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Re: statistics in Roberts' paper on Miller

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Hello, Tom:

> I am the author of the arxiv paper 0608238v2.pdf, from
> 1986.

I find your statistical procedure in section IV described
somewhat hurriedly so that I, as well as some other readers,
had trouble understanding it. Below I describe in detail and
with equations, yet with maximum concision, my best
understanding of your transformations of the raw Miller
data. Please, let me know whether I interpolate them
correctly. I hope it will enable statisticians to see your
procedure with better clarity.

The raw data is a series of 20 runs, or interferometer
revolusions (r), with fringe shift observations (S) taken at
sixteen equidistant azimuths (a): S[r,a], where 1<=r<=20 and
1<=a<=16. You propose a model expressing the observations as
a combination of aether drift D and systematic error E:

                   S[r,a] = D[a] + E[t] ,

where the drift is a function of orientation and the error a
function of time.  Time, in turn, may be considered equal to
the observation number within the entire run, expressed in
the number of revolutions:

                  t[r,a] = r + (a-1)/16 ,

so that a function of (r,a) is also a function of time, and:

             E[r,a] = E(t) = E[r + (a-1)/16] .

You then observe that the signal D[a] may be cancelled out
by subtracting the first run form the rest for each azimuth.
Taking advantage, however, of the half-periodic symmetry in
the predicted effect, you combine the observations half a
cycle apart, defining eight interleaved sequences Ed[a] of
systematic-error differences:

          Ed[r,a]             = E[r,a] - E[1,a]
          Ed[a](r + (a-1)/16) = S[r,a] - S[1,a] ,

each of which evaluates twice per revolution.  From now on,
the azimuth of the folded data is in [1,8].  These eight
Ed[a]'s are plotted in your figure 10.

Whereas Ed[a] are interlevaed in time, it is reasonable to
assume they should combine into a single smooth function of
systematic-error difference Edc(t) with 8*2*20 = 320
equidistant samples:

          Edc(t) = Ed[a](t) + B[a], 1 <= a <= 8 .

Edc(t) is specified with eight degrees of freedom,
corresponding to the baselines B[a] of the error-differences
for individual combined orientations.  Since the whole model
is invariant to a constant additive, you fixed the baseline
of the first sequence at zero:

                          B[1] = 0

ending up with seven degrees of freedom, wich you fitted on
a computer to obtain as smooth a Edc(t) as possible.
Knowing B[a], the systematic error can be restored:

          E[r,a  ] = S[r,a  ] - S[1,a  ] + B[a]
          E[r,a+8] = S[r,a+8] - S[1,a+8] + B[a] .

And the ether-drift is calculated by subtracting the error
from the raw data:

                   D[a] = S[r,a] - E[r,a] .
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statistics in Roberts. Was: RAW vs. raw image format Rich Ulrich <rich.ulrich@comcast.net> - 2023-02-27 16:00 -0500
  Re: statistics in Roberts. Was: RAW vs. raw image format davidd02@tpg.com.au (David Duffy) - 2023-02-28 07:47 +0000
    Re: statistics in Roberts. Was: RAW vs. raw image format davidd02@tpg.com.au (David Duffy) - 2023-02-28 09:12 +0000
      Re: statistics in Roberts. Was: RAW vs. raw image format davidd02@tpg.com.au (David Duffy) - 2023-03-13 01:23 +0000
        Re: statistics in Roberts. Was: RAW vs. raw image format Anton Shepelev <anton.txt@g{oogle}mail.com> - 2023-03-13 10:32 +0300
        Re: statistics in Roberts. Was: RAW vs. raw image format Anton Shepelev <anton.txt@g{oogle}mail.com> - 2023-03-13 10:58 +0300
          Re: statistics in Roberts. Was: RAW vs. raw image format davidd02@tpg.com.au (David Duffy) - 2023-03-14 04:13 +0000
  Re: statistics in Roberts. Was: RAW vs. raw image format "David Jones" <dajhawkxx@nowherel.com> - 2023-02-28 12:42 +0000
    Re: statistics in Roberts. Was: RAW vs. raw image format nospam@de-ster.demon.nl (J. J. Lodder) - 2023-02-28 14:22 +0100
      Re: statistics in Roberts. Was: RAW vs. raw image format "David Jones" <dajhawkxx@nowherel.com> - 2023-02-28 14:01 +0000
        Re: statistics in Roberts. Was: RAW vs. raw image format nospam@de-ster.demon.nl (J. J. Lodder) - 2023-02-28 15:12 +0100
          Re: statistics in Roberts. Was: RAW vs. raw image format "David Jones" <dajhawkxx@nowherel.com> - 2023-02-28 14:57 +0000
            Re: statistics in Roberts. Was: RAW vs. raw image format nospam@de-ster.demon.nl (J. J. Lodder) - 2023-02-28 21:36 +0100
    Re: statistics in Roberts. Was: RAW vs. raw image format Anton Shepelev <anton.txt@gmail.moc> - 2023-03-04 00:46 +0300
      Re: statistics in Roberts. Was: RAW vs. raw image format nospam@de-ster.demon.nl (J. J. Lodder) - 2023-03-03 23:40 +0100
    Re: statistics in Roberts. Was: RAW vs. raw image format "David Jones" <dajhawkxx@nowherel.com> - 2023-03-09 17:07 +0000
    Re: statistics in Roberts. Was: RAW vs. raw image format Anton Shepelev <anton.txt@gmail.moc> - 2023-03-19 00:41 +0300
      Re: statistics in Roberts. Was: RAW vs. raw image format "David Jones" <dajhawkxx@nowherel.com> - 2023-03-19 00:08 +0000
        Re: statistics in Roberts. Was: RAW vs. raw image format Anton Shepelev <anton.txt@gmail.moc> - 2023-03-20 01:08 +0300
          Re: statistics in Roberts. Was: RAW vs. raw image format "David Jones" <dajhawkxx@nowherel.com> - 2023-03-19 23:14 +0000
            Re: statistics in Roberts. Was: RAW vs. raw image format "David Jones" <dajhawkxx@nowherel.com> - 2023-03-19 23:48 +0000
            Re: statistics in Roberts. Was: RAW vs. raw image format Anton Shepelev <anton.txt@g{oogle}mail.com> - 2023-03-20 11:58 +0300
              Re: statistics in Roberts. Was: RAW vs. raw image format "David Jones" <dajhawkxx@nowherel.com> - 2023-03-20 17:23 +0000
                Re: statistics in Roberts. Was: RAW vs. raw image format Anton Shepelev <anton.txt@gmail.moc> - 2023-03-24 00:11 +0300
                Re: statistics in Roberts. Was: RAW vs. raw image format "David Jones" <dajhawk18xx@@nowhere.com> - 2023-03-23 23:59 +0000
      Re: statistics in Roberts. Was: RAW vs. raw image format Tom Roberts <tjoberts137@sbcglobal.net> - 2023-03-19 14:22 -0500
        Re: statistics in Roberts. Was: RAW vs. raw image format Anton Shepelev <anton.txt@g{oogle}mail.com> - 2023-03-20 14:07 +0300
          Re: statistics in Roberts. Was: RAW vs. raw image format nospam@de-ster.demon.nl (J. J. Lodder) - 2023-03-20 14:35 +0100
            Re: statistics in Roberts. Was: RAW vs. raw image format Anton Shepelev <anton.txt@g{oogle}mail.com> - 2023-03-20 17:01 +0300
          Re: statistics in Roberts. Was: RAW vs. raw image format Tom Roberts <tjoberts137@sbcglobal.net> - 2023-03-20 16:20 -0500
        Re: statistics in Roberts. Was: RAW vs. raw image format nospam@de-ster.demon.nl (J. J. Lodder) - 2023-03-20 14:35 +0100
          Re: statistics in Roberts. Was: RAW vs. raw image format Anton Shepelev <anton.txt@g{oogle}mail.com> - 2023-03-20 17:33 +0300
  Re: statistics in Roberts. Was: RAW vs. raw image format Anton Shepelev <anton.txt@gmail.moc> - 2023-03-03 23:33 +0300
  Re: statistics in Roberts' paper on Miller Tom Roberts <tjoberts137@sbcglobal.net> - 2023-03-05 12:48 -0600
    Re: statistics in Roberts' paper on Miller nospam@de-ster.demon.nl (J. J. Lodder) - 2023-03-05 21:47 +0100
      Re: statistics in Roberts' paper on Miller Tom Roberts <tjoberts137@sbcglobal.net> - 2023-03-08 21:26 -0600
    Re: statistics in Roberts' paper on Miller Rich Ulrich <rich.ulrich@comcast.net> - 2023-03-06 23:08 -0500
      Re: statistics in Roberts' paper on Miller Tom Roberts <tjoberts137@sbcglobal.net> - 2023-03-08 21:40 -0600
    Re: statistics in Roberts' paper on Miller Anton Shepelev <anton.txt@gmail.moc> - 2023-03-08 15:33 +0300
      Re: statistics in Roberts' paper on Miller Tom Roberts <tjoberts137@sbcglobal.net> - 2023-03-08 23:11 -0600
        Re: statistics in Roberts' paper on Miller Anton Shepelev <anton.txt@gmail.moc> - 2023-03-10 01:04 +0300
          Re: statistics in Roberts' paper on Miller Tom Roberts <tjoberts137@sbcglobal.net> - 2023-03-11 12:54 -0600
            Re: statistics in Roberts' paper on Miller Anton Shepelev <anton.txt@gmail.moc> - 2023-03-12 14:25 +0300
              Re: statistics in Roberts' paper on Miller Tom Roberts <tjoberts137@sbcglobal.net> - 2023-03-17 13:27 -0500
                Re: statistics in Roberts' paper on Miller Anton Shepelev <anton.txt@gmail.moc> - 2023-03-19 00:32 +0300
                Re: statistics in Roberts' paper on Miller Tom Roberts <tjoberts137@sbcglobal.net> - 2023-03-19 14:45 -0500
            Re: statistics in Roberts' paper on Miller Tom Roberts <tjoberts137@sbcglobal.net> - 2023-03-20 16:06 -0500
              Re: statistics in Roberts' paper on Miller Anton Shepelev <anton.txt@g{oogle}mail.com> - 2023-03-21 13:00 +0300
    Re: statistics in Roberts' paper on Miller Anton Shepelev <anton.txt@gmail.moc> - 2023-03-08 19:11 +0300
      Re: statistics in Roberts' paper on Miller Anton Shepelev <anton.txt@g{oogle}mail.com> - 2023-03-09 11:48 +0300
        Re: statistics in Roberts' paper on Miller Anton Shepelev <anton.txt@g{oogle}mail.com> - 2023-03-09 11:57 +0300
    Re: statistics in Roberts' paper on Miller Tom Roberts <tjoberts137@sbcglobal.net> - 2023-03-08 21:09 -0600
    Re: statistics in Roberts' paper on Miller "David Jones" <dajhawkxx@nowherel.com> - 2023-03-09 17:13 +0000
      Re: statistics in Roberts' paper on Miller Anton Shepelev <anton.txt@gmail.moc> - 2023-03-09 23:26 +0300
        Re: statistics in Roberts' paper on Miller "David Jones" <dajhawkxx@nowherel.com> - 2023-03-10 01:37 +0000
          Re: statistics in Roberts' paper on Miller nospam@de-ster.demon.nl (J. J. Lodder) - 2023-03-10 11:25 +0100
          Re: statistics in Roberts' paper on Miller Anton Shepelev <anton.txt@gmail.moc> - 2023-03-11 00:13 +0300
          Re: statistics in Roberts' paper on Miller Tom Roberts <tjoberts137@sbcglobal.net> - 2023-03-10 15:42 -0600
            Re: statistics in Roberts' paper on Miller Anton Shepelev <anton.txt@gmail.moc> - 2023-03-11 01:32 +0300
              Re: statistics in Roberts' paper on Miller - Data Tom Roberts <tjoberts137@sbcglobal.net> - 2023-03-11 12:18 -0600
                Re: statistics in Roberts' paper on Miller - Data Anton Shepelev <anton.txt@gmail.moc> - 2023-03-12 00:13 +0300
                Re: statistics in Roberts' paper on Miller - Data Anton Shepelev <anton.txt@gmail.moc> - 2023-03-20 00:38 +0300
        Re: statistics in Roberts' paper on Miller Tom Roberts <tjoberts137@sbcglobal.net> - 2023-03-10 15:25 -0600
          Re: statistics in Roberts' paper on Miller Anton Shepelev <anton.txt@gmail.moc> - 2023-03-12 00:01 +0300
      Re: statistics in Roberts' paper on Miller Tom Roberts <tjoberts137@sbcglobal.net> - 2023-03-10 15:10 -0600
    Re: statistics in Roberts' paper on Miller Anton Shepelev <anton.txt@gmail.moc> - 2023-03-11 01:14 +0300
      Re: statistics in Roberts' paper on Miller Tom Roberts <tjoberts137@sbcglobal.net> - 2023-03-11 13:08 -0600
        Re: statistics in Roberts' paper on Miller Anton Shepelev <anton.txt@gmail.moc> - 2023-03-12 01:51 +0300
    Re: statistics in Roberts' paper on Miller "David Jones" <dajhawkxx@nowherel.com> - 2023-03-15 17:10 +0000
      Re: statistics in Roberts' paper on Miller nospam@de-ster.demon.nl (J. J. Lodder) - 2023-03-15 22:36 +0100
        Re: statistics in Roberts' paper on Miller "David Jones" <dajhawk18xx@@nowhere.com> - 2023-03-15 22:18 +0000
          Re: statistics in Roberts' paper on Miller nospam@de-ster.demon.nl (J. J. Lodder) - 2023-03-15 23:28 +0100
      Re: statistics in Roberts' paper on Miller Tom Roberts <tjoberts137@sbcglobal.net> - 2023-03-19 13:46 -0500
        Re: statistics in Roberts' paper on Miller "David Jones" <dajhawkxx@nowherel.com> - 2023-03-19 22:57 +0000
          Re: statistics in Roberts' paper on Miller Anton Shepelev <anton.txt@g{oogle}mail.com> - 2023-03-20 12:05 +0300

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