[vox-tech] OT math question

[Peter Jay Salzman]

Dylan,

If your goal is to characterize the relative "flatness" of a numerical
Gaussian, you might want to use the 2nd derivative.

I assume you're interested in some kind of measure that determines whether a
Gaussian looks like a skyscraper versus a gentle rolling hill.

The 1st derivative at the Gaussian's maximum will be zero (since the slope at
the Gaussian's max will be zero).

The 2nd derivative of your Gaussian's maximum will be negative since the
maximum is concave down.  The question is, just how concave down.  For
skyscrapers, the 2nd derivative will be very large and negative.  For rolling
hills, the 2nd derivative will be nearly zero and negative.


For numerical data, you might want to consider the following procedure:

1. Find the maximum of your Gaussian.
2. Calculate a numerical 2nd derivative.  Use the 3-point method:

                 f(max + dx)  -  2 * f(max)  +  f(max - dx)
   f''(max)  =   ------------------------------------------
                                ( dx )^2

where "max" is the "x-value" of the datapoint with the largest "y-value", and
"dx" is the distance between "x-values".


That would be for one Gaussian.  If you wanted to characterize this for a set
of graphs, then you might want to think about using the standard deviation,
which gives you information about how many Gaussians have a 2nd derivative
between xavg+delta and xavg-delta.

Hope I understood the question correctly!

Pete





On Wed 10 Nov 04, 12:28 AM, Dylan Beaudette <dylan at iici.no-ip.org> said:
> I know this is quite off topic... however it deals with numbers generated
> by a computer, so....
> 
> So I have some solar insolation values- 1 value per day for an entire year.
> the graph of solar insolation (y-axis) vs. day of year (x-axis) vary in
> shape - sometime looking very similar to a Gaussian distribution. I would
> like to characterize the relative fatness of these "Gaussian-like" graphs
> with something like a coefficient of variation... however, to the best of
> my knowledge things like the CV operate on frequency distributions, which
> in this case would only be characterizing the variation within the
> insolation values with respect to eachother.
> 
> just for reference a sample image can be found here:
> http://169.237.35.250/~dylan/solar/index.html#more
> 
> and if anyone is interested this is part of a dataset that was created and
> is being visualized with the open source GIS package called GRASS. (yeah
> for OSS!!)
> 
> thanks in advance, and sorry about the slightly off topic (and probably
> bone headed) question...

-- 
The mathematics of physics has become ever more abstract, rather than more
complicated.  The mind of God appears to be abstract but not complicated.
He also appears to like group theory.  --  Tony Zee's "Fearful Symmetry"

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