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Area under lethality curve

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    Posted: 07 Apr 2011 at 11:11pm
I have taken 7 sets of time temperature data and calculated Fo using the Trapezoid rule, Simpson's rule and the Romberg method. The results are summarised below.
 
For 3 of the data sets the Trapezoid rule slightly overestimated Fo compared with the more accurate methods. However, it underestimated Fo for 4 of the datasets.
 

Fo  values obtained using published data and three methods of numerical integration

Integration method

Data set 1

Data set 2

Data set 3

Data set 4

Data set 5

Data set 6

Data set 7

Simpson's rule

21.322

9.865

6.783

15.828

9.833

3.908

11.020

Trapezoid rule

21.588

9.821

6.906

15.276

9.319

3.872

11.412

Romberg

21.242

9.851

6.783

15.828

ND

ND

11.083

 
I will provide the full analysis including all the data,  the trend line equations and the spreadsheets as a download at a future time if there is enough interest.
 
So Beachgirl it would be reasonable to predict that there will be instances when your company's  F values have been underestimated when  the Trapezoid rule has been used. BUT unless you have used large time intervals and few data points this underestimation should be too small to cause any concern.
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Post Options Post Options   Thanks (0) Thanks(0)   Quote Admin Quote  Post ReplyReply Direct Link To This Post Posted: 30 Mar 2011 at 8:52pm
I have been wondering why you have kept asking questions about calculating the area under the lethality curve!
 
I am aware that others have solved a range of equations (with known solutions) using several numerical methods; based on their findings the order of accuracy in obtaining the area under the curve is 1, Romberg, 2 Simpson's and 3 the trapezoidal rule. Based on most of the work published the trapezoidal rule is slightly less accurate and sometimes overestimates area (F value) compared with more accurate methods (e.g. Simpson's rule) - in thermal processing.
 
Having calculated a few F values using the three methods (using actual data) I have found that as expected  the trapezoidal rule gave lower F values compared with Simpson's rule for many data sets. 
Thank you for being persistent.
 
 
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Post Options Post Options   Thanks (0) Thanks(0)   Quote Admin Quote  Post ReplyReply Direct Link To This Post Posted: 27 Mar 2011 at 8:42pm
You certainly are interested in this topic!
 
I will prepare a spreadsheet over the next week or so and answer your question more fully.
 
The trapezoidal rule is expected to slightly underestimate area with concave down curves. However if the spacing (or interval between readings) is small and a large number of readings are taken the error becomes very small and to all practical purposes (thermal processing calculations) is not significant.
 
Now to theory. The theoretical error of the simple trapezoidal rule is proportional to sample interval raised to the third power. The error for the composite rule is proportional to the sample interval squared. If you increase the number of samples so as to decrease the interval by a factor of 2, then the error will decrease by 2 x 2 , a  factor of 4.  If you are using 4 min intervals then moving to 2 or 1 will markedly affect accuracy. With a small number of test results and larger time intervals Simpson's rules (there are 2) will give a more accurate estimate. But in many practical situations this does not really matter.
 
You have succeeded in me wanting to close this topic! So over the next few weeks I will prepare a spreadsheet showing F values obtained using trapezoidal rule, Simpson's two rules and the equation for a lethality curve. Hopefully this will give you the information you need?
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Post Options Post Options   Thanks (0) Thanks(0)   Quote beachgirl Quote  Post ReplyReply Direct Link To This Post Posted: 27 Mar 2011 at 11:28am
Hi u might said this bfor what are the rrors with the t rule & how u control?
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Post Options Post Options   Thanks (0) Thanks(0)   Quote Admin Quote  Post ReplyReply Direct Link To This Post Posted: 22 Mar 2011 at 10:00pm
Yes welcome back.
 
I think I have responded to this query under thermal processing before? Perhaps the easiest way to get the equation you want is to use the trendline function in Excel. If the r value is > 0.9 (you can use lower r values) and your knowledge of calculus is good you can integrate the equation to get the area. The

Wolfram|Alpha: Computational Knowledge Engine

 can be used to check your calculation.
 
There are a few really good Excel textbooks written by scientists for scientists and engineers and provide a good introduction to the computing power of Excel. Look up the  LINEST function it can used instead of the trendline function to find the equation for a curve.
 
What's wrong with numerical integration using e.g. the trapezoidal rule; the error is usually less than

0.5% and can be reduced further?

 
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Post Options Post Options   Thanks (0) Thanks(0)   Quote beachgirl Quote  Post ReplyReply Direct Link To This Post Posted: 21 Mar 2011 at 10:20pm
Hi back again! How do you get equation for lethal curve and how to solve it?
 
Thnks!
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