From: "Marken, Richard S." <Marken@COURIER4.AERO.ORG>
Subject: Hierarchical control research
To: Multiple recipients of list CSG-L <CSG-L%UIUCVMD.bitnet@vm42.cso.uiuc.edu>
Status: RO
[From Rick Marken (940202.1330)]
The following is a LONG discussion of a little research I've
been doing on hierarchical control. Non-PCT-fanatics (ie. normal
human beings) might want to skip this, though I would really
appreciate ANY comments, questions and, especially, suggestions.
---------
I have just completed some preliminary experiments (written in
Hypercard-- I hate myself for doing it but it's sooo easy) that
suggest a potentially nice new way of demonstrating hierarchical
control in action. The basic experimental setup can be diagrammed
as follows:
dt-->t
dc-->c<--m
There is a display with a moving target (t) above a movable cursor (c).
Target position at any instant is determined by the current value
of a number series called dt (disturbance to target); cursor position
at any instant is determined by the sum of two values 1) the current
value of a number series called dc (disturbance to the cursor) and the
current output of the mouse port, m (which is the integral of mouse
movement on a surface).
The subject is asked to keep the cursor, c, aligned with with the
target, t. This is a simple example of a combined pursuit and
compensatory tracking task. Pursuit tracking is involved in keeping
c aligned with t despite changes in t (this is like keeping the image
of the hood of your car in its lane on a winding road). Compensatory
tracking is involved in keeping c in its intended position despite
disturbances to its position (this is like keeping the car in its
intended position despite invisible disturbances such cross winds).
Long ago I thought that this task might provide a basis for revealing two
levels of control: one level of control handles the "pursuit" aspect
of the task, controlling the difference between t and c (so the
controlled variable is t-c); the other (lower) level of control just
controls c. It is possible to build a hierarchical control model that
will perform this task: the higher level t-c controller acts by
setting the reference for the c controller. But I could find no evidence
in the data that two levels of control were actually involved in this
task: all aspects of the data were handled by a simple, single level
control system controlling t-c (by varying the output variable, m,
directly).
A post from Martin Taylor suggested to me a new way to approach this
problem. Martin (for reasons other than revealing hierarchical control)
suggested doing a combined pursuit/compensatory task where in one condition
target movement was predictable and in another not. I knew that pursuit
tracking would be better with the predictable; I suspected that the
compensatory aspect of the tracking would remain unchanged. If this
were the case, it would reveal two levels of control very clearly;
control of both the "pursuit" variable (the "pattern" of target
motion) and the compensatory variable (t-c) could be observed
simultaneously AND control of the compensatory variable could be seen
as the MEANS of controlling the pursuit variable.
The basic "levels revealing" experiment is pretty simple. The subject
does the tracking task as usual (trying to keep c = t). In one
condition of the experiment, dt is a smoothed random series (the
usual PCT kind); in the other condition , dt is a sine wave (of
about the same frequency as the center frequency of the random
series-- about .3Hz). In both conditions, the disturbance to the
cursor, dc, is exactly the same.
My approach to analyzing the results was based on measuring
control of t-c and c in the two conditions using the "stability
factor". I would have used RMS error but I realized that my
selection of the reference level for c would be arbitrary.
The stability factor measures control in terms of the ratio
of expected to observed variance in a hypothetical controlled
variable. The stability factor for the "pursuit" variable
(t-c) was computed as follows:
S = [var(dt)+var(c)]/var(t-c)
The numerator is the "expected variance" of t-c if there is
NO control. The expected variance would be proportional to
the sum of the variances of dt -- var(dt) -- and c-- var(c).
If there is control the actual, observed variance of t-c --
var(t-c) -- should be much smaller than the expected variance,
and S should be much greater than 1. If there is no control,
the expected and actual variance of t-c will be the same and S=1.
The stability factor for the "compensatory" variable (c)
was computed as follows:
S = [var(dc)+var(m)]/var(c)
Again, the expected variance of the cursor is proportional
to the sum of the variances of the variables that affect it,
in this case dc and m. If there is control, the observed var(c)
will be less than expected and S will be greater than 1.
What's nice about this stability calculation is that is does
not depend on estimates of reference levels. In fact, the
reference level for c was shifting a great deal during the
experiment but the stability factor showed that c was under
control.
Here are some representative results with yours truly as subject:
Type of Target Movement (dt)
Random Sine Wave
Pursuit 6.88 15.8
Comp 3.7 2.1
The numbers are stability factors (S) indicating the ability
to control the pursuit (t-c) and compensatory (c) variables
when target movement was random vs predictable (sine).
Very nearly the same S values were obtained every time I
did the experiment. The interesting result is that, while pursuit
control (the ability to control t-c) improves substantially
(it more than doubles, going from 6.88 to 15.8) when the
target is predictable (a well known fact), compensatory control
(control of the position of the cursor relative to a changing
reference level) remains the same (or even declines). In fact,
the apparent decline in compensatory control is an artifact
of the difference in the variance of the reference level in the
random and sine wave target movement conditions. I found this
out by running a single level control model (with parameters
adjusted for best fit) in both conditions. The single level model
controlled only t-c and it did so by varying m using a pure
integrator in the output. The computer version of the "system
equation" for the model was:
m := m + g* (t-c)
where t-c is the perceptual signal, the reference signal value is
implicitly zero and m is the simulated mouse output.
The results of running this model under the same conditions as
the subject (same dt and dc) were are follows:
Type of Target Movement (dt)
Random Sine Wave
Pursuit 6.67 6.26
Comp 3.78 2.5
Note that the model's ability to control the pursuit variable
is exactly the same with both random and sine disturbances. This
is expected since the model has no way to take advantage of the
predictability of the sine disturbance. But note that the model's
ability to control the compensatory variable is almost exactly
the same as the subject's IN BOTH CONDITIONS. This is strong
evidence that we are looking at two levels of control when the
SUBJECT is controlling with the predictable target.
Stronger evidence comes from my preliminary attempts to develop
a two level model of this task. The lowest level is almost
identical to the single level of the first model but the reference
signal is now the predicted position of the target, t', rather than
the target itseld (as it was, functionally, in the first model). I
haven't really modelled the second level system yet (the one that
produces the predicted target positions -- which are reference inputs
to the lower level system). I got the predicted target positions, t',
for the model by sampling ahead in the sine distrubance table.
The model that generated stability factors closest to the one's
observed sampled the equivalent of [ ] ahead. [ Martin: If you are
reading this, can you intuit, based on IT, how far ahead the model had to
predict the since disturbance in order to match the subject's
performance?]. Here are the results for this "two level" model:
Type of Target Movement (dt)
Random Sine Wave
Pursuit 6.67 15.85
Comp 3.78 2.4
(The one level model was used again to get the data in the
random condition).
There is actually a third level of control evident in the
behavior in this experiment. It is the level that notices
that the target has become predictable. The subject is clearly
not using predictive control with the random target -- there is
nothing to predict. The third level system "switches" in the second
level system that let's the purpuit control system control (t'-c)
rather than (t-c). I became personally aware of this third
level while I was blithly testing my program. At one point, though
inattention, I failed to notice the change from random to predictable
target (marked only by the posting of data from the end of the run).
So I just kept tracking the target position as though I did not
know here it was going to be at the next instant; in other words,
I kept tracking the sine wave target as though it was still the
random target. When the program printed out the results I was
alarmed becuase it shows that my ability to control with the
predictable target was EXACTLY the same as my ability to control
with the random target. At first, I though that perhaps results
like those shown in the first table above did not occur reliably. In
fact, the results only occur reliably if the subject is actually
controlling a higher level variable (t'-c) rather than (t-c) in the
predictable target condition. I loved it; my little lapse showed (once
again) that it is the subject, not the "stimulus situation", who controls
what happens in this experiment. And with PCT you can tell exactly what the
subject is controlling.
There is still much more detailed model testing to do -- and I
have a nice idea for some future experiments -- but I feel like I've
nudged the door to hierarchical control open just a crack -- maybe.
Best
Rick
Thought is the thought of thought -- James Joyce, "Ulysses"
