 “e”), a contractingdescending bear cycle (ID  “d”), and a bear connector cycle(ID  “i”) (Figure 21-2).From Table 21-2, we find the frequencies shown in Table 21-3for the connector cycle.Thus there is over a 65 percent likelihood that the fourth wavewill touch or descend below the lowest point in the third wavewhen using a 15-pip reversal amount.
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Properties of Four-Wave Cycles
Table 21-2 Four-Wave Cycle PercentagesSorted by Cycle IDCycle
5-Pips
15-Pips
25-Pips
AGAHAIBBBCBFCACDCEDBDCDFEAEDEEFGFHFIGBGCGFHGHHHIIAIDIE
3.113.476.765.841.994.973.053.461.075.021.536.543.452.620.853.392.607.031.944.051.521.010.825.07806.847.015.00
1.339.4618.690.140.130.351.240.140.340.370.742.4110.081.163.230.131.242.200.110.850.810.322.8710.3918.162.2210.89
0.5712.2818.930.000.000.040.420.000.040.040.230.7211.640.263.780.000.380.790.000.230.420.084.2712.5919.730.7211.87
Figure 21-1 Three-Wave Bull Connector Cycle.
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Part 7: Four-Wave Cycles
Figure 21-2 Possible Fourth-Wave Continuations.
Table 21-3 Connector Cycle PercentagesCycle IDIAIDIE
Raw Percent
Adjusted Percent
18.162.2210.89
58.087.1034.82
8
PARTFive-WaveCycles
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Chapter
22
Properties ofFive-WaveCyclesPRUNINGAs the minimum reversal amount is increased, five of the ninebasic three-wave cycles decrease in frequency of occurrenceowing to the equality relationship. We will arbitrarily refer tothese cycles as secondary cycles, indicating that their low frequency equates to less importance. Specifically, these cycles arelisted in Table 22-1.The four remaining cycle types will be referred to as primarycycles because their greater frequency of occurrence impliesgreater importance. They are listed in Table 22-2.The purpose for making this distinction is to lessen the numberof cycle permutations that will occur when examining cycles composed of more than four waves. Using the overlapping methodology to name all five-wave cycles, there are 81 possibilities(9  3  3  81).However, by removing all the secondary cycles, this number isreduced significantly to only 16 permutations (4  2  2  16). In
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Part 8: Five-Wave Cycles
Table 22-1 Secondary Cycle TypesCycle ID
Cycle Name
BCDF
RectangleContracting descending triangleContracting ascending triangleExpanding ascending triangle
G
Expanding descending triangle
other words, we are dropping the “equals” condition from theheight ratio criteria. We are now concerned only with “greaterthan” or “less than” height ratios. This form of pruning is justifiedmathematically because it keeps the data and the results moremanageable.
FREQUENCIESIn Table 22-3, cycles are sorted in descending order of the fourthcolumn.There is no surprise that the trending cycles “IAI” and “AIA” leadthe list. It is interesting to note, though, that the 5- and 15-pipreversal amounts do not exactly match the order of the 25-pipreversal amount.
Table 22-2 Primary Cycle TypesCycle ID
Cycle Name
AEH
ImpulseContracting symmetrical triangleExpanding symmetrical triangle
I
Connector
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Properties of Five-Wave Cycles
Table 22-3 Five-Wave Cycles Sorted by 25-PipReversal AmountCycle ID
5-Pips
15-Pips
25-Pips
IAIAIAAHIIEAHIAIAHAIEEAIEAHHIEHHIAHHIEEEEAHHH
15.9211.7810.9210.776.847.526.537.254.128.052.911.932.861.910.30
14.9114.408.979.437.017.117.597.864.175.263.012.703.223.060.55
13.1412.709.469.107.847.647.126.795.385.303.723.603.403.320.85
EEE
0.39
0.73
0.65
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Chapter
23
Forecasting theFifth WaveOVERVIEWTable 23-1 is sorted by cycle ID and provides the information necessary to calculate probabilities for the binary relationships“greater than” and “less than” for height ratios.
EXTENDED IMPULSE CYCLEIn our naming convention, we define a four-wave extendedimpulse cycle as “Ai” because it has the potential to become a fivewave Elliot impulse cycle, as shown in Figure 23-1.Using a 15-pip reversal amount and referring to Table 23-1, wecan calculate the likelihood that a full five-wave bull Elliott impulsecycle (“AiA”) will come to fruition:AiA = 14.40AiE = 7.59Probability of AiA = 100 × 14.440/(14.40 + 7.59)Probability of AiA = 65.5 perccentThis 66 percent likelihood gives some credibility to the ancientmaxim, “A trend continues until it ends.”143Copyright © 2007 by The McGraw-Hill Companies. Click here for terms of use.
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Part 8: Five-Wave Cycles
Table 23-1 Five-Wave Cycle Frequencies Sorted by Cycle IDCycle ID
5-Pips
15-Pips
25-Pips
AHHAHIAIAAIEEAHEAIEEAEEEHHHHHIHIAHIEIAHIAIIEAIEE
1.9310.9211.786.534.127.251.910.390.302.916.848.057.5215.9210.772.86
2.708.9714.407.594.177.863.060.730.553.017.015.267.1114.919.433.22
3.609.4612.707.125.386.793.320.650.853.727.845.307.6413.149.103.40
POTENTIAL HEAD AND SHOULDERSHead and shoulders patterns represent trend reversals and, assuch, should be examined closely. The head and shoulders patternthat we will analyze has the critical descending neckline (cycleID  “Ah”), as shown in Figure 23-2.The first indication that the highest point in the third wave is infact a major reversal point is that the fifth wave does not exceedthis point. Thus we need to calculate the probability that the cycle
Figure 23-1 Potential Five-Wave Elliott Impulse Cycle.
Forecasting the Fifth Wave
145
Figure 23-2 Head and Shoulders with Descending Neckline.
“AhI” will occur. Again, refer to Table 23-1, this time using a 5-pipreversal amount.AhI = 10.92AhH = 1.93Probability of AhI = 100 × 10.992/(10.92 + 1.93)Probability of AhI = 85.0 perccentThis is a very promising result. There is an 85 percent likelihoodthat the fifth wave will be the beginning of a downward trend. Notethe relation between this pattern and the previous one, theextended impulse cycle, where the height of the fourth wave is theonly difference. The extended impulse cycle has the potential tobe a head and shoulders pattern with an ascending neckline, butthe probability is far less.
EXTENDED CONTRACTING TRIANGLEFrom classical pattern-recognition theory, we recall that trianglesnormally are considered to be continuation harbingers. Thismay be an optimistic appraisal that we feel requires testing. Wewill examine the bull contracting symmetrical triangle(cycle ID  “Ee”) using a 25-pip reversal amount, as shown inFigure 23-3.
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Part 8: Five-Wave Cycles
Figure 23-3 Extended Contracting Symmetrical Triangle.
Our premise is that the trend of the first wave is the current overall trend. Therefore, we want to calculate the likelihood that thefifth wave will be greater than the fourth wave.EeA = 3.32EeE = 0.65Probability of EeA = 100 × 3.32/(3.32 + 0.65)Probability of EeA = 83.6 percenttThis implies that there is an 84 percent likelihood that the existing trend prior to a double contracting symmetrical triangle willcontinue. Contracting triangles harbor their own unique fascination, not unlike increasing the torque on a metal spring. Sharpbreakouts frequently follow these triangles.
EXTENDED EXPANDING TRIANGLEExtended expanding triangles are a sign that the market is bothhighly volatile and very confused. Perhaps indecisive is a betterword (Figure 23-4).Expanding triangles are the only pattern that cannot be maintained indefinitely. In fact, in the 29 million  EURUSD databasecovering the tick data for calendar year 2005, there was only one
Forecasting the Fifth Wave
147
Figure 23-4 Extended Expanding Symmetrical Triangle.
occurrence of an “HhHhH” cycle, a seven-wave expanding triangle.Therefore, we will calculate the probability that the height of the fifthwave in this cycle will be less than the height of the fourth wave, thusterminating the expanding phenomenon. We will employ a 5-pipreversal amount in this example.HhI = 2.91HhH = 0.30Probability of HhI = 100 × 2.91/(2.91 + 0.30)Probability of HhI = 90.7 percenttThis result is not surprising, merely good to know, particularlywhen dealing with higher swing reversal amounts. We also shouldnote that contracting triangles theoretically can be maintainedover time simply by entering a phase of zero activity where the lastprice lingers indefinitely (such as weekends).This 91 percent probability represents about as close to a “surething” as is possible when forecasting with wave theory.
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9
PARTSix-WaveCycles
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Chapter
24
Properties ofSix-Wave CyclesOVERVIEWUsing the three logical operators , , and  for height-ratio comparisons of six-wave cycles generates 243 permutations (9  3  3 3  243). By excluding the equality relationship, the number of permutations is reduced to 32 possibilities (4  2  2  2  32).
FREQUENCIESTable 24-1 lists six-wave cycle frequencies sorted in descendingorder on the fourth column.Note the visible decrease in frequency between the second andthird rows. Also again we find the extended contracting andexpanding triangles at the bottom of the frequency list.
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Part 9: Six-Wave Cycles
Table 24-1 Six-Wave Cycles Sorted by 25-Pip Reversal AmountCycle
5-Pips
15-Pips
25-Pips
AIAIIAIAIEAIIAHIAIEAIAIEEAIAAHIAHIAIAIAHAHIEHIEAIEAHEAHIIEEAAHHIEAIEHIAHEEAIHHIAAIEEIAHHHIEEHHIEEAHHEEAHIEEEEEEAHHHIAHHHHHHH
8.839.177.507.674.854.913.904.825.244.155.475.293.934.062.062.192.342.421.181.161.001.351.801.690.760.830.430.300.360.240.03
9.709.336.335.735.665.365.404.945.034.943.923.663.183.222.482.342.382.081.871.751.741.611.441.211.111.130.690.650.530.470.06
9.269.076.025.715.555.275.205.115.074.953.853.713.403.362.542.472.352.211.901.851.791.711.431.301.221.160.680.640.590.530.06
EEEE
0.05
0.06
0.05
Chapter
25
Forecasting theSixth WaveOVERVIEWWhen analyzing cycles composed of six waves, we must note thatthese are not just randomly connected cycles. Our overlappingwave scheme ensures that the four central three-wave cycles aretightly linked to the adjacent cycles using the logical height-ratiocomparisons.Table 25-1 lists six-wave cycles that have been sorted by cycle IDto facilitate forecasting the sixth wave.
EXTENDED IMPULSE CYCLEGiven the five-wave impulse cycle shown in Figure 25-1, we wantto determine the likelihood that the sixth wave will be shorter thanthe fifth wave, thereby facilitating still another bull wave continuing the upward trend.The frequencies from Table 25-1 using a 15-pip reversal amount areAiAi = 9.70AiAh = 4.94Probability of AiAi = 100 × 9.70/(9.70 + 4.94)Probability of AiAi = 66.3 perrcent153Copyright © 2007 by The McGraw-Hill Companies. Click here for terms of use.
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Part 9: Six-Wave Cycles
Table 25-1 Six-Wave Cycles Sorted by Cycle IDCycle
5-Pips
15-Pips
25-Pips
AHHHAHHIAHIAAHIEAIAHAIAIAIEAAIEEEAHHEAHIEAIAEAIEEEAHEEAIEEEAEEEEHHHHHHHIHHIAHHIEHIAHHIAIHIEAHIEEIAHHIAHIIAIAIAIEIEAHIEAIIEEAIEEE
0.242.194.825.474.158.834.851.000.764.063.902.340.831.180.300.050.030.361.161.692.425.245.291.801.357.679.174.913.937.502.060.43
0.472.344.943.924.949.705.661.741.113.225.402.381.131.870.650.060.060.531.751.212.085.033.661.441.615.739.335.363.186.332.480.69
0.532.475.113.854.959.265.551.791.223.365.202.351.161.900.640.050.060.591.851.302.215.073.711.431.715.719.075.273.406.022.540.68
Thus there is a two out of three chance that a five-wave impulsecycle will continue trending in the same direction. This formationshould be confirmed by using different swing reversal amounts.
HEAD AND SHOULDERSA five-wave head and shoulders formation with a descending necklineis designated “AhI.” We want to calculate the probability that the sixthwave will fall below the vertex of the fourth and fifth waves, thus confirming that a reversal is actually in progress (Figure 25-2).
Forecasting the Sixth Wave
Figure 25-1 Extended Impulse Cycle.
Figure 25-2 Head and Shoulders Forecast.
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Part 9: Six-Wave Cycles
Using a 25-pip reversal amount, we obtain the following fromTable 25-1:AhIa = 5.11AhIe = 3.85Probability of AhIa = 100 × 5.11/(5.11 + 3.85)Probability of AhIe = 57.0 perrcentWe also will calculate the probability using a 5-pip reversal amount:AhIa = 4.82AhJe = 5.47Probability of AhIa = 100 × 4.82/(4.82 + 5.47)Probability of AhIe = 46.8 perrcentFortunately, this inconsistency where probabilities lie on bothsides of the 50 percent median when using different reversalamounts is very rare. When in doubt, stay out of the market.
CHIMERA CYCLEIn this example, we will analyze a five-wave chimera cycle so namedbecause of its changeling properties (Figure 25-3). It starts as acontracting symmetrical triangle and ends as an expanding symmetrical triangle (cycle ID  EaH). This cycle occurs frequentlyduring lateral congestion.We want to test to determine if the sixth wave will drop belowthe vertex of the fourth and fifth waves. This time we will use a15-pip reversal amount in Table 25-1:AhIa = 4.82AhJe = 5.47Probability of AhIa = 100 × 4.82/(4.82 + 5.47)Probability of AhIe = 46.8 perrcent
Forecasting the Sixth Wave
157
There is a 74 percent likelihood that the sixth wave will notdescend below the vertex of the fourth and fifth waves.
Figure 25-3 Chimera Cycle Forecast.
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Chapter
26
Double-WaveForecastingOVERVIEWIn previous chapters we used the height ratios of two or more wavesto extrapolate the height ratio of the next single wave in the swingdata. In this chapter we will illustrate examples of double-wave forecasting. The object is to determine the likelihood that a future point(the final point in the extrapolated two waves) will fall above orbelow the last known point in the swing data (Figure 26-1).
METHODOLOGYTo accomplish this task, we need to use the actual raw frequencycounts rather than the adjusted frequency percentages. In allcases, a 15-pip minimum reversal amount will be employed.First, we will examine the four-wave cycle “Ah” (Figure 26-2 andTable 26-1).
DOUBLE-WAVE ARITHMETICTwo of the cycles (“AhHi” and “AhIe”) have final points higherthan the last point in the “Ah” cycle. The other two cycles (“AhHh”159Copyright © 2007 by The McGraw-Hill Companies. Click here for terms of use.
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Part 9: Six-Wave Cycles
Figure 26-1 Double-Wave Extrapolation.
Figure 26-2 Four-Wave “Ah” Cycle.
Table 26-1 Raw Frequencies for “Ah” CycleCycle ID
Frequency
AHHHAHHIAHIAAHIE
41203428340
Double-Wave Forecasting
161
Figure 26-3 Double-Wave Forecast for “Ah” Cycle.
and “AhIa”) have final points lower than the last point in the“Ah” cycle. ThusHigher than last point = 100 × (203 + 340)/(41 + 203 + 428+340)Higher than last point = 53.7 percentThere is a 54 percent likelihood that an “Ah” cycle will befollowed by two waves whose final price will be higher than the finalprice in the “Ah” cycle (Figure 26-3).
RESULTS (Table 26-2)Unfortunately, the percentages of likelihood for price advancesor price declines are insufficient in themselves to warrant anymarket entry orders because they do not deviate significantly fromthe 50 percent median. Nonetheless, this type of information maybe useful when combined with other indicators. Interesting to noteis that the height of the vertex between the fifth and sixth waveshas an entirely neutral effect in this type of analysis.
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Part 9: Six-Wave Cycles
Table 26-2 Double-Wave Forecast PercentagesCycle ID
Higher
Lower
AhAiEaEeHhHiIaIe
53.751.946.252.049.053.050.355.4
46.348.153.848.051.047.049.744.6
10
PARTAdvancedTopics
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Chapter
27
Data OperationsTICK DATAIn order to compile an analytical study of this depth, it was necessary to acquire massive amounts of raw historical quotes. Forthese efforts, we wish to express our appreciation to Disk Trading,Ltd. (www.disktrading.com) for their extensive and well-organizedarchive of historical currency prices (both spot and futures) dating back to the early 1970s.Forex data can be packaged either as streaming data or as interval data. Streaming data consist of every single price change as itoccurs regardless of the time lapse between ticks. This creates hugeamounts of data. For example, all the tick quotes for the EURUSDcurrency pair from January 1, 2005, to December 31, 2005, provide the statistical sample for numerous analyses in this book witha sample size of 7,974,098 prices.Streaming data use the comma-delimited field conventionsshown in Table 27-1 for tick data.Data are shipped on compact disks and DVDs because the sheervolume of data is too large to download at current modem speeds.Comma-separated values (CSV) files must be unzipped and thenread as flat ASCII files.
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Part 10: Advanced Topics
Table 27-1 EURUSD Streaming Tick DataDate
Time
Price
2/3/20062/3/20062/3/20062/3/20062/3/20062/3/20062/3/20062/3/20062/3/20062/3/20062/3/20062/3/20062/3/20062/3/20062/3/2006
083008300830083008300830083008300830083008300830083008300830
1.20621.20641.20701.20661.20661.20871.20971.21011.21011.20991.21021.21021.21011.21051.2102
INTERVAL DATAInterval data, on the other hand, are compiled from the streaming data by coercing them into the standard open, high, low, close(OHLC) format for equal-interval time periods. Disk Trading, Ltd.,packages these types of data as 1-, 5-, 10-, and 30-minute, hourly,and daily data.Interval data are stored in the convention shown in Table 27-2(1-minute interval example).Dates are always expressed using the standard conventionMM/DD/YYYY, whereas the time field uses a four-digit integer torepresent the 24-hour convention (i.e., 2030  8:30 p.m.).Owing to the lack of centralization, Forex currency data do nothave volume and open-interest fields as in commodity futuresquotes. The last two fields above are upticks and downticks. Thesetwo fields can be used to calculate two indicators specific to currencytrading, the activity oscillator and the direction oscillator.
UNRAVELING INTERVAL DATAMost currency data vendors provide historical quote data in manydifferent time intervals. Most typical are streaming ticks (closes
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Data Operations
Table 27-2 EURUSD 1-Minute Interval DataDate
Time
Open
High
Low
Close
Upticks
Downticks
1/6/20061/6/20061/6/20061/6/20061/6/20061/6/20061/6/20061/6/20061/6/20061/6/20061/6/20061/6/20061/6/20061/6/20061/6/20061/6/2006
0830083108320833083408350836083708380839084008410842084308440845
1.20951.20941.21311.21231.21241.21081.21231.20981.20941.21121.21121.21161.21111.21181.21141.2112
1.20981.21421.21341.21321.21291.21281.21241.21101.21121.21191.21191.21171.21241.21221.21171.2116
1.20921.20941.21231.21181.21121.21081.20961.20871.20901.21051.21091.21081.21091.21021.21091.2108
1.20941.21231.21241.21251.21151.21231.20981.20951.21121.21121.21121.21131.21231.21141.21141.2113
110225260274246264229288228215171182216192126145
120228311284280267263240250203167232224179128165
only), 1-minute OHLC, 10-minute OHLC, 30-minute OHLC,hourly OHLC, and daily OHLC. However, most traders rarely havethe need or the patience to deal directly with tick data (exceptduring live trading sessions) because of their pure bulk and storage overhead. Critical analysts and statisticians do this job for theaverage trader.When dealing with swing reversal algorithms, tick data mayseem superfluous because the analyst is trying to filter out minorfluctuations anyway. Thus equispaced interval data are almostalways employed.All OHLC interval data have an intrinsic flaw by the mere factthat it is unknown which extreme occurred first, the high or thelow. In our algorithms that convert raw data to both swing dataand P&F data, we found it necessary to interject a preliminary