Scaling of Relative Dispersion

The two nonlinear analysis approaches being examined in Phase Two include using relative dispersion to identify fractal scaling and recurrence analysis to detect changes in stationarity and memory within the time series. Both methods are thought to be appropriate for application to time series of relatively short length and time series that are nonstationary.

When a process or object does not have a characteristic scale it is said to obey a scaling law. When this phenomenon is a time series, it is believed to be fractal in time (West, Hamilton and West, 1999). Thus, the scaling relation ties together what happens at the shortest time scale with what happens at the longest time scale and the underlying stochastic phenomenon is said to have a long-term memory. We call a phenomenon that obeys this type of scaling a fractal random process (FRP). Any social process described by an FRP is not simple but consists of social agents and organizations that interact across a variety of time scales.

The team utilized a technique that reveals the degree of variability observed in the time series as a function of the unit time used in the analysis. The research team developed the software used in this analysis. Using all the data we first calculate the relative dispersion (RD) of the time series

(1)

RD=SD/Mean     

where SD is the standard deviation. Next, we aggregate each two nearest neighboring data points to form a sequence of length N/2. Once again we calculate RD and then aggregate the three nearest neighbors and continue the process of aggregation of r nearest neighbors. The value of r is dependent on the size of the data set. As the number of aggregates approaches N, the statistics become unstable.

If the time series is a simple fractal process, the RD of the point process scales as

(2)                                                                         

                                                                                RD (r) a rb .

The relations among the power law index B, is the Hurst exponent (H) and the fractal dimension (D) are given by

(3)

                                                                                D = 2 – H, and

(4)                                                                           

                                                                                b = H – 1.

For a complete discussion of the methods and additional results see West, Hamilton and West, 1999. Presented here are new results in which eleven public health regions in Texas were examined and a FRP was identified for each. These results suggest that scaling exists in all regions. These findings show that the values in the series are temporally clustered and have long term memory. The Hurst exponent (H) has promise as a means of quantifying the degree of tenacity of the process. By tenacity we mean the tendency of the values in the time series to demonstrate a trend (either increasing or decreasing) over short time scales. Of course a trend over a long time scale would also be detected by the calculation of the Hurst exponent but its value is primarily in measuring the more subtle short-term tendencies in the data. The Hurst exponent is .5 for uncorrelated noise, >.5 for a persistent process and <.5 for an anti-persistent process. An anti-persistent process is one in which the values in the series are negatively correlated over short time scales. In other words, if a value is high at time step 1 it will be low at time step 2, deterministically.

Elsewhere we argue that these results are indicative of a renormalization group process (West, Hamilton and West, 1999). A renormalization group process is one based on contingencies - everything depends on everything else. There is no characteristic scale; the process has feedback across all time scales (days, weeks, months, years and decades). Such a system certainly applies to the phenomenon of teen births. This phenomenon represents a process in which coupled systems monitor internal and external environmental responses to their behavior and accommodate themselves to information about the behavior of other systems.

RD was carried out on data from the eleven public health regions of Texas. All regions would be classified as persistent in their time series as the Hurst values all were greater than .5. However, there is interesting variance of H among the regions. Results from region one (the Texas Panhandle) and region eleven (the Rio Grande Valley) are shown in Figure 4.

                Region 1 B= -.33, H=.67, FD= 1.33                     Region 11 B= -.14, H=.86, FD=1.14

Figure 4. RD for two public health regions in Texas.

The investigators plan a systematic examination of sociodemographic factors that may be associated with the differences in H values across the state. Of course, the fractal dimension (D) or the scaling parameter (b ) also could be used as we have shown in (3) and (4) above the relationship among them. However, the notion of persistence has intuitive significance for individuals developing policy and evaluating interventions and is currently utilized in other social contexts such as economics and stock pricing (May, 1999).