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ex-5: add the plot 5-fit.pdf to the paper

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Giù Marcer 2020-06-01 16:20:34 +02:00 committed by rnhmjoj
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commit aa676cf9a9
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@ -69,6 +69,11 @@ this can result in an underestimation of the error.
In this case $f(x) = e^{x}$ and $\Omega = [0,1]$, hence $V = 1$.
![Estimated values of $I$ obatined by Plain MC technique with different
number of function calls; logarithmic scale; errorbars showing their
estimated uncertainties. As can be seen, the process does a sort o seesaw
around the correct value.](images/5-MC_MC.pdf){#fig:plain-mc-iter}
Since the distance from $I$ of $I_N$ is related to $N$, the accuracy of the
method lies in how many points are generated, namely how many function calls
are executed when the iterative method is implemented. In @fig:plain-mc-iter
@ -77,11 +82,6 @@ red. In @tbl:plain-mc-res, some of them are listed: the estimated integrals
$I^{\text{oss}}$ are compared to the expected value $I$ and the differences
between them are given.
![Estimated values of $I$ obatined by Plain MC technique with different
number of function calls; logarithmic scale; errorbars showing their
estimated uncertainties. As can be seen, the process does a sort o seesaw
around the correct value.](images/5-MC_MC.pdf){#fig:plain-mc-iter}
---------------------------------------------------------------------------
calls $I^{\text{oss}}$ $\sigma$ diff
------------------ ------------------ ------------------ ------------------
@ -99,24 +99,23 @@ Table: Some MC results with three different numbers of function calls.
As can be seen, $\sigma$ is always of the same order of magnitude of diff,
except for very low numbers of function calls. Even with \num{5e7} calls,
$I^{\text{oss}}$ still differs from $I$ at the fifth decimal place, meaning
that this method shows a really slow convergence. In fact, since the $\sigma$
dependence on the number $C$ of function calls is confirmed:
that this method shows a really slow convergence.
The $\sigma$ dependence on the number $C$ of function calls was checked with a
least square minimization by modeling the data with the function:
$$
\begin{cases}
\sigma_1 = \num{6.95569e-4} \longleftrightarrow C_1 = \num{5e6} \\
\sigma_2 = \num{6.95809e-5} \longleftrightarrow C_1 = \num{5e8}
\end{cases}
\sigma = \frac{a}{x^b}
$$
$$
\thus
\frac{\sigma_1}{\sigma_2} = 9.9965508 \sim 10 = \sqrt{\frac{C_2}{C_1}}
$$
As can be seen in @fig:err_fit, the obtained result confirmes the expected value
of $b^{\text{exp}} = 0.5$, having found $b \sim 0.499$.
Given this dependence, for an error of $10^{-n}$, a number $\propto 10^{2n}$ of
function calls is needed. To compute an integral within double precision, an
impossibly large number of $\sigma \sim 10^{32}$ calls is needed, which makes
this method unpractical for high-precision applications.
For an error of $10^{-n}$, a number $\propto 10^{2n}$ of function calls is
needed. To compute an integral within double precision,
an impossibly large number of $\sigma \sim 10^{32}$ calls is needed,
which makes this method unpractical for high-precision applications.
![Plain MC uncertainties estimations $\sigma$ as a function of the
number of function calls $C$. Observed values in red, predicted
dependence in gray.](images/5-fit.pdf){#fig:err_fit}
## Stratified sampling
@ -381,6 +380,10 @@ calls $I^{\text{oss}}$ $\sigma$ diff
Table: Some VEGAS results with different numbers of
function calls. {#tbl:vegas-res}
![Only the most accurate results are shown in order to stress the
differences between VEGAS (in gray) and MISER (in black) methods
results.](images/5-MC_MI_VE.pdf){#fig:vegas-iter}
As can be appreciated in @fig:vegas-iter, the VEGAS algorithm manages to compute the
integral value more accurately compared to MISER. The $\chi_r^2$ turns out to
be close enough to 1 to guarantee a good estimation of $I$, goodness which is
@ -388,10 +391,6 @@ also confirmed by the very small difference shown in @tbl:vegas-res.
In fact, with a number of \num{5e7} function calls, the difference is
smaller than \num{1e-10}.
![Only the most accurate results are shown in order to stress the
differences between VEGAS (in gray) and MISER (in black) methods
results.](images/5-MC_MI_VE.pdf){#fig:vegas-iter}
In conclusion, between a plain Monte Carlo technique, stratified sampling and
importance sampling, the last turned out to be the most powerful mean to
obtain a good estimation of the integrand.