ex-7: started writing the test part

notes/sections/0.md

@@ -42,6 +42,10 @@ header-includes: |
   \DeclareMathOperator*{\et}{%
   \hspace{30pt} \wedge \hspace{30pt}
   }
+  %% "if" in formulas
+  \DeclareMathOperator*{\incase}{%
+  \hspace{20pt} \text{if} \hspace{20pt}
+  }
 
   \makeatletter         
   \renewcommand\maketitle{

notes/sections/7.md

@@ -199,11 +199,11 @@ this case were the weight vector and the position of the point to be projected.
 ![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
+Aerial 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
+Results obtained for the same sample in @fig:points are shown in
 @fig:fisher_proj. The weight vector $w$ was found to be:
 
 $$
@@ -227,22 +227,21 @@ output value. The inferred function can be used for mapping new examples. The
 algorithm will be generalized to correctly determine the class labels for unseen
 instances.
 
-The aim is to determine the threshold function $f(x)$ for the dot product
-between the (in this case 2D) vector point $x$ and the weight vector $w$:
+The aim is to determine the bias $b$ such that the threshold function $f(x)$:
 
 $$
-  f(x) = x \cdot w + b
+  f(x) = x \cdot w + b \hspace{20pt}
+  \begin{cases}
+    \geqslant 0 \incase x \in \text{signal} \\
+    < 0         \incase x \in \text{noise}
+  \end{cases}
 $$ {#eq:perc}
 
-where $b$ is called 'bias'. If $f(x) \geqslant 0$, than the point can be
-assigned to the class $C_1$, to $C_2$ otherwise.
-
-The training was performed as follow. The idea is that the function $f(x)$ must
-return 0 when the point $x$ belongs to the noise and 1 if it belongs to the 
-signal. Initial values were set as $w = (0,0)$ and $b = 0$. From  these, the
-perceptron starts to improve their estimations. The sample was passed point by
-point into a reiterative procedure a grand total of $N_c$ calls: each time, the
-projection $w \cdot x$ of the point was computed and then the variable $\Delta$ was defined as:
+The training was performed as follow. Initial values were set as $w = (0,0)$ and 
+$b = 0$. From  these, the perceptron starts to improve their estimations. The
+sample was passed point by point into a reiterative procedure a grand total of
+$N_c$ calls: each time, the projection $w \cdot x$ of the point was computed
+and then the variable $\Delta$ was defined as:
 
 $$
   \Delta = r * (e - \theta (f(x))
@@ -254,15 +253,15 @@ where:
     larger $r$, the more volatile the weight changes. In the code, it was set
     $r = 0.8$;
   - $e$ is the expected value, namely 0 if $x$ is noise and 1 if it is signal;
-  - $\theta$ is the Heavyside theta function;
+  - $\theta$ is the Heaviside theta function;
   - $o$ is the observed value of $f(x)$ defined in @eq:perc.
 
 Then $b$ and $w$ must be updated as:
 
 $$
-  b \longrightarrow b + \Delta
+  b \to b + \Delta
   \et
-  w \longrightarrow w + x \Delta
+  w \to w + x \Delta
 $$
 
 <div id="fig:percep_proj">
@@ -270,12 +269,12 @@ $$
 ![Gaussian of the samples on the projection
   line.](images/percep-proj.pdf){height=5.7cm}
 
-Aeral and lateral views of the projection direction, in blue, and the cut, in
+Aerial and lateral views of the projection direction, in blue, and the cut, in
 red.
 </div>
 
 It can be shown that this method converges to the coveted function.  
-As stated in the previous section, the weight vector must finally be normalzied.
+As stated in the previous section, the weight vector must finally be normalized.
 
 With $N_c = 5$, the values of $w$ and $t_{\text{cut}}$ level off up to the third
 digit. The following results were obtained:
@@ -289,3 +288,47 @@ this case, the projection line does not lies along the mains of the two
 samples. Plots in @fig:percep_proj.
 
 ## Efficiency test
+
+A program was implemented in order to check the validity of the two
+aforementioned methods.  
+A number $N_t$ of test samples was generated and the
+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.
+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:
+      
+  - 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 $\alpha$
+and efficiency $\beta$ for the employed statistics:
+
+$$
+  \alpha = 1 - \frac{\text{mean}(N_{fn})}{N_s} \et
+  \beta = 1 - \frac{\text{mean}(N_{fp})}{N_n}
+$$
+
+Results for $N_t = 500$:
+
+-------------------------------------------------------------------------------------------
+                  $\alpha$       $\sigma_{\alpha}$        $\beta$        $\sigma_{\beta}$
+----------- ------------------- ------------------- ------------------- -------------------
+Fisher       0.9999              0.33                0.9999              0.33
+
+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.}