"Thomas Smid" writes:
Craig Markwardt wrote:
"Thomas Smid" writes:

Shortly after the first release of the WMAP data three years ago, I
made already the point that the angular power spectrum of the CMB
fluctuations is at least partly due to a systematic error (see my
webpage http://www.physicsmyths.org.uk/wmap.htm ).

As noted on a separate thread on sci.astro by Greg Henessy and myself,
the claims you make on that page are erroneous. Briefly,
* you assign "Poissonian" uncertainties to the data when the
limiting uncertainties are not Poissonian (Jarosik et al 2003
(statistics); Bennet et al 2003 (foregrounds));

My argument should apply to Poissonian as well as Gaussian
uncertainties:
as an illustration, consider a coin which you toss a large number n
times. If one side of the coin has the value 2 and the other the value
4, then the actual average value of the tosses will not be 3 but be
distributed with a standard deviation +-1/sqrt(n) around this.

It's not clear why your analogy applies, or why you brought up yet
another statistical distribution (Binomial). You claimed on your web
page that the uncertainty is proportional to square root of
*intensity* (not "n" as you state above). I believe that is incorrect,
and provided references to back it up. I note that you did not.


* you fundamentally misinterpret estimated uncertainties as *biases*
when they are not (your Fig. 5 shows subtraction of uncertainties,
which is not a relevant quantity);

The uncertainties (standard deviations) become biases by means of the
differential technique used and their display as a power spectrum.
If you extend the example above to the difference of the average value
of two coins, then you have for both an expected mean of 3 with an
uncertainty of +-1/sqrt(n) and the difference of this would be an
expected mean of 0 with an average deviation of +-sqrt(2)/sqrt(n). If
you repeat this procedure often enough, then you will find that the
amplitude of this fluctuation is sqrt(2)/sqrt(n) (which thus
corresponds to the amplitude of the power spectrum)

Non sequitur. If the expected mean difference is zero as you say,
then there is no bias! Thus your claim is erroneous.

There will indeed be some scatter expected in the data, because of the
statistical variations, but that is exactly what the error bars
represent in the published CMB spectrum! And it is immediately
obvious that the CMB signal is much larger than the error bars.


* you assume that the authors don't account for the WMAP
instrumental beam pattern, when in fact they do (Hinshaw et al
2003);

They account for the beam pattern, but not correctly if the signal for
both telescopes is identical apart from a random intensity fluctuation.

Really? So which precise step in the analysis (such as described by
Hinshaw et al 2003) is incorrect? If you cannot provide a detailed
critique of the published analysis technique, then how is your own
analysis relevant?

As previously noted on the thread in sci.astro, the beam pattern was
determined by measuring the response of the double-feed instrument to
signal in one feed alone (Jupiter; Page et al 2003). This is
precisely what is needed when one is computing the differential
response to a spatially varying cosmic signal.


This is in my opinion now confirmed by the latest data release:
the difference between the 3- and 1-year maps shows residuals that have
about the same amplitude as the second peak near 0.3 degree in the
power spectrum. This is evident from Figs.3 and 9 in
http://map.gsfc.nasa.gov/m_mm/pub_pa...p_3yr_temp.pdf
(PDF file, 2.3 MB) which reveals a residual temperature fluctuation for
the difference map of about =20microK.
[ note corrected value ]

And do you have evidence that these differences appear at the relevant
angular frequencies? Since the maximum differences appear near the
galactic plane, and much smaller differences appear on very broad
spatial scales (i.e. low l), the answer would probably be, "no."

If you look at Fig.9 in the above paper, you can see that the galactic
plane produces an enhanced signal of about 30 microK for the difference
map. As I understand, this has something to do with the different
method to treat the foreground radiation in the 3-year analysis and is
completely unrelated to the apparently random fluctuation of +-20
microK over the whole sky.
If the WMAP team had released a power spectrum of the difference map
for the unsmoothed data, then I could tell where this random
fluctuation in the difference map would appear in the spectrum. The
size of the fluctuation suggests very much that this will be at the
location of the second peak.


Actually, the WMAP team *did* provide a comparison of the differences
between the one- and three-year power spectra (Figure 19, Hinshaw et
al 2006). It's pretty obvious that the new analysis did not
appreciably change the amplitude of the second peak (at ~0.3 deg).
They also provide an explanation of the differences, and it does *not*
mention the effects you describe. Threfore your claims are still erroneous.



Also, your "Update April 2006" section does not account for several
factors. First, while the exposure *per observation* is a small
amount (as perhaps the 77 msec that you quote), the WMAP analysis
involves averaging many observations together. Thus, the standard
error of the mean will be much smaller than the value you quote for
the statistical error for one measurement. Even if your estimated
statistical uncertainty had been correct, it is a statistical
uncertainty and *not* a bias, and so neighboring angular frequency
bins would not be correlated, as the real data are.


Using the 'coin tossing' example from above again: if you repeat a
sequence of n tosses of two coins M times, then you will obtain a
difference of the average scores for each sequence of sqrt(2)/sqrt(n)
with a relative uncertainty of +-1/sqrt(M). Increasing M will therefore
not change the average value sqrt(2)/sqrt(n) but merely its uncertainty
(this affects thus only the error bars of the data in the power
spectrum).

However, your statistical reasoning is flawed. As you point out
above, the expected difference value is zero. [ Yes, the measured
values will have a distribution about the mean, but the mean will be
zero. ] Furthermore, you are using a non-Gaussian example (coin flip)
to describe primarily Gaussian data (WMAP data). This is not really
appropriate. In fact, for a sample of N gaussian observations, {x_i},

the sample mean value will be xav = SUM(x_i)/N,
the sample standard deviation will be std = SQRT( SUM((x_i-xav)^2)/(N-1) )
and the standard error of the mean will be stderr = std / SQRT(N-1)

(I invite you to consult any basic statistic textbook to verify this
result). Thus, considering that the number of observations per pixel
is between 560 and 5000 (Hinshaw et al 2006, Fig. 2), one can see that
the standard error on the mean is between 24 and 71 times smaller than
the sample standard deviation (which you attempted to calculate). The
standard error on the mean is the relevant quantity because it is the
mean difference maps that are used to compute the angular power
spectrum. Thus, your claims continue to be erroneous.

CM