# Spreadsheet Demo of PCT Explanation of Power Law

[From Rick Marken (2016.08.17.1650)]

RM: Attached is an Excel spreadsheet that demonstrates the PCT explanation of the power law using both behavioral data and modeling. It’s still a work in progress so suggestions regarding how to make it more usable and clear would be welcome. Here’s a list of instructions if you want to try using the spreadsheet.

1. The spreadsheet works only in versions of Excel that allow macros.
2. When you open the spreadsheet you will be asked to “Enable macros”, Do so; there are no viruses in the spreadsheet.
3. The spreadsheet has a list of buttons on the left, two graphs in the middle and the results of a power law regression analyses on the right.
4. The buttons allow you to collect and analyze data; the graphs display the data collected and the power law regressions show the results of analyzing the data shown in the graphs.
5. First I’ll describe the buttons:
• The “Collect Data” button allows you to collect movement data in a tracking task.
• The “Analyze Human Input” button allows you to analyze your own movement data once you’ve complete collecting it in the tracking task; in this case it allows you to analyze the movement of the cursor (the input variable) in the tracking task.
• The “Analyze Human Output” button also allows you to analyze your own movement data; in this case the movement of the mouse (the output variable) in the tracking task.
• The “Analyze Model Input” and “Analyze Model Output” buttons allow you to analyze the movements of the input and output variables produced by a PCT model doing the same tracking task as you did.
• The “Squiggle” button makes a squiggly input movement pattern which can be analyzed and compared to your own squiggly movement.
• The graphs that you see when you open the spreadsheet contain my own movement data; the top graph is the X,Y pattern of cursor movements I made and the lower graph is a segment of the temporal variations in the X and Y movement.
• The analyses on the right are the power law regressions relating log® to log (V) and log© to log (A) for the tracking data shown in the graphs. The top analysis shows that the estimate of the power coefficient (beta) relating R to V for the cursor (input) movement pattern shown in the graph is .32 (very close to the expected 1/3 power law for V vs R) with an R^2 of .56; the analysis that is second from the bottom shows that the estimate of the power coefficient relating C to A for the same cursor movement pattern is .68 (very close to the expected 2/3 power law for A vs C) with a more impressive R^2 of .85. These are the two analysis to keep your eye on when you collect your own data.
• To collect your own data for analysis, press the “Collect Data” button. You will be taken to a new page and a “Start” message will appear. When you press “OK” the tracking task begins. The goal is to keep the red dot inside the moving ring.
• Once you have completed the tracking task you will be returned to the main screen, now with the graphs blank and the analysis tables set to 0.0.
• Now you can do the following analyses on your and the PCT model’s data:
• Press the “Analyze Human Input” button and it will display graphs of the cursor (red dot) movements that you made during the tracking task and the results of the power law regression for log® on log(V) and log© on log(A) for that movement pattern.
• Press the “Analyze Human Output” button and it will display graphs of the mouse movements that you made during the tracking task and the results of the power law regression for log® on log(V) and log© on log(A) for that movement pattern.
• Press the “Analyze Model Input” button and it will display graphs of the cursor movements that the model made during the same tracking task and the results of the power law regression for log® on log(V) and log© on log(A) for that movement pattern.
• Press the “Analyze Model Output” button and it will display graphs of the mouse movements that the model made during the tracking task and the results of the power law regression for log® on log(V) and log© on log(A) for that movement pattern.
• What to look for:
• The target in the tracking task moves in an elliptical path. When you are able to track that path accurately you should find that the result of the power law regression of R on V for that movement is close to .33. The more accurate the tracking, the closer the power law coefficient is to .33 (for R vs V and to .67 for C vs A).
• If you are having difficulty doing the tracking accurately you can slow down the target by changing the entry in the “Display Rate” cell. It’s currently set at 50% of top speed. You can reduce the speed of the target by entering a number in the yellow cell (B20) less than 50 (but not 0) and speed it up by entering a number greater than 50 (but not greater than 100).
• Note that when you analyze the PCT model’s input (the cursor movement produced by the model) the power law regression coefficient relating R to V is .32 (and the one relating C to A is .68). The estimate would be exactly .33 (or .67) but noise is added to prevent that. We can talk more about this if anyone is actually interested in this demo. It should be noted that the model produces this result without having a reference that varies elliptically; the model has a fixed reference of 0 for the distance between the center of the cursor (red dot) and the center of the target (ring); so the model is controlling for zero distance between cursor and target (as, presumably, are you when you do the tracking task). The elliptical movement of the cursor is, in both cases, a side effect of controlling for keeping this distance as close to zero as possible. This demonstrates the fact that the power law observed in power law research could be the result of regressing two measures of the same movement pattern on each other.
• This last conclusion is reinforced by the finding that cursor (input) movement patterns that conform to the 1/3 (or 2/3) power law can be produced by outputs that look nothing like the resulting cursor movement pattern. You can see the very non-elliptical mouse movement pattern that produces the elliptical cursor movement pattern by pressing the “Analyze Human Output”. You will see that the power law doesn’t fit the human output (mouse) movement at all, even when it does fit the cursor movement produced by this mouse movement. This shows that the power law does not represent any kind of “constraint” on the outputs that produce the observed input (cursor) movement patterns. A power law does fit the output movements of the PCT model, however, probably because the model movements are very smooth. A two level control model is probably needed to capture the non-power law conforming output variations produced by humans when they produce power law conforming elliptical cursor movements.
• Note that both the human and the model can produce power-law conforming elliptical cursor movements at many different target speeds. I can produce power law conforming elliptical movements at 100% or 30% of the fastest rate of target movement. I can also produce power law conforming elliptical movements by varying the speed of the cursor as it moves through the ellipse (And, thus, leading or following the moving ring sometimes during the tracking task).
• Finally, you can use the demo to show that random “scribble” movements also conform to the power law, but the power coefficient of these movements is sometimes close to .33 and sometimes closer to .25, depending on the squiggle pattern itself. You can see this by creating your own squiggly cursor pattern using the “Collect Data” button and moving the cursor (red dot) around the screen in a arbitrary pattern (and changing speeds all the time, of you like). When you then press the “Analyze Human Input” button you will get an estimate of the power coefficient for that movement pattern. It will probably be a number between .25 and .35 for the R vs V regression (and between .65 and .75 for the C vs A regression). Then press the “Scribble” button and watch the computer create its own random squiggly patterns and note the power coefficients for these. You are sure to find a randomly produced squiggle pattern that has a power coefficient that corresponds to the power coefficient for the pattern you produced yourself.
• The results of this demo are strong evidence that the power law found in power law research says nothing about how movement is controlled. Movement is controlled as described by PCT – by control of input. The power law is a side effect of measuring properties of a controlled variable without knowing that it is a controlled variable.
Best

Rick

···

Richard S. Marken

“The childhood of the human race is far from over. We
have a long way to go before most people will understand that what they do for
others is just as important to their well-being as what they do for
themselves.” – William T. Powers

[Martin Taylor 2016.08.18.10.50]

[From Rick Marken (2016.08.17.1650)]

RM: Attached is an Excel spreadsheet that demonstrates the PCT explanation of the power law using both behavioral data and modeling. It's still a work in progress so suggestions regarding how to make it more usable and clear would be welcome.

I can't actually run this spreadsheet because my version of Excel doesn't do macros, and both Libre Office and Neo Office fail on it. So I can go only by what you say about it and what I see.

From what I see, there's only one reference track, and that track is created with the power-law relation between V and R built-in, because it is an ellipse generated by sine waves over time in x and y. Of course, as I have pointed out, the output of any good tracker will conform to the power law if the reference movement does. We assume that both your model and the human are good trackers.

To test your theory, you have to provide a reference track in which the reference motion does not follow the 1/3 power law, for example, one that is slow on the flatter curves and quicker on the sharp ends of an ellipse. Or allow the user to create a basic track by drawing at arbitrary speeds with the mouse. Or take your forced-power-law ellipse and randomly disturb the alnong-track velocity to create a reference track for the subject. There are lots of ways to avoid providing the answer you want in the question you ask.

Whatever scientific question you ever want to address, it's best not to use questions of the form "If X = 3, what is the value of X?".

If I have misinterpreted your reference track, I apologise. I'm only going by what you say and what I see in a static version of your spreadsheet.

Martin
PS. Are you going to answer any of the questions I posed in other threads? I'll repeat some in a separate message on the "Laputian" thread in case you erased the originals. I'll have a couple of comments in that message, as well.

[From Bruce Abbott (2016.08.18.1210 EDT]

BA: Like you, Martin, I am still awaiting a reply (cogent or otherwise) to
my last post (in my case The Parable of the Rectangles, which unambiguously
demonstrates the fallacy of Rick's approach of including D in the
regression.)

Martin Taylor 2016.08.18.10.50 --

[From Rick Marken (2016.08.17.1650)]

RM: Attached is an Excel spreadsheet that demonstrates the PCT
explanation of the power law using both behavioral data and modeling.
It's still a work in progress so suggestions regarding how to make it
more usable and clear would be welcome.

I can't actually run this spreadsheet because my version of Excel doesn't do
macros, and both Libre Office and Neo Office fail on it. So I can go only by
what you say about it and what I see.

From what I see, there's only one reference track, and that track is
created with the power-law relation between V and R built-in, because it is
an ellipse generated by sine waves over time in x and y. Of course, as I
have pointed out, the output of any good tracker will conform to the power
law if the reference movement does. We assume that both your model and the
human are good trackers.

To test your theory, you have to provide a reference track in which the
reference motion does not follow the 1/3 power law, for example, one that is
slow on the flatter curves and quicker on the sharp ends of an ellipse. Or
allow the user to create a basic track by drawing at arbitrary speeds with
the mouse. Or take your forced-power-law ellipse and randomly disturb the
alnong-track velocity to create a reference track for the subject. There are
lots of ways to avoid providing the answer you want in the question you ask.

Whatever scientific question you ever want to address, it's best not to use
questions of the form "If X = 3, what is the value of X?".

If I have misinterpreted your reference track, I apologise. I'm only going
by what you say and what I see in a static version of your spreadsheet.

Martin
PS. Are you going to answer any of the questions I posed in other threads?
I'll repeat some in a separate message on the "Laputian" thread in case you
erased the originals. I'll have a couple of comments in that message, as
well.

BA: I CAN run Rick's spreadsheet. Not only does it have the user track an
ellipse, even the "squiggle" pattern it produces is created by sines and
cosines, in which movement slows with approach to the peaks and speeds up as
the point moves toward the middle of the excursion.

BA: The so-called "PCT" model assume that users draw by decomposing target
movement into X- and Y-axis directions separately tracking in those two
directions. As we have previously noted, this system simply reproduces a
reference pattern that already conforms to the power law. Making someone
trace it does nothing to change that, other than to introduce "noise"
(disturbance effects).

BA: Even the disturbance waveform is created by sine or cosine waves. If
you do an experimental run but do not move the mouse at all during the run,
the analysis finds a nice power law relationship with the usual exponent.

BA: Rick persists in demonstrating the results of regressions in which log
D is entered as a predictor. Not surprisingly he gets a perfect fit.
However, when using log A and log C, the exponent for log D is wrong (.333
rather than .667). (Will Rick understand why?)

BA: The whole exercise reveals that Rick has paid not the least attention
to our repeated explanations (some offered with empirical proof) and
admonitions that his approach is completely off track.

BA: He simply refuses to look through the telescope!

[From Rick Marken (2016.08.19.1040)]

···

Bruce Abbott (2016.08.18.1210 EDT)–

BA: I CAN run Rick’s spreadsheet. Not only does it have the user track an

ellipse, even the “squiggle” pattern it produces is created by sines and

cosines, in which movement slows with approach to the peaks and speeds up as the point moves toward the middle of the excursion.

RM: As I said in the write up, you can move the cursor faster or slower than the movement of the target and you will still find a power law relationship (close to 1/3) between curvature (R) and velocity (V).

BA: The so-called “PCT” model assume that users draw by decomposing target movement into X- and Y-axis directions separately tracking in those two directions.

RM: Why “so-called”? It’s a one level control model with two control systems at the same level. If you think it’s not a correect PCT model how about posting the correct one.

BA: As we have previously noted, this system simply reproduces a

reference pattern that already conforms to the power law.

RM: Actually, it doesn’t. The control systems in the spreadsheet model are controlling for distance between cursor and target and both references are fixed at zero. The elliptical pattern made by the model is a side effect of controlling for cursor-target distance.

BA: Rick persists in demonstrating the results of regressions in which log

D is entered as a predictor. Not surprisingly he gets a perfect fit.

RM: You can ignore those if they make you uncomfortable. The important results are the regressions of log (R) on log(V) and of log(C ) on log (A). Those are the ones calculated by power law researchers. They are the data that the model accounts for.

BA: However, when using log A and log C, the exponent for log D is wrong (.333 rather than .667). (Will Rick understand why?)

RM: Yes, I understand why. It’s because A = D1/3 *C2/3. The regression of log(D) and log(C) on log (A) will find a coefficient of .33 for log (D), and this this applies to all curves.

BA: The whole exercise reveals that Rick has paid not the least attention

to our repeated explanations (some offered with empirical proof) and

admonitions that his approach is completely off track.

RM: Oh, I have definitely payed attention to them.

BA: He simply refuses to look through the telescope!

RM: I have looked through the telescope, all right. From both ends. I’m trying to get you to look through it from the other end. I’m fully aware of the fact that that’s never going to happen.

Best

Rick

Richard S. Marken

“The childhood of the human race is far from over. We
have a long way to go before most people will understand that what they do for
others is just as important to their well-being as what they do for
themselves.” – William T. Powers