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Stocks & Commodities V. 14:11 (483-487): Stay In Phase by John F. Ehlers

Stay InPhase

TRADING TECHNIQUES

A

This cycles software specialist discusses anindicator based on cyclical analysis.

by John F. Ehlers

cycle is one market characteris-tic that can be scientificallymeasured. Although they can bemeasured, they are still mad-dening because they are in es-

sence ephemeral; they come and they go. Ourrecent research, however, indicates there is afundamental cycle parameter that leads us tothe correct trading strategy for any currentmarket mode. To find out more, we must startby defining two possible market modes, thetrend mode and the cycle mode. In the trendmode, the correct strategy is to buy (or sell, fordowntrends) and hold. In the cycle mode, thecorrect strategy is to buy and sell on the cyclicvalleys and peaks.

The parameter we use is the phase of thecycle. The measured phase tells us with greatsensitivity when we are in the trend mode,enabling the capture of a large fraction of thetrend movement. Typically, this capture rangeis far larger than can be obtained with acrossing moving average or other usual trendidentification techniques. In the cycle mode,the measured phase pinpoints the cyclic turnsin advance, with the further advantage that thefalse whipsaw signals of typical oscillatorsignals are avoided.

THE NATURE OF PHASETo use phase, we must first understand what itis. Simply, it is a description of where we arein the cycle. Are we at the beginning, middleor end of the cycle? Phase is a quantitativedescription of that location. Each cycle passesthrough 360 degrees to complete the cycle.One basic definition of a cycle is that it con-sists of an action having a uniform rate changeof phase. For example, a 10-day cycle passesthrough 360 degrees every 10 days. There-fore, a perfect cycle must change phase at the

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Copyright (c) Technical Analysis Inc.

Stocks & Commodities V. 14:11 (483-487): Stay In Phase by John F. Ehlers

rate of 36 degrees per day each day throughout the cycle.How does this help us see a trend mode? By reverse logic.

In a trend mode, there is no cycle, or at most a very weak one,and therefore, there is no rate change of phase. If we comparethe rate change of measured phase to the theoretical ratechange of phase of the weak dominant cycle in the trendmode, we get a correlation failure. This failure to correlate thetwo cases of the rate change of phase enables us to define thepresence of a trend. By knowing we have a trend, it is easy toset our strategy to a simple buy-and-hold method until thetrend disappears.

Θ

360˚

0˚

term averages to zero, with theresult that the correlation has anamplitude of π. The horizontalcomponent is Sin(θ)*Cos(θ) =0.5*Sin(2 θ). This term averagesto zero over the full cycle, withthe result that there is no horizon-tal component. The ratio of thetwo components goes to infinitybecause we are dividing by zero,and the arctangent is therefore 90degrees. This means the arrow ispointing straight up, right at thepeak of the sinewave.

One additional step in our cal-culations is required to clear the ambiguity of the tangentfunction. In the first quadrant, both the sine and cosine havepositive polarity. In the second quadrant, the sine is positiveand the cosine is negative. In the third quadrant, both arenegative. Finally, in the fourth quadrant, the sine is negativeand the cosine is positive. The phase angle is obtainedregardless of the amplitude of the cycle. For more informa-tion on the calculation, see sidebars, “BASIC code for phasecalculation” and “TradeStation code for phase calculation.”

PUTTING THE PHASETO WORKWe can make an outstanding cyclic indicator simply byplotting the sine of the measured phase angle. When we arein a cycle mode, this indicator looks very much like asinewave, but when we are in a trend mode, the sine of the

FIGURE 1: PHASE. Consider acycle to be one trip around a circle,or a 360-degree movement. Phasedescribes the location within acycle in degrees.

FIGURE 2: ADDING TIME. A sinewave in the time domain can be generated byplacing a pen on the arrowhead and drawing the paper along at a uniform rate, justlike a seismograph.

FIGURE 3: CYCLE BEGINNING. Phase varies uniformly throughout the cycle, andis drawn as reset to show the beginning of a new cycle.

PICTURE THISOne easy way to picture a cycleis as an indicator arrow bolted toa rotating shaft, as can be seen inthe phasor diagram of Figure 1.Each time the arrowhead sweepsthrough one complete rotation,a cycle is completed. The phase

increases uniformly throughout the cycle, as shown in Figure2. The phase continues for the next cycle but is usually drawnas reset to zero to start the next cycle.

If we also place a pen on the arrowhead and draw on a sheetof paper below the arrowhead at a uniform rate, much in theway like a seismograph, the pen draws a theoretical sinewave.The relationship between the phasor diagram and the theo-retical sinewave is shown in Figure 3. The sinewave is thetypical cycle waveform we recognize in the time domain onour charts. The phase angle of the arrow uniquely describeswhere we are in the time domain waveform.

BUT FIRST, SOME TRIGThe position of the tip of the arrow in Figure 1 can bedescribed in terms of the length of the arrow, L, and the phaseangle, θ. If we let the arrow be the hypotenuse of a righttriangle, we can convert the description of the arrow fromlength and angle to two orthogonal components — the othertwo legs of the right triangle. The vertical component isL*Sin(θ) and the horizontal component is L*Cos(θ). Theratio of these two components is the tangent of the phaseangle. So if we know the two components, all we have to doto find the phase angle is to take the arctangent of their ratio.This is something that may not be easy for you to do by hand,but it’s a breeze for your computer.

We measure the phase of the dominant cycle by establish-ing the average lengths of the two orthogonal components.This is done by correlating the data over one full cycle periodagainst the sine and cosine functions. Once the two orthogo-nal components are measured, the phase angle is establishedby taking the tangent of their ratio.

A simple test is to assume the price function is a perfectsinewave, or Sin(θ). The vertical component would be Sin2(θ)= 0.5*(1-Cos(2 θ)) taken over the full cycle. The Cos(2 θ)

Copyright (c) Technical Analysis Inc.

Stocks & Commodities V. 14:11 (483-487): Stay In Phase by John F. Ehlers

measured phase angle tends to wander slowly because thereis only an incidental rate change of phase. A clear, unequivo-cal indicator can be generated by plotting the sine of themeasured phase angle advanced by 45 degrees.

Such a case is depicted for the phasor diagram and the time

REAL WORLD EXAMPLESWhile theory is nice — it’s alwayshow I initially attack each tradingproblem — it is also possible fortheory to be a laboratory curiositywith no practical application. Hap-pily, this is not the case for the useof phase in trading.

FIGURE 4A: PLOTTING THE SINE OF THE MEASURED PHASE ANGLE. When ina trend mode, the sine of the measured phase angle tends to wander around slowlybecause there is only an incidental rate change of phase.

FIGURE 4B: PHASOR DIAGRAM AND TIME DOMAIN. Sinewave indicator iscreated by advancing the measured phase by 45 degrees.

domain in Figure 4B. The two lines cross shortly before thepeaks and valleys of the cyclic turning points, enabling theuser to make his trading decision in time to profit from theentire amplitude swing of the cycle. A significant advantage isthat the two indicator lines don’t cross except at cyclic turningpoints, avoiding the false whipsaw signals of most oscillatorswhen the market is in a trend mode. The two lines don’t crossbecause the phase rate of change is nearly zero in a trend mode.Since the phase does not change, the two lines separated by 45degrees in phase never get the opportunity to cross.

If the rate of change of the measured phase does notcorrelate with the theoretical phase rate change of the domi-nant cycle, then a trend must be in force. A workable defini-tion is that a trend exists when the measured phase rate ofchange is less than 67% of the theoretical phase rate of thedominant cycle. This is a very sensitive detector for the trendmode, enabling you to capture high percentages of the trendmovement.

The parameter we use is the phaseof the cycle. The measured phasetells us with great sensitivity whenwe are in the trend mode, enablingthe capture of a large fraction ofthe trend movement.

Figure 5 is a display for the September 1996 Deutschemarkcontract. The price bars are displayed in the top segment withtwo adaptive moving average overlays. The second segmentis the sinewave indicator, plots of the sine of the measuredphase angle and the phase angle advanced by 45 degrees.The measured phase is displayed below the sinewave indica-tor, and the measured dominant cycle and spectrum aredisplayed in the bottom segment.

From the way the phase varied uniformly in March, it isclear that Deutschemark was in the cycle mode during that

month. As a result, the sinewave indicator looks like asinewave and gives two buy signals and one sell signal wherethe sinewave indicator lines cross.

The phase stopped changing at a uniform rate in April andMay because two cycles identified by the spectrum displaywere present simultaneously. Since the phase hardly changedfrom day to day, the Deutschemark went into a trend modeduring these two months. The trend mode is identified, inTradeStation lingo, by “PaintBars,” where the price bars areviolet.

The correct trading strategy during April and May was tohold a short position (a move worth more than $2,500 percontract) because the faster adaptive moving average was

FIGURE 5: SEPTEMBER 1996 DEUTSCHEMARK. Here’s the cycle and phaseresponse of the September 1996 Deutschemark.

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Copyright (c) Technical Analysis Inc.

Stocks & Commodities V. 14:11 (483-487): Stay In Phase by John F. Ehlers

below the slower one. Another way to identify the downtrendis that the phase remained near zero degrees (or near 360degrees) during these two months. Note that the sinewaveindicator does not give false whipsaw signals during Apriland May. Whipsaws in the trend mode are common foroscillator indicators such as the stochastic and the relativestrength indicator (RSI).

The phase resumed its uniform rate of change during Juneand into July because the dominant cycle settled down to arelatively stationary value. As a result, the cycle mode ap-peared, the sinewave indicator again looks like a sinewave,and four excellent sell signals and three excellent buy signalsresulted.

Figure 6, showing the cycle and phase response of theSeptember contract of US Treasury bonds, is another ex-ample of how phase can be used to sharpen your tradingskills. In February and March, bonds were in a decline. Thephase hovered near zero degrees, clearly identifying thedowntrend. The correct trading strategy during this periodwas to hold a short position until the first cyclic buy signalgiven by the sinewave indicator early in April.

From that first cyclic buy signal, there were three morecyclic buy signals and three cyclic sell signals in the ensuingthree months. Bonds didn’t stay exclusively in the cyclemode during that time because the cycle length tended towander around. However, the cycle wandering only intro-duced distortions in the shape of the sinewave indicator. Thecrossover signals it produced were unequivocal and wouldhave produced substantial profits in every case.

CONCLUSIONPhase is an exciting new parameter to help techniciansanalyze the market, and it can help in several regards. First,it enables you to establish your trading strategy to fit the trend

BASIC CODE FOR PHASE CALCULATIONThis BASIC code finds the real part of the phasor (thehorizontal component) and the imaginary part of thephasor (the vertical component) by summing the productof the price and the two sinusoids over one full cycle ofthe dominant cycle. The arctangent function locates thephase to be in the first or fourth quadrant. The quadrantambiguity is removed by adding π to the phase angle is thereal part is negative. A value of π/2 is arbitrarily added tothe computed phase so the start of the cycle is referencedto a sinewave. The computed phase angle is then tested tofall within the range from zero to 2π. The phase is thenconverted to degrees from radian measure.

An interesting observation is that if the price is a linearslope, summing the product of the price and a sine over acycle is the discrete equivalent of the integral ∫ x Sin(x)dx. Correspondingly, the real part is the equivalent of theintegral ∫ x Cos(x) dx. Working through these theoreticalexamples, we find that the phase is 180 degrees for atrending upslope and zero degrees for a trendingdownslope. Thus, phase can be a sensitive way to detecta trend.

The dominant cycle is a required parameter in the code.This can be a constant, obtained by measuring the numberof bars between significant lows or significant highs. Aswith the TradeStation code, you can use a default value of15 for the dominant cycle. In MESA96, the dominantcycle is measured for every bar and can change from barto bar.

Pi=3.1415926TwoPi=2*PiFor I=FirstRecord to LastRecord RealPart = 0 ImagPart = 0 For J = 0 To DominantCycle(I) - 1 If I > DominantCycle(I) Then Weight = Close(I - J) RealPart = RealPart + Cos(TwoPi * J /DominantCycle(I)) * Weight ImagPart = ImagPart + Sin(TwoPi * J /DominantCycle(I)) * Weight Next If Abs(RealPart) > .001 Then Phase(I) = Atn(ImagPart / RealPart) Else Phase(I) = Pi / 2 * Sgn(ImagPart) End If If RealPart < 0 Then Phase(I) = Phase(I) + Pi Phase(I) = Phase(I) + Pi / 2 If Phase(I) < 0 Then Phase(I) = Phase(I) + TwoPi If Phase(I) > TwoPi Then Phase(I) = Phase(I) -TwoPi Phase(I) = 180 * Phase(I) / PiNext

—J.F.E.

FIGURE 6: CYCLE AND PHASE RESPONSE OF SEPTEMBER 1996 TREASURYBONDS. Bonds were in a decline in February and March. The phase hovered nearzero degrees, clearly identifying the downtrend. The correct trading strategy in thisperiod was to hold a short position until the first cyclic buy signal given by thesinewave indicator early in April.

Copyright (c) Technical Analysis Inc.

Stocks & Commodities V. 14:11 (483-487): Stay In Phase by John F. Ehlers

TRADESTATION CODE FOR PHASE CALCULATIONThe following code is written in EasyLanguage for usewith TradeStation. The major differences from the BASIC

code are that the angles are computed in degrees ratherthan radians and the input to the routine is the dominantcycle. We have let the default dominant cycle be 15 bars.

inputs: DomCycle(15);vars: RealPart(0), ImagPart(0), Weight(0), Phase(0),J(0);

for J = 0 to DomCycle -1 Beginweight = close[J];If DomCycle <>0 then BeginRealPart = RealPart + Cosine(360 * J / DomCycle) *Weight;ImagPart = ImagPart + Sine(360 * J / DomCycle) *Weight;

mode or the cycle mode. If you aren’t comfortable trading thecycle mode, you can always stand aside for a while until a newtrend mode is established. If you want to trade the cycle mode,the sinewave indicator, which is created by plotting the sine ofthe phase angle and the sine of the phase angle advanced by 45degrees, gives clear buy and sell signals in advance of eachcyclic turn. Getting the signal in advance enables you to make †See Traders’ Glossary for definition S&C

end;end;

If AbsValue(RealPart) > .001 Then BeginPhase = ArcTangent(ImagPart/RealPart);endelse BeginPhase = 90 * Sign(ImagPart);end;

If RealPart < 0 then Phase = Phase +180;Phase = Phase + 90;If Phase < 0 then Phase = Phase + 360;If Phase > 360 then Phase = Phase - 360;

plot1(Phase, “Phase”);—J.F.E.

your entry and exit right at the cyclic turning point withoutgiving up a piece of the market movement.

John Ehlers is an engineer and developer of the MESA96trading software program.

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