not Effective Measuring Market Cycles

One tool in most toolbox software packages is the Fast Fourier Transform (FFT). Using FFTs for market analysis is analogous to using a chainsaw at a wood-carving convention. FFTs are subject to several constraints. One of these constraints is that there can be only an integer number of cycles in the data window. For example, if we have 64 data samples in our measurement window (a 64-point FFT), the longest cycle length we can measure is 64 bars. The next longest length has 2 cycles in the window, or 64/2 = 32 bar cycle. The next longest lengths are 64/3 = 21.3 bars, 64/4 = 16 bars, and so forth. Therefore, the integer constraint means that there is a lack of resolution (i.e., a large gap between the measured cycle lengths that can be produced, right in the length of cycle periods that we wish to work). We can't tell if the real cycle is 14 or 19 bars in length.

The only way to increase the FFT resolution is to increase the length of the data window. If the data length is increased to 256 samples, we reach a one-bar resolution for cycle lengths in the vicinity of a 16-bar cycle. However, obtaining this resolution highlights another constraint. The cycle measurement is valid only if the data is stationary over the entire data window. That means that a 16-bar cycle must have the same amplitude and phase over a total of 16 full cycles.

In other words, using daily data, a 16-day cycle must be consistently present for more than a full year for the measurement to be valid. Can this happen? I don't think so! By the time, a 16-bar cycle occurs for more than several cycles, it will be observed by every trader in the world and they will destroy that cycle by jumping all over it. Its potential long-term existence is the very cause of its demise!

Why the Maximum Entropy Spectral Analysis (MESA) Method Is the Effective Solution to Measuring Market Cycles

The only way to obtain a high-resolution cycle measurement that is valid is to select a technique where only a short amount of data is required. MESA fills this requirement.

Still not convinced? Perhaps we can demonstrate our point with some measurements. Figure 1 shows how we have converted the amplitude of a conventional bell-shaped spectrum display to colors according to the amplitude of the spectral components. Think of the colors as ranging from white hot to ice cold. Colorizing the amplitude enables us to plot the spectrum contour below the price bars in time synchronization. A spectrum that is basically a yellow line has a sharp, well-defined cycle. A spectrum that has a wide yellow splotch means that the top of the bell-shaped curve is very broad and the measurement has poor resolution.

Figure 2 is a 64-point FFT measurement of a theoretical 24-bar sinewave. Because this is a theoretical cycle with no noise, the measurement should be precise. But, it is not! The spectral contour shows the measurement has very poor resolution. The measured length could just as easily be 15 bars as 30 bars. Figure 3 is a 64-point FFT taken on real market data. Here, one can barely determine that the cycle is moving around but cannot definitively identify the cycle. We will revisit these data again using the MESA measurement technique.

Figure 1.  Spectrum Amplitude to Color Conversion

 

Figure 2.  64-Point FFT of a Theoretical 24-Bar Cycle

 

Figure 3.  64-Point FFT of Market Data

The notional schematic for the way MESA measures the spectrum is shown in Figure 4. The data sample is fed into one input of a comparitor. This data sample can be any length, even less than a single dominant cycle period. The other input into the comparitor comes from the output of a digital filter. The signal input to the digital filter is white noise (containing all frequencies and amplitudes). This digital filter is tuned by the output of the comparitor until the two inputs are as nearly alike as possible.

In short, what we have done is pattern matching in the time domain. With some artistic license, we have removed the signal components with the filter, leaving the residual with maximum entropy (maximum disarray). Once the filter has been set, we can do several things with it. First, we can connect a sweep generator to the filter input and sense the relative amplitude of the output as the frequency band is swept. This produces the bell-shaped spectral estimate similar to the one shown in Figure 1. This spectral estimate is, in fact, the cycle content of the original data sample within the measurement capabilities of the digital filter. Secondly, because we have a digital filter on a clock, we can let the clock run into the future and predict futures prices on the assumption that the measured cycles will continue for a short time.

The MESA cycle measurement is notable in several regards. Most importantly, only a small amount of data is required to make a high-quality measurement. This means that  there is a higher probability of making a measurement using nearly stationary data because the data need to remain stationary over only a short span. As previously indicated, cycle measurements are valid only if the data is stationary. Secondly, the short amount of data used enables us to exploit the short-term coherency of the market. This is entirely consistent with the Telegrapher's Equation solution to the Drunkard's Walk problem, so the measured cycle, when the market is in the Cycle Mode, has predictive capability. Thirdly, the MESA approach allows high-resolution spectral estimates to be made.

The high-quality measurement of the theoretical 24-bar cycle is shown in Figure 5, where only one cycle's worth of data is used in the measurements. Here, the spectral contour is a single line, meaning that the bell-shaped curve is just a spike centered at the 24-bar cycle period. Figure 6 shows the ebb and flow of the measured cycle for the same data used in Figure 3. This cycle characteristic was only inferred in the FFT measurement.

 

 

Figure 5.  MESA Measurement of a Theoretical 24-Bar Cycle

Figure 6.  MESA Cycle Measurement of Market Data

THE IMPORTANCE OF PHASE          

To use phase, we must first understand what it is. Put simply, phase is a description of where we are in the cycle. Are we at the beginning, middle or end of the cycle? Phase is a quantitative description of that location. Each cycle passes through 360 degrees to complete the cycle. One basic definition of a cycle is that it consists of an action having a uniform rate-change of phase. For example, a 10-day cycle passes through 360 degrees every 10 days. For it to be a perfect cycle, it must change phase at the rate of 36 degrees per day each day throughout the cycle.

How does this help us see a Trend Mode? Easy. By reverse logic. In a Trend Mode, there is no cycle, or at the very least, a very weak one. Therefore, there is no rate change of phase. So, if we compare the rate change of measured phase to the theoretical rate change of phase of the weak dominant cycle present in the Trend Mode, we get a correlation failure. This failure to correlate the two cases of the rate change of phase enables us to define the presence of a trend. Because we know that we have a trend, it is easy to set our strategy to a simple buy-and-hold until the trend disappears.

One easy way to picture a cycle is as an indicator arrow bolted to a rotating shaft as shown in the phasor diagram of Figure 7. Each time the arrowhead sweeps through one complete rotation, a cycle is completed. The phase increases uniformly throughout the cycle, as shown in Figure 8. The phase continues on for the next cycle but is usually drawn as being reset to zero to start the next cycle.

Additionally, if we place a pen on the arrowhead and draw a sheet of paper below the arrowhead at a uniform rate, as they do for seismographs, the pen draws a theoretical sinewave. The relationship between the phasor diagram and the theoretical sinewave is shown in Figure 9. The sinewave is the typical cycle waveform we recognize in the time domain on our charts. The phase angle of the arrow uniquely describes where we are in the time domain waveform.

The position of the tip of the arrow in Figure 7 can be described in terms of the length of the arrow, L, and the phase angle, q. If we let the arrow be the hypotenuse of a right triangle, we can convert the description of the arrow from length and angle to two orthogonal components -- the other two legs of the right triangle. The vertical component is L*Sin(q) and the horizontal component is L*Cos(q). The ratio of these two components is the tangent of the phase angle. So, if we know the two components, all we have to do to find the phase angle is to take the arctangent of their ratio. This is something that may be tough for you, but it's a piece of cake for your computer.

We measure the phase of the dominant cycle by establishing the average lengths of the two orthogonal components. This is done by correlating the data over one full cycle period against the sine and cosine functions. Once the two orthogonal components are measured, the phase angle is established by taking the tangent of their ratio.

A simple test is to assume the price function is a perfect sinewave, or Sin(q). The vertical component would be Sin²(q) = .5*(1-Cos(2q)) taken over the full cycle. The Cos(2q) term averages to zero, with the result that the correlation has an amplitude of Pi. The horizontal component is Sin(q)*Cos(q) = .5*Sin(2q). This term averages to zero over the full cycle, with the result that there is no horizontal component. The ratio of the two components goes to infinity because we are dividing by zero, and the arctangent is, therefore, 90 degrees. This means the arrow is pointing straight up, right at the peak of the sinewave.

One additional step in our calculations is required to clear up the ambiguity of the tangent function. In the first quadrant, both the sine and cosine have positive polarity. In the second quadrant, the sine is positive and the cosine is negative. In the third quadrant both are negative. Finally, in the fourth quadrant, the sine is negative and the cosine is positive. The phase angle is obtained regardless of the amplitude of the cycle. 

An interesting observation is that, if the price is a linear slope, summing the product of the price and a sine over a cycle is the discrete equivalent of the integral  òx Sin(x) dx. Correspondingly, the real part is the equivalent of the integral  òx Cos(x) dx. Working through these theoretical examples, we find that the phase is 180 degrees for a trending upslope and is zero degrees for a trending downslope. Thus, phase can possibly be an additional way to determine the direction of the trend.

THE SINEWAVE INDICATOR AND HOW IT HELPS YOU MAKE TRADING DECISIONS

We can make an outstanding cyclic indicator simply by plotting the Sine of the measured phase angle. When we are in a Cycle Mode, this indicator looks very much like a sinewave. When we are in a Trend Mode, the Sine of the measured phase angle tends to wander around slowly because there is only an incidental rate change of phase. A clear, unequivocal indicator can be generated by plotting the Sine of the measured phase angle advanced by 45 degrees. This case is depicted for the phasor diagram and the time domain in Figure 10. The two lines cross SHORTLY BEFORE the peaks and valleys of the cyclic turning points, enabling you to make your trading decision in time to profit from the entire amplitude swing of the cycle.

A significant additional advantage is that the two indicator lines don't cross except at cyclic turning points, avoiding the false whipsaw signals of most "oscillators" when the market is in a Trend Mode. The two lines don't cross because the phase rate of change is nearly zero in a trend mode. Since the phase is not changing, the two lines separated by 45 degrees in phase never get the opportunity to cross.

Figure 10.  Generation of the Sinewave Indicator

If the rate of change of the measured phase does not correlate with the theoretical phase rate-change of the dominant cycle, a Trend must be in force. We will describe another workable definition for a Cycle mode in the next section.

USING MOVING AVERAGES WITH CYCLES

All moving averages smooth the input data and all moving averages suffer lag. The more smoothing you perform, the more lag you incur. Those are the facts of life. Within these parameters, some moving averages have unique characteristics. For example, a weighted moving average tends to have a delay response similar to a Bessel Filter. That is, many cycle lengths all have the same delay. This minimizes distortion of the filtered output. The amount of lag a moving average causes is calculated as the "center of gravity" (cg) of its weighting function. Because the weighting function of a conventional weighted moving average is a triangle, the induced lag is just one third of the window length.

Simple Averages are of more interest for use with cycles because they can be used to completely eliminate the dominant cycle component. The transfer response of a simple average is Sin(X) / X, which is the Fourier Transform of its rectangular weighting function. X is p times the frequency being filtered relative to the cycle length that just fits in the average window.

Consider an average length that is exactly one cycle long. Within this averaging window, there are exactly as many sample points above the center as below it. The result is that the average is zero, and the cycle within this window is completely eliminated by the averaging. We can make the simple average length just the length of the dominant cycle on any given day. This eliminates the dominant cycle at the output of the filter. If we repeat this everyday and connect the filter output values, we have an adaptive moving average from which the dominant cycle is completely eliminated. This adaptive moving average then becomes an instantaneous trendline because we asserted our model of the market could only have a Cycle Mode and a Trend Mode. Since the cyclic components are eliminated, the residual must be the instantaneous trendline. Creating an instantaneous trendline is a significant result of our cyclic analysis.

If we use a Zero Lag Kalman Filter, this filter line will cross the Instantaneous Trendline every half cycle when the market is in a Cycle Mode. If the Zero Lag Kalman filter fails to cross the Instantaneous Trendline within the last half cycle period, this is another way of declaring a Trend Mode is in force. The Trend Mode ends when the Zero Lag Kalman Filter line again crosses the Instantaneous Trendline.

By examining the peak-to-peak swing of the Zero Lag Kalman Filter, we can make an estimate of the peak swing of the dominant cycle. In general, if the peak-to-peak swing of the Zero Lag Kalman Filter is greater than twice the average range of the price bars, we have sufficient cycle amplitude to trade the short-term cycle in the Cycle Mode. If the peak swing of the cycle is less than twice the average bar height, getting a good entry and exit for the trade becomes a crapshoot. It is best to stand aside if the market is in a Cycle Mode and the cycle amplitude is low.

TRADING STRATEGIES AND TACTICS

Figure 11 is the MESA screen for a theoretical 24-bar cycle. There are four display segments on the screen. These are:

1.      The price bars, with the overlay of the instantaneous trendline and the Zero Lag Kalman Filter

2.      The Sinewave Indicator, consisting of the Sine of the measured phase and the LeadSine where the phase is advanced by 45 degrees

3.  The phase measurement, where phase varies between 0 and 360 degrees

4.  The Dominant Cycle

Figure 11.  MESA Display of a Theoretical 24-Bar Cycle

Because the data is a theoretical 24-bar cycle, the high-resolution cycle measurement in the bottom segment is essentially a straight line centered at the correct 24-bar cycle length. Similarly, the phase increases uniformly across the perfect cycles, snapping back to zero degrees to begin a new cycle when reaching 360 degrees at the end of a cycle. These two displays are uninteresting for the theoretical waveform other than to confirm the correct measurement of the data cycle.

The Sinewave Indicator segment has the cyan line as the Sine of the measured phase and is exactly in phase with the cycle in the price data. The LeadSine curve (in red) crosses the Sine curve with just enough advance notice to enable an entry or exit at the exact peak and exact valley of the price data.

The price bar segment shows the theoretical 24-bar cycle bars, having a swing of ±5, centered at 40. This chart has data only in the Cycle mode because the phase is changing uniformly. Because this theoretical waveform has no trend, the instantaneous trendline is a straight line at the 40 level.

We can make some observations about the indicators. Because we are in a Cycle Mode, the Sinewave indicator gives far and away the best signals. The half dominant cycle adaptive moving average crossing the instantaneous trendline gives exactly the wrong signals in this Cycle Mode condition. However, the half dominant cycle adaptive moving average indicates the cycle amplitude is sufficient to trade in the Cycle Mode.

COMBINING CYCLE-BASED INDICATORS

We will describe all the MESA indicators with the real-world example of the Treasury Bonds Futures continuous contract shown in Figure 12. As an overview, we see Bonds were mostly in a trend mode from the left side of the chart until mid-October because the Kalman Filter line stays above the Instantaneous Trendline most of the time. There were two short periods where the Cycle Mode occurred, but these quickly reverted back to the Trend Mode. Similarly, Bonds were mostly in the cycle mode for the right side of the chart. The market modes are indicated in subgraph 4, where a one indicates trend and a zero indicates cycle. The major uptrend ended on 11 October when the Kalman Filter line crossed the Instantaneous Trendline.

Although a cycle mode is given due to this crossing, it is clear that the rate change of phase and the Sine/LeadSine indicators in subgraph 2 indicate that there is no measured cycle. However, on 14 November the Sine/LeadSine indicators cross, correctly indicating a short entry. They cross again on 4 December, correctly indicating a reversal to the long side. The next cycle mode reversal to the short side is indicated by the Sine/LeadSine indicator crossing on 17 December. This position is quickly reversed to long on 23 December because the market reverted to a Trend Mode. Conflicting signals occur when the Kalman Filter line crosses the Instantaneous Trendline on 6 January. The most likely scenario is that the trader would have reversed to a short position on that date. In any event, the Sine/LeadSine indicators cross on 10 January, indicating a long side position.

The general guidance is to first look at the Mode Indicator in the fourth subgraph. A one of this indicator indicates a trend and a zero indicates cycle. Trade the relationship of the Kalman Filter line and the Instantaneous Trendline when the market is in a Trend Mode. Trade the crossings of the Sine/LeadSine indicator when the market is in a Cycle Mode.

The phase indicator can be used as a more detailed analysis tool to determine whether the market is cyclic or trending. When cyclic, the phase display will look like a sawtooth waveform. When trending, the phase display tends to meandering. There is no specific rate change of phase when the trend is dominant.

Figure 12.  MESA Display of Treasury Bonds

The dominant cycle display can be used in several ways. First, the cycle period should be relatively flat, indicating that the data is stationary. If the cycle period is decreasing, the average period will be too long, making the Sine/LeadSine indicator cross late. Conversely, if the cycle period is increasing the average period will be too short, making the Sine/LeadSine indicator cross early. You can use this to adjust the timing of your entry or position reversal.

This examination of the Treasury Bonds continuous contract was given to illustrate how all the philosophy, cycle-based indicators, and strategies and tactics all play together. Even the logic to break the rules generated by the automatic analysis was given. We hope the perspective on trading given in this tutorial has been educational and inspirational for you. Now, go get 'em. Good Trading!