From f74bd2b71388983cb7641a0e850a0c83c2ad817e Mon Sep 17 00:00:00 2001 From: rnhmjoj Date: Mon, 6 Jul 2020 18:22:31 +0200 Subject: [PATCH] notes: fix a few small issues in ex-6 --- notes/sections/6.md | 8 ++++---- 1 file changed, 4 insertions(+), 4 deletions(-) diff --git a/notes/sections/6.md b/notes/sections/6.md index 66e1df9..ba95a04 100644 --- a/notes/sections/6.md +++ b/notes/sections/6.md @@ -114,7 +114,7 @@ implemented for discrete arrays of numbers, such as histograms or vectors: (f * g)(x) &= \int \limits_{- \infty}^{+ \infty} dy f(y) (R \, g)(y-x) \\ &= \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*} 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 estimate for $f(\xi)$ can be generated, using the observed sample {$x_i$} to 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) \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 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 -@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 close to the one found for $\sigma = 0.5$. 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 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 histogram is shown in @fig:noisy. The following three values of $\sigma_N$ were tested: