Tom Roberts:
> Anton Shepelev:
>
> > The raw data is a series of 20 runs, or interferometer
>
> 20 TURNS, not "runs". There are 67 runs, each consisting
> of of 20 turns. Turn = rotation. These are Miller's
> terms, and I followed him.

This is a mental slip recurring throughout my entire post.
I beg your pardon.

> > [Roberts reanalysis exressed in equations]
>
> That looks correct. I don't see what use it might be.

a) Equations do not have the ambiguity of natural language,
b) their writing requires complete understaning c) they may
be useful to other participants having trouble understanding
your statistical model.

> I _DID_ "sew them together". Miller recorded the value at
> orientation 1 just before the adjustment turn, and again
> just after it. For all data thereafter I added (before-
> after) to every value, thus canceling the effect of his
> adjustment, as best as can be done. This was done just
> after reading the data file, before any analysis or
> plotting.

Thanks, that's right.

> While I was at CWRU in 2006, after giving a colloquium on
> this analysis, Prof. Fickinger and I visited the archives
> and spent an hour or two scanning Miller's data sheets for
> runs without adjustments, indicating the instrument was
> more stable than usual. We found several dozen, but I
> never got around to analyzing them. I did look at them,
> and many of them are just a monotonic drift from start to
> finish -- no signal at all.

To your visual estimation? Well, OK...

Please, notice that I answered to your generous offer of the
digitized data of the 67 runs that you have on your HDD.
Let me know how you should like to share it, or how you want
me to take it.

> > These calibration-caused irregularities may have a
> > negative effect upon the fitting of combined systematic
> > drift.
>
> Hmmm. The instability of the instrument is at fault. The
> procedure I used is the best that can be done, given
> Miller's methods.

Since you sewed the observation turns across the calibration
turns, my suspicion does not hold. But thinking your
procedure the best possible one is somewhat immodest of you
:-) Have it been formally proven to be the best?

> > Not all the errors are systematic, as Miller himself
> > noticed the action of sound in disturbing the air in the
> > interferometer light path, let alone those due to the
> > hypothetical aether wind, which, if partially entrained,
> > will be affected by atmospheric turbulances, as well as
> > show the typical instabilities occuring when a laminar
> > flow meets with obstacles.
>
> None of those are anywhere close to the magnitude of the
> drift.

No, but they are larger than the magnitude of the alleged
signal.

> Moreover, if they are in Miller's data then they are in my
> model of the systematic.

Only as long as well-behaved noise, being symmetrical, does
not affect the optimal combination of the drift curves,
because the upward and downward spikes cancel out. Squared
differences, though, do not cancel out as well as if they
were L1:

s2 = s1 + s2 => s2^2 != s1^2 + s2^2

L2 needs more samples for the same stability.

> For about $50,000 and a year of effort you could build a
> pair of them and instrument the heterodyne between lasers
> locked to each. Point one arm straight up so it behaves
> differently with orientation than the other one (with two
> horizontal arms). Dedicate another year or two of your
> life to taking data....
>
> Attempting to put them on a rotating table is hopeless, as
> you can never get the vertical arm to be vertical
> accurately enough; microradians matter.

Indeed. It is much more practicable to let the Earth do the
rotation!

Tom Roberts:
> Anton Shepelev:
> > Tom Roberts:
> >
> > > By taking advantage of the 180-degree symmetry of the
> > > instrument, only 8 orientations are used.
> >
> > No, I think you are taking advantage of the 180-degree
> > symmetry of the hypothesised effect rather than of the
> > instrument, which itself may be asymmetrical due to many
> > factors, including an asymmetrical air flow and
> > temperature in the aether house.
>
> The INSTRUMENT is exactly 180-degree symmetrical, as light
> does not care if it goes east then west, or west then
> east;

You are talking about light, not about the instrument.
Reading on:

> deviations from exactly 90 degrees between the arms do not
> change this.

No, they do not change the behavior of light, not the half-
cycle symmetry of the device.

> Sources of error need not be symmetric, but most of them
> have a symmetric effect on the symmetric instrument.

One can easily imagine many faults that will disrupt the
half-period symmetry of the MMI interferometer, for
example -- a kink in the rotation mechanism causing an bump
at certain orientation, or a different thermal inertia of
one of the arms.

> > The subtraction of the first turn has but one
> > effect -- that of offsetting each of the eight error-
> > difference curves by a constant value, equal to the
> > observation in the first turn at the corresponding
> > azimuth. It has /no/ effect on the forms of those
> > curves. Since your fitting consists in finding the seven
> > relative vertical offsets between these curves, it may
> > safely be applied to the raw drifts at each combined
> > mark, in which case the seven fit parameters will
> > represent the pure signal, if any!
>
> No! The EIGHT fit parameters represent the signal PLUS THE
> VALUE OF THE SYSTEMATIC AT THE START OF THE RUN (for each
> orientation), with the entire run offset to start at zero.

Please, wait a minute. In your paper, where you operate with
the partial error-differences, the eight fit parameters
represent the initial drift (at the first rotation) at each
of the eight combined orientations. Consider a simple
situation of a sine signal and no drift. All the partial
error-differences are constantly zero and coincide. All the
fit parameters are zero -- because the drift is zero. It is
as I said -- in your paper the eight parameters represent
the pure value of the systematic drift!

If, however, you do the same thing sans subtracting the
first rotation from the rest, the eight fit parameters will
show the pure negative signal, because the fitting model
will in effect try to cancel the signal by aligning the
values at adjecent orientations. The two methods are
equivalent because, as you write in a footnote, "The chi^2
is made up of differences, so any constant can be added to
all 8 parameters without changing chi^2."

My point was the subtracting the first turn from the rest
was a redundant operation.

> > So you used a weighted form the of least-squares. But
> > then a complete enumeration is unnecessary, becuase
> > least-squares is designed to be an analitical method
> > with linear complexity: you simply write the smoothness
> > function as a sum of weighted squared differences over
> > the tabulated data and optimise it the usual way via
> > partial derivatives.
>
> It makes no sense to fit continuous parameters to
> quantized data,

At least, it would have save you from the brute-force
enumeration and have let you use the least-squares method as
it was intended. Also, you would have been able to avoid
combining opposite orientaions and analyse the entire turns,
with 15 degrees of freedom. With the half-turns combined,
the error differences beween opposite orientations are
"baked" into the partial curves and uncapable of smoothing
out.

> so the parameters are quantized like the data. Partial
> derivatives of the parameters are not possible, and
> enumeration is the only method I found.

The other method is not to quantize the seven parameters
before fitting. If you must, quantize them after fitting,
or better not a tall, taking advantage of the higher
precision of the exact values.

> > Notice, however, that large discontinuitites between
> > runs due to interferomenter calibration are likely to
> > dominate the fitting.
>
> I never combined runs, so as stated this is a non issue.
> If by "run" you mean turn, it is also a non issue because
> I corrected the data for the offset in each recalibration
> turn.

Yes, I meant a turn, or rotation. Understood.

> So look at my Fig. 2 and say with a straight face that you
> think a signal with amplitude ~ 0.1 fringe can be
> extracted from the data.

I do not have that Oscilloscopic, Harmonic-analysing,
Fourier-transforming vision that you seem to take for
granted :-) Yes, it looks awful.

> > What is your opinion regarding the claimed galactic
> > orientation of the measured drift, as plotted in fig. 22
> > of the 1933 paper? Can an instumental error have a
> > concistent half-periodic dependency on 1) time of day
> > and 2) the season of the year so as to point into a
> > fixed direction in the galaxy?
>
> Computing an average always yields a value, so it's no
> surprise that he came up with an answer.

Of course. Any noise or drift will have a Fourier spectrum.

> Had he computed errorbars on it, they would have been
> larger than 360 degrees, probably much larger.

I cannot comment upon your estimation of the errorbars, yet.

> Look at my Fig. 5. The phase of a fitted sinewave clearly
> does not determine any direction whatsoever.

The phase would indicate the direction, and the
amplitude -- the velocity of the aether wind speed as
projected upon the plane of the interferometer. The
galactic motion of the Earth is dervied from observations at
four different epochs. This is a relatively simple
astronomical calculation using linear algebra. Regardless of
the enormous errorbars, Miller's curves seem to agree with
the hypothesis that the Solar system is moving toward the
constellation of the Dragon. Both the phrases and ampitudes
of their curves seem to correspond closely with those
calculated astronomically. I have not (yet) analysed them
myself.

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