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Re: [Phys-l] Significant figures -- again



On 03/30/2012 03:15 AM, Brian Blais wrote:
Bob knows more than just the uncertainty,

That's the right general idea.

Let me suggest a slightly different way of expressing the idea.

Much depends on what we mean by "the" uncertainty. I claim we
need to emphasize the uncertainty of the situation _as a whole_.

In the example we have been considering, there are four variables.
Let's call them A, B, C, and D. There is some /correlated/
uncertainty in the situation. For simplicity, let's assume the
fluctuations follow an N-dimensional normal Gaussian distribution.
Therefore, to my way of thinking, to express the uncertainty of
the situation, we need the entire 4x4 _covariance matrix_.

The four _variances_ are the diagonal elements of the covariance
matrix.

If we write the four numbers as A ± σA, B ± σB, C ± σC, and
D ± σD, each σ represents the square root of a variance ... but
this approach leaves us with no way to express the off-diagonal
elements of the covariance matrix.

Non-experts tend to think of σA as supposedly "the" uncertainty
supposedly "associated" with A ... but alas, this is grossly
misleading. It is not a satisfactory way to describe the
uncertainty of the situation as a whole.

This is a fundamental limitation of the whole "±σ" concept.
It is a limitation of the concept, not merely the notation,
and it cannot be fixed merely by changing the notation.

This is one of the most pernicious things about the sig-figs
doctrine, namely that it /requires/ you to associate exactly N
uncertainties with N numbers ... when in fact the uncertainty
involves N^2 things, namely the covariances.

You can sometimes get away with only N uncertainties, provided
a) the uncertainties are known, and
b) they are known to be uncorrelated.

Proviso (b) covers the case where N=1, in which case you don't
need to worry about correlations.

However, in the real world, far more often than not,
a) the uncertainties are NOT KNOWN at the time the numbers
are written down, and/or
b) there are multiple variables with correlated uncertainties.

Let's be clear: It pays to think about "the" uncertainty of
the situation _as a whole_. With isolated exceptions, this
cannot be done on a variable-by-variable basis. You need the
NxN covariances, not just the N variances.

Correlated uncertainties can be visualized as an elongated
multi-dimensional Gaussian that is rotated relative to your
chosen variables.
http://www.av8n.com/physics/uncertainty.htm#fig-grav-correlated

It is always possible to perform a change of variables in such
a way that the new variables are uncorrelated. Conceptually
and pictorially, this corresponds to rotating the Gaussian
so that it aligns with the new axes. Mathematically, this
corresponds to the singular value decomposition that I mentioned
yesterday. Since the covariance matrix is symmetric, the singular
value decomposition is also the eigenvalue decomposition.

This does not mean you can get away with only the N variances;
in the SVD representation you need the N diagonalized variances
*and* the rotation matrix that maps the old variables to the
new variables.