RMS and RSS in Error AnalysisDate: 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. |
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