ex-7: FLD terminated

notes/sections/7.md

@@ -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
 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
-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
 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
-`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.
 
+<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.