Info about disturbance

CSGnet Archive CSG1993
1

[From Bill Powers (930331.1030 MST)]

Rick Marken (930330.2100) --

I don't think Allan is going to use information about o (the
o generated in the real, closed loop run) at all. In fact, the
way I conceive of Allan's study, you don't need to save o at
all; all you need is d and p from the closed loop run.

As I understood it, Allen was going to assume that control was
perfect, so o = -d with a constant reference signal. So the
procedure depends on knowing o, which amounts to knowing d.

But let's wait to see what Allen says.

···

-----------------------------------------------------------
Martin Taylor

(930330.1615) --

It tends to be ignored that we have other ways than movement of
affecting our environment, and that these ways can also be
important in controlling perception -- very, very important.
Chemical output, which can be terribly important in mate
selection, for example.

There are different kinds of "important." I think it's unlikely
that there are control systems that vary pheromone output in
order to attract a mate, even though it's true that emitting
pheromones does attract mates and makes continuation of the
species possible. The importance here is from the perspective of
the scientist who realizes the role that phenomones play, and the
essential effect on survival of the species. Some human females
are attracted by the smell of sweat, and thus sweating
contributes to the preservation of the jock species. But the
control systems involved with perspiration control for
temperature, not sex or survival.

It's just as much an error to attribute purpose to nonpurposive
effects as it is to do the opposite.
--------------------------
(930330.1120) --

The diagram you gave, below, won't work:

             ----> Signal X (which should match the disturbance)
            >
       mystery function M(r, p)
        ^ ^
        > >
        > > V (reference signal R(t) into ECS)
        > > >
        > <-------|
        > V
        >---------->comparator------- error = P-R
        > >
    perceptual output
     signal P(t) function O(error)
        ^ |
        > V
        > output signal
        > (accepted as mirroring the disturbance)

If the reference signal is zero, the signal X won't mirror the
disturbance. In general, it won't. What you need is

             ----> Signal X (which should match the disturbance)
            >
       mystery function M(O,P) <-------------
        ^ ^
        > >
        > ----------------<------------------ <-----
        > V (ref sig R(t) into ECS) |
        > > >
        > V |
        >---------->comparator------- error = P-R |
        > > >
    perceptual output |
     signal P(t) function O(error) |
        ^ | |
        > V |
        > output signal O -->-
        > (accepted as mirroring the disturbance)

The output signal does NOT follow the reference signal, so must
be sensed directly.

If we accept the perceptual signal as following the external
controlled quantity qi, we have simply

      p(t) = O + D, exactly

Thus, D = p(t) - O, exactly. If the signal X is to match D, the
higher system must sense the output itself, which requires a
sensor that the illustrated control system does not have.

In all our examples so far, the reference signal has been fixed
at 0. According to your diagram, the mystery function would then
receive the perceptual signal and a null signal, with only the
perceptual signal variations then producing signal X. This would
say that when the reference signal is zero, the perceptual signal
is the same as the disturbance, which is not true. Check it out
with Simcon.

Furthermore, the sensor should not really detect the output
signal directly, but the EFFECT of the output signal as
translated through the output transducer and through the
environmental link between the physical output and the controlled
quantity qi. But that's a side-issue.

I think that QE is not quite yet D.
---------------------------
In these developments, "information" is being bandied about
loosely. From Garner, information is simply the negative log of
the number of values that a variable can attain. If the
disturbance can take on any value between 0 and R, and this range
is divided into n intervals r, the information in the disturbance
is just -blog(R/r). Let's assume a continuous disturbance with a
bandwidth less than that of the control system.

The controlled variable, on the other hand, with a constant
reference signal, has a range of R', which equals R/(1+G), G
being the loop gain at the maximum frequency contained in the
disturbance. The quantum of measurement (in a tracking
experiment) is the same, r, the screen resolution. So the
information in the input quantity is -blog(R'/r), which is
equivalent to -blog(R/[(1+g)*r]).

The ratio of the information in the perceptual signal to the
information in the disturbance is

                 blog(R/r) - blog(1+G)
                 --------------------
                       blog(R/r)

This ratio is self-evidently less than 1 with any loop gain of 0
or greater. With very large loop gains, the ratio can become very
small: there is much less information in the perceptual signal
than in the disturbance. In Ashby's terms, information from the
disturbance has been "blocked" from reaching the controlled
variable. In fact, the ratio can be negative: the control system
itself can be putting information into the perceptual signal
(oscillations due to the quantized nature of the controlled
quantity).

This only treats the AMOUNT of information; it says nothing about
the "contingent uncertainty," which depends on the correlation
between the perceptual signal and the disturbance. The maximum
information would flow if the perceptual signal were perfectly
correlated with the disturbance. So the actual fraction of the
information about the disturbance present in the perceptual
signal must be even less than the ratio computed above. The
greater the loop gain, the less the information about the
disturbance that exists in the perceptual signal.

Clearly, if the perceptual signal can attain only n values while
the disturbance can attain m values, and m > n, it is impossible
to reconstruct the disturbance from the information contained in
the perceptual signal. The larger m is in comparison with n, the
poorer the reconstruction would be. When the range of the
perceptual signal has been reduced to 1/50 or less of the range
of the disturbance, practically nothing could be said about the
disturbance from knowledge of the perceptual signal.

It seems to me that this argument, which hardly takes one step
away from first principles, supports the case that Rick and I are
making in a simple and direct way.
--------------------------------------------------------------
Best,

Bill P.

2

[Martin Taylor 930331 14:15]
(Bill Powers 930331.1030)

The diagram you gave, below, won't work:

            ----> Signal X (which should match the disturbance)
           >
      mystery function M(r, p)
       ^ ^
       > >
       > > V (reference signal R(t) into ECS)
       > > >
       > <-------|
       > V
       >---------->comparator------- error = P-R
       > >
   perceptual output
    signal P(t) function O(error)
       ^ |
       > V
       > output signal
       > (accepted as mirroring the disturbance)

If the reference signal is zero, the signal X won't mirror the
disturbance. In general, it won't.

Have another look. There are two inputs to function M. But there is
only one input to function O. The input to function O is (P-R).
M is defined as equivalent to O except that it incorporates (R-P)
as a first stage. It is functionally identical to O(R-P). If you
want to be even more general about it, you can remove the requirement
for odd symmetry in O, and define M as -O(P-R).

Why does the value R = 0 have any special quality? In what way will
X not mirror O?

Notice that I never claimed O mirrors the disturbance. That's Rick's
claim (I don't remember you making it without the necessary simplifying
assumptions). The diagram and thought experiment is intended to show
the inconsistency of simultaneously maintaining the two claims:

(1) The output mirrors the disturbance,
(2) There is no information about the disturbance in the perceptual signal.

In all our examples so far, the reference signal has been fixed
at 0. According to your diagram, the mystery function would then
receive the perceptual signal and a null signal, with only the
perceptual signal variations then producing signal X. This would
say that when the reference signal is zero, the perceptual signal
is the same as the disturbance, which is not true.

Neither is it claimed. Mystery function M is not a unity operator.
It has every characteristic that the output function has, including
any leaky integrators and nonlinearities. It has access, apart from
the specific inputs described in the figure, only to that information
available to the output function. If the output function contains or
is affected by some kind of homunculus that can look to see whether
gain should be changed, so does (is) the M function.

Function M simply takes the reference signal, which no-one has claimed
to contain information about the disturbance, together with the perceptual
signal, which some people have claimed to have no information about the
disturbance, and reproduces a mirror image of the output signal, which
some people have claimed reproduces the disturbance exactly.

There are three possibilities: (1) The reference signal contains information
about the disturbance; (2) The perceptual signal contains information
about the disturbance; (3) The output signal does not mirror the
disturbance, and is uncorrelated with the disturbance. At least one of
these three must be true. Rick's claim is that all are false. The
Q that ED is that Rick's claim is false; accepting (1 and 3) as false
requires (2) to be true.

QED.

Furthermore, the sensor should not really detect the output
signal directly, but the EFFECT of the output signal as
translated through the output transducer and through the
environmental link between the physical output and the controlled
quantity qi. But that's a side-issue.

For the purpose of the argument, it's a side issue, which is why I
so labelled it in my original posting; but it will loom large in the
information-theoretic argument yet to come, once we get this roadblock
out of the way.

ALL THE INFORMATION IN THE DISTURBANCE CAN BE RECOVERED FROM THE
PERCEPTUAL SIGNAL (There--I can CAPITALIZE too).

In these developments, "information" is being bandied about
loosely.

For the purpose of arguing with Rick, I use Rick's definition. Besides,
if we can recover the disturbance exactly out of the perceptual signal
by a demonstrable mechanism, there seems little need to fluster about
the abstractions of information theory, which are often misunderstood.

Your argument from information-theoretic principles is, so far as I can
see, correct in demonstrating that the current value of the disturbance
is not recoverable from the current value of the perceptual signal. But
then, I know of nobody who thought it was. That's red herring that
Rick keeps asking Allan and me to accept as a fish fresh for cooking.
I won't, and I don't think Allan will, either. (Though I really like
a good kipper, it isn't so nice after it has been dragged across the
fox's trail).

I should actually compliment you on the IT analysis, because it saves me
a bit of work later. I was intending to go through the points you raised,
but maybe now I need only refer to your posting.

Martin

3

[From RIck Marken (930331.2100)]

Martin Taylor (930331 14:15) --

ALL THE INFORMATION IN THE DISTURBANCE CAN BE RECOVERED FROM THE
PERCEPTUAL SIGNAL (There--I can CAPITALIZE too).

That's your position, all right. And I agree that this is true
as long as p = o + d and you have access to both p (the
perceptual signal) AND o. But no information about the disturbance
can be recovered from the perceptual signal all by itself (or, as
you will see from the simulation, with the help of the output function).

For the purpose of arguing with Rick, I use Rick's definition [of
information]. Besides, if we can recover the disturbance exactly out
of the perceptual signal by a demonstrable mechanism, there seems little
need to fluster about the abstractions of information theory, which are
often misunderstood.

This is a very helpful way to formulate the problem; it let's us get
past definitions and into working simulations of control systems. I
have developed a HyperCard stack that implements your ingenious method
of testing whether or not one can recover the disturbance exactly out of
the perceptual signal. Your method does not use knowledge of o to extract
the information about d from p -- so I consider it quite a fair method.
It does use information about the output function (that transforms error
into the variable that affects the perception). But I think you imagine
that it is the output function that extracts (recovers) the information
from p. In fact it doesn't -- but that's what the simulations will show.
You will find when you run the simulations that there is only one case
in which you can recover the disturbance from the perceptual input
(recover it from the output os what Bill calls the Mystery function. This
occurs ONLY when R = 0 and P = O + D. Even in this case, however, when
you introduce the realistic possibility that there will be neural and
integration error added to the computation, the closed loop output, o,
still approximates -D (with a very small RMS deviation); while the
open loop output deviates from D by a huge amount.

The HyperCard stack lets you compute the match of the open loop output of
the Mystery function and the closed loop output of the output function
(both computed simultaneously, of course) under various conditions:
fixed or variable reference signal (fixed at 0 or 50), linear or non-
linear connection between o and p (ie. p = o + d is linear; p = o^3 + d
is non-linear) and with noise (the same noise, just to make it fair)
added or not added to the input to the output and Mystery function.

The results are quite interesting; but they do show that in one very
specific condition (p = o + d, r = o, no error in the system) you
are right -- the output of the Mystery function can be considered to
be the extracted value of the disturbance. So this means that there
IS information in the perception when the reference for that perception
is 0, there is no error contributing to the signal going into the
output function -- ie. no neural noise -- and the output of the system,
o contributes linearly (and with a coeficient of 1) to the perecptual
input variable. Given those conditions, you can fairly claim that in-
formation about the disturbance can be extracted from the perceptual
signal. But there is no such information available when any of these con-
ditions don't prevail -- that is, when the reference is not a constant = 0,
when there is neural noise or when o has non-linear (or linear with
a coefficient not equal to 1) effects on p.

Would you like the stack?

Best

Rick

4

[From Bill Powers (930806.0745)]

Allan Randall (930805.1525) --

At the meeting, Martin Taylor and I cleared up the impasse about
"information in the disturbance." There's nothing like a face-to-
face discussion to cut through the crap.

As I understand it, you and Martin have been using "disturbance"
to mean the actual fluctuation in the perceptual signal due to a
change in the CEV that is induced by some external cause. Because
the CEV is explicitly represented in perception, such a
fluctuation does, of course, have an immediate and unique effect
on the perceptual signal. Knowing the organization of all the
components of the closed loop, one could deduce the amount of
that fluctuation due to the system's own output, and thus the
portion of it (unaccounted for by the system's output) that must
have resulted from some external influence. The control system
itself would not perform this calculation, but an informed
observer of the system could do so.

What Rick Marken and I have been referring to as "the
disturbance" is the external CAUSE of the fluctuation. Clearly,
the fluctuation in the CEV contains no information about what
caused it. Identically the same fluctuation in perceived light
intensity, for example, could be caused by a weak light source
close to the receptor or a powerful one far away. It could be
caused by a multiplicity of light sources, each contributing some
changing part of the total sensed flux. There is and can be no
information in the perceptual signal about the causes of
fluctuations in the CEV.

Does this take care of the problem for everyone?

Best,

Bill P.

5

[Allan Randall (930806.1520)]

Bill Powers (930806.0745)

Hey, this posting actually clears quite a bit up! However, it raises
new concerns and also I want to be sure I've understood you. So:

As I understand it, you and Martin have been using "disturbance"
to mean the actual fluctuation in the perceptual signal due to a
change in the CEV that is induced by some external cause.
...
What Rick Marken and I have been referring to as "the
disturbance" is the external CAUSE of the fluctuation.

Just to make sure I understand, is the following a proper paraphrase?:

Disturbance(1) (Taylor & Randall): that part of the CEV fluctuation
that is not caused by the control system itself, and thus is due to an
external cause.

Disturbance(2) (Powers & Marken): the external cause of that part of
the CEV fluctuation that is not caused by the control system itself.

Assuming this is an accurate rewording, then I am happy with your
depiction of my use of the word "disturbance." If it is not accurate, then
you can ignore the rest of this posting.

It sounds like you agree that the percept contains information about
disturbance(1), but not about disturbance(2)? Is this right?

However, I do not think I see why anything changes when we switch to
definition (2). Perhaps the reason we have been talking past each other is
that you are making a distinction that I do not really see. Is there any
*meaningful* way in which disturbance(1) and disturbance(2) are different?
Isn't "that part of the fluctuation caused externally" simply a way of
describing "the external cause of the fluctuation"? How can something be
said to have information about (1) and not about (2)? Unless you can
provide some reasons to make this distinction, I have to say my arguments
about information in disturbance apply to both definitions.

If there *is* a meaningful distinction here, I'm not sure how definition
(2) fits into the arguments Rick Marken has been making. Rick and I have
been talking about reconstruction of the disturbance in a computer
simulation. What are the values Rick was looking for me to reconstruct if
he was using (2) as opposed to (1)? To Rick: do you concur with Bill's
analysis? Or do you feel that nonlinearities erase information about (1)
as well as (2)? If not, in what sense do the nonlinearities have anything
at all to do with erasing information about (2)?

... Identically the same fluctuation in perceived light
intensity, for example, could be caused by a weak light source
close to the receptor or a powerful one far away.

If they both really do give identical fluctuations, then there is no point
in making any distinction at all between them. Certainly calling this extra
aspect (which doesn't even affect the CEV) part of the "disturbance" seems
a tad bizaare, considering that it's actually the part that doesn't disturb
the CEV at all! Why call it "disturbance," then? Why include it as part
of the external cause of the fluctuation if it has no effect on the
fluctuation whatsoever? What is it even doing in our control system
diagrams, for that matter? Why do you even care enough about it to label it?

I won't say any more until I find out if I've paraphrased you correctly or
not (I haven't talked to Martin yet to get his version of things).

Anyway, it sounds like the Durango meeting was productive with respect to
the information issue. I hope it was equally useful in other areas.

···

-----------------------------------------
Allan Randall, randall@dciem.dciem.dnd.ca
NTT Systems, Inc.
Toronto, ON

6

[Martin Taylor 930817 17:10]

(Bill Powers 930806.0745, Allan Randall 930806.1520, Bill P 930806.1945)

I'm back at work, after a delightful meeting (with all my e-mail friends
in the flesh) and a wonderful view of parts of the USA I've never before
seen. So now it's back to electronic communication. As Dag Forsell noted,
it's easier when you can point to pictures while you talk.

I agree wholly with Bill in the cited exchange with Allan on the "information
in the disturbance". In all of our previous discussions, Allan and I
started from the premise that there is nothing in the perceptual
signal except for the state of the CEV (possibly a noisy representation
of that state, which is the basis for the computational argument). That
is what I cleared up, as Bill says, in face-to-face conversation. I had
thought that this point had been made quite clear in several postings, but
apparently it hadn't.

Allan, in his posting noted above, was not clear on the importance of the
distinction between "disturbance (1)" and "disturbance (2)", (1) being the
fluctuation that would occur in the CEV if it were not opposed by the output
of the ECS, and (2) being the external cause of that fluctuation. I agree
wholly with Bill that the distinction is very important, because whatever
perceives the disturbance (2), it is not the ECS that counters disturbance (1).
It may be an analyst; it may be another PIF that observed somewhat more
than the CEV of interest; but it is not the ECS that is the object of our
analysis.

Bill suggested in our conversation that we should use the term "fluctuation"
in the CEV to express what we wanted to say when we used "disturbance."
That may be better, but I can imagine quite a few ways in which that wording
could also lead to misunderstandings. But we can try it. I'd like, some
day, to get on with the development of the information analysis. Not now,
pressures back at work being what I knew they would be.

Martin

7

Hi, Martin --
(copy to Rick)

Repeating myself, I remind you that we have agreed to
distinguish between "disturbance" (an external variable that
contributes to the state of the CEV via some physical link)
and "fluctuation"...

Yes. Did I forget? I thought I was keeping to that, but one
can never be sure. In responding to Rick, I should have made
things clearer. I was referring to the effect of the
disturbance on the CEV, not to the disturbance itself or to the
fluctuation in the CEV, when I mentioned "information about the
disturbance" or whatever term I used.

I was speaking to both of you, really.

You now seem to have converted my dichotomy into three terms:

The disturbance itself

The effect of the disturbance on the CEV

The fluctuation in the state of the CEV.

The middle term is unobservable. We can observe the state of a
disturbing quantity (the mass of a disturbing planet and its
relative distance) and the fluctutation in the CEV (the deviation
of a spacecraft from the intended trajectory in the presence of
all disturbances and thrust from the rocket engine). We can't,
however, observe the effect of one planet on the CEV. That can
only be calculated on the basis of the inverse square law of
gravitational attraction and the distance to a particular
disturbing planet of known mass. The Earth, Moon and Sun are also
contributing disturbing influences in different directions; only
the momentary vector sum of all influences, including the rocket
engine, determines the state of the CEV, the trajectory of the
spacecraft.

Fluctuations of the CEV result through physical laws from the
vector sum of all forces acting on the mass of the spacecraft.
The control system can sense only the state of the CEV, measured
relative to the reference signal and in units of perceptual
signal. From knowing only the deviations of the CEV from the
reference state, nothing inside the spacecraft can deduce the
masses or distances of all perturbing bodies (or even one of
them), or separate from the total fluctuation the part of it due
to the rocket's own thrust. To use information about the rocket's
own thrust would require something to sense that thrust, and a
computing device that would apply the appropriate physical law to
construct the vector contribution of the thrust to the total
state of the CEV and subtract it to leave only the component due
to the vector sum of all disturbances. And that would still not
provide any information about the state of any or all
disturbances: the mass and distance of the perturbing body or
bodies. Further, the equipment necessary for doing this
calculation is not part of the basic control system. The basic
control system works without any extra equipment.

We, of course, know the position and mass of every perturbing
body and we know the thrust vector of the rocket engine. We also
know the construction of the control system which operates the
rocket engine to maintain the sensed trajectory near its
reference state. But the control system can't operate on the
basis of what WE know. We have to try to understand how the
control system can operate successfully by itself without any of
this extra information that is available to us -- in fact, how it
can operate strictly on the basis of the state of its own
perceptual signal, representing the current state of the CEV.

We know from direct observation and from modeling that the only
components a control system needs in order to operate properly
are the ones in the canonical model. If you are going to find a
role for information theory in this model, everything pertaining
to informational calculations (that makes a difference in the
system's operation) must take place in the existing components,
without adding any new capabilities. It's incumbent on you to
show that the required calculations are done somehow by the
functions already provided in the existing system, rather than
being done by you, outside the control system, using information
not available to the control system.

I don't see how you're going to do this. But perhaps you do -- I
guess we'll see the result one of these days.

Bill

8

Hi Bill & Martin --

Have you guys been talking behind my back?

Good.

I thought we had a solution, Martin. The perceptual signal
has information about the CEV. If you lose information
about the CEV, control deteriorates. So information is real
important in PCT -- it's what is controlled.

If you had bought this idea, you could call the paper you
are wrorking on "Behavior: The control of information".
Catchy title.

Best

Rick