OpenEuphoria

Re: Calculating with imprecise values

• Posted by Juergen Luethje <j.lue at gmx.de> Dec 17, 2005
• 621 views

Al Getz wrote:

[Sensitivity Analysis]

> Hi Juergen,
> 
> Sorry i dont have a link in mind, but a quick google turned up several
> sites including a 'forum' for Sens A.  I didnt want it to sound too
> complicated though, because basically it's not,

Much appreciated. 

> so here's a quick example to help show how simple it can really be....
> 
> First a pure approach then a numerical approach that can be used in
> an Eu program...
> 
> I think a good example would be finding the volume of a rather long
> box with rectangular cross section we'll say the length of is 'a',
> and the two sides are 'b' and 'c'.  This would look like a rectangular
> pipe.  We want to find the volume, but also the change in volume so
> we can understand what happens when either of the measurements for
> a,b, and c are either off to begin with (due to measurement inaccuracies)
> or just to see what happens when a given dimension changes.
> 
> For this example we'll say
>   a=100
>   b=10
>   c=10
> in inches.
> 
> The volume would be v=a*b*c which we'll rewrite
>   y=a*b*c
> 
> First we take first partials:
>   dy/da=b*c
>   dy/db=a*c
>   dy/dc=a*b
> 
> and now we can call the sensitivities:
>   Sya=dy/da=b*c
>   Syb=dy/db=a*c
>   Syc=dy/dc=a*b
> 
> Where 'Sya' is read: "The sensitivity of y with respect to a".
> 
> Now we look at the actual numerical value of these to see what
> we can find out:
>   Sya=b*c=100
>   Syb=a*c=1000
>   Syc=a*b=1000
> 
>  From this we can quickly see that Sya is ten times less than Syb or
> Syc, so the measurement in 'a' isnt as important as the measurement
> for 'b' or 'c'.
> 
> Going back to the original equation with the chosen values for this 
> example, we can see that a change of say 1/16 inch causes a much
> bigger change in volume when the change occurs in either b or c, while
> not as big of a change in volume when the change occurs in a.

I see. I can imagine that this kind of approach to a problem often can
be very useful.

> Writing this out as a 'formula' we get an approximate numerical
> method:
>   Sya=(f(a+inc,b,c)-f(a,b,c))/inc
>   Syb=(f(a,b+inc,c)-f(a,b,c))/inc
>   Syc=(f(a,b,c+inc)-f(a,b,c))/inc
> where
>   f(a,b,c)=a*b*c (the original volume equation), and
>   'inc' is a small number like 0.001.
> 
> The above can be used in a program quite easily, but usually a slightly
> more accurate method is used in programs (the 'Central Means' formula):
>   Sya=(f(a+inc,b,c)-f(a-inc,b,c))/(inc+inc)
>   Syb=(f(a,b+inc,c)-f(a,b-inc,c))/(inc+inc)
>   Syc=(f(a,b,c+inc)-f(a,b,c-inc))/(inc+inc)
> All that's involved here is calling the function f twice for each
> sensitivity: once using a positive inc and once a negative inc,
> then subtracting the results and dividing by twice the inc.

Al, thanks a lot for the trouble writing down this comprehensive and
good understandable explanation!

Regards,
   Juergen

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