Hello, Martin –

I was replying to your “triangulation” post to CSGnet, so I

copy this reply to your latest to CSGnet.

[From Bill Powers (2009.02.02.1046 MST)]

Bill,

I have a response to an earlier message of yours under construction, but

I thought I should answer this one first, because it suggests some

conceptual issues remain unresolved.

Yup.

Any sequence of N

real numbers can be seen as a point in a space of N dimensions, or as a

vector from the origin to that point.

I’m really

struggling with this idea; something keeps nagging at me that there’s

something wrong here. It’s not that I dispute the calculations; dot

product and all that. But now I’m wondering if this is a valid way to

deal with the physical situation as opposed to the

mathematics.Well, physicists have been doing it for a long time, so it can’t be too

bad!

Obviously, I am wondering about them, too.

But what if the first vector

simply disappears before we start drawing the second one? Where do we

start the second vector?This question comes completely out of left field for me. Are you asking

what we do if the coordinates of the first vector are forgotten, or

determined tobe invalid? How does the first vector manage to

disappear?

Yes, I thought I might get a rise out of you with that. But now, the next

day, I’m still of the same mind…

When you make series of observations of a variable through time, that

variable has only one value at a time. If you plot the state of that

variable in some N-dimensional space as you are suggesting, recording the

values in all N dimensions each time you enter a new point, at any given

instant only one of those dimensions will have a possibly non-zero value.

All the others will be zero. The next time you sample the data, a

different dimension will be possibly non-zero, and the one that was

nonzero is back to zero. The values of those separate observations don’t

add either directly or in quadrature because they don’t exist at the same

time. Such an addition would have no physical meaning.

So we should not

imagine an N-dimensional space with each axis marked by a point

representing one observation, and a single point that represents the sum

of all the observations.No, we should imagine an N-dimensional space in which there is one point

that represents the entire trace.

The values it represents do not physically exist at the same time. The

trace itself, of course, is a function of the successive values of the

variable, so it grows with time as you say and, after the last

observation, continues to exist as a construct based on all the past

values of the variable that were observed. You can then take the numbers

represented as a trace and do anything you like with them, but what you

do with them does not represent any physical entity. Relationships among

those recorded numbers never existed in the physical world. No variable

is directly affected either by its past or its future values. Any

proposed effects involve imagined relationships which are in the eye of

the beholder, or involve other variables and their values that do exist

at the same time.

Its projections on the N

axes represent the values of the trace at successive instants. Let’s

follow the development of the beginning of the trace. Assume that the

successive values of a trace are -3, 2, 5, 7, … After one sample, we

have a line, on which there is a point at -3. After two, we have a point

at {-3, 2}.

No. You are confusing the *record* of the past values of a variable

with the *values*, only one of which can ever exist at a given

time. So at the time when the second of two values of the variable is

about to be recorded, you have only one value of the variable and one

record of a past value; after the current value is recorded, you have two

numbers stored in the record of past values, but still only one current

value of the variable.

After three samples, we

have a 3D space that contains a point at {-3. 2. 5}. After 4,

… And so on.

You’re speaking only of the record, the perceptual sequence being built

as successive values of the variable are recorded. The record contains no

hint of a relationship between successive values of the variable, because

at no time does the variable have more than one value.

… only one axis

has a point on it somewhere other than at the

origin.So where would all the previous sample values go? The trace is

represented by a sequence of numbers. The values don’t vanish because the

trace happened yesterday or last year, do they? We do, after all, analyze

traces that were taken last year, don’t we?

Yes, and this is my point. The traces or samples are not the variable

they represent: the map is not the territory. Once the traces exist, you

can read all kinds of functions of many variables into them because now

you are looking, in present time, at a set of different numbers all of

which coexist. But you can never see those relationships, those functions

of many variables, when you are simply watching the variable itself. In

that case all you ever see is one variable with one value.

The proper

visualization of the vector representation of a variation through time is

not that of a set of values that exist at the same time and add in

quadrature: those values do not all exist at the same time. The final

value at the end of the series is not the square root of the sum of the

squares of all the individual values; it is simply the value of the last

point.The value at any sample moment is the projection of the point on the axis

that represents that moment. I’m not at all clear what you mean by

“the proper visualisation”, as if there were only one

“proper” way to visualize a variation through

time.

If you are observing a variation through time, then you do not see any

time but the present and you never see more than one value of the

variable. Only if you can remember one value, so as to compare it in

present time with the value currently being observed, can you see some

relationship between these values (such as rate of change). I believe you

have said as much yourself, more than once. The traces of which you speak

require memory to be seen, and any relationship you see between the

current value and a currently remembered value has no physical

significance.

A Fourier transform is an

equally proper way to do it, in my book. A Fourier transform is just one

of an infinite number of possible rotations of this basis space, in which

the axes don’t represent moments in time, but represent well-defined

functions of the corrdinates on those axes. That’s one beauty of the

geometric approach, because this kind of relation between time and

frequency analysis is not obvious from the formulae.

Fourier analysis (as well as all other ways of characterizing a series of

changes in a single variable) is an artifact of memory and calculation,

and does not imply any corresponding physical reality. It’s true that a

given series of variations of a single variable could be generated by a

set of harmonic oscillators, but there is no need for this to be the

case; an infinite number of other physical explanations of the same

waveform is equally possible; in fact, any one of them is probably more

plausible than the idea that there is some physical set of synchronized

oscillators behind the waveform. Fourier analysis should never be taken

as prima facie evidence that there is a set of physical oscillations

being added together. That would be begging the question.

Incidentally, the final value at

the end of the series is exactly what you say it is. Why would you think

differently? It’s the length of the vector that is the square root of the

sum of the individual elements, just as it is for any

vector.

That is exactly the point that got me onto this track. You say “the

final value at the end of a series”, but the final value of what?

Not the final value of the variable that’s being observed! If the

variable happens to be constant at a magnitude of 100, and you take 100

observations of it, the final value of the variable will be 100, not

10,000. What you mean is “the final value of some function of

X”, not “the final value of X”. In the cases we’ve been

looking at, the function is sqrt(sum-of-squares(X)). The value obtained

is a perception in an observer, not a property of X. The observer could

apply any other computation to get a final value; it would still be an

arbitrary way of perceiving memories of X, with no necessary relationship

to any physical variable.

That the vector represents

a time series doesn’t affect that fact. If the series is the voltage

across a 1 ohm resistor, the sum of squares is simply the total energy

dissipated, not the voltage at the last moment.

True, but “energy” is a product of perception and imagination,

not a physical – by which I mean observational – variable. It 's

a possible way of combining a record of many observations, but it is not

the thing being observed. The reading on the voltmeter is what we

observe. There is only one reading at a time.

The final error is

not the square root of the sum of the squares of all sampled error

values, when there are two vectors; the final error is simply the final

error, which is something completely, physically, different from the RMS

error or a correlation or any other measure that refers to some

cumulative function of the data set as a whole.Yes. So???

So much for my great insight.

Note that in this

cumulative function, we lose all information about the sequence in which

the observations appeared. Each X goes with its own Y, each X1, X2 pair

goes with its Y1, Y2 pair, but we get the same RMS error or the same

correlation regardless of the order in which these pairs are observed. If

there is any physical reason why the 32nd pair is not observed before the

1st pair, it is not apparent from the cumulative measure. So these mass

measures lose a critically important aspect of the physical phenomenon:

its progress through time.Yes. So???

I guess this insight isn’t catching. I notice the lack of comment on the

fact that reversing the order in which we use a disturbance table will

not simply reverse the waveform of the error signal or the output of a

real control system. The temporal information is important, but can’t be

found by a statistical analysis.

From this I think

we have to conclude that a purely statistical analysis of data can never

substitute for a system model.Never any question about that. Again, I ask why you need to mention

this.

Because you seem to be trying to evaluate a model on the basis of

probability calculations, which I thought I was showing to be

inadequate.

Failure of the null

hypothesis on statistical grounds is not sufficient grounds for

acceptance of a hypothesis;Oh dear! Are we back to significance testing again? I thought we

had gone past that, long ago.Contrary to Popper and others,

when it is observed that the elements of the predicted data set do occur

with the correct magnitudes and in the correct order, we can count that

as a validation of the hypothesis (on a scale from 0 to

1).

I guess I should have said “on probabalistic grounds.” I love

that “oh, dear.” Perhaps you would be less distressed if

you associated with a better class of thinkers.

No we can’t, not if you mean by “validation” that no other

hypothesis can possibly do the same.

I don’t. I mean that no other hypothesis of which we know can predict as

well. I have my own criteria for how good a prediction has to be to count

as verification; most people seem to think I am too strict, you included

if I recall correctly.

If you mean it in the

Bayesian sense, that the hypothesis is or is close to the maximum

likelihood solution among those you have considered, then what you say is

true (but so far as I can see, is not contrary to Popper and

others).

Except that if you use probabilistic calculations, you lose all

information that depends on the sequence in which the probable events

occur, and for that reason can’t distinguish two theories that predict

different sequences where probability calculations can’t

distinguish sequences.

I am comfortable

with measuring RMS values and correlations in the usual manner. All the

conclusions that can be reached on that basis, other than the validity of

a model, can be accepted. But I am fairly well convinced, right now, that

the only way to verify a model is to show that it behaves through time

sufficiently like the real observations to pass reasonably stringent

requirements.

Yes. But again, I ask what is it

that you feel you are contesting?Though I said “Yes”, I would argue that there is no way to

verify a model, if by that you mean to show that it is the one and only

possible model. All you can show is that the model is consistent with

whatever tests you can throw at it with the data at hand. You can’t show

that no other as yet unimagined model must therefore be inconsistent with

the data. If you do mean the latter when you say “verify”, then

I have to disagree.

I mean that the Bayesian kind of analysis, or any analysis in which a

record of past events is characterized only by measures that span time,

cannot determine the model that is most consistent with the

data.

I am quite conscious that my opinions may be the result of nothing more

significant (if you’ll pardon the term) than my mathematical limitations.

I would appredicate being shown where my errors are.

Best.

Bill P.

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At 11:07 AM 2/2/2009 -0500, Martin Taylor wrote: