Is this error analysis equation correct for a simple geometric mean?

[imabug]

Any error analysis gurus out there? I'm trying to work out the uncertainty for the following equation:
C = \sqrt{(A-B_A)(P-B_P)}

A and P are counts from a radioactive source from two opposing detectors. B_P and B_A are background counts with no source present.

It's a simple geometric mean equation. When I use the error analysis techniques I learned way back in my undergrad days (Intro to Error Analysis by J Taylor), the result I end up with is
\frac{\delta C}{C} = \frac{1}{2} \sqrt{\frac{A+B_A}{{(A-B_A)}^2}+\frac{P+B_P}{{(P-B_P)}^2}}

Pretty reasonable solution, but when I plug in the numbers, I get much smaller numbers for the fractional uncertainty than I think I should be getting. So either the equation is correct and my expectations are way off, or I've done something wrong in my derivation. Someone out there help me verify this equation or point out where I may have gone wrong?

Thanks

 

[reilly]

Until you specify the mean, you cannot find an error, typically the standard deviation of the mean, which also is dependent upon the number of measurements. Generally speaking you need to use computer for such a complicated situation. But if the errors are very small, then you can work with a Taylor's series approach. Google would be a good place to start to get it right.

Regards,
Reilly Atkinson

 

[Dr Transport]

Try calculating \frac{\delta C}{C} again. I came up with something different.

 

[imabug]

I'm thinking my error comes from not using the right value for my \delta{A}, \delta{P}, \delta{B_A} and \delta{B_P}. Since these are counts from a radioactive source, my first thought is to use \delta{A} = \sqrt{A}. But I also have a bad history of doing steps in my head and not writing them down and getting into trouble there.

Back to the drawing board...

 

[Dr Transport]

Take the derivative of C with respect to each of the variables, square , sum and take the square root then divide by C again and simplify. Each of the variables you have listed will have some error associated with their measurement. Take multiple measurements of each then assuming that the distribution of the measurements is Gaussian take the average and standard deviation, if you have the software available, calculate the skew and the kurtosis from the data, if the distribution is normal both will be found in a statistics book. The CRC Standard Math Tables has a chapter on probability and statistics which isn't too bad in helping to determine which test to use. My first indication is to start with students-t and work from there to find the tests of significance.

In taking the data, take many samples, the minimum I'd take is 25 for each of the measurements that way you'll get significance when you take the average, standard deviation of the mean, etc. it is always advisable to have a lot of data to get better statistics.

If you have not already thought of it, when you make the measurements, are the detectors fixed in position? and how else will they be changed during the experiment. Writing down all of this will help you formulate the correct measurement equation and all its components. Learning this now will save you later in life, in many industrial applications NIST is requiring, or should I say the customers you make measurements for require NIST traceability, i.e. correlation to some standard material the has been measured by NIST or another national lab. Part of this traceability is doing a complete measurement analysis and statistical analysis on your measurement apparatus to be able to assign a measurement uncertainty.

 

[reilly]

Your major problem is the non-linearity of C. As I'm sure you know, the mean of, say,
X/Y is not the ratio of the individual means -- the exception is when the variation of the individual variables is small relative to their mean. Even if X and Y, in the previous example, are normally distributed, their ratio is not normally distributed -- ratios are quite difficult when it comes to statistics and significance tests. Your C will almost certainly not be normally distibuted -- although with a big enough sample, probably 200 or more, the mean will approach normality (Law of Large numbers).

For the Taylor expansion to work you need 1. SD/Mean (coefficient of variation) to be very small, where SD is the standard variation, and 2. No correlations between the independent variables(measurements) -- generally, finding such correlations is an empirical task, not a theoretical one.

Even in survey research, a sample of 25 is small, 100 is often the smallest that's acceptable; 50 once in a while. And, for survey results, the statistics are very straightforward, and linear -- unlike your situation. (In fact, there's a small industry devoted to dealing with sample size determinations.)


Students t-Test is a comparative test: you can test whether a mean is non-zero, or whether two means differ. You could, for example, test whether your A and P counts differ significantly from background, whether A & B differ, and so on. ) Do go to a statistics text; CRC assumes a background in statistics and probability -- there's a lot to worry about; correlations, paired comparisons or not, and so on.

Your best bet, in my opinion as a statistician, is to do simulations. Your problem is very complex from a statistical perspective, way beyond normal theory. With simulations then you can see how everything works.

Regards,
Reilly Atkinson

 

[imabug]

gee, who knew this would get so complex. All I was trying to do was an error propagation analysis and see how the individual measurement uncertainties propagated into the computed quantity C. Still working on it.

Take the derivative of LaTeX graphic is being generated. Reload this page in a moment. with respect to each of the variables, square , sum and take the square root then divide by LaTeX graphic is being generated. Reload this page in a moment. again and simplify.

Yeah, this is the general method that I also saw in my Taylor book. Tried it that way and ended up with an equation that gave me uncertainties that were several orders of magnitude larger than the numbers I fed into the equation. That makes me think that I'm not subbing in the right expression for the \deltas of my measurements.

unfortunately, because of time and equipment accessibility constraints, I'm only able to make one measurement at each time point. Definitely not ideal, but sufficient for the needs of this experiment. Of course, now that I mention this, I'm suddenly struck by the thought that there's probably no point in even doing this kind of error analysis anyway...

I appreciate all the comments.

 

[Dr Transport]

\frac{\delta C}{C} = \sqrt{\sum_i (\frac{\partial C}{\partial x_i})^2 <br /> \frac{(\delta x_i )^2}{C^2}} which in your case comes out to be (if I took the derivatives correctly)

\frac{\delta C}{C} = \sqrt{\frac{(\delta A)^2 + (\delta B_A)^2}{(A-B_A)^2} <br /> + \frac{(\delta P)^2 + (\delta B_P)^2}{(P-B_P)^2}}

Now this expression can be used to calculate the uncertainty of a measurement, given the constituent parts. As Reilly has said, 100 data points is really the minimum, I suggested 25 because for some of the apparatus' I use I can get 25 fairly quickly and check the distribution against a normal curve. If they are not normal. I take an additional 25 points to see how that comes out, then get into a full blown analysis using other distributions to check the errors involved. In my case, we are working towards placing confidence limits on measurements to ensure that they fall in a certain band for consistentency. We spend an inordinate amount of time and money calibrating, measuring errors and calculating error estimates etc...The full program that covers this is called 6-sigma and has become the standard methodology in industry.

In your case, the manufacturer of the detector system you are using may assign a number to the error in counts i.e. X \pm \delta X, you can use this in the equation above to get the uncertainty of the measurement. This may simplify the analysis somewhat.

Take this with a grain of salt, I am a theorist working in an experimental group. It is my job to stare at every data point and doubt its value. Remember, the guy who calculates a quantity does not believe it, and the guy who measures it doesn't believe his measurement.

 

[imabug]

well, at least I know I was on the right track. that's similar to what I ended up getting. Guess my problem now is to figure out what my \deltas really are.

Thanks a bunch dr transport