medpy.features.histogram.gaussian_membership

medpy.features.histogram.gaussian_membership(bin_center, bin_width, smoothness)

Create a gaussian membership function for a fuzzy histogram bin.

Parameters:

bin_centernumber

The center of the bin of which to compute the membership function.

bin_widthnumber

The width of a single bin (all expected to be of equal width).

smoothnessnumber, optional

The smoothness of the function; determines the neighbourhood affected. See below and fuzzy_histogram for a more detailed explanation

Returns:

gaussian_membershipfunction

The cumulative density function of the desired gaussian.

Notes

Since the gaussian membership function is infinite, it is not actually true that it does not contribute to bins outside of the neighbourhood range. But the contribution is so marginal (\(eps <= 0.001\) per value) that it can be safely ignored.

The gaussian membership function is defined as

\[\mu_{gauss}(x) = \frac{1}{\sigma\sqrt{2\pi}} e^{-\frac{(x-\zeta)^2}{2\sigma^2}}\]

Since the gaussian distributions can not be formed to sum up to one at each point of the x-axis, their cumulative density functions (CDF) are used instead. For more details on CDF see http://en.wikipedia.org/wiki/Normal_distribution .

The gaussian and therefore the CDF are centered above the requested value instead of the bin center. Then the CDF value for the left side of the bin is subtracted from the CDF value returned for the right side. The result is the integral under the gaussian with \(\mu/\zeta = value\) with the bin-sides as the integral borders.

This approach might seem a little bit unintuitive, but is the best possible for gaussian membership functions. The following graph gives a graphical example of the computation of each values bin membership

“Trapezium functions (1)”

where the bin_width is 1, one bin between each of the x tics (e.g. [-1, 0], [0, 1], etc.). The value which membership should be computed is marked by a yellow bar at \(x = 0.3\). Its membership in each bin is defined by the integral under the gaussian centered at the value (i.e. 0.3). The purple area therefore defines its membership in the [-2,-1] bin, the red area its membership in the [-1,0] bin, etc. Since the gaussian is guaranteed to have an infinite integral of 1, the some of the contributions of a value to all bins is one.

For computation the function normalizes all values to a bin_width of 1, which can introduce marginal rounding errors.