[Numpy-discussion] Generating Bell Curves (was: Using normal() )
David Huard
david.huard@gmail....
Fri Apr 25 10:33:17 CDT 2008
Other suggestions for bounded bell-shaped functions that reach zero on a
finite interval:
- Beta distribution: http://en.wikipedia.org/wiki/Beta_distribution
- Cubic B-splines:http://www.ibiblio.org/e-notes/Splines/Basis.htm
2008/4/25 Bruce Southey <bsouthey@gmail.com>:
> Rich Shepard wrote:
> > Thanks to several of you I produced test code using the normal
> density
> > function, and it does not do what we need. Neither does the Gaussian
> > function using fwhm that I've tried. The latter comes closer, but the
> ends
> > do not reach y=0 when the inflection point is y=0.5.
> >
> > So, let me ask the collective expertise here how to generate the
> curves
> > that we need.
> >
> > We need to generate bell-shaped curves given a midpoint, width (where
> y=0)
> > and inflection point (by default, y=0.5) where y is [0.0, 1.0], and x is
> > usually [0, 100], but can vary. Using the NumPy arange() function to
> produce
> > the x values (e.g, arange(0, 100, 0.1)), I need a function that will
> produce
> > the associated y values for a bell-shaped curve. These curves represent
> the
> > membership functions for fuzzy term sets, and generally adjacent curves
> > overlap where y=0.5. It would be a bonus to be able to adjust the skew
> and
> > kurtosis of the curves, but the controlling data would be the
> > center/midpoint and width, with defaults for inflection point, and other
> > parameters.
> >
> > I've been searching for quite some time without finding a solution
> that
> > works as we need it to work.
> >
> > TIA,
> >
> > Rich
> >
> >
> Hi,
> You could use a Gamma distribution to get a skewed distribution. But to
> extend Keith's comment, continuous distributions typically go from
> minus infinity or zero to positive infinity and, furthermore, the
> probability of a single point in a continuous distribution is always
> zero. The only way you are going to get this from a single continuous
> distribution is via some truncated distribution - essentially Keith's
> reply.
>
> Alternatively, you may get away with a discrete distribution like the
> Poisson since it very quickly approaches normality but is skewed. A
> multinomial distribution may also work but that is more assumptions. In
> either case, you have map the points into the valid space because it is
> the distribution within the set that is used not the distribution of the
> data.
>
> I do not see the requirement for overlapping curves because the expected
> distribution of each set should be independent of the data and of the
> other sets. In that case, you just find the mean and variance of each
> set to get the degree of overlap you require. The inflection point
> requirement is very hard to understand as it different meanings such as
> just crossing or same area under the curve. I don't see any simple
> solution to that - two normals with the same variance but different
> means probably would. If the sets are dependent then you need a
> multivariate solution. Really you probably need a mixture of
> distributions and/or generate your own function to get something that
> meets you full requirements.
>
> Regards
> Bruce
>
>
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