From 05e97580b2ee00b0dae20c94a8e7c51352d064ef Mon Sep 17 00:00:00 2001 From: Shudipto <32040453+shudipto-amin@users.noreply.github.com> Date: Fri, 3 Dec 2021 08:25:04 -0500 Subject: [PATCH 01/36] New chapter "Metropolis" with Python Implementation (#929) * Add new chapter: metropolis_hastings; in python * Add markdown and python files for metropolis * Add image for metropolis * Update .gitignore, SUMMARY.md, and metropolis * Untrack .ipynb_checkpoints * Untrack ipynb_checkpoints * Fix algorithm steps list * Really fix markdown list * Add plot of P to chapter * Add metropolis animation and update plot of P(x) * Add minor update to chapter text * Generate gif and mp4 for random walk * Complete first draft! * Final version before Pull Request * Add metropolis citation * Remove unnecessary lines from code and bib file * Fix display of code in md * Fix random walk capitalization. * Apply Amaras' suggestions from code review This commit makes minor edits which resolve conflicts with PEP8 and improve readability of code. It also makes chapter sub-chapter of Monte Carlo. Co-authored-by: James Schloss Co-authored-by: Sammy Plat * Fix the code import lines in md file. * Change to in metropolis.py * Add probability section to metropolis.md However, this is temporary, as this section needs to go into a separate chapter. * Move Probability section in metropolis to own chapter. * Fix SUMMARY.md spelling mistake from previous commit * Add figures for probability distribution chapter * Update image of normal distribution * Finish first draft of probability chapter * Minor edits to distributions.md. * "Minor changes to distributions.md" * Complete the Example/Application section of metropolis. * Address most issues in review of PR#929 * Application sections created. * g() definition moved up to a better place * More discussion on g. * Missing citations provided * Clarity provided (hopefully) * Add image of 1D_particles * Update citations in md file * Add testing function to metropolis python code. Also modifies markdown to describe test. Also adds citation for RMSD. * Numpyfy metropolis f function and fix errors. * Implement generator to iterate. * Reformat output of test and nrmsd error reporting. * Add description of video and fix code display. * Update contents/probability/distributions/distributions.md Co-authored-by: James Schloss * Update contents/probability/distributions/distributions.md Co-authored-by: James Schloss * Update contents/probability/distributions/distributions.md Co-authored-by: James Schloss * Update contents/probability/distributions/distributions.md Co-authored-by: James Schloss * Update contents/probability/distributions/distributions.md Co-authored-by: James Schloss * Update contents/probability/distributions/distributions.md Co-authored-by: James Schloss * Update contents/probability/distributions/distributions.md Co-authored-by: James Schloss * Update contents/probability/distributions/distributions.md Co-authored-by: James Schloss * Update contents/probability/distributions/distributions.md Co-authored-by: James Schloss * Put sentences in `metropolis.md` on separate lines. * Put sentences in distributions.md chapter on separate lines. * Addresses issues raised by Leios' 1st review of probability chapter. * Adds intro to beginning. * Fixes plurality of die. * Clarifies some notation in discrete section. * Removes explanation of calculus. * Other minor edits. * Add minor edits. * Update contents/metropolis/metropolis.md title Co-authored-by: James Schloss * Simplify intro to contents/metropolis/metropolis.md Co-authored-by: James Schloss * Minor formatting to contents/metropolis/metropolis.md Co-authored-by: James Schloss * Fix spelling contents/metropolis/metropolis.md Co-authored-by: James Schloss * Simplify line 71 of contents/metropolis/metropolis.md Co-authored-by: James Schloss * Update contents/metropolis/metropolis.md Co-authored-by: James Schloss * Update example application in metropolis according to Leios comments. * Minor edit: contents/probability/distributions/distributions.md Co-authored-by: James Schloss * Minor edit: contents/probability/distributions/distributions.md Co-authored-by: James Schloss * Minor edit: contents/probability/distributions/distributions.md Co-authored-by: James Schloss * Minor edit: contents/probability/distributions/distributions.md Co-authored-by: James Schloss * Minor edit: contents/probability/distributions/distributions.md Co-authored-by: James Schloss * Apply minor suggestions from Leios' code review Mostly typos, spelling, punctuation, and grammar. Co-authored-by: James Schloss * Add minor edits from code Leios' review * Add minor edit * Apply minor edit suggestions from Leios Mostly punctuation and grammar. Co-authored-by: James Schloss * Add minor formatting, such as periods after equations. * Fix integer interval notation using hack. * Add minor edits missed previously in probability chapter. * Add minor edits to metropolis, mostly punctuation. * Apply minor edits from Leios Mostly grammar and sentence structure. Co-authored-by: James Schloss * Add intro line to Algorithm section of metropolis. * Add name to contributor.md * Apply suggestions from Leios * Minor edits for grammar and clarity. * Applies to both metropolis and probablity chapters. Co-authored-by: James Schloss * Apply suggestions from code review Co-authored-by: Shudipto <32040453+shudipto-amin@users.noreply.github.com> Co-authored-by: Kazi Shudipto Amin Co-authored-by: James Schloss Co-authored-by: Sammy Plat --- .gitignore | 2 +- CONTRIBUTORS.md | 1 + SUMMARY.md | 2 + contents/metropolis/code/python/metropolis.py | 90 ++++++ contents/metropolis/metropolis.md | 283 ++++++++++++++++++ contents/metropolis/res/1D_particles.png | Bin 0 -> 23564 bytes .../metropolis/res/animated_metropolis.gif | Bin 0 -> 1647776 bytes .../metropolis/res/animated_metropolis.mp4 | Bin 0 -> 577837 bytes .../metropolis/res/animated_random_walk.gif | Bin 0 -> 42487 bytes .../metropolis/res/animated_random_walk.mp4 | Bin 0 -> 20682 bytes .../metropolis/res/multiple_histograms.png | Bin 0 -> 191289 bytes contents/metropolis/res/plot_of_P.png | Bin 0 -> 158010 bytes .../distributions/distributions.md | 214 +++++++++++++ .../res/double_die_frequencies.png | Bin 0 -> 96012 bytes .../distributions/res/normal_distribution.png | Bin 0 -> 206149 bytes literature.bib | 101 +++++++ 16 files changed, 692 insertions(+), 1 deletion(-) create mode 100644 contents/metropolis/code/python/metropolis.py create mode 100644 contents/metropolis/metropolis.md create mode 100644 contents/metropolis/res/1D_particles.png create mode 100644 contents/metropolis/res/animated_metropolis.gif create mode 100644 contents/metropolis/res/animated_metropolis.mp4 create mode 100644 contents/metropolis/res/animated_random_walk.gif create mode 100644 contents/metropolis/res/animated_random_walk.mp4 create mode 100644 contents/metropolis/res/multiple_histograms.png create mode 100644 contents/metropolis/res/plot_of_P.png create mode 100644 contents/probability/distributions/distributions.md create mode 100644 contents/probability/distributions/res/double_die_frequencies.png create mode 100644 contents/probability/distributions/res/normal_distribution.png diff --git a/.gitignore b/.gitignore index 09a7ab178..5e1e79300 100644 --- a/.gitignore +++ b/.gitignore @@ -478,7 +478,7 @@ paket-files/ # Python Tools for Visual Studio (PTVS) __pycache__/ *.pyc - +*.ipynb_checkpoints* # Cake - Uncomment if you are using it # tools/** # !tools/packages.config diff --git a/CONTRIBUTORS.md b/CONTRIBUTORS.md index fc3113ecc..ca16b50f9 100644 --- a/CONTRIBUTORS.md +++ b/CONTRIBUTORS.md @@ -61,4 +61,5 @@ This file lists everyone, who contributed to this repo and wanted to show up her - Hugo Salou - Dimitri Belopopsky - Henrik Abel Christensen +- K. Shudipto Amin - Peanutbutter_Warrior diff --git a/SUMMARY.md b/SUMMARY.md index 4d4597b1c..644fccc56 100644 --- a/SUMMARY.md +++ b/SUMMARY.md @@ -20,9 +20,11 @@ * [Multiplication as a Convolution](contents/convolutions/multiplication/multiplication.md) * [Convolutions of Images (2D)](contents/convolutions/2d/2d.md) * [Convolutional Theorem](contents/convolutions/convolutional_theorem/convolutional_theorem.md) + * [Probability Distributions](contents/probability/distributions/distributions.md) * [Tree Traversal](contents/tree_traversal/tree_traversal.md) * [Euclidean Algorithm](contents/euclidean_algorithm/euclidean_algorithm.md) * [Monte Carlo](contents/monte_carlo_integration/monte_carlo_integration.md) + * [Metropolis](contents/metropolis/metropolis.md) * [Matrix Methods](contents/matrix_methods/matrix_methods.md) * [Gaussian Elimination](contents/gaussian_elimination/gaussian_elimination.md) * [Thomas Algorithm](contents/thomas_algorithm/thomas_algorithm.md) diff --git a/contents/metropolis/code/python/metropolis.py b/contents/metropolis/code/python/metropolis.py new file mode 100644 index 000000000..a4fc29794 --- /dev/null +++ b/contents/metropolis/code/python/metropolis.py @@ -0,0 +1,90 @@ +import numpy as np + + +def f(x, normalize=False): + ''' + Function proportional to target distribution, a sum of Gaussians. + For testing, set normalize to True, to get target distribution exactly. + ''' + # Gaussian heights, width parameters, and mean positions respectively: + a = np.array([10., 3., 1.]).reshape(3, 1) + b = np.array([ 4., 0.2, 2.]).reshape(3, 1) + xs = np.array([-4., -1., 5.]).reshape(3, 1) + + if normalize: + norm = (np.sqrt(np.pi) * (a / np.sqrt(b))).sum() + a /= norm + + return (a * np.exp(-b * (x - xs)**2)).sum(axis=0) + +def g(): + '''Random step vector.''' + return np.random.uniform(-1,1) + +def metropolis_step(x, f=f, g=g): + '''Perform one full iteration and return new position.''' + + x_proposed = x + g() + a = min(1, (f(x_proposed) / f(x)).item()) + + x_new = np.random.choice([x_proposed, x], p=[a, 1-a]) + + return x_new + +def metropolis_iterate(x0, num_steps): + '''Iterate metropolis algorithm for num_steps using iniital position x_0''' + + for n in range(num_steps): + if n == 0: + x = x0 + else: + x = metropolis_step(x) + yield x + + +def test_metropolis_iterate(num_steps, xmin, xmax, x0): + ''' + Calculate error in normalized density histogram of data + generated by metropolis_iterate() by using + normalized-root-mean-square-deviation metric. + ''' + + bin_width = 0.25 + bins = np.arange(xmin, xmax + bin_width/2, bin_width) + centers = np.arange(xmin + bin_width/2, xmax, bin_width) + + true_values = f(centers, normalize=True) + mean_value = np.mean(true_values - min(true_values)) + + x_dat = list(metropolis_iterate(x0, num_steps)) + heights, _ = np.histogram(x_dat, bins=bins, density=True) + + nmsd = np.average((heights - true_values)**2 / mean_value) + nrmsd = np.sqrt(nmsd) + + return nrmsd + + + +if __name__ == "__main__": + xmin, xmax = -10, 10 + x0 = np.random.uniform(xmin, xmax) + + num_steps = 50_000 + + x_dat = list(metropolis_iterate(x0, 50_000)) + + # Write data to file + output_string = "\n".join(str(x) for x in x_dat) + + with open("output.dat", "w") as out: + out.write(output_string) + out.write("\n") + + + # Testing + print(f"Testing with x0 = {x0:5.2f}") + print(f"{'num_steps':>10s} {'NRMSD':10s}") + for num_steps in (500, 5_000, 50_000): + nrmsd = test_metropolis_iterate(num_steps, xmin, xmax, x0) + print(f"{num_steps:10d} {nrmsd:5.1%}") diff --git a/contents/metropolis/metropolis.md b/contents/metropolis/metropolis.md new file mode 100644 index 000000000..961f1b14d --- /dev/null +++ b/contents/metropolis/metropolis.md @@ -0,0 +1,283 @@ +# The Metropolis Algorithm + +The [Monte Carlo Integration](../monte_carlo_integration/monte_carlo_integration.html) method uses random numbers to approximate the area of pretty much any shape we choose. +The Metropolis algorithm {{ "metropolis1953equation" | cite }} is a slightly more advanced Monte Carlo method which uses random numbers to approximate a [probability distribution](../probability/distributions/distributions.md): + +$$ +P(\mathbf{x}) = \frac{f(\mathbf{x})}{\displaystyle\int_D f(\mathbf{x})d\mathbf{x}}, +$$ + +where $$D$$ is the domain of $$P(\mathbf{x})$$, i.e., all possible values of the $$\mathbf{x}$$ for which $$P(\mathbf{x})$$ is defined. +$$f(\mathbf{x})$$ is a function that is proportional to $$P(x)$$, such as a statistical frequency distribution which counts the number of occurences of each $$\mathbf{x}$$. +The integral in the denominator is the __normalization factor__ which ensures that the sum of all probabilities is unity, i.e., +$$ +\int_D P(\mathbf{x}) d\mathbf{x} = 1. +$$ +A one-dimensional example is the __normal distribution__, or __Gaussian distribution__, given by + +$$ +P(x) = \frac{e^{-x^2}}{\displaystyle\int_{-\infty}^{\infty} e^{-x^2} dx} = \frac{1}{\sqrt{\pi}} e^{-x^2}. +$$ + + +In practice, it's often easy for us to know $$f(x)$$, but the integral in the denominator can be quite difficult to calculate, even numerically. +This is especially true when the coordinates ($$\mathbf{x}$$) are multidimensional, and $$f(\mathbf{x})$$ is an expensive calculation, as we shall see in the examples below. + +## An example application + +One example of a complicated probability function arises when considering a physical system of $$N$$ particles. +These could be atoms, molecules, or even star systems! +For such systems, we can usually describe the __potential energy__ {{ "potential_energy_wiki" | cite }} of the system as a function of the coordinates of all particles, $$\mathbf{x}$$, + +$$ +E(\mathbf{x}) = E(x_1, y_1, z_1, x_2, y_2, z_2, ... ,x_N, y_N, z_N), +$$ + +where $$x_i, y_i, z_i$$ are the spatial coordinates of particle $$i$$. +So altogether there are $$3N$$ coordinates – making $$E(\mathbf{x})$$ a $$3N$$ dimensional function, which can be a computationally intensive calculation on it's own. But it doesn't end there! + +The physicist Ludwig Boltzmann {{ "ludwig_boltzmann_wiki" | cite }} discovered that when such a system is in equilibrium at some temperature $$T$$, you can describe the probability density of the system for any set of coordinates $$\mathbf{x}$$ using, {{ "boltzmann_distribution_wiki" | cite }} + +$$ +P(\mathbf{x}) = \frac{\displaystyle \exp\left[{\displaystyle\frac{-E(\mathbf{x})}{T} } \right]} {Q}, +$$ + +where the numerator is called the __Boltzmann factor__, and $$Q$$ is the [normalization constant](../probability/distributions/distributions.md), + +$$ +Q = \int_D \exp\left[{\displaystyle\frac{-E(\mathbf{x})}{T} } \right] d\mathbf{x}. +$$ + +We can see now that the probability density function is a difficult calculation, particularly because of $$Q$$. +Almost always, no analytical solution exists to the integral in $$Q$$, and the numerical integration is unfeasible. + +To see that $$Q$$ is unfeasible to calculate, imagine there are just $$10$$ particles which all exist in a $$1$$D world, restricted to a line segment. + +

+ <FIG> 1D particles +

+ +Let's assume that the particles _interact_, meaning that the position of one particle affects that of another. +This could be the case, for example, if all the particles were charged, and so they would be repelling or attracting each other. +This means that the energy $$E(\mathbf{x}) = E(x_1,...,x_{10})$$ of the system is a $$10$$D function, and it would not be possible to simplify it any further due to the interactions. +Thus, the Boltzmann factor, $$\exp\left[-E(\mathbf{x})/T\right]$$, is also a $$10$$D function. To calculate $$Q$$, we would have to integrate the Boltzmann factor $$10$$ times, one for each coordinate, + +$$ +Q = \int_{x_1} \dots \int_{x_{10}} \exp\left[\frac{-E(x_1,\dots x_{10})}{T}\right]\ dx_1\dots dx_{10}. +$$ + +In most cases, there is no known analytical expression for the above integral, so it has to be done numerically. +To do so, imagine that we divide the $$1$$D line segment into only $$50$$ different intervals, allowing each particle to take on $$50$$ different positions. +This is equivalent to dividing the length of a football field into intervals of about $$2$$ meters – not a resolution you'd wanna watch a game in! +Even with such poor resolution, the number of different combinations of positions is $$10^{50}$$ – a colossal number indeed. +To see how large this number is, imagine that a single computation of $$E(\mathbf{x})$$ took only $$1$$ nanosecond on a single processor, which is much faster than most energy calculations for physical systems in practice. + With that speed, it would require $$10^{41}$$ seconds on a single processor to calculate $$Q$$ – which means that _even_ with all the processors in the world running in parallel (there could be billions or trillions of them), calculating $$Q$$ would still take longer than the age of the universe – by many orders of magnitude! + +What's really powerful about the Metropolis approach is that you don't need to know the probability function itself. +Instead, you just need a function which is _proportional_ to it. +What this means for the Boltzmann distribution is that you only need to know the term, + +$$ +f(\mathbf{x}) = \exp\left[{\displaystyle\frac{-E(\mathbf{x})}{T} } \right]. +$$ + +The Metropolis algorithm can bypass the calculation of $$Q$$ altogether and use $$f(x)$$ to generate a distribution of $$x$$ which follows the probability density $$P(x)$$. +In other words, it can sample values of $$x$$ in such away that the probability of sampling $$x$$ will follow the actual distribution $$P(x)$$. +Thus, if Metropolis was used to sample from $$x$$, the number of occurences of $$x$$ would be proportional to $$P(x)$$. +Numerical normalization can then be done by using the total number of samples instead of performing an integration. +This fact dramatically reduces the number of calculations needed to approximate the probability distribution. + +Finally, the Metropolis algorithm can be modified or implemented in other methods, and forms the basis of many advanced sampling algorithms. +The most popular is probably the Metropolis-Hastings algorithm {{ "hastings1970monte" | cite }} which is fundamentally the same. +Some other algorithms that use this method are Metropolis-adjusted Langevin algorithm {{ "mala_wiki" | cite }}, and Hamiltonian Monte Carlo {{ "hmc_wiki" | cite }}, to name a few. +They are often used for physical systems that follow a Boltzmann distribution. + + +## A Random Walk in One Dimension + +In the rest of this chapter, we will look at $$1$$D examples to understand the Metropolis algorithm. +Although the algorithm is not particularly efficient in just one dimension, it is much easier to understand in one dimension than in multiple dimensions. +The Metropolis algorithm is very similar to a random walk, so let's first see how we can get a distribution from a random walk. + +
+ +
+ +The dot in the figure above is a "walker", whose initial position is $$x=0$$. +The step size, $$g$$, is a random number in the interval $$(-1, 1)$$. +To get the next position of the walker, we simply generate $$g$$ and add it to the current position. +To get a distribution of $$x$$ from this walk, we can divide the domain into discrete locations or "bins" and count how often the walker visits each bin. +Each time it visits a bin, the frequency for that bin goes up by one. +Over many iterations, we get a frequency distribution of $$x$$. + +## A Random Walk With an Acceptance Criterion + +The Metropolis algorithm works in a similar way to the random walk, but differs crucially in one way – after choosing a random step for the walker, a decision is made about whether to __accept__ or __reject__ the step based on the function $$f(x)$$. +To understand how this works, let's call $$x_t$$ the position before the step, and $$x'$$ the position after it. +We then define the probability of __accepting the step__ to be + +$$ +A = \min \left(\frac{f(x')}{f(x_t)}, 1\right). +$$ + +The $$\min$$ function above implies that $$A=1$$ if $$f(x') \gt f(x_t)$$, which means that the move will __always__ be accepted if it is toward a higher probability position. +Otherwise, it will be accepted with a probability of $$f(x') / f(x_t)$$. +If we create a histogram of this walk for some arbitrary target function $$P(x)$$, we can see from the figure below that the frequency starts to look very much like it after many iterations! + +
+ +
+ +Although convergence occurs eventually, not all parts of the distribution achieve convergence quickly. +Note from the animation above, that the walker very quickly replicates the distribution of the two peaks on the left, but takes quite a while to even reach the third peak to the right. +This is because there is a long low probability region between the third peak and second peak that acts as a "barrier." +This may not necessarily be a bad thing – sometimes one might want to estimate how long something takes to transition from one state to another, and often these peaks represent such 'states'. +So averaging over many metropolis runs may give some estimate of these transition times. +If global sampling is the goal, the process of exploration could be sped up by choosing larger step sizes for the walker, for example by choosing step size $$g$$ from an interval like $$(-3,3)$$ instead of $$(-1,1)$$. + + +## The Algorithm for a One Dimensional Example + +Now let's dive into the actual algorithm with some example code! + +### The Initial Setup + +Let our target distribution be +$$ +P(x) = \frac{f(x)}{\int_{-10}^{10} f(x)}, +$$ + +where $$f(x)$$ is the same function we have shown above and is given by +$$ +f(x) = 10e^{-4(x+4)^2} + 3e^{-0.2(x+1)^2} + e^{-2(x-5)^2}. +$$ + +The code for defining this function is given below. + +{% method %} +{% sample lang="py" %} +[import:4-18, lang:"python"](code/python/metropolis.py) +{% endmethod %} + +Since this is an easy function to integrate, and hence get our target distribution $$P(x)$$ directly, we can use it to verify the Metropolis algorithm. +The plot of $$P(x)$$ in the figure below shows three different peaks of varying width and height, with some overlap as well. + +

+ <FIG> Plot of P(x) +

+ +Next, we define our walker's symmetric step generating function. +As in the random walk example, we will use a random real number between $$-1$$ and $$+1$$ as the step size. + +{% method %} +{% sample lang="py" %} +[import:20-22, lang:"python"](code/python/metropolis.py) +{% endmethod %} + +However, $$g$$ can be any function symmetric about $$0$$ for the above algorithm to work. +For example, it can be a number chosen randomly from a discrete list, such as $$[ -3, -1, -1, +1, +1, +3]$$. +It can also be a number chosen from a symmetric continuos distribution, like the Gaussian, $$e^{-x^2}$$. +In higher dimensions, the function should be spherically symmetric, such as a multidimensional Gaussian function, $$e^{-(x^2 +y^2 + ...)}$$. +Whatever function you choose, there are at least a couple of things to note: +1. If the function $$g$$ is discrete, you will only sample discrete values. +For example, if $$g$$ returns only $$-1$$ or $$+1$$, and nothing in between, you will sample only integer steps away from the initial $$x_0$$. +2. The average step size really matters! +A small step-size means the walker will carefully sample nearby regions more, but will walk more slowly, so might not be good at exploring far and wide. +On the other hand, a walker with a large step size may not sample nearby regions accurately – and actually has a higher chance of being rejected if the walker is already in a high probability region, since the acceptance ratio is more drastic for large steps. +The effect of step-size on the walker's efficiency is far from obvious! + +The question of how to choose an optimal $$g$$ is a research area on its own, and depends largely on what the goal of the sampling is. +Some techniques even use an "adaptive" method where $$g$$ is "trained" on-the-fly using a learning algorithm! +Some of these methods and others are discussed in Ref. {{ "rosenthal2011optimal" | cite }} and Ref. {{ "gareth2001optimal" | cite }}. +In a lot of cases, people just use trial and error, as the algorithm is not too difficult to implement. + +After chosing $$g$$, we are almost ready to iterate. +We just need to choose the domain of $$x$$, and an initial point for $$ x_0 $$ ($$x_t$$ at $$t = 0$$) chosen randomly from the domain of $$x$$. + +{% method %} +{% sample lang="py" %} +[import:70-71, lang:"python"](code/python/metropolis.py) +{% endmethod %} + +### How to Iterate + +1. Generate new proposed position $$x' = x_t + g$$. +2. Calculate the acceptance probability, +$$ +A = \min\left(1, \frac{f(x')}{f(x_t)}\right). +$$ +3. Accept proposal, $$x'$$ with probability $$A$$. If your programming language doesn't have a built-in method for this, + * Generate a random number $$u$$ between $$0$$ and $$1$$. + * If $$ u \leq A $$, then __accept__ move, and set new position, $$x_{t+1} = x' $$. + * Otherwise, __reject__ move, and set new position to current position, $$x_{t+1} = x_t $$. +4. Increment $$t \rightarrow t + 1$$ and repeat from step 1. + +The code for steps 1 to 3 is: + +{% method %} +{% sample lang="py" %} +[import:34-42, lang:"python"](code/python/metropolis.py) +{% endmethod %} + +The following plot shows the result of running the algorithm for different numbers of iterations ($$N$$), with the same initial position. +The histograms are normalized so that they integrate to $$1$$. +We can see the convergence toward $$P(x)$$ as we increase $$N$$. + +

+ <FIG> multiple histograms +

+ + +## Example Code +The following code puts everything together, and runs the Metropolis algorithm for a number of steps given by `num_steps`. +All the positions visited by the algorithm are then written to a file, which can be later read and fed into a histogram or other density calculating scheme. +The code also incorporates a few tests of the algorithm using the `test_metropolis_iterate` method. +This test will create a normalized density histogram from the generated data, and compare it to $$P(x)$$ using the Root Mean Square Deviations metric {{ "rmsd_wiki" | cite }}. + +{% method %} +{% sample lang="py" %} +[import, lang:"python"](code/python/metropolis.py) +{% endmethod %} + + + + + +### Bibliography + +{% references %} {% endreferences %} + +## License + +##### Code Examples + +The code examples are licensed under the MIT license (found in [LICENSE.md](https://github.com/algorithm-archivists/algorithm-archive/blob/master/LICENSE.md)). + + +##### Images/Graphics +- The animation "[Animated Random Walk](res/animated_random_walk.gif)" was created by [K. Shudipto Amin](https://github.com/shudipto-amin) and is licensed under the [Creative Commons Attribution-ShareAlike 4.0 International License](https://creativecommons.org/licenses/by-sa/4.0/legalcode). + +- The animation "[Animated Metropolis](res/animated_metropolis.gif)" was created by [K. Shudipto Amin](https://github.com/shudipto-amin) and is licensed under the [Creative Commons Attribution-ShareAlike 4.0 International License](https://creativecommons.org/licenses/by-sa/4.0/legalcode). + +- The image "[Plot of P(x)](res/plot_of_P.png)" was created by [K. Shudipto Amin](https://github.com/shudipto-amin) and is licensed under the [Creative Commons Attribution-ShareAlike 4.0 International License](https://creativecommons.org/licenses/by-sa/4.0/legalcode). + +- The image "[Multiple Histograms](res/multiple_histograms.png)" was created by [K. Shudipto Amin](https://github.com/shudipto-amin) and is licensed under the [Creative Commons Attribution-ShareAlike 4.0 International License](https://creativecommons.org/licenses/by-sa/4.0/legalcode). + +##### Text + +The text of this chapter was written by [K. Shudipto Amin](https://github.com/shudipto-amin) and is licensed under the [Creative Commons Attribution-ShareAlike 4.0 International License](https://creativecommons.org/licenses/by-sa/4.0/legalcode). + +[

](https://creativecommons.org/licenses/by-sa/4.0/) + +##### Pull Requests + +After initial licensing ([#560](https://github.com/algorithm-archivists/algorithm-archive/pull/560)), the following pull requests have modified the text or graphics of this chapter: +- none diff --git a/contents/metropolis/res/1D_particles.png b/contents/metropolis/res/1D_particles.png new file mode 100644 index 0000000000000000000000000000000000000000..3d8ab7d99efb58e5368cd06942138a4f2135f08e GIT binary patch literal 23564 zcmeFZWn7hA^frhs76Q@|lG5E}5Yo~O(jX<>pn^&xhY~5JrMpX|Q$V`(020!|p<&kT z^F04~XXZCw=F5C}`QT6G#J=}lajk1zYX?16l)iq2^a>Uh)^(Xj5-M0&7ntGi{g*Go zpS0%|ji_xA^BmeT|NTmW^s^})?ted4XQN{N=Qrut z@)~&m{U(zBe|^*YrvLq>zx(ueBO=M%f4(B3lEFKB?i_hV&*|~H-&Djz}tkVs+U#l8jL`^BS$pgVU2^ z^>tx1aiPd?vank(Zi|cD;qJP$mKHIE*9xb*yZZ-^KfkcP+ip*}%~V%Kk6SPe2L(Rz zdY4B_?RZXJBZpI4Mu*d^^A>K%RtV9+oV>ioQ(Zm1+p;lC8A4L+r(N-RW!I5YZn*@1 z&N==b&VN|`bO=^oS6^?d9wwNnPi8`u(zf{SF}^yFuvp?`jO+18fLx-o%nY 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