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ex:7 - fixe some conceptual errors (false positives and false negatives)

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Giù Marcer 2020-05-30 09:37:36 +02:00 committed by rnhmjoj
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notes/sections/7.md
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@ -366,6 +366,10 @@ $$
Similarly for the case with $e = 1$ and $f(x) = 0$. Similarly for the case with $e = 1$ and $f(x) = 0$.
![Weiht vector and threshold value obtained with the perceptron method as a
function of the number of iterations. Both level off at the third
iteration.](images/7-iterations.pdf){#fig:iterations}
As far as convergence is concerned, the perceptron will never get to the state As far as convergence is concerned, the perceptron will never get to the state
with all the input points classified correctly if the training set is not with all the input points classified correctly if the training set is not
linearly separable, meaning that the signal cannot be separated from the noise linearly separable, meaning that the signal cannot be separated from the noise
@ -406,7 +410,7 @@ samples was generated and the points were divided into noise and signal
applying both methods. To avoid storing large datasets in memory, at each applying both methods. To avoid storing large datasets in memory, at each
iteration, false positives and negatives were recorded using a running iteration, false positives and negatives were recorded using a running
statistics method implemented in the `gsl_rstat` library. For each sample, the statistics method implemented in the `gsl_rstat` library. For each sample, the
numbers $N_{fn}$ and $N_{fp}$ of false positive and false negative were obtained numbers $N_{fn}$ and $N_{fp}$ of false negative and false positive were obtained
this way: for every noise point $x_n$, the threshold function $f(x_n)$ was this way: for every noise point $x_n$, the threshold function $f(x_n)$ was
computed, then: computed, then:
@ -430,9 +434,9 @@ false-positive than false-negative, being also more variable from dataset to
dataset. dataset.
A possible explanation of this fact is that, for linearly separable and normally A possible explanation of this fact is that, for linearly separable and normally
distributed points, the Fisher linear discriminant is an exact analytical distributed points, the Fisher linear discriminant is an exact analytical
solution, whereas the perceptron is only expected to converge to the solution solution, the most powerful one, according to the Neyman-Pearson lemma, whereas
and is therefore more subject to random fluctuations. the perceptron is only expected to converge to the solution and is therefore
more subject to random fluctuations.
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$\alpha$ $\sigma_{\alpha}$ $\beta$ $\sigma_{\beta}$ $\alpha$ $\sigma_{\alpha}$ $\beta$ $\sigma_{\beta}$