Characterization of Patterns

denote the times of onset of three sets of (lowest) consecutive local minima up to time /. To obtain a head and shoulders pattern, the heights of the local minima in the first and third sets must be (approxi ­mately) the same, say M*. In addition, the levels of the local minima in the second set must be significantly higher, say M**. Then a (sell) signal is issued the first time X, falls below M* once such a pattern takes shape. That is to say,

t, = inf{*,/„}.

Since {/,k\, the local minima defined in (8) are measurable. Hence an event describing head and shoulders becomes measurable once a formal way of subdividing the three sets of local extrema shown in (8) is selected. Such a criterion is needed in order to decide when the local minima in the middle exceed significantly the local minima in the first and third sets. This requires an a priori selection of a lower bound on the difference M** - M*, although the levels of M* and M** need not be specified individually. If all these conditions are met, a head and shoulders pattern becomes measurable.

This construction shows that actual signals generated using observed head and shoulders patterns are not Markov times. For example, in figure 3, the first occurrence of such an event is illustrated by the line AB rather than the suggested head and shoulders pattern CD selected by technicians. The only way one would select CD is if one anticipated that a local minimum such as D would occur at time /. Accordingly, in this example, the decision of whether С‚ = r or not depends on future values of the underlying series. The stopping time illustrated in figure 3a cannot be a Markov time.

Further, head and shoulders patterns defined formally as above are likely to be probability-zero events if one insists that the minima in the first and second sets have the same height M*. Conversely, remov ­ing this requirement will impose further a priori restrictions on the sets of local minima shown in (8).

Figure 36 illustrates a triangle. In principle, this pattern can also be defined using consecutive local minima and maxima. To generate such a triangle, consecutive local maxima have to be in descending, and consecutive local minima in ascending order. Thus let

{' « «. !<•••< W*} and {Max, > Max2 > ... > Max*}   (9)

represent the times of onset and the heights of Рє consecutive local maxima of X, during Рў;_, < /. Similarly, let {/mjnii < . . . < 1^тк) and (Min, < . . . < Min*} be the times of onset and heights of Рє lowest local minima during the same period. Then, a buy or sell signal is generated as soon as observed X, exceeds the last local maxima or falls below the latest local minima (see fig. 3/ »). More precisely,

t, = inf{Max, > ... > МахА<ЛГ,огМт, < ... A'I}, (10) forT,_, < /.

Clearly, this is an /,-measurable event, hence a prediction method using triangles defined this way will generate Markov times. Yet this does not mean that, in practice, the signals generated using triangles are non anticipatory. In fact, what makes the above signals /,-measur ­able is the a priori specification of the parameter A, namely, the number of minima or maxima that one has to observe before the crossing occurs. If this information is omitted from the definition, the use of triangles will cease to generate Markov times. Without this parameter, even two consecutive local extrema can generate a triangle. Clearly, this is not what technical analysts have in mind, as shown in figure 36. If two extrema are not sufficient, then how many does one need? Obviously, the answer to these questions necessitates a priori selection of some parameter such as k.

Finally, in figure 3c, we show another pattern, namely, gaps in daily price ranges. In contrast to other patterns, the use of gaps does generate Markov times since a signal is issued the first time three consecu ­tive gaps are observed. This constitutes a first entry into an interval and leads to Markov times.