# Rubber Band Variation - Reposted

[From Fred Nickols (2007.02.26.1100 EST)] --

Oops. I forgot to date-time stamp my post. So, here it is with the stamp.

I anticipate doing another rubber band demo in the not too distance future
and I am planning on doing a variation of it. I'd be curious about others'
reactions to what I plan and I need some help with words.

In the past, I have used two rubber bands tied to a small metal washer. I
place several dots on the easel sheet and ask my subject/partner to pick one
dot and to take on the goal of keeping the washer positioned over the
selected dot - without informing me which dot has been selected. I then
move my rubber band and, of course, the washer moves and my subject/partner
compensates for or counters the "disturbance."

Next time, I'm going to suspend the easel sheet itself by tape and rubber
bands. Then, after doing the regular demo in which I move my rubber band, I
intend to hold my rubber band stationary and move the easel sheet itself.

My aim here is to show that the first instance is an example of a
disturbance to the position of the washer whereas the second illustrates a
disturbance to the position of the dot (which is like the cursor tracking
exercise).

I have the sense that this is an important distinction in terms of the
nature of the "disturbance" but I'm struggling for words to express it. My
initial intention was to use the second instance as an example of tracking a
moving target instead of simply countering interfering factors. And, of
course, I could do a third variation in which I move the easel sheet and my
rubber band.

Help with words, please. In PCT terms, what am I doing with these three
demos?

1) leaving the dot stationary and moving the rubber band
2) leaving the rubber band stationary and moving the dot
3) moving both the rubber band and the dot

Maybe all it illustrates is an increasing degree of complexity but I'd be
glad to hear from anyone who wishes to comment.

Regards,

Fred Nickols
www.nickols.us
nickols@att.net

[From Rick Marken (2007.02.26.0910)]

I think this is a great idea. I think what you are showing is that there are various ways to disturb (influence the state of) the variable being controlled, which is something like the perceived distance between washer and dot. You can disturb this variable by pulling on a rubber band, which moves the washer, or you can disturb the same variable by moving the sheet with the dot. You can show that this variable (perceived distance between washer and dot) can also be disturbed by the subject’s own movements relative to the washer and board (due to parallax). So if the subject moves to the side he or she will have to move their end of the rubber band to return the perceived distance to the reference state. This latter disturbance (change in the subject’s position relative to the washer/dot) shows that it is clearly not the objective distance between washer and dot that is controlled but, rather, the distance as perceived by the subject.

I think this is a brilliant demonstration of control of perception (in all cases using the same means – pulling on the rubber band). I wish I had thought of it for the class;-)

Best

Rick

···

Fred Nickols (2007.02.26.1100 EST)–

Oops. I forgot to date-time stamp my post. So, here it is with the stamp.

I anticipate doing another rubber band demo in the not too distance future
and I am planning on doing a variation of it. I’d be curious about others’
reactions to what I plan and I need some help with words.

In the past, I have used two rubber bands tied to a small metal washer. I
place several dots on the easel sheet and ask my subject/partner to pick one
dot and to take on the goal of keeping the washer positioned over the

selected dot - without informing me which dot has been selected. I then
move my rubber band and, of course, the washer moves and my subject/partner
compensates for or counters the “disturbance.”

Next time, I’m going to suspend the easel sheet itself by tape and rubber
bands. Then, after doing the regular demo in which I move my rubber band, I
intend to hold my rubber band stationary and move the easel sheet itself.

My aim here is to show that the first instance is an example of a
disturbance to the position of the washer whereas the second illustrates a
disturbance to the position of the dot (which is like the cursor tracking

exercise).

I have the sense that this is an important distinction in terms of the
nature of the “disturbance” but I’m struggling for words to express it. My
initial intention was to use the second instance as an example of tracking a

moving target instead of simply countering interfering factors. And, of
course, I could do a third variation in which I move the easel sheet and my
rubber band.

Help with words, please. In PCT terms, what am I doing with these three

demos?

1. leaving the dot stationary and moving the rubber band
2. leaving the rubber band stationary and moving the dot
3. moving both the rubber band and the dot

Maybe all it illustrates is an increasing degree of complexity but I’d be

glad to hear from anyone who wishes to comment.

Regards,

Fred Nickols
www.nickols.us
nickols@att.net

Richard S. Marken
rsmarken@gmail.com

[Martin Taylor 2007.02.26.14.10]

[From Fred Nickols (2007.02.26.1100 EST)] --

Help with words, please. In PCT terms, what am I doing with these three
demos?

1) leaving the dot stationary and moving the rubber band
2) leaving the rubber band stationary and moving the dot
3) moving both the rubber band and the dot

1. Compensatory tracking. The washer is like the cursor on a screen based tracking task.
2. Pursuit tracking. The washer is still like the cursor, chasing a target moving on the screen.
3. Mixed. I don't know if there's a word, but in real life it's probably the most common case.

Martin

[From Erling Jorgensen (2007.02.26 1600 EST)]

Fred Nickols (2007.02.26.1100 EST) --

Martin Taylor 2007.02.26.14.10

Help with words, please. In PCT terms, what am I doing with these three
demos?

1) leaving the dot stationary and moving the rubber band
2) leaving the rubber band stationary and moving the dot
3) moving both the rubber band and the dot

1. Compensatory tracking. The washer is like the cursor on a screen
2. Pursuit tracking. The washer is still like the cursor, chasing a
target moving on the screen.
3. Mixed. I don't know if there's a word, but in real life it's
probably the most common case.

A follow-up demo occurs to me, once the observers get the idea of
the tracking, which might bring out the hierarchical nature of the
control tasks, & their time relationships.

You could ask the subject/partner to choose three or four of the dots
on the easel. Then to spend 15 seconds with the washer over each one
in sequence. If you asked them to consider three or four target dots
that would form a triangle or rectangle, then you get the possibility
of three layers of control --

a) Compensatory tracking of a disturbed washer over any given dot.
b) Pursuit tracking of a changing reference standard, of a series of
dots in sequence. The sequence of first one target, then the next,
occurs at a slower rate of speed (and a slower rate of observation)
than the washer-over-dot relationship.
c) Higher level tracking of forming a virtual triangle or rectangle,
by the sequence of target dots. This perception is being controlled
at an even slower rate of speed than moving from one target dot to
another. I believe a way to bring out that time relationship would be
to ask the observers how long it took them by observation to realize,
first that there was a changing series of target dots, and second that
the full series of target dots formed a triangle or rectangle. The
rate of recognition, I believe, ought to roughly correlate with the
layer of control, with higher levels being controlled (and recognized)
at a slower rate.

All the best,
Erling

[From Bill Powers (2007.02.26.1440 MST)]

Martin Taylor 2007.02.26.14.10 –

I concur with your reply to Fred Nickols. Also, good suggestions from
Erling and Rick,

I’ve been thinking about your statement that a correlation is the same as
the cosine of the angle between two vectors. I still can’t figure out how
that works.

Suppose we just have two vectors in 2-space, extending from the origin

The two vectors form a triangle with sides s1, s2, and s3. The length of
s1 is, for example,

s1 = sqrt(x1^2 + y1^2).

The cosine of the angle A between the vectors is

``````     s1^2 + s2^2 -
``````

s3^2

cos(A) = ---------------------

2(s1*s2)

Now consider the formula for the correlation between s1 and s2, the
lengths of the two vectors (S stands for Sigma, the summation
sign):.

r = S(s1s2) - N(s1Avgs2Avg)

···

to: x1,y1 and x2,y2.

``````-------------------------
``````

NSigX*SigY

where SigX is

``````        S(s1 -
``````

s1Avg)

SigX = sqrt(------------)

N

and SigY is similar.

Are you saying that after all manipulations are done, we will find that r
= Cos(A) regardless of the values of s1 and s2? If so, I’d like to see
the derivation.

Best,

Bill P.

Re: Rubber Band Variation -
Reposted
[Martin Taylor 2007.02.26.22.00]

[From Bill Powers (2007.02.26.1440
MST)]

Martin Taylor 2007.02.26.14.10 –

I concur with your reply to Fred Nickols. Also, good suggestions from
Erling and Rick,
that a correlation is the same as the cosine of the angle between two
vectors. I still can’t figure out how that works.

Are you saying that after all
manipulations are done, we will find that r = Cos(A) regardless of the
values of s1 and s2? If so, I’d like to see the
derivation.

We have been talking about waveforms with a zero dc component, so
it applies.

Martin

Message
[From David Goldstein (2007.02.27.0254 EST)]

Hello Bill and listmates,

Take a look at the following url for a derivation;

http://www.mega.nu/ampp/rummel/uc.htm#C5

It is part of a chapter on the topic of correlation coefficient.

David

···

-----Original Message-----
From: Control Systems Group Network (CSGnet) [mailto:CSGNET@LISTSERV.UIUC.EDU] ** On Behalf Of** Bill Powers
Sent: Monday, February 26, 2007 6:13 PM
To: CSGNET@LISTSERV.UIUC.EDU
Subject: Re: Rubber Band Variation - Reposted

[From Bill Powers (2007.02.26.1440 MST)]

Martin Taylor 2007.02.26.14.10 –

I concur with your reply to Fred Nickols. Also, good suggestions from Erling and Rick,

I’ve been thinking about your statement that a correlation is the same as the cosine of the angle between two vectors. I still can’t figure out how that works.

Suppose we just have two vectors in 2-space, extending from the origin to: x1,y1 and x2,y2.

The two vectors form a triangle with sides s1, s2, and s3. The length of s1 is, for example,

s1 = sqrt(x1^2 + y1^2).

The cosine of the angle A between the vectors is

``````       s1^2 + s2^2 - s3^2
``````

cos(A) = ---------------------
2(s1*s2)

Now consider the formula for the correlation between s1 and s2, the lengths of the two vectors (S stands for Sigma, the summation sign):.

r = S(s1s2) - N(s1Avgs2Avg)
-------------------------
NSigX*SigY

where SigX is

``````          S(s1 - s1Avg)
``````

SigX = sqrt(------------)
N

and SigY is similar.

Are you saying that after all manipulations are done, we will find that r = Cos(A) regardless of the values of s1 and s2? If so, I’d like to see the derivation.

Best,

Bill P.

Hi, David –

Thanks for the reference, but it seems to me that the correlation
obtained in this way is simply the regression line, not a measure of the
scatter in the relationship. I have been understanding the correlation
coefficient to measure the difference between

Y = f(X) (a noiseless relationship)

and

Y = F(X) + R(X) where R is a random functiom with a certain RMS amplitude
and a zero mean.

What we measure is the second expression, from which we can obtain the
first expression and an estimate of the mean and standard deviation of R,
under the assumption that the underlying relationship is linear.

I’ll study the reference some more.

Bill

[Martin Taylor
2007.02.26.22.00]

[From Bill Powers
(2007.02.26.1440 MST)]

Martin Taylor 2007.02.26.14.10 –

I concur with your reply to Fred Nickols. Also, good suggestions from
Erling and Rick,

I’ve been thinking about your statement that a correlation is the same as
the cosine of the angle between two vectors. I still can’t figure out how
that works.

Are you saying that after all manipulations are done, we will find that r
= Cos(A) regardless of the values of s1 and s2? If so, I’d like to see
the derivation.

http://en.wikipedia.org/wiki/Correlation

We have been talking about waveforms with a zero dc component, so it
applies.

I defined two vectors extending from the origin to generalized points
(x1, y1) and (x2,y2). The angle between the two lines is fixed and the
lengths are L1 and L2. If the angle of L1 to the X axis is A and the
angle of L2 is B, x1 is L1 cos(A) and y1 is L1 Sin(A), while x2 is L2
cosB and y2 is L2 Sin(B). The angle between the vectors is B -
A.

Now we add random numbers to the lengths of the vectors L1 and L2 while
we record samples of the values (x1,x2) and (y1, y2); Since the direction
of L1 and L2 in space remains the same, the cosine of the angle between
the two vectors remains constant while their amplitudes fluctuate. If the
amplitudes fluctuate exactly together, the correlation between the
amplitudes will be 1 (as will the correlations between x1, x2, y1, and
y2). If the random fluctuations are unrelated and the mean values are
zero, all the correlations will be zero. But the angle between the
vectors remains unchanged, as does the cosine of the angle.

That is why I am having a problem seeing any relation between the
correlation coefficient and the angle between two vectors. Can you clear
this up for me? What am I doing wrong?

Best,

Bill P.

The correlation between the amplitudes can be calculated from the formula
for correlation I cited in my last post.

···

At 10:03 PM 2/26/2007 -0500, you wrote:

[From Richard Kennaway (2007.02.27.1230 GMT)]

Bill Powers writes:

Now we add random numbers to the lengths of the vectors L1 and L2 while we record samples of the values (x1,x2) and (y1, y2); Since the direction of L1 and L2 in space remains the same, the cosine of the angle between the two vectors remains constant while their amplitudes fluctuate. If the amplitudes fluctuate exactly together, the correlation between the amplitudes will be 1 (as will the correlations between x1, x2, y1, and y2). If the random fluctuations are unrelated and the mean values are zero, all the correlations will be zero. But the angle between the vectors remains unchanged, as does the cosine of the angle.

That is why I am having a problem seeing any relation between the correlation coefficient and the angle between two vectors. Can you clear this up for me? What am I doing wrong?

Those aren't the vectors whose cosine is being taken.

Suppose you have 1000 observations (x1,1) ... (x1000,y1000).
The x values can be represented as a single vector in 1000-dimensional space: (x1,x2,...,x1000). Similarly the y values. These are the vectors in question.

If the means of both the x values and the y values are zero, then the lengths of those vectors are the standard deviations of the x values and the y values.

Now scale the values to make the standard deviations both equal to 1. This is equivalent to projecting both 1000-dimensional vectors to the surface of the unit 1000-dimensional sphere.

The formula for the correlation reduces to x1*y1 + x2*y2 + ... + x1000*y1000. This is the dot product of the two unit vectors, which is equal to the cosine of the angle between them.

···

--
Richard Kennaway, jrk@cmp.uea.ac.uk, Richard Kennaway
School of Computing Sciences,
University of East Anglia, Norwich NR4 7TJ, U.K.

Those aren’t the vectors whose
cosine is being taken.

Suppose you have 1000 observations (x1,1) … (x1000,y1000).

The x values can be represented as a single vector in 1000-dimensional
space: (x1,x2,…,x1000). Similarly the y values. These are the
vectors in question.

If the means of both the x values and the y values are zero, then the
lengths of those vectors are the standard deviations of the x values and
the y values.
[From Bill Powers (2007.02.28.0540 MST)]

Richard Kennaway (2007.02.27.1230 GMT) –

Comes the dawn. Thank you so much – couldn’t be clearer. The vector is a
static picture of a collection of values of two variables, so the order
in which the pairs occur is irrelevant, even if the pair of values
represent a time function. It just happens that the calculation of r
takes the form of computing the cosine of an angle in hyperspace between
vectors of this kind. Has nothing to do with the regression line, which
makes sense only when the successive pairs are taken in their original
order. Have I got it right?

This is one reason, it occurs to me, why statistical analyses provide so
little information about what happened – they eliminate time or
ordering. Plotting the data from first to last is completely meaningless,
statistically! That’s done by the analyst, not by the
statistics.

Best,

Bill

[from Gary Cziko 2007.02.28 07:50 CST]

[From Bill Powers (2007.02.28.0540 MST)]

Comes the dawn. Thank you so much – couldn’t be clearer. The vector is a
static picture of a collection of values of two variables, so the order
in which the pairs occur is irrelevant, even if the pair of values
represent a time function. It just happens that the calculation of r
takes the form of computing the cosine of an angle in hyperspace between
vectors of this kind. Has nothing to do with the regression line, which
makes sense only when the successive pairs are taken in their original
order. Have I got it right?

No matter how you calculate the (Pearson) correlation coefficient, the coefficient is the slope of the regression line if the two variables are adjusted to have the same standard deviation. So a correlation of -.90 between disturbance and cursor in a tracking task means that a disturbance one standard deviation above the mean predicts a cursor position of .90 SD below its mean.

This is one reason, it occurs to me, why statistical analyses provide so
little information about what happened – they eliminate time or
ordering. Plotting the data from first to last is completely meaningless,
statistically! That’s done by the analyst, not by the
statistics.

Well, you could make time one of the variables to see, for example, if there is between time of the year and size of the ozone hole over Antarctica (I have read it is growing again).

–Gary

[Martin Taylor 2007.02.18]

[From Bill Powers (2007.02.28.0540 MST)]

Richard Kennaway (2007.02.27.1230 GMT) --

Those aren't the vectors whose cosine is being taken.

Comes the dawn.

Great! Most things are clearer in sunlight!

This is one reason, it occurs to me, why statistical analyses provide so little information about what happened -- they eliminate time or ordering. Plotting the data from first to last is completely meaningless, statistically! That's done by the analyst, not by the statistics.

That's an overgeneralization. Time can appear in statistical analyses in many ways. For example, it might occur as an overt variable, or it might occur because the quantities in the analysis are Fourier components. There are lots of other ways.

Consider the Fourier transform. If you take the vector in the hyperspace Richard described, the different basis vectors can be labelled by the time the relevant sample was taken. Do that, rotate the space appropriately, and you can get the Fourier transform. The signal vectors don't change, and neither does their correlation.

Time enters quite explicitly here.

Martin

No matter how you calculate the
(Pearson) correlation coefficient, the coefficient is the slope of the
regression line if the two variables are adjusted to have the same
standard deviation. So a correlation of -.90 between disturbance and
cursor in a tracking task means that a disturbance one standard deviation
above the mean predicts a cursor position of .90 SD below its mean.
[From Bill Powers (2007.02.28.1010 MST)]

Gary Cziko 2007.02.28 07:50 CST

also Martin Taylor 2007.02.18

Adjusting one variable to have the same standard deviation as the other
means multiplying one data set by a constant relative to the other data
set, which changes the slope just enough to make it equal to r.

Consider two cases:

Case 1: Y = a(X- XBar) + b.

The slope of the linear relationship of Y to X is a. The correlation of Y
with X is is 1.000… Note that this involves the standard
deviations of X and Y going to zero, making the correlation formula
indeterminate.

Case 2: Y = a(X - Xbar) + b + R where R is a zero-mean random variable
with some standard deviation. The slope, it seems to me, is still a, but
the correlation of Y with X is now less than 1.000…

What do we make of that???

Suppose (this just occurred to me) that the regression formula
is

Y = a(X - Xbar + R) + b

This says that the “input” variable X (or its measurement) has
a random component but the assumed linear relationship itself is free of
noise (the standard deviations of a and b are zero). Now Y will have a
standard deviation that is a times the standard deviation of R. Adjusting
the slope of the regression line to make the standard deviation of Y
equal that of X is then the same thing as dividing by the slope a. You
can then “prove” that the correlation coefficient is the same
as the slope of the regression line by applying a fudge factor that just
happens to equal the ratio of the calculated correlation to the
calculated slope.

I see now that the first way of writing the regression equation makes the
random variable add to Y after it is computed from X, so it represents
the error in measuring Y. The second version represents an actual
randomness in the value of X or a measurement error relative to X. I
don’t know how adding a second random variable to Y would work out.
Probably it will do the same thing in reverse.

I think we can conclude that without the fudge factor, the regression
coefficient is not the same as the slope of the regression
line.

This is one reason, it occurs to me, why statistical
analyses provide so little information about what happened – they
eliminate time or ordering. Plotting the data from first to last is
completely meaningless, statistically! That’s done by the analyst, not by
the statistics.
Well, you could make time one of the variables to
see, for example, if there is between time of the year and size of the
ozone hole over Antarctica (I have read it is growing
again).

Making time one variable makes the whole data set simultaneous, unless
you go along with J. W. Dunne and propose that there are two time scales,
with time1 going at a rate determined by its coordinate in time2.

My main point was that when Y is a function of X, a statistical analysis
of the data set [X(t), Y(t)] throws away the time dimension. If you
scramble the order in which you write down the (X,Y) pairs, you will get
the same correlation and regression line. If it happened that two
different physical phenomena actually created the same data pairs in a
different order, the statistical treatment would not show any
difference.

Best,

Bill P.

Martin Taylor 2007.02.28.14.08]

[From Bill Powers (2007.02.28.1010 MST)]

Making time one variable makes the whole data set simultaneous,

Doesn't ANY analysis do that? If you take a Fourier transform of a waveform, you are dealing at any one moment with the whole duration of the waveform. If you compare the results of a dynamic simulation with the waveform, you are treating the whole waveform at the moment of analysis... I can't see how you can do an analysis of any kind without including the data in the analysis, which means you are making the whole data set simultaneous. That includes the timing data of when each sample was taken.

unless you go along with J. W. Dunne and propose that there are two time scales, with time1 going at a rate determined by its coordinate in time2.

I don't know the reference, but it sounds like the effect of doing an analysis with a sliding time window, a very common procedure.

My main point was that when Y is a function of X, a statistical analysis of the data set [X(t), Y(t)] throws away the time dimension. If you scramble the order in which you write down the (X,Y) pairs, you will get the same correlation and regression line. If it happened that two different physical phenomena actually created the same data pairs in a different order, the statistical treatment would not show any difference.

That's simply because you deliberately chose a statistical treatment that ignored the time sequence. You are free to do that, but you are not free then to complain that the analysis did what you asked it to do, rather than what you wanted it to do.

Sorry this has to be short; I'm running out of time before we leave on Monday.

Martin

Making time one variable makes
the whole data set simultaneous,

Doesn’t ANY analysis do that?
[From Bill Powers (2007.02.28.1315 MST)]

Martin Taylor 2007.02.28.14.08 –

No. How can you deal with sequence without time remaining explicit?

unless you go
along with J. W. Dunne and propose that there are two time scales, with
time1 going at a rate determined by its coordinate in
time2.

I don’t know the reference, but it sounds like the effect of doing an
analysis with a sliding time window, a very common
procedure.

Oh, pooh pooh yourself. If you want a treat, look up “An Experiment
with Time” by J. W. Dunne. 50 years ago or so. About prescient
dreaming, which at one time I thought worth paying attention to.

That’s simply
because you deliberately chose a statistical treatment that ignored the
time sequence.

Well, yes, that’s what I wanted to talk about. I was thinking about all
the data in the literature in which you see charts of before and after
scores, in which the author doesn’t seem to realize that the correlations
are irrelevant to the order in which the data points are plotted. The
author plots the points from left to right because that’s the order in
which the data were obtained. But they could have been plotted in any
other order with the same statistical result.

Besgt,

Bill P.