Example

As an example, let's examine the Karnaugh map for our subject. The variables chosen are BFAM, BEMOT and BSTRE. It is shown on figure 4.17 ^4.15.

Figure 4.17: Karnaugh map for BFAM, BEMOT and BSTRE variables

At first the visualization of such figure leads more to confusion than understanding. We mainly see that all configuration states were encountered, except one (configuration#3523#> #010, $BFAM=0$, $BEMOT=1$ and $STRE=0$). More interesting, transitions seem to appear from each 7 configural states towards all other 7. This implies a diversified period for this subject.

Plotting all configuration#3524#> transitions is probably a waste of time - and ink. In almost all cases for which we plotted such graphic, a blurred picture appears. There is a better way to synthesize the information#3526#> about the dynamics without overwhelming or confusing investigators and readers of such map.

A simple, efficient proposition is to consider only the MFTs, the most frequent transitions. As explained before, instead of taking into account all transitions from each configuration#3527#> state, only the transition with the highest probability is retained and plotted. This solution allows to retain the most "predictive" state given each state. It represents what the system is most likely to be found if we know in which state it is in. Figure 4.18 ^4.16 represents the MFTs of the last Karnaugh map.

Figure 4.18: Karnaugh map of MFTs for BFAM, BEMOT and BSTRE variables

From this simplified Karnaugh map#3530#> emerges a much more interesting picture. The apparent fuzziness of the dynamics was produced by close to irrelevant transitions. There is an order, an organization behind the sequence of observations. The Karnaugh map#3531#> of most frequent transitions reveals that this subject spent most of her time in a "comfortable" configuration, #100, exhibiting a familiar situation (BFAM=1), low emotionality (BEMOT=0) and low stress (BSTRE=0). Considering transitions from each configural state, we see that although she encountered other kinds of experiences than this easy #100 configuration, she (almost) always would return to that configural state the next moment (2 hours later, the interval between each self-observation).

But there are two exceptions to this "dynamics rule": from state #010 she mostly went through state #110, implying she would first try to find a familiar situation, before reducing her stress (indicating an external attributional style?); the other exception is for state #111, from which she equally stayed in that same state than transited to the #100, easy configuration.

Considering the description given above configuration#3536#> #100 may well receive a particular name: an attractor . Complex dynamical systems theory describes an attractor "a state towards which a system returns to, even after some perturbations" [HakenHaken1983]. The metaphor of a ball in a bowl is illustrative of the type of attractor we encounter in our data. Throw the ball in the bowl, it will roll and move around for a while, and sooner or later it will gradually slow down and immobilize at the bottom. Shake the bowl a little, the ball will move but return to the bottom. The bottom of the bowl is the attractor of the system.

The same happens here with our subject. The attractor configuration#3537#> is the familiar situation with low stress and emotionality. Bring some (external) perturbations, source of potential stress and emotions, the person will momentarily experience something else, but through diverse coping strategies she will quickly return to this central configural state.

As an example of a quadrivariate dynamical Karnaugh map, a fourth variable is added to the preceding graphics: PRESS. The dynamical Karnaugh map of the most frequent transitions is shown on figure 4.19 ^4.17. The resulting picture makes it clear that the configuration #1000, characterized by a familiar situation (BFAM=1), low environmental pressure (BPRESS=0), low emotionality (BEMOT=0) and low stress (BSTRE=0). The subject really tended to stay in this configuration or return to it if changes occurred.

Figure 4.19: Karnaugh map for BFAM, BPRESS, BEMOT and BSTRE variables (subject: FA7)

This same kind of affairs is found for all of our subjects; they all return to this same configuration #1000. As an example, we show on figures 4.20 ^4.18 and 4.21 ^4.19 the dynamical Karnaugh map of the most frequent transitions for two subjects. A few transitions do not go directly to the #1000 attractor, but they occurred from very infrequent (n=2) configurations.

Figure 4.20: Karnaugh map for BFAM, BPRESS, BEMOT and BSTRE variables (subject: FG9)

Figure 4.21: Karnaugh map for BFAM, BPRESS, BEMOT and BSTRE variables (subject: XR3)

 ^4.20

A last example will show a Karnaugh map of a random process (for each of the four variables 261 values were generated by a binomial random process of mean 0.5). The resulting Karnaugh map of MFTs is shown on figure 4.22 ^4.21. As expected no particular patterns of transition emerges from the map. It is surely not a proof that the other subjects are truly self-organized, by it nevertheless constitutes an interesting comparison.

Figure 4.22: Karnaugh map for BFAM, BPRESS, BEMOT and BSTRE variables (subject: Random)