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notes/sections/5.md
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@ -2,12 +2,15 @@
The following integral is to be evaluated comparing different Monte Carlo
techniques.
\vspace{20pt}
$$
I = \int_0^1 \!\!\! dx \, e^x
$$
\vspace{-70pt}
\begin{figure}
\hypertarget{fig:exp}{%
\centering
\begin{tikzpicture}
\definecolor{cyclamen}{RGB}{146, 24, 43}
% Integral
\filldraw [cyclamen!15!white, domain=0:5, variable=\x]
(0,0) -- plot({\x},{exp(\x/5)}) -- (5,0) -- cycle;
@ -21,19 +24,16 @@ techniques.
% Plot
\draw [domain=-2:7, smooth, variable=\x,
cyclamen, ultra thick] plot ({\x},{exp(\x/5)});
% Equation
\node [above] at (2.5, 2.5) {$I = \int\limits_0^1 dx \, e^x$};
\end{tikzpicture}
\caption{Plot of the integral to be evaluated.}
}
\end{figure}
whose exact value is 1.7182818285...
The three most popular Monte Carlo (MC) methods where applied: plain MC, Miser
and Vegas. Besides being commonly used, these were chosen for also being
implemented in the GSL libraries `gsl_monte_plain`, `gsl_monte_miser` and
`gsl_monte_vegas`, respectively.
`gsl_monte_vegas`.
## Plain Monte Carlo
@ -44,24 +44,21 @@ $$
I = \int\limits_{\Omega} dx \, f(x)
\with V = \int\limits_{\Omega} dx
$$
the simplest MC method approach is to sample $N$ points $x_i$ in $V$ and
approximate $I$ as:
$$
I \approx I_N = \frac{V}{N} \sum_{i=1}^N f(x_i) = V \cdot \avg{f}
$$
If $x_i$ are uniformly distributed $I_N \rightarrow I$ for $N \rightarrow +
\infty$ by the law of large numbers, whereas the sample variance can be
\infty$ by the law of large numbers, whereas the integral variance can be
estimated as:
$$
\sigma^2_f = \frac{1}{N - 1}
\sum_{i = 1}^N \left( f(x_i) - \avg{f} \right)^2
\et
\et
\sigma^2_I = \frac{V^2}{N^2} \sum_{i = 1}^N
\sigma^2_f = \frac{V^2}{N} \sigma^2_f
$$
Thus, the error decreases as $1/\sqrt{N}$.
Unlike in deterministic methods, the error estimate is not a strict bound:
random sampling may not cover all the important features of the integrand and
@ -99,15 +96,15 @@ 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.
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:
$$
\sigma = \frac{a}{x^b}
$$
As can be seen in @fig:err_fit, the obtained result confirmes the expected value
of $b^{\text{exp}} = 0.5$, having found $b \sim 0.499$.
The obtained result (@fig:err_fit) confirms the expected result
$b^{\text{exp}} = 0.5$, with an observed value of $\sim 0.499$.
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
@ -125,11 +122,11 @@ partitioned into subpopulations. Stratification, indeed, is the process of
dividing the primary sample into subgroups (strata) before sampling
within each stratum.
Given a sample $\{x_j\}_i$ of the $i$-th strata, its mean $\bar{x}_i$ and
variance ${\sigma^2_x}_i$, are given by
variance ${\sigma^2_x}_i$, are given by:
$$
\bar{x}_i = \frac{1}{n_i} \sum_j x_j
$$
and from:
and
$$
\sigma_i^2 = \frac{1}{n_i - 1} \sum_j \left( x_j - \bar{x}_i \right)^2
\thus
@ -137,8 +134,8 @@ $$
$$
where:
- $j$ runs over the points $x_j$ of the sample
- $n_i$ is the size of the sample
- $j$ runs over the points $x_j$ of the sample,
- $n_i$ is the size of the sample,
- $\sigma_i^2$ is the variance associated to every point of the $i$-th
stratum.
@ -169,12 +166,11 @@ population. For examples, see [@ridder17].
The MISER technique aims at reducing the integration error through the use of
recursive stratified sampling.
As stated before, according to the law of large numbers, for a large number of
As stated before, according to the law of large numbers, for a great number of
extracted points, the estimation of the integral $I$ can be computed as:
$$
I= V \cdot \avg{f}
$$
Since $V$ is known (in this case, $V = 1$), it is sufficient to estimate
$\avg{f}$.
@ -191,26 +187,23 @@ is given by:
$$
\sigma^2 = \frac{\sigma_a^2}{4n_a} + \frac{\sigma_b^2}{4n_b}
$$
It can be shown that this variance is minimized by distributing the points such
that:
$$
\frac{n_a}{n_a + n_b} = \frac{\sigma_a}{\sigma_a + \sigma_b}
$$
Hence, the smallest error estimate is obtained by allocating sample points in
proportion to the standard deviation of the function in each sub-region.
The whole integral estimate and its variance are therefore given by:
$$
I = V \cdot \avg{f} \et \sigma_I^2 = V^2 \cdot \sigma^2
$$
When implemented, MISER is in fact a recursive method. First, all the possible
bisections of $\Omega$ are tested and the one which minimizes the combined
variance of the two sub-regions is selected. In order to speed up the
algorithm, the variance in the sub-regions is estimated with a fraction of the
total number of available points (function calls), in GSL it default to 0.1.
The remaining points are allocated to the sub-regions using the formula for
total number of available points (function calls), in GSL it is default set to
0.1. The remaining points are allocated to the sub-regions using the formula for
$n_a$ and $n_b$, once the variances are computed.
This procedure is then repeated recursively for each of the two half-regions
@ -219,15 +212,18 @@ from the best bisection. When the allocated calls for a region running out
The final individual values and their error estimates are then combined upwards
to give an overall result and an estimate of its error [@sayah19].
![Estimations $I^{\text{oss}}$ of the integral $I$ obtained
for the three implemented method for different values of
function calls. Errorbars showing their estimated
uncertainties.](images/5-MC_MC_MI.pdf){#fig:miser-iter}
![Estimations $I^{\text{oss}}$ of the integral $I$ obtained for plain MC and
MISER for different values of function calls. Errorbars showing their
estimated uncertainties.](images/5-MC_MC_MI.pdf){#fig:miser-iter}
Results for this particular sample are shown in black in @fig:miser-iter and
some of them are listed in @tbl:miser-res. Except for the first very little
number of calls, the improvement with respect to the Plain MC technique (in
red) is appreciable.
compared with the plain MC results (in red). Some of them are listed in
@tbl:miser-res. Except for the first very little number of calls, the
improvement with respect to the Plain MC technique is appreciable.
The convergence is much faster than a plain MC: at 500'000 function calls, the
estimate agrees with the exact integral to the fifth decimal place. Once again,
the standard deviation and the difference share the same magnitude.
--------------------------------------------------------------------
calls $I^{\text{oss}}$ $\sigma$ diff
@ -242,55 +238,45 @@ calls $I^{\text{oss}}$ $\sigma$ diff
Table: MISER results with different numbers of function calls. Differences
between computed and exact values are given in diff. {#tbl:miser-res}
The convergence is much faster than a plain MC: at 500'000 function calls, the
estimate agrees with the exact integral to the fifth decimal place. Once again,
the standard deviation and the difference share the same magnitude.
## Importance sampling
In Monte Carlo methods, importance sampling is a technique which samples points
from distribution whose shape is close to the integrand $f$ itself. In this way
from distribution whose shape is close to the integrand $f$ itself. This way,
the points cluster in the regions that make the largest contribution to the
integral $\int f(x)dx$ and consequently decrease the variance.
In a plain MC the points are sampled uniformly, so their probability
density is given by
density is given by:
$$
g(x) = \frac{1}{V} \quad \forall x \in \Omega
$$
and the integral can be written as
and the integral can be written as:
$$
I = \int_\Omega dx f(x) = V \int_\Omega f(x) \frac{1}{V}dx
\approx V \avg{f}
$$
More generally, consider a distribution $h(x)$ and similarly do
More generally, consider a distribution $h(x)$ and similarly do:
$$
I
= \int_\Omega dx f(x)
= \int_\Omega dx \, \frac{f(x)}{h(x)} \, h(x)
= \Exp \left[ \frac{f}{h}, h \right]
$$
where $\Exp[X, h]$ is the expected value of $X$ wrt $h$.
Also note that $h$ has to vanish outside $\Omega$ for this to hold.
As anticipated, to reduce the variance $h$ must be close to $f$.
Assuming they are proportional, $h(x) = \alpha |f(x)|$, it follows
that:
where $\Exp[X, h]$ is the expected value of $X$ wrt $h$. Also note that $h$ has
to vanish outside $\Omega$ for this to hold.
To reduce the variance, as anticipated, $h$ must be close to $f$. Assuming they
are proportional, $h(x) = \alpha |f(x)|$, it follows that:
$$
\Exp \left[ \frac{f}{h}, h \right] = \frac{1}{\alpha}
\et
\Var \left[ \frac{f}{h}, h \right] = 0
$$
For the expected value to give the original $I$, the proportionality constant
must be taken to be $I^{-1}$, meaning:
$$
h(z) = \frac{1}{I}\, |f(z)|
$$
The sampling from this $h$ would produce a perfect result with zero variance.
Of course, this is nonsense: if $I$ is known in advance, there would be no need
to do a Monte Carlo integration to begin with. Nonetheless, this example serves
@ -316,7 +302,7 @@ different subregions with an iterative method, namely:
- the volume $V$ is divided into $N$ intervals of width $\Delta x_i =
\Delta x \, \forall \, i$, where $N$ is limited by the computer storage
space available and must be held constant from iteration to iteration.
(In GSL this default to $N = 50$);
(In GSL this defaults to $N = 50$);
- each interval is then divided into $m_i + 1$ subintervals, where:
$$
@ -326,8 +312,8 @@ different subregions with an iterative method, namely:
where $j$ runs over all the intervals and $\bar{f}_i$ is the average value
of $f$ in the interval. Hence, $m_i$ is therefore a measure of the
"importance" of the interval with respect to the others: the higher
$\bar{f}_i$, the higher $m_i$. The constant $K$ is called stiffness.
It is defaults 1.5 in GSL;
$\bar{f}_i$, the higher $m_i$. The constant $K$, called stiffness, defaults
to 1.5 in GSL;
- as it is desirable to restore the number of intervals to its original value
$N$, groups of the new intervals must be merged into larger intervals, the
@ -337,10 +323,10 @@ different subregions with an iterative method, namely:
- the function is integrated with a plain MC method in each interval
and the sum of the integrals is taken as the $j$-th estimate of $I$. Its
error is given the sum of the variances in each interval.
error is given by the sum of the variances in each interval;
- the new grid is used and further refined in subsequent iterations.
By default, the number of iterations 5 in GSL.
- the new grid is further refined in subsequent iterations. By default, the
number of iterations is 5 in GSL.
The final estimate of the integral $I$ and its error
$\sigma_I$ are made based on weighted average:
@ -359,27 +345,13 @@ $$
\chi^2_i = \sum_{j \le i}
\frac{(I_j - \avg{I})^2}{\sigma_j^2}
$$
While performing the iterations, if the value of $\chi_r^2$ exceed 1.5, the
routine stops since is not making progress.
While performing the iterations, if the value of $\chi_r^2$ exceeds 1.5, the
routine stops since it is not making progress.
Clearly, a better estimation is achieved with a greater number of function
calls. For this particular sample, the most accurate results are shown in
@fig:vegas-iter and some of them are listed in @tbl:vegas-res.
----------------------------------------------------------------------------------------------
calls $I^{\text{oss}}$ $\sigma$ diff $\chi_r^2$
------------------ ------------------ ------------------ ------------------ ------------------
\num{5e5} 1.7182818281 0.0000000012 0.0000000004 1.457
\num{5e6} 1.7182818284 0.0000000000 0.0000000001 0.632
\num{5e7} 1.7182818285 0.0000000000 0.0000000000 0.884
----------------------------------------------------------------------------------------------
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}
@ -391,6 +363,19 @@ 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}.
----------------------------------------------------------------------------------------------
calls $I^{\text{oss}}$ $\sigma$ diff $\chi_r^2$
------------------ ------------------ ------------------ ------------------ ------------------
\num{5e5} 1.7182818281 0.0000000012 0.0000000004 1.457
\num{5e6} 1.7182818284 0.0000000000 0.0000000001 0.632
\num{5e7} 1.7182818285 0.0000000000 0.0000000000 0.884
----------------------------------------------------------------------------------------------
Table: Best VEGAS results with different numbers of
function calls. {#tbl:vegas-res}
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.