Inverted T Ratio Analysis, again

(From Bruce Abbott [950221.1120 EST])

Rick Marken (950220.1900)

My model is the same as yours except for this peculiar method of computing
model output:

h = h - k * (e*horiz);

Why multiply the error by horiz? What's horiz anyway? I don't think this
was the reason you had problems distinguishing the models, though.

Somehow you are not communicating to me what steps your model takes, with
each variable defined. I tried what I think you (and Bill) suggest is the
proper model and it didn't seem to work (see my post 950220.1540 EST). I
have a feeling that my version and yours are mathematically equivalent.
Horiz is the length of the horizontal line. (In my version the vertical
line is controlled.) H (for "handle") is actually the model's vertical line
length. Because e is an error in the RATIO, you have to multiply by horiz
to convert this to an error in vertical line length. How are you doing this
differently? As a guess, I'd say that you are converting your ratio
reference to an X line length and then just working with X. But if this is
correct, you never made that explicit, or at least I didn't pick up on it.

By the way, who said I had any trouble distinguishing the models? My
results agree with yours in that the model/actual correlation and RMS were
both better for the V/H model, and I said that. I was only suggesting that
comparing correlations is probably not the best way to evaluate the relative
merits of these two models.

If you'd like to try an interesting variation on this task, move the
horizontal line upward on the screen so that is bisects the vertical
one.

Before we do this, I would rather see how changes in r affect the
relative fits of the two model. Just try runs where you ask people (or
yourself) to "keep x equal to y", "keep x a constant amount shorter
than y", "keep x a constant amount longer than y" and then compare
the x-y and x/y model fits in each case. Doing this requires no change in
the program -- just in the subject.

Sounds interesting. I'm not surprised that you had more trouble trying to
maintain a constant offset. How does this change the regression equation
for X on Y? If you were successful in adding a constant offset to your
performance, this should change the intercept without much affecting the
slope. Because the X/Y model does not take the offset into account, the
model/actual RMS error would be expected to suffer (assuming use of the
optimal reference value based on the data). If you merely changed the ratio
reference, the slope of the regression of X on Y should change. I'll give
it a try and report my results.

Regards,

Bruce

[From Rick Marken (950221.1100)]

Bruce Abbott (950221.1120 EST])--

In my version the vertical line is controlled.

I think you mean that vertical line length is INFLUENCED by the subject's
mouse movements; the goal of this whole exercise is to figure out what
perceptual variable(s) is (are) CONTROLLED, right?

Because e is an error in the RATIO, you have to multiply by horiz to convert
this to an error in vertical line length.

e is a measure of the difference between two neural signals, p and r.
Variations in p represent variations in a ratio IF the perceptual
function computes the ratio of its inputs, ie. if p = x/y. But p could
also represent variations in some other variable if the perceptual function
computes a different function of its inputs, like, p = x-y or
p = sqrt(x^2+y^2)/ y or p = cos(x+ y) etc.

r and p are just scalar variables; their "meaning" is given to them by the
perceptual function. A properly designed control system keeps p = r; that's
it. What the system is doing from the point of view of an outside observer is
controlling the function of variables computed by the perceptual function but
what the system is "really" doing is just keeping one number (the perceptual
signal) equal to another (the reference signal).

How are you doing this differently? As a guess, I'd say that you are
converting your ratio reference to an X line length and then just working
with X. But if this is correct, you never made that explicit, or at least I
didn't pick up on it.

I think you are making the mistake of thinking that the variables in a
control loop differ in some way depending on what they represent. In fact, p,
r, and e are always just scalar variables; what they "mean" in terms of what
the control loop "does" is determined by the perceptual function.

Maybe I can make this clear by showing you my model: it's VERY simple:

1. p = x/y or p=x-y

The first step on each iteration is to compute the value of the perceptual
signal. In one model, the perceptual signal is proportional to the ratio of
line lengths (x and y); in the other model p is proportional to the
difference between line lengths.

2. h = h + k (r-p)

The second step is to compute the output. The value of k must be adjusted to
stabilize the behavior of the control loop. This value depends on what
function of the environmental variables (x and y) is being controlled; the
value of k was .5 when the model was controlling x-y; it was 10 when the
model was controlling x/y. The value of r can be ANY number; if r is 1 and
the model is controlling x-y then p will be 1 and there will be a 1 unit
differnece maintained between x and y. If r is 1 and the model is controlling
x/y then p will still be kept at 1 but now this means that the ratio of x to
y will be kept equal to 1 meaning that x will be equal in length to y. I used
different reference values for the two models so that the models would keep
the environmental correlates if the controlled percpetion (x-y or x/y) at
values that matched (as closely as possible) the corresponding values
maintained by the subject (and observed in the subject's data).

3. x = h

One input to the perceptual function (x in my case) is a direct function
of the output. This closes the control loop and we go back to one and
computer the perceptual variable.

(By the way, remember that what is a sequential process on a computer is not
sequential in a real control loop; all variables, p, h and x are changing
at the same time. The non-sequentiality of control is simulated MUCH more
accurately in Wolfgang Zocher's sublime SimCon program).

By the way, who said I had any trouble distinguishing the models?

I meant that the difference between the models was VERY small. I like to see
more dramatic differences and I also like to see the successful model
accounting for 99% of the variance before I think I know what's going on.

As I play with this horizontal/vertical task I realize that it is probably
possible to learn to control either x-y or x/y at will. There are also other
variables that I can control; I can control the shape of the triangle
formed by the ends of the x line and the tip of the y line. I can control the
angle made by the imaginary line running from the tip of one end of the x
line to the free end of the y line. I think it might be interesting to see
whether we can tell which aspect of the display any particular subject is
controlling at any time.

Best

Rick