Recurrence Analysis

The second method of nonlinear analysis examined in the feasibility study was recurrence analysis (RA). Eckmann, Kamphorst and Ruelle (1987) first introduced the recurrence plot as a graphical tool to locate hidden rhythms, patterns and nonstationarities in experimental data. Recurrence analysis is the quantification of features of recurrence plots. Such features include correlation among the data points in multidimensional embedding space, percent determinism in the correlations, entropy, and an estimate of the Lyapunov exponent that measures the sensitive dependence to initial conditions. More information on RA as well as excellent freeware can be found at http://home.netcome.com/~ eugenek/download.html

When applied to physiological data RA is shown to detect shifts in physiological states, such as onset of fatigue and anesthesia-induced respiratory changes, earlier than competing methods (Webber & Zbilut, 1994). The successful use of RA in a social science context may provide information on shifts in dynamics in the process (teen births) that precede changes in the mean or variance of the series. Such an early indicator of state shift in the system might allow for more sensitive evaluation of the effects of policy and intervention and might also be useful for targeting the timing of interventions to decrease births to teens.

In the color version of RA shown in Figure 5 a point in the plot is color-coded. Vectors in the series correlated within the same tolerance are represented in the matrix with the same color. The x and y axes represent starting points of vectors formed from the series. On the diagonal, points in the series are perfectly correlated with themselves. Points off-diagonal indicate the presence or absence of correlation of each vector with other vectors in the series. Points that are white or red are highly correlated; those that are blue or black are less strongly correlated. Just as with a linear correlation matrix, the information contained in the graph is a mirror image above and below the diagonal.

Figure 5. Recurrence plot for births to teens from 1964 –1998 (On the color bar to the right of the graph 0.00 means no distance between vectors [perfect correlation] and 123.00 refers to maximum distance between vectors [weakest correlation].)

The plot shows changes in correlation in the time series, some roughly coinciding with changes in policy influencing reproductive behavior of teens. Notably, the period from 1964 to 1970 (points 1 to 2556 on the x and y axes) seems to be one of strong correlation in the series (more white points in the matrix). In 1970 (point 2556) Title X legislation made contraceptives available at all federally funded clinics. In a reversal of policy, The Adolescent Family Life Act or Chastity Bill was passed in 1981 (point 6574). There is insufficient evidence to claim causal relationships between the policies noted and the RA results but these findings do warrant further investigation. The findings are consistent with the evidence of a renormalization group process mentioned earlier. Across the time period social systems and agents appear to have been adapting to changes within their environment. Atwood and Kasindorf (1992) describe the complexity of adolescent behavior this way, "…multiple inconsistent inputs lead to inconsistent responses by many adolescents" (p. 355). It remains to be seen whether linear analysis of teens’ responses to inconsistent policies across time can capture the complex patterns that emerged in RA. Currently, investigation is underway to further explore specific quantities derived from RA and to determine their sensitivity to policy changes.

Another use of RA is the comparison of data sets collected over identical time periods. The data could be from geographic regions such as the public health regions mentioned above or from demographic groups such as Hispanic and Black and White teens. Differences in correlation among the data points, percent determinism in the correlations, entropy, and Lyapunov exponent could then be determined and linked to sociodemographic factors accounting for those differences.