# 4-level model of tracking

Bill Powers wrote:

Hello, all --

Attached is the Vensim sketch for a 4-level model, starting with a
contractile element that generates a force, followed by a velocity control,
a position control, and finally a visual control of the distance between a
cursor and a target. Most of the equations are shown in red.

# Maximum of simulations/optimizations found at:   *delay = 0.00333438    Refdist = 7.92461    Kv = 1.57547    Kr = 6.6916    Kx = 1.1044    Kd = 7.61076    J = 0.0215284 Simulations = 1658 Optimizations = 5 Payoff = -636.836

The "payoff" is just a measure of goodness of fit. The RMS error between
the model's output and the real output from an experimental run is about
3.5 units, with the peak-to-peak range of movement being about 350 units.

A most interesting result is that the transport lag is only 3 milliseconds.
I believe this is because a second- or third-order system produces a lag
that behaves a great deal like a transport lag, much more so than a
first-order lag (leaky integrator) does. The response to a step input is an
initial acceleration to a maximum upslope, the velocity starting at zero
and only gradually becoming positive. That looks a lot like a shift of the
response in time -- a "reaction time."

Of course this probably means we will have to find some other way to
distinguish between transport lags and the double or triple integral type
of lag. Three milliseconds, I wager, is much too short for a loop involving
vision, although we must also consider that the traditional "200
millisecond" lag could well be an effect of multiple integrals in a
hierarchy of systems, and not really a true reaction time at all.

Also, the value of J (the moment of inertia about the shoulder joint in
this case) is probably way too low. Using so many parameters for optimizing
is not a good idea; as many as possible ought to be determined by
measurement and not included in the optimization. I have yet to learn how
to do a sensitivity analysis with Vensim, but when I do we can find out
just what the error bars are for each parameter.

I haven't heard from anyone yet about purchasing the Pro version of Vensim
or getting a break if no funds are available. Is anyone in this group
working with me, or are you all just smiling and nodding? You have to know
the joke about the comedian in the Chinese theater to get that one.

I'm happy doing my modeling in Excel, since I plan to stay on a Mac (as my
main computer) for some time. If Vensim runs on a Mac then I'd be happy to
get a copy but I don't like paying much for software. It just all seems very
complex and confusing at the moment. I am certainly following your modeling
with _great_ interest and I hope to see a draft of the paper on the
"Adaptation Illusion" very soon (to which I'll be happy to contribute; I'd
also be happy to make an excel version of your model (augmenting mine, I
suppose) so John can see how it works.

Best regards

Rick

···

---
Richard S. Marken, Ph.D.
The RAND Corporation
PO Box 2138
1700 Main Street
Santa Monica, CA 90407-2138
Tel: 310-393-0411 x7971
Fax: 310-451-7018
E-mail: rmarken@rand.org

Dear Bill and fellow PCTers,

I would appreciate going over the 4-level model.

First of all, what sort of physical setup did you use to generate the data?

Did you use a mouse or did you have some other kind device for controlling
the cursor?

I was struck by the fact that the four level model could be represented in a
Vensim diagram.

Also, you say that we should measure rather than estimate the parameters.
Can you give an example of how you would do this for one of the parameters.

The psychological interpretation of the parameters is a question I have. For
example, in the previous model, not this one, you said that the leakage
parameter might be related to the muscle tension at the shoulder. Would
measuring the EMG at some location be of value here?

By the way, I would appreciate receiving the Vensim program for the
four-level model. I would like to see the equations. Maybe that would help
me understand better.

Thanks,
David M. Goldstein, Ph.D.

···

----- Original Message -----
From: "Bruce B.Abbott" <abbott@ipfw.edu>
To: <powers_w@frontier.net>
<john.flach@wright.edu>
Sent: Wednesday, September 25, 2002 4:37 PM
Subject: Re: 4-level model of tracking

Hi Bill,

As you know, I do have a copy of the pro version of Vensim, thanks to Fred
Nichols. I've been following along, too, between other activities that are
taking up too much of my time.

Bruce A.

-----Original Message-----
From: Bill Powers <powers_w@frontier.net>
kurtzer@BINAH.CC.BRANDEIS.EDU
Date: Tue, 24 Sep 2002 16:24:51 -0600
Subject: 4-level model of tracking

Hello, all --

I haven't heard from anyone yet about purchasing the Pro version of Vensim
or getting a break if no funds are available. Is anyone in this group
working with me, or are you all just smiling and nodding? You have to know
the joke about the comedian in the Chinese theater to get that one.

Best,

Bill P.