Nova Statistics

It's nice to find another coloring method that gives a genuinely new and interesting look to the intense detail of the nova fractal. I recently discovered Kerry Mitchell's 'Statistics' coloring (in lkm.ucl) which gives some really interesting structures that blend well with Exponential Smoothing Normalized.

Here is Kerry's description from the 'Statistics' coloring source code, it says it's better suited to 'inside' coloring which is great because I spend s lot of my time in the lakes ;)

Quoted from lkm.ucl:

Statistics

This started out with an article written by Stephen Ferguson on an analysis of fractal dimension, which he based on an algorithm by Holger Jaenisch. The formulas here modify Stephen's analysis and add other standard statistical measures. Since the measures are typically used with bounded datasets, this coloring method may be more applicable as an "inside" scheme, but there's nothing stopping a user from employing it as an "outside" scheme as well.

The statistical measures implemented are: minimum, maximum, range, mean, standard deviation, coefficient of variation, and fractal dimension. All are defined only for real variables, so there are 4 choices for reducing the complex #z to a real number: real(#z), imag(#z), the magnitude of #z, and imag(#z)/real(#z). The last method is an attempt to capture the polar angle of #z, without the discontinuities involved in actually using the angle arg(#z).

Once the choice of variable has been made, the first 3 measures are simple enough to compute. Simply monitor the orbit, updating the minimum and maximum as they change. Once the iteration has ceased, the range is just (maximum - minimum). The mean is computed by keeping a running sum of the variable, then dividing it by the number of iterations.

The standard deviation is a measure of the spread of the data, and is defined in terms of the sum of the squared differences between each datum and the mean. This can also be computed by keeping a running sum of the square of the variable. This sum is used with the sum for the mean to determine the standard deviation. The coefficient of variation is a normalized standard deviation: it's the ratio of the standard deviation to the mean. Finally, the "fractal dimension" computed here is not the true fractal dimension, but an approximation to it. It's the standard deviation normalized by the range.

Aside from being texturally interesting, this setup has a gorgeous gold/blue balance to it:













Further exploration:















Blue set:









Green set: