Figure 4.7 A plot of the NASDAQ Composite Index showing support and resistance levels. Support and resistance levels fall outside congestion regions and are present where the PDF is small. This reflects the observation that the price or index spends little time near these levels. Expect a bounce or a break through near these levels, but not stagnation.

Another common reference is the simple moving average. This is simply an average of some number of data points, say M data points, that is plotted at the last time value. Mathematically, the average best corresponds to the time at the center of the interval of M points, and by plotting it at the last time value, a delay is introduced of half the interval size. While smoothing that is accomplished by the average is good, the delay is often viewed as undesirable. Nonetheless, the simple moving average is widely used and quite often seems to provide support or resistance to prices.

Why is this? One possible explanation might be that market congestion plays a role in price behavior. It turns out that for short periods, simple PDF structures usually result. The simple moving average is an approximate means of defining the location of congestion for this situation. The delay that is introduced is appropriate and even desirable for comparing the congestion to current price. For longer time periods and complex PDF structures, the average may be less meaningful. For example, consider a PDF with two congestion peaks. These peaks then define congestion levels, whereas the average might fall between the peaks at the wrong level. For short periods,

the PDF, in most cases, has only one peak and the simple moving average is a often a reasonable approximation to the location of this peak.

One method that I have found convenient for observing congestion I refer to as "Price Layers." A layer plot is simply a trace forward of a price. Extend (draw horizontally) each closing price forward for a fixed time interval. The extension should be long enough to represent a cycle period, as identified by inspection of the prior history or by Fourier analysis or some other frequency analysis. If the drawn lines are closely spaced, then there is clustering that reflects the presence of congestion. If the lines are widely spaced, then there is evidence of a lack of congestion. The line plots also give some representation of the range of prices over a brief period by observing the extreme top and extreme bottom lines at any given time.

Several patterns are frequently evident in layer plots, flat tops, flat bottoms, and flat steps. An example of steps is seen in Figure 4.8. A clustering of layers often

Figure 4.8 A time history for the NASDAQ presented as a Layer Plot and

corresponding PDF for the 59-day period. the layers, which are simply horizontal extensions of the daily closes, provide a graphical picture of congestion. groupings or clustering of these layers correspond to congestion, and wide spaces, the lack of congestion. this plot shows a strong uptrend with three major steps and congestion peaks. the

LAST STEP IS A TOP FORMATION WITH SUCCESSIVELY LOWER HlGHS WITH THE PRICE MOVING BELOW THE UPPER CONGESTION PEAK.

represents a value close to the mode of the PDF and spaced layers correspond to valleys in the PDF. Breaking through the layers gives an advance warning of an upcoming test of support or resistance. Other variations are also possible as extensions of combinations of the high, low, and close. This approach is a means of assessing congestion in a very qualitative way. It can be plotted by either programming graphics in Visual Basic or by repeatedly plotting price values with varying incremental delays.

In summary, PDFs provide additional references for analysis such as the peak or peaks and the minimum values of the distribution. There are alternative ways of viewing congestion such as price layering. PDFs also complement other analysis references such as support and resistance and averages. A PDF can identify regions between support and resistance levels as being either high congestion or low congestion, an important piece of information.

Mobility Oscillators

As mentioned, direct viewing of PDF functions for mobility analysis of a time history is possible, but can be difficult since the structure can be complex. What is needed is to boil down the PDF into an oscillator that gives a precise measure of congestion and mobility and reflects previous market movement. Since the PDF implicitly includes this and other reference information on price, it is a matter of sorting out the information and framing it in a useful form.

The first step in construction of an oscillator is to find suitable reference points. In this case, there are several possibilities. There are price references for the look-back period: the high, the low, the average, and the mode. For example, a comparison between the current price with the high, low, and average or mode for the period could be made. In fact, I tried this with some success, resulting in an oscillator similar to the Stochastic Oscillator with the mode as an internal reference point to give direction. While this price reference method was somewhat useful, it did not give a precise comparison of congestion levels. Consequently, the analysis was reformatted using congestion as the basis for comparison and for the references.

This resulted in what is referred to as the Mobility Oscillator. That is, the maximum congestion level of a look-back period is the peak value of the PDF that occurs at the price mode value and is the upper reference point. The lower reference is the minimum possible congestion level and is zero. The current congestion level, the PDF value at the current price value, is compared with these references. A simple, linear proportionality of the current congestion level to the upper and lower reference congestion levels gives the gauge. Next, a direction can be assigned by comparing the current price with the mode. This is described mathematically as follows.

The value of the price at the peak of the distribution function is the mode. The mode is the most likely value for the price during the look-back period, the value where the price spent the most time. It represents the price where there was maximum