RMS and RSS in Error Analysis

Date: 10/28/2005 at 09:39:42
From: mike
Subject: When do you use RMS vs. RSS in error analysis

In error analsysis, sometimes an RMS equation is used and sometimes an 
RSS equation is used to calculate overall error given a list of 
contributing variables.  Often in the literature I've read it says to
use RSS if the contributing variables are uncorrelated.  Does this
mean we should use RMS if the variables are correlated?  Can you
explain both concepts (RMS and RSS) and the stipulation of the 
dependencies between the contributing variables in an intuitive manner?

Date: 10/28/2005 at 15:24:45
From: Doctor Douglas
Subject: Re: When do you use RMS vs. RSS in error analysis

Hi Mike.

I believe that the terms you are referring to are RMS ("root mean
square") and RSS ("root of sum of squares").  These two are closely
related and are used to estimate the variation of some quantity
about some typical behavior.  However, they are used to answer
different questions and are not alternatives to each other.

The sample standard deviation in statistics is a root mean square:

   sdv = sqrt{ (1/N) * [(x1-u)^2 + (x2-u)^2 + ... + (xN-u)^2] }.

You can see that we have obtained the sum of the squares, and then
taken their mean (by dividing by N), and then taken the square root.
This "root-mean-square" is a _typical_ error of each of the N 
measurements; it gives an estimate of how far each measurement is from 
the mean u.  This is not so much an "overall" error as it is a 
"typical" error.

The RSS ("root of sum of squares") is closely related, differing only 
in that there is no division by N.  This type of combination is used 
for when you are combining sources of errors (not trying to determine 
a typical error for a sample, as we did above).  For example, suppose 
you are trying to pinpoint your latitude/longitude coordinates.  Let's 
suppose that your uncertainty in the east-west measurement is 1 km, 
and the uncertainty in the north-south measurement is only 0.4 km, 
maybe because you have a very good sighting of the North Star.  Notice 
that instead of having a circle of uncertainty around the location, 
you have an ellipse.  How could you estimate the typical distance 
error in this case?  It is

  sqrt[(1 km)^2 + (0.4 km)^2] = 1.08 km

Notice that the total error is always larger than any of the 
individual errors, and it is dominated by the largest error source (in 
this example the east-west error).  If the two error sources were 
equal then this calculation would have become sqrt(2)*(1 km) = 
1.41 km.

Finally notice that this "root of sum of squares" or "addition in 
quadrature" requires that the error sources be uncorrelated.  If they 
are correlated, then we should take this into account.  For example, 
suppose that our error sources are

  east-west         1.0 km
  north-south       0.4 km
  NE-SW             0.1 km, where the N/S component of this error is
                            always in the opposite direction as the
                            0.4 km error, and the E/W component of
                            this error is uncorrelated with the
                            1.0 km error.

In this case the calculation leads to

   sqrt[(1.0)^2 + (0.4 - 0.1*sin(45 deg))^2 + (0.1*cos(45))^2]
    = sqrt[1.0^2 + 0.2586^2 + 0.0707^2]
    = 1.0353 km

By including the diagonal NE-SW error, our total error decreased!  
This may seem surprising, but remember that a component of it is
anti-correlated with one of our original error sources, and partially
cancelled it out.  Notice that in order to treat correlated errors we 
require information about their correlations or covariances.  
Sometimes this is difficult information to obtain, and sometimes 
there's good reason to expect that the error sources are uncorrelated, 
in which case the simple RSS or add-in-quadrature formula is used.

- Doctor Douglas, The Math Forum
  http://mathforum.org/dr.math/ 

Date: 10/28/2005 at 16:23:35
From: mike
Subject: Thank you (When do you use RMS vs. RSS in error analysis)

Awesome!!!  Thank you very much.  I've been surfing the web to no
avail trying to compile information to render an answer.  Your answer
and examples provide just the explanation and intuition I needed.

Thank you very much.