Image:LogDirichletDensity-alpha 0.3 to alpha 2.0.gif

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Image:LogDirichletDensity-alpha 0.3 to alpha 2.0.gif

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LogDirichletDensity-alpha_0.3_to_alpha_2.0.gif‎ (512 × 512 pixels, file size: 3.35 MB, MIME type: image/gif)

[edit] Summary

We illustrate the log of the density function:

$\log (f(x_1,\dots, x_{K-1}; \alpha_1,\dots, \alpha_K)) = \log(\frac{1}{\mathrm{B}(\alpha)} \prod_{i=1}^K x_i^{\alpha_i - 1}) = + \sum_{i=1}^K \alpha_i \log(x_i) - \sum_{i=1}^K \log(x_i) - \sum_{i=1}^K \log(\Gamma(\alpha_i)) + \log(\Gamma(\sum_{i=1}^K \alpha_i))$

for $K = 3$ . In other words, we have two parameters $x 1, x 2$ varying on the two axes, and an implicit $x 3 = 1 - x 1 - x 2$ .

The picture illustrates the case where $α 1 = α 2 = α 3 = α$ and we vary over time the parameter $α$ from 0.3 to 2.0.

[edit] Maple Code

The animated plot was generated using Maple 11, with the following code:


restart; 

with(plots);

B := (a1, a2, a3) -> (GAMMA(1.0*a1) * GAMMA(1.0*a2) * GAMMA(1.0*a3)) / GAMMA(1.0*a1+1.0*a2+1.0*a3);

f := (x1, x2, a1, a2, a3) ->  (x1^(a1-1)) * (x2^(a2-1)) * ( (1-x1-x2)^(a3-1)) /B(a1,a2,a3) ;

plotsetup(gif, plotoutput=`LogDirichletDensity-alpha_0.3_to_alpha_2.0.gif`); 

animate ( plot3d,  [eval(log(f(x1, x2, a1, a2, a3)), {a1=a, a2=a, a3=a}), x1=0.00..1, x2=0.00..1, 
axes=BOXED, grid=[25,25],gridstyle=triangular,orientation=[-135, 60], shading=zhue, contours=20, style=surfacecontour, view=-3..2 ],
a=0.3..2.0, frames=100);

[edit] Licensing

This work is licensed under the Creative Commons Attribution 3.0 License.

File history

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	Date/Time	Dimensions	User	Comment
current	04:10, 28 October 2007	512×512 (3.35 MB)	Ipeirotis (Talk \| contribs)	(== Summary == We illustrate the log of the density function: <math>\log (f(x_1,\dots, x_{K-1}; \alpha_1,\dots, \alpha_K)) = \log(\frac{1}{\mathrm{B}(\alpha)} \prod_{i=1}^K x_i^{\alpha_i - 1}) </math> for <math>K=3</math>. In other words, we have two par)