September 6th, 2012, 09:29 AM  #1 
Newbie Joined: Sep 2012 Posts: 2 Thanks: 0  Math function to fit into the
Dear All, Could anyone please suggest a function which would describe the following data set. I am trying to calibrate my pyrometer (noncontact, infrared temp. meter) and I want to interpolate the Voltage data within the given range of 7 to 600 deg. Obviously, I need R^2 as close to 1 as possible. I just need a mathematical function which describes these values (power function? exponential?). Measurement results: Temp [deg. C] Output voltage [uV] 7 20494 600 20755 700 20974 800 21285 900 21696 1000 22282 1100 22871 1200 23534 1300 24333 1400 25161 1500 26078 1600 27074 1680 28031 1700 28251 I cannot use polynomials. I have to stick to the theory (Planck's law). Would prefer a power function, exponential growth function or any other function resembling Planck's Law. Thank you very much in advance for any comments/suggestions/criticism/anything. Peter Drawa 
September 6th, 2012, 10:55 AM  #2 
Math Team Joined: Sep 2007 Posts: 2,409 Thanks: 6  Re: Math function to fit into the
The "slope" seems to increase sharply as temperature increases so I certainly think an expoential would be appropriate. If you want to use something like then a "quick and dirty" solution would be to use the lowest and highest temperatures to get two equations to solve for C and a. That is, with T= 7, V= 20494 so we have the equation . With T= 1700, V= 28251, so we have the equation . Of course, there is no way of telling whether it will fit the other points. Another thing you could do is use, say, , which gives you three values to determine and so you can fit it to three points. I recommend using three points as far apart as possible: the first, T= 7, V= 20494, the middle, T= 1200, F= 23534, and the last, T= 1700, V= 28251. The hardest thing to do would be a "least squares" exponential, which would not necessarily exactly match any point but would be close to all of them.

September 6th, 2012, 11:43 AM  #3  
Global Moderator Joined: Nov 2006 From: UTC 5 Posts: 16,046 Thanks: 938 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms  Re: Math function to fit into the Quote:
 
September 6th, 2012, 12:22 PM  #4 
Global Moderator Joined: Dec 2006 Posts: 20,464 Thanks: 2038 
This doesn't look like exponential growth or a power function to me. The data (except for (7, 20494)) seem to correspond to a parabola (as the "slope graph" is roughly linear), but it would be unwise to use that parabola to interpolate for temperatures between 7° and 600°. Is at least some data available for temperatures within that range?

September 6th, 2012, 01:31 PM  #5 
Global Moderator Joined: Nov 2006 From: UTC 5 Posts: 16,046 Thanks: 938 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms  Re: Math function to fit into the
I think a parabola is too lowdegree. A cubic fits much better. In any case you should convert to Kelvins and set the constant term to something like 0.

September 6th, 2012, 02:56 PM  #6 
Global Moderator Joined: Dec 2006 Posts: 20,464 Thanks: 2038 
Are you sure? 0.004308T˛  3.08T + 21024 works really well (except for T = 7).

September 11th, 2012, 07:34 AM  #7 
Newbie Joined: Sep 2012 Posts: 2 Thanks: 0  Re: Math function to fit into the
Thank you all for your comments (very useful!). I have carried out some curve fitting over the last few days. Here is what I discovered:  modified StefanBoltzman (y(x) = a*T^b + c) fits well for T < 300 deg. of C.  I have come across a function based on Planck's Law, which has a form y(x) = A/(exp(B/T)  C) and it becomes very accurate for T > 400 deg. of C. (So one can use it complementary with StefanBoltzman for Temps > 400 deg. C).  surprisingly, the formula for the cumulative distribution function of the Weibull distribution (y(x) = a  b*exp(c*T^d)) fits very well especially for lower temperatures (T < 500 deg. of C). What I also like about this function is that it is sigmoidal so in contrast to the previous function (the one related to Planck's law), it flattens out for T >> inf. instead of breaking down. I am not sure if it is legitimate to use such a function to calibrate my pyrometer, but I guess I can treat it as a purely empirical function. Thank you again for all your comments. Peter 
October 14th, 2012, 04:56 AM  #8  
Global Moderator Joined: Dec 2006 Posts: 20,464 Thanks: 2038  Quote:
 

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