[From Bill Powers (960914.0530 MDT)]
Hans Blom, 960912 --
In MCT, the notion "relative humidity" -- as any other notion --
would be a notion _of the model_, not something with an independent
existence in the environment. In MCT, there is no need for such a
thing as an external reality (although in simulations we may posit
one, because we need a generator of the measurements).
There's one place where an external reality is needed: where the output u
has an effect on the perceptual signal y. In PCT this is called "the
environmental feedback function;" in MCT it's called "the plant function."
We have no direct knowledge of this function (speaking as the experiencing
system, not the external engineer), but there must be some reasonably
regular connection between the output actions and the input changes if
control is to be possible under any scheme, MCT or PCT. Martin Taylor has
pointed this out frequently.
What is known
of the environment is only the measurements, the outputs of the
sensors. Nothing else. If more is "known", it is _relations_ that the
model has somehow been able to extract out of those measurements. So,
in a sense, the model takes the "engineering point of view" in that
it imposes some kind of order on the measurements -- in the form of
?certain relationships, not between the measurements per se (although
this can be done) but between "idealizations" (model states).
Yes, this is true in PCT, too. In PCT the idealization comes in the
perceptual representations built on what you call "measurements" and what we
call the "intensity level" of perception. Each sensory neuron emits a signal
that is a function of the amount of stimulation of the nerve-ending. So the
environment is represented at the lowest level by a set of signals which
reflect only the intensity of stimulation of local areas around each sensory
ending.
Sensory signals generated by the lowest level are the basis of all higher
perceptions. A taste-sensation, it is believed, is a function of signal
magnitudes from four types of chemosensors. Color sensations are a function
of signals fron three types of retinal light sensors. In PCT, each
identifiable type of perception is assumed to be a specific function of
inputs from a common set of lower-level signals; that is, in standard color
vision theory a blue-green color sensation would arise from an input
function that computes a weighted sum of input intensities of the three
types, while an orange sensation would arise from a different input function
that weights the inputs differently. At a given level of perception there
may be mutual inhibition among different input functions which gives the
input function with the largest output a suppressive effect on input
functions that generate a lesser output. However, many of the names we use
for color suggest that more than one color signal is present at the same
time: "a reddish brown with a tinge of purple." This hints at a model in
which color sensations are reported in parallel and independently by
specialized input functions.
The "idealization" of sensations in PCT arises from the fact that one input
function gives maximal output for one combination of its inputs, and a
smaller output for combinations that deviate from the one that yields
maximum output. But one sensation goes with one input function, so even with
a continuum of input variations, the world is represented only in terms of a
smaller number of variables, corresponding to the number of different input
functions that the particular brain has developed.
If you consider the set of all intensity signals (at the first level) as a
vector, then each sensation signal represents the projection of that vector
into a space where the basis vectors are the weightings of the inputs to a
particular input function.
If an MCT-style control system were constructed at the sensation level, the
input variable y that would be used for adjusting the parameters of the
world-model would itself be a weighted sum of intensity signals. The
world-model itself would be cast in terms of sensations, not intensities.
Since sensations are already an idealization of the more detailed world at
the intensity level, the model would begin with idealized variables, and it
would further idealize the situation by proposing a particular kind of
effect of the sensation level's outputs (u) on the world-model's output (x')
via the hypothetical environment.
When you go through a door, do you walk through it before you open
it, or open it before you walk through it? Can you tell the
difference? If so, you can perceive the temporal relationship
"before" as in "open _before_ walking through."
I can adequately go through a door, thank you.
Goodness, I wasn't impugning your abilities. Let me put it this way: can you
tell whether someone else opens a door before or after passing through it?
If you can, then you are able to perceive in terms of the before-after
dimension of experience. And if you can perceive in those terms, you have a
perceptual function that can report a relationship between independent
events in terms of this temporal dimension.
But I'm not aware that
I need a perception of "beforeness" in order to be able to do so.
Maybe my model parametrizes differently from yours ;-).
Possibly. But you need _some_ parameterization of this dimension of
experience in order to be able to control it. You may prefer to buy your
tickets to a concert just before it starts, comfortably before it, or weeks
before it, but definitely not after it. This perception is of a high level,
being derived from many lower-level perceptions, but it is still a
controllable perception in its own right. If you're like most people, you
don't call this dimension of experience a "perception." It's just a feature
of the world, that some things can happen before or after others and by
different amounts. But as a modeler, you must realize that what you know of
the world has to exist in the form of perceptions, and if perceptions exist
they must be derived by some kind of input function.
Can we translate someone's "intellictual schemes" with that person's
way of ordering the world? Would a cat have a notion of "beforeness"?
A flea? A bacterium? A virus?
For a person, yes: an "intellectual scheme" is in part a way of perceiving.
Does a cat perceive and control beforeness? I don't know -- possibly, but
I'd have to devise an experiment to see if a cat would correct a sequence
error. A flea or a virus? I doubt it. While sequence makes a difference in
the world, it's not every organism that can perceive and control sequence.
I doubt
somehow that all these species have the intellectual ability -- or
the need -- to develop a "beforeness" concept.
It may be a notion in _your_ model, it need not be one in mine.
Agreed. But you seem to know what I'm talking about when I refer to one
event occurring "before" another, so I would guess that you can perceive
your world in such terms. Suppose I said "Hans, I want you to agree that my
statement is right, and then I'll tell you what the statement is." I suspect
that you might object to telling me I'm right BEFORE you hear my statement,
and you might make some move toward reversing the sequence. So far every
person I know seems to understand what I mean by "before," but I'll admit
that I haven't spoken with everyone.
···
---------------------------------------------------------------------------
Hans Blom, 960912--
So dewpoint and relative humidity are measurement-derived abstract
concepts that are part of our scientific theories. Do they (also)
have an "independent existence in the environment"? How about
force, acceleration, mass?
Dewpoint and relative humidity are useful perceptions of the environment,
derived from more detailed perceptions. As to force, acceleration, and mass,
I would say that force and acceleration can be independently observed, but
mass can't, so force and acceleration are lower levels of perceptions than
mass. All mass-detectors, as far as I know, depend on measurements of force
and/or acceleration, but measures of force and acceleration don't depend on
measures of mass.
As to the independent existence of such things in the environment, we have
no direct way of verifying or disproving such a proposition. We know that
the length of a horse-hair, the conductivity of a certain kind of plastic,
and the feeling in our skin correlate highly with a certain function of
wet-bulb and dry-bulb temperature that we call relative humidity. But is
there "really" such a thing, Out There, as relative humidity? There's no way
to tell. I would guess not, since all the effects of relative humidity seem
to depend on other measurable variables.
---------------------------------------------------------------------------
Martin Taylor 960912 11:00--
In fact, when the human controls accurately, almost ANY control system that
controls accurately will provide a good fit to the human data. It is only
when the human has problems that different simulated control systems will
have appreciably different fit values.
If all control models fit the data equally accurately, there would be no
basis for preferring one model over another. The model we now use for
tracking experiments was developed over quite some time; although it seems
very simple, it works better than other simple models we tried.
Also, even for the model we use, the parameters make a lot of difference. It
is possible to adjust the output factor and the delay so that the model
performs distinctly better than the person. I suggested adjusting the
difficulty (bandwidth) of the disturbances in the sleep experiment so that
each subject, fully practiced, showed about an 8% RMS tracking error. When
showing this much error, most subjects will admit that they found the task
quite difficult, and this is independent of the level of skill. When there
is 8% tracking error, subjects can still perform consistently, if they are
practiced, and the control model will still fit the performance very
accurately. The control model will show tracking errors that match those of
the subject fairly well, and more important, it is possible to tell when the
control model is tracking _too_ well. The parameters can be adjusted on
either side of the best-fit value and the fit will reliably get worse. Under
these conditions, it is possible to measure k, the output sensitivity
factor, to within about 5%, meaning that a variation of +/- 5% will give a
reliably worse fit. If two subjects have best-fit values of k that differ by
5 to 10 percent, the values will not be interchangeable; that is the k-value
obtained from one subject will not serve equally well to represent the
behavior of the second subject in the same experiment.
With a very easy disturbance (low bandwidth), the tracking errors a
well-practiced subject makes are within the noise variations of the data.
That, of course, makes alternative models difficult to distinguish, and
hardly distinguishes the behavior from ideal or perfect control, except for
the noise level. Easy disturbances are useful for demonstrating the fact of
control, but they aren't of much use for choosing one model over another.
What is an easy disturbance depends on how skillful the controller is with
asymptotic practice; some people get very good, while others can never reach
the same degree of skill. But for any controller, there is always a
bandwidth of disturbance at which the errors start to mount, and we can
standardize on a level of error just by adjusting the bandwidth. This does
two things for us: it gives us a measure of skill, and it puts enough
tracking errors into the performance to permit finding the best values of
parameters.
... it would seem that a good criterion for the fit of
a model to human tracking data (given, _a priori_, that it is accepted that
the human is a control system) would be something along the lines of:
sqrt((sum(model-actual)^2)/(sum(model error)^2 + sum(actual error)^2))
Yes, I like this better than the correlation measure. The correlation
measure is not very sensitive to errors of fit when it gets into the high
regions, and it's extremely nonlinear as a measure of performance. Your
measure at least goes rapidly to zero when the match approaches perfection,
instead of being squeezed up against an asymptotic value of 1.
----------------------------
You have to deal with averages. It is impossible to equate any single track
with one disturbance type to any single track with another disturbance type,
even within a particular task type. I treated whole-block averages because
within each block every disturbance difficulty level and type is used with
every task, and this is the minimum unit over which this is true. If you
go to smaller units, you wind up with incommensurable data.
Perhaps what I'm asking for will be more evident if I say what application I
have in mind. Suppose you had a new person to whom you wanted to apply the
results of this study. Suppose that all you wanted to predict is whether the
model would fit the person's behavior better with the J or the U
disturbance, at a given level of difficulty and on a given task. The basic
question I'm asking is, what are your chances of predicting the outcome
correctly for this person on any particular task? Or more to the point, what
is the range of expected error of prediction for this person (at, say, the
90% probability level)?
Since we're talking about one trial, one task, one level of difficulty, and
one person, the sources of error include those across runs, tasks, and
levels of difficulty, as well as across people. So for the prediction I have
in mine, you have to plot a frequency curve showing the degree of fit over
all runs, tasks, conditions, and people for each of the J and the U
disturbances. The general statement being investigated is, "the model fits
performance better for the U disturbances than for the J disturbances." From
the plot I have in mind, you can find out how many runs showed a worse fit
for the J tasks than for the U tasks. This will enable to you calculate the
probability that your general statement will be true of this individual on
the next run of the task with the disturbance difficulty that is to be used.
If you use averaged data for any part of this, you reduce the range of
variation considerably, and spuriously increase the probability that your
prediction will be correct. This is not "spurious" if your prediction
concerns a group doing repetitions of tasks; averaging only makes the
conclusion incorrect when you're trying to predict what a specific person
will do in one specific run of one task. It's the latter I'm asking about --
not the usual sort of question that is asked about statistical results.
You really would like to compare
all the fits for, say, task A, difficulty level 1, G disturbance, against
task A, difficulty level 1, U disturbance. And then you would want to
average those differences across all the tasks and difficulty levels, or
at least to plot the Gaussian curve for all those differences.
This would be true if my objective were to say something about the entire
group of 34 surviving subjects, over repeated instances of the same task and
conditions. But I am asking about a single randomly-selected person on a
single trial of a single randomly-selected task (from the set tested). Even
though this is not a conventional question, there must be some way to answer
it. Given a conclusion drawn from the group statistics, what are the chances
that the conclusion will be true of the next person who does a run of the
experiment, assuming no sampling error?
--------------------------------------------------
That's about all I feel like commenting on for now. We walked around the
Hovenweep ruins up and down canyon walls on Thursday and camped that night
at Sand Island, 40 miles west in Utah, that night. We got up the next day,
drove to Mesa Verde, climbed down and up cliffs, and then drove home last
night. We are a little stiff, but happy. We saw a lot of artifacts, the
purpose of all of which was not self-evident. But there were some
interesting guesses.
-----------------------------------------------------------------------------
Best to all,
Bill P.