ex-7: completed

notes/sections/7.md

@@ -296,20 +296,18 @@ points were divided into the two classes according to the selected method.
 At each iteration, false positives and negatives are recorded using a running
 statistics method implemented in the `gsl_rstat` library, being suitable for
 handling large datasets for which it is inconvenient to store in memory all at
-once.
+once.  
 For each sample, the numbers $N_{fn}$ and $N_{fp}$ of false positive and false
-negative are computed with the following trick:
-
-Every noise point $x_n$ was checked this way: the function $f(x_n)$ was computed
-with the weight vector $w$ and the $t_{\text{cut}}$ given by the employed method,
-then:
+negative are computed with the following trick: every noise point $x_n$ was
+checked this way: the function $f(x_n)$ was computed with the weight vector $w$
+and the $t_{\text{cut}}$ given by the employed method, then:
       
   - if $f(x) < 0 \thus$ $N_{fn} \to N_{fn}$
   - if $f(x) > 0 \thus$ $N_{fn} \to N_{fn} + 1$ 
 
 Similarly for the positive points.  
-Finally, the mean and the standard deviation were obtained from $N_{fn}$ and
-$N_{fp}$ computed for every sample in order to get the mean purity $\alpha$
+Finally, the mean and the standard deviation were computed from $N_{fn}$ and
+$N_{fp}$ obtained for every sample in order to get the mean purity $\alpha$
 and efficiency $\beta$ for the employed statistics:
 
 $$
@@ -317,7 +315,16 @@ $$
   \beta = 1 - \frac{\text{mean}(N_{fp})}{N_n}
 $$
 
-Results for $N_t = 500$:
+Results for $N_t = 500$ are shown in @tbl:res_comp. As can be observed, the
+Fisher method gives a nearly perfect assignment of the points to their belonging
+class, with a symmetric distribution of false negative and false positive,
+whereas the points perceptron-divided show a little more false-positive than
+false-negative, being also more changable from dataset to dataset.  
+The reason why this happened lies in the fact that the Fisher linear
+discriminant is an exact analitical result, whereas the perceptron is based on
+a convergent behaviour which cannot be exactely reached by definition.
+
+
 
 -------------------------------------------------------------------------------------------
                   $\alpha$       $\sigma_{\alpha}$        $\beta$        $\sigma_{\beta}$
@@ -329,6 +336,4 @@ Perceptron   0.9999              0.28                0.9995              0.64
 
 Table: Results for Fisher and perceptron method. $\sigma_{\alpha}$ and
        $\sigma_{\beta}$ stand for the standard deviation of the false
-       negative and false positive respectively.
-
-\textcolor{red}{MISSING COMMENTS ON RESULTS.}
+       negative and false positive respectively. {#tbl:res_comp}