WMAP Data Analysis Revisited

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 ).
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 =B120microK. Considering the circumstance
that the difference map was smoothed with a 1 degree radius (which
should have about halved the amplitude) this corresponds thus to the
amplitude of the second peak (which is 50microK). The latter proves
therefore to be due due to statistical fluctuations both in space and
time which, as shown on my webpage, lead to an angular bias resulting
in a residual signal at about 0.3 deg.

Thomas

Just a typo correction (apparently the system wasn't able to deal with
the +-sign I used and put something else in):

it should read above: 'temperature fluctuation for the difference map
of about +-20microK'.

Thomas

"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));
* you fundamentally misinterpret estimated uncertainties as *biases*
when they are not (your Fig. 5 shows subtraction of uncertainties,
which is not a relevant quantity);
* you assume that the authors don't account for the WMAP
instrumental beam pattern, when in fact they do (Hinshaw et al
2003);
* you make other erroneous assumptions about how the angular power
spectrum is made; despite having been referred to the Hinshaw
paper several times, you apparently have not read it.

Thus, based on shaky premises, your conclusions are highly suspect.

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."

... Considering the circumstance
that the difference map was smoothed with a 1 degree radius (which
should have about halved the amplitude) this corresponds thus to the
amplitude of the second peak (which is 50microK). The latter proves
therefore to be due due to statistical fluctuations both in space and
time which, as shown on my webpage, lead to an angular bias resulting
in a residual signal at about 0.3 deg.

Unlikely (see above).

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.

CM

References
C.L. Bennett, et al., 2003, ApJS, 148, 97
G. Hinshaw, et al., 2003, ApJS, 148, 135
N. Jarosik, et al., 2003, ApJS, 148, 29

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.

* 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)

* 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.

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.



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).


Thomas

"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

Craig Markwardt wrote:
"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));

The noise issue covered in the Jarosik paper relates to the internal
noise of the radiometers, not any noise in the actual signal. The
former is being eliminated in the data analysis by taking the cross
power spectrum between different receivers, but the latter should not
be affected by this as it is likely to be correlated in all receivers
(a scintillation in the microwave background is related to concrete
physical causes and should be correlated at a wide range of
frequencies).



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.

I don't know why you think that the exact statistical distribution
function is important here. The standard deviation should be in any
case proportional to 1/sqrt(n) for discrete events. There is
furthermore nothing to prevent you from describing the intensity in
terms of photons i.e. using a discrete variable n in this context.



* 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.

WMAP is not interested in the mean value. Its whole purpose and design
is to measure deviations from it. What they effectively plot in the
power spectrum is the average absolute value of the deviations from the
mean. If this happens to be variable within the beam, then the power
spectrum will become biased unless this variation is being taken into
account accordingly.


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.

As I said above already, the error bars merely represent the
uncertainties in the average absolute value of the deviations from the
mean (i.e. the uncertainties of the bias in this case).


* 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?

I thought I had made this clear already: the window functions have been
effectively assumed too narrow in the analysis. The beam widths should
have been assumed a factor sqrt(2) larger for those parts of the map
corresponding to the CMB.


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.

They merely subtracted the power spectra for the 1- and 3-year maps.
They did not produce a power spectrum of the differential map. If you
subtract two statistically independent random distributions from each
other, then the standard distribution of the resulting distribution is
not zero but the same as the original (or rather even larger by a
factor sqrt(2)). Obviously, this statistical feature is completely lost
by just subtracting the two power spectra.




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.

I don't understand your point: the error bars are indeed consistent
with the number of repeated observations of the features, but this
holds irrespective of the nature of the features i.e. also if they are
due to some kind of bias in the data.

Thomas

"Thomas Smid" writes:

Craig Markwardt wrote:
"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));

The noise issue covered in the Jarosik paper relates to the internal
noise of the radiometers, not any noise in the actual signal. The
former is being eliminated in the data analysis by taking the cross
power spectrum between different receivers, but the latter should not
be affected by this as it is likely to be correlated in all receivers
(a scintillation in the microwave background is related to concrete
physical causes and should be correlated at a wide range of
frequencies).

Let's return to your web page, which is after all what you opened this
thread with. You made the claim, "the difference between count rates
will not be zero but on average be equal to the square root of count
rates". However this is false. As noted above, the receivers are not
in a Poissonian regime but in a Gaussian one, with specific
non-Poissonian noise characteristics (Jarosik et al 2003).
Furthermore, there are foregrounds that make the analysis non-ideal
and certainly non-Poissonian (i.e. Bennett et al 2003). More on this
below.

You rely on your "square root of intensity" claim to further your
subsequent claim, but since this initial premise is false, your
conclusions are largely irrelevant.


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.

I don't know why you think that the exact statistical distribution
function is important here. The standard deviation should be in any
case proportional to 1/sqrt(n) for discrete events. There is
furthermore nothing to prevent you from describing the intensity in
terms of photons i.e. using a discrete variable n in this context.

However, it was *you* who claimed that the uncertainty is proportional
to the square root of intensity (your web page first and second
equations). This is unfounded. The Jupiter observations that form
the basis of the Page et al (2003) paper are averages of many
observations. Thus, there is a complex combination of statistical and
systematic errors that come into play.


* 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.

WMAP is not interested in the mean value. Its whole purpose and design
is to measure deviations from it. ...

Your claim is erroneous. A simple examination of either of the
Hinshaw et al 2003 papers demonstrates that the first step in the
analysis process is to make a mean sky map (with signed values of
course!).

... What they effectively plot in the
power spectrum is the average absolute value of the deviations from the
mean. ...

This is also erroneous. Again, consultation of Hinshaw et al (2003)
shows that while the procedure is complicated, the angular power
spectrum is a sum of squared *Spherical harmonic coefficients*
(Hinshaw et al 2003, eqn A3). Those in turn are based on the mean sky
maps, not "average absolute value of deviations from the mean" (eqn
A2).

... If this happens to be variable within the beam, then the power
spectrum will become biased unless this variation is being taken into
account accordingly.

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.

As I said above already, the error bars merely represent the
uncertainties in the average absolute value of the deviations from the
mean (i.e. the uncertainties of the bias in this case).

However, since what you said above is erroneous, your claim is
irrelevant. In fact, the reported uncertainties of the angular power
spectrum are based on an optimal covariance matrix estimate (Hinshaw
et al. 2003, sec 5).


* 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?

I thought I had made this clear already: the window functions have been
effectively assumed too narrow in the analysis. The beam widths should
have been assumed a factor sqrt(2) larger for those parts of the map
corresponding to the CMB.

Again, this is erroneous. The smoothing effect of the beam size of
both feeds of the radiometer was accounted for (see Hinshaw et al
2003, eqn 6 and sect 2.1).


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.

Note, no response.


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.

They merely subtracted the power spectra for the 1- and 3-year maps.
They did not produce a power spectrum of the differential map. If you
subtract two statistically independent random distributions from each
other, then the standard distribution of the resulting distribution is
not zero but the same as the original (or rather even larger by a
factor sqrt(2)). Obviously, this statistical feature is completely lost
by just subtracting the two power spectra.

However, it was *you* who claimed that the residual differences
between the 1- and 3-year maps were associated with the peak at ~0.3
deg (see above). However, it is clear that the 0.3 deg feature is
strongly present in both 1- and 3-year maps, so whatever small
differences there are between the two analyses are essentially
negligible for your purposes.


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.

I don't understand your point: the error bars are indeed consistent
with the number of repeated observations of the features, but this
holds irrespective of the nature of the features i.e. also if they are
due to some kind of bias in the data.

However, it was *you* who claimed on your "updated" web page that the
size of the 0.3 deg angular power spectrum feature is comparable to
the statistical noise at a single pixel. However, as I noted, you
failed to account for the multiple observations at a given pixel.
Furthermore, one needs to account for the fact that the angular power
spectrum is effectively a weighted average over all sky pixels. The
resulting statistical uncertainties are far smaller than the amplitude
of the 0.3 deg feature, and your "update" is therefore erroneous.

CM

Craig Markwardt wrote:
[Mod. note: entire quoted, unedited article deleted -- mjh]

Craig, you keep on repeating the same points which I have shown
previously already to be inappropriate or simply incorrect in this
context.
Please read the very references you have been quoting more carefully
and try to set them in a proper context to my claims. As I have said
before, there is nothing in the data analysis that would take a
possible random fluctuation in the signal itself into account (given
the differential nature of the experiments) and as such the data must
be considered at least as dubious. In this sense, my interpretation of
certain features in the power spectrun are at least as conclusive as
any cosmological interpretation (in fact it should be more conclusive
as it does not depend on the adjustment of any free parameters in order
to fit the theoretical spectrum to the observed one).

Thomas

I trust someone here has noticed this week's new abstract at
http://arxiv.org/abs/astro-ph/0605135 which treats exactly this
problem and finds very troubling (their words) foreground systematic
effects, apparently related to the plane of the solar system.

Eric

(Eric Flesch) writes:

I trust someone here has noticed this week's new abstract at
http://arxiv.org/abs/astro-ph/0605135 which treats exactly this
problem and finds very troubling (their words) foreground systematic
effects, apparently related to the plane of the solar system.

The Copi et al paper you refer to considers the low-l components of
the CMB (i.e. quadrupole, octupole), which correspond to large-scale
spatial fluctuations.

Smid's claims refer to the high-l components of the spectrum, which
Copi does not consider. Thus, Copi et al are not really considering
"exactly this problem."

CM