[vsnet-alert 11034] Re: re CSS090201:090210-113032 : problematic 2MASScolours

varposts at Safe-mail.net varposts at Safe-mail.net
Tue Feb 3 03:52:23 JST 2009


-------- Original Message --------
From: Michael Linnolt <linnolt at hawaii.edu>
To: varposts at Safe-mail.net
Cc: vsnet-alert at ooruri.kusastro.kyoto-u.ac.jp
Subject: Re: [vsnet-alert 11030] re CSS090201:090210-113032 : problematic 2MASScolours
Date: Mon, 02 Feb 2009 09:41:45 -0800

> Need some clarification on this. Malmquist effect is a sampling error due to arbitrary lower-limit flux cutoff of the instrument (selection effect, affecting a number of observations of objects). Not really biasing any given observation to appear brighter...<

No the Malmquist Effect is something else altogether.

But often astronomical measurement biases due to limits are named after it, but the original Malmquist Effect has nothing to do with instrumental limit fluxes directly.  The Malmquist Effect was to do with a flux limited _object_sample_, which gives a selection effect biasing towards only the brighter objects being preferentially seen near the limit, but often folk will use it re instrumental flux detection too.

The thing you describe, the lower limit flux cutoff, is what causes a tendency on the whole for data in a dataset to be brighter on average than they should be.  You usually readily see this when you match two datasets of differing limiting magnitudes, here's an old one

ftp://ftp.lowell.edu/pub/bas/VJT2mVJL.gif

Tycho2 V as per Tycho2 from VT and BT conversion tested against LONEOS.PHOT literature sourced Vs for t'same stars, which goes far deeper, you can see the trend there as Tycho2 limit is approached.

Basically, and very lamely, consider a measure not to be a point with errors, but a roundish fuzzy statistical region in a 2D 'magnitude space'.  If you make the threshold limit, the flux limit, a rigid line (in actuality it too will be an extended region, not just a 1D line, but it's complicated enough already) then think in terms of photons you have it.

In each fuzzy round magnitude measure the photons going towards making up the spread, each of different energy, are evenly distributed around said fuzz.  Make the top half the more energetic, higher intensity, photons, the bottom half the lower intensity ones.  Now think of a round fuzz bissected by the flux detection limit.  The ones below the threshold just are not seen.  But the ones above are.  The photons that are on average bright get kept, the ones on average faint are totally lost, this biases the object towards looking brighter than it is.

Not a fantastic analogy, but it's best I can do.


Note again that Malmquist illustrated the bias about eighty years ago to show a bias for brighter galaxies to be detected over faint ones.  Granted this is somewhat due to the limits of the telescopes used in any observing generation, ie is connected to flux limited survyes, but is not strictly about limitations of measures from a specific instrumental setup.

I can't find the Malmquist reference now.  I found it once, and memory's telling me it was first noted by him in an Astronomy Encyclopedia in 1928... ...however that memory could well be wrong and I could be thinking about some other thing.

A quick google shows a nice review of Malmquist Bias in general and it causing a selection effect here :-

http://astroprofspage.com/archives/81

Note I said "Malmquist Effect"-esque in my mail.  I've asked around at times, pros and ams, as to what the actual _mathematical_ term is for generic limit generated selection effects, but everyone's just said that they tend to call it the Malmquist Effect/Bias.

Summary : for an object at the faint limit remember that you don't measure it's magnitude, you measure a probability space of summed photons of varying intensity, and the stuff below the faint limit doesn't get picked up, that above the limit does, making the object appear brighter than is.  In practice it's a sliding scale, a simple graph above illustrates the increase in spread at faint mags, and the "handedness" towards falsely bright of the spread due to this effect.  Not only error increases, accuracy gets affected too.

And since I've been writing this, there's been Brian Skiff's post too, and his end sentence "and the bias works in the sense of picking up only on the statistically-spurious high values." summarises it quite nicely.

Cheers

John


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