# Assessing rate-control

[From Bruce Abbott (9705.1220 EST)]

In the prototypical tracking experiment, a participant seated at a computer
is asked to keep a cursor aligned with a target by moving the computer's
mouse. Meanwhile, the cursor's position is being affected by a disturbance,
which in the absence of counteracting mouse movements causes the cursor to
drift from the target's position. The disturbance, mouse position, and
cursor position are recorded during the run and then the the correlations
between each possible pair of these three variables is computed. The
classic result is a near-zero correlation between variation in cursor and
mouse positions and a nearly perfect negative correlation (generally in
excess of -.995) between variations in disturbance and mouse position. A
typical example from an actual run appears below:
Cursor Mouse
Mouse 0.093
Dist. -0.011 -0.997

This matrix demonstrates excellent control of cursor position; a plot of
mouse position and disturbance as functions of time would produce two lines
that were nearly mirror images of each other, whereas cursor position would
appear to vary little and with no apparent relationship to either mouse
position or disturbance. This is because mouse movements are essentially
cancelling the effect of the disturbance on the position of the cursor.

This analysis assumes that the participant was simply controlling cursor
position. However, there is another strategy participants could use in this
task to help keep the cursor on-target. Because the disturbance was made to
vary relatively smoothly, a participant could adjust the rate of
mouse-travel as required to reduce the rate of cursor-travel to zero. If
the cursor were already on-target, this strategy would tend to keep it
there. Of course, if the cursor were elsewhere, the rate of mouse-travel
would have to be adjusted so as to permit a residual drift of the cursor
back to target and then changed so as to zero out further drift. This
strategy would amount to a combination of position _and_ rate control of the
cursor.

Is there evidence that the participant who generated the correlation matrix
above used rate control in addition to position control? One way to get at
this question is to take the difference between successive disturbance,
cursor, and mouse positions; these differences provide the rates of change
in each of the three variables. Here are the results:

dCursor dMouse
dMouse 0.786
dDist. 0.382 -0.270

These correlations suggest that control over cursor _rate_ was poor: cursor
rate correlates moderately high with the rate of mouse movement and
moderately low with the rate of disturbance change, indicating that
mouse-rate was not very effective in zeroing out the rate of cursor
movement, and the correlation between the rates of disturbance change and
mouse movement, although negative, was very low. Nevertheless, I would
suggest that this matrix provides some small measure of evidence for some
very poor degree of rate control.

The analysis is complicated by the fact that positional control-actions
would tend to occur at the expense of rate-control. In addition, it seems
likely that positional control would entail a negative correlation between
disturbance and mouse rates. To the extent that positional mouse-action
fell behind (or got ahead of) the disturbance, the participant would have to
play "catch-up" (or "fall-back") and this would reduce the correlation
between disturbance rate and mouse rate. What these data show is that very
high correlations between disturbance and mouse-position can be achieved in
the absence of good control over cursor rate.

Regards,

Bruce

[From Bill Powers (970505.1238 MST)]

Bruce Abbott (9705.1220 EST)--

Is there evidence that the participant who generated the correlation
matrix above used rate control in addition to position control? One way
to get at this question is to take the difference between successive
disturbance, cursor, and mouse positions; these differences provide the
rates of change in each of the three variables. Here are the results:

dCursor dMouse
dMouse 0.786
dDist. 0.382 -0.270

...

The analysis is complicated by the fact that positional control-actions
would tend to occur at the expense of rate-control. In addition, it >seems

likely that positional control would entail a negative correlation >between
disturbance and mouse rates. To the extent that positional >mouse-action
fell behind (or got ahead of) the disturbance, the >participant would have
to play "catch-up" (or "fall-back") and this >would reduce the correlation
between disturbance rate and mouse rate.

This is a very good example of applying the Test. When you hypothesize that
velocity is the controlled variable, you define the disturbance and the
output in terms of their effects on velocity, which is the proper procedure.
Your statistical analysis then shows that the correlation of mouse velocity
with cursor velocity is almost 0.8, while mouse versus disturbance velocity
is only -0.270. Both of these correlations argue for rejecting the hypothesis.

When you change hypotheses so it is position that is proposed to be under
control, the correlations change radically. Mouse vs cursor position is
0.093, and mouse vs. disturbance position is -0.997. The Test is clearly
passed even if we set the criterion level (mouse-disturbance) to -0.95.

If you wanted to be formal about this, you would verify that the controlled
variable is actually affected by the mouse (not very difficult), and that
the controlled variable must be sensed in order to be controlled (slightly
more difficult, but still easy).

What these data show is that very high correlations between disturbance
and mouse-position can be achieved in the absence of good control over

cursor rate.

I wouldn't put it quite this way. If position is perfectly controlled, then
rate is also controlled perfectly. If you were to construct a position
control model without any noise in it, this is what you would find. However,
there is noise in the real system, which amounts to uncorrelated
disturbances of both rate and position. The controller brings the error to
the level where it is comparable to the noise level, but that is true only
when you measure position. When you measure velocity, the velocity error is
larger than the noise component because in fact it isn't being directly
controlled.

Your description of "playing catchup", however, is probably quite right: the
catchup is due to position errors, which are corrected without regard to the
effect on velocity.

The next thing to try is an experiment with actual velocity control. Can you
think of an experiment in which a particular velocity rather than a
particular position has to be maintained? If you're controlling velocity,
then velocity errors can lead to cumulative changes in position, so the
challenge is to arrange a display in which changes in position don't
eventually run you off the screen!

Once you have the experiment, you can apply the same analysis as above to
discriminate position control from velocity control, and you should then
find the opposite outcomes, with position control being rejected.

This is what I call basic PCT research!

Best,

Bill P.

[From Rick Marken (970505.2130 PDT)]

Bruce Abbott (9705.1220 EST) --

dCursor dMouse
dMouse 0.786
dDist. 0.382 -0.270

These correlations suggest that control over cursor _rate_ was
poor: cursor rate correlates moderately high with the rate of
mouse movement and moderately low with the rate of disturbance
change, indicating that mouse-rate was not very effective in
zeroing out the rate of cursor movement, and the correlation
between the rates of disturbance change and mouse movement,
although negative, was very low.

Bill Powers (970505.1238 MST) --

This is a very good example of applying the Test...This is what
I call basic PCT research!

Yes. It also might provide a nice way for Bruce to illustrate the
use of some fairly complex statistical methods in PCT research.
If you did a multiple regression on the position control data,
using Dist and Mouse to predict Cursor, you would probably find
a regression equation like Cursor' = 1.0*Mouse-1.0*Dist. The
coefficients would probably be slightly different than 1.0 and
-1.0 and the intercept would probably be slightly different from 0.0.
But they would all be close to these values, and R^2 would probably
be around .98. This is the result expected if Cursor position is
under control.

But it might be that the actual controlled variable is rate of
change in Cursor position (dCursor) rather than Cursor position.
This could be tested by adding dMouse and dDist to the prediction
equation and making dCursor the dependent variable. The complete
regression equation becomes:

dCursor = a*b1*Cursor+b2*Mouse+b3*Dist+b4*dMouse+b5*dDist

We are only interested in the partial regression of dMouse and
dDist on dCursor but the other variables have to be included in
the solution of the multiple regression in order to factor out
their contribution to the variance in dCursor, dMouse and dDist.

If dCursor is under control, then the R^2 of the partial
regression of dMouse and dDist on dCursor should be high
(greater than .98) and, more importantly, the coefficients
of dMouse and dDist should be about equal (1.0) and of
_opposite_ sign.

I think this could be a nice way to do The Test when you can't
actively manipulate disturances. In this tracking task, Dist
and dDist are probably confounded (the correlation between them
is >0.0) and partial regression (like analysis of covariance)
is a way to eliminate this confound statistically.

Best

Rick

[Martin Taylor 970516 10:30]

Bill Powers (970505.1238 MST)] to Bruce Abbott (9705.1220 EST)--

If position is perfectly controlled, then
rate is also controlled perfectly. If you were to construct a position
control model without any noise in it, this is what you would find. However,
there is noise in the real system, which amounts to uncorrelated
disturbances of both rate and position. The controller brings the error to
the level where it is comparable to the noise level, but that is true only
when you measure position. When you measure velocity, the velocity error is
larger than the noise component because in fact it isn't being directly
controlled.

Yes, all that is true, and there's another effect, too.

Velocity is the derivative of position, and to take a derivative is
equivalent to putting a 6 dB per octave high frequency emphasis onto
the spectrum. The higher the frequency of the disturbance, the poorer
the control (in a control loop with finite transport lag). So even if
the velocity were to be independently controlled (if that were possible),
it would still show poorer stability factor than would positional control
against the same disturbance. (Here I mean the same physical disturbance
affecting the physical entity whose position or velocity might be the
complex environmental variable corresponding to the controlled perception).
You could test this assertion merely by telling the subjects in one run to
keep the cursor still at all times, wherever it might be, and then in another
run with the same disturbance ask them to keep the cursor aligned with
the target at all times. You can ask them to do both at once, but they
won't be able to do it!

Independent parallel control of position and velocity is impossible, but
positional control could easily be effected by altering the reference signal
to a velocity control system (or vice-versa). See the Arm demo. What that
means is that even if velocity is precisely controlled, the data you get
will not show it in the correlations, because the velocity reference value
is changing all over the lot as a function of the position perceptual error.
(But if position were being controlled in order to implement velocity
control, you would see relatively good correlations in the velocity data
but not in the position data, allowing for the spectral high-frequency
emphasis mentioned above).

Martin

[From Bill Powers (970506.1018 MST)]

Martin Taylor 970516 10:30--

Velocity is the derivative of position, and to take a derivative is
equivalent to putting a 6 dB per octave high frequency emphasis onto
the spectrum. The higher the frequency of the disturbance, the poorer
the control (in a control loop with finite transport lag). So even if
the velocity were to be independently controlled (if that were possible),
it would still show poorer stability factor than would positional control
against the same disturbance.

I'm not sure if this is the right direction in hyperspace to generalize
about rate control. If you do have rate control, it can seem just as good as
position control over short periods of time, given the same basic components
from which to build a control system and the same external disturbance. The
difference is that a pure rate control system will drift under a steady
disturbance (or the steady component of a disturbance that does not have a
zero average value). Transport lag, as you say, limits how much gain a rate
control system can have at the highest frequency that can be transmitted
through the system, but that limit is the same in a proportional system or
an integrating system. The main difference is in how the _low_ frequency
components are handled.

Independent parallel control of position and velocity is impossible, but
positional control could easily be effected by altering the reference
signal to a velocity control system (or vice-versa). See the Arm demo.

Yes, you're quite right about that. In fact, this brings up a point that
we've touched on very briefly now and then: is there one rate control level,
or two? In the Arm demo, as modeled, the position control system in the
spinal control systems does set a reference velocity, but this is really an
imaginary hierarchy. In fact, the annulospiral length sensors have a rate
plus proportional response; we can separate these responses out into a rate
signal and a proportional signal for computational convenience, but both
signals actually reach the same motor neuron, the same comparator, so there
is no physically separate level where a pure rate signal is compared against
a rate reference signal. Instead, the effective rate reference signal is
always zero. The gamma reference signal is a position reference signal, and
the alpha reference signal is a force (or torque) reference signal. There is
no separate rate reference signal that I know of.

This means, I think, that we don't really have a distinct velocity control
level below the position control level. What we have is rate plus
proportional sensing, which puts damping into the position control system.
The result is not to add a rate component to the resulting closed-loop
perceptual signal, but exactly the opposite: the damping _suppresses_ the
high frequencies in the whole loop. The result is that the perceptual signal
follows rapid changes in the position reference signal more closely than it
would do without the rate component. Confusing.

However, when the Arm model's parameters are adjusted for the fastest stable
response to square-wave reference signals, what we find is that the static
loop gain is only 2 to 4, while the dynamic loop gain is anywhere from 50 to
100. As a result, when gravity is turned on, the arm sags! So it seems that
these spinal control loops could be viewed (as a whole) as rate control
systems, with only a little proportional gain. In order to eliminate the
sag, it is necessary to add another control loop using joint-angle
perceptual signals, with an integrating output function. I have seen
evidence that in the brainstep loops there are such integrations in the
output functions (the motor nuclei of the brain stem).

Obviously, all these conjectures call for a much more extensive
investigation of these peripheral control systems. I hope someone will take
it up, once the Arm paper appears in IJHMS.

(But if position were being controlled in order to implement velocity
control, you would see relatively good correlations in the velocity data
but not in the position data, allowing for the spectral high-frequency
emphasis mentioned above).

The _second_ level of velocity control is what I have called the fourth
level, the transition level. The strongest evidence I know of for this level
is the phenomenonon of stroboscopic motion. Here we have a series of static
configurations which, presented in rapid sequence, give rise to a perception
of motion. It's just as though there were a higher level of perception which
takes first derivatives (or first differences?) of signals from the
configuration level and constructs a velocity signal from them (and perhaps
acceleration signals too?). In this case we would have what you describe: a
control system maintaining perceived rates of change by varying
configuration reference signals. This level is actually _slower_ than the
configuration or position control level: I believe that the impression of
motion begins to disappear when alternating positions appear 300
milliseconds apart or less. That's a long time; the reaction time for
position control is much less than that. If we were talking about true rate
feedback at a lower level, the rate signal should _increase_ as the
frequency of changes gets higher, and we should see a rate response at much
higher frequencies.

This obviously needs a lot of experimental sorting out. Rick Marken's
experiments with hierarchical perceptions are one good way to go; Bruce
Abbott's statistical analysis approach is another.

Incidentally, transport lag in the peripheral systems (or anywhere in the
brain, for that matter) is not really a simple phenomenon. Every control
loop is actually made up of many parallel control loops, and in each loop
the delays will be a little different. Some loops might have very short
transmission times; others abnormally long ones. In addition, there's the
"recruitment" effect (I don't know where neurologists get these irrelevant
images): some paths will not begin transmitting until the signal is very
strong, while others have nearly zero thresholds. The other end of this
effect is that some pathways will saturate long before others. The result is
that the composite response in the whole control loop can be very different
from that of any one microloop (I read somewhere that in the stretch reflex,
the resulting overall response is far more linear than what is found in any
individual pathway). The dynamic response to a step input would not be a
delayed step output, but a smeared-out rise along some curve.

These are _continuous_ systems: the transport lags do not all start at the
same instant, so in fact there is no clear single delay. Any one signal in
the loop would appear to vary continuously, at frequencies all the way up to
the saturation limit and all the way down to where the "grain" of a signal
carried by a train of impulses begins to matter.

Best,

Bill P.

[From Bruce Abbott (970506.1430 EST)]

Rick Marken (970505.2130 PDT) --

If you did a multiple regression on the position control data,
using Dist and Mouse to predict Cursor, you would probably find
a regression equation like Cursor' = 1.0*Mouse-1.0*Dist. The
coefficients would probably be slightly different than 1.0 and
-1.0 and the intercept would probably be slightly different from 0.0.
But they would all be close to these values, and R^2 would probably
be around .98. This is the result expected if Cursor position is
under control.

The equation is Cursor' = 1.00*Mouse + 1.00*Dist, and R^2 = 100.0.

This equation tells us that the cursor position was the sum of the the mouse
and disturbance values (which were zero-centered).

But it might be that the actual controlled variable is rate of
change in Cursor position (dCursor) rather than Cursor position.
This could be tested by adding dMouse and dDist to the prediction
equation and making dCursor the dependent variable. The complete
regression equation becomes:

dCursor = a*b1*Cursor+b2*Mouse+b3*Dist+b4*dMouse+b5*dDist

We are only interested in the partial regression of dMouse and
dDist on dCursor but the other variables have to be included in
the solution of the multiple regression in order to factor out
their contribution to the variance in dCursor, dMouse and dDist.

If dCursor is under control, then the R^2 of the partial
regression of dMouse and dDist on dCursor should be high
(greater than .98) and, more importantly, the coefficients
of dMouse and dDist should be about equal (1.0) and of
_opposite_ sign.

Actual results:

dCursor' = 0.0*Cursor + 0.0*Mouse + 0.0*Dist + 1.00*dMouse + 1.00*Dist,
and R^ = 1.000.

also, dCursor' = - 4.94 + 0.0155 Cursor3 +0.000276 Mouse, Dist having been
removed from the equation because of its redundancy with the other variables
entered first. This equation's R^2 was 0.008: positional values account for
essentially none of the variance in cursor rate.

Basically, what linear regression is finding is that Cursor is the sum of
Mouse and Dist, and that dCursor is the sum of dMouse and dDist, as in fact
they are. So what we are seeing are the feedback functions (mouse effect on
cursor) and disturbance functions (disturbance effect on cursor). These are
captured whether there is excellent control (position) or not (rate).

By the way, I tried to improve the correlations for rate control by
consciously attempting to null out the cursor's motion without attempting to
correct positional error. After two such attempts, I was still getting an
indication of excellent position control, without seeing much improvement in
the rate-control numbers. I found it very difficult to keep the rate of
cursor motion at zero given the disturbance waveform I was using, which
included some relatively fast changes in rate and direction.

Regards,

Bruce

[From Bill Powers (970506.1615 MST)]

Bruce Abbott (970506.1430 EST)--

Basically, what linear regression is finding is that Cursor is the sum of
Mouse and Dist, and that dCursor is the sum of dMouse and dDist, as in fact
they are. So what we are seeing are the feedback functions (mouse effect on
cursor) and disturbance functions (disturbance effect on cursor). These are
captured whether there is excellent control (position) or not (rate).

Nice to know it works!

By the way, I tried to improve the correlations for rate control by
consciously attempting to null out the cursor's motion without attempting to
correct positional error. After two such attempts, I was still getting an
indication of excellent position control, without seeing much improvement in
the rate-control numbers. I found it very difficult to keep the rate of
cursor motion at zero given the disturbance waveform I was using, which
included some relatively fast changes in rate and direction.

This is another interesting subject. Try doing a run while you imagine that
your hand is pushing the mouse through very thick syrup. You can change your
tracking characteristics considerably in this way. This is another reason I
think we should deal only with practiced subjects trying to do their best:
there are many ways to fake suboptimum control, but I can't think of any
ways to pretend to control better than you possibly can!

Best,

Bill P.

[From Rick Marken (970506.2050)]

Me:

If you did a multiple regression on the position control data,
using Dist and Mouse to predict Cursor, you would probably find
a regression equation like Cursor' = 1.0*Mouse-1.0*Dist.

Bruce Abbott (970506.1430 EST) --

The equation is Cursor' = 1.00*Mouse + 1.00*Dist, and R^2 = 100.0.

Boy, is my face red. That's what I get for trying to conceptualize
control in terms of conventional statistical models. The regression
model led me to think of the variance in the dependent variable (Cursor
position) as being partly the result of random error. That's why I
thought R^2 would be only about .98. After all, the correlation between
the behavior of control model and subject is usually no greater than
.99 and .99^2 = .98.

I also thought the regression model would reveal the nature of the
control law in the same way that it supposedly reveals cause-effect laws
in conventional research; that's why I thought the coefficients would be
about equal and of opposite sign. What I forgot was the physical fact
that Cursor = Mouse + Dist. So if Mouse and Disturbance are used to
predict Cursor position then the coefficients should be _exactly_ 1.0
(and the same sign). Also, since this weighted sum
is the exact equation used to determine Cursor position, it should
predict Cursor position perfectly: R^2 should = 1.0 (assuming no
rounding errors in computing the values of the variable) and it does.

So the moral (for me) is: the best was to study control is still
the old fashioned way; Testing for Controlled Variables and
comparing observed behavior to the behavior of models that control those
same variables.

Best

Rick

[From Bill Powers (970507.0500 MST)]

Rick Marken (970506.2050) (replying to Bruce Abbott)--

Mouse and Disturbance are used to
predict Cursor position then the coefficients should be _exactly_ 1.0
(and the same sign). Also, since this weighted sum
is the exact equation used to determine Cursor position, it should
predict Cursor position perfectly: R^2 should = 1.0 (assuming no
rounding errors in computing the values of the variable) and it does.

This is interesting. Given output and disturbance, we can predict cursor
perfectly; in fact, given any two, we can predict the other. So a multiple
regression analysis in this case amounts to an analysis of linear physical
relationships in the environment (no organism required).

Hold that thought.

Now, what is it that psychologists try to do when they vary a stimulus and
try to predict a response? They are varying disturbance and trying to
predict output. This can't be done, because there is a missing term: cursor,
the controlled variable. You must know two out of the three variables to
predict the remaining one. However, it is possible to assume a value of
cursor, namely, zero. Under that assumption, one can do a multiple
regression and solve for the response (responses) given the stimulus
(stimuli). This will yield non-zero regression coefficients. The intercept,
in fact, will yield an assumed constant value of cursor that will depend on
the zeros and signs of the measurement scales used. Is this right so far?

To assume a constant value of cursor is to assume a constant value of the
reference signal (and no uncontrolled disturbances). This can mean only the
_average_ value of the reference signal, over some period of time. If the
reference signal actually varies in an unpredictable way with time, or
across subjects, or both, the multiple correlations will be less than 1.0.
The regression coefficients will show the equal-and-opposite effects of
disturbance and output on the cursor, but only on the average. The signs of
the regression coefficients and correlations will depend on how disturbance
and output are measured; since measurement scales are arbitrary, either sign
could be found (for example, one could measure "proximity" or "distance").

Conclusion: this seems to me an elegant way to discuss the behavioral
illusion in language familiar to psychologists.

Let's turn now to operant conditioning. Here we have measures of output
(responses) and cursor (reinforcements), and we know the relationship
between them (the schedule). From these we should be able to calculate the
disturbance signal, although not the disturbance itself. There are even
measures that are related to the disturbance signal (deprivation, the
"establishing conditions" if that's the right term). So it should be
possible to calculate (roughly) the disturbance signal and relate it to
measures like the degree of deprivation. A multiple regression should then
show the expected correlations: a positive correlation between deprivation
and responses (since positive deprivation has an effect on reinforcements
opposite to the effect of responses under a normal schedule). The same
assumption of a constant or average reference signal holds.

The results of a correlation analysis would then give us a picture of the
environmental relationships involved. The organism would still not be in the
picture, except for the value of the average reference signal which shows up
as the intercept(s).

For operant behavior, there are some additional problems. The most popular
schedules are the variable ones, which put a large random component into the
dependence of reinforcement on behavior, as well as making the average
relationship nonlinear. This greatly reduces the correlation coefficients
that are to be expected, and requires taking much more data to make any
stable relationship visible. Also, there have been unwarranted assumptions
about the actual measure of behavior that is appropriate. Bruce Abbott has
found that the most common measure -- rate of bar-pressing -- is not
actually a variable in the experiments he has investigated. Instead, the
rate of pressing is almost constant, while time away from the lever, meal
size, and interval between meals are much more likely to vary (when
possible). These alternative variables are hidden by the practice of
counting total bar presses per session and dividing by session duration to
get "responses per unit time." This crude measure does give us a behavioral
variable to measure that has a positive effect on reinforcement rate, so a
correlation analysis would still be possible, although less informative than
it might be.

Since we observe the total responses and the total reinforcements, it should
be possible to find the mean relationship between them even for
variable-ratio or variable-interval schedules. A plot of one against the
other, with long periods of observation for each point, should exactly match
the theoretical response of the apparatus to lever-pressing at various mean
rates. In fact, this response can be measured without the organism present,
by generating artificial lever presses at various rates and observing the
resulting rates of reinforcement, all in simulation but using the same
program that is used to operate the real apparatus.

Once we have the relationship between mean pressing rate and mean rate of
reinforcement, we can calculate, in real experiments, the magnitude of the
disturbing signal under the assumption of a constant reference signal, or
the magnitude of the reference signal assuming a constant disturbance (I
suspect that the two will be the same). What we can't do is separate the
effects of disturbances from the effects of changes in the reference signal.
Or maybe we can -- that would be a most interesting result!

Anyway, this too would be another useful way to apply standard statistical
methods to show the relationship between PCT and an accepted way of
analyzing behavior.

I propose that Rick and Bruce (and anyone else who wants to volunteer) put
their heads together and develop a rigorous statistical analysis for both of
these cases. This has been accomplished in part already; what we need is the
rest of the story. It would make a valuable addition to the PCT literature
even if it never gets published anywhere else.

Best,

Bill P.

[From

Rick Marken (970506.2050)

So the moral (for me) is: the best was to study control is still
the old fashioned way; Testing for Controlled Variables and
comparing observed behavior to the behavior of models that control those
same variables.

Rick,

Thought you might resonate with this quote from an article
entitled "Does the Cosmos Have a Direction" in the 26 April
_Science News_:

"I really think this is much ado about nothing," says
cosmologist Michael S. Turner ... "The number one rule in
astronomy is that you can't reanalyze someone else's data to
look for an effect that [the observations] were not designed to
measure."

Regards,

Bruce

[From Rick Marken (970508.1215 PDT)]

Bruce Grergory (970508) --

Thought you might resonate with this quote from an article entitled
"Does the Cosmos Have a Direction" in the 26 April _Science News_:

says cosmologist Michael S. Turner ... "The number one rule in astronomy
is that you can't reanalyze someone else's data to look for an effect
that [the observations] were not designed to measure."

I'll put it in the frontpiece of my book, if I am ever able to write one.
This is what I've been trying to say for years (with respect to PCT, of
course); maybe some people will believe it now that it comes from (what
I presume is) a _real_ scientist.

Best

Rick