ex-7: revised and typo-fixed

In addition, the folder ex-7/iters was created in order to plot the results
of the Perceptron method as a function of the iterations parameter.

ex-7/iters/iter.txt

@@ -0,0 +1,11 @@
+#iters w_x          w_y          b 
+1      0.4265959958 0.9044422902 1.7487522878
+2      0.5659895562 0.8244124103 1.4450529880
+3      0.6540716938 0.7564325610 1.2125805084
+4      0.6540716938 0.7564325610 1.2125805084
+5      0.6540716938 0.7564325610 1.2125805084
+6      0.6540716938 0.7564325610 1.2125805084
+7      0.6540716938 0.7564325610 1.2125805084
+8      0.6540716938 0.7564325610 1.2125805084
+9      0.6540716938 0.7564325610 1.2125805084
+10     0.6540716938 0.7564325610 1.2125805084

notes/sections/7.md

@@ -377,13 +377,28 @@ As in the previous section, once found, the weight vector is to be normalized.
 With $N = 5$ iterations, the values of $w$ and $t_{\text{cut}}$ level off up to the third
 digit. The following results were obtained:
 
+Different values of the learning rate were tested, all giving the same result,
+converging for a number $N = 3$ of iterations. In @fig:iterations, results are
+shown for $r = 0.8$: as can be seen, for $N = 3$, the values of $w$ and
+$t^{\text{cut}}$ level off.  
+The following results were obtained:
 $$
   w = (0.654, 0.756) \et t_{\text{cut}} = 1.213
 $$
 
-where, once again, $t_{\text{cut}}$ is computed from the origin of the axes. In
+In this case, the projection line is not parallel with the line joining the
-this case, the projection line does not lies along the mains of the two
+means of the two samples. Plots in @fig:percep_proj.
-samples. Plots in @fig:percep_proj.
+
+<div id="fig:percep_proj">
+![View of the samples in the plane.](images/7-percep-plane.pdf)
+![View of the samples projections onto the projection
+  line.](images/7-percep-proj.pdf)
+
+Aerial and lateral views of the samples. Projection line in blue and cut in
+red.
+</div>
+
+## Efficiency test {#sec:7_results}
 
 <div id="fig:percep_proj">
 ![View from above of the samples.](images/7-percep-plane.pdf){height=5.7cm}
@@ -396,40 +411,37 @@ red.
 
 ## Efficiency test {#sec:7_results}
 
-A program was implemented to check the validity of the two
+Using the same parameters of the training set, a number $N_t$ of test
-classification methods.  
+samples was generated and the points were divided into noise and signal
-A number $N_t$ of test samples, with the same parameters of the training set,
+applying both methods. To avoid storing large datasets in memory, at each
-is generated using an RNG and their points are divided into noise/signal by
+iteration, false positives and negatives were recorded using a running
-both methods. At each iteration, false positives and negatives are recorded
+statistics method implemented in the `gsl_rstat` library. For each sample, the
-using a running statistics method implemented in the `gsl_rstat` library, to
+numbers $N_{fn}$ and $N_{fp}$ of false positive and false negative were obtained
-avoid storing large datasets in memory.  
+this way: for every noise point $x_n$, the threshold function $f(x_n)$ was
-In each sample, the numbers $N_{fn}$ and $N_{fp}$ of false positive and false
+computed, then:
-negative are obtained in this way: for every noise point $x_n$ compute the
-activation function $f(x_n)$ with the weight vector $w$ and the
-$t_{\text{cut}}$, then:
 
-  - if $f(x) < 0 \thus$ $N_{fn} \to N_{fn}$
+  - if $f(x) = 0 \thus$ $N_{fn} \to N_{fn}$
-  - if $f(x) > 0 \thus$ $N_{fn} \to N_{fn} + 1$ 
+  - if $f(x) \neq 0 \thus$ $N_{fn} \to N_{fn} + 1$ 
 
 and similarly for the positive points.  
-Finally, the mean and standard deviation are computed from $N_{fn}$ and
+Finally, the mean and standard deviation were computed from $N_{fn}$ and
-$N_{fp}$ of every sample and used to estimate purity $\alpha$
+$N_{fp}$ for every sample and used to estimate the purity $\alpha$ and
-and efficiency $\beta$ of the classification:
+efficiency $\beta$ of the classification:
 
 $$
   \alpha = 1 - \frac{\text{mean}(N_{fn})}{N_s} \et
   \beta = 1 - \frac{\text{mean}(N_{fp})}{N_n}
 $$
 
-Results for $N_t = 500$ are shown in @tbl:res_comp. As can be seen, the
+Results for $N_t = 500$ are shown in @tbl:res_comp. As can be seen, the Fisher
-Fisher discriminant gives a nearly perfect classification
+discriminant gives a nearly perfect classification with a symmetric distribution
-with a symmetric distribution of false negative and false positive,
+of false negative and false positive, whereas the perceptron shows a little more
-whereas the perceptron show a little more false-positive than
+false-positive than false-negative, being also more variable from dataset to
-false-negative, being also more variable from dataset to dataset.  
+dataset.  
-A possible explanation of this fact is that, for linearly separable and
+A possible explanation of this fact is that, for linearly separable and normally
-normally distributed points, the Fisher linear discriminant is an exact
+distributed points, the Fisher linear discriminant is an exact analytical
-analytical solution, whereas the perceptron is only expected to converge to the
+solution, whereas the perceptron is only expected to converge to the solution
-solution and thus more subjected to random fluctuations.
+and is therefore more subject to random fluctuations.
 
 
 -------------------------------------------------------------------------------------------
@@ -442,4 +454,4 @@ 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. {#tbl:res_comp}
+       negatives and false positives respectively. {#tbl:res_comp}