examining-bolocat.html

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         title="Permalink to Examining bolocat">Examining bolocat</a></h2>
 
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    <abbr class="published" title="2011-07-27T22:51:00-06:00">
      2011/07/27
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      By <a class="url fn" href="https://keflavich.github.io/blog/author/adam-adamgginsburggmailcom.html">Adam (adam.g.ginsburg@gmail.com)</a>
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    <p>Just in case we were wondering, the V1 bolocat is completely
inconsistent with a lognormal distribution, but is perfectly consistent
(or... at least reasonably consistent...) with a power law distribution.</p>
<img alt="" src="http://2.bp.blogspot.com/-kKaOXdf9hW4/TjBRmgiNmqI/AAAAAAAAGUI/4H0KJk70F_o/s320/bolocat_flux_cdf.png" />
<img alt="" src="http://1.bp.blogspot.com/-3T9mrt5U39E/TjBRm5p7inI/AAAAAAAAGUQ/ARP6EKQDYo0/s320/bolocat_flux40_cdf.png" />
<img alt="" src="http://4.bp.blogspot.com/-ytrgQo5XYEg/TjBRndOQqOI/AAAAAAAAGUY/9g--hpa5X0Y/s320/bolocat_flux80_cdf.png" />
<img alt="" src="http://3.bp.blogspot.com/-TqVY1B1uGLw/TjBRnt6uLEI/AAAAAAAAGUg/FhXct0wpEio/s320/bolocat_flux120_cdf.png" />
<p>These plots show power-law fits (red) and lognormal fits (blue) to the
data (black). It's pretty obvious that the lognormal is a bad fit, but
in case you're unconvinced, the ks test for the &quot;source flux&quot; has a
probability 1.6e-7, and it is the highest likelihood by 17 orders of
magnitude out of the 4 flux types.
By contrast, the simulations are on average (though not uniformly) more
consistent with lognormal than powerlaw distributions:</p>
<img alt="" src="http://2.bp.blogspot.com/-lpOaHHV3e3Q/TjCTzm9RwFI/AAAAAAAAGWI/o57oH5SfdqE/s320/simulations_ksvalues_lognormvspowerlaw.png" />
<p>Even in those examples where the KS test is slightly more favorable for
the powerlaw distribution, the lognormal is a pretty good fit, and the
failure points for the two distributions are in about the same place.
The smoothness of the lognormal distribution is required to reproduce
the observed distribution.</p>
<img alt="" src="http://1.bp.blogspot.com/-8_WCGM_8AjM/TjB2tW_JZeI/AAAAAAAAGU4/SDR1guNcNq0/s320/exp10_ds2_astrosky_arrang45_atmotest_amp1.0E%252B01_sky07_seed00_peak100.00_smooth_bolocat_cdf.png" />
<p>Note that the first 4 plots are for the whole BGPS survey. What about an
individual field? For obvious reasons, I choose l30 again.</p>
<img alt="" src="http://1.bp.blogspot.com/-vHQoJtAe-vU/TjCIiVcRHsI/AAAAAAAAGVo/04fA4WSCnlA/s320/bolocat_flux40_L30_cdf.png" />
<img alt="" src="http://1.bp.blogspot.com/-kIpe2v9_YpQ/TjCIimC8K2I/AAAAAAAAGVw/-o5GZ7DoLOw/s320/bolocat_flux80_L30_cdf.png" />
<img alt="" src="http://2.bp.blogspot.com/-Shg66Vp6sUI/TjCIi2HEe4I/AAAAAAAAGV4/na0n21atNsI/s320/bolocat_flux120_L30_cdf.png" />
<img alt="" src="http://1.bp.blogspot.com/-R2zq58aXXyI/TjCIjeTdjHI/AAAAAAAAGWA/D3xEDRAgY7M/s320/bolocat_flux_L30_cdf.png" />
<p>This gets to be a little more interesting - apparently the &quot;source flux&quot;
has a tendency to pick up the power-law distributed background
structure, since it is consistent with a lognormal (but note that it is
also consistent with a powerlaw! The ks test doesn't really say
definitively which is better). Although the fits look bad at high flux,
note that this is a log-log plot and therefore the difference in
probability is rather small.
What does this all indicate? It's not entirely clear whether individual
fields are genuinely more lognormally-distributed or whether the number
statistics are just worse. However, even the source flux is consistent
with a power-law, while many realizations of the simulations are not.
Therefore, we should perform the next logical test - add point sources
drawn from a power-law distribution (and a log-normal distribution?) and
see what bolocat retrieves. We can at least say now that the point
source contribution cannot be ignored, since there is no power-law
distribution that can reproduce the observed bolocat flux distribution.</p>

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