[Bill Powers (950215.1240 MST)]
RE: E. coli analysis
Actually if the step is taken to be a step in direction rather than
distance, only one added calculation needs to be made: to project the
direction until it intersects a radius to the target at right angles,
and use that as the origin for the next step.
Yes, that's exactly the rationale I used in saying that you could
get the probability by taking half the length of step that gave you
the desired probability of improving on the initial state.
Sorry, I don't see any resemblance between my paragraph and your
paragraph. I think you're skipping too many steps; how about trying
again?
Let me put my proposition in different terms.
x2,y2
* *
* /\ *
* / \ *
* / a \ *
x1,y1*/<----R----->\*x0,y0
In this crude diagram, the stars are supposed to be a semicircle with
_diameter_ R. x0,y0 is the target position, x1,y1 is the present
position of a general point x,y. The angle between the two lines made
with / and \ is a right angle no matter where the line from the left
intersects the semicircle.
The "random move" is a choice of angle a between +/- pi measured from
the direction of R. After an angle is chosen, and if it is less than
pi/2, the point x,y moves from the position x1,y1 at angle a and at some
constant velocity until it intersects the semicircle at x2,y2. At this
point the radius from x,y to x0,y0 just begins to increase again. Note
that there are no new random changes in direction before this
intersection occurs. Then there is a single new random choice of
direction, and if it lies within the new semicircle with a diameter
between x2,y2 and x0,y0, analogous to the one shown, movement continues
until there is another intersection.
If the angle a is greater than +/- pi/2, the distance from the moving
point x,y to x0,y0 will immediately increase. We could specify some
amount of increase i at which it becomes detectable. As soon as it is
detectable, there will be another random change in direction. So
clearly, the smaller i is, the less the distance to x0,y0 will increase
before there is another random change of direction. In the limit, with i
= 0, there will be no progress away from the target x0,y0 for any angle
a greater than pi/2. The average speed of movement toward the target
will be the average of R(1 - cos(a)) for angles a from -pi/2 to pi/2,
times the probability of a choice of any angle in that range.
If the direction is chosen as a set of direction cosines, each of which
varies at random between 1 and -1, we can easily extend this to n
dimensions, provided we know how, which I don't.
I hope this shows the link between my conception of the E. coli problem
and the problem you and your colleague appear to be working on. If you
think of a "jump" as being a choice of a new set of direction cosines,
you can then convert to choice of an angle in hyperspace. The general
point moves in this direction until its distance from the origin reaches
a minimum, and only then is a new random choice of direction cosines
made.
For two dimensions, I believe that the average speed toward the target
(with i = 0) turns out to be half the lineal speed. But I don't trust my
integrations.
In your approach, the radius M = sqrt(dx^2 + dy^2) comes into play, so
you compute the probability of an improvement of magnitude M on a single
step. In my approach, it is an angle, not a distance, that is selected
at random, so the magnitude of the step in position is of no
consequence. Once the angle is selected, movement continues uniformly in
that direction until the radius to the target begins to increase. Thus
the distance travelled after each random selection depends on the
direction.
It occurs to me that we are probably re-inventing Newton's method.
ยทยทยท
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Martin Taylor (950214 11:10)--
Because in some places the wind is relatively calm, and leaves stay
there if they get there, whereas in other places they are blown
away by the wind. There's no "goal" to reorganization.
I beg to differ, as least as far as the PCT _theory_ of reorganization
is concerned. There is a goal: it is zero intrinsic error, which means
that some set of intrinsic variables are at their genetically-given
reference levels. Of course there is no goal for the _means_ by which
this is accomplished. A "stable society" is any social organization that
reduces all intrinsic errors to zero. That's probably why we haven't
seen any stable societies yet (on the historical time-scale).
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Rick Marken (950214.1040)--
RE: post of Lars Christian Smith (950214 12:40 CET) about method of
levels --
Nice description, Rick. Let me put in a few clarifying observations.
The next step in going up a level would be to figure out WHY you
have a control system operating with the goal of keeping the cursor
on the target.
This can too easily be taken to mean "think up a logical rationalization
or hypothesis." When you're really doing the Method, you don't have to
look for the higher-level process: it's already there and working. What
you have to do is _notice_ it. When you're tracking, there's always
something there looking at the performance and evaluating it: "Oops,
that was a tough one; now I'm doing better, hope I can keep it up; how
much longer does this run go on?" and so forth. Maybe you're pleased
with yourself for doing so well; maybe you're anxious because you're not
doing well. Or maybe you're just waiting for the run to be over so you
can see how the analysis comes out.
No matter what you're doing or thinking, there's always some sort of
background activity going on ABOUT what you're thinking or doing. That's
where you find the next level. After you've done this a few times, the
interesting question arises as to how many times in a row you can do it.
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Martin Taylor (950214 12:10)--
More horning in:
(1) It is through the perceptual function that the information
needed for control is obtained.
I would say that it is through the perceptual function and the
comparator that the information needed for control is obtained. The
reference signal changes the meaning of perceptual inputs. And as far as
I can see, nobody knows how to analyze the information situation in a
closed-loop system (at least I've never seen it done), so it's a moot
point anyway.
(2) There are more perceptual functions involving controllable
perceptions than can be controlled at any moment, and there is a
need for some mechanism to shift what is controlled when some
currently uncontrolled perception is disturbed sufficiently to
matter.
If an "uncontrolled" perception matters, it is a controlled perception.
[No claim that]... the signals from the more alerting senses
(touch, hearing, peripheral vision) "cause" people to shift their
attention.
What senses are not "alerting" senses? My mouse is not on the mouse pad
and the moment, as I can see. But I don't care about that, so I'm not
"alerted" to the fact. However, in another circumstance (when I bring up
Windows), the very same visual image becomes an "alerting" stimulus,
telling me that I have to move the mouse to the pad. Or is it just that
when I decide to use Windows, one of the sub-goals involved is to put
the mouse on the mouse pad, if it's not already there? I don't think
it's possible to define a class of "alerting" stimuli. That's a holdover
from an older theoretical explanation of behavior, in which certain
stimuli are given inherent behavior-causing properties.
Perhaps I should ask you to take the other position, and show,
using only the core PCT assumptions, how it is possible to avoid
the necessity for the existence of some alerting mechanism, given
the larger number of potentially controllable perceptions than
available outputs.
We control the perceptions that we control; we don't control those we
are not controlling. When we pick a perception to control, we do so for
a higher-level reason; there is no compulsion to control everything.
Perceptions don't "need" controlling. If we aren't monitoring a
perception and have no goal for it, it's just part of the background.
Before it can matter to us, we have to incorporate it into a control
loop. It may "matter" in a different sense, so ignoring it can lead to
problems for us. But if that is true, we then develop a control system
to keep it under control.
We control perceptions when they matter. If the telephone rings, I
generally answer it, which involves a higher-level system adjusting the
reference levels for whatever I am doing now and turning on the
reference levels which lead to answering the phone (when it's not
important to finish what I'm doing right now). The higher system is
always ready to do this when the phone rings. But I don't have to have
the phone-answering mechanisms turned on when the phone isn't ringing; I
can use them for other things, like typing.
While I'm talking on the phone, I may realize that I haven't stored what
I was writing for some time. So the higher-level systems will tell the
caller to wait, go to the keyboard, store the text, and go back to the
talking-on-the-phone system. No problem. This isn't some special
phenomenon called "alerting." It's just time-sharing.
Maybe you should offer an example that you think PCT can't handle
without the "alerting" hypothesis.
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Gary Cziko (950214.2355 GMT) --
But right now it is much easier for me to put documents into our
Gopher format for which ASCII is the way to go.
I hope we're not going to shift over to a system that I can't use. I am
connected via phone lines to Ft. Lewis' VAX cluster, which is not a
Gopher server and does not run Mosaic or any other special software for
connecting to WWW. I hope to upgrade pretty soon from 2400 Baud to
14,400, but even with the upgrade, looking at graphics would take
forever, and storing it in my PC would quickly fill my hard disk. And
anyway, I share the computer with 75-100 other terminal users and there
are only 14 phone lines for remote access. I can't sit for hours in
front of a screen monopolizing a phone line. My method is to read and
compose posts off-line, connecting only long enough to upload or
download.
You guys who are connected directly to mainframes may have a distorted
idea of how the rest of us operate!
What, by the way, is RTF and HTML? What sort of word-processor is
involved?
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Best to all,
Bill P.