analistica/notes/sections/5.md

Exercise 5

The following integral is to be evaluated comparing different Monte Carlo techniques.

\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; \draw [cyclamen] (5,0) -- (5,2.7182818); \node [below] at (5,0) {1}; % Axis \draw [thick, <-] (0,4) -- (0,0); \draw [thick, ->] (-2,0) -- (7,0); \node [below right] at (7,0) {$x$}; \node [above left] at (0,4) {$e^{x}$}; % 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.

Plain Monte Carlo

When an integral I over a $n-$dimensional space \Omega of volume V of a function f has to be evaluated, that is:

  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 estimated as:

  \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 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.

{#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 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.

---

calls I^{\text{oss}} \sigma diff

---

\num{5e5} 1.7166435813 0.0006955691 0.0016382472

\num{5e6} 1.7181231109 0.0002200309 0.0001587176

\num{5e7} 1.7183387184 0.0000695809 0.0000568899

Table: Some MC results with three different numbers of function calls. Differences between computed and exact values are given in 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 place, meaning 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.
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.

{#fig:err_fit}

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 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

  \bar{x}_i = \frac{1}{n_i} \sum_j x_j

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 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.

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:

• 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 fact, if measurements within strata have lower standard deviation, the final 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. For examples, see [@ridder17].

MISER

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 \avg{f}

Since V is known (in this case, V = 1), it is sufficient to estimate \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 \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}:

  \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}

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 n_a and n_b, once the variances are computed.

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].

{#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.

---

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: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 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

  g(x) = \frac{1}{V} \quad \forall x \in \Omega

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

    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:

  \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 to prove how variance reduction is achieved by sampling from an approximation of the integrand.

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:

• a fixed number of points (function calls) is generated uniformly in the whole region;

• 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);

• 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}
  

  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;

• 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 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 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:

  \avg{I} = \sigma_I^2 \sum_i \frac{I_i}{\sigma_i^2}
  \with
  \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}

While performing the iterations, if the value of \chi_r^2 exceed 1.5, the routine stops since 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}

{#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 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}.

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.