OpenEuphoria

Re: Calculating with imprecise values

• Posted by Al Getz <Xaxo at aol.com> Dec 16, 2005
• 614 views

Juergen Luethje wrote:
> 
> Al Getz wrote:
> 
> > I've used so called 'sensitivity' analysis in the past to measure changes
> > in functions with respect to a given variable and this helped understand
> > things better sometimes especially when human error is involved.
> > 
> > For example....
> > 
> > for the equation with some function F:
> >   y=F(a,b,c)
> > 
> > the sensitivity with change of parameter 'a' might be 100 times as great
> > as with parameter 'b', and perhaps the sensitivity with respect to 'c'
> > might be 1000 times lower than with 'a', meaning y hardly changes
> > with 'b' or 'c' but changes quite a bit with parameter 'a'.  Thus,
> > the error in measurement in 'b' or 'c' doesnt matter as much as with
> > the error in 'a'.  This would mean great care should be used in measuring
> > 'a' while with 'b' or 'c' you might get away with estimating from other
> > parameters or whatever.
> 
> Interesting!
> Do you have a link at hand that explains sensitivity analysis more in
> detail? 
> 
> Regards,
>    Juergen

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, 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.

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.


Take care,
Al

And, good luck with your Euphoria programming!

My bumper sticker: "I brake for LED's"