Text Clustering with Mixture of von Mises-Fisher Distributions - Text Mining: Classification, Clustering, and Applications - page 127

6.5.2 Experimental Study of the Approximation
In this section we provide a brief experimental study to assess the qual-
ity of our approximation of the concentration parameter κ . Recall that our
approximation (6.17) attempts to solve the implicit non-linear equation

I d/ 2 ( κ )
I d/ 2 − 1 ( κ ) = r.

(6.18)

We note that for large values of r ( r close to 1), approximation (6.14) is
reasonable; for small values of r (usually for r< 0 . 2) estimate (6.15) is quite
good; whereas (6.17) yields good approximations for most values of r .
Since a particular value of r may correspond to many different combinations
of κ and d values, to assess the quality of various approximations, we need
to evaluate their performance across the ( κ, d ) plane. However, such an as-
sessment is dicult to illustrate through 2-dimensional plots. To supplement
Table 6.1 , which showed how the three approximations behave on a sampling
of points from the ( κ, d ) plane, in this section we present experimental results
on some slices of this plane, where we either keep d fixed and vary κ ,orwe
keep κ fixed and vary d . For all our evaluations, the r values were computed
using (6.18).

Comparison of various kappa approximations (d = 1000)

6000

True κ

(1.14)
(1.15)
(1.17)

5000

4000

3000

2000

1000

0

0 1000 2000 3000 4000 5000 6000

Kappa ( κ )

FIGURE 6.1 : Comparison of true and approximated κ values, with d =
1000.