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ness. The skewness of a distribution is the degree of its asymmetry around “stable Paretian.” Stable Paretian distributions have higher peaks at the
the mean. Positive skewness indicates a distribution with an asymmetric
tail extending more toward positive values. Negative skewness indicates a
distribution with an asymmetric tail extending more toward negative val-
ues. Next, we observe the kurtosis of the distribution. Kurtosis charac-
terizes the relative peakedness or flatness of a distribution, compared to
the normal distribution. A positive kurtosis indicates a relatively peaked
distribution. A negative kurtosis indicates a relatively flat distribution.
When we look at the distribution of financial data, we see some inter-
esting things. First, financial data have a distribution that is leptokur-
totic: large moves occur more than they should for a normal distribution.
6.2 The distribution of 5.day returns for the D-Mark
This property of being leptokurtotic˜is very important. The fact that large FIGURE
98 Statistically Based Market Prediction 99
A Trader™s Guide to Statistical Analysis


STATISTICAL TESTS™ VALUE TO TRADING
mean and fat tails, and they are characterized by a tendency to trend as
SYSTEM DEVELOPERS
well as to be cyclical. They also have discontinuous changes, and they
can be adjusted for skewness. They are different from simple leptokur-
totic gaussian distributions in that they have an infinite or undefined var- Many statistical testing methods are valuable for analyzing a market or
ante. These types of distributions are now called “fractal distributions.” for developing and testing a trading system, and you should learn how to
Because financial markets do not follow a gaussian distribution, using use some of these methods. Most of the statistical methods used in mar-
standard statistics can give us only an imperfect estimate, but that esti- ket analysis will tell (1) whether the distributions of two data sets are
mate is good enough to give us an edge. different or (2) whether the means of the populations are different. An
understanding of what each method is trying to prove is necessary be-
cause, when using statistical methods, you need to formulate both a hy-
THE CONCEPT OF VARIANCE AND STANDARD DEVIATION pothesis and its inverse, called the null hypothesis.
Hypothesis testing works as follows. We formulate a statistical method
The variance and standard deviation of a data set are very important. The so that the null hypothesis is used in the formulation. For example, if we
variance is the average deviation of the data set from its mean. The vari- had a trading system with an average trade of $200.00 and wanted to
ance is calculated as follows: prove statistically that our average trade was greater than 0, we would
formulate our statistical measure to assume a zero average trade. We
would then calculate our statistical measure and use that value to decide
whether to accept or reject the null hypothesis. If we reject the null hy-
pothesis, we show the original hypothesis to be true. This decision is
where N is the number of elements, M is the same mean, and D is the cur- based on a gaussian distribution of the statistical values for a given test.
rent value. The standard deviation is simply the square root of the vari- The relevant charts and values are available in every textbook on statis-
ance. The standard deviation has some very interesting trading tics. If we are two standard deviations from the mean value of a statistic
applications and is used in many different indicators. A Bollinger band, based on a standard distribution, then we can reject the null hypothesis
for example, is simply a price band drawn two standard deviations away at the 95 percent confidence level.
from the mean. To fully understand how to formulate a statistical test, let™s take a
closer look at some of the tests we could use. We will start with something
called the Z test, which is calculated as follows:
HOW GAUSSIAN DISTRIBUTION, MEAN, AND STANDARD
DEVIATION INTERRELATE M-D
z=yIIIIz
dV/,Y
The interaction between the mean and the standard deviation has many
where:
trading applications. First, there is a basic relationship between the
mean of a data set and the standard deviation. This relationship states M is the mean of the sample,
that, for a normal or standard distribution, 68 percent of the data is
D is the value based on the null hypothesis,
contained within one standard deviation and 95 percent of the data is
Vis the variance and its root is the standard deviation of the sample,
within two standard deviations. Almost all of the data is contained
and
within three standard deviations. For a normal distribution, this num-
ber is 99.5 percent. iV is the number of cases.
A Trader™s Guide to Statistical Analysis 101
100 Statisticallv Based Market Prediction



Let™s now see how we would use this in a trading application. Say we
Chi-rquare(*˜)=(134-100)1 =1,156=I1,56
want to know whether an S&P500 trading system had greater than a
100 100
$250.00 average trade. When we collect the system™s results, we find that
In our simple case, the Chi-square value is 11.56. For a two-tail Chi-
the average trade was $275.00 with a standard deviation of $30.00, and
there were 100 trades. We use the Z test to see whether this is true. Our square test, the 99.9 percent confidence level is above 10.83. We can now
first step is to formulate our hypothesis. Because our average trade is conclude that, for our sample, our results are predictive.
Another test, called student f, is often used. This test can indicate
$275.00, our null hypothesis can be that our average trade is $250.00 be-
cause $275.00 is greater than $250.00. We would then state as our hy- whether two distributions have the same mean or variances. The formula
is as follows:
pothesis: The average trade is greater than $250.00.
Our Z calculation is as follows:

%
z= 275-250


where ˜ =
JiGziG


I
z=25
N, +N,-2
30/10
2 = 8.33
so is the standard error of the difference of the means. Each sum is
over the points in one sample, the first or second. Likewise, each mean
Based on a normal distribution, 1 percent of the scores will have a Z
refers to one sample or the other, and N, and N? are the numbers of points
value greater than 2.33. Because 8.33 is greater than 2.33, we can con-
in the first and second samples, respectively. N, + N2 - 2 would be our
clude that our average trade was greater than $250.00.
degrees of freedom.
Another valuable statistical measure is called the “Chi-square test.”
Once we calculate t, we need to get the critical values for I based on
This test is used to judge whether the hypothesis results are valid for a
the degrees of freedom and a standard distribution.
small sample and is often used to test hypotheses based on discrete out-
comes. The formula for Chi-square is as follows:
CORRELATION ANALYSIS

One of the most powerful tools for developing and testing trading indi-
cators and systems is a statistical measure called Pearson™s correlation,
where Oi is the observed frequency in the ? class and E, is the expected a measure of the correlation between two data series. A 1 is a perfect
frequency in the ifh class. positive relationship and a -1 would be a perfect negative relationship.
Let™s look at an example of how we would use this, Suppose we have The formula for Pearson™s correlation r is as follows:
a pattern, in that the market rises 134 times out of 200 cases. We would
like to know whether this is statistically significant. Because 134 cases
out of 200 is greater than 50 percent or 100 cases, which is our expected
frequency based on a random population, Chi-square will tell us whether
our pattern is predictive of rises in the market. In this case, we would
calculate Chi-square as follows:
Statistically Based Market Prediction
102


where (Xi, Y;), i =I, , N is the pair of quantities whose correlation
we want to estimate, and X and L are the means of the Xi™s and Y!˜s,
respectively.
7
Pearson™s correlation is useful in many trading applications, for ex-
ample, in evaluating the current strength of different intermarket rela-
tionships. Pearson™s correlation can also be used to evaluate inputs for
Cycle-Based Trading
neural networks or other machine learning methods.
These examples are only a few of the many trading-based applications
for simple correlation analysis. We will be using correlation analysis
many times in later chapters of the book.

This chapter has given an overview of some simple statistical methods
that are valuable to traders. Other valuable methods are in use, but these
were chosen for discussion because we will use them in later chapters.


Cycles are recurring patterns in a given market. The area of cycles has
been intensely researched for almost 100 years. In fact, there are orga-
nizations dedicated to the study of cycles. Cycle-based trading has be-
come a hot topic because of the software now available for analyzing
financial data and for developing trading systems. Because of this soft-
ware, we can now see how a market is really composed of a series of dif-
ferent cycles that together form its general trading patterns. We can also
see that the markets are not stationary. This means that the cycles in a
market change over time. Change occurs because the movements of
a market are not composed solely of a series of cycles. Fundamental
forces, as well as noise, combine to produce the price chart.
Cycle-based trading uses only the cycle and the noise part of the sig-
nal. There are many tools that can be used for cycle analysis. The best
known are the mechanical cycle tools that are laid over a chart. An ex-
ample is the Stan Ehrlich cycle finder, a mechanical tool that is overlaid
on a chart to detect the current dominant cycle.
Among the several numerical methods for finding cycles, the most well
known is Fourier analysis. Fourier analysis is not a good tool for finding
cycles in financial data because it requires a long, stationary series of
data-that is, the cycle content of the data does not change. The best au-
merical.method for finding cycles in financial data is the maximum en-
trophy method (MEM), an autoregressive method that fits an equation

103
Statisticah Based Market Prediction
104 Cycle-Based Trading 105


THE NATURE OF CYCLES
by minimizing error. The original MEM method for extracting cycles
from data was discovered by J. P. Burg in 1967. Burg wrote a thesis on
MEM, which was applied to oil exploration in the 1960s. The method Let™s start our discussion by probing the nature of cycles. A cycle has
was used to analyze the returning spectra from sound waves sent into three major components: (1) frequency, (2) phase, and (3) amplitude. Fre-
rock to detect oil. There are several products that use the MEM algo- quency is a measure of the angular rate of change in a cycle. For exam-
rithm. The first cycle-based product was MESA, by John Ehlers. It is ple, a IO-day cycle has a frequency of 0.10 cycle per day. The formula
now available as a stand-alone WindowsTM product as well as an add-in for frequency is:
for TradeStation. Another powerful product is TradeCyclesTM, codevel-
Frequency = l/Cycle length
oped by Ruggiero Associates and Scientific Consultant Services. There
are other products, such as Cycle Finder by Walter Bresser, but they do
The phase is an angular measure of where you are in a cycle. If you had
not offer the ability to backtest results. If you cannot use a tool to back-
a 20.day cycle and were 5 days into the cycle, you would be at 90 degrees:.
test your results, then, in my opinion, you should be very careful trying
a complete cycle is 360 degrees, and you are 25 percent into the cycle.
to trade it.
The last major characteristic of a primitive cycle is amplitude. Am-
Using MEM to develop trading applications requires some under-
plitude is the power of a cycle and is independent of frequency and phase.
standing of how MEM works. The MEM algorithm was not originally
All three of these features make up a simple cycle. Let™s now use them
designed for financial data, so the first thing that must be done to make
to plot a simple sine wave in Omega TradeStation, using the following
MEM work on financial data is de&end it. There are many ways to de-
formula:
trend data. We used the difference between two Butterworth filters, one
with a period of 6 and the other with a period of 20. (A Butterworth fil-
Value1 = (Sine ((360 x Current Bar)/Period)*Amplitude) + Offset;
ter is a fancy type of moving average.) Once the data has been detrended
and normalized, the MEM algorithm can be applied. The MEM algo-
rithm will develop a polynomial equation based on a series of linear pre- Using a period of 30, an amplitude of 20, and an offset of 600, these
dictors. MEM can be used to forecast future values by recursively using parameters produce the curve shown in Figure 7.1, which looks a little
the identified prediction coefficients. Because we need to preprocess like the S&P500 and shows how phase interacts with a simple sine wave.
our data, we are really predicting our d&ended values and rxX the real Note in Figure 7.1 that, by using the TradeCycles phase indicator, tops
price. MEM also gives us the power of the spectra at each frequency. occur at 180 degrees and bottoms occur at 0 degrees.
Using this information, we can develop cycle-based forecasts and use A simple sine wave is not very interesting, but by adding and sub-
MEM for trading. MEM requires us to select (1) how much data are used tracting harmonics we can produce a pattern that looks a little like an
in developing our polynomial and (2) the number of coefficients used. Elliott Wave. Our formula is as follows:
The amount of data will be referred to as window size, and the number
of coefficients, as poles. These numbers are very important. The larger Elliott Wave = Sine(Period x 360) x Amplitude
the window size, the better the sharpness of the spectra, but the spectra -.5 x Sine@ x Period x 360) x Amplitude
+ h x Sine(3 x Period x 360) x Amplitude
will also then contain false peaks at different frequencies because of
noise in the data. The number of poles also affects the sharpness of the
This simple curve resembles as Elliott Wave pattern using a period of
spectra. The spectra are less defined and smoother when fewer poles are
30 and an amplitude of 20 (see Figure 7.2).
used. TradeCycles allows adjustment of both of these parameters, as well
The figure is an example of how chart patterns are actually made up
as others, because different applications of MEM require different op-
of combinations of cycles. Let™s use our sine wave to test the relationship
timal parameters.
Cycle-Based Trading 107
Statistically Based Market Prediction
106


between cycles and a simple moving average. We can start with a simple
half-cycle moving average. The lag in a half-cycle moving average is 90
degrees. Figure 7.3 shows a sine wave curve, a half-period moving aver-
age, and a full-period moving average. The full-period moving average is
always zero because it has as many values below zero as above zero.
If we buy when the half-period moving average crosses below the full-
period moving average, and sell when it crosses above it, we have the per-
fect system. We will buy every bottom and sell every top. These rules are
the opposite of the classic moving-average system.
Let™s now build a simulated Elliott Wave with a period of 30 and an
amplitude of 20. If we trade the classic moving-average crossover system,
we see that a 2-period and a H-period moving average produce the best
results (about 16.90 points per cycle). This system bought just as our fake
wave three took out our fake wave one to the upside, and it sold about
one-third of the way down on the short side. On the other hand, if we use
the reverse rules and a 15period and 30-period moving average, we then
make over 53.00 points per cycle. This system would buy at the bottom
Fet
91
NW DES
SW M

A 30.day cycle versus its phase angle.
FIGURE 7.1




No”
sip oa FIGURE. 7.3 The interaction between a 30-day period sine wave and
both a full-period and half-period moving average.
An example of a fake Elliott Wave composed of sine waves.
FIGURE 7.2
108 Statisticallv Based Market Prediction Cycle-Based Trading 109


and sell about one-third before the top, or about at the point where wave discussed this mode in Chapter 4.) When using MEM for cycle-based
five begins. These results show that, in a pure cycle-based simulated mar- trading applications, the higher the signal-to-noise ratio, the more reli-
ket, the classic moving-average system would lose money. In real life, able the trading results. If we are using MEM for trend-based trading,
moving-average systems only make money when a market moves into a then the signal-to-noise ratio relationship is less clear. This is currently
trend mode or because of shifting in cycles that causes changes in the one of my main areas of research.
phase of the moving averages to price. We cannot predict these shifts in
phase. Unless a market trends, a system based on moving-average opti-
mization cannot be expected to make money into the future. USING CYCLES TO DETECT WHEN A MARKET IS TRENDING
Let™s now look at another classic indicator, the RSI, which expresses
the relative strength of the momentum in a given market. Once again, we Let™s start our discussion of cycle-based trading by investigating how to
use our simulated Elliott Wave, based on a 30-day dominant cycle. We tell when a market is in a trend. We begin by studying the dominant cycle
have found that combining a 16-day RSI with the classic 30 and 70 lev- and signal-to-noise ratio for the Yen during the period from 3/20/96 to
els produces good results, showing that RSI is really a cycle-based indi- 7126196. Figure 1.4 shows how the dominant cycle increases at the start
cator. Using a simulated market over 5 complete cycles, we produced over of a major trend.
59.00 points per cycle and the signals produced bought 1 day after the We will start by studying the spectral intensity of a market over dif-
bottom sold and 3 days after the top. ferent time frames, beginning with an uptrend followed by a major top.
Another classic system that actually works well in cycle mode and will Figure 7.5 shows the spectra for the D-Mark on October 23, 1995, the
continue to work in trend mode is a consecutive close system. Channel
breakout also would work, but you would want to use a 2-day high or low
to maximize the profit based on our simulated Elliott Wave. If you use a
l-day high, your trades would be whipsawed in waves two and four, as
they would be in a real market, because the channel length is too short.


CYCLE-BASED TRADING IN THE REAL WORLD

Let™s now talk about using the MEM algorithm to trade cycles in the real
world. The MEM algorithm requires tuning of many technical parame-
ters. MESA 1996 adjusts these parameters for you. TradeCycles adjusts
many of these parameters too, but it also lets you set the window and
the number of poles. Real financial data are noisy, so it is important to
know the power of the dominant cycle. We call the power, relative to the
power of the rest of the spectra, the signal-to-noise ratio. The higher the
signal-to-noise ratio, the more reliable the cycle. In TradeCycles, we cal-
culate this ratio by determining the scaled amplitude of the dominant Ji
J&l
dr w
cycle and dividing it by the average strength of all frequencies. The level
of the signal-to-noise ratio tells us a lot about the markets. Many times, FIGURE 7.4 The dominant cycle, signal-to-noise ratio, and price for the
Yen, for April to July 1996.
the signal-to-noise ratio is lower when a market is in breakout mode. (We
layjeur aql Kes us3 am ˜aIaLa-3leq B ueql JaZuol103 ai?eela.m %U!AOO˜U aIDK3
-I˜IJ a* a.toqe sKels I! suo˜lgmo˜ iaym Ieal u! 31 ˜alaLa e 30 3Ieq 103 a%
˜1™˜ a[qeJ U! paanpoJda1 s! %npoa aqJ .eyal!m ua@ e slaaru -Ja,te %˜AOUI poFJad-11n3 e Mo[aq JO a˜oqe Km I[? auna aqi ˜ar\e-M auls
qa!qm Jeq 1? Mo[aq 10 akoqe lop e smd amMoqs v ˜ammoqs pearl a[dw!s laa3rad e 103 ˜Ieql pony noL˜˜akeern auis aIdur!s e 01 saielar po!Iad a%elaAe
s,uogeisapeIL pm sala,Qape˜˜
slql3o apoa aql pU!3 aa ˜a%nih$se˜ -SU!AOUJ e .4t0q JO uo!ssnas!p aqi Zqe3a˜ .puaJi 01 paiws seq iaveu
%u˜sfl ˜spuawwiop 103 sJeq alah mt˜˜yop aql30 maxad 0˜ ]se[ aql3o e uaq,v, p&y ,eql sroleatpul do[ahap 01 idwaw s,ial ˜%urpuarl s! iax
q8g Isa&q aql JOU spuaJ)dn ˜03 sreq a[30 meu!luop aql30 maxad 0s -1eu.x e uaqm sa%ueqa eJ,aads aql ˜oq MOU %˜MOU˜ ˜layJew %u!puari e
lse1 aql30 hoI isaM aql ueql aJocu pua4 aql w!eZe aaerlaJ 10˜ plnoqs 11 u! UOWUIO˜ ale spo!Jad JaE?uoI 01 ahow e pue eaaads aqi 30 %upIap!M
˜puaxi 01 weis laylam e uaqM ˜apea 01 qsnoua @a puau e Eh@luap! (˜9˜ arn%g aas) ˜pasealau! a$3 lueuyop aq puno.re eJi
Jo3 [tarn yJom sapu asaqL .aro3aq sKep 2 aye.4 aqi se u@s atues aql seq -aads aql30 ease aql30 q]p!h aqi pue ˜SZ™EZ 01 pasealau! apK3 meu!uIop
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aauala33Fp hp-2 e (z) pue ˜a1aLa-˜alJenb e ueql alom 103 a4eIaAe %!AOUI -UMOP Jorem aql30 weis aql Synp yIem-a aqi ie yool Mou s,la7
alaLa-lln3 e Mo[aq Jo a.hoqe sAels iayxeur e (1) uaqfi s1n330 puall v .layJeur a8uer &npeIl e u! UOWUIO˜ SF yead MODU v .sKep SZ™LI IV yead
˜JoieaFpu! ue qans plyq MOU s,ia˜ .Jolexpu! puan ˜002 e a*eq MOLIBU e aas UK a,tt ˜ewads aql II? %!yoo˜ .sKep SZ™LI SBM a13K3 lueu!
p[noM afi ˜a13h31eq e ueqi a.rom 103 akane %goom aql tio˜aq IO a,ioqe -crop manna aql ˜m!od s!ql IV ˜TJem-a aql IIF do1 maaal ISOU aql3o awp
u!euIaI II+ saa!ld IaqlaqM aledyyue pIno:, aM 31 .spuaJl %q saqama
Kpo pue aleI KJaA spuali Kuem olu˜ sla2 idaauos a[dur!s s!qJ %!puaIl SF
112 Statistically Based Market Prediction


TABLE 7.1 CYCLE BASED TREND INDICATOR.
Van: DCycle(O),Osc(O)TrFlag(O);
DCycle=RSMemCyclel .Z(C.6,50,30,12,0):
Osc=Close-Average(Close,DCycles);
TrFlag==O;
If MRO(sign˜Osc˜osign˜Osc˜l]˜,.25*valuel,l˜=-1 and sign(Osc-Osc[21)=
sign(Osc[21-Osc[41) then TrFlag=l;
IfOsc<O and High=Highest(High,.j*Dcycle) then TrFlag=O;
If 0˜00 and Low=Lowest(Low,.S*Cycle) then TrFlag=O;
If TrFla,q.=l then PloTl(High,“CycleTrend”);



over the period from 12/I/94 to 5/l/95. Note in Figure 7.7 that the mar- I
ket has a major uptrend that is detected near the bottom in December No” Dee 95 FCb MB,
Od m
1994, as shown by the dots at the top of the bars.
FIGURE 7.7 How the dominant cycle-based trend indicator shows the
How can the phase between the price data and the current dominant
major uptrend in the D-Mark that started in December 1994.
cycle be used to detect when a market is trending or is in cycle mode? If
we assume that we have a perfect 30-day cycle, the phase should change
360/30, or 12 degrees per day. The more this rate of change of phase dif-
fers from the ideal, the less the market is acting as if it is in cycle mode.
Using this theory, let™s develop a cycle and trend mode indicator based on
phase. We will compare the ideal rate of change with the actual rate of
change calculated using our price data. When the rate of change of the
phase is the same when using real data as when using theoretical data the
market is in a perfect cycle mode. When the rate of change is less than 1
by some threshold, then the market is trending. If it is greater than 1 by
some threshold, the market is consolidating. When we are within +-our
threshold of 1, then the market is in cycle mode. Because real data are
noisy, we smooth this ratio by using a quarter-cycle moving average. Let™s
take a look at how this indicator worked for the D-Mark during the pe-
riod from 111196 to 711196 (see Figure 7.8).
During this period, the D-Mark started to trend several times. We can
tell it was trending because the RSPhaseMode indicator described above
was significantly below 1, When this indicator stays at about 1, the mar- 96 Fit Msr MbY &
Ab
ket is moving in cycle mode. If it is significantly above 1, the market is
FIGURE 7.8
consolidating. When we use this indicator, we normally use below 0.67 The phase mode indicator points to a trending market
for trending and above 1.33 for the consolidation mode. If this indicator whem it is much below 1, a cycle market when it is near 1, and a
consolidation mode market when it is much above 1.
is between these levels, we can say the market is in cycle mode.

113
Statistically Based Market Prediction
114 Cvcle-Based Tradine 115


USING PREDICTIONS FROM MEM FOR TRADING
ADAPTIVE CHANNEL BREAKOUT

Predictions from MEM can be used to develop trading strategies. These
Let™s now look at another method for using cycle analysis to confirm not
only whether the market is trending but also its direction. This method, predictions are good for predicting turning points but not for predicting
magnitude. The MEM prediction works only when the market is in a
called adaptive channel breakout, defines an uptrend as occurring when
cycle mode. Let™s now look at an example of how to use the prediction
the market makes the highest high of the past dominant cycle bars. The
trend is then up, and you should buy on a stop. When the market makes from MEM. We will learn later in this book, when we develop systems
based on any autoregressive method or even on neural networks, that it is
the lowest low of the past dominant cycle bars, you should sell on a stop.
This definition of both trend and direction is so good that it can be traded easier to develop models on weekly data than on daily data. Once you
as a stand-alone system. My research in the area of developing trading can get a prediction method to work well on weekly data, you can then
move to daily data and then intraday data. Our basic model is shown in
systems using MEM has shown that the size of the window used to cal-
culate MEM and the number of poles used in the calculations have an Table 7.3.
We tested this model on weekly D-Mark data from l/1/80 to 6/22/96
effect on how MEM performs for different applications. We have opti-
and deducted $50.00 for slippage and commissions. Based on our test-
mized the number of poles using a fixed window size of 30 and have
found that when using MEM in a trend-following application, fewer poles ing, we decided that, in normal conditions, our MEM prediction should
look ahead only 4 bars. Our optimal window size was 24 bars, and we
produce more reliable performance, because the output from both the
used 12 poles for both the MEM cycle calculation and the predictions.
dominant cycle and the MEM predictions is smoother. For example, using
Table 7.4 shows how our simple model, using these parameters, per-
a window size of 30, we found that having 6 poles produces the best
formed over our analysis period.
results on daily data for currencies for our adaptive channel breakout sys-
The results show that this simple model is predictive. We can improve
tem. The results for the D-Mark, Yen, and Swiss Franc for the period
the results if we trade the model only when the market is in acycle mode.
from l/1/80 to 6/22/96 are shown in Table 7.2. (A deduction of $50.00
For example, from May 22, 1992, to September 18, 1992, the system
was made for slippage and commissions.)
traded six times and lost about $2,000.00 overall. The D-Mark was in a
These results across the three most popular currencies show the power
major uptrend, rising almost 8.00 full points in this short period of time.
of this method. It can be used to trade a basket of commodities by opti-
During this type of period, the method will perform badly. This is a good
mizing window size and poles over the complete basket and selecting the
most robust pair of parameters. We will learn later that this concept can example for your own use of MEM to develop trading systems and filters
for when a market is in a cycle mode.
be made even more adaptive by optimizing the window size and poles in
The predictions from MEM can also be used in intermarket analysis.
line, based on the current performance of each set of parameters over re-
We can use the classic intermarket relationships discussed in Chapter 1,
cent history. This concept is called optimal adaptive channel breakout.
and then filter our trades by requiring that the prediction from MEM
RESULTS OF ADAPTIVE CHANNEL BREAKOUT.
TABLE 7.2
D-Mark TABLE 7.3 SYSTEM USING MEM PREDICTIONS.
$131,325.00
$99.237.50 $161,937.50
Net profit If MEMPred>MEMPred[21 and MEMPred[4l>MEMPred[21 then buy at open;
105
94 95
Trades
If BarsSinceEntry>Dominate Cycle*.25 then exitlong at open;
44
47
49
Win%
If MEMPred<MEMPredlZl and MEMPred[4l<MEMPredI21 then sell open;
$1,250.71
51,704&l
$1,055.72
Average trade
-$12,412.00 If BarsSinceEntry>Dominate Cycle*.25 then exitshort open:
Drawdown -$11,312.50 -$&775.00
Statistically Based Market Prediction
116 Cycle-Based Trading 117


TABLE 7.4 MEM PREDICTION RESULTS fewer trades than when using divergence without the MEM prediction
WEEKLY D-MARK. filter. We did not optimize the other parameters. They were selected
based on other research that recommended a window size of 30 and 6
$75,562.50
Net profit
255 poles for many trading applications. We found that using 20 for the TrLen
Trades
49
Win% and 30 for InterLen produced reliable and stable results. Our results over
$296.32
Average trade our test period are shown in Table 7.6. (A deduction of $50.00 was made
-$10,787.50
Drawdown for slippage and commissions.)
Table 7.6 shows that this system produced better overall results than
even the best intermarket divergence system for the S&P500. The concept
of confirming intermarket relationships using MEM predictions does
must confirm the system. Let™s look at an example of this concept. Sup-
have value and is an important area of future research.
pose we are trading the S&P500 using T-Bonds. If bonds are in an up-
One fact is little discussed: Predictions from MEM change when dif-
trend and the S&P500 is in a downtrend, we would then buy the S&P500.
ferent parameters are used. One method for improving performance when
If T-Bonds are in a downtrend and the S&P500 is in an uptrend, we would
using MEM would be to combine different predictions made with dif-
then sell the S&P500. We showed in Chapter 1 that this type of model
ferent parameters. A consensus or even an average method is a possible
works very well. We could filter this model by requiring the MEM pre-
choice. A lot of computer power would be required, but the performance
diction to be positive for both markets to buy the S&P500, and negative
would improve.
for both markets to sell the S&P500. This type of filter can help solve
This chapter has shown how cycles can be used to create a trading sys-
the problem of an intermarket change of direction just as the signal is
tem, or can become part of a trading system, or can even be inputted into
generated. Table 7.5 shows this concept, coded in TradeStation™s Easy-
a neural network or genetic algorithm. These issues are addressed further
Language and using TradeCycles. Note that the S&P500 is in Data1 and
in Chapter 18, but I will give you one example now. If you look at the rate
T-Bond futures are in Data2.
of change of a dominant cycle, it will give you an idea of whether the pre-
We tested this system from April 21, 1982 to June 28, 1996. We opti-
dicted turning points would be too early, or late, or on time. When the
mized the moving-average lengths for both TrLen and InterLen, using the
range of 10 to 30 in steps of 2. We used a window size of 30, 6 poles, and
a lookahead of 6. On average, we realized a higher winning percentage and
TABLE 7.6 RESULTS OF INTERMARKET CYCLE
BASED SYSTEM S&P500 USING T-BONDS
AS THE INTERMARKET.
TABLE 7.5 INTERMARKET BASED CYCLE ANALYSIS SYSTEM.
Net profit $326,775.00
Inputs: LK1(6),LK2(6),TrLen(20),lnterLen(30),Win(30˜,Poles˜6˜; Profit long $269.100.00
Vars: TrOsc(O),InterOsc˜0),TrPred˜0˜,lnterPred˜0˜; Profit short $57,675.00
Trades
TrPred=RSMemPred(Close of Datal,Win,Poles,LKl); 54
Win% 80
InterPred=RSMemPred(Close oi DataZ,Win,Poles,LK2);
Win% long 93
TrOsc=Close of Datal-Average(Close of Data1 ,TrLen);
Win% short 67
InterOsc=Close of Data-Average(Close of Data2,lnterLen); Average trade $6,051.39
If InterPred>O and TrPred>O and TrOsc<O and InterOsoO then buy at open; Drawdown -$27,600.00
Profit factor
If InterRed<O and TrPred<O and TrOsoO and InterOsc<O then sell at open; 8.02
Statisticallv Based Market Prediction
118


dominant cycles are getting longer, the turning points you predict would
be too early. When the cycles are getting shorter, the turning points would
be too late.
8
This chapter is a starting point for future work on using spectral analy-
sis to develop trading systems. This area of research should be one of the
most important technologies in developing the next generation of trading
Combining Statistics and
systems.

lntermarket Analysis


In Chapter 1, we discussed many different intermarket relationships that
are valuable for developing trading systems. If you actually have pro-
grammed some of the examples in Chapter 1, you have learned that these
systems work very well during some periods, but do have long periods of
drawdown.
Our research over the past few years has shown that analysis of
intermarket relationships, based on current correlations between the in-
termarket and the market you are trading, is a very valuable tool in de-
veloping trading strategies.


USING CORRELATION TO FILTER INTERMARKET PATTERNS

Let™s now show how Pearson™s correlation can be used to improve clas-
sic intermarket relationships. In Chapter 1, we showed that you can trade
crude oil using the Dollar index (see Table 1.3). You go long when the
dollar is below its 40.day moving average, and you go short when it is
above that average. This relationship has been steady over the years, but
it did have problems during 1991 and 1992. During those years, this
model lost $3,920.00 and recorded its maximum drawdown.
Byusing Pearson™s correlation as a filter for this simple intermarket re-
lationship, we are able to improve our model™s performance. We will still

119
Combining Statistics and Intermarket Analysis 121
120 Statistically Based Market Prediction


TABLE 8.2 RESULTS USING UTY TO TRADE
enter and exit our trades using a 4C-day moving average of the Dollar, but
T-BOND WITH CORRELATION AS A FILTER.
we now also require a 40-day correlation between the Dollar and crude oil
to be less than -.5. Net profit $108,037.50
Using this new filter, we more than doubled our average trade and cut Trades 68
Win%
our drawdown in half. Our first trade was on March 17, 1986, and ourre-
Average trade ::.5@3.79
salts, after deducting $50.00 for slippage and commissions, are as shoivn
Maximum drawdown -$6,593.75
in Table 8.1.
Profit factor 5.2R
This model did not make as much as the original one, but the average
trade, drawdown, and profit factor all improved. In addition, the model,
using the correlation filter, made $1,800.00 during 1991 and 1992, when
the original model suffered its largest drawdown. Use of the filter re- The use of correlation as a filter improved almost every part of this
system™s performance. Net profit increased and drawdown dropped by
duced the number of trades from 167 to 55, and increased the profit fac-
about 30 percent. We also won 75 percent of our trades.
tor from 2.07 to 3.07.
Let™s now apply a simple intermarket pattern filter to the use of
Let™s look at another example of using correlation analysis to develop
day-of-week analysis. We will buy the S&P500 on Monday™s open when
intermarket trading models. We will use the relationship between
T-Bonds are above their 26.day moving average, and we will exit this po-
T-Bonds and UTY, which was discussed in Chapter 1. Our model was
sition on the close. This pattern has performed well from 1982 to date.
based on prices relative to a moving average. We used an g-day period
Table 8.3 shows the results from 4/21/82 to 7126196, with $50.00 deducted
for T-Bonds and a 24-day period for UTY. We took trades when T-
for slippage and commissions.
Bonds and UTY diverged. If UTY was rising and T-Bonds were falling,
The results of our buy-on-Monday pattern are good, but we can im-
we bought T-Bonds. If UTY was falling and T-Bonds were rising, we
prove them by using correlation. We first use a simple correlation be-
sold T-Bonds. For the period from 6/l/87 to 6/18/96, this model pro-
tween T-Bonds and the S&P500. Because we base this pattern on the
duced a little over $98,000.00, with 64 percent winning trades and about
relationship between T-Bonds and the S&P500, we want to filter out
a -$9,500.00 drawdown. Can filtering our trades, using the correlation
trades when the link between the S&P500 and T-Bonds is weaker than
between UTY and T-Bonds, help even a system that performed this
well? If we filter our signals, we require a 12.day correlation between normal. We therefore take trades only when the 20.day correlation be-
tween T-Bonds and the S&P500 is greater than .40.,This filter improves
UTY and T-Bond to be greater than .50. Our results for the period from
the performance of our original pattern. Table 8.4 shows our results
611187 to 7126196 are shown in Table 8.2.



TABLE 8.3 RESULTS OF BUY MONDAY
T A B L E 8 . 1 RESULTS OF TRULONC CRUDE OIL
WHEN T-BONDS ARE IN AN UPTREND.
USING THE DOLLAR INDEX AND CORRELATION.
Net profit $89,100.00
$39,499.00
Net profit
Trades 417
$34,319.00
Profit long
Average trade
$5,180.00 $213.67
Profit short
Win% 55
49
Win%
Profit factor 1.57
$718.16
Average trade
Drawdown -818.975.00
-$5,930.00
Drawdown
122 Statisticah Based Market Prediction Combining Statistics and Intermarket Analysis 123


TABLE 8.4 RESULTS OF BUY MONDAY WHEN PREDICTIVE CORRELATION
T-BONDS ARE IN AN UPTREND AND S&P500
AND T-BONDS ARE STRONGLY LINKED. A correlation between two markets does not always mean that the current
movement in one market can be used to predict the movement in the other.
Net profit $88,200.00
To address this issue, I have developed a concept calledpredicrive corre-
Trades 268
him. The idea behind predictive correlation requires taking a correla-
Average trade $329.10
Win% 58 tion between an indicator N periods ago and a change in a given market
Profit factor 2.01
over the last N periods. For example, on daily data, we can take a corre-
Drawdown -$7,775.00
lation between T-Bonds[5]-T-Bonds[lO] and the S&PSOO-S&P500[5].
This correlation will tell us how predictive a simple momentum of
T-Bonds has been over the length of the correlation. The predictive cor-
for the period from 4/21/82 to 7/26/96, allowing $50.00 for slippage and relation curve is much different from the curve generated by standard
commission. correlation, but it does seem that they both trend in the same direction.
Filtering the trades by using correlation not only improves our aver- The power of predictive correlation is that we can correlate an indicator
age trade by 54 percent, but also improves our percentage of wins, draw- or intermarket market relationship to future price movements in the mar-
down, profit factor, and win/loss ratio. We were able to filter out about ket we are trying to trade. This allows us to use relationships and indica-
160 trades that averaged about $6.00 each. We could have used higher tors in rules, and to trade these rules only when these indicators are
thresholds for our trigger, but that tactic would have filtered out too many currently predictive. Let™s txlw add predictive correlation to our modified
trades. For example, using an g-day correlation and a .90 trigger yielded S&P500 pattern.
over $500.00 per trade but produced only 36 trades in 13 years. We use Close[ l] of T-Bonds-Open[ l] of T-Bonds as our independent
Let™s start again with our buy-on-Monday strategy when T-Bonds are variable, and Close-Open of the S&P500 as our dependent variable. We
above their 26-day moving-average pattern and we have an additional fil- go long on Mondays only when a 35.day predictive correlation is above
ter. We now buy at the open on a Monday only when T-Bonds are above 0. The amazing results, from 4/21/82 to 7/26/96, are shown in Table 8.6.
their 26-day moving average and when T-Bonds closed higher than they This system produced over $600.00 per trade, after deducting $50.00
opened on Friday. for slippage and commissions. We won 66 percent of our trades and had
The new requirement, that T-Bonds had to close higher than they a profit factor of 3.75. These numbers are much better than any of the
opened on Friday, improved the performance of our original pattern. The variations that did not use predictive correlation, and they should prove
results are shown in Table 8.5. the power of predictive correlation.


TABLE 8.5 RESULTS OF MONDAY RISK TABLE 8.6 RESULTS OF ADDING
WITH T-BONDS HEAVY AND UP FRIDAY. PREDICTIVE CORRELATION.
Net profit $75,825.00 Net profit $55,050.00
Trades 244 Trades 88
Average trade $310.76 Average trade $625.57
57 Win% 66
Win%
1.86 Profit factor 3.75
Profit factor
Drawdown -$13,800.00 Drawdown -$4,400.00
124 Statistically Based Market Prediction Combining Statistics and Intermarket Analysis 125


USING THE CR6 AND PREDICTIVE CORRELATION TABLE 8.7 COLDjCRB RATIO SYSTEM.
TO PREDICT COLD
Vars: IntRatio˜O˜,lntOsc˜O˜,Correl˜O˜;
Vars: IndfObDepiO);
In Chapter 1, we discussed many of the classic methods for expressing in-
lntRatio=Close of data2lClose;
termarket relationships. One of the most powerful methods is a ratio be-
Ind=lntRatiol51;
tween the intermarket and the commodity being traded. I will ww show
Dep=Close-CLoseiS];
you how to combine the ratio between the CRB and gold with predictive
Correl=RACorrel(Ind,Dep,24);
correlation to develop a very profitable and reliable system for trading
IntOsc=Average(lntRatio,l2)-Average(lntRatio,30);
gold.
If IntOso and Correl>.6 then buy at XAverageKlose,BO) Limit;
The Commodity Research Bureau index (the CRB) is a basket of 21
If IntOsc<O and Correl>.h then sell at XAverage(Close,80) Limit;
commodities.* This index has been traded as a futures contract since mid-
RACorrel is a user iunction developed by Ruggiero Associates. It calculates the standard
1986. It has had an inverse correlation to T-Bond prices and it has been
Pearson™s correlation found in any statistics textbook.
positively correlated to gold during its history.
On the basis of this relationship, I decided to use the ratio between the
CRB and gold to develop a trading system for gold. When this ratio is
The model made over $48,000.00 during this period, and the system
moving up, the CRB is outperforming gold, and gold should catch up,
was profitable on both the long and short sides. Another important point:
Another fact about gold was revealed in my research. Often, when the
The entry method (buy on a limit set at the SO-day exponential moving
CRB starts moving, gold will first move in the opposite direction and
average of the close) increased the average trade by over $500.00 when
test support, before moving in the direction of the CRB.
compared to the method of entering at the next open when the signal first
On the basis of my understanding of the gold market, I am proposing
occurs.
a system that (1) uses a moving-average crossover of the ratio between the
The system does have some problems, however. For example, the aver-
CRB and gold to generate its signals and (2) enters the market on a limit
age winning trade lasted 45 days but the average losing trade lasted 144
order set at the level of an N-day exponential moving average. This con-
days. We can help solve this problem by developing better exits for the
cept was tested on backadjusted contracts over the period from 1 l/18/86
model. Even with this problem, the model is fundamentally sound and
to 7/26/96. The system is based on a sound premise. If inflation increases,
could be the core for a system for trading gold futures or gold mutual
so will the price of gold. Still, it performed badly and made only
funds.
$4,000.00 in profit over the test period. The reason the system did so
badly is that it had large drawdown during periods when the CRB and
gold decoupled. We can filter these periods out by using correlation
TABLE 8.8 GOLD/CRB RATIO
analysis. Let™s now add the predictive correlation between the ratio of
SYSTEM RESULTS.
the CRB/gold 5 days ago and the past 5&y change in gold. This simple
$48,794.70
Net profit
gold model, coded in TradeStation™s EasyLanguage with parameters se-
35
Trades
lected based on my research, is shown in Table 8.7.
27
Wins
This simple model has performed very well in the gold market over a
LOS%S
the past decade. We tested the model using continuous backadjusted con- Win% 77
Average trade $1,394.13
tracts for the period from 1 l/18/86 to 7/26/96, and deducted $50.00 for
Drawdown -$11,250.00
slippage and commissions. The results are shown in Table 8.8.
Win/loss ratio 1.56
Profit factor 5.26
* The CRB was reformulated in December 1995.
126 Statistically Based Market Prediction 127
Combining Statistics and Intermarket Analysis


INTERMARKET ANALYSIS AND PREDICTING THE This link between correlation and trend also occurs during major
EXISTENCE OF A TREND downtrends. An example, the last important stock market correction, in
February 1994, is shown in Figure 8.2.
Intermarket analysis is another powerful tool for predicting when a mar- One of the few downtrends in the S&P500 occurred during the period
ket will trend. My research has shown that many markets will trend when from 1993 to 1995. During this short time, the S&P500 dropped almost
well-known intermarket linkages are strong-for example, the link be- 40.00 points during just a 45-day trading period. When trading the S&P500,
tween the S&P500 and T-Bonds. I have found that the S&P500 trends correlation analysis can tell you when the trend is really your friend.
when the 50-day correlation between the S&P500 and T-Bonds is high. This relationship between trend and intermarket linkage does not exist
Let™s look at some examples of how intermarket linkage relates to a solely for the S&P500 using T-Bonds. Gold almost never trends without
market™s trending. Figure 8.1 shows that the 50-day correlation between a strong link to the CRB index. Using a 50.day correlation between gold
the S&P500 and T-Bonds started to fall in mid-July of 1995, at just about and the CRB, I have found that almost every major up or down move in
the time when the S&P500 moved into a trading range. The correlation gold started when the correlation between the CRB and gold was above
bottomed in early September and then rose rapidly. During this rapid rise .40 and rising, or was stable above .6. The only major trend in gold that
and a stabilization at higher levels, the S&P500 rose 57.55 points in about this relationship missed was the rally in gold that started in June 1992.
70 trading days without recording more than two consecutive days on Let™s review an example of how the CRB can be used to predict trends
which the market closed lower than it opened, in gold. One of the last explosive rallies in the gold market was in early




.T,..
RACurd -0.03-x o.mm osmo




FIGURE 8.1 FIGURE 8.2
The correlation between the S&P500 and T-Bonds can Another example of how the correlation between the
predict when a market will trend. In 1995, as correlation rose, the S&P500 and T-Bonds predicts trends. The last major stock market
S&P500 exploded and rose 57.55 points in only about 70trending days correlation occurred in February 1994.
Statistically Based Market Prediction
128 Combining Statistics and Intermarket Analysis 129


November of 1989. The correlation between the CRB and gold started to
increase in mid-October and rose steadily until November 27,1989. Dur-
ing this time, the gold market rallied over $50.00 per ounce. The corre-
lation broke above SO on November 7, 1989, and did not drop below SO
until December 21, 1989. During this time, gold rallied $26.70 per ounce
in only 31 trading days. (See Figure 8.3.)
On the basis of my research, the gold market will almost never have a
major move until the SO-day correlation between gold and the CRB rises
above SO. This means that the great bull market that many believe will
happen soon in gold will most likely not start until the 50-day correla-
tion between the CRB and gold rises above .50 while the CRB is in a
major uptrend. Without the intermarket link between the CRB and gold,
most breakouts to the upside will fail within several trading days.
In early 1996, gold had a breakout to $420.00 per ounce, and many ex-
perts predicted that gold would go to $450.00 per ounce within a few
.“,I”
months. Using correlation analysis, we could see that this breakout would s Feb MB, m

FIGURE 8.4 During the upside breakout in gold in early 1996, it
decoupled from the CR6 and then failed.



fail and that the correlation from $420.00 down to $400.00 would be a
very strong trending move. Figure 8.4 shows that the breakout during
early 1996 occurred as the gold decoupled from the CRB. The move was
not a long-term move that would produce a rally of $50.00 or more. When
gold collapsed in early February 1996, it had a 50.day correlation with the
CRB of greater than .50. Once gold started its collapse, it dropped 24.00
points in one month. After that, the correlation between gold and the
CRB dropped, and gold once again moved into a trading range.
The correlation between intermarkets is a valuable tool for developing
trading systems and can even be a tool for discretionary traders. We will
learn later that it is even useful as an input for a neural network. Corre-
lation analysis and predictive correlation, combined with intermarket
analysis, can be used to develop trading systems as well as to improve ex-
isting ones. Later in the book, we will use this type of analysis to develop
systems based on advanced technologies such as neural networks.
FIGURE 8.3 We can use the correlation between gold and the CRB
to predict when gold will trend. The correlation broke above .50 on
January 7, 1989, and gold rallied $26.70 per ounce in only 3˜1 days.
Using Statistical Analysis to Develop Intelligent Exits 131



TABLE 9.1 ENTRY RULE TESTS.
Exit after holding a position for N bars.
1.
2. Exit on an N bar low for longs or an N bar high for shorts.
9 3. Exit after N consecutive bars in which the trades moves against you.
4. Exit at a target profit oi N.

Using Statistical Analysis
to Develop Intelligent Exits there are many reasons to exit a trade. The exit development process re-
quires mixing money management and technical information about a sys-
tem for a given market. For example, you can exit a losing trade using a
$500.00 stop, or you can exit a long position when the market reaches a
5-bar low. You should exit a trade when the assumptions that caused you
to enter the trade are proven wrong. Let™s suppose you entered a long
trade because the price bounced off a support line. You should then exit
the trade if the price closes below that support level. You might also exit
a trade if you have IX) opinion about the future direction of the market you
When most traders develop mechanical trading systems, they spend 90
are currently trading.
percent of their time developing the entry signals. The exit signals are
Now that you have a better understanding of the logic used in devel-
usually less complex and are tested only in combination with the entries.
oping exits, I can show you how to develop your own intelligent exits.
Unfortunately, this process does n+X develop optimal exits for a given sys-
tem and market. This chapter discusses how to develop properly designed
exit signals, using various statistical methods.
DEVELOPING DOLLAR-BASED STOPS

THE DIFFERENCE BETWEEN DEVELOPING One of the mosr frequently used methods for exiting a trade is triggered
when the market moves some given level against investors. These types
ENTRIES AND EXITS
of exits, called “stops,” are used in two ways:
The underlying logic between developing entry and exit signals is differ-
1. If a trade is losing money, exit at a given loss level to protect the
ent. When developing entry signals, we are trying to find a set of condi-
trading capital.
tions that statistically produces a good risk-reward ratio when taking a
position in a given direction. To judge how predictive an entry rule is, I 2. If a trade is winning, use money-level stops. Exit the trade after
use the set of primitive exits shown in Table 9.1. reaching a minimum profit level and retracing a given percentage of
Test your entry rules using different values of N for each of the prim- the maximum profit during that trade. This is called a “trailing
stop.”
itive exits defined above. This will allow you to evaluate how well agiven
entry rule works in predicting future market direction. It is easy to un-
derstand the logic needed to develop and test entry rules. The problem Scatter charts can be used to develop intelligent Dollar-based stops
with developing exitrules is thatthe logic is tx)t as easy to define because for a given system. In Chapter 4, we showed a simple channel breakout

130
Using Statistical Analysis to Develop Intelligent Exits 133
Statistically Based Market Prediction
132


TABLE 9.2 CODE TO GENERATE MAXIMUM ADVERSE
MOVEMENT SPREADSHEET.

Input: Length(l0);
Vars: Direct(O);
Buy Highest(High,Length)+l point Stop;
point Stop;
Sell Lowest(Low,Length)-1
Direct=MarketPosition;
If Current&u=1 then
P,int(file(“d:\book\chap9\dmadver.txt”),”EntryDate”.“,“,“MarketPosition”,“,˜,˜
MaxPositionLoss”,“,“,“PositionProfit”);
If DirectoDirect[l] then begin
Print(file(“d:\book\chap9\dmadver,txt”),EntryDate(l ).“.“,MarketPosition(1),“,“,
MaxPositionLoss(lL”,“,PositionProfit(l));
end;



FIGURE 9.1 Adverse movement versus final trading profit.
system that used a 20-day high or low to set the breakout. The system
was profitable over many different commodities. Let™s look at how this
system performed on the D-Mark and how we can use intelligent exits to
maximum adverse movement of more than -$1,800.00 also made more
improve the system. than $2,000.00 when the trade was closed. During the trade history, 29
The original system on the D-Mark, shown in Table 4.8, made a little
trades finished with a profit of more than $1,500.00. There were 25 trades
over $56,000.00 and had a drawdown of over -$22,000.00 from l/1/80
with a maximum adverse movement of more than -$1,800.00.
to 5/17/96. Let™s develop an intelligent money management stop for this
On the basis of our analysis using the scatter chart, we can set a pro-
system.
tective stop of -$1,800.00. This stop level is based on the fact that only
We start by collecting the maximum adverse movement and final profit
three winning trades had adverse movement of -$1,800.00 or more. We
for each trade. The code (in TradeStation™s EasyLanguage) for collecting
used this -$1,800.00 level and then added $50.00 for commissions. The
this information in an ASCII file is shown in Table 9.2.
protective stop produced the results shown in Table 9.3.
This code first saves the current market position for each bar in the
Drawdown was reduced by over 50 percent. Net profit increased
variable “Direct.” When the market position changes from one bar to the
slightly and the winning percentage decreased by only 1 percent!
next, we have just closed out a position. We then output the following to
an ASCII file: entry date, market position, and maximum position loss,
which we refer to as maximum adverse movement.
TABLE 9.3 SYSTEM RESULTS
USING NEW STOPS.
Net profit
USING SCATTER CHARTS OF˜ADVERSE MOVEMENT TO $72,462.50
Percent profitable 49%
DEVELOP STOPS
Average winner $2,964.42
Average loser -.$1,512.61
Figure 9.1 shows a scatter plot of maxitium˜adverse movement on the X Maximum drawdown -$10,087.50
axis and final trade profit on the Y axis. Only three trades that had a
,._


Statistically Based Market Prediction Using Statistical Analysis to Develop Intelligent Exits
134 135


then check to see whether the position in the market has changed and
You might ask: Why not just use the optimizer in TradeStation to set
whether the trade has lasted Nor more bars. If so, we output the entry
the stops? Analyzing scatter charts will produce better results, for several
date, Position, P/L at bar N, and final profit. We can then chart these re-
reasons. First, we want to improve overall system performance, not just
sults using a scatter chart by plotting profit on bar N on the X axis and
net profit or drawdown. The TradeStation optimizer can optimize on only
final trade profit on the Y axis.
one factor. This can cause a major problem because protective stops al-
Let™s analyze our simple system on the D-Mark for bar 5 of our trade.
ways lower the winning percentage of a system. The optimal stop based
Figure 9.2 shows a scatter chart of adverse movement on day 5 versus
on profit or drawdown could reduce the winning percentage by 30 to 70
final trade profit.
percent. For many trend-following systems, such as our D-Mark example,
Only 3 trades that were losing more than $500.00 during day 5 are
the winning percentage could drop from 40 or 50 percent down to 25 per-
profitable. Only 5 trades closed profitably that were not profitable by day
cent or less. Second, and more important, when stops are developed based
5. Based on this analysis, we can use either a $500.00 stop on day 5 or a
on the distribution of trades, they are more robust and should continue to
breakeven stop on day 5. The results for both possibilities are given in
work into the future. When stops are developed based on optimization,
Table 9.5.
it is possible that a few isolated events could have produced the improve-
Table 9.5 shows that using a breakeven stop on day 5 cuts the draw-
ments. Because these improvements are not based on the distribution of
down by over 50 percent from the original system and cuts the average
trades, they might not be as robust.
losing trade from -$1,899.50 to -$961.72. Because we can use a
We also can analyze current trading profits versus final trading profits
breakeven or a $500.00 stop on day 5, we have greatly reduced the risk
on a bar-by-bar basis, TradeStation™s EasyLanguage code for creating these
statistics in an ASCII file, for any bar in a trade, is shown in Table 9.4.
This code generates the trades and saves the current market position
and the bars since entry. We save the open position profit for bar N. We


TABLE 9.4 CODE TO OUTPUT ADVERSE MOVEMENT OF BAR N.
Input: LengthUO),BarNo(S);
Vars: MarkPos(O),TradeLen(O),OpProf(O):
Buy Highest(High,Length)+l point Stop;
sell Lowest(Low,Length)-1 point Stop;
MarkPos=MarketPosition;
TradeLen=BarsSinceEntry;
if BarsSinceEntryzBarNo then OpProf=OpenPositionProfit:
if CurrentBar=l then
Print(file(“d:\book\chap9\chap9B.txt”),”EntryDate”,“.“.“MarketPOSition”,“˜“,”
CurrentProfIt “,“,“,“PositionProfit”˜;
if MarkPosoMarkPos[ll and TradeLenll˜l>=BarNo then begin
Print(file(“d:\book\chap9\chap9B.txt”),EntryDate(l ),“,“,MarketPosition(l),“,“,OP IMuerse HO”ml.?“, rmg 5

Prof,“,“. PositionProfit(l
FIGURE 9.2 Adverse movement on day 5 versus final trading profit and
end; loss.
Using Statistical Analysis to Develops lntellieent Exits
Statistically Based Market Prediction 137
136


ADAPTIVE STOPS
TABLE 9.5 RESULT BASED ON ADVERSE MOVEMENT
ON DAY 5 OF THE TRADE.
To develop a simple adaptive example, we start with an S&P500 pattern
Breakeven
$500.00 stop
we have discussed in Chapter 1 and Chapter 8. This pattern buys on Mon-
$70,562.50
$72,475.50
Net profit days when T-Bonds futures are above their 26-day moving average. This
38%
45%
Percent profitable pattern made a little over $89,000.00 from April 21, 1982, to July 27,
53,072.92
$3,042.00
Average winner
1996, with a drawdown of about -$19,000.00. We will now develop an
-8961.72
-$1,284.27
Average loser
$551.27 adaptive stop for this pattern. On every bar, we will find and keep track
$647.10
Average trade
-$9,637.50
-$11,312.50
Maximum drawdown of the adverse movement on winning trades only. We will then calculate
both the average and the standard deviation. We will place our stop at
one standard deviation from the mean below the open.
To develop an adaptive intelligent exit, we simulate taking trades with-
out stops and we collect the adverse movement information. If we use
attached to using the $1,800.00 stop we discussedearlier. Using this type
TradeStation to manage the trades once we begin to use the stops, we then
of analysis, we can set an ideal stop or even a trailing stop for the com-
have a different statistical profile. For this reason, we must simulate the
plete life of the average trade. The analysis can be repeated for at least the
trades without stops and then apply the information to rules that actually
number of bars in the average winning trade or average losing trade,
take these trades in TradeStation. We also must make sure that we have
whichever is greater. Our D-Mark trader would run this analysis from
enough information before applying it to our system. In our example, we
bar 1 to bar 60 because an average winning trade lasted 60 days and an
waited and collected 30 winning trades. Until then, we used a 2.point
average losing trade lasted only 19 days.
($l,OOO.OO) stop. The TradeStation code for this system and stop is shown
With the filter, trades that would have been stopped out on previous
in Table 9.6.
bars can be excluded from analysis. This method offers the possibility of
The results for this system for the period from 4/21/82 to 7/26/96, with
developing both protective stops and trailing stops, based on the statis-
$50.00 deducted for slippage and commissions, are shown in Table 9.7.
tics of the system and market we are trading. Using the bar-by-bar analy-
We have improved our net profit and still cut our drawdown in half. We
sis method, it is possible to exit trades using much smaller stops, or even
have also won about the same percentage of our trades (55%), and cut
to exit at a profit trades that would normally have been exited at a loss.
our largest losing trade from -$9,425.00 to only -$1,925.00.
Suppose we have a system in which trades that had an open position profit
This simple example shows the power of developing a system using
of more than $200.00 on bar 20 average a final profit of $l,OOO.OO, and
adaptive intelligent exits. This example made it easy to show how to col-
trades that had an open position profit of less than $200.00 produce an av-
lect the adverse movement information and simulate the trades. You can
erage loss of $200.00 per trade. We could use a profit floor of $200.00 for
apply this type of technology to any system, as long as you can write code
bar 20 and improve our system™s performance. This is only the beginning
to simulate the trades and collect the statistics.
of the uses of bar-by-bar analysis to develop stops for a mechanical trad-
ing system.
This chapter has introduced the concepts of developing statistically
Using a scatter chart to develop intelligent exits is a valuable method
based intelligent exits. This technology can also be used to develop exits
for creating stops, but it does have some drawbacks. Primarily, it is very
based on the potential profit and risk for your trade. Other technologies,
time-consuming and must be rtm for each system and commodity being
such as neural networks and machine induction, which are discussed later
traded. Using TradeStation, let™s try to develop self-adjusting exits based
in this book, can also be used to develop intelligent exit methods.
on this technology.
Using Statistical Analysis to Develop Intelligent Exits
Statistically Based Market Prediction 139
138


TABLE 9.7 RESULTS OF BUY MONDAY
TABLE 9.6 S&P500 MONDAY SYSTEM WITH ADAPTIVE STOPS.
WITH ADAPTIVE STOPS.
Vars: WinNo(O),AveWin(O),StopLev(2);
Net profit $92,950.00
Vars: stdwin(O),Count(O);
55
Win%
{ If we had signal to buy Monday and it was a winning trade store adverse
$224.52
Average trade
movement1 Drawdown -$9,175.00
if (DayOfWeek(Date)=S and Close of DataZ>Average(Close of Data2,26))ll I and 1.62
Profit factor
close>Open then begin AdvWinIWinNol=Open-Low;
WinNo=WinNo+l;
end;
We also learned how to develop a very simple trade simulator and col-
{ Calculate the Average Adverse movement] lect the statistics based on these signals. The next chapter addresses the
if WinNo> then begin development of these simulators in more detail. What we have learned
For Count=0 to WinNo begin here will be used later to develop system examples using advanced tech-
AveWin=AveWin+AdvWinICountl; nologies such as neural networks and genetic algorithms.
end;

AveWin=AveWin/WinNo+l;
( Calculate the Standard Deviation)
for Count=0 to WinNo begin
stdwin=(˜vWin[Count]-AveWin)*(AdvWinLCountl-AveWin)+stdwin;


stdwin=SquareRoot(stdwin/(WinNo+l));
end;

if DayOfWeek(Date)=S and Close of Data2>Average(Close of Data2.26) then
buy at open;
exitlong at close;
( Use Adaptive Exit After 30 Trades and 2 point exit before1
( Using one Standard Deviation from the mean will only stop out 5% of the
trades based on a normal distribution}
if WinNo> then exitlong (“Adaptive”) at NextOpen-AveWin-stdwin stop
else exitlong f”Adaptive2”) at NextOpen-2.00 stop;
Usine Svstem Feedback to lmwove Tradine Svstem Performance 141


HOW TO MEASURE SYSTEM PERFORMANCE
FOR USE AS FEEDBACK


10 We measure the performance of a system based on its trade history. We
can look at the trading results from two different perspectives: (1) on the
basis of only closed-out positions, or (2) on a bar-by-bar basis, by record-
Using System Feedback ing the opening equity of each position. In this book, we will study only
the equity of closed-out positions.
to Improve Trading We also can consider how technical and intermarket analysis can be
combined with equity analysis to improve system performance. When we
System Performance analyze which component of a trading system model generated a signal
and combine this knowledge with trading performance feedback, we are
able to see relationships that are important in modeling the markets but
would not otherwise be observed.


METHODS OF VIEWING TRADING PERFORMANCE
FOR USE AS FEEDBACK

The first method of viewing performance is to build an equity curve,
which is a moving cumulative value of all closed out trades over the eval-
A mechanical trading system is really a predictive model of a given mar-
uation period for a particular system. A modified version of this method
ket. Many predictive models suffer from the effect of noise or an inade-
is to view not all trades, but only closed-out trades, over a period of N
quate understanding of the problem being modeled. In most fields outside
of trading, the errors of the last few forecasts are analyzed in order to days-usualiy, the past 100 days.
Another method is to view a trade history, which can be generated in
improve future forecasts. This concept is calledfeedback. Most predictive
a program like SuperCharts TM. The trade history would have a trade on
models output numerical values. A simple trading system has four dis-
each row and would show the type of signal, entry date and price, exit
crete values: (1) long, (2) exit long, (3) short, and (4) cover.
date and price, trade profit and loss, and current equity. SuperCharts al-
lows exporting this type of history to a spreadsheet.
HOW FEEDBACK CAN HELP MECHANICAL We can also view closed trade equity versus technical or intermarket
TRADING SYSTEMS relationships. For example, a loo-day change in equity can be plotted on
the Y axis, and volatility on the X axis. Figure 10.1 is an example of this
type of chart.
In a mechanical trading system, feedback is valuable in efforts to iden-
Alternatively, current volatility can be plotted versus the next N-day
tify which signals from the system have the highest probability of being
change in equity.
profitable. Feedback also helps in selecting which system to trade when
The interrelationships between the trading systems and the market
multiple systems are trading in the same market. Later in this book, we
price actions are very complex and are well hidden from simple analy-
will discuss how to implement this application using advanced technolo-
sis. When a system is complex and uses many indicators, it becomes
gies such as machine learning. :

140
142 Statisticah Based Market Prediction Using System Feedback to Improve Trading System Performance 143



develop stops for the actual trading system in a walk forward adaptive
manner. The same idea can be used to adapt system parameters and rules
based on a moving window of equity.
Let™s look at an example of how this approach can improve system per-
formance. We will use a 20-bar channel breakout, as in Chapter 4, and
will apply this analysis to the D-Mark.
Channel breakout systems can be helpful using equity feedback be-
cause there are dependencies for trades. For example, if the last trade
on the buy side was winning, the chances of the next trade on the buy
side, winnings are increased. Let™s now develop code for TradeStation,
which can track a simulated equity curve for both the long and the short
sides. We need to simulate these trades because once we change the
trades based on our analysis, we have, in effect, changed the system. If
we want to develop statistics on our original system and use them to
modify a new system, we need to simulate the trade. The code for sim-
IS 2 3 35
-l-O 05 I ulating both the long and short sides for a channel breakout system is
uotatiaty shown in Table 10.1.
These two simple user functions-TrendChanShort and TrendChan-
A loo-day change in equity versus 1 N-day volatility.
FIGURE 10.1 Long-keep track of the equity on the long side and the short side, respec-
tively. They work well, but in order to simplify this example, we avoid
handling several special cases. First case is that, if you are using backad-
impossible to judge how it will perform in any given market condition.
Feedback analysis is a valuable tool for making this impossible job man- justed contacts, we do not handle cases in which prices of a commodity go
negative. Another unaddressed issue is when both a long and a short signal
ageable. All mechanical systems have a given trading pattern or “foot-
print.” When we test most trend-following methods over a long enough are generated on the same day. These issues are not a problem with com-
time period, a common trading pattern emerges. The classic channel modities like the currency, the S&P500, and T-Bonds, but they are a prob-
lem with crude oil because this market often moves in a tight range. The
breakout system will win 35 to 50 percent of its trades, and the average
code for handling these issues exists but is beyond the scope of this book.
winner will be much larger than the average loser. The winner will also
have more bars in its winning trades than in its losing trades. Another Let™s now use these routines in the channel breakout trading system.
often overlooked fact is that most trading systems are directly or indi- We ran this system on the D-Mark, using data from l/l/SO to 5/17/96.
Using equity analysis, we found a pattern that worked well on the
rectly based on cycles, System performance will change as the different
D-Mark: Take only short trades when the system has been profitable on
dominant cycles in the market appear, disappear, and shift. We will also
the long side but has been losing money on the short side. Using the same
see changes in the trading results as changes occur in the composite of
logic, we take only long trades when the system is making money on the
multiple cycles present in the market.
short side and losing on the long side. We also wait 10 trades before ap-
plying equity analysis. The code for our feedback channel breakout sys-
WALK FORWARD EQUITY FEEDBACK tem is shown in Table 10.2.
We tested this code on the D-Mark data from the period l/l/SO to
5/17/96. Our results are shown in Table 10.3, along with the results for
In Chapter 9, we showed how we can simulate a trading system and calcu-
the original system.
late its adverse excursion on a system. We can then use this information to \
Using System Feedback to improve Trading System Performance 145
144 Statistically Based Market Prediction


TABLE 10.2 CHANNEL BREAKOUT WITH
TABLE 10.1 CODE TO SIMULATE EQUITY
SIMULATED EOUITY CODE.
ON LONG TRADES ONLY.
Input: SLen(1 80),LLen(l20);
Inputs: ChanLen(Numeric);
Vars: LongE˜“(OUhortEq”iO),TBars(O˜;
Vars: BuyLev(Ol,SellLev˜O˜,BuyEntry˜O˜,SellEntry˜0˜,Position˜0˜;
ShortEqu=TrendChanShort(20,0):
˜Jars: Returns(O);
LongEqu=TrendChanLong(20,0);
if CurrentBar= then Returns=O;
If TotalTrades<lO then begin
BuyLev=Highest(High,ChanLen)[ll;
Buy Highest(High,ZO)+l point Stop;
sellLev=Lowest(Low,ChanLen)[ll:
Sell Lowest(Low,20)-1 point Stop:
If High>BuyLev and position<>1 then begin
end;
BuyEntry=MaxList(BuyLev,Open);
If ShortEqu-ShortEqu[SLen]<O and LongEqu-LongEqu[LLenl>O then Sell
Position=l;
Lowest(Low,ZO) -1 point Stop;
end:
ExitShort at Highest(High,ZO)+l point stop;
If Low<SellLev and position<>-I then begin
If LongEqu-LongEqu[LLenl<O and ShortEqu-ShortEqu[SLenl>O then Buy
SellEntry=MinList(SellLev,Open);
Highest(High,20) -1 point Stop;
Position=-1;
ExitLong at Lowest(HiRh,ZO)+l point stop;
end;
if Position=1 and Position[l]=-1 then Returns=Returns+(SellEntry-BuyEntry);
TrendChanLong=Returns;
By using this method of equity feedback to filter our trades, we reduced
{ Code to simulate equity on short trades only1
both the number of trades and the drawdown by almost half. We made 11
Inputs: ChanLen(Numeric);
percent less money but increased our average trades by 84 percent.
V X BuyLev˜O),SellLev˜O˜,BuyEntry˜O˜,SellEntry˜O˜,Position˜O˜;
Our second example, a 20-day channel breakout system applied to the
Vars: Returns(O); D-Mark, used a moving-average crossover of equity as a filter for both
if CurrentBar=l then Retums=O;
long and short trades. We take only long trades when a faster period mow
BuyLev=Highest(High,ChanLen)ill;
ing average of long equity is greater than a slower period one. We take
sellLev=Lowest(Low,ChanLen)[ll;
only short trades when a faster period moving average of equity on the
If High>BuyLev and position<>1 then begin
BuyEntry=MaxList(BuyLev,Open);
Position=l;
TABLE 10.3 CHANNEL BREAKOUT ON D-MARK WITH
end;
AND WITHOUT EQUITY ANALYSIS.
If Low&llLev and position<>-1 then begin
Original System Equity Feedback Demo 1
SelIEntry=MinList(SetlLev,Open);
Position=-I; Net profit $56,663.75 $50,945.00
Trades 104 50
end;
Win% 50 54
if Position=-? and Position[l]=l then Returns=Returns+(SellEntry-BUyEntry);
Average trade $544.84 $1 ,018.90
TrendChanShort=Returns
Drawdown -$22,075.00 -$12,245.00
Profit factor 1.57 2.53
146 Statistically Based Market Prediction Usina System Feedback to Improve Trading System Performance 147


TABLE 10.4 CODE FOR MOVING AVERAGE OF EQUITY FILTER Using this moving average filter, we cut the drawdown by over 200
WITH CHANNEL BREAKOUT SYSTEM. percent and filtered out a little more than one-third of the trades. We did
this while slightly increasing net profit.
Input: SLen(l30),LLen(l50);
Van: LongEqu(O),ShortEqu(O),TBars(O);
ShortEqu=TrendChanShort(20,0);
HOW TO USE FEEDBACK TO DEVELOP ADAPTIVE
LongEqu=TrendChanLong(20,0˜;
SYSTEMS OR SWITCH BETWEEN SYSTEMS
If TotalTrades<lO then begin
Buy Highest(High,ZO) + 1 point Stop;
Equity curve analysis is a powerful tool for improving the performance
Sell Lowest(Low,20) 1 point Stop;
of trading systems. The examples presented are only a sample of what
end;
cab be done using this technology. In another application, this technol-
If Average(ShortEqu,SLen)>Average(ShortEqu,LLen) then Sell Lowest(Low,ZO) 1
ogy can be used to switch between different trading systems. Suppose we
point Stop;
had a channel breakout system, a system based on intermarket analysis,
ExitShort at Highest(High,ZO)+l point stop;
and a countertrending type of system. We could simulate the equity
If Average(LongEqu,SLen)>AveragefLongEqu,LLen) then Buy Highest(High.20)
I point Stop; curves for each of these systems, and then, based on our analysis, select
ExitLong at Lowest(High,ZO)+l point stop; the one that currently has the highest probability of producing the best
trade. When developing these types of models, we can combine technical
or fundamental factors with our equity curve analysis to further improve
performance.
short side is above a slower period one. The code for this system is shown
Another application of this technology is to adjust parameters for a
in Table 10.4.
given system in a walk forward manner. This can be done by simulating
On the basis of our analysis, we found that using a 13Gday average of
the equity curve of many different sets of parameters for the same sys-
equity minus a 150-day average produced good results. It might be sur-
tem and then, based on our analysis, selecting the best one to currently
prising to know that these two moving averages produce good™results
trade.
when they are so close together. We were surprised, but almost all of the
top sets of parameters had moving-average lengths very close together.
Using lengths this close reduced our analysis to a mechanical way of
WHY DO THESE METHODS WORK?
saying that the last trade was profitable, and we did not have a quick l-
or 2-day whipsaw. The results based on these parameters, over the same
For a channel breakout system to work well, two conditions must be true:
period used earlier, are shown in Table 10.5.
1. The market must be trending.
TABLE 10.5 RESULTS OF CHANNEL 2. The performance of a channel breakout system is linked to a rela-
BREAKOUT WITH EQUITY FILTER. tionship between the period used in breakout and the dominant
$57,886.25 cycle.
Net profit
69
Trades
Win% If the cycle length becomes longer, then the channel breakout system will
:L3.93
Average trade get whipsawed during Elliott Waves two and four. ADX can be used to
Drawdown -$10˜,335.00
see whether the market is trending. The problem is that the interaction
2.11
Profit factor
between the dominant cycle and how much the market is trending is
Statistically Based Market Prediction
148


complex and has a major effect on the performance of a channel break-
out system. Modeling this interaction would require developing complex
rules that will make the system less robust. Equity curve analysis can be
11
used to solve this problem because the equity curve contains information
about this interaction that can be used to improve the system without a lot
of complex analysis.
An Overview of
The concept of system feedback is a powerful trading tool that can im-
prove profit, drawdown, and the winning percentage for most trading
Advanced Technologies
strategies. Trading a system without at least being aware of the equity
curve is like driving a car at night without lights-possible, but dangerous.




Advanced technologies are methods based on machine learning or on
analysis of data and development of models or formulas. I have used many
different advanced technologies in developing market timing systems.
These technologies include neural networks, machine induction methods,
and genetic algorithms. This chapter gives a technical overview of each
of these methods and introduces chaos theory, statistical pattern recog-
nition, and “fuzzy logic.”


THE BASICS OF NEURAL NETWORKS

Neural networks are loosely based on the human brain but are more sim-
ilar to standard regression analysis than to neurons and synapses. Neural
networks are much more powerful than regression analysis and can be
programmed for many complex relationships and patterns that standard
statistical methods can not. Their effectiveness in pattern recognition
makes them ideal for developing trading systems.
Neural networks “learn” by using examples. They are given a set of
input data and the correct answers for each case. Using these examples,
a neural network will learn to develop a formula or model for solving a
given problem.

149
150 Statistically Based Market Prediction 151
An Overview of Advanced Technologies


Let us now discuss how a simple neural network works. Artificial is set to a random value. We call this process “initializing the weights.”
neural networks, like the human brain, are composed of neurons and Once we have initialized the weights, we can adjust them during the
synapses. Neurons are the processing elements of the brain, and synapses learning process.
connect them. In our computer-simulated neural networks, neurons, also A perceptron neural network learns by repeatedly producing an an-
called nodes, are simply elements that add together input values multi- swer for each case, using the current value of its weights and compar-
plied by the coefficients assigned to them. We call these coefficients ing that value to the correct answer. It then adjusts the weights to try to
weights. After we have added together these values, we take this total and better learn the complete set of data. We call this process “supervised
apply a decision function. A decision function translates the total pro- learning.”
duced by the node into a value used to solve the problem. With simple two-layer perceptrons, the method or algorithm used to
For example, a decision function could decide to buy when the sum is adjust these weights could not solve a very important type of problem.
In 1969, Minsky and Papart, in a book entitled The Perceptron, proved
greater than 5 and sell when it is less than or equal to 5. Figure 11.1 shows
a simple example of a neural network. that a simple perceptron with two layers could not solve “non-linearly
The rows of one or more nodes are called layers. The first row of nodes separable problems” such as “Exclusive OR.” An example of an Exclusive
is the input layer, and the last row is the output layer. When only these OR problem is: You can go to the store or to see a movie, but you cannot
simple neurons are connected, we call it a two-layer perceptron. do both.
During the early 196Os, Bernard Widrow™used two-layer perceptrons This flaw in two-layer perceptron neural networks killed funding for
to solve many real-world problems-for example, short-range weather neural network research until the mid-1980s. Many researchers still con-
forecasts. Widrow even developed a weather forecasting neural network tinued working on neural networks, but, without funding, progress was
that was able to perform as well as the National Weather Service. slow.
How do we get the value of the weights used to solve a problem? Be- In 1974, Dr. Paul Werbos developed a method for using a three-layer
fore a neural network starts learning how to solve a problem, each weight neural network to solve nonlinearly separable problems such as Exclusive
OR. Rumelhart popularized a similar method and started the neural ex-
plosion in the mid-1980s. This method, called “backpropagation,” is the
most widely used neural network algorithm today.
Let™s see how this method differs from two-layer perceptrons.
Figure 11.2 sh0ws.a simple backpropagation neural network. The sec-
ond row of nodes is called the hidden layer. The first and third layers are
called the input layer (inputs) and the output layer (outputs), respectively.
A backpropagation neural network will have one or more hidden layers.
There are two major differences between a backpropagation neural net-
work and a simple two-layer perceptron. The first difference is that the
decision functions must now be more complex and nonlinear. The second
difference is in how they learn. In general, a backpropagation neural net-
˜work learns in the same way as the two-layer perceptron. The main dif-
R a w Output=tnputl”Weighfl+Input2*Weight2+lnpti3*Weight3 ference is that, because of the hidden layer(s), we must use advanced
Then we apply the decision function mathematics to calculate the weight adjustments during learning.
The classic backpropagation algorithm learns slowly and could take
thousands ofipasses through the data to learn a given problem. This is
FIGURE 11 .l A simple two-layer pe%eptton.
An Overview of Advanced Technologies 153
152 Statistically Based Market Prediction


network can be used just like any other indicator to build trading systems.
I -
inputs
The basics Neural networks™ predictions don™t need to have high correlation with fu-
ture price action; a correlation of 2 or .3 can produce huge returns.
sap500


MACHINE INDUCTION METHODS
Bonds
S&P500 % change
+5 days Machine induction methods are ways to generate rules from data. There
are many different machine induction methods, and each one has its own
CRB
strengths and weaknesses. The two that I use are called C4.5, which is a
descendant of an algorithm called ID3, and rough sets.
This diagram shows a simple
neural network™s processing

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