Behavioral illusion, based on Powers (1978)

[rsmarken]

RM: I base the distinction I make between causal and non-causal functions on Bill’s reply to my question “but is o really a causal function of d”? Bill replied:

AM: Notice that he uses the term “causal path”, not “causal function”, and generally never used “causal function”.

RM: Well, he didn’t seem to mind my use of the term. I think it’s pretty clear that he understood that by “causal function” I meant “a function that corresponds to the causal path between variables”. And when he talked about a causal path between variables, such as the causal path from qd to qi, he referred to it as a function, such as “disturbance function”.

RM: I think you are trying to make a mountain of horribleness out of a molehill of word choice in your efforts to make it seem like Bill’s concept of PCT was different from mine. It works well as propaganda – note the effectiveness of “Hillary’s emails” – but only on your base;-)

RM: and the feedback function, qi= g(o).
RM: The feedback function, qi = g(qo) is the only “backward” function
is actually a mathematical approximation to the inverse of the feedback function that relates the DV (qo) to the controlled variable, qi (not to the IV, qd).

AM: There is a causal path, or causal link, causal something, from qo to qi, passing trough the environment, through the feedback function. The environment is usually modeled with two functions, one is G, and the other is the adding of G(qo) and H(d), or qd and qf (or whatever you want to name it), to finally make qi. Well, H is a third environmental function. What you wrote (four, five times) is not the feedback function as defined in PCT.

RM: Actually it is. G() is the only feedback function in the PCT model (and in reality).

RM: So what you are observing as the relationship between qo and qd is actually the inverse of the relationship between qo and qi.

AM: It looks to me like your causal analysis led to you to a mistaken conclusion. As I’m sure you know from the test for the controlled variable, the relationship between qi and qo in a good control system should be a low, near zero correlation. If there is no correlation, you can’t describe the relationship as a function.

RM: In the test for the controlled variable you are looking for a zero correlation between qi and qd, not qi and qo. It’s true that when control is good the observed correlation between qi and qo will be zero (because qo = -qd), which means that the forward causal path from qi to qo – which corresponds to the system function qo = f(qi) – is not visible when control is good (see my paper When causality does not imply correlation:

Dropbox - CauseCorrelation2011.pdf - Simplify your life.

RM: However, the feedback function qi = g(qo) is visible (as its inverse) in the observed relationship between qd and qo. That’s what Bill called the “behavioral illusion” in the 1978 paper.

RM: We seem to draw very different conclusions from Powers’ 1978 paper regarding what the behavioral illusion is and what its implications are for behavioral science research. This means we have very different ideas about what PCT is all about. My main takeaway from Bill’s paper is that you can’t correctly understand the behavior of organisms without understanding what perceptual variables they are controlling. So some version of the test for controlled variables should be the essential component of all behavioral science research. Your takeaway seems to be rather different, at least in terms of seeing the determination of controlled variables as being essential to behavioral research. Your takeaway from Bill’s paper seems similar to the ideas of non-PCT control theorists who pay no attention to controlled variables at all.

RM: So I’m wondering why you think PCT is relevant to your work? Why not base your work on the theories of non-PCT control theorists? An excellent description of the non-PCT application of control theory to human behavior is in the book Control theory for humans: Quantitative approaches to modeling performance by R. Jagacinski and J. Flach (NJ: Erlbaum, 2002).

RM: I would like to know why you prefer PCT to non-PCT versions of control theory?

Best

Rick

[amatic]

Ouch, Hillary’s emails are on the floor! I think when you calm down, you will see that in PCT, the feedback function is defined as a function relating the output quantity to the feedback quantity.

Just look at this nice diagram from LCSIII:

The problem, it seems, is that the feedback quantity is not often named, and the summing function is not often named. Easy to miss and misunderstand. But they are right there. The other problem is that you invented a silly expression, the “causal function”, and you get silly conclusions when you follow the “causal function” method.

RM: However, the feedback function qi = g(qo) is visible (as its inverse) in the observed relationship between qd and qo. That’s what Bill called the “behavioral illusion” in the 1978 paper.

The environmental function in 1978 paper is (2) qi = G(qo) + H(d)
The feedback function is: qf = G(qo).
The disturbance function is: qd = H(d)

RM: I would like to know why you prefer PCT to non-PCT versions of control theory?

It’s not me who prefers non-PCT, Rick. Why do you think there is a need to invent new kinds of functions, “causal functions”, and add them to the PCT vocabulary, where they were never present?

[rsmarken]

AM: I think when you calm down, you will see that in PCT, the feedback function is defined as a function relating the output quantity to the feedback quantity.

AM: Just look at this nice diagram from LCSIII:

RM: I think that’s the only PCT diagram I’ve seen that labels the output of the feedback function “Feedback Quantity”. Since Bill didn’t create this display I’m pretty sure that he didn’t intend that the label of the output of the feedback function be “Feedback Quantity”. But whether he did or not, I think that label is quite misleading. It implies that the output of the feedback function is an entity in the environment that is separate from the controlled variable. In fact there is no such entity; what is called the “Feedback Quantity” in this diagram is just the effect of qo on the controlled variable, qi (called the Input Quantity" in the diagram). So it would be better to have labeled that variable “Feedback Effect”. And the feedback effect of qo on qi is g(qo). That’s why in the 1978 paper Bill wrote qi = g(qo)+h(qd) and not qi = fq + h(qd)

RM: In Powers’ 1978 paper (and in virtually every other paper on PCT that I know of), the output of the feedback function is shown as directly affecting the controlled variable. In the 1978 paper there is no Feedback Quantity, as shown in Figure 3 from that paper:

AM: The problem, it seems, is that the feedback quantity is not often named,

RM: I consider that a feature, not a bug. There is no such thing as a feedback quantity in the environment of a control system; there is just the feedback effect of output on teh controlled variable.

AM: and the summing function is not often named.

RM: Actually, it is always named. As I said above, Bill names it in his equation 2) as qi = g(qo)+h(qd).

AM: The other problem is that you invented a silly expression, the “causal function”, and you get silly conclusions when you follow the “causal function” method.

RM: That silly expression corresponds to the arrowheads in Bill’s Figure 2 which show the direction of causality of the functions h, f,and g. Note that all arrows point in one direction – the direction of causality. There are two functions that relate qi to qo and they differ in the direction of causality. The direction of causality between the same two variables can go in two different directions at the same time because the functions go through two different causal paths: the causal path of function f goes through the organism from qi to qo while the causal path of function g goes through the environment from qo to qi. Note that the causal path of the disturbance function, h, goes in only one direction, from qd to qi.

RM: It’s important to know the direction of causality of these functions in order to understand the behavioral illusion. When an experimenter manipulates qd (under controlled conditions) and sees concomitant variation in qo he (or she or they) assumes that qd is the cause of qo. The relationship between qd and qo is assumed to reflect the causal path from qd to qo; qo is assumed to be a causal function of qd of the form qo = f(h(qd)) (the last equation on p. 425 of Powers, 1978).

RM: But the 1978 paper shows that, when the subject of study is a control system, what is actually being observed is not the function qo = f(h(qd)) but, rather, the function qo = g-1(-h(qd)) (the second to last equation on p. 425 of Powers, 1978 with the reference for the controlled variable, qi*, set to 0). This is an illusion because the experimenter is seeing qo = g-1(-h(qd)) and taking it for qo = f(h(qd)).

RM: It is also an illusion because the experimenter is taking a non-causal function-- qo = g-1(-h(qd)) – for a causal function – qo = f(h(qd)). The function qo = g-1(-h(qd)) is non-causal because it does not correspond to an actual causal path between qd and qo. The feedback function, g, describes the causal path from qo to qi, not from qo and qd. So the function qo = g-1(-h(qd)) describes the inverse of a causal path that doesn’t exist.

RM: However, the feedback function qi = g(qo) is visible (as its inverse) in the observed relationship between qd and qo. That’s what Bill called the “behavioral illusion” in the 1978 paper.

AM: The environmental function in 1978 paper is (2) qi = G(qo) + H(d)
The feedback function is: qf = G(qo).
The disturbance function is: qd = H(d)

RM: As long as you remain aware of the fact that qf and qd are the effects of qo and qd, respectively, on the controlled variable and that these effects are not separately detectable quantities in the environment I guess there is no problem. But I have never seen them included in a PCT analysis or model of behavior. All you really need to know is that qi = G(qo) + H(qd)

RM: I would like to know why you prefer PCT to non-PCT versions of control theory?

AM: It’s not me who prefers non-PCT, Rick.

RM: Yes, I know. I’m the enemy of PCT. But could you tell me what you see as the difference between non-PCT and PCT models of behavior that leads you to prefer the latter.

Best

Rick

[amatic]

You can find “qf” in Bill’s old SIMCON models all over the place. You know the simulator of analog computers? It can still be downloaded and run in dosbox. Bill worked a lot on electronic analog computers when developing PCT back in the fifties and sixties, you must have heard of that.

There, the feedback quantity is an entity, an electric signal going out of the feedback function, entering a summing function together with the disturbing quantity to form the input quantity. All of them are “environmental quantities”, as opposed to organism “neural signals”. Functions are “boxes” that convert one signal to another, or one quantity to another, or sometimes several signals to one signal, like the summing function and the comparator. Interestingly, no functions have two different output signals.

RM: So it would be better to have labeled that variable “Feedback Effect”. And the feedback effect of qo on qi is g(qo). That’s why in the 1978 paper Bill wrote qi = g(qo)+h(qd) and not qi = fq + h(qd)

As if the name is important. “Feedback effect” is a perfectly fine name for that variable. You can call it whatever you want, x, y, z, alpha, omega. The important point is that it is a variable in the PCT model of a closed loop system, and that the feedback function relates qo to that variable, and then the summing function relates qd, qf and qi. The feedback function can be defined as fe = G(qo). Using that definition, whatever you call the feedback effect variable, gives you correct results in a quasi-static analysis, as I’ve demonstrated a few days ago, consistent with simulations and experiments. Also, you don’t have to call it anything, perfectly fine if it leads to results consistent with experiments and simulations.

A misleading definition of the feedback function is your: qi = G(qo), because it leads to wrong conclusion, as you have demonstrated in the previous post, saying that qd-qo relationship will be the inverse of a zero correlation relationship qi-qo. Causal path =/= function.

RM: [blah blah] non-causal function [blah blah] causal function [blah]

What an abomination. The PCT approach to modelling behavior is meant to be an improvement over the old causal approach of looking for causes and effects; an improvement over box-and-arrow diagrams that just list what flows where. They are meant to represent quantitative relationships independent of the causal flow.

BP: This is why models in PCT are not just lists of causes and their immediate behavioral effects, like a description of stimuli and the responses they supposedly cause. They are not “flow charts.” PCT models are more like a circuit diagram for a radio or a television set or the insides of a computer chip. The elements of a PCT model are not results to be accomplished in a certain order, but physical devices that continually do whatever they do as long as the power is turned on,. We call those devices “functions” because we try to represent their input-output relationships using mathematical functions, expressions that describe operations like multiplying, dividing, adding, subtracting,. integrating, differentiating, limiting, amplifying, recording, or playing back. A function can be completely defined without saying what the magnitudes of its inputs are going to be or how they will change.

BP: Causation describes the order in which events reliably occur, but it does not involve a description of the mechanism that links cause and effect. Describing behavior in terms of causes and effects is very unparsimonious, in that every new cause has to be separately linked to its effects. The functional approach can do better than that.
http://www.pctweb.org/Cause.pdf?LMCL=YtmNfR

But let’s forget all that and say you invented a useful new thing, functions that ALSO take causal flow into account. You presented your reasoning relating to causal function, you got a result inconsistent with know functional relationship, you predicted a functional relationship between variables that are known to have a near zero correlation. What does that say about the usefulness of the analysis and its assumptions?

[rsmarken]

AM: You can find “qf” in Bill’s old SIMCON models all over the place…

AM: There, the feedback quantity is an entity, an electric signal going out of the feedback function, entering a summing function together with the disturbing quantity to form the input quantity.

RM: Yes, and I can see that qf could be considered a separate quantity in a pursuit tracking task. Cursor position could be called qf as long as cursor position was affected only by the subject’s output (mouse position) and not by any disturbances.

RM: So it would be better to have labeled that variable “Feedback Effect”. And the feedback effect of qo on qi is g(qo). That’s why in the 1978 paper Bill wrote qi = g(qo)+h(qd) and not qi = fq + h(qd)

AM: As if the name is important. “Feedback effect” is a perfectly fine name for that variable. You can call it whatever you want, x, y, z, alpha, omega. The important point is that it is a variable in the PCT model of a closed loop system, and that the feedback function relates qo to that variable,

RM: I think it depends on the actual situation. But I don’t think it matters much. I could add qf to my models and it would make no difference at all in the performance of the models. It certainly makes no difference to the control system itself or to the analysis of the behavioral illusion.

AM: A misleading definition of the feedback function is your: qi = G(qo), because it leads to wrong conclusion, as you have demonstrated in the previous post, saying that qd-qo relationship will be the inverse of a zero correlation relationship qi-qo. Causal path =/= function.

RM: My misleading definition of the feedback function is the exact same one used by Powers in the 1978 paper. There was no qf in his analysis. And my wrong conclusion is the same as Bill’s in that paper: if the system under study is an N-system then the observed relationship between qd and qo will be the inverse of the feedback function, g, relating qo to qi, and not the forward organism function, f, relating qd to qo.

RM: [blah blah] non-causal function [blah blah] causal function [blah]

AM: What an abomination.

RM: So I’ve graduated from nonsense to abomination. Being an atheist I consider it a compliment.

AM: The PCT approach to modelling behavior is meant to be an improvement over the old causal approach of looking for causes and effects; an improvement over box-and-arrow diagrams that just list what flows where. They are meant to represent quantitative relationships independent of the causal flow.

RM: This is news to me. It seems to contradict what Bill said in answer to my question about whether qo is really caused by qd:

BP: There is clearly a causal path from d to qi (the disturbance function), another from qi to e (perceptual function and comparator with an input from r), another from e to o (output function), and finally another from o to qi (environmental feedback function). All but the last work in the “forward” direction.

RM: Bill seems to be pretty comfortable talking about quantitative relationships between variables (disturbance function, etc) as being dependent on the direction of causal flow (all but the environmental function going in the forward direction).

AM: quoting BP: Causation describes the order in which events reliably occur, but it does not involve a description of the mechanism that links cause and effect. Describing behavior in terms of causes and effects is very unparsimonious, in that every new cause has to be separately linked to its effects. The functional approach can do better than that.
http://www.pctweb.org/Cause.pdf?LMCL=YtmNfR

RM: I don’t think Bill means what you think he means. He certainly doesn’t mean that the functions in PCT represent quantitative relationships independent of the causal flow. What I think he means is that a functional description of the causal connection between variables gives a better description of the causal mechanisms involved in behavior than does saying that event A causes event B. Bill wasn’t replacing causality with functional analysis; he was saying that a functional analysis gives a better representation of the causal processes - the mechanisms — that produce what we call behavior.

AM: But let’s forget all that and say you invented a useful new thing, functions that ALSO take causal flow into account.

RM: Actually, I didn’t invent it. It’s been part of PCT analysis from the get go.

AM: You presented your reasoning relating to causal function, you got a result inconsistent with know functional relationship,

RM: Actually I got a result that was not inconsistent with anything (except possibly your ideas about causality); a result that is exactly the same as the one Bill presented in the 1978 paper.

AM: you predicted a functional relationship between variables that are known to have a near zero correlation.

RM: No, like Bill I predicted that the observed relationship between qd and qo for an N- system would be the inverse of the feedback function connecting qo to qi. And that prediction is confirmed in Experiment 4 of the 1978 paper. Anyone can also confirm it for themsevels by doing my “Behavioral Illusion” demo at Behavioral Illusion.

AM: What does that say about the usefulness of the analysis and its assumptions?

RM: I think it says it is very useful because it shows what I believe Bill showed in his 1978 paper using the same analysis: that the only proper way to study the behavior of living control systems is to do experiments aimed at determining the variables they control and how they control them. But apparently, since you consider my analysis (which I think is the same as Bill’s) to be an abomination, you have a different idea about what Bill showed in that paper. I don’t really know what you think it showed but perhaps I’ll find out when you publish some research based on your understanding of the implications of that paper.

RM: But for now I think it’s “most likely you go your way (and I’ll go mine)”.

Best

Rick

[amatic]

RM: So what you are observing as the relationship between qo and qd is actually the inverse of the relationship between qo and qi

Right. Completely consistent.

[amatic]

RM: So I’ve graduated from nonsense to abomination. Being an atheist I consider it a compliment.

Nothing religious implied, I just mean that the expression “causal function” looks horrible to me. Especially horrible when applied to the quasi-static analysis, which explicitly ignores transients (small changes of variables in time), and thus the causality. Sure, the arrows in diagrams do represent causal flow, it is not like PCT completely ignores causality, and I did not say correctly that functional representations are independent of causal flow. For example, we usually write the output (effect) variable on the left side of the equation, and input or inputs on the right.

Still, functional descriptions are very much pitted against causal descriptions in the cause vs mechanism paper. That is why the term “causal function” looks so horrible and that is why it was never used in PCT. Maybe the “non causal function” is worse. Each function in the loop is causal in the sense that changes in that function will cause changes in the behavior of the system and the qd-qo relationship.

There are no “non-causal” functions in that sense. You change the Kf, qd-qo changes. You change Ko, qd-qo changes. Change Kd, qo-qd changes.

On top of that, what you call the causal and non-causal functions in the context of the behavioral illusion, are really solutions for the qd-qo relationship in an N and Z system. That is, those are solutions to different systems. I wrote the whole analysis of N and Z systems, derivations of the solutions for them, and you agreed and said that that was what you and Bill were saying all along, even though you were not saying it exactly. At that time you were calling those two functions “observed” and “actual”, and later you changed the name to “causal” and “non-causal”.

Well, anyway, as I said, it is very strange talking to you. You seem to have very different understanding of what “function” or “relationship” or “A relates to B”, or “equation”, or “solution” means than I do, and it seems you kind-of shift the meaning depending on the point you’re trying to make.

[amatic]

For example, it is more-less ok to say that the feedback function relates qo to qi, The problem is the vagueness. “Relates” implies a function between qo and qi, and can be misleading, and has mislead you to write that the feedback function is qi=G(qo), several times.

There is a function in the diagram between variables “qi” and “qo”. Still, qo and qi are not “related by a function”, because in an N system qi is stable and not correlated with qo, so there is no function describing the relationship. Same goes for qd-qi.

“The feedback function relates qo to an effect (or contribution) on the qi” is more correct. If we call this effect “feedback_effect”, then we can also write the function relating the output quantity to the feedback effect as: feedback_effect = G(qo). Now, that is a nice, real function you can see by plotting “feedback_effect” against qo. You can see its inverse by plotting qo against the “feedback effect” (just by switching the x and y variable in the plot).

Incidentally, the “feedback effect” is going to be negative of qd, so the plotted inverse of “feedback_effect” vs “qo” is going to be very close to qd vs qo. I might have switched some of the sides there, I should verify in by actually plotting the thing.

[rsmarken]

Hi Adam

RM: So what you are observing as the relationship between qo and qd is actually the inverse of the relationship between qo and qi

AM: Right. Completely consistent.

RM: It’s not a matter of consistency; it’s a matter of fact; when you are studying a living control system what you are observing as the relationship between qo and qd is actually the inverse of the relationship between qo and qi. That is what you are seeing in my little demo of the “Behavioral Illusion”: Behavioral Illusion

Here are the typical results:

RM: What you are seeing here is a plot of the observed relationship between qd (Stimulus, S) and qo (Response, R) that occurs in a tracking task with two different feedback functions (functions which relate qo to qi). The feedback functions are both linear: qi = k*qo. So the observed relationship between qd and qo is expected to be qo = -1/k (d) (per the first of the two equations at the end of p. 425 of Powers, 1978).

RM: And, indeed, when the feedback function is weak (qi = .5 * qo) the observed relationship between qd and qo is qo = - 2* qd (2 = 1/.5) and when the feedback function is strong (qi = 1.5 * qo) the observed relationship between qd and qo is qo = - .67* qd (.67 = 1/1.5). So the observed functional relationship between qd and qo (between S to R) does not reflect the function that corresponds to the actual causal path from S to R (from qd to qo) through the organism. Rather it reflects the function that corresponds to the inverse of the causal path from R to qi (from qo to qi) through the environment, qi, of course, being the controlled variable, the variable missing from all models of organisms in conventional psychology.

RM: So the functional relationships between qd and qo that you see in the figure above have nothing to do with the actual causal relationship between S and R (qd and qo). The observed relationships between S and R shown above are non-causal functions in the sense that they are functional relationships that do not correspond to the actual causal path that exists between S and R. The actual causal path between S and R (qd and qo) is characterized by some causal function (called f in the 1978 paper), the nature of which is determined by characteristics of the organism and is almost certainly the same in both feedback conditions, though we can’t be sure of that because we can’t see the actual causal function relating qd to qo.

RM: The relationship between S and R that you are seeing in the figure above is a mathematical side-effect of the behavior of a closed-loop system; the two functional relationships between S and R (qd and qo) that are shown in the figure are non-causal functions in the sense that they are purely mathematical and have nothing at all to do with the actual causal path that connects qd to qo.

RM: If my distinction between causal and non-causal functions gives you heartburn you are getting a sense of how more than 99% of the scientific psychology community felt (and still feels) about the basic message of a PCT analysis of behavior. They felt that way because what Bill was demonstrating in that paper is essentially what he said in his Foreword to my book MIND READINGS: If the phenomenon [control] you see here really works as the model [PCT] shows it to work, then a whole segment of the scientific literature needs to be deposited in the wastebasket. In other words, it says that all of your scientific work in psychology has been a waste of time.

RM: Another way to describe the conclusion of the 1978 paper is that you can’t learn much about how living systems work by studying relationships between qds (IVs) and qos (DVs). In order to understand how behavior works you have to determine, at the very least, what perceptual variables the organism is controlling. This is why the title of Bill’s main book on PCT is Behavior: The Control of Perception and not Behavior: Based on Functions, not Causes.

RM: I know you have drawn a very different conclusion from the 1978 paper regarding what the behavioral illusion is and what its implications are for the study of the behavior of living systems. So I imagine you will want to answer this post. If you do, you will get to have had the last word in this discussion because I really have no more interest in continuing it. It has been just too depressing. Unlike Inigo Montoya (of Princess Bride fame) I just haven’t been able to get used to disappointment.

RM: I will keep following it though and my interest in participating in the discussion would likely be rekindled if you presented some of the research you have done based on your understanding of the behavioral illusion and/or on what you learned from Powers’ 1978 paper. You are obviously a very smart guy; a highly skilled researcher, mathematician and computer scientist. So while I think you couldn’t be more wrong about what the behavioral illusion is and why it’s important, I have the greatest respect for your skills and would very much like to see what research you have done based on your understanding of PCT.

Best regards

Rick

[amatic]

RM: It’s not a matter of consistency; it’s a matter of fact; when you are studying a living control system what you are observing as the relationship between qo and qd is actually the inverse of the relationship between qo and qi

That plot is showing the relationship between qo and qd (R and S) and is not showing the inverse of the relationship between qo and qi, because that relationship is a zero correlation, and if you show it on a plot, it looks like a cloud of points, not like these nice lines.

RM The feedback functions are both linear: qi = k*qo.
RM when the feedback function is strong (qi = 1.5 * qo)

That is not the feedback function. The feedback function in your code that you used to make the plot for the behavioral illusion is defined differently.

xpos = (-d[session][n]) + mx*1.5

The feedback function here is G(x) = x * 1.5, and qi = qd + G(qo), and not qi=G(qo), as you wrote.

Why you keep insisting on changing the definitions of functions are their names, even though your own code shows you know how to put them in a model is… strange.

RM relationship between qd and qo (between S to R) does not reflect the function that corresponds to the actual causal path from S to R (from qd to qo) through the organism. Rather it reflects the function that corresponds to the inverse of the causal path from R to qi (from qo to qi) through the environment […]

Good. Both of those are causal paths, while you used call one of them non-causal. And also you don’t call them causal functions, but correctly call them causal paths. Functions on the diagram do not correspond to causal paths, though. There is a function in the path between qo and qi, but qo and qi are not related by a function, because there are other causal paths ending in qi.

---

I imagine it is depressing and embarrassing to you when you realize that you made these sorts of basic mistakes, so that’s why you keep on claiming you were right, and also getting tired of discussion.

One thing I got from this is, even though you are the first to jump on other people - either PCT-ers or scientists from other areas, and point out mistakes in their reasoning or accuse them of not understanding control theory, or policing their diagrams for mislabeled variables, etc - when it comes to your understanding of control systems and the underlying relationships between variables, you could work on it a bit more. I hope in your future papers and books you double and triple check your claims.

I think we got the same general message from 1978 paper on scientific psychology and the importance of control systems as models of living systems, and controlled variables as crucial phenomena. That was never under discussion on this topic.

This discussion was specifically about the scope of the behavioral illusion. It is one of many possible mistakes, illusions and artifacts, and if someone claims a phenomenon is a behavioral illusion, he needs to demonstrate, just like the 1978 paper demonstrates in three examples, that the inverse of the feedback function mostly determines the qd-qo relationship.

Best, Adam

[rsmarken]

RM: Darnit, Adam, I just can’t quit ya;-)

RM: By the way, before I start I want to mention that I got a message from Discourse suggesting that I invite others into this conversation. So if anyone besides Adam and I are watching this please feel free to join in.

RM: It’s not a matter of consistency; it’s a matter of fact; when you are studying a living control system what you are observing as the relationship between qo and qd is actually the inverse of the relationship between qo and qi

AM: That plot is showing the relationship between qo and qd (R and S) and is not showing the inverse of the relationship between qo and qi, because that relationship is a zero correlation, and if you show it on a plot, it looks like a cloud of points, not like these nice lines.

RM: That plot is showing the inverse of the actual feedback function relating qo to qi, not of the observed relationship between qo and qi. What I am plotting is precisely the same as what Bill plotted in Figure 6 in the 1978 paper:

RM: The only difference between the graph in Figure 6 above and my plot (below) is that Bill’s feedback function was a polynomial – g(qo) = Aqo + B qo^2 – whereas mine is simply a linear function – g(qo) = A*qo.

RM: In my tracking task the feedback function changes from qi =.5 * qo to qi = 1.5 * qo so you see a change in the slope of the relationship between qd (S) and qo (R ), a change that clearly reflects the change in the feedback function, not the forward causal path from qd to qo.

RM The feedback functions are both linear: qi = k*qo.
RM when the feedback function is strong (qi = 1.5 * qo)

AM: That is not the feedback function. The feedback function in your code that you used to make the plot for the behavioral illusion is defined differently.

xpos = (-d[session][n]) + mx*1.5

AM: The feedback function here is G(x) = x * 1.5, and qi = qd + G(qo), and not qi=G(qo), as you wrote.

RM: As you note in the equation that defines qi, G(qo) is the same as G(x) since the argument of the feedback function is qo. And qi = G(qo) is, indeed, the feedback function. I write it as a separate equation without including qd because the feedback function, G(), defines the independent effect of the system on the controlled variable. It is the inverse of G() that shows up as the observed relationship between qd and qo in a negative feedback control system. That’s the behavioral illusion: What you observe as the relationship between qd and qo is the inverse of the feedback function, G(), relating qo to qi rather than the organism function, F(), relating qd to qo.

AM: Why you keep insisting on changing the definitions of functions are their names, even though your own code shows you know how to put them in a model is… strange.

RM: Perhaps it seems strange because it’s not happening. I think I’ve been pretty consistent in my description of functions and their names. I think you are trying too hard to see that I’m not.

RM: the relationship between qd and qo (between S to R) does not reflect the function that corresponds to the actual causal path from S to R (from qd to qo) through the organism. Rather it reflects the function that corresponds to the inverse of the causal path from R to qi (from qo to qi) through the environment […]

AM: Good. Both of those are causal paths, while you used call one of them non-causal.

RM: I don’t think so. I believe I have consistently referred to the feedback function as a causal function representing a causal path from qo to qi. Maybe you are confused because, when the inverse of the feedback function shows up as the observed relationship qd and qo (as in the Figures above), I have referred to this observed relationship between qd and qo as a non-causal functional relationship. I refer to the observed qd - qo relationship this way – the relationship that is the inverse of the feedback function – even though the feedback function corresponds to a real causal relationship between qo and qi, because, as Bill shows in the 1978 paper, the inverse of the feedback function relationship between qd and qo that is observed when studying an N-system is simply a mathematical side effect of the disturbance resisting characteristic of such systems.

AM: And also you don’t call them causal functions, but correctly call them causal paths. Functions on the diagram do not correspond to causal paths, though.

RM: Maybe not for you but for Bill they certainly did.

AM: There is a function in the path between qo and qi, but qo and qi are not related by a function, because there are other causal paths ending in qi.

RM: Maybe this is the source of our apparently irreconcilable differences. This is an invention of yours; it has nothing to do with Bill’s 1978 paper. What you are saying is that qo and qi are not related by a function because there are other causal paths ending in qi. In Bill’s analysis in the 1978 paper the only other causal path ending in qi besides the one from qo is the one from qd. So you are saying that because qi = h(qd) + g(qo), qo and qi are not related by the function g(). Nothing like that shows up in Bill’ analysis probably because it is not true; qd and qo have independent effects (via independent functions) on qi so the functional relationship between qo and qi – the feedback effect of qo on qi – can be treated – indeed, must be treated – independently from the effect of qd on qi.

AM: I imagine it is depressing and embarrassing to you when you realize that you made these sorts of basic mistakes, so that’s why you keep on claiming you were right, and also getting tired of discussion.

RM: Nope, I am neither embarrassed nor depressed. Just disappointed. And I’m not really tired of the discussion. I just think that I could understand your position better (and I could stop repeating mine) if we had some actual data to discuss. I think I could get a better handle on what you think was Bill’s message in the 1978 paper if you could show me some research that you think is consistent with the points Bill was trying to make in that paper.

AM: One thing I got from this is, even though you are the first to jump on other people - either PCT-ers or scientists from other areas, and point out mistakes in their reasoning or accuse them of not understanding control theory, or policing their diagrams for mislabeled variables, etc - when it comes to your understanding of control systems and the underlying relationships between variables, you could work on it a bit more. I hope in your future papers and books you double and triple check your claims.

RM: I’m sorry if it seems like I’m “pouncing” but I have been working on PCT for a heck of a long time and when I see what seem like clearly mistaken ideas about it I point them out. I do feel like I am qualified to teach PCT but I am also aware of the fact that I can be wrong. So while I don’t like being told that I am wrong I try not to become immediately defensive but, rather, I try to look at the criticism carefully and see whether I am indeed wrong.

RM: When Bill said I was wrong, which wasn’t often but he certainly did on occasion, it turns out that he was almost always right and I accepted his correction (as in the case you posted). (Actually, I can think of only one time he was wrong and I was right; it was quite exciting. I can’t remember what it was about but I can remember exactly where I was when it happened).Others have also pointed out when I was wrong and sometimes they were right and I was not. When I can see that I am being corrected correctly I will change my understanding and welcome it as a learning experience.

RM: I guarantee you that I am willing to be convinced that your understanding of Bill’s 1978 paper is correct and mine is wrong. But so far you haven’t convinced me. That’s why I want to see some research that you see as being consistent with the analysis in the 1978 paper. That might convince me that I am either wrong or just haven’t been understanding you.

AM: I think we got the same general message from 1978 paper on scientific psychology and the importance of control systems as models of living systems, and controlled variables as crucial phenomena. That was never under discussion on this topic.

RM: I’m not sure we did. If we did I don’t believe we would still be disagreeing.

AM: This discussion was specifically about the scope of the behavioral illusion. It is one of many possible mistakes, illusions and artifacts, and if someone claims a phenomenon is a behavioral illusion, he needs to demonstrate, just like the 1978 paper demonstrates in three examples, that the inverse of the feedback function mostly determines the qd-qo relationship.

RM: First of all, if the system under study is an N-system then you have to know what variables the system is controlling in order to know whether any observed relationship between environmental and behavioral variables is one between qd and qo. That’s because qd is defined as a disturbance to a controlled variable, qi, and qo is defined as a behavioral output that compensates for the effect of that disturbance to qi. So you can’t do your proposed demonstration until after you have done some form of the test for the controlled variable.

RM: Once you have identified a controlled variable so that you can identify a disturbance to that variable and an output that has opposing effects on that variable – that is, once you have identified a qd and qo – then there is no need to do your proposed demonstration because, per Bill’s analysis in the 1978 paper, you know that the observed relationship between qd and qo will be qo = g-1[-h(qd)] – that is, it will be mostly determined by the inverse of the feedback connection between qo and qi. If h() is a multiplier of 1 then qo = g-1(qd) and the observed relationship between qo and qd will be completely determined by the inverse of the feedback connection between qo and qi.

RM: So you have demonstrated that the relationship between qd and qo is a behavioral illusion as soon as you have identified the variable, qi, that is being kept under control by the opposing effects of qo and qd. There is no need to identify the feedback function to do this demonstration but it might be fun to verify it when you can measure the actual feedback function.

RM: Sorry I didn’t keep my promise about not responding but, hey, I’m a control system and your posts are apparently disturbances to the variables I am controlling so here’s my output. I guess we’ll probably just go on until one of us drops. But this has actually been very helpful to me; I think I’ve improved my understanding of the illusion described in Bill’s 1978 paper considerably. And I am pretty sure that I would get a better idea of what you take away from the 1978 paper if you could at least describe some of your research that you think is the kind that is consistent with the analysis in that paper.

Best regards

Rick

[amatic]

RM: That plot is showing the inverse of the actual feedback function relating qo to qi, not of the observed relationship between qo and qi.

After all this talk I know what you mean by “actual feedback function relating qo to qi”, but that is not a good way of expressing the relationship between qo and qi, because the feedback function is not relating qo to qi.

The feedback function is relating qo to an effect on qi, what Bill in programs for the analog computer and Bill and Bruce Abbott in LCSIII called the feedback quantity. The name is appropriate, I think, because it is another environment quantity, along with input, output and disturbance quantities, in contrast to perceptual, reference and error signals. I’m not saying that is the only correct name, I just think it is nicely chosen. ‘Feedback effect’ is also good.

As you can see here, and also in equation (1) in 1978 paper, there is a function relating qo and qi, that is the organism function F, qo = F(qi). The feedback function is relating qo to qf. It can be very misleading to mix the two definitions. Another environmental function is relating qi to qd and qf. This is what Bill calls “linear combination” in the 1978 paper, which just means “sum”, as in qi = qd + qf. This can also be written as qi = S(qd, qf), which means that S is a function of two variables.

I find plots of these relationships illuminating and it is nice to see how they are consistent with the verbal descriptions and with math equations. If they are not consistent, there is something wrong in either math, verbal description or plots and simulations (or diagrams). In either case an opportunity to dig deeper.
Here is a plot of a simulation of a control system with Kf = 5, and Ko = -50, time constant 3 seconds.
python code

Next, the observed input-output relationship, or disturbance - output relationship.

That is the observed in blue and predicted in red. The red line was predicted as a solution of equations describing an negative feedback system for Kf = 5. The observed and predicted match very nicely. I don’t see anything else that could be called “actual qd-qo”, there is just this one, that is the observed and actual qd-qo relationship.

Next, the qo-qf relationship.

Notice how the inverse of a function is the same as rotating the plot on the side (well, flipping trough a diagonal would be a more correct description, in any case, just switching the x and y variables in the plot). The inverse of multiplication by 5 is dividing by 5, so: qf = 5*qo, and qo = 0.2 qf.

The inverse of the feedback function is qo = 0.2 * qf, but we have qo = - 0.2 * qd. We can loose the minus if we use a comparator instead of a summer in the other environment function, but Ok. We do get that the qd-qo relationship is the inverse of the function relating qf to qo. And the qd-qo function is the same as function relating qo to qf (changing the sign).

Now for the qi-qo relationship. The function that relates qi and qo can also be plotted, along with its inverse:

That is the “relationship between qo and qi” and the “inverse of the relationship between qo and qi”. I know you meant the one going from qo toward qi through the feedback function, but the relationship between qi and qo is plotted here. You also wrote in the previous post (twice) that the organism function is relating qd and qo, that is not correct. The organism function is relating qi and qo.

---

Some notes: in the simulation, the F function is an integrator, it accumulates values, meaning that its output value depends not only on the input, but also on previous values of output. This could be further explored, to see the relationship between qi and the derivative of qo, it gives nice plots, too, in some simulations. But, that is also the reason why qi-qo is not a nice linear plot.

Another direction for exploring starts from the fact that the inverse of integrating is derivating. Putting an integrator in the feedback path can make the qd vary as a derivative of qo.

This is a discussion strictly about the behavioral illusion, as the title says, not about all the other possible blunders or artifacts or general consequences of mistaking N systems for Z systems, etc, etc, so I’m sticking to the topic of the behavioral illusion and not discussing controlled variables or other things.

RM: First of all, if the system under study is an N-system then you have to know what variables the system is controlling in order to know whether any observed relationship between environmental and behavioral variables is one between qd and qo. That’s because qd is defined as a disturbance to a controlled variable, qi, and qo is defined as a behavioral output that compensates for the effect of that disturbance to qi. So you can’t do your proposed demonstration until after you have done some form of the test for the controlled variable.

Yes, I perfectly agree with that, especially the bold part.

[rsmarken]

Hi Adam

AM: The feedback function is relating qo to an effect on qi, what Bill in programs for the analog computer and Bill and Bruce Abbott in LCSIII called the feedback quantity. The name is appropriate, I think, because it is another environment quantity, along with input, output and disturbance quantities, in contrast to perceptual, reference and error signals. I’m not saying that is the only correct name, I just think it is nicely chosen. ‘Feedback effect’ is also good.

RM: I agree that it’s important to keep in mind that the feedback function, g(), converts system output, qo, into an effect on the controlled variable, qi. But it’s also important to keep in mind that the same is true of the disturbance; the disturbance function, h(), converts the disturbance quantity, qd, into an effect on the controlled variable. This is shown in Figure 1 from the the 1978 paper:

RM: The effects of output and disturbance on the controlled variable are represented as g(qo) and h(qd) in this diagram. If you added the variable qf to this diagram to explicitly show the feedback effect on qi I think you should also add a variable such as qe to the diagram to explicitly show the effect of the disturbance on qi: qe = h(qd). This might make things clearer from your perspective but i think it’s unnecessary.

AM: You also wrote in the previous post (twice) that the organism function is relating qd and qo, that is not correct. The organism function is relating qi and qo.

RM: Actually, we are both correct. The organism function is qo = f(qi), per equation 1 in Powers (1978). However Powers’ paper is a critique of scientific psychology in general and the experimental method used in the field in particular. In conventional scientific psychology the experimentally determined relationship between qd (IV) and qo (DV) Is thought to tell us about the nature of the organism. This is done under the assumption that the system under study is either a Z- system (no feedback effect of qo on qi) or, if an N-System, a system where the feedback makes little difference.

RM: So in Powers analysis the organism function assumed by scientific psychologists is taken to be qo = f[h(qd)] (last equation on p. 423). This is derived from the organism function above and the environment function qi = g(qo) + h(qd) (equation 2 in Powers (1978)) under the assumption that g(qo) - the feedback effect of output – either doesn’t exist (Z -system) or doesn’t matter.

RM: The assumption in scientific psychology is that the function h() is essentially a multiplier of one. So in the conventional psychology experiment, when qd is manipulated and concomitant variation in qo is observed, the researcher assumes that what is being seen is an approximation to the organism function, f(), since qo = f(qd) (or DV = f(IV)). Powers (1978) shows that what is actually being seen is an approximation to qo = g-1(-qd).

---

AM: Some notes: in the simulation, the F function is an integrator, it accumulates values, meaning that its output value depends not only on the input, but also on previous values of output. This could be further explored, to see the relationship between qi and the derivative of qo, it gives nice plots, too, in some simulations. But, that is also the reason why qi-qo is not a nice linear plot.

RM: All this assumes that qi is a simple scalar variable. In most tracking tasks this is probably close to being the case. But the kinds of variables controlled by living organisms – particularly people – can be pretty complex functions of simpler perceptual or environmental variables. So whatever relationships between qi and qo you find to hold for scalar qi may not hold for more complex qi’s. Why not try modeling control of some of the qi in STEP H: BEYOND TRACKING of your beautiful reproductions of Bill’s demos at PCT Tutorial 1. See if the type of variable controlled makes a difference in your conclusions about the relationships between variables in a control loop.

AM: Another direction for exploring starts from the fact that the inverse of integrating is derivating. Putting an integrator in the feedback path can make the qd vary as a derivative of qo.

RM: THe feedback path shouldn’t affect the disturbance. I think what you mean is that it would make qo vary as the derivative of qd, which should be true.

AM: This is a discussion strictly about the behavioral illusion, as the title says, not about all the other possible blunders or artifacts or general consequences of mistaking N systems for Z systems, etc, etc, so I’m sticking to the topic of the behavioral illusion and not discussing controlled variables or other things.

RM: I don’t see how you can make any sense of the behavioral illusion discussed in Powers (1978) without discussing controlled variables. The illusion turns on the fact that the existence of a controlled variable is being ignored or simply missed. It seems to me that discussing the behavioral illusion without discussing controlled variables is like discussing the bent stick illusion without discussing the differential refraction of light in air and water.

RM: And I also wonder why you are fixated on the S-R behavioral illusion described in Powers (1978). It doesn’t seem relevant to your power law of movement research. It’s only relevant to research where an environmental variable (qd) is manipulated under controlled conditions to determine whether there is concomitant variation in a behavioral variable (qo). In the power law research both variables involved in the power law – curvature and speed of movement – are behavioral variables. So there is no controlled variable being disturbed by one of those variables and protected from that disturbance by the other.

RM: First of all, if the system under study is an N-system then you have to know what variables the system is controlling in order to know whether any observed relationship between environmental and behavioral variables is one between qd and qo. That’s because qd is defined as a disturbance to a controlled variable, qi, and qo is defined as a behavioral output that compensates for the effect of that disturbance to qi. So you can’t do your proposed demonstration until after you have done some form of the test for the controlled variable.

AM: Yes, I perfectly agree with that, especially the bold part.

RM: Super!

Best

Rick

[amatic]

Adding the H function that converts a disturbance to the disturbance effect on qi is definitely a thing to explore. The solution for the qo-qo in an ideal negative feedback system is:

qo = G^-1 [ H (qd) ]

The interesting phenomenon is when the disturbance function is the same as the feedback function, G=H. The inverse of G cancels out the inner G, and the equation simplifies to just

qo = qd

If H and G are nonlinear, the nonlinearities cancel out, we get a linear relationship.

AM: You also wrote in the previous post (twice) that the organism function is relating qd and qo, that is not correct. The organism function is relating qi and qo.
RM: Actually, we are both correct.

Nice damage control. You were referring to some other organism function.

Also, correct for the qo varying as derivative of qd.

Let me get back to this:

RM: Maybe this is the source of our apparently irreconcilable differences. This is an invention of yours; it has nothing to do with Bill’s 1978 paper. What you are saying is that qo and qi are not related by a function because there are other causal paths ending in qi. In Bill’s analysis in the 1978 paper the only other causal path ending in qi besides the one from qo is the one from qd. So you are saying that because qi = h(qd) + g(qo), qo and qi are not related by the function g(). Nothing like that shows up in Bill’ analysis probably because it is not true; qd and qo have independent effects (via independent functions) on qi so the functional relationship between qo and qi – the feedback effect of qo on qi – can be treated – indeed, must be treated – independently from the effect of qd on qi.

I’m saying that because qi = h(qd) + g(qo), it is not correct to say “qi and qo are related by function G”, because it is misleading. It implies that you can write qi = G(qo), but that is not correct, because that is not what the plots are going to show.

If you make a plot for an ideal negative feedback system, qi is going to be just one constant value (emphasize - in IDEAL negative feedback systems, in real systems it is going to be some low correlation cloud). So, qi is really not related (correlated) to any other variable in the ideal negative feedback system, even though there are at least two causal paths ending in qi.
(edited the upper two paragraphs a few times, not yet sure it is too clear. Qi is related to qo by an integral, which is a function too…)

Which brings me to the answer to what happens if the type of the controlled variable is different - nothing much. For good control, qi is always going to be more stable (have less variance) than some disturbing quantity, because behavior will vary to oppose the disturbance and keep qi near the reference value. To the extent any real system is well approximated by an ideal negative feedback system, the relationships hold. For low gain control, or difficult disturbances, or very nonlinear input functions, many things are possible, like your demo area vs perimeter shows.

RM: It seems to me that discussing the behavioral illusion without discussing controlled variables is like discussing the bent stick illusion without discussing the differential refraction of light in air and water.

Not at all. Qi represents all possible controlled variables. All the same to math or simulation. Just trying to stay on topic.

RM: And I also wonder why you are fixated on the S-R behavioral illusion described in Powers (1978).

I think it is a very interesting phenomenon, not just as the possible illusion in behavior research, but as a general property of negative feedback systems. It was used in analog computers to build inverse functions, and it is still used in control engineering and in designing amplifier circuits, etc, etc. Wherever there are feedback systems, there is input-output determined by the function in the feedback path, and mostly independent of the amplifier gain in the forward path. It is important to understand it right, can be useful for research.

RM: It doesn’t seem relevant to your power law of movement research. It’s only relevant to research where an environmental variable (qd) is manipulated under controlled conditions to determine whether there is concomitant variation in a behavioral variable (qo). In the power law research both variables involved in the power law – curvature and speed of movement – are behavioral variables. So there is no controlled variable being disturbed by one of those variables and protected from that disturbance by the other.

That sounds like something I would say (almost). Highly suspicious to hear it from you.

Ok, I’ll bite. What brings you to that conclusion after your previous conviction that the power law is an example of the behavioral illusion?

[rsmarken]

Hi Adam

AM: The interesting phenomenon is when the disturbance function is the same as the feedback function, G=H. The inverse of G cancels out the inner G, and the equation simplifies to just

qo = qd

RM: Yes, this is the situation in most of the tracking tasks we use in demos since G and H are typically multipliers of 1.0. That’s why we see the nice “mirror image” relationship between qo and qd.

AM: If H and G are nonlinear, the nonlinearities cancel out, we get a linear relationship.

RM: Yes!.

AM: You also wrote in the previous post (twice) that the organism function is relating qd and qo, that is not correct. The organism function is relating qi and qo.

RM: Actually, we are both correct.

AM: Nice damage control. You were referring to some other organism function.

RM: It wasn’t “damage control”. It was simply pointing out an essential fact about what Powers was referring to as the “organism function” in the article we have been discussing. You were right that qo = f(qi) is the “organism function” in a control diagram. But because Powers paper was about how scientific psychologists go about trying to determine the organism function, he wrote the organism function as qo = f[h(qd)]. This is because experimental psychologists implicitly assume that the proximal cause of behavior – what we call qi – is directly proportional to the distal cause – qd – which is the independent variable in conventional psychology experiments. That is, experimental psychologists have assumed tha h(qd) is essentially a multiplier of 1 so that qi = qd. So while conventional psychologists are actually looking at the relationship qo = f[h(qd)] they assume they are looking directly at the system function qo = f(qi).

RM: It is not “damage control” to point out that Powers considered qo = f[h(qd)] --rather than qo = f(qi) – to be the relevant organism function because seeing the organism function as qo = f[h(qd)] is crucial to understanding the behavioral illusion, as can be seen in this except from Powers (1978):

RM: I highlighted “organism function” to make it clear that Bill is referring to the second equation – qo = f[h(qd)] – as the organism function, the one for a Z- system.

AM: Let me get back to this:

RM: Maybe this is the source of our apparently irreconcilable differences…What you are saying is that qo and qi are not related by a function because there are other causal paths ending in qi…

AM: I’m saying that because qi = h(qd) + g(qo), it is not correct to say “qi and qo are related by function G”, because it is misleading. It implies that you can write qi = G(qo), because that is not what the plot are going to show.

RM: Well, I don’t think it is misleading. But I no longer think this is the reason for our disagreement.

RM: It seems to me that discussing the behavioral illusion without discussing controlled variables is like discussing the bent stick illusion without discussing the differential refraction of light in air and water.

AM: Not at all. Qi represents all possible controlled variables. All the same to math or simulation. Just trying to stay on topic.

RM: I meant that when the topic of discussion is the behavioral illusion as described in Powers (1978) – an illusion that results from failure to take into account the fact that qd and qo have opposing effects on a controlled variable, Qi, whatever that variable may be – it seems like the discussion should always focus on the fact that the illusion occurs because the existence of controlled variables is being ignored or missed.

RM: And I also wonder why you are fixated on the S-R behavioral illusion described in Powers (1978)…
RM: It doesn’t seem relevant to your power law of movement research…

AM: That sounds like something I would say (almost). Highly suspicious to hear it from you.

AM: Ok, I’ll bite. What brings you to that conclusion after your previous conviction that the power law is an example of the behavioral illusion?

RM: I never thought the power law was an example of the behavioral illusion described in Powers (1978). It was obvious from the get go that the power law was not that kind of illusion; there was no independent variable that could be the qd causing a qo via the organism. But it is an illusion in the same way that the observed relationship between qd and qo is an illusion; it is an observed side effect of control that is taken to reflect something about how the organism works – about the organism function – when it doesn’t. I explained this in both of my papers on the power law. You (and almost everyone else involved in that debate) didn’t care much for my explanation, to say the least. So let’s just leave it at our agreeing that the power law is not an example of the behavioral illusion described in Powers (1978).

Best

Rick

[amatic]

RM: It wasn’t “damage control”. It was simply pointing out an essential fact about what Powers was referring to as the “organism function” in the article we have been discussing. You were right that qo = f(qi) is the “organism function” in a control diagram. But because Powers paper was about how scientific psychologists go about trying to determine the organism function, he wrote the organism function as qo = f[h(qd)].

Sorry, I thought it was damage control.

To me, it seems like “the organism function” is always F, defined in equation (1) qo = F(qi). Bill is very consistent about it in the paper. The equation qo = f(h(qd)) is not the organism function, and he doesn’t refer to it as the organism function. It is the solution for a zero-feedback system for qo, meaning “how will qo depend on other variables in the system”.

In the text you highlighted, he is saying that the organism function F is replaced by the feedback function G’s inverse in the solutions. He is comparing to equations, solution for qo for an N system, and solution for qo for a Z system. He is also saying that qi* can be taken as zero. Let’s also put a name to H(qd), to simplify the expressions:

(1) qi* = 0
(2) qe = H(qd)

(3) qo = G^-1 [ qi* - H(qd) ]
(4) qo = G^-1 [ H (qd) ]
(5) qo = G^-1 ( qe )

Equation (3) is a solution for qo for a negative feedback system taken from the paper, and equation (5) is a slightly simplified form.

---

(6) qo = F [ H (qd) ], using (2):
(7) qo = F ( qe )

Equation (6) is a solution for qo for a zero feedback system taken from the paper, and equation (7) is a slightly simplified form.

---

Specifically in the sentence where you highlighted “organism function”, he is referring to these two equations:

N-system equation: (5) qo = G^-1 ( qe )
Z-system equation: (7) qo = F ( qe )

In the N-system equation, qo is determined by the feedback function’s inverse of qe. In the Z-system equation, the qo is determined by the F function of qe. That is why he says “the organism function F in the z-system equation is replaced by the feedback function g^-1 in the N system equation”.

RM: So while conventional psychologists are actually looking at the relationship qo = f[h(qd)] they assume they are looking directly at the system function qo = f(qi).

If they are looking at an N system, they are looking at qo = G^-1 ( H (qd) ], or simplified (5) qo = G^-1(qe), but they assume they are looking at (7) qo = F (qe). They have misidentified an N system, thinking it is a Z system.

If they measure this in an experiment:
qo = 0.2 * qe

They will think that F(x) = 0.2 * x. That is the behavioral illusion, because F might really be F(x)= integral (1000x) or whatever, and the 0.2 was determined by their experimental setup where they had some G function G(x) = 5 * x.

RM: I meant that when the topic of discussion is the behavioral illusion as described in Powers (1978) – an illusion that results from failure to take into account the fact that qd and qo have opposing effects on a controlled variable , Qi, whatever that variable may be – it seems like the discussion should always focus on the fact that the illusion occurs because the existence of controlled variables is being ignored or missed.

I think that ignoring the existence of controlled variables is a consequence of having a wrong model of organism behavior. There is no such thing as a controlled variable if the model of behavior is a lineal causation, stimulus-response, R = O(S), which is equivalent to (7) qo = F(qe).

The behavioral illusion happens because of adopting (7) as the model of organism behavior and trying to find the organism function O by relating stimuli to responses. Sure, they are also ignoring controlled variables and dynamics of feedback systems and so on, but I think the core of the illusion is the R = O(S) model. An error in system identification.

RM: I never thought the power law was an example of the behavioral illusion described in Powers (1978).

No?

M&S (2018): In the present paper we answer these claims and show that the power law of movement is, indeed, an example of a behavioral illusion.
[…] we showed that this assumption is likely to be based on what Powers (1978) called a behavioral illusion

The title of the paper - “Power law as behavioral illusion” - really sounds like you think (or thought at the moment of writing) that the power law is an example of the behavioral illusion, as defined in 1978 paper. Now it sounds like you’ve changed your mind.

As for the side effects of control not revealing something about the organism, we might disagree there too, as side-effects often do reveal “something” about the organism (reaction time reveals something about the level of control, maybe), but that is a whole different discussion not related to the behavioral illusion.

RM: So let’s just leave it at our agreeing that the power law is not an example of the behavioral illusion described in Powers (1978).

Great, fine with me.

[rsmarken]

Hi Adam

AM: To me, it seems like “the organism function” is always F, defined in equation (1) qo = F(qi). Bill is very consistent about it in the paper. The equation qo = f(h(qd)) is not the organism function, and he doesn’t refer to it as the organism function.

RM: The function qo = f[h(qd)] is the organism function from the point of view of scientific psychologists who view organisms as Z-systems. This equation is equivalent to the organism function in equation (1) because qi = h(qd) and experimental psychologists assume that h() is a multiplier of 1. And Bill certainly does refer to the f function in the equation for a Z-system – qo = f[h(qd)] – as “the organism function” when he says " …the organism function f in the Z-system equation is replaced by the inverse of the feedback function g-1 in the N-system equation".

AM: Specifically in the sentence where you highlighted “organism function”, he is referring to these two equations:

N-system equation: (5) qo = G^-1 ( qe )
Z-system equation: (7) qo = F ( qe )

AM: In the N-system equation, qo is determined by the feedback function’s inverse of qe. In the Z-system equation, the qo is determined by the F function of qe. That is why he says “the organism function F in the z-system equation is replaced by the feedback function g^-1 in the N system equation”.

RM: That’s mathematically correct because you have defined qe = h(qd). This hides the fact that it is qd – not qe – that is manipulated in a psychology experiment. Thus, you have effectively “buried the lede” of Bill’s paper, which is as follows: The foundation of experimental psychology is that DV = f(IV) or, in Bill’s terms, qo = f(qd), where f() represents functional characteristics of the organism under study.This foundational assumption is wrong if organisms are N-Systems.

RM: So while conventional psychologists are actually looking at the relationship qo = f[h(qd)] they assume they are looking directly at the system function qo = f(qi).

AM: If they are looking at an N system, they are looking at qo = G^-1 ( H (qd) ], or simplified (5) qo = G^-1(qe), but they assume they are looking at (7) qo = F (qe). They have misidentified an N system, thinking it is a Z system.

RM: Conventional psychologists can’t be looking at qo = G^-1(qe) or qo = F(qe) because they can’t see qe! The whole point of the 1978 paper is that experimental psychologists have been studying relationships between qd’s and qo’s (IV’s and DV’s) thinking that those relationships, when they are found, tell them something about the nature of the organism’s under study; about the organism function, f. But, in fact, they are looking at the inverse of the feedback function that relates qo to qi.

RM: By writing your equations in terms of qe, you have managed to conceal (from your readers as well as from yourself) the main point of Powers’ 1978 paper, which is: You can’t tell anything about the behavior of a living control system – an N-System – by studying relationships between independent (stimulus) and dependent (response) variables. In order to understand the behavior of a living control system you have to determine what variables it is controlling. This is the truly revolutionary point of PCT.

AM: If they measure this in an experiment:
qo = 0.2 * qe

RM: They can’t (and don’t) measure that in an experiment because they can’t measure qe. If they could, they would be measuring qi and they would, thus, already be doing PCT based experiments. But what conventional experimental psychologists can (and do) measure in an experiment is qd, the disturbance (IV) and qo, system output (DV).

AM: The behavioral illusion happens because of adopting (7) as the model of organism behavior and trying to find the organism function O by relating stimuli to responses. Sure, they are also ignoring controlled variables and dynamics of feedback systems and so on, but I think the core of the illusion is the R = O(S) model. An error in system identification.

RM: The behavioral illusion doesn’t result from an error in system identification; it reveals an error in system identification. And it shows that this error in system identification results from failure to see that the system is a control system – an N- System – controlling perceptual aspects of its own environment. Once you know that you have made this error you can stop studying the system as though it were a cause - effect system – a Z- System - and start studying it knowing that is is an N - System. And that means you can start doing research aimed at identifying the variables the system controls rather than the variables that “control” (or cause the behavior of) the system. Because there are no such variables.

RM: I never thought the power law was an example of the behavioral illusion described in Powers (1978).

AM: No?

M&S (2018): In the present paper we answer these claims and show that the power law of movement is, indeed, an example of a behavioral illusion.
[…] we showed that this assumption is likely to be based on what Powers (1978) called a behavioral illusion

RM: In English (and, according to linguist John McWhorter, in all European languages) articles are very important. So notice that I referred to “a behavioral illusion” (using the indefinite article “a”) rather than “the behavioral illusion” (using the definite article “the”). Once you understand that organisms are N- rather than Z- Systems you can see that experimental psychologists have been subject to several illusions, all of which result from failure to notice controlled variables. Another obvious illusion – one that Powers has discussed (though not in the 1978 paper) – Is the reinforcement illusion; the illusion that consequences select actions when, in fact, actions control consequences.

AM: The title of the paper - “Power law as behavioral illusion” - really sounds like you think (or thought at the moment of writing) that the power law is an example of the behavioral illusion, as defined in 1978 paper. Now it sounds like you’ve changed your mind.

RM: That time I left out the article. In English that implies the indefinite article. I did that on purpose because I was well aware of the fact that the power law is not an example of the S-R illusion that Powers describes in the 1978 paper. In both of our papers on the power law I tried to make clear what we meant by “behavioral illusion”. Here’s a quote from the second paper: " …a behavioral illusion occurs when an observed relationship between variables is seen as revealing something about the mechanisms that produce a behavior when, in fact, it does not". The power law (like the S-R illusion) is a behavioral illusion in this sense.

AM: As for the side effects of control not revealing something about the organism, we might disagree there too, as side-effects often do reveal “something” about the organism (reaction time reveals something about the level of control, maybe), but that is a whole different discussion not related to the behavioral illusion.

RM: Reaction time is not a side effect of control; it is as aspect of control system operation.

RM: So let’s just leave it at our agreeing that the power law is not an example of the behavioral illusion described in Powers (1978).

AM Great, fine with me.

RM: But I do admit that I am interested in knowing what you think the power law tells us about how an N-System produces curved movements. Well, maybe not that interested;-)

Best

Rick

[amatic]

RM: The function qo = f[h(qd)] is the organism function from the point of view of scientific psychologists who view organisms as Z-systems.

Well,that is the solution of a Z system, in terms of the 1978 paper, it is NOT the organism function. From the point of view of scientific psychologists, it is something like R = O (S), they don’t usually speak about disturbances or quantities, etc.

RM: This equation is equivalent to the organism function in equation (1) because qi = h(qd) and experimental psychologists assume that h() is a multiplier of 1.

Good. For no feedback effects in a Z-system, qi = h(qd) is correct.

RM And Bill certainly does refer to the f function in the equation for a Z-system – qo = f[h(qd)] – as “the organism function” when he says " …the organism function f in the Z-system equation is replaced by the inverse of the feedback function g-1 in the N-system equation".

Yes, he refers to F as the organism function.

He does not refer to qo = f[h(qd)] as the organism function, as you claimed two posts ago. There is no other organism function in the 1978 paper. Glad we have that settled. You were not doing damage control, you really misunderstood the text.

RM: That’s mathematically correct because you have defined qe = h(qd). This hides the fact that it is qd – not qe – that is manipulated in a psychology experiment. Thus, you have effectively “buried the lede” of Bill’s paper, which is as follows: The foundation of experimental psychology is that DV = f(IV) or, in Bill’s terms, qo = f(qd), where f() represents functional characteristics of the organism under study.This foundational assumption is wrong if organisms are N-Systems.

In Bill’s terms, it is qo = F(H(qd)), so bold is not quite correct. If you put H(x) = x, then it is correct.

Also, for H(x) = x, you get a nice: qe = qd, so you can replace qe with qd in the equations of Z-systems.

RM: Conventional psychologists can’t be looking at qo = G^-1(qe) or qo = F(qe) because they can’t see qe !

Sure they can, qe is H(qd), by definition. You claim explicitly “RM: experimental psychologists assume that h() is a multiplier of 1.”, so qe = qd, and all the equations hold, and the experimental psychologists are really seeing qo = G^-1(qd), while they think they are seeing qo = F(qd).

If the H is not really the multiplier with 1, but some other function, they will absolutely see qo = G^-1(qe), or equivalently qo = G^-1( H(qd)), while they might think qo = F(qd).

But you’ve missed the point:

They are looking at qo = G^-1 ( H (qd)).
They think they are looking at qo = F(H(qd)).

There, now with H. The only difference between these two equations is that the F is replaced by G^-1. That is the behavioral illusion. Taking a wrong model, a zero feedback system, as the model of organism behavior. That is the point of the whole section of the 1978 paper that talks about the behavioral illusion, although not the general point of the paper.

RM: inverse of the feedback function that relates qo to qi.

Nope. The feedback function relates qo to effects on qi. The organism function F relates qo and qi.

Organism function: qo = F(qi)
Feedback function: qf = G(qo)
Disturbance function: qe = H(qd)
Environment function: qi = G(qo) + H(qd), or equivalently: qi = qf + qe

RM the main point of Powers’ 1978 paper, which is: You can’t tell anything about the behavior of a living control system – an N-System – by studying relationships between independent (stimulus) and dependent (response) variables. In order to understand the behavior of a living control system you have to determine what variables it is controlling. This is the truly revolutionary point of PCT.

That is good, the bold part. I agree that is one of the main points of the paper. There are many other minor points, one of which is the behavioral illusion (topic of the current discussion), and others are various blunders and mistakes.

The preceding part is complete nonsense:

RM You can’t tell anything about the behavior of a living control system – an N-System – by studying relationships between independent (stimulus) and dependent (response) variables.

Qd and qo are stimuli and responses. The tracking task is a careful examination of the relationship between a continuous stimulus and continuous response. Any test for the controlled variable is going to be conducted by applying a stimulus that possibly disturbs the controlled variable and examining the relationship between the stimulus and the response, and other variables we can calculate from the stimulus, the response and our model of the system, including the hypothetical controlled variable qi. If the response ‘cancels’ the stimulus, and qi is near zero, that is a good hint we found the controlled variable, for example. But we are still doing stimulus-response experiments. The difference is that stimuli and responses are continuous, so we allow feedback from the behavior to the controlled variable, and we study individuals and their behavior, and not averages of stimuli and responses in groups.

You need to find a better way of saying the sentence I quoted. Maybe: “You can’t find the organism function by looking at the S-R plot”. or “S-R experiments focused at finding controlled variables can reveal important things about the organism”

RM: The behavioral illusion doesn’t result from an error in system identification; it reveals an error in system identification. And it shows that this error in system identification results from failure to see that the system is a control system

“System identification” in this context means identifying the system, organism, as a Z or N system. so your second sentence is “And it shows that this error in system identification results from a failure of system identification.”

Yes, I agree. An error in system identification is an error in system identification. They did not notice there is feedback in behavior, or they did not think it is important. Their error is possibly going to result with the behavioral illusion, if they claim they found the organism function in the S-R relationship, and then we can reveal the error by pointing out “hey, that is a behavioral illusion, your S-R relationship will change when you change the environment feedback function G”.

RM: I did that on purpose because I was well aware of the fact that the power law is not an example of the S-R illusion that Powers describes in the 1978 paper. In both of our papers on the power law I tried to make clear what we meant by “behavioral illusion”. Here’s a quote from the second paper: " …a behavioral illusion occurs when an observed relationship between variables is seen as revealing something about the mechanisms that produce a behavior when, in fact, it does not". The power law (like the S-R illusion) is a behavioral illusion in this sense.

Hm. So, you quote the 1978 paper for people to look up the definition of the behavioral illusion, which is taking the observed S-R relationship as the organism function F when the S-R relationship reveals the inverse of the environmental feedback function G.

Then you redefine behavioral illusion to be different from the 1978 definition. Instead of the “organism function” you put “something about the mechanisms that produce behavior”.

Why the redefinition while still quoting the 1978 paper? Why even call it “a” behavioral illusion, instead of just “statistical artifact”. This way is sounds like the the 1978 behavioral illusion is just some statistical artifact.

RM: Reaction time is not a side effect of control; it is as aspect of control system operation.

Sure it is. The main effect of control is keeping the controlled variable stable or at reference level. All other effects are side-effects. And many times, they do reveal something about the organism or the “mechanisms that produce behavior”.

I don’t know where you get this silly idea that side effects don’t tell you anything about the system. They don’t tell you what is the controlled variable, and that is the main thing to discover, I agree, but the various side effects can certainly be useful in finding other parameters of the system. After finding the controlled variable, you can look at responses at different frequencies, examine delays, speeds, amplitudes… All reveal something about the system, while not being the main effect.

[amatic]

It seems that main cause of our disagreement have been your repeated mistakes in identifying and defining functions in the control loop. You also have non-standard notation for functions, like f or g, and also for the function inverse you write g-1, and you often mistake solutions of systems of functions.

The organism function in PCT is always defined as the function that takes qi as input, and gives qo as output. The feedback function is not always defined, because it often assumed it doesn’t change the organism output qo, but in the 1978 paper it is defined as the function that takes qo as input, and gives “contribution” to qi as output, which I’ve named qf.

Your first reaction was that giving the output of the feedback function a name is somehow wrong and misleading. Your second reaction was to claim that the output of the feedback function was not “a real entity”. Your third reaction was to define the feedback function as qi = G(qo), and for the rest of the topic you keep writing that the feedback function relates qi to qo. This is not only misleading but also very wrong.

Take a look here, for example: What is a Function , or some other math source that describes what are functions and what are relationships, etc.

Relating
A function relates an input to an output.

Formal Definition of a Function

A function relates each element of a set with exactly one element of another set (possibly the same set).
The Two Important Things!
1.“…each element…” means that every element in X is related to some element in Y.
2. “…exactly one…” means that a function is single valued. It will not give back 2 or more results for the same input. “One-to-many” is not allowed, but “many-to-one” is allowed:

When a relationship does not follow those two rules then it is not a function … it is still a relationship, just not a function.

AM: The feedback function does not relate qi to qo, it relates qo to qf.

Or take a look again at the source of your conviction that functions correspond to causal paths.

RM My impression was that the relationship between o and d is a side effect of the system acting to to keep error at zero. Indeed, I thought that was one way of looking at the “behavioral illusion”; the illusion being that the relationship between d and o appears to reflect the causal path from stimulus (disturbance) to response (output) when, in fact, no such causal path exists. Do I have that wrong?

BP: Yes . There is clearly a causal path from d to qi (the disturbance function), another from qi to e (perceptual function and comparator with an input from r), another from e to o (output function), and finally another from o to qi (environmental feedback function). All but the last work in the “forward” direction.
Nothing mysterious happens when the loop is closed. The forward functions do not disappear. They are still there and are always working.

You were saying that the possible explanation of the behavioral illusion is that the causal path does not exist at all. The causal path does exist, Bill replied, and it goes trough all those function in a loop. Nothing about “causal functions”, and certainly nothing about functions corresponding to causal paths.

[rsmarken]

RM: Conventional psychologists can’t be looking at qo = G^-1(qe) or qo = F(qe) because they can’t see qe !

AM: Sure they can, qe is H(qd), by definition. You claim explicitly “RM: experimental psychologists assume that h() is a multiplier of 1.”, so qe = qd, and all the equations hold, and the experimental psychologists are really seeing qo = G^-1(qd), while they think they are seeing qo = F(qd).

RM: Assuming h() is a multiplier of 1 doesn’t make h() a multiplier of 1. Bill’s analysis is based on the fact that the independent variable in an experiment, qd, is what the experimenter sees as the cause of the behavior. So you correctly state that what psychologists think they are seeing is qo = F(qd), since experimental psychologists implicitly assume that h() is a multiplier of 1.0. But what that are really seeing (if the system under study is a control system) is qo = G^-1 [h(qd)] not qo = G^-1(qd).

AM: If the H is not really the multiplier with 1, but some other function, they will absolutely see qo = G^-1(qe), or equivalently qo = G^-1( H(qd)), while they might think qo = F(qd).

RM: And that fact is captured by writing what they actually see as qo = G^-1 [h(qd)] . Your introduction of the variable qe just confuses things.

AM: But you’ve missed the point:
They are looking at qo = G^-1 ( H (qd)).
They think they are looking at qo = F(H(qd)).

RM: I think it’s better to say that they think the are looking at qo = f(qd) since scientific psychologists have no notion that qd is a disturbance to a controlled variable, which is what H(qd) implies.

AM: There, now with H. The only difference between these two equations is that the F is replaced by G^-1.

RM: Yes, it is the behavioral illusion experienced by psychologists doing conventional psychological experiments.

AM: That is the behavioral illusion. Taking a wrong model, a zero feedback system, as the model of organism behavior. That is the point of the whole section of the 1978 paper that talks about the behavioral illusion, although not the general point of the paper.

RM: Yes, indeed.

AM: The preceding part is complete nonsense:

RM You can’t tell anything about the behavior of a living control system – an N-System – by studying relationships between independent (stimulus) and dependent (response) variables.

RM: I take it that you mean the “succeeding part” that you quoted as what is complete nonsense. Would it be less nonsensical if I had said: When the organism under study is an N-System the form of the dependence of qo (DV) on qd (IV), which appears to reveal something about the organism under study, actually reflects only properties of the local environment. Maybe it sounds less nonsensical when Bill says it;-)

AM: Qd and qo are stimuli and responses. The tracking task is a careful examination of the relationship between a continuous stimulus and continuous response. Any test for the controlled variable is going to be conducted by applying a stimulus that possibly disturbs the controlled variable and examining the relationship between the stimulus and the response, and other variables we can calculate from the stimulus, the response and our model of the system, including the hypothetical controlled variable qi. If the response ‘cancels’ the stimulus, and qi is near zero, that is a good hint we found the controlled variable, for example. But we are still doing stimulus-response experiments. The difference is that stimuli and responses are continuous, so we allow feedback from the behavior to the controlled variable, and we study individuals and their behavior, and not averages of stimuli and responses in groups.

RM: I don’t think it’s helpful to call experiments based on an understanding of organisms as control systems stimulus-response experiments. The main goal of PCT-based experiments is to identify controlled variables. You do this by manipulating IVs (S) which could be called stimuli. But what you are looking for as the DV (R) is lack of effect S on the hypothetical controlled variable. If S has an effect then you try a new hypothesis about the controlled variable, testing it using different Ss. It’s an iterative process. Of course, when you have identified a controlled variable there will be S-R relationships between disturbances to that variable and system outputs that compensate for those disturbances. But these S-R relationships are only of incidental interest. Once you know what variable(s) the system is controlling you know how the system will respond (R) to any disturbances (S) to those variables.

AM: You need to find a better way of saying the sentence I quoted. Maybe: “You can’t find the organism function by looking at the S-R plot”. or “S-R experiments focused at finding controlled variables can reveal important things about the organism”

RM: I think I’ll just stick with describing the research as testing for controlled variables and leave the S-R out of it.

RM: The behavioral illusion doesn’t result from an error in system identification; it reveals an error in system identification. And it shows that this error in system identification results from failure to see that the system is a control system

AM: “System identification” in this context means identifying the system, organism, as a Z or N system. so your second sentence is “And it shows that this error in system identification results from a failure of system identification.”

RM: You’re right. The way I said it could be rephrased as a tautology. It would have been better to say: the behavioral illusion shows that the error in system identification that you are making – taking an N-system for a Z system – results from failure to notice the existence of controlled variables: the fact of control.

RM: I did that on purpose because I was well aware of the fact that the power law is not an example of the S-R illusion that Powers describes in the 1978 paper.

AM: Then you redefine behavioral illusion to be different from the 1978 definition. Instead of the “organism function” you put “something about the mechanisms that produce behavior”.
Why the redefinition while still quoting the 1978 paper? Why even call it “a” behavioral illusion, instead of just “statistical artifact”. This way is sounds like the the 1978 behavioral illusion is just some statistical artifact.

RM: So you’re complaint is about me calling the power law a behavioral illusion? What do you think it is?

AM: I don’t know where you get this silly idea that side effects don’t tell you anything about the system. They don’t tell you what is the controlled variable, and that is the main thing to discover, I agree, but the various side effects can certainly be useful in finding other parameters of the system. After finding the controlled variable, you can look at responses at different frequencies, examine delays, speeds, amplitudes… All reveal something about the system, while not being the main effect.

RM: Some of these are not side effects, from my point of view. A side effect is an observed aspect of the behavior of a control system that has nothing to do with its operation; it’s something you don’t have to put into a model of the control system in order to produce the observed behavior. Behavioral illusions are side effects of control in that sense. The fact that the observed relationship between qd and qo is the inverse of the feedback function (the S-R behavioral illusion described in the 1978 paper) is a side effect of control in this sense. Same for the power law;you don’t have to put anything into the control model to make the power law appear.