Giu/analistica
1
0
Fork 0
Code Issues Pull Requests Releases Wiki Activity

ex-5: review

Browse Source
This commit is contained in:
Michele Guerini Rocco 2020-05-31 17:17:00 +02:00
parent 686d2d4554
commit d0d03d521e
1 changed files with 164 additions and 174 deletions
Show Stats Download Patch File Download Diff File Expand all files Collapse all files

322
notes/sections/5.md
View File

@ -31,9 +31,9 @@ techniques.
whose exact value is 1.7182818285...
The three most popular Monte Carlo (MC) methods where applied: plain MC, Miser
and Vegas. Besides this popularity fact, these three method were chosen for
being implemented in the GSL libraries `gsl_monte_plain`, `gsl_monte_miser` and
`gsl_monte_vegas` respectively.
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.
## Plain Monte Carlo
@ -45,39 +45,42 @@ $$
\with V = \int\limits_{\Omega} dx
$$
the simplest MC method approach is to sample $N$ points $x_i$ evenly distributed
in $V$ and approx $I$ as:
the simplest MC method approach is to sample $N$ points $x_i$ in $V$ and
approximate $I$ as:
$$
I \sim I_N = \frac{V}{N} \sum_{i=1}^N f(x_i) = V \cdot \langle f \rangle
I \approx I_N = \frac{V}{N} \sum_{i=1}^N f(x_i) = V \cdot \avg{f}
$$
with $I_N \rightarrow I$ for $N \rightarrow + \infty$ because of the law of
large numbers, whereas the sample variance can be estimated as:
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
estimated as:
$$
\sigma^2_f = \frac{1}{N - 1} \sum_{i = 1}^N \left( f(x_i) - \langle f
\rangle \right)^2 \et \sigma^2_I = \frac{V^2}{N^2} \sum_{i = 1}^N
\sigma^2_f = \frac{1}{N - 1}
\sum_{i = 1}^N \left( f(x_i) - \avg{f} \right)^2
\et
\sigma^2_I = \frac{V^2}{N^2} \sum_{i = 1}^N
\sigma^2_f = \frac{V^2}{N} \sigma^2_f
$$
Thus, the error decreases as $1/\sqrt{N}$.
Unlike in deterministic methods, the estimate of the error is not a strict error
bound: random sampling may not uncover all the important features of the
integrand and this can result in an underestimate of the error.
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
this can result in an underestimation of the error.
In this case $f(x) = e^{x}$ and $\Omega = [0,1]$, hence $V = 1$.
Since the proximity of $I_N$ to $I$ is related to $N$, the accuracy of the
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:MC and
@fig:MI, results obtained with the plain MC method are shown in red. In
@tbl:MC, some of them are listed: the estimated integrals $I^{\text{oss}}$ are
compared to the expected value $I$ and the differences diff between them are
given.
are executed when the iterative method is implemented. In @fig:plain-mc-iter
and @fig:miser-iter, results obtained with the plain MC method are shown in
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:MC}
around the correct value.](images/5-MC_MC.pdf){#fig:plain-mc-iter}
---------------------------------------------------------------------------
calls $I^{\text{oss}}$ $\sigma$ diff
@ -91,12 +94,12 @@ calls $I^{\text{oss}}$ $\sigma$ diff
Table: Some MC results with three different numbers of function calls.
Differences between computed and exact values are given in
diff. {#tbl:MC}
diff. {#tbl:plain-mc-res}
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 digit, meaning that
this method shows a really slow convergence. In fact, since the $\sigma$s
$I^{\text{oss}}$ 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:
$$
\begin{cases}
@ -110,46 +113,48 @@ $$
\frac{\sigma_1}{\sigma_2} = 9.9965508 \sim 10 = \sqrt{\frac{C_2}{C_1}}
$$
if an error of $\sim 1^{-n}$ is required, a number $\propto 10^{2n}$ of
function calls should be executed, meaning that for $\sigma \sim 1^{-10}
\rightarrow C = \num{1e20}$, which would be impractical.
For an error of $10^{-n}$, a number $\propto 10^{2n}$ of function calls is
needed. To compute an integral within double precision,
an impossibly large number of $\sigma \sim 10^{32}$ calls is needed,
which makes this method unpractical for high-precision applications.
## Stratified sampling
In statistics, stratified sampling is a method of sampling from a population
partitioned into subpopulations. Stratification, indeed, is the process of
dividing the primary sample into subgroups (strata) before sampling random
dividing the primary sample into subgroups (strata) before sampling
within each stratum.
Given the mean $\bar{x}_i$ and variance ${\sigma^2_x}_i$ of an entity $x$
sorted with simple random sampling in the $i^{\text{th}}$ strata, such as:
Given a sample $\{x_j\}_i$ of the $i$-th strata, its mean $\bar{x}_i$ and
variance ${\sigma^2_x}_i$, are given by
$$
\bar{x}_i = \frac{1}{n_i} \sum_j x_j
$$
and:
and from:
$$
\sigma_i^2 = \frac{1}{n_i - 1} \sum_j \left( x_j - \bar{x}_i \right)^2
\thus
{\sigma^2_x}_i = \frac{1}{n_i^2} \sum_j \sigma_i^2 = \frac{\sigma_i^2}{n_i}
$$
where:
- $j$ runs over the points $x_j$ sampled in the $i^{\text{th}}$ stratum,
- $n_i$ is the number of points sorted in it,
- $\sigma_i^2$ is the variance associated with the $j^{\text{th}}$ point.
- $j$ runs over the points $x_j$ of the sample
- $n_i$ is the size of the sample
- $\sigma_i^2$ is the variance associated to every point of the $i$-th
stratum.
then, the mean $\bar{x}$ and variance $\sigma_x^2$ estimated with stratified
sampling for the whole population are:
An estimation of the mean $\bar{x}$ and variance $\sigma_x^2$ for the whole
population are then given by the stratified sampling as follows:
$$
\bar{x} = \frac{1}{N} \sum_i N_i \bar{x}_i \et
\sigma_x^2 = \sum_i \left( \frac{N_i}{N} \right)^2 {\sigma_x}^2_i
= \sum_i \left( \frac{N_i}{N} \right)^2 \frac{\sigma^2_i}{n_i}
$$
where:
where $i$ runs over the strata, $N_i$ is the weight of the $i^{\text{th}}$
stratum and $N$ is the sum of all strata weights.
- $i$ runs over the strata,
- $N_i$ is the weight of the $i$-th stratum
- $N$ is the sum of all strata weights.
In practical terms, it can produce a weighted mean that has less variability
than the arithmetic mean of a simple random sample of the whole population. In
@ -158,34 +163,31 @@ result will have a smaller error in estimation with respect to the one otherwise
obtained with simple sampling.
For this reason, stratified sampling is used as a method of variance reduction
when MC methods are used to estimate population statistics from a known
population [@ridder17].
population. For examples, see [@ridder17].
### MISER
The MISER technique aims to reduce the integration error through the use of
The MISER technique aims at reducing the integration error through the use of
recursive stratified sampling.
As stated before, according to the law of large numbers, for a large number of
extracted points, the estimation of the integral $I$ can be computed as:
$$
I= V \cdot \langle f \rangle
I= V \cdot \avg{f}
$$
Since $V$ is known (in this case, $V = 1$), it is sufficient to estimate
$\langle f \rangle$.
$\avg{f}$.
Consider two disjoint regions $a$ and $b$, such that $a \cup b = \Omega$, in
which $n_a$ and $n_b$ points are respectively uniformly sampled. Given the
Monte Carlo estimates of the means $\langle f \rangle_a$ and $\langle f
\rangle_b$ of those points and their variances $\sigma_a^2$ and $\sigma_b^2$, if
the weights $N_a$ and $N_b$ of $\langle f \rangle_a$ and $\langle f \rangle_b$
are chosen unitary, then the variance $\sigma^2$ of the combined estimate
$\langle f \rangle$:
Monte Carlo estimates of the means $\avg{f}_a$ and $\avg{f}_b$ of those points
and their variances $\sigma_a^2$ and $\sigma_b^2$, if the weights $N_a$ and
$N_b$ of $\avg{f}_a$ and $\avg{f}_b$ are chosen unitary, then the variance
$\sigma^2$ of the combined estimate $\avg{f}$:
$$
\langle f \rangle = \frac{1}{2} \left( \langle f \rangle_a
+ \langle f \rangle_b \right)
\avg{f} = \frac{1}{2} \left( \avg{f}_a + \avg{f}_b \right)
$$
is given by:
$$
\sigma^2 = \frac{\sigma_a^2}{4n_a} + \frac{\sigma_b^2}{4n_b}
@ -201,146 +203,122 @@ 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:
$$
I = V \cdot \langle f \rangle \et \sigma_I^2 = V^2 \cdot \sigma^2
I = V \cdot \avg{f} \et \sigma_I^2 = V^2 \cdot \sigma^2
$$
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) which is default set to 0.1.
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
$n_a$ and $n_b$, once the variances are computed.
The same procedure is then repeated recursively for each of the two half-spaces
from the best bisection. When less than 32*16 (default option) function calls
are available in a single section, the integral and the error in this section
are estimated using the plain Monte Carlo algorithm.
This procedure is then repeated recursively for each of the two half-regions
from the best bisection. When the allocated calls for a region running out
(less than 512 in GSL), the method falls back to a plain Monte Carlo.
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:MI}
![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}
Results for this particular sample are shown in black in @fig:MI and some of
them are listed in @tbl:MI. Except for the first very little number of calls,
the improvement with respect to the Plain MC technique (in red) is appreciable.
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.
---------------------------------------------------------------------------
--------------------------------------------------------------------
calls $I^{\text{oss}}$ $\sigma$ diff
------------------ ------------------ ------------------ ------------------
----------- ------------------ ------------------ ------------------
\num{5e5} 1.7182850738 0.0000021829 0.0000032453
\num{5e6} 1.7182819143 0.0000001024 0.0000000858
\num{5e7} 1.7182818221 0.0000000049 0.0000000064
---------------------------------------------------------------------------
--------------------------------------------------------------------
Table: MISER results with different numbers of function calls. Differences
between computed and exact values are given in diff. {#tbl:MI}
between computed and exact values are given in diff. {#tbl:miser-res}
This time the convergence is much faster: already with 500'000 number of points,
the correct results differs from the computed one at the fifth decimal digit.
Once again, the error lies always in the same order of magnitude of diff.
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 statistics, importance sampling is a method which samples points from the
probability distribution $f$ itself, so that the points cluster in the regions
that make the largest contribution to the integral.
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
the points cluster in the regions that make the largest contribution to the
integral $\int f(x)dx$ and consequently decrease the variance.
Remind that $I = V \cdot \langle f \rangle$ and therefore only $\langle f
\rangle$ is to be estimated. Consider a sample of $n$ points {$x_i$} generated
according to a probability distribution function $P$ which gives thereby the
following expected value:
In a plain MC the points are sampled uniformly, so their probability
density is given by
$$
E [x, P] = \frac{1}{n} \sum_i x_i
g(x) = \frac{1}{V} \quad \forall x \in \Omega
$$
with variance:
and the integral can be written as
$$
\sigma^2 [E, P] = \frac{\sigma^2 [x, P]}{n}
\with \sigma^2 [x, P] = \frac{1}{n -1} \sum_i \left( x_i - E [x, P] \right)^2
I = \int_\Omega dx f(x) = V \int_\Omega f(x) \frac{1}{V}dx
\approx V \avg{f}
$$
where $i$ runs over the sample.
In the case of plain MC, $\langle f \rangle$ is estimated as the expected
value of points {$f(x_i)$} sorted with $P (x_i) = 1 \quad \forall i$, since they
are evenly distributed in $\Omega$. The idea is to sample points from a
different distribution to lower the variance of $E[x, P]$, which results in
lowering $\sigma^2 [x, P]$. This is accomplished by choosing a random variable
$y$ and defining a new probability $P^{(y)}$ in order to satisfy:
More generally, consider a distribution $h(x)$ and similarly do
$$
E [x, P] = E \left[ \frac{x}{y}, P^{(y)} \right]
I
= \int_\Omega dx f(x)
= \int_\Omega dx \, \frac{f(x)}{h(x)} \, h(x)
= \Exp \left[ \frac{f}{h}, h \right]
$$
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.
which is to say:
As anticipated, to reduce the variance $h$ must be close to $f$.
Assuming they are proportional, $h(x) = \alpha |f(x)|$, it follows
that:
$$
I = \int \limits_{\Omega} dx f(x) =
\int \limits_{\Omega} dx \, \frac{f(x)}{g(x)} \, g(x)=
\int \limits_{\Omega} dx \, w(x) \, g(x)
$$
where $E \, \longleftrightarrow \, I$ and:
$$
\begin{cases}
f(x) \, \longleftrightarrow \, x \\
1 \, \longleftrightarrow \, P
\end{cases}
\Exp \left[ \frac{f}{h}, h \right] = \frac{1}{\alpha}
\et
\begin{cases}
g(x) \, \longleftrightarrow \, y = P^{(y)} \\
w(x) = \frac{f(x)}{g(x)} \, \longleftrightarrow \, \frac{x}{y}
\end{cases}
\Var \left[ \frac{f}{h}, h \right] = 0
$$
Where the symbol $\longleftrightarrow$ points out the connection between the
variables. The best variable $y$ would be:
For the expected value to give the original $I$, the proportionality constant
must be taken to be $I^{-1}$, meaning:
$$
y = \frac{x}{E [x, P]} \, \longleftrightarrow \, \frac{f(x)}{I}
\thus \frac{x}{y} = E [x, P]
h(z) = \frac{1}{I}\, |f(z)|
$$
and even a single sample under $P^{(y)}$ would be sufficient to give its value:
it can be shown that under this probability distribution the variance vanishes.
Obviously, it is not possible to take exactly this choice, since $E [x, P]$ is
not given a priori. However, this gives an insight into what importance sampling
does. In fact, given that:
$$
E [x, P] = \int \limits_{a = - \infty}^{a = + \infty}
da \, a P(x \in [a, a + da])
$$
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
to prove how variance reduction is achieved by sampling from an approximation
of the integrand.
the best probability $P^{(y)}$ redistributes the law of $x$ so that its samples
frequencies are sorted directly according to their weights in $E[x, P]$, namely:
$$
P^{(y}(x \in [a, a + da]) = \frac{1}{E [x, P]} a P (x \in [a, a + da])
$$
In conclusion, since certain values of $x$ have more impact on $E [x, P]$ than
others, these "important" values must be emphasized by sampling them more
In conclusion, since certain values of $x$ have more impact on $\Exp[f/h, h]$
than others, these "important" values must be emphasized by sampling them more
frequently. As a consequence, the estimator variance will be reduced.
### VEGAS
The VEGAS algorithm [@lepage78] of G. P. Lepage is based on importance
sampling. As stated before, it is in practice impossible to sample points from
the best distribution $h(x)$: only a good approximation can be achieved. The
VEGAS algorithm attempts this by building a histogram of the function $f$ in
different subregions with an iterative method, namely:
The VEGAS algorithm of Lepage is based on importance sampling. It aims to
reduce the integration error by concentrating points in the regions that make
the largest contribution to the integral.
- a fixed number of points (function calls) is generated uniformly in the
whole region;
As stated before, in practice it is impossible to sample points from the best
distribution $P^{(y)}$: only a good approximation can be achieved. In GSL, the
VEGAS algorithm approximates the distribution by histogramming the function $f$
in different subregions with a iterative method [@lepage78], namely:
- a fixed number of points (function calls) is evenly generated in the whole
region;
- the volume $V$ is divided into $N$ intervals with width $\Delta x_i =
- 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.
Default $N = 50$;
(In GSL this default to $N = 50$);
- each interval is then divided into $m_i + 1$ subintervals, where:
$$
m_i = K \frac{\bar{f}_i \Delta x_i}{\sum_j \bar{f}_j \Delta x_j}
@ -349,34 +327,46 @@ in different subregions with a iterative method [@lepage78], namely:
where $j$ runs over all the intervals and $\bar{f}_i$ is the average value
of $f$ in the interval. Hence, $m_i$ is therefore a measure of the
"importance" of the interval with respect to the others: the higher
$\bar{f}_i$, the higher $m_i$. The stiffness $K$ is default set to 1.5;
$\bar{f}_i$, the higher $m_i$. The constant $K$ is called stiffness.
It is defaults 1.5 in GSL;
- as it is desirable to restore the number of intervals to its original value
$N$, groups of the new intervals must be amalgamated into larger intervals,
the number of subintervals in each group being constant. The net effect is
$N$, groups of the new intervals must be merged into larger intervals, the
number of subintervals in each group being constant. The net effect is
to alter the intervals sizes, while keeping the total number constant, so
that the smallest intervals occur where $f$ is largest;
- the new grid is used and further refined in subsequent iterations until the
optimal grid has been obtained. By default, the number of iterations is set
to 5.
At the end, a cumulative estimate of the integral $I$ and its error $\sigma_I$
are made based on weighted average:
- the function is integrated with a plain MC method in each interval
and the sum of the integrals is taken as the $j$-th estimate of $I$. Its
error is given the sum of the variances in each interval.
- the new grid is used and further refined in subsequent iterations.
By default, the number of iterations 5 in GSL.
The final estimate of the integral $I$ and its error
$\sigma_I$ are made based on weighted average:
$$
I = \sigma_I^2 \sum_i \frac{I_i}{\sigma_i^2}
\avg{I} = \sigma_I^2 \sum_i \frac{I_i}{\sigma_i^2}
\with
\sigma_I^2 = \left( \sum_i \frac{1}{\sigma_i^2} \right)^{-1}
\frac{1}{\sigma_I^2} = \sum_i \frac{1}{\sigma_i^2}
$$
where $I_i$ and $\sigma_i$ are are the integral and standard deviation
estimated in each iteration.
The reliability of the result is asserted by a chi-squared per degree of
freedom $\chi_r^2$, which should be close to 1 for a good estimation. At a
given iteration $i$, the $\chi^2_i$ is computed as follows:
$$
\chi^2_i = \sum_{j \le i}
\frac{(I_j - \avg{I})^2}{\sigma_j^2}
$$
where $I_i$ and $\sigma_i$ are are the integral and standard deviation
estimated in each interval $\Delta x_i$ of the last iteration using the plain
MC technique.
For the results to be reliable, the chi-squared per degree of freedom
$\chi_r^2$ must be consistent with 1. While performing the eiterations, if
the $\chi_r^2$ value exceed 1.5, the cycle breaks.
While performing the iterations, if the value of $\chi_r^2$ exceed 1.5, the
routine stops since is not making progress.
Clearly, once again a better estimation is achieved with a greater number of
function calls. For this particular sample, the most accurate results are shown
in @fig:MI_VE and some of them are listed in @tbl:VEGAS.
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 $\chi_r^2$
@ -389,19 +379,19 @@ calls $I^{\text{oss}}$ $\sigma$ diff
----------------------------------------------------------------------------------------------
Table: Some VEGAS results with different numbers of
function calls. {#tbl:VEGAS}
function calls. {#tbl:vegas-res}
As can be appreciated in @fig:MI_VE, the VEGAS algorithm manages to compute
the integral value in a most accurate way with respect to MISER. The $\chi_r^2$
turns out to be close enough to 1 to guarantee a good estimation of $I$,
goodness which is also confirmed by the very small difference between estimation
and exact value, as shown in @tbl:VEGAS: with a number of \num{5e7} of function
calls, the difference is smaller than \num{1e-10}.
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
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:MI_VE}
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 one turned out to be the most powerful mean to
importance sampling, the last turned out to be the most powerful mean to
obtain a good estimation of the integrand.