Consider a particle with a one dimensional periodic trajectory given by z(t) for all t. 0≤t≤T. Suppose for the matter of definiteness the trajectory takes the particle within the limits of a and b; i.e., a≤z(t)≤b for all t. Suppose also that the particle completes its trajectory within the limits of 0≤t≤T. The question to be addressed is what proportion of the time does the particle spend in the vicinity of any value of z. It is presumed that the particle is always moving so it is never at rest for any value of z. However the amount of time in which the particle is between z(t)−Δz/2 and z(t)+Δz/2 is Δz/|z'(t)|. Since the total amount of time the particle spends travesing its circuit or orbit is T the proportion of time spent in that interval is (1/T)[Δz/|z'(t)|]. It may be the case that there is more than one time that the particle is passing through the point z.

If the width of the intervals of z is made increasingly small the result is a probability densit function

p(z)dz = dz
so the probability of the particle
being in the interval [c,d] is
∫_c^dΣ_z(t)=z[(1/T)/|z'(t)|]dz

This is the same formula that applies for determining the probability distribution for y when y=f(x) and the probability distribution function for x is known; i.e.,

q(y)dy = Σ_f(x)=y[p(x)/|f'(x)|]dx

It is as thought the probability density function for t is 1/T and this is transformed into the probability density function for z of [(1/T)/|z'(t)|].

For example, take the case of z(t)=A*sin(ωt). The period of this function is T=2π/ω. The values of z are between −A and +A. For every value of z between those limits there are two values of t such that A*sin(ωt)=z.

Since z'(t)=ωAcos(ωt) and t=(1/ω)sin^-1(z/A) the probability density function for z is

p(z) = 2(1/T)/[ωAcos(sin^-1(z/A))]
which reduces to
p(z) = (1/π)/[Acos(sin^-1(z/A))]

Notice that the frequency ω has disappeared from the formula.

The graph of p(z) as a function of z is shown below.

For this function the variable z spends about 20 percent of its time in the range from -A to -0.9A, another 20 percent in the range 0.9A to A and only 6.5 percent of its time within ±5 percent of z=0. Thus altogether the variable spends 40 percent of its time within 10 percent of the extremes. On average the z variable is zero but it spends a forty percent of its time within 10 percent of the extremes.

The hours of daylight follows a sinusoidal pattern, increasing from a low value of the day of the winter solstice to a high value for the summer solstice. If one were to tabulate a frequency diagram for the hours of daylight the result would be bimodal. A random selection of a day in the year would be much more likely to come up with one that is long or one that is short than one that has approximately equal daytime and night-time hours.

The lesson learned from this example is that variables spend an inordinate amount of their time near their critical points; i.e., relative maximums or minimums or inflection points.

If the growth rate for a variable is cyclical then the growth rate will appear to be occurring at the maximum and minimum an inordinate amount of the time. If the variable has no long term trend then the variable will typically appear to be growing at a maximum rate or declining at a maximum rate.

Consider the following simulation of the temperature of an object as the cumulative sum of random levels of net heat input to the object. The average value of the net heat input is zero so there is no long term trend. The simulation is based upon a sample of 2000 random values for the net heat inputs. Each time the REFRESH button is clicked a new sample of 2000 random values is chosen.

A few refreshes of the display will probably show many examples of the phenomenon of rapid growth and decrease. For more on this topic see Apparent Trends.

(To be continued.)