# Triangulation

Hello, Martin –

[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

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 to

be 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

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.

···

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