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notes: fix a few small issues in ex-6

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Michele Guerini Rocco 2020-07-06 18:22:31 +02:00
parent 527f1fcfce
commit f74bd2b713
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1 changed files with 4 additions and 4 deletions
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notes/sections/6.md
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@ -114,7 +114,7 @@ implemented for discrete arrays of numbers, such as histograms or vectors:
(f * g)(x) (f * g)(x)
&= \int \limits_{- \infty}^{+ \infty} dy f(y) (R \, g)(y-x) \\ &= \int \limits_{- \infty}^{+ \infty} dy f(y) (R \, g)(y-x) \\
&= \int \limits_{- \infty}^{+ \infty} dy f(y) (T_x \, R \, g)(y) \\ &= \int \limits_{- \infty}^{+ \infty} dy f(y) (T_x \, R \, g)(y) \\
&= (f, T_x \, R \, g)(y) &= (f, T_x \, R \, g)
\end{align*} \end{align*}
where: where:
@ -368,7 +368,7 @@ a known $P(x | \xi)$, @eq:first can be used to calculate an estimate for
$Q (\xi | x)$. Then, taking the hint provided by @eq:second, an improved $Q (\xi | x)$. Then, taking the hint provided by @eq:second, an improved
estimate for $f(\xi)$ can be generated, using the observed sample {$x_i$} to estimate for $f(\xi)$ can be generated, using the observed sample {$x_i$} to
give an approximation for $\phi$. give an approximation for $\phi$.
Thus, if $f^t$ is the $t-th$ estimate, the next is given by: Thus, if $f^t$ is the $t$-th estimate, the next is given by:
$$ $$
f^{t + 1}(\xi) = \int dx \, \phi(x) Q^t(\xi | x) f^{t + 1}(\xi) = \int dx \, \phi(x) Q^t(\xi | x)
\with \with
@ -544,7 +544,7 @@ merely a fact of floating-points precision) and the best result is obtained for
$r = 2$, meaning that the convergence of the RL algorithm is really fast and $r = 2$, meaning that the convergence of the RL algorithm is really fast and
this is due to the fact that the histogram was only slightly modified. this is due to the fact that the histogram was only slightly modified.
In @fig:rless-0.5, the curve starts to flatten at about 10 rounds, whereas in In @fig:rless-0.5, the curve starts to flatten at about 10 rounds, whereas in
@fig:rless-1 a minimum occurs around \num{5e3} rounds, meaning that, whit such @fig:rless-1 a minimum occurs around \num{5e3} rounds, meaning that, with such
a large kernel, the convergence is very slow, even if the best results are a large kernel, the convergence is very slow, even if the best results are
close to the one found for $\sigma = 0.5$. close to the one found for $\sigma = 0.5$.
The following $r$s were chosen as the most fitting: The following $r$s were chosen as the most fitting:
@ -625,7 +625,7 @@ follows.
For each bin, once the convolved histogram was computed, a value $v_N$ was For each bin, once the convolved histogram was computed, a value $v_N$ was
randomly sampled from a Gaussian distribution with standard deviation randomly sampled from a Gaussian distribution with standard deviation
$\sigma_N$, and the value $v_n \cdot b$ was added to the bin itself, where $b$ $\sigma_N$, and the value $v_N \cdot b$ was added to the bin itself, where $b$
is the count of the bin. An example with $\sigma_N = 0.05$ of the new is the count of the bin. An example with $\sigma_N = 0.05$ of the new
histogram is shown in @fig:noisy. histogram is shown in @fig:noisy.
The following three values of $\sigma_N$ were tested: The following three values of $\sigma_N$ were tested: