[From Rick Marken (930104.1800)]
Greg Williams (920103 - 2)--
It will not amaze those who think PCTers have identified incorrect "stimuli"
and "responses."
In a compensatory tracking task all the subject can see is the time
trace of the cursor -- c(t). In the "different stimuli/same responses"
experiment there are two "stimuli" -- c1(t) and c2(t); the output is
the traces, o1(t) and o2(t), which are the same to c1(t) and c2(t),
even though the stimulus traces are different each time. If the
outputs were proportional to some non-linear function of the inputs
then c1(t) and c2(t) would be the same -- we would not know the nature of
the function, just that some function would work; but, since c1(t)<>c2(t)
there is no need to look for such a function. The same is true for the
possibility that the output is based on a delay with respect to
c(t) -- for example, o(t) = f(o(t-tau)); but this possibility is
rejected by the result as well -- since the same delay would
characterize the response to both c1(t) and c2(t) so c1(t) should
equal c2(t) -- but they don't. So the "different stimuli/same
responses" experiment shows that the lack of correlation between
cursor (c(t)) and output (o(t)) in Bill's experiment is NOT because
o(t) = f(c(t)) but f is highly non-linear and it is not because
o(t) = f(c(t-tau)).
The experiment does NOT rule out the possibility that there may be
a relationship between derivatives or integrals of c1(t) and c2(t)
and, indeed, if you compute the indefinite integral of the cursor
traces -- call it int(c(t)) -- you DO find a correlation between
the integrals -- and there IS a high correlation between
int(c1(t)) and o1(t), for example. So now the clever nonPCTer
can get excited and say -- AH HA!! int(c1(t)) IS THE STIMULUS
that guides responses in a tracking task -- the INPUT-OUTPUT
MODEL IS SAVED!!!
This is where quantitative modelling is needed again (one little
demo can't shut the non-PCTer up forever -- if at all). If int(c(t))
is the stimulus for tracking then we should be able to build a
model using int(c(t)) as the stimulus. I did this -- I have not been
able to make the model work. Maybe I havn't tried hard enough. The
model I used is as follows
c(t) = o(t) + d (t) ; cursor position at any instant depends on
the output (handle position) and the disturbance
(this is just a physical fact).
p (t) = int (c(t)) ; the stimulus (perception) that causes the response
is the indefinite integral of cursor position
(where cursor position is measured relative to
the target, which is 0).
o (t) = -k* p(t) ; the input-output equation; handle output is
proportional to perceptual input (k is negative
because the correlation between int(c(t)) and
o(t) is negative -- because it's a negative
feedback loop.
One can diddle with k to try to make the model work. I couldn't make
it work. I also tried making the integration "leaky"; this didn't
work either. I would prefer an analytic proof that the integral of
cursor position cannot be the "stimulus" for control -- but I can't
do it; this would be a nice job for a real mathematician.
So why does int(c(t)) correlate so highly with o(t); because the output
of the control system is int(c(t)) -- generated in a closed loop; the
integral is picking up the component of c(t) that is the result of o(t) --
remember c(t) = o(t) + d(t) -- and in a control loop o(t) is the result
of an integration of the error signal (I think this is the explanation,
anyway).
Where is the person who is claiming that there is ONE function which maps all
of the different i's to the same o? They will say that each i has a different
function mapping to the same o, which is perfectly possible, mathematically.
Then I hope they will also say HOW the system knows which function to
pick each time in order to map the different c(t)'s into the same o(t).
It looks as though they might contest PCTers' claims QUANTITATIVELY, too. It
all hinges on their claim that your i and o are straw variables.
I think it all hinges on what they mean by QUANTITATIVELY.
Best
Rick