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ex-7: FLD terminated

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Giù Marcer 2020-04-03 23:28:29 +02:00 committed by rnhmjoj
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commit d3e0be657b
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notes/sections/7.md
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@ -35,7 +35,12 @@ In the code, default settings are $N_s = 800$ points for the signal and $N_n =
samples were handled as matrices of dimension $n$ x 2, where $n$ is the number samples were handled as matrices of dimension $n$ x 2, where $n$ is the number
of points in the sample. The library `gsl_matrix` provided by GSL was employed of points in the sample. The library `gsl_matrix` provided by GSL was employed
for this purpose and the function `gsl_ran_bivariate_gaussian()` was used for for this purpose and the function `gsl_ran_bivariate_gaussian()` was used for
generating the points. generating the points.
An example of the two samples is shown in @fig:fisher_points.
![Example of points sorted according to two Gaussian with
the given parameters. Noise points in pink and signal points
in yellow.](images/fisher-points.pdf){#fig:fisher_points}
Assuming not to know how the points were generated, a model of classification Assuming not to know how the points were generated, a model of classification
must then be implemented in order to assign each point to the right class must then be implemented in order to assign each point to the right class
@ -185,8 +190,25 @@ $$
$$ $$
The projection of the points was accomplished by the use of the function The projection of the points was accomplished by the use of the function
`gsl_blas_ddot`, which computed a dot product between two vectors, which in `gsl_blas_ddot()`, which computed a dot product between two vectors, which in
this case were the weight vector and the position of the point to be projected. this case were the weight vector and the position of the point to be projected.
<div id="fig:fisher_proj">
![View from above of the samples.](images/fisher-plane.pdf){height=5.7cm}
![Gaussian of the samples on the projection
line.](images/fisher-proj.pdf){height=5.7cm}
Aeral and lateral views of the projection direction, in blue, and the cut, in red.
</div>
Results obtained for the same sample in @fig:fisher_points are shown in
@fig:fisher_proj. The weight vector $w$ was found to be:
$$
w = (0.707, 0.707)
$$
and $t_{\text{cut}}$ is 1.323 far from the origin of the axes. Hence, as can be
seen, the vector $w$ turned out to be parallel to the line joining the means of
the two classes (reminded to be $(0, 0)$ and $(4, 4)$) which means that the
total covariance matrix $S$ is isotropic, proportional to the unit matrix.