Misc angels on selected pinheads

[William_T_Powers4]

[From Bill Powers (950504.0530 MDT)]

Martin Taylor (950502 21:00) --

As to playing Ping-Pong by moonlight, the primary limitation is ...
signal-to-noise ratio and the integration time-constant needed to
smooth the signal.

     Aha, so now you DO agree that the information rate is critical.
     Great!

No, I didn't say that. I don't know how to define information rate,
because I don't know what assumptions about the message to make. You
have convinced me that there is no single way to measure information.

     Yes, That's exactly why it takes time to perceive where the ball
     is/was. The perceptual system takes time to integrate and smooth
     the signal. That's part of the transport lag, and is incompatible
     with the idea that the effect would remain if transport lags were
     zero.

We seem to have different definitions of transport lag and integration
lag. Mine is the engineering definition:

···

*********************
                                  *
                                  *
                                  *
                                  *
Input *******************
                                  >
                                  > *******************
                                  > *
                                  > *
                                  > *
Output with | * *
Transport lag **************************** *
                                             *
                                  > *
                                  > *
                                  > *
                                  > *
Output with *
Integral lag ********************

Note that the output with an integral lag starts to rise at the same
instant the input rises, while the output with a transport lag remains
at zero until after a delay, and then the input waveform is reproduced
exactly. Both types of lag can be present, in which case the integral
response would not start rising until after the transport lag. The
integral "lag" is the time it takes the integrator output to reach an
arbitrary level, and depends on where that level is set. The transport
lag is absolute. Only true transport lags cause stability problems in
control systems; single integral lags are stabilizing.

In the low-light-level case, the integral lag is what slows the control
process. The transport lag is the same at all light levels.

     It's not an output problem or an error problem, but something the
     perceptual system does to delay the perception when the signal is
     noisy. That's transport lag in the visual system, in my book.

Get another book. It's integral lag, which is not transport lag.

     The mechanical-inertial limitations are in the part of the loop
     between the output of the ECU and the input to the perceptual
     system, aren't they? Don't they assure that there is a lag between
     the output and ANY possible effect on the controlled perception?

You're confusing integration lags, which can help to stabilize a system,
with transport lags, which help destabilize it. The phase shift in an
integrator is 90 degrees at all frequencies. The phase shift with a
transport lag rises without limit as frequency increases, reaching 180
degrees when frequency = 0.5/(transport lag). In systems with a single
integration but no transport lag, the feedback effect begins to appear
at the same moment the disturbance occurs. The effective delay decreases
as loop gain increases. In systems with transport lags alone, effective
delay is constant and independent of loop gain.

     Which component, perceptual processing or mechanical inertial lag,
     is more important seems to me rather a red herring.

You missed the point. In the model of which I spoke, it happened that
the perceptual lag was a true transport lag. That is why it had a
different effect from the integral lag in the output function. The
difference was not due to WHERE the lag appear, but to the KIND of lag
that was used.

     So where do the mechanical-inertial effects get off being special?

They are not transport lags. The instant a force is applied to a mass,
it begins to accelerate; the instant an acceleration occurs, the
velocity departs from zero. There is no transport lag between the
application of the force and the change in velocity.

     Nyquist sampling is the rate at which the sampling permits an exact
     (and I mean EXACT) representation of the continuous signal that is
     sampled. That's the ONLY meaning of a Nyquist sampling rate.

The Nyquist frequency is also the frequency at which the phase shift in
the output wave becomes greater than 90 degrees relative to the input
wave. Sampling at the Nyquist frequency does not give an exact
reproduction of an input sine wave; it gives a square-wave
"reproduction" of it.

     There's no sense in which the notion of Nyquist sampling can be
     applied to a discrete system "in which events take place at
     discrete and synchronized instants."

Even to speak of sampling rate is to speak of a synchronized system with
samples occuring at regular discrete intervals.

I see no evidence that human organization is optimized in any way:

     Oh, so you don't believe Rick's "Hierarchy of Perception" paper?

I don't know what you mean.

     If you need to stop and eat every five minutes, you are going to
     have more problems than someone who can go for a few days without
     food, at a pinch. At least if food is in short supply, as it
     sometimes is. But yes, we need some waste in an optimized system.
     Systems that are optimized for a fixed and stable environment are
     terribly vulnerable to changes in that environment.

You can make any statement appear true if you use examples only from the
extremes of a range, and stick to verbal logic. And anyway, what does
what we need have to do with what evolution has provided? We have just
barely enough to survive exactly as well as we do, and no more.

The actual bandwidth of behavioral control systems is far less
than one would guess just from looking at transport lags.

     That's an interesting observation. It would be nice to have data,
     and if you have published it somewhere I have forgotten, I'd like
     to be pointed to it.

The transport lag in the tendon-stretch reflex system is about 9
milliseconds. The implied bandwidth is roughly 1/0.018 or 55 Hz. The
actual control bandwidth is about 2.5 Hz. The main limitation comes from
mechanical properties of muscles and inertial properties of the limb
mass, which involve no transport lags.
-------------------------------
     To Rick:
     >But plausible explanations aren't enough: we need to have
     >enough data so that NO OTHER EXPLANATION IS POSSIBLE.

     That's an infinite amount of data.

That was my post. I should have said "so that no other explanation is
known to be possible." If you weren't in your lawyer mode, that's how
you would have read it, especially considering what immediately followed
it:

It's only when the
data rule out every explanation we can think of -- all but one -- that
we can claim to have a real scientific explanation ...

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

...your continuing insistence that this _component_ of perception has a
separate enabling effect to which we object -- your insistence that
somehow it is this _component_ that provides the information that makes
control possible.

     And from my side, the reason for the continuing friction is that
     you keep asserting that I say this. I _INSIST_ that I do not
     consider the perceptual signal to be breakable into components, and
     I _INSIST_ that at no time have I so assumed. However, time after
     time, you have in one way or another tried to push me into a
     position where I am presumed to take this nonsensical view, and
     then you have told me over and over how nonsensical it is. You are
     preaching to the choir, not to a Devil's Associate.

Then please explain how information about the disturbance can be "passed
through the perceptual signal" and "used" to "make control possible,"
without any ability in the system to react specifically to that
information rather than just to the total perceptual signal. Whatever
image of these processes you may have in mind, your writing is not
revealing it. The impression that your words leave in my mind (and
evidently not mine alone) is that it is something originating in the
disturbance that makes control possible.
-----------------------------

The final form of f(tau) is the same one we get using a uniformly
distributed random disturbance pattern. It represents approximately the
inverse of the transfer function of the external load.

     Nice. And not what I thought, in the case of repetitive
     disturbances. I would have expected the performance with the AC
     trained to that pattern to be better than with the AC trained to a
     variety of disturbance waveforms.

I suspect that this is a holdover from ideas in the back-propagation
field, where the training is concerned with specific signals rather than
with structural (physical) properties. The AC is not "trained to
disturbance waveforms." It is trained to minimize the error signal by
means of changing its structural properties to match those of the
external feedback loop. A random disturbance (actually introduced via
the reference signal in the model) is used because it introduces errors
over a wide range of frequencies, thus exciting the system in all the
ways that other more regular disturbances would excite it. The errors at
these various frequencies contribute to changes in the f(tau) function
at various delays; thus during adaptation we want to be sure that errors
are experienced at all frequencies within the operating bandwidth. When
that is done, we get the best possible error-correction for any
disturbance waveform.

I should mention that the AC has some limitations; if a pure integrator
is called for, the f(tau) function would have to have infinite length.
But the AC also works when used as a trimming function operating in
parallel with fixed functions, so many limitations can be handled in
that way.
--------------------------------
     Not to argue the point (see the opening of this posting) but the
     predictable world I was trying to imagine was not one with no
     noise, but one in which all the possible disturbing influences were
     known for all future time. In such a world there is no need for
     at-the-time computation, so the issue of whether a control system
     needs less processing power does not arise.

Let's argue the point. In order for all possible disturbing influences
to be known, the system that knows them would have to have infinite
storage capacity. And even if it knew them, it would still have to
compute what actions to produce in order to provide itself with food,
water, mates, shelter, etc. in the presence of all these known
disturbances. The only way to avoid at-the-time computation would be to
precompute all possible actions and their interactions with the
environment and store the results for ready reference in the future: if
I want to consume my daily 638 grams of food of various compositions, I
should press button A which executes the stored program for acquisition
of the required food elements, given my current position in the
environment, all the disturbances that are present, the physical
properties of myself and other objects, and so forth. There's no way to
avoid computation.

It would be far more feasible to use control systems instead.
----------------------------
     I agree that such an argument is perilously close to the "angels on
     pin-head" kind of discussion. My reason for bringing it up is
     simply to put control systems in the context of an otherwise
     chaotic environment--an environment that is neither totally random
     (where control would not work) nor totally predictable over
     indefinite periods (where control would not be required). We live
     in a world in which control seems to be the only option, regardless
     of any other benefits it might offer in terms of efficiency and
     processing requirements.

I don't know what it will take to persuade you that a totally
predictable environment would still need control systems. I guess I
should give up on that goal.
---------------------------------------------------------------------
Best,

Bill P.

[_Martin_Taylor]

[Martin Taylor 950504 10:30]

Bill Powers (950504.0530 MDT)

Once again, I'll try to be concise, and selective as to which issues I treat.

Low-light vision:

The (conscious) perceptual effect is not of a Ping-Pong ball smeared over
its track, as your graph would suggest, but of a ball in a precise position
appreciably delayed as compared with the tactile or auditory perceptions.
When you look at a swinging pendulum when one eye is covered by a dark
filter, the pendulum seems to swing in an elliptical path in depth, not
to have a blurry bob. This suggests that the perceptual effect at a very
low level (nothing to do with consciousness) is also a simple delay. That
the phenomenon is integration in the perceptual function (not the output
function, where we usually talk about integration) seems to have nothing
to do with whether the perceptual signal shows the effect of a straight
transport lag (using the definition in your diagrams).

In the low-light-level case, the integral lag is what slows the control
process. The transport lag is the same at all light levels.

Quite apart from the above argument, I'm not at all sure that this is true
neurologically. But not being a neurophysiologist, I won't argue strongly,
if you have evidence that it is so.

There's quite another point about the integral lag issue. The point at issue
is the effect of current output on current perception. Even in the physically
impossible case of zero transport lag, an integral output has zero effect at
zero time, and negligible effect for small time. The effect of a prior
output on perception grows continuously over time, and the total effect is
the summation of the prior effects over all time. The phrase to which
you initially objected was something like "from any prior outputs as are
still effecting the perception." (Not a real quote). If you have an arm
that has been accelerated over the last 100 msec, it has a velocity that
continues to affect the perception of its position, and that effect will
probably still be evident for several hundreds of msec. The output that
caused it may have stopped long before the effect on perception stops. And
that's a general statement about the real world.

    To Rick:
    >But plausible explanations aren't enough: we need to have
    >enough data so that NO OTHER EXPLANATION IS POSSIBLE.

    That's an infinite amount of data.

That was my post. I should have said "so that no other explanation is
known to be possible." If you weren't in your lawyer mode, that's how
you would have read it, especially considering what immediately followed
it:

As I noted in my foollowing comment. I suppose my comment was unfair, since
it came from a background unknown to you. Sherlock Holmes made the same
comment as you did, and when I first read it I complained (to my father,
I think) about its illogicality. Many times over the intervening 50 years I
have been tweaked by people saying the same thing. If you hadn't capitalized
it, I probably would have let it pass, especially as you properly weakened
it in your next sentence or two.

···

==================

    Nyquist sampling is the rate at which the sampling permits an exact
    (and I mean EXACT) representation of the continuous signal that is
    sampled. That's the ONLY meaning of a Nyquist sampling rate.

The Nyquist frequency is also the frequency at which the phase shift in
the output wave becomes greater than 90 degrees relative to the input
wave. Sampling at the Nyquist frequency does not give an exact
reproduction of an input sine wave; it gives a square-wave
"reproduction" of it.

Sampling does not "give" any kind of reproduction, square or otherwise.
Sampling provides data. Reproduction is created by some process using
these data. Any kind of sampling gives a square wave reproduction if the
process is designed to create a square wave from the samples. But with
an appropriate filter process (i.e. a filter with the exact bandwidth of
the original waveform for which the sampling was at Nyquist rate), Nyquist
sampling produces an exact reproduction of the original wave. Sampling at
any lower rate cannot, and sampling at any higher rate is unnecessary, as it
gives only redundant information. Sampling is normally done at higher then
Nyquist rates because filter processes are not mathematical ideals and it's
much easier to get a close (not necessarily exact) reproduction of the
original wave if you use an easily constructed filter process and a high
sampling rate.

Even to speak of sampling rate is to speak of a synchronized system with
samples occuring at regular discrete intervals.

A sampling system is a system that takes samples. No more, no less. If the
samples are synchronized when taken from different signal sources, it is
a synchronized system. If the samples are taken at regular intervals, it
is a periodic sampling system. Variable rates are quite feasible and
possibly sometimes useful. There is value in taking samples at Poisson
intervals, if the signal to be sampled has no defined upper frequency limit.
To do so can reduce aliasing problems. There are all sorts of sampling
systems. How the samples are taken determines the maximum accuracy with
which the reconstructor can be assured of matching the original waveform.
----------------

Then please explain how information about the disturbance can be "passed
through the perceptual signal" and "used" to "make control possible,"
without any ability in the system to react specifically to that
information rather than just to the total perceptual signal. Whatever
image of these processes you may have in mind, your writing is not
revealing it.

Obviously so, but I con't figure out how to change the writing so as to
tune into your models in a way that will solve the problem. But perhaps
if you try to force an interpretation in which there is no "componentization,"
and worry it around like a puppy with a new toy bone, at some moment there
might come a glimmer. Then it might become possible for us to have a
true dialogue on the matter.

I also will (as I have) worry the problem. But so far, my glimmer has not
come. I thought it had, several times, but each new attempt to say things
in a different way has run up against the identical misunderstanding.

The impression that your words leave in my mind (and
evidently not mine alone) is that it is something originating in the
disturbance that makes control possible.

You see, part of my problem is that I simply cannot see any way in which
anything I have ever written would lead you to say something like this. And
since I can't, I find it hard to know how to say anything to correct it.

I have a possibly silly notion. I know you have written several miles of
paper on the subject, most of which I have read. Always I think that what
I write is totally consistent with everything you have written on how
control works. So why don't you try to write it in a different way, somehow,
in a way I can't imagine, that might make ME see it in a different light
and perhaps would let me see how to get my view across so that it can be
criticised for its faults rather than for my writing style.

There's a related problem, a matter of a triangle relationship. I think I
write things that agree with your position. So does Rick. When I read his
things, I often think he is in substantial disagreement with you, and
criticise him for what I think are his errors. He thinks, and apparently
so do you, that his writings agree with you and you just reinforce each
other's words. How is it that I often think you are right and he is wrong,
just as he thinks you are right an I am wrong? One possibility is that I
have a total misunderstanding of your writing, but that doesn't explain
why what Rick writes about some aspects of PCT often seems opposed to what
you write, or why it is that what you write seems correct and self-consistent
(most of the time).

---------------------
On a totally predictable universe.

Let's argue the point.

I'd rather not, but if you insist.

In order for all possible disturbing influences
to be known, the system that knows them would have to have infinite
storage capacity.

Yes, presumably. Or at least very large (is the universe infinite?).

And even if it knew them, it would still have to
compute what actions to produce in order to provide itself with food,
water, mates, shelter, etc. in the presence of all these known
disturbances.

Yes.

The only way to avoid at-the-time computation would be to
precompute all possible actions and their interactions with the
environment and store the results for ready reference in the future:

That's what I meant, yes. That's what it would be able to do.

if
I want to consume my daily 638 grams of food of various compositions, I
should press button A which executes the stored program for acquisition
of the required food elements, given my current position in the
environment, all the disturbances that are present, the physical
properties of myself and other objects, and so forth. There's no way to
avoid computation.

I didn't suggest that computation would be avoided. I suggested that
the computation could be done at leisure, long in advance, so that the
"right thing" could be done exactly at the time it was necessary, without
at-the-time observation or control.

It would be far more feasible to use control systems instead.

I agree, but for the third(?) time, that's not the point. The point is
whether control systems would be required. And they wouldn't.

I don't know what it will take to persuade you that a totally
predictable environment would still need control systems. I guess I
should give up on that goal.

So far as I can see, you haven't yet introduced any arguments in favour
of that position, so you haven't started to try to persuade me of it.
What you said in the paragraph I quoted was all in support of the idea that
control systems would NOT be needed, but would be much simpler and easier
to construct and use. You've said that before, and I have agreed.

Personally, I don't think any angels can dance on the head of a pin. It's
too slippery and either they'd fall off or they'd be flying, which can't
count as dancing.

My son, at around the age of 7, pointed out that the egg did indeed come
before the chicken, if birds descended from dinosaurs.

Martin