Editing Information bottleneck method (section)

=== An example ===
The following case examines clustering in a four quadrant multiplier with random inputs <math>u, v \,</math> and two categories of output, <math>\pm 1 \,</math>, generated by <math>y=\operatorname{sign}(uv) \,</math>. This function has two spatially separated clusters for each category and so demonstrates that the method can handle such distributions.

20 samples are taken, uniformly distributed on the square <math>[-1,1]^2 \,</math> . The number of clusters used beyond the number of categories, two in this case, has little effect on performance and the results are shown for two clusters using parameters <math>\lambda = 3,\, \beta = 2.5</math>.

The distance function is <math>d_{i,j} =  \Big| x_i - x_j \Big |^2</math> where <math>x_i = (u_i,v_i)^T \, </math> while the conditional distribution <math>p(y|x)\, </math> is a 2&nbsp;×&nbsp;20 matrix

: <math>\begin{align} & Pr(y_i=1) = 1\text{ if }\operatorname{sign}(u_iv_i)=1\, \\
& Pr(y_i= -1) = 1\text{ if }\operatorname{sign}(u_iv_i)= -1\,
\end{align}</math>

and zero elsewhere.

The summation in line 2 incorporates only two values representing the training values of +1 or &minus;1, but nevertheless works well. The figure shows the locations of the twenty samples with '0' representing ''Y'' = 1 and 'x' representing ''Y'' = &minus;1. The contour at the unity likelihood ratio level is shown,

: <math>L= \frac{\Pr(1)}{\Pr(-1)} = 1</math>

as a new sample <math>x' \,</math>is scanned over the square. Theoretically the contour should align with the <math>u=0 \,</math> and <math>v=0 \,</math> coordinates but for such small sample numbers they have instead followed the spurious clusterings of the sample points.
[[Image:BottleCateg 1.jpg|thumb|Decision contours]]