5 76 77 78 79 80- YearCopper prices 1963-1979
 
be.
UNCOVERING THE CYCLE 201
 
Cycle Finder mentioned in a previous section. A single-frequency curve can be created sim-ply by identifying the most dominant peaks, averaging the distance (period), and applyingthe single-frequency formula.
‘Two-Frequency Trigonometric Regression
The combination of more than one set of sine and cosine waves of varying amplitudes andfrequencies will create a better fit than a single-frequency solution. This is analogous to theuse of a second-order (curvilinear) solution instead of the first-order linear. The equationfor the two-frequency cycle is
 
  
Jr =A, COS Of + b, Sin Wt + a2 COS Mzt + b; Sin Wot
Yo find the results of this complex wave, apply the same techniques used in the single-frequency approach to the detrended copper data. The algebra for solving this problem isan expanded form of the previous solution, and the use of a computer is a requirement.The programs necessary to solve this one appear in Appendix 4. The frequencies @, and ©,are found by solving the quadratic equation:
2x? — ox — (1 + 0/2) =Owhere x = cos @, using the standard formula:
oy + Von? + BCI + 0/2)4
x
 
‘The same least-squares method as before can be used, derived from the general form:
(Mn + Yn 43) + C2 Pav =In-1 + Pnss
‘The least-squares equations for finding 04 and 02 are:> +0, > ed => opaD ed+0 > d?=> dp
where C=¥n+Invn d=Insa, and p =In-1+Jn +3 These equations can be solved for 0 ando using:
_LweSe-SeSeaDede-(Sa)S p-e% ¥ edYe
‘Then, @, and @, are calculated from the two solutions x, and x; of the quadratic equation.‘The next step is to solve the normal equations to find the amplitudes ay, b1, 42, and bp:
Oo
a=
a, > cos? at + by D, cos Wit sin wt + az & cos wrt cos wot
+B, >, cos ait sin o,f = > 9, cos Ot 
- 202CYCLEANALYSIS
4 > sin ot cos tt by S) sin? ant + ay S sin wrt cos wrt+b, > sin o,f sin ot = >” y, sin oe4, > 608 wt cos @it +, cos wnat sin wt + az 5” cos? wat+62 > cos wnt sin wt = Sy, cos Oat4, > sin @st cos wt + 6, > sin wnt sin wyt + a, S° sin wt cos @:t+b, > sin? wt = Sy, sin ontOnce the sums are obtained, the final step is to create a 4x 5 matrix to solve the four nor-
mal i i Whmal equations for the coefficients a), 6,, a2, and b;. When plotting the answer it will be best0 p original two-frequency equation in its component forms as well as in combina-
  
I= @ COS Ort +b, Sin Oti = Gz COS Ont + b; Sin WtWaVAH
where a, = 3.635, b, = ~.317, a = -.930, and b; = iwre a 53.695, bet 317, a2 = 930, ind 6, = .762. The solution to the two-frequency
= .535 x, = .830
and= .133 x) =~.764
and finally the frequencies:,=33.9 and @, = 139.8
 
 
correspond to 10.6 and 2.6 calendar quarters (Figure 8-9).
Fourier Analysis: Complex Trigonometric Regression
peweioped by the french mathematician John Baptiste Joseph Fourier, Fourier analysis isSmet oF cor plex rigonometric regression, which expresses any data set as a series ofand cosine w ves of the same type as discussed in the previous section.cycle then i ere is a cycle and that there are N data points in each repetition of this, ‘ourier method of analysis shows that the N’ points lie on the regression curve:wa
_ 2nki 2nkiyal+ (« aki in 2TREi= 1+ > (ue cos Gay * SING ‘)
where the regression coefficients u, and v are given by:
 
 
13 2nki N
y= y aki =
Cay 28 ay BEN ry1 2nki oN
m= > in —— = =
Ry 2 py BER
vwa = 0
UNCOVERING THE CYCLE 203
 
 
FIGURE 8-9 Two-frequency trigonom:70
Two-frequencyapproximation
Trendline trom
50 linear regression
Price (cents/pound)
       
2s es 65 66 678 «69 «70:«71~«72:~«73:« 747576 «7778 79 «BO BL
Year (in quarters)
 
It is important to see that the mean of all the points on one cycle is equal to 1. The NV val-ues of y, will have the property
w
Swen
Applying the Fourier series to the seasonal component will help clarify this method.Seasonal data form the most obvious cycle. Using average monthly prices, detrended toavoid letting the trend overwhelm the cycle, let N = 12. It is also known that seasonallyadjusted prices will vary about the mean; hence, the weighting factors will have the sameproperty as the above equation. With this information, the trigonometric curve, whichapproximates the seasonals, can be generated and compared with the results of other
methods.” -
Spectral AnalysisDerived from the word spectrum, spectral analysis is a statistical procedure that isolatesand measures the cycles within a data series. The specific technique used is the Fourierseries as previously discussed, although other series have also been used.
‘When studying the cycles that comprise a data series, it is important to refer to theirphase with respect to each other. Phase is the relationship of the starting points of differ-‘ent cycles. For example, if one cycle bas the same period as another but its peaks and val-
74 continuation of this development can be found in Warren Gilchrist, Statistical Forecasting John Wiley &Sons, London, 1976, pp. 139-148); a more theoretical approach is to be found in C. Chatfield, The Analysis of aTime Series: Theory and Practice (Chapman and Hall, London, 1975, Chapter 7). 
204 CYCLE ANALYSIS
 
leys are exactly opposite, it is 180° out-of-phase. If the two cycles are identical in phase,they are coincident. Cycles with the same period may lead or lag the other by being out-of-phase to various degrees.
A tool used in spectral analysis to visualize the relative significance of a series’ cycliccomponents is the periodogram. Weighting the cyclic components in the periodogram willgive the more popular spectral density diagram, which will be used to illustrate the resultsof the spectral analysis. Density refers to the frequency of occurrence. Figures 8-10a and8-106 show the spectral density of a series composed of three simple waves (D is theFourier series made up of waves A, B, and C).* The cycle length, shown at the bottom of thespectral density chart, corresponds exactly to the cycle length of the component waves A,B, and C. The spectral density, measured along the left side of Figure 8-106, varies with theamplitude squared of the cycle and the magnitude of the noise, or random price move-ments, which obscures the cycle. In Figure 8-108, the result is based on a series composedof only three pure waves. Had there been noise of the same magnitude as the underlyingcycle amplitude, those cycles identified by the spectral analysis. would have been com-pletely obscured. Readers who have studied ARIMA will recognize the similarity betweenthe spectral density and the correlogram.
As in trigonometric regression analysis, the other basic price components can distortthe results. A noticeable trend in the data must be removed or it will be interpreted as thedominant cycle. The familiar methods of first differencing or linear regression can be usedto accomplish this. The seasonal component is itself a cycle and does not need to beremoved from the series. Because spectral analysis will identify both the seasonal andcyclic components, the success of the results will depend on the strength of these wavescompared with the noise that remains. In applying this technique to real data, it would notbe surprising to see the results demonstrated in Figure 8-11. Three subcycles of length 10,20, and 40 days are shown as part of a 250-day (seasonal) cycle. Notice that, as the cyclelengthens, the width of the spectral density representation widens. This does not meanthat the wider peaks are more important.
‘The trader is most interested in those cycles with greater spectral density, correspondingto a larger price move. The minimum amount of data necessary to find these cycles mustinclude the full cycle that might be identified. For example, to see any seasonal pattern, a min-imum of 12 months is needed. More data is better when using spectral analysis to confirm theconsistency of the cycle. A single year is not adequate to support any seasonal findings.
Weighting Factors
The most important part of spectral analysis is finding the proper estimators, or weightingfactors, for the single-frequency series of cosine waves. When looking for long-term cycles,it is worth being reminded that the trend and seasonal components must be removed,because the method of spectral analysis will consider these the dominant characteristics,and other cycles may be obscured.
As in the other trigonometric formulas, the basic time'series notation is used, where 9,1=1,2,...,N are the data points and 9, will be the resulting estimated points on the spec.tral analysis. Then
 
   
Flo) = Hen + 2’ Ce COS ot)
where
 
 
UNCOVERING THE CYCLE 205
formed fromm throw primary waves, A,FIGURE 8-10 Spectral density. (2) A compound wave D, formed from three primary waves, A,B, and C. (b) Spectral density of compound wave D.
 
 
     
 
TT TIT TT TTT TTT500| TTT TTT TTT475450]425|400]3753503253002 275= 250225200175150125100750]25)ql ey pity py0 10 20.30 405060708090 110 130 150 170 190 210 230 250Trading daysfa)290000267000184000= 161000‘2 1300003*B 1150003B 92000590004600029000
~ ° {___ LIoO 10 20 30 40 50 60 70 80 90 100cycle length
()
  
206 CYCLE ANALYSIS
FIGURE 8-|
  
0-, and 40-day cycles, within a 250-day seasonal.
Spectral density
10 40 250 500Cycle length
 
Methods of performing spectral analysis vary due to the choice of weighting functionsthat compensate for the fact that the accuracy of ¢, decreases as k increases. The two mostpopular techniques for adjusting for this problem introduce an estimator Ax called a lagwindow and a truncation point M < N so that the values of c, for M < k < N are no longerused and the values of c, for k < M are weighted by Ay.
‘The spectral analysis approximation is then written:
 
where }, can be either of the following:
 
(a) Tukey window
1 mkMas (1+ cos *#) k=0,1,...,M
(b) Parzen window ‘ky k\ mM1-6(— = M(E}+6(t) ocnek\ - Mm2(1-= a( £) 7 sks
Using a Fast Fourier Transform Program = —
There are computer programs that apply a Fast Fourier Transform to perform a spectralanalysis and create a Fourier power spectrum such as the one in Figure 8-106. Anthony‘Warren's approach’ can be found in Appendix 5 written in BASIC program code. The pro-
° “A Mini GuiAnthony Warren, “A Mini Guide to Fourier Spectrum Analysis,” Technical Analysis of Stocks & Commodities (Jan-
Pad 1983). ‘Avery useful series of articles on spectral analysis has been published in Technical Analysis beginning,
in January 1983, authored by both Anthony W. Warren and Jack K. Hutson. Much of the inf in this
tion was drawn from that material. ontne tnformacion in tis sec:
UNCOVERING THE CYCLE 207
gram detrends the data and reduces endpoint discontinuity, which can produce largeunwanted cycles. This is accomplished by multiplying the data by a bell-shaped windowand extending the endpoints to give a more definitive structure to the detrended data,without affecting the results (as discussed in the previous section).
‘A second filter is applied using selected moving averages. The moving average willreduce or eliminate the importance of those cycles, which are equal to or shorter than twotimes the length of the moving average, letting the more dominant cycles appear. Forexample, the use of a 10-day moving average will eliminate cycles of length less than 20days (frequencies greater than 12.5 per year). Figure 8-12 shows the output of the com-puter program.
‘Subsequent works by Warren and Hutson” present a computer program to calculatemoving average weighted filters using linear, triangular, and Hanning weights.
Interpreting the Results of the Fourier Power SpectrumBoth Figures 8-106 and 8-12 show a power spectrum resulting from a Fourier transform.Figure 8-106 is an ideal representation, in which the cycles stand out with no ambiguity;Figure 8-12 is more realistic, showing both the dominant cycles and a certain amount ofvariance around those values. In the power spectrum, the cycle power shown along the y-axis is the cycle amplitude squared. In Figure 8-10a, cycle D peaks at a price of about 425,which yields a spectral density, or spectral power, of 180,625 when squared, correspond-ing roughly to the 40-day cycle in Figure 8-10b.
Using the information from the beginning of this chapter, the frequency is the inverseof the cycle length; therefore, if the cycle length is 40 days, the frequency F = 360/40 = 9.
© Anthony Warren and Jack K. Hutson, “Finite Impulse Response Filter,” Technical Analysis of Stocks & Com-modities (May 1983).
FIGURE 8-12 Output of spectral analysis program.
 
        
TITLE T BILLS $03/83 01/07/83FOURIER ANALYSIS OF 7-DAY MOVING AVG.9.88 FourierN= 256 om
 
J.OAY MOVING AVERAGEWF DAILY CLOSES (CENTERED!
 
’oct NOV DEC
Source: Jack K. Hutson, “Using Fourier.” Technical Analysis of Stocks & Commodities, |, n0. 2 (january & February1983). © 1983 Technical Analysis, Inc. Used with permission. 
208 CYCLE ANALYSIS
The sine wave changeg phase at the rate of 9 degrees per day, completing one full cycleevery 40 days.
A fast method for observing the possible results is to use weekly rather than daily data.‘This will be a close approximation for low-frequency waves but will be less representativefor the high frequencies. Averaging the data points can yield results very similar to the dailyanalysis.
MAXIMUM ENTROPY
Maximum Entropy Spectral Analysis (MESA) is a technique that filters noise (entropy)from a time series and exposes the useful cycles;" it provides a very practical alternative toFourier analysis that makes it possible to find cycles using a very small amount of data. Theuse of Fourier transforms requires at least 256 data points and a minimum of 16 consistentcycles of 16 bars. That would eliminate the possibility of uncovering cycles for the short-term trader.
John Ehlers describes the existence of short-term cycles as a natural phenomenon.” Itis part of the process that causes rivers to meander back and forth as water seeks to flow ina straight line, or a drunkard who walks through an alley bumping against the walls butmoving steadily forward. From these patterns, useful cycles can be found about 20% of thetime.
Using the Phase Angle
In an ideal situation, in which the market cycle can be shown as a pure sine wave, thephase angle constantly increases throughout the cycle, beginning at 0 and ending at 360°.The phase angle then drops to 0 when the new cycle begins and increases again at a con-stant rate until it ends at 360°. This repeated pattern forms a sawtooth chart, as shown inFigure 8-13. Although the cycle goes from peak to value, the phase angle moves constantlyin one direction.
In the practical analysis of short-term cycles, Ehlers compresses tick data into bars ofequal numbers of ticks, then examines the phase for uniformity. Once found, the uniformphase, which appears as a sawtooth chart, will become erratic as the short-term cycle
begins to break down, marking the end ofa current market event.
xEhlers’s Lateral Shift in ThinkingAt first glance, the use of only a small amount of data needed by MESA seems to contradictthe basic rules of statistics, which demand that results be based on as much data as possi-ble to be reliable. But Ehlers, who has been the dominant influence in cycles since about1990, is too knowledgeable to have made a mistake so simple. His book MESA and Trad-ing Market Cycles" focuses on the use of short-term cycles based on short sample timeperiods. Instead, he has used this very attribute to apply cycles inside out.
Ehlers’s objective is to find very short-term cycles. By definition, these cycles must bethe result of human behavior, rather than based on market economics, because funda-mentals are not usually relevant to periods of only a few days, and they are not likely tohave a regular pattern when they make a rare appearance. If very short-term cycles exist,
™ Anthony Warren, “An Introduction to Maximum Entropy Method (MEM),” Technical Analysis of Stocks & Com-modities (February 1984). See the bibliography for other articles on this topic.
® John F. Ehlers, “How to Use Maximum Entropy,” Technical Analysis of Stocks & Commodities (November1987).
8 John E. Ehlers, MESA and Trading Market Cycles John Wiley & Sons, New York, 1992).
CYCLE CHANNEL INDEX 209
FIGURE 8-13 The phase angle forms a sawtooth pattern.
tH| \
 
  
|408 +385 |me'i
 
 
 
 
 
meSE EAE
 
     
Source: john Ehlers, “Cycle Analysis and Intraday Trading,” Technical Analysis of Stocks & Commodities, 11, no.(February 1993).© 1993 Technical Analysis, Inc. Used with permission.
i i i ize them quickly if they are tothey will not continue for long periods, and you must recognize 1be useful, therefore, short-term cycles are found by analyzing only a small amount of
nt data. .ieee Then how does it help to find a short-term cycle based on a small amount of data, if it
is not statistically dependable? In a lateral shift, Ehlers uses the existence of a short-termcycle to tell if prices are in a sideways pattern or trending. Ifa short-term cycle exists, erthe market trend should be weak. Ehlers has no interest in trading the cycle, which is sut-prising for a cycle expert, but prefers the dependability of the trend. He has, instead,itempted to solve one of the most difficult problems facing the analyst, trying to distin:guish between a trending and sideways market. Ifa short-term cycle exists, then we cannorely on the trend. Ehlers develops this method throughout his book.
CYCLE CHANNEL INDEX
‘A trend-following system that operates for a market with a well-defined eyctc attenshould have specific qualities that do not necessarily exist in a generalize’ smoothi nemodel. ‘To confirm the cyclic turning points, which do not often occur precisely where ©?are expected, a standard moving average should be used, rather than an exponents lySmoothed one. Although exponential smoothing always includes some residual ECC 0Older data, the determination of a cyclic turning point must be limited to data that is HEATto one-fourth of the period, combined with a measure of the relative noise in the s Syich may obscure the turn.
~ “Thece features have been combined by Lambert(CCD, which is calculated as follows:
4s into a Commodity Cycle Index
Mponald R. Lambert, ‘Commodity Channel Index Tools for Trading Cyelic Trends,” Commodities (1980),
reprinted in Technical Analysis of Stocks & Commodities. 
240 CYCLE ANALYSIS
 
x,-%, :.015MDwhere x, = (H, +L, + C,)/3 is the average of the daily high, low, and close
> xis the moving average over the past V days
CCL, =
x,
 
MD=
Nis the number of days selected (less than % cycle)
|x; ~.x| is the mean deviation over the past N days
Because all terms are divided by N, that value has been omitted. In the CCI calcula-tions, the use of .015MD as a divisor scales the result so that 70% to 80% of the values fallwithin a +100 to -100 channel. The rules for using the CCI state that a value greater than+100 indicates a cyclic turn upward; a value lower than —100 defines a turn downward.Improvements in timing rest in the selection of N as short as possible but with a mean devi-ation calculation that is a consistent representation of the noise. The CCI concept of iden-tifying cyclic turns is good because of the substantial latitude in the variance of peaks andvalleys, even with regular cycles.
PHASING
One of the most interesting applications of the cyclic element of a time series is presentedby J.M. Hurst in The Profit Magic of Stock Transaction Timing (Prentice-Hall); it is thephasing or synchronization of a moving average to represent cycles. This section will high-light some of the concepts and present a simplified example of the method. It is alreadyknown that to isolate the cycle from the other elements, the trending and seasonal factorsshould be subtracted, reducing the resulting series to its cyclic and chance parts. In manycases, the seasonal and cyclic components are similar, but the trend is unique. Hurst treatsthe cyclic component as the dominant component of price movement and uses a movingaverage in a unique way to identify the combined trend-cycle.
‘The system can be visualized as measuring the oscillation about a straight-line approx-imation of the trend (centered line), anticipating equal moves above and below. Priceshave many long- and short-term trends, depending on the interval of analysis. Because thistechnique was originally applied to stocks, most of the examples used by Hurst are long-term trends expressed in weeks. For commodities the same technique could be used byapplying the nearest futures contract on a continuous basis.
Asa simple example of the concept, choose a moving average of medium length for thetrending component. The full-span moving average may be selected by-averaging the dis-tance between the tops on a price chart. The balf-span moving average is then equal to halfthe days used in the full-span average.
The problem with using moving averages is that they aways lag. A 40-day moving aver-age is always 20 days behind the price movement. The current average is plotted under themost recent price, although it actually represents the price pattern if the plot were laggedby one-half the value of the average. This method applies a process called phasing, whichaligns the tops and bottoms of the moving average with the corresponding tops and bot-toms of the price movement. To phase the full- and halfspan moving averages, lag eachplot by half the days in the average; this causes the curve to overlay the prices (Figure8-14). Then project the phased full- and half-span moving averages until they cross. A lineor curve connecting two or more of the most recent intersections will be the major trend-line. The more points used, the more complicated the regression formula for calculating
auPHASINGa variety of linear and nonlinear techniques for finding the
5 discusses ¢Te ee oan nterse Once the trendline is calculated, it is projected as the cen-
best fit for these intersections.
if the next price cycle. |“r “with the trend identified and projected, the next step is to reflect the cycle about the
trend, When the phased half-span average turns down at point A igure #15), measure: i | prices above the projected trendline. The systhe greatest distance D of the actual p! ‘ i ne. The system theni i i dline at point X and declining an eq
anticipates the actual price crossing the tren ¢
tance D below the projected trendline. Once the projected crossing, becomes an actualGrossing, the distance D can be measured exactly and the price objective firmed. Rules forusing this technique can be listed as follows:
1. Calculate the full-span moving average for the selected number of days; lag the
plot by half the days. If the full-span moving average uses F days, the value of the
Rverage is calculated at t ~ F/2, where f is the current day. Call this phased point
PH,2. The half-span moving average is cal3. Record the points where the two p!
oints X, Xp-ay +++ |4. Pind the irend by performing a linear regression on the crossing
X,-1.... Ifa straight line, then ¥-= a + 6Xr.
me
FIGURE 8-14 Phasing.
Jculated for H days and plotted at ¢ - H/2 +PH,.jhased averages PH, and PF, cross and call these
points X,,
 
Lag Plot 20 Days Before
  
 
212CYCLE ANALYSIS
    
 
   
            
Half-Span
Moving Average Projected Cycle
Lee
So Neat Projected Crossing
Crossing X,
   
Projected Trendpr ae bhp
 
Price Obdjective(Target)
®!5
 
5. Record the highest (or lowest) values of the price since the last crossing, X,6. Calculate the projection of the halespan by creating a straight line from the highest (or lowest) half-span value since the last crossing (A) to the last calculated half.5, Spal value, This equation will be Yo= e+ de ". Find the point at which the projected trendline crosses the proj isetting we equations equal to one another and solving for Ravar ace i 2rossing (X;, ¥7) = Xe, ¥.), giving ions i ich is easicrossing Oy MD = Oe, %0. ivi ne so equations in two unknowns, which is easily. If the halfspan is moving down, the maximum price reached by the commodisince the last crossing is subtracted from the ¥ coordinate of the projected cross,ing. This distance D is subtracted again from. the ¥ coordinate to determine theprice objective. If the half span is moving up, the price objective uses the minimumprice and reflects the distance above the projected crossing. It should be notedthat this calculation of distance is simplified because the trend is established b:straight line; for nonlinear fits, the measurement of D will be more complicated. .9. Recalculate the moving averages, the half-span projection (6), the projected cross-
ing (7), and price objecti i iing @). and pe jective (8) each day until the actual crossing occurs. At that10. Follow the trading rules: .
@
a. Enter a new long position when the halfspan moving average turns up; cover1, 2nY existing short postions regardless ofthe price objective ,. Enter a new short position when the halfspan moving aveclose out any long positions. Pas moving average turns down;€. Close ous both long and short postions if the price abjectve is reached. Amallowable error factor is considered as 10% of th (Lowable ero fa ¢ height of the full cycle (lowestsem wis approach to cycles should be studied carefully as an example of a complex prob-tem folved using elementary mathematics. There are many techniques for determiningtrends and a number of seasonally oriented systems, but acyclic approach is rare, Whereasxplanation is more complete and more sophisticated, the ini ati. ny " y terpretati 7sented in this section should be considered only a reasonable approximation.
ee
 
Charting
ries and complex formulas may ultimately be successful, but the loss of perspec-tive is rarely corrected without a simple chart. We should remember the investorwho, anxious after a long technical presentation by a research analyst, could only blurtout, “But is it going up or down?” Even the most sophisticated market strategies mustcapture the obvious trends or countertrends. Before any trading method is used, thepast buy and sell signals should be plotted on a chart. Those signals should appear at log-ical points; otherwise, the basis of the strategy or the testing method should be ques-tioned.| Through the mid-1980s technical analysis was considered chart interpretation. In theequities industry that perception is still strong. Most traders begin as chartists, and manyreturn to it or use it along with their other methods. William L. Jiler, a great trader andfounder of Commodity Research Bureau, wrote:
Ne can a picture be more valuable than in price forecasting. Elaborate theo-
One of the most significant and intriguing concepts derived from intensive chartstudies by this writer is that of characterization, or habit. Generally speaking,charts of the same commodity tend to have similar pattern sequences whichmay be different from those of another commodity. In other words, charts ofone particular commodity may appear to have an identity or a character pecu-liar to that commodity. For example, cotton charts display many round tops andbottoms, and even a series of these constructions, which are seldom observedin soybeans and wheat. The examination of soybean charts over the years re-veals that triangles are especially favored. Head and shoulders formationsabound throughout the wheat charts. All commodities seem to favor certain be-havior patterns."
In addition to Jiler’s observation, the cattle market is recognized as also having theunusual occurrence of “V” bottoms. Both the silver and pork belly markets have tenden-cies to look very similar, with long periods of sideways movement and short-lived, violentprice shocks, where prices leap rather than trend to a new level. The financial markets haveequally unique personalities. The S&P traditionally makes new highs, then immediatelyfalls back; it has fast, short-lived drops and slower, steady gains. Currencies show interme-diate trends bounded by noticeable major stopping levels, while long-term interest rates
have long-term direction. _Charting remains a most popular and practical form for evaluating commodity price
movement, and numerous works have been written on methods of interpretation. Thischapter will summarize some of the accepted approaches to charting, and then consideradvanced concepts of both standard charting methods and systems designed to take advan-tage of behavioral patterns found in charts.