diff --git a/notes/sections/7.md b/notes/sections/7.md index c977686..bba7a20 100644 --- a/notes/sections/7.md +++ b/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}