Control and correlation

[Martin_Taylor4]

[Martin Taylor 980303 10:20]

(The date stamp is when I created the first draft of this message. I have
witheld
posting it because I am not 100% sure of its correctness. I post it now so that
people more mathematically adept than I can check its correctness, and so that
those with simulation data can see whether in fact the prediction is accurate,
even if it is mathematically flawed).

I have about 100 messages unread since Saturday, but I thought I'd better
send this
out anyway.

···

-----------------------
This message was stimulated by the following exchange between Bill Powers
and Bruce Abbott.

+[From Bill Powers (980226.0148 MST)]
+
+Bruce Abbott (980225.2005 EST)--
+
+>>You haven't yet replied to my description of the correlations that are
+>>actually observed, which contradict what your common sense is telling you.
+>
+>I thought that this was it:
+>
+>>>In an
+>>>ideal control system, this immediate response means that e would never get
+>>>large enough even to detect, even though d and o varied over a wide range.
+>>>At these levels of variation, _observed_ changes in e would constitute
+>>>nothing but measurement error, and variation in e would appear uncorrelated
+>>>(except by chance) with d and with o.
+
+This isn't the actual explanation of the correlation phenomenon to which I
+refer. It's not a question of measurement error in observing an ideal
+control system. We see this effect in real data, measures of definitely
+non-ideal human control. We can measure the cv and the output more
+accurately than they can be controlled (cv) or varied (output) by the
+organism. The real explanation lies in system noise -- the fact that
+outputs and other variables show small spontaneous variations uncorrelated
+with changes in the disturbance.

I wanted to see if it was possible to say anything about the correlation
between the
perceptual signal and the disturbance signal of a control loop in the
absence of system
noise. It turned out to be possible, unless I've made some serious mistake.
So here's
the message as originally drafted.

-----------------------

The recent reincarnation of the discussion on the relation between the
disturbance signal and the perceptual signal has made possible an
interesting analysis. We can compute, at
least for the simplest control loop with a perfect integrator output
function, the maximum
possible correlation between the two signals as a function of the degree of
control.
If my analysis is correct, this maximum is 1/CR, where CR is the control
ratio--the
ratio between the magnitude of the amplitude of the variation in the perceptual
signal in the absence of control to its amplitude when control is in operation.
The correlation would be lower, if there is noise in the system.

I'll make a large caveat here. To me, this analysis seems too simple, and
I'd be
grateful if anyone can find an error that _increases_ the maximum
correlation possible
between p and d in this idealized maximally simple control loop, or show by
simulation
that the actual correlation is higher than the analysis shows. I submit the
analysis
because I can't by myself, see anything wrong with it, other than
approximations that
work in the direction of increasing the maximum.

----------------------------

We start with the classic diagram of a control system:

  perceptual signal (p) ^ | reference signal (r)
                         > V
                         >____________-___error signal(e)__
                         > comparator |
   perceptual signal (p) | |
                         > output function (G)
                 input function |
                         > >output signal (o)
                    CCEV + -----environment function-------|
                         > >
   disturbance signal (d)| |output signal (o)
                         ^ V

And we will make the usual assumptions that are made when using the
expression "p = o + d". That is to say, all the functions in the loop
except the output function are unit transforms.

Since it is short, I will repeat the loop derivation I wrote for Rick.

p = o + d = Ge + d = G(r-p) + d = Gr - Gp + d (1)

which yields d = p + Gp - Gr

That's starting point 1.

In the equation, d = p + Gp - Gr, all the "variables" are actually Laplace
transforms, which can be treated as if they were algebraic variables if the
system is linear (an integrator is a linear system). A Laplace transform is
a way of describing a time function, to put it crudely. The output
function, G, is a pure integrator in the example I
analyze below.

------------------------

Starting point 2 comes from an interchange between Bill Powers and me,
summarized by Bill (980224.1755 MST):

If the output function is a pure integrator, all frequency components are
phase-shifted by >90 degrees, which reduces the correlation to zero. For a
system with a large leakage term, >the correlation would be high unless
there was noise added in the output function or >comparator.

For now, I'll consider only the case with a perfect integrator output
function, but the
argument works (giving a different final result) with any function. One
just has to
know the correlation between the function's input and output. For G a pure
integrator,
x is uncorrelated with Gx if Bill's and my speculation is correct. For
other functions,
the correlation is likely to depend on the frequency spectrum of the input
of the function.

-----------------------

The analysis:

Firstly, it may help you to understand the analysis if you note that
symbols like "p"
and "d" represent waveforms extended over a long (notionally infinite)
time. I use
the same symbol also to represent the Laplace transform of the signal. When
the signals
are treated as vectors and we talk of correlation, the time variation (the
waveform) is
what you should think of. In developing the equations (e.g. "d = p + Gp -
Gr") the
Laplace transforms are used. The two are essentially interchangeable in the
current
context, which is why I haven't worried about using a script font for the
transform
(as if I could, in an e-mail message:-).

If G = 0 (no control), then p = d. Clearly then the correlation between p
and d is 1.0.

To start the next phase of the derivation, consider the case of r = 0
forever. In this
case, Gr (the integral of r) is also zero forever, and we have

d = p + Gp

Now we use the assertion that a variable is uncorrelated with its integral.
The
variable "d" is composed of two orthogonal components, p and Gp (remember,
"G" is
the output function, which is a perfect integrator).

      Gp * *d
         > /
         > /
         > /
         > /
         > /
         > /
         > /
         >/_______* p

The squares of the lengths of the vectors represents the mean-square
amplitude variation
in the signal values. When you think of the variations on this diagram that
occur as
the output gain changes, or as the reference signal is allowed to vary,
remember that
it is the "d" vector that stays constant, while the others may change their
magnitude
and direction--not the other way around.

The correlation between any two vectors is the cosine of the angle between
them. That
is why Gp and p are drawn at right angles. Their correlation is zero. If Gp
is large
compared to p, d has a correlation of nearly 1.0 with Gp and nearly zero
with p. Since
we are dealing only with the case in which the reference signal is fixed at
zero, the
output signal is Ge, which is -Gp. So the disturbance signal is correlated
almost
-1.0 with the output signal--as we know to be the case for good control.

The control ratio (CR) is the ratio between the fluctuations that would
occur in p in
the absence of control and the fluctuations in p when the perception is
controlled. In
other words, CR = d/p when the transform between qi and p is the unit
transform.

The cosine of an angle in a right-angled triangle is the length of the
opposite side
over the hypotenuse. That is to say Corr(d:p) = p/d.

From this, the correlation between d and p is 1/CR.

-------------------
Variation in the reference signal

Why did I say this was a maximum correlation rather than the precise
correlation?
In part this is because there is always noise, as Bill pointed out. But
more because
of the role of the reference signal. Above, we considered the case of r
permanently
zero. Now we let r vary, independently of d, of course.

The control system is linear. What this means is that the superposition
theorem holds--
the contributions of different components can be added in the time domain,
in the
frequency domain, and in the Laplace domain. In particular, the relation d
= p + Gp
can be used as a starting point, onto which can be added any effects due to the
variation of r. Most importantly, p and Gp will change when r varies.

Gp (the integral of p in the simple system we are analyzing) can, because
of superposition,
be divided into two parts, which I will label G_d.p and G_r.p. G_d.p is
just what
we had before, when r was fixed permanently at zero. G_r.p is the variation
in Gp that
is extra, due to the variation in r.

Now we can look at the full expressions that was shown at the head of this
message:

d = p + Gp - Gr

and rewrite it

d = p + G_d.p + (G_r.p - Gr)

The first part of this is exactly what we had before, when r was
permanently fixed
at zero. The part in brackets is the contribution of variation in r.

What does the part in brackets contribute to the correlation between d and
p? Since the
reference signal varies independently of the variation in the disturbance
signal, and
any contribution of (G_r.p - Gr) is due to the reference signal, that
contribution is
orthogonal to d (and to G_d.p). It cannot increase the correlation between
d and p,
except by accident over the (very) short term. Furthermore, the better the
control,
the more nearly does p match r, and therefore the more nearly does G_r.p
match Gr.
The two tend to cancel one another, so if they have an effect, it tends to
become
small when control is good.

What this means is that even when the reference signal is allowed to vary
freely, the
maximum correlation that should be observed between the perceptual signal
and the
disturbance signal is 1/CR.

-------------------------
Extension to more realistic control loops.

The above applies mathematically only to a control system that is linear,
has no loop
transport delay, has a perfect integrator as its output/feedback function,
and has no
other time-binding functions in the loop (i.e. all the other functions are
simple
summations or multiplications by constant factors). Most control systems
are not
like that. What then?

Some cases can be examined heuristically. For example, if the output
function G is a
leaky integrator rather than a perfect integrator, the angle between p and
Gp depends
on the frequency spectrum of p. If low frequencies dominate, then Gp
correlates well
with p, but if high frequencies dominate, G acts like a good integrator. So the
correlation between d and p can be greater than 1/CR if the output function
is a
leaky integrator.

If there is some loop transport delay, it has to be incorporated into the
initial
expression for d. It could all be included in the output function, G. With loop
delay, the correlation between the input to G and its output will vary between
positive and negative correlation as a function of the frequency of p. This
will
show up in a correlation between Gp and p that varies with the spectrum of
p. In the
diagram, Gp leans left and right as the frequency of d varies. If the
magnitude of
G is large enough, this can lead at some frequencies to p being larger than
d--there
is no control, and the cosine takes on an imaginary (or at least complex)
value; the
loop is oscillating, not controlling.

Other cases can be examined similarly, provided the system is composed of
linear
components that allow the use of Laplace transforms. And for some
non-linear systems
one can make reasonable heuristic approximations by appealing to
small-amplitude
linearity.

--------------------------

All of the above depends on the correctness of two things: (1) the
statement that the
output of an integrator is, in general, uncorrelated with its input, and
(2) whether
I have done the analysis correctly. As I said up front, the result seems
too simple
not to be well known, and from prior discussion it clearly is not well
known. I'm
sure Bill has masses of simulation data that could be used to test the result
claimed, even though simulation data necessarily include transport loop delay
(which I think tends on balance to lower the correlation, but in a way that
depends on the spectrum of p). Perhaps Bill's data could show that the
correlation
control loops between the disturbance signal and the perceptual signal in such
"perfect integrator" is characteristically greater than 1/CR. If it is, it
would
show that there is a flaw in the analysis somewhere, even though I can't see it
right now, and I've asked a couple of control engineers, who also seem to
think it
is OK.

Martin

[Bill_Powers1]

[From Bill Powers (980312.0104 MST)]

Martin Taylor 980303 10:20--

If G = 0 (no control), then p = d. Clearly then the correlation between p
and d is 1.0.

To start the next phase of the derivation, consider the case of r = 0s
forever. In this
case, Gr (the integral of r) is also zero forever, and we have

d = p + Gp

Now we use the assertion that a variable is uncorrelated with its integral.

Have you checked my assertion? As I think about it, I'm not sure it would
be true in all cases. For example, suppose the variable varies as exp(-kt).
In that case, the integral varies as -k*exp(-kt), which is proportional to
the variable, and (it seems to me) would correlate highly with the
variable. I was thinking of sinusoidal waveforms, on the basis that the
sine and cosine of a variable ought to be uncorrelated. But that's not the
general case.

I'm sure you realize that the case you're analyzing is NOT the case that
leads to the behavioral illusion. What you're calling d here is really DS,
the disturbance signal, which is not directly manipulable. The only way you
can physically disturb qi is to alter some other physical variable on which
qi depends.

Best,

Bill P.

[Martin_Taylor4]

[Martin Taylor 980312 17:03]

Bill Powers (980312.0104 MST)]

Martin Taylor 980303 10:20--

If G = 0 (no control), then p = d. Clearly then the correlation between p
and d is 1.0.

To start the next phase of the derivation, consider the case of r = 0s
forever. In this
case, Gr (the integral of r) is also zero forever, and we have

d = p + Gp

Now we use the assertion that a variable is uncorrelated with its integral.

Have you checked my assertion? As I think about it, I'm not sure it would
be true in all cases. For example, suppose the variable varies as exp(-kt).
In that case, the integral varies as -k*exp(-kt), which is proportional to
the variable, and (it seems to me) would correlate highly with the
variable. I was thinking of sinusoidal waveforms, on the basis that the
sine and cosine of a variable ought to be uncorrelated. But that's not the
general case.

It's the general case except for the DC component, isn't it? All Fourier-
decomposable signals (i.e. all signals producible in practice) can be
described in terms of a sum of a DC component and a set of sinusoids.
The assertion fails if the DC component is non-zero, but should hold
for the other components. What this does in respect of the correlation
between p and DS (I use this symbol in replying to you, though I dislike
2-letter symbols when doing algebra) is to increase the possible correlation
between them by an amount that depends on the relative magnitudes of the
DC component and the rest of the DS waveform.

So long as the integration is over a time-span that is either an integer
number of cycles of the lowest frequency component, or is very long
compared to the period of the lowest frequency component, the assertion
ought to be valid--or so I think, and so think a couple of engineers I
sprung this on. But the fact we all think it is so does not prove it to
be so. I don't know how to prove or disprove it, at present.

It's hard to integrate over a time-span that is an integer number of
cycles of a DC signal:-)

Martin

[Bill_Powers1]

[From Bill Powers (980312.2049 MST)]

Martin Taylor 980312 17:03--

Have you checked my assertion? As I think about it, I'm not sure it would
be true in all cases. For example, suppose the variable varies as exp(-kt).
In that case, the integral varies as -k*exp(-kt), which is proportional to
the variable, and (it seems to me) would correlate highly with the
variable. I was thinking of sinusoidal waveforms, on the basis that the
sine and cosine of a variable ought to be uncorrelated. But that's not the
general case.

It's the general case except for the DC component, isn't it?

No, I think it's peculiar to sines and cosines to fail to correlate with
each other. I misstated the integral above, but the idea is correct. The
integral of exp(-kt) would be of the form [1 - exp(-kt)] which with the
constant term removed, is the negative of exp(-kt). So if you had a
repetitive waveform of the form exp(-kt), its integral would correlate with
it perfectly (and negatively).

All Fourier-
decomposable signals (i.e. all signals producible in practice) can be
described in terms of a sum of a DC component and a set of sinusoids.

Yes, but we're talking about the sum of the series, not the components. I
think this whole approach is untenable.

Best,

Bill P.

[Martin_Taylor4]

[Martin Taylor 980312 17:03]

Bill Powers (980312.0104 MST)]

Martin Taylor 980303 10:20--

If G = 0 (no control), then p = d. Clearly then the correlation between p
and d is 1.0.

To start the next phase of the derivation, consider the case of r = 0s
forever. In this
case, Gr (the integral of r) is also zero forever, and we have

d = p + Gp

Now we use the assertion that a variable is uncorrelated with its integral.

Have you checked my assertion? As I think about it, I'm not sure it would
be true in all cases.

It's the general case except for the DC component, isn't it?

Now I _have_ analyzed it, and I think I have proved the assertion to be
true for signals with a zero DC component. Here is the proof, unless I
have made some silly mistake.

···

--------------------

The simplest form for the correlation between any two vectors
X = {x1, x2, ...xi, ..., xn) and Y = {y1, y2, ...yi, ..., yn} is
r = Sum(xi*yi)/sqrt(sum(xi^2)*sum(yi^2)). We are concerned here with vectors
that are the successive samples of some signal over time. The samples
may be taken at infinitesimal intervals, if the signals vary continuously,
but it doesn't matter--the same analysis applies to continuous as to
discretely sampled signals.

Ignoring the denominator, the numerator is the sum of the cross-products
of the values of X times the corresponding values of Y. This can be
generalized to continuous waveforms by using integrals rather than
sums, taking the cross-product of the values of the two waveforms at the
same moment in time.

X may be decomposed into its Fourier components (or any other orthogonal
decomposition), X = a0 + a1*W1 + a2*W2 + ... Likewise, Y may be decomposed
into its components, Y = b0 + b1*W1 + b2*W2, ...If it is a Fourier
decomposition, W1, W2, ...Wn are simes and cosines of some base frequency
and its harmonics. Whatever the decomposition, the W1, W2, ...Wn have the
property that they are uncorrelated, meaning that Sum(Xmi * Xni) = 0.
The numerator of the expression for the correlation coefficient r is
rnom = Sum((a0 + a1*W1i + a2*W2i + ....) * (b0 + b1*W1i + b2*W2i + ...))
= Sum(a0*b0 + a0*(b1*W1i+b2*W2i+ ...) + a1*W1i*(b0+b1*W1i b2*W2i+ ...))

This sum has a large number of components, but they can be divided into
groups. Group 1 is a0*b0. Group 2 consists of all the elements of the
form a0*bj*Wji or b0*aj*Wji. Group 3 consists of the elements of the form
ak*Wki*bkWki. Group 4 has elements of the form ak*Wki*bj*Wji where k != j.

Summed over all i, Wji = 0, because that is a requirement for the
orthogonal decomposition. It is trivially true of a sine or a cosine summed
(or integrated) over an integer number of periods. So the Group 2 elements
have a total contribution of zero to the sum. Likewise, group 4 elements
each sum to zero, because they are of the form c*Wki*Wji, and by definition
the sum over i of Wki*Wji is zero for an orthogonal decomposition. That
leaves Group 1, which is a single constant, and Group 3 to deal with. The
question is: if Y is the integral of X, is the sum of the Group 3 elements
zero?

In a Fourier decomposition, if W(2n-1) = sin(x), then W(2n) is cos(x). The
integral of sin(x) is cos(x) and vice-versa, give or take a minus sign.
So, if X = a0 + a1*W1 + a2*W2 + ..., then Y = integral X = a1*W2 - a2*W1 +...
Group 3 elements are therefore of the form ami*ami*Wmi*Wni (where m and n
are consecutive integers. But this is exactly the form of a Group 4 element,
for which the sum over i is zero, since the decomposition is orthogonal.

The sum of Group 3 elements is therefore zero, meaning that the correlation
between a variable and its integral is zero apart from the contribution of
the DC component.

This justifies the computation (Martin Taylor 980303 10:20) of the maximum
correlation between the perceptual signal and the disturbance signal in a
control loop with zero loop delay and a perfect integrator output function.
The maximum correlation is 1/(control ratio), where the control ratio is
the ratio of the fluctuation amplitude in the perceptual signal if the
environmental feedback were to be cut off to its amplitude when control is
functioning.

Martin

[Bill_Powers1]

[From Bill Powers (980314.1751 MST)]

Martin Taylor 980312 17:03--

I think I have proved the assertion to be
true for signals with a zero DC component. Here is the proof, unless I
have made some silly mistake.

I don't feel competent to check your proof, so I'll await the verdict of
our mathematicians.

Could you remind me of why we're going through this?

Best,

Bill P.

[Martin_Taylor4]

[Martin Taylor 980315 00:03
Sorry, I didn't provide a date stamp on the original, which should have
been [Martin Taylor 980314 17:24]

There's a typo in the presentation (at least one...)

The simplest form for the correlation between any two vectors
X = {x1, x2, ...xi, ..., xn) and Y = {y1, y2, ...yi, ..., yn} is
r = Sum(xi*yi)/sqrt(sum(xi^2)*sum(yi^2)).

X may be decomposed into its Fourier components (or any other orthogonal
decomposition), X = a0 + a1*W1 + a2*W2 + ... Likewise, Y may be decomposed
into its components, Y = b0 + b1*W1 + b2*W2, ...If it is a Fourier
decomposition, W1, W2, ...Wn are simes and cosines of some base frequency
and its harmonics. Whatever the decomposition, the W1, W2, ...Wn have the
property that they are uncorrelated, meaning that Sum(Xmi * Xni) = 0.
The numerator of the expression for the correlation coefficient r is
rnom = Sum((a0 + a1*W1i + a2*W2i + ....) * (b0 + b1*W1i + b2*W2i + ...))
= Sum(a0*b0 + a0*(b1*W1i+b2*W2i+ ...) + a1*W1i*(b0+b1*W1i b2*W2i+ ...))

The last line should be

= Sum(a0*b0+a0*(b1*W1i+b2*W2i+ ...)+a1*W1i*(b0+b1*W1i b2*W2i+ ...)+a2*W2i*...)

Sorry.

Martin

[Martin_Taylor4]

[Martin Taylor 980315 0:08]

Bill Powers (980314.1751 MST)]

Martin Taylor 980312 17:03--

I think I have proved the assertion to be
true for signals with a zero DC component. Here is the proof, unless I
have made some silly mistake.

I don't feel competent to check your proof, so I'll await the verdict of
our mathematicians.

Could you remind me of why we're going through this?

Yes, it happened because I was intrigued by your comment to Bruce Abbott
that the reason for the decorrelation between the perceptual signal and
the disturbance signal was noise. I wondered whether the correlation could
be computed in the absence of noise, and our little interchange in which
we speculated that the correlation between a waveform and its integral
would be zero provided the clue to how to go about it.

+Bill Powers (980304.1600 MST)

+You (and Martin) are neglecting system noise. The value of qi, as has been
+shown experimentally many times, does not show a high correlation with d.
+Neither, therefore, would the perceptual signal. If we, as external
+observers, can see that qi has a 0.2 correlation with d, we can conclude
+that no useful reconstruction of d from qi could be made. Martin, I am
+sure, will agree.

I wanted to see whether I would agree. Now, if my analysis is correct, we
can show that in the linear zero-loop-delay perfect-integrator-output
control loop, we get a maximum correlation 1/CR between p (or qi, since
they are the same in the loop I analyzed) and d (or rather DS). Even so,
one can make a perfect reconstruction of d (or rather, DS) from p or qi.

Before I did the analysis, I might have agreed with you. Now I hope you
will agree with me that I would have been wrong to do so.

If my result is correct, and right now I'm pretty confident it is, then
it is a fact that was not well known (to us at least) about control
systems of a kind much-used in our modelling. New facts about control
systems have to be welcome in PCT, I should think.

Martin

[Bill_Powers1]

[From Bill Powers (980315.0339 MST)]

Martin Taylor 980315 0:08--

Could you remind me of why we're going through this?

Yes, it happened because I was intrigued by your comment to Bruce Abbott
that the reason for the decorrelation between the perceptual signal and
the disturbance signal was noise.

As I recall, that was not the subject. It was the lack of correlation
between d and qi, and between qi and qo, as measured. The perceptual signal
is not measurable, as far as I know.

What we observe is that the disturbing variable d correlates highly with
the output quantity qo, but that neither d nor qo correlates highly with
qi, under our customary definitions of these terms.

In the experiments in question, the effect of d on qi is purely
proportional, so there is no question that the observed low correlation
could be due to the phase-shift of integration. The modeled relation
between qi and qo does involve a leaky integration. However, the
decorrelation is the greatest at the lowest frequencies, where the phase
shift due to the integration is the least and the output function behaves
most nearly like a proportional function. Thus the integration can't be the
correct explanation for the lack of correlation between qi and qo, either.

If your analysis holds up, you will have shown that a lack of correlation
is to be expected even in the absence of noise when the functions relating
the variables are pure integrators (or, I presume, differentiators). This
does not, however, imply that if a lack of correlation is observed, the
explanation must be that the pertinent functions are pure integrators, as
you seem to be saying:

I wanted to see whether I would agree. Now, if my analysis is correct, we
can show that in the linear zero-loop-delay perfect-integrator-output
control loop, we get a maximum correlation 1/CR between p (or qi, since
they are the same in the loop I analyzed) and d (or rather DS). Even so,
one can make a perfect reconstruction of d (or rather, DS) from p or qi.

Before I did the analysis, I might have agreed with you. Now I hope you
will agree with me that I would have been wrong to do so.

Not at all. If I had been talking about the kind of system you describe,
you would have been wrong to agree. However, I was talking about the real
system in which d affects qi through a proportional function Fd, and qi
affects o through a leaky integrator that is almost proportional at the
frequencies where the decorrelation is the greatest. Your analysis would
predict a high correlation between d and qi, and between qi and qo, under
those conditions, which is not what we observe. The correct explanation for
the low correlations we observe is, as I said to Bruce Abbott, system noise
(which includes uncorrelated variations in the reference signal).

Best,

Bill P.

[Bill_Powers1]

[From Bill Powers (980317.0754 MST)]

Martin Taylor 980316 0:55--

As I recall, that was not the subject. It was the lack of correlation
between d and qi, and between qi and qo, as measured. The perceptual signal
is not measurable, as far as I know.

It is, in a simulation. Neither qo nor qi are measurable in an experiment
on humans, though the experimenter may measure the effects on one or more
of his/her own perceptual variables.

qi and qo are strictly observer-centered quantities used to construct a
model of the behaving system. In a tracking experiment, qi is the distance
between cursor and targer, and qo is the mouse position. So I don't know
what you're talking about.

So your comment is irrelevant,
especially since toward the end of your message you appeal to changes
in the reference signal (presumably measured, since you seemed to intend
to be quantitative). Apparently, the reference signal value is measurable
but the perceptual signal value is not.

No, the reference signal is inferred. See my chapter on experimental
investigation of purpose, in Hershberger's volume, _Volitional Action_. I
didn't "appeal" to changes in the reference signal; I mentioned them as one
source of variations of output that are uncorrelated with disturbances.

I am quite unclear as to what all this has to do with the statement whose
correctness I was trying to check. The relevant interchange was as follows:

If we, as external
observers, can see that qi has a 0.2 correlation with d, we can conclude
that no useful reconstruction of d from qi could be made. Martin, I am
sure, will agree.

Whenever "d" is encountered as a variable in a PCT model, it means the
remote disturbing quantity, not the derived quantity you call the
disturbing signal. To refresh your memory, what we observe, typically, is

  d --->[Fd] --> qi <---[Fe]<-----qo

  <---- 0.2 ---> <---0.2 ----->
                                       (correlations)
  <---------- -0.99 -------------->

We know that Fd is a simple constant of proportionality, so your "integral"
explanation does not apply to that correlation. We know that Fe is also a
proportional multiplier, not an integral, so that low correlation is not
explained by the "integral" proposition. The only explanation we are left
with is that there is system noise comparable in magnitude to the
fluctuations in qi. Thus from observing the magnitudes of qi, one cannot,
even knowing the form of Fd, reconstruct the waveform of d with any accuracy.

Before I did the analysis, I might have agreed with you. Now I hope you
will agree with me that I would have been wrong to do so.

Not at all. If I had been talking about the kind of system you describe,
you would have been wrong to agree.

What has it to do with "the kind of system?" You made a blanket statement
with which I might have agreed, before doing the analysis. Having done
the analysis I don't agree with it, and neither should you.

You assumed a "kind of system" in which the connecting functions were leaky
integrals. The actual "kind of system" we are dealing with in the
experiments where the apparent anomaly is observer involves only
proportional functions, to which your argument applies only in the limit of
a very short time constant (and in which the two low correlations above are
expected to be high).

I have shown that there exists a system in which qi has a correlation
of 1/CR with d (which could be much less than 0.2), and in which d can be
exactly reconstructed from qi. This shows that I would have been wrong to
agree with your statement, a statement that is quite independent of what
system was being tested.

When this came up before, you presented much data for the correlation
between p and d.

I did not. I presented data for a correlation between qi and d. The
perceptual signal is not observable in tracking experiments, although we
usually assump p = qi.

Recently you have been asserting that qi is really p
because there is no CCEV.

This is all going on in your head, Martin. If qi is a perception, it's a
perception in the OBSERVER, not in the control system being investigated.
But as observers, we project qi into the environment; that's basically how
we model the environment.

Now you say you can measure qi, but not p.

Yes, of course. The observer measures qi as if it's an objective variable
in the environment, although we know it is a perception in the observer.
The observer, however, cannot measure p, because that is a hypothetical
perceptual signal IN THE OTHER SYSTEM and has no corresponding perceptual
signal in the observer (except as the observer imagines it). My diagram
above shows what the observer can see, and what he attributes to the
environment.

It seems to me that what you are really doing is finding any way you
can to say that whatever I contribute to PCT is wrong.

Don't be petulant. If what you're saying looks wrong to me, I'll say it's
wrong. I'm not going to say it's right just to stay in your good graces, if
I think it's wrong.

If your analysis holds up, you will have shown that a lack of correlation
is to be expected even in the absence of noise when the functions relating
the variables are pure integrators (or, I presume, differentiators). This
does not, however, imply that if a lack of correlation is observed, the
explanation must be that the pertinent functions are pure integrators, as
you seem to be saying:

It would be nice if you would be a little less selective in your quoting,
and include the bits where I said that your noise explanation was valid
as well. Here's one from the message in which I presented the initial
analysis (Martin Taylor 980303 10:20):

+Why did I say this was a maximum correlation rather than the precise
+correlation? In part this is because there is always noise, as Bill
+pointed out.

But you started out trying to explain the _low_ correlations, not the high
one, and you introduced the idea of the integral to show how a non-noisy
function could still result in a low correlation. That is the argument I am
saying is irrelevant, because in the case in question we do NOT have
integral functions. Thus in the case in question, the ONLY remaining
explanation is the system noise explanation. It is not the high correlation
that needs explanation, but the low one, and for THE CASE IN QUESTION, your
explanation is the wrong one.

That's not the only time I concurred with you that noise is important.

It's not just important in the case we're talking about, it's the only
tenable explanation for the apparently paradoxical set of correlations.

What I wanted was to see whether I would agree that noise was the _only_
explanation of the low correlation between perceptual and disturbance
signals. To do this it was necessary to examine the behaviour of a
noiseless system. And I examined the case of the pure integrator
output function in a loop with no transport delay partly because
that is a loop often used as an example of what PCT is all about,
and partly because it is a case I found I could analyze.

But it is an irrelevant case: see diagram above.

And, in this case, we are talking not about "the functions relating the
variables" but about the behaviour of a complete control loop.

The behavior of the whole loop makes no difference to the diagram above. No
matter what the control system is (or even whether it exists), qi = Fd(d) +
Fe(qo). That is a stand-alone relationship, independent of what goes on
inside the behaving system.

Now we come to some interesting comments, in which you mention an
experimental fact that I have never seen or heard you mention before:
that the decorrelation is greater for low frequency disturbance signals
than for high.

Yes, of course. When the disturbance is easy (only very low frequencies),
the person can keep qi very nearly constant, so essentially ALL the error
is system noise. When the disturbances contain substantial amounts of
higher frequencies, around 0.5 to 1 Hz, the error gets much larger, and the
correlations of d and qi, and of qi and qo, increase substantially -- for
example, to 0.5 or 0.6. The overall correlation of d and qo falls, to
perhaps 0.9.

That is an extremely interesting fact, and I'd love to see the data,
to see how the change in correlation with frequency (or upper cut-off?)
of the disturbance signal relates to the change in slope with frequency
of the leaky integrator. That would tell us quite a lot, I think. And
if you also have data that tie this into loop delay, it would be really
fascinating. Please post at least some of these data. I don't know what
you have, but presumably it involves simulations using disturbance signals
of varying low-pass bandwidths, though it would be a better test if it
involved disturbances of constant bandwidth but a varying centre frequency.
Perhaps you have it with sinusoidal disturbances at different frequencies
compared to the leaky integrator cutoff frequency?

I'll have to construct a simple tracking experiment to show this -- old
data has a tendency to disappear as I houseclean my disk. I'm sure you have
many examples of this from the sleep study. It's not a tiny effect; it's
obvious and robust.

The correct explanation for
the low correlations we observe is, as I said to Bruce Abbott, system noise
(which includes uncorrelated variations in the reference signal).

Which can be observed only in simulation models.

NO, dammit. These observations are of real data with real subjects. I keep
saying that and you keep ignoring me. In fact, with a simulation model we
DON'T see this decorrelation, because we don't put any noise into the
models. In a noise-free model all the correlations are high.

I _love_ the way you substitute assertion for analysis or data in
directing what we must all believe to be "correct". What would
constitute a proof of what you say? Clearly it would be the data you are
going to post, since the correlations will be high when you simulate
without system noise and low when you add system noise, and will
decrease at lower as compared to higher frequencies. It's the high
correlations in noiseless conditions, coupled with the frequency-
dependent decline, that will be interesting.

Right. That is what will happen. I will construct a simple experiment and
post the data.

To be fair, I think we both got thrown off the track: the form of the
system's output function is irrelevant. The relation between qo and qi that
counts is the environment feedback function.

If we get this right, we should be able to use your results to infer
(by modelling) the location and spectrum of noise in the control loop
for human subjects, by testing with appropriate spectra of the disturbance
signal.

This is exciting!

Considering that I've been saying these same things for about 20 years, the
thrill has worn off for me.

Best,

Bill P.

[Richard_Kennaway1]

[From Richard Kennaway (980319.2256 JST)]

I see a number of problems with the analysis of correlations in a
control loop.

Firstly, as Bill Powers pointed out, not all functions have zero
correlation with their integrals. Exponentials are exactly proportional
to their integrals.

Secondly, the correlation between two quantities does not necessarily
have anything to do with the degree to which one gives information about
the other. The correlation between the functions f(x) = x and g(x) = x*x
over the interval -1 <= x <= 1 is zero, yet knowing f(x) tells you g(x)
exactly.

Thirdly, the correlation between disturbance and perception has in
general very little to do with the quality of control. For example, a
control loop whose output is proportional to the current error will
exhibit a very high correlation between disturbance and perception
(assuming constant reference and no noise), but if the gain is high,
then the perception will be held nearly constant while the disturbance
varies over a wide range. A better measure of control would be some
sort of rejection ratio: the ratio between the amplitude of variations
in the disturbance and the amplitude of variations in the perception.
In the case of linear control loops, you can apply pure sine wave
disturbances and plot the rejection ratio in dB against frequency. What
you would want to see in a well-functioning control loop is a plot that
stays above, say, 20dB (a ratio of 100:1) over the range of frequencies
you want the control loop to handle.
In the general nonlinear case, it's not clear to me what to measure,
but the ratio of standard deviations of disturbance and perception
would be one possibility. Rick, if you're reading this, would it be
useful to add such a measure to your control demos?

-- Richard Kennaway, jrk@sys.uea.ac.uk
   (The etl.go.jp address is temporary until 7 April 1998.)

[Bill_Powers1]

[From Bill Powers (980319.0907 MST)]

Richard Kennaway (980319.2256 JST)--

I see a number of problems with the analysis of correlations in a
control loop.

Firstly, as Bill Powers pointed out, not all functions have zero
correlation with their integrals. Exponentials are exactly proportional
to their integrals.

That's what I thought. So there must be something wrong with Martin
Taylor's conclusion that all functions show zero correlation with their
integrals.

Secondly, the correlation between two quantities does not necessarily
have anything to do with the degree to which one gives information about
the other.

...

Thirdly, the correlation between disturbance and perception has in
general very little to do with the quality of control. ...

Thanks much for the useful comments. It's ironic that I started calculating
correlations because I thought this would be a measure that psychologists
would find more familiar than loop gain, RMS error signal, and resistance
to disturbance. I agree with you that correlations are a poor measure of
control. It would be nice if I could just put this albatross down somewhere...

I hope your sojourne in Japan is proving fruitful, or at least profitable.

Best,

Bill P.

[Richard_Kennaway1]

[From Richard Kennaway (980320.1053 JST)]

Bill Powers (980319.0907 MST):

That's what I thought. So there must be something wrong with Martin
Taylor's conclusion that all functions show zero correlation with their
integrals.

I've thought some more about this and I believe I know where his
analysis breaks down.

Warning: the rest of this is all mathematics.

If f is a periodic function, and if the integral of f over one period is
zero, then the integral of f is also a periodic function, and f and its
integral will have zero correlation. This can be proved by analysis
into Fourier components: for sine waves whose periods are an integer
fraction of some period T, the correlation between sines of different
frequency is zero, regardless of phase, and between a sine and its
integral is zero. I think this is essentially Martin Taylor's
argument.

If the integral of f over one period is some non-zero value A, and the
period is T, then the integral of f from zero to x will have the form
F(x) + Ax/T, where F is periodic. f and F will have zero correlation,
and f and Ax/T will have arbitrarily low correlation if measured over a
large enough interval. So again, the correlation between f and its
integral will be arbitrarily low if measured over a long enough
interval.

So it looks as if every periodic function has zero correlation with its
integral. Then by letting the period tend to infinity, this looks as if
it should apply to all functions.

(In hindsight, this is the point at which one can detect the first
inkling that something might go wrong when the period is allowed to tend
to infinity. The zero correlation of f with Ax/T depends on integrating
over many periods of f. Over a single period, the correlation could be
anything at all. When the period goes to infinity, you only have one
period of f left.)

What goes wrong for the exponential f(x) = P exp(Q x), which over any
finite interval has a correlation of 1 with its integral (P/Q) exp(Q x),
is that the analysis into Fourier components (on which the proof of zero
correlation depends) fails. The integral over the real line of f(x)
sin(kx) is undefined.

That's not quite the full story. You can define the Fourier transform
of P exp(Q x) by other means than that divergent integral. The
component of frequency omega has amplitude 2P/sqrt(Q^2 + omega^2) and
phase arctan(omega/Q) (or something like that -- I may have made various
calculation errors). However, the proof of zero correlation with the
integral will still fall down, at a point where the argument would
require an invalid reversal of the order of integrating over two
variables.

-- Richard Kennaway, jrk@sys.uea.ac.uk
   (The etl.go.jp address is temporary until 7 April 1998.)

[Martin_Taylor4]

[Martin Taylor 980316 0:55]

My mailer seems to indicate that the following message was never sent out.
In case it remains relevant, I send it now.

Bill Powers (980315.0339 MST)]

Martin Taylor 980315 0:08--

Could you remind me of why we're going through this?

Yes, it happened because I was intrigued by your comment to Bruce Abbott
that the reason for the decorrelation between the perceptual signal and
the disturbance signal was noise.

As I recall, that was not the subject. It was the lack of correlation
between d and qi, and between qi and qo, as measured. The perceptual signal
is not measurable, as far as I know.

But at the end of this message, you say you have measured the reference
signal, so I suppose you are talking about simulations, for which the
perceptual signal is definitely measurable.

When this came up before, you presented much data for the correlation
between p and d. Recently you have been asserting that qi is really p
because there is no CCEV. It seems to me that what you are really doing
is finding any way you can to say that whatever I contribute is wrong.
Well, so be it. I don't intend to stop trying to contribute. PCT is worth
more than personalities.

If your analysis holds up, you will have shown that a lack of correlation
is to be expected even in the absence of noise when the functions relating
the variables are pure integrators (or, I presume, differentiators). This
does not, however, imply that if a lack of correlation is observed, the
explanation must be that the pertinent functions are pure integrators, as
you seem to be saying:

It would be nice if you would be a little less selective in your quoting,
and include the bits where I said that your noise explanation was valid
as well. What I wanted to see whether I would agree with was whether it
was the _only_ explanation. To do this it was necessary to examine the
behaviour of a noiseless system. And I examined the case of the pure
integrator output function in a loop with no transport delay because
that is a loop sometimes cited, and one I found I could analyze.

However, I was talking about the real
system in which d affects qi through a proportional function Fd, and qi
affects o through a leaky integrator that is almost proportional at the
frequencies where the decorrelation is the greatest.

I am quite unclear as to what this has to do with the statement whose
correctness I was trying to check. The relevant interchange was as follows:

If we, as external
observers, can see that qi has a 0.2 correlation with d, we can conclude
that no useful reconstruction of d from qi could be made. Martin, I am
sure, will agree.

Before I did the analysis, I might have agreed with you. Now I hope you
will agree with me that I would have been wrong to do so.

Not at all. If I had been talking about the kind of system you describe,
you would have been wrong to agree.

I have shown that there exists a system in which qi has a correlation
of 1/CR with d (which could be much less than 0.2), and in which d can be
exactly reconstructed from qi. This shows that I would have been wrong to
agree with your statement, a statement that is quite independent of what
system was being tested.

I would be much happier if you would do as I earlier suggested, and
present the results you have, showing the actual correlations you obtained
in modelling these systems. I know you have them, because you have posted
samples before. I would believe the analysis much better if your simulations
showed it to give the right answer.

Now, since you say that the decorrelation is worse at frequencies low
compared to the cut-off of the leaky integrator, I gather that you have
those data as well, and can post the results not only as a function of
control ratio, but also as a function of the relation of the disturbance
waveform frequency to the integrator cut-off frequency.

I'm assuming, of course, that your disturbance signals in these experiments
were low-pass random noise, or sinusoids.

Your analysis would
predict a high correlation between d and qi, and between qi and qo, under
those conditions, which is not what we observe.

Let us all see what "we" observe, then. I'm happy to take your word for the
qualitative results, but for this purpose, quantitative is much more
useful. And persuasive. What we need is just a tabulation of correlations
for a single simple loop with a leaky integrator output function that has
a specified cut-off frequency, where the rows of the table are the upper
frequency limit of the disturbance signal.

The correct explanation for
the low correlations we observe is, as I said to Bruce Abbott, system noise
(which includes uncorrelated variations in the reference signal).

Which can be observed only in simulation models. From this I know that you do
have the data and can present it.

I _love_ the way you substitute assertion for analysis or data. What would
constitute a proof of what you say? Clearly it would be the data you are
going to post, since the correlations will be high when you simulate
without system noise and low when you add system noise. It's the high
correlations in noiseless conditions that will be interesting.

As you will have noted if you read the messages in this thread (which I have
to assume you did, since you commented on them), the analysis gives only
a _maximum_ value of the correlation. Any noise effects will reduce the
correlation below this maximum, but they can't increase it above the
maximum. The analysis will be proved wrong if your data for a loop with
a perfect integrator and minimal loop delay (you can't simulate zero) give
correlations much above 1/CR. (Statistical fluctuations may bring particular
samples a bit above 1/CR for short periods of control).

Anyway, let's see the data rather than verbal quibbles.

Martin

[Bill_Powers1]

[From Bill Powers (980324.1206 MST)]

Martin Taylor 980316 0:55--

But at the end of this message, you say you have measured the reference
signal, so I suppose you are talking about simulations, for which the
perceptual signal is definitely measurable.

I don't know which comment you're talking about. Did I say I had _deduced_
the reference signal? (That is, referring to the paper in Hershberger). Or
measured the reference _level_ (which, although model-dependent, can be
measured from the data)? If I spoke about measuring the reference signal, I
misspoke myself. Only the reference _level_ is measurable. Nothing in a
simulation is "measured," because all values there are hypothetical.

The rest of this seems to be from an earlier post. The above may be also,
but I don't recognize it.

Best,

Bill P.

[Martin_Taylor4]

[Martin Taylor 980325 10:25]

Bill Powers (980324.1206 MST)]

Martin Taylor 980316 0:55--

But at the end of this message, you say you have measured the reference
signal, so I suppose you are talking about simulations, for which the
perceptual signal is definitely measurable.

I don't know which comment you're talking about. Did I say I had _deduced_
the reference signal? (That is, referring to the paper in Hershberger). Or
measured the reference _level_ (which, although model-dependent, can be
measured from the data)? If I spoke about measuring the reference signal, I
misspoke myself. Only the reference _level_ is measurable. Nothing in a
simulation is "measured," because all values there are hypothetical.

Here's the relevant quote from Bill Powers (980315.0339 MST):

+However, I was talking about the real
+system in which d affects qi through a proportional function Fd, and qi
+affects o through a leaky integrator that is almost proportional at the
+frequencies where the decorrelation is the greatest. Your analysis would
+predict a high correlation between d and qi, and between qi and qo, under
+those conditions, which is not what we observe. The correct explanation for
+the low correlations we observe is, as I said to Bruce Abbott, system noise
+(which includes uncorrelated variations in the reference signal).

Since you had made a point earlier in the message about fluctuations in
the perceptual signal not being measurable, I took this statement to be
inconsistent with the idea you were talking about human experiments. In
simulations, the uncorrelated variations in the reference signal _are_
measurable, as are fluctuations in the perceptual signal. Hence, I took
you to be talking about simulations.

It is in accurate simulations that the adequacy of the analysis can be
tested.

Martin

[Bill_Powers1]

[From Bill Powers (980325.0947 MST)]

Martin Taylor 980325 10:25--

But at the end of this message, you say you have measured the reference
signal, so I suppose you are talking about simulations, for which the
perceptual signal is definitely measurable.

+The correct explanation for
+the low correlations we observe is, as I said to Bruce Abbott, system noise
+(which includes uncorrelated variations in the reference signal).

Since you had made a point earlier in the message about fluctuations in
the perceptual signal not being measurable, I took this statement to be
inconsistent with the idea you were talking about human experiments.

The perceptual signal is part of a model. Fluctutations in a perceptual
signal are therefore unmeasurable; they are imaginary. Same goes for the
reference signal, the error signal, and the output signal. All parts of a
model; all figments of our imaginations; none is measurable.

I am talking about human experimental measurements, not models. There is no
need for models in explaining the effects I'm talking about. Between qi and
qo there are two paths: the forward and feedback paths. The feedback path
is proportional in our experiments; we can deduce the form of the forward
path by comparing variations in qi with variations in qo. The net relation
between qi and qo is described by the amplitude and phase relation between
qi and qo.

On the other hand, there is only one path between d and qi; it is strictly
proportional at all frequencies. So we can measure (statistically)
everything needed to determine whether the low correlations are due to
phase shifts of repetitive waveforms, or to noise.

In simulations, the uncorrelated variations in the reference signal _are_
measurable, as are fluctuations in the perceptual signal. Hence, I took
you to be talking about simulations.

Our problem is with the word "measurable." I trust you see what I mean. You
can make anything in a simulation be anything you want.

It is in accurate simulations that the adequacy of the analysis can be
tested.

That's one way. It's not needed here.

Best,

Bill P.