{ "cells": [ { "cell_type": "markdown", "id": "7711bd97", "metadata": {}, "source": [ "# Exercise sheet\n", "\n", "Some general remarks about the exercises:\n", "* For your convenience functions from the lecture are included below. Feel free to reuse them without copying to the exercise solution box.\n", "* For each part of the exercise a solution box has been added, but you may insert additional boxes. Do not hesitate to add Markdown boxes for textual or LaTeX answers (via `Cell > Cell Type > Markdown`). But make sure to replace any part that says `YOUR CODE HERE` or `YOUR ANSWER HERE` and remove the `raise NotImplementedError()`.\n", "* Please make your code readable by humans (and not just by the Python interpreter): choose informative function and variable names and use consistent formatting. Feel free to check the [PEP 8 Style Guide for Python](https://www.python.org/dev/peps/pep-0008/) for the widely adopted coding conventions or [this guide for explanation](https://realpython.com/python-pep8/).\n", "* Make sure that the full notebook runs without errors before submitting your work. This you can do by selecting `Kernel > Restart & Run All` in the jupyter menu.\n", "* For some exercises test cases have been provided in a separate cell in the form of `assert` statements. When run, a successful test will give no output, whereas a failed test will display an error message.\n", "* Each sheet has 100 points worth of exercises. Note that only the grades of sheets number 2, 4, 6, 8 count towards the course examination. Submitting sheets 1, 3, 5, 7 & 9 is voluntary and their grades are just for feedback.\n", "\n", "Please fill in your name here:" ] }, { "cell_type": "code", "execution_count": 1, "id": "e39c230e", "metadata": {}, "outputs": [], "source": [ "NAME = \"Kees van Kempen\"\n", "NAMES_OF_COLLABORATORS = \"\"" ] }, { "cell_type": "markdown", "id": "2e7caa84", "metadata": {}, "source": [ "---" ] }, { "cell_type": "markdown", "id": "0c4606a8", "metadata": { "deletable": false, "editable": false, "nbgrader": { "cell_type": "markdown", "checksum": "1ba146ece4effd3e8dcf33fe63ec69dc", "grade": false, "grade_id": "cell-455926422ebe4f85", "locked": true, "schema_version": 3, "solution": false, "task": false } }, "source": [ "**Exercise sheet 4**\n", "\n", "Code from the lectures:" ] }, { "cell_type": "code", "execution_count": 2, "id": "6d3bb610", "metadata": { "deletable": false, "editable": false, "nbgrader": { "cell_type": "code", "checksum": "ac562d84531ae19e1d920c10a8861b8e", "grade": false, "grade_id": "cell-6b2d0465c2abaaa6", "locked": true, "schema_version": 3, "solution": false, "task": false } }, "outputs": [], "source": [ "import numpy as np\n", "import matplotlib.pylab as plt\n", "import networkx as nx\n", "\n", "rng = np.random.default_rng()\n", "%matplotlib inline\n", "\n", "def draw_transition_graph(P):\n", " # construct a directed graph directly from the matrix\n", " graph = nx.DiGraph(P) \n", " # draw it in such a way that edges in both directions are visible and have appropriate width\n", " nx.draw_networkx(graph,connectionstyle='arc3, rad = 0.15',width=[6*P[u,v] for u,v in graph.edges()])\n", "\n", "def sample_next(P,current):\n", " return rng.choice(len(P),p=P[current])\n", "\n", "def sample_chain(P,start,n):\n", " chain = [start]\n", " for _ in range(n):\n", " chain.append(sample_next(P,chain[-1]))\n", " return chain\n", "\n", "def stationary_distributions(P):\n", " eigenvalues, eigenvectors = np.linalg.eig(np.transpose(P))\n", " # make list of normalized eigenvectors for which the eigenvalue is very close to 1\n", " return [eigenvectors[:,i]/np.sum(eigenvectors[:,i]) for i in range(len(eigenvalues)) \n", " if np.abs(eigenvalues[i]-1) < 1e-10]\n", "\n", "def markov_sample_mean(P,start,function,n):\n", " total = 0\n", " state = start\n", " for _ in range(n):\n", " state = sample_next(P,state)\n", " total += function[state]\n", " return total/n" ] }, { "cell_type": "markdown", "id": "2f7814d8", "metadata": {}, "source": [ "## Markov Chain on a graph\n", "\n", "**(50 points)**\n", "\n", "The goal of this exercise is to use Metropolis-Hastings to sample a uniform vertex in a (finite, undirected) connected graph $G$. More precisely, the state space $\\Gamma = \\{0,\\ldots,n-1\\}$ is the set of vertices of a graph and the desired probability mass function is $\\pi(x) = 1/n$ for $x\\in\\Gamma$. The set of edges is denoted $E = \\{ \\{x_1,y_1\\}, \\ldots,\\{x_k,y_k\\}\\}$, $x_i,y_i\\in\\Gamma$, and we assume that there are no edges connecting a vertex with itself ($x_i\\neq y_i$) and there is at most one edge between any pair of vertices. The **neighbors** of a vertex $x$ are the vertices $y\\neq x$ such that $\\{x,y\\}\\in E$. The **degree** $d_x$ of a vertex $x$ is its number of neighbours. An example is the following graph:" ] }, { "cell_type": "code", "execution_count": 3, "id": "071e5617", "metadata": {}, "outputs": [ { "data": { "image/png": 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\n", 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" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "edges = [(0,1),(1,2),(0,3),(1,4),(2,5),(3,4),(4,5),(3,6),(4,7),(5,8),(6,7),(7,8),(5,7),(0,4)]\n", "example_graph = nx.Graph(edges)\n", "nx.draw(example_graph,with_labels=True)" ] }, { "cell_type": "markdown", "id": "952c97cd", "metadata": { "deletable": false, "editable": false, "nbgrader": { "cell_type": "markdown", "checksum": "a9ae9f2c0eb7e99b85fd90ae657a9202", "grade": false, "grade_id": "cell-a8413209ddb70ab0", "locked": true, "schema_version": 3, "solution": false, "task": false } }, "source": [ "A natural proposal transition matrix is $Q(x \\to y) = \\frac{1}{d_x} \\mathbf{1}_{\\{\\{x,y\\}\\in E\\}}$. In other words, when at $x$ the proposed next state is chosen uniformly among its neighbors.\n", "\n", "__(a)__ Write a function `sample_proposal` that, given a (networkX) graph and node $x$, samples $y$ according to transition matrix $Q(x \\to y)$. _Hint_: a useful Graph member function is [`neighbors`](https://networkx.org/documentation/stable/reference/classes/generated/networkx.Graph.neighbors.html). **(10 pts)**" ] }, { "cell_type": "code", "execution_count": 4, "id": "449da919", "metadata": { "deletable": false, "nbgrader": { "cell_type": "code", "checksum": "db0642dd7c9f53225d13fb9674c89a56", "grade": false, "grade_id": "cell-df840ef2dee49b9f", "locked": false, "schema_version": 3, "solution": true, "task": false } }, "outputs": [], "source": [ "def sample_proposal(graph,x):\n", " '''Pick a random node y from the neighbors of x in graph with uniform\n", " probability, according to Q.'''\n", " y_list = np.fromiter(graph.neighbors(x), dtype=int)\n", " return rng.choice(y_list)" ] }, { "cell_type": "code", "execution_count": 5, "id": "604ce3f4", "metadata": { "deletable": false, "editable": false, "nbgrader": { "cell_type": "code", "checksum": "690a774380e6a4383ccb5da2dee8a737", "grade": true, "grade_id": "cell-c70fec7da167b7be", "locked": true, "points": 10, "schema_version": 3, "solution": false, "task": false } }, "outputs": [], "source": [ "from nose.tools import assert_almost_equal\n", "assert sample_proposal(nx.Graph([(0,1)]),0)==1\n", "assert_almost_equal([sample_proposal(example_graph,3) for _ in range(1000)].count(4),333,delta=50)\n", "assert_almost_equal([sample_proposal(example_graph,8) for _ in range(1000)].count(5),500,delta=60)" ] }, { "cell_type": "markdown", "id": "f7a2e658", "metadata": { "deletable": false, "editable": false, "nbgrader": { "cell_type": "markdown", "checksum": "99c1e08eed2976f4625452d9eed6a5d5", "grade": false, "grade_id": "cell-4c74e7dae57e50e3", "locked": true, "schema_version": 3, "solution": false, "task": false } }, "source": [ "__(b)__ Let us consider the Markov chain corresponding to the transition matrix $Q(x \\to y)$. Produce a histogram of the states visited in the first ~20000 steps. Compare this to the exact stationary distribution found by the function `stationary_distributions` from the lecture applied to the transition matrix $Q$. _Hint_: another useful Graph member function is [`degree`](https://networkx.org/documentation/stable/reference/classes/generated/networkx.Graph.degree.html). **(15 pts)**" ] }, { "cell_type": "code", "execution_count": 6, "id": "f52fcbd5", "metadata": { "deletable": false, "nbgrader": { "cell_type": "code", "checksum": "44377fd9fa8cd0c3bc78ab03753db1ce", "grade": false, "grade_id": "cell-ca4f5c3685b72c8a", "locked": false, "schema_version": 3, "solution": true, "task": false } }, "outputs": [ { "data": { "image/png": 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\n", "text/plain": [ "
" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "def chain_Q_histogram(graph,start,k):\n", " '''Produce a histogram (a Numpy array of length equal to the number of \n", " nodes of graph) of the states visited (excluding initial state) by the \n", " Q Markov chain in the first k steps when started at start.'''\n", " n = graph.number_of_nodes()\n", " number_of_visits = np.zeros(n)\n", " x = start\n", " \n", " for _ in range(k):\n", " x = sample_proposal(graph, x)\n", " number_of_visits[x] += 1\n", " \n", " return number_of_visits\n", "\n", "def transition_matrix_Q(graph):\n", " '''Construct transition matrix Q from graph as two-dimensional Numpy array.'''\n", " n = example_graph.number_of_nodes()\n", " Q = np.zeros((n, n))\n", " for x in range(n): \n", " for k in example_graph.neighbors(x):\n", " Q[x, k] = 1/example_graph.degree(x)\n", " return Q\n", "\n", "# Compare histogram and stationary distribution in a plot\n", "x_start = 1\n", "k = 100000\n", "\n", "plt.figure()\n", "x_list = list(range(example_graph.number_of_nodes()))\n", "plt.bar(x_list, chain_Q_histogram(example_graph, x_start, k)/k, color=\"lightblue\", label=\"sampled\")\n", "plt.scatter(x_list, stationary_distributions(transition_matrix_Q(example_graph)), s=200, marker=\"_\", color=\"black\", label=\"theoretical\")\n", "plt.ylabel(\"visiting density\")\n", "plt.xlabel(\"node $x$\")\n", "plt.legend()\n", "plt.show()" ] }, { "cell_type": "code", "execution_count": 7, "id": "812cd2ef", "metadata": { "deletable": false, "editable": false, "nbgrader": { "cell_type": "code", "checksum": "c3e08e0577c953ca68fe8159cadc0634", "grade": true, "grade_id": "cell-bdea40714e10d0e8", "locked": true, "points": 15, "schema_version": 3, "solution": false, "task": false } }, "outputs": [], "source": [ "assert_almost_equal(transition_matrix_Q(example_graph)[3,4],1/3,delta=1e-9)\n", "assert_almost_equal(transition_matrix_Q(example_graph)[3,7],0.0,delta=1e-9)\n", "assert_almost_equal(transition_matrix_Q(example_graph)[2,2],0.0,delta=1e-9)\n", "assert_almost_equal(np.sum(transition_matrix_Q(example_graph)[7]),1.0,delta=1e-9)\n", "assert chain_Q_histogram(nx.Graph([(0,1)]),0,100)[1] == 50\n", "assert len(chain_Q_histogram(example_graph,0,100)) == example_graph.number_of_nodes()" ] }, { "cell_type": "markdown", "id": "8ebe4946", "metadata": { "deletable": false, "editable": false, "nbgrader": { "cell_type": "markdown", "checksum": "f4338397c60be0fc0069e30bec121faa", "grade": false, "grade_id": "cell-5a9debbf863b9e3b", "locked": true, "schema_version": 3, "solution": false, "task": false } }, "source": [ "__(c)__ Determine the appropriate Metropolis-Hastings acceptance probability $A(x \\to y)$ for $x\\neq y$\n", "and write a function that, given a graph and $x$, samples the next state with $y$ according to the Metropolis-Hastings transition matrix $P(x \\to y)$. **(10 pts)**" ] }, { "cell_type": "code", "execution_count": 8, "id": "37804448", "metadata": { "deletable": false, "nbgrader": { "cell_type": "code", "checksum": "639522f9ff68136165c0d9daa174b4ba", "grade": false, "grade_id": "cell-92e3b11a3af7eb99", "locked": false, "schema_version": 3, "solution": true, "task": false } }, "outputs": [], "source": [ "def acceptance_probability(graph,x,y):\n", " '''Compute A(x -> y) for the supplied graph (assuming x!=y).'''\n", " assert x != y\n", " \n", " Q = transition_matrix_Q(graph)\n", " n = graph.number_of_nodes()\n", " # We want to sample a uniform mass distribution pi:\n", " pi = np.ones(n)/n\n", " \n", " if Q[x, y] == 0:\n", " return 0\n", " else:\n", " A = pi[y]*Q[y, x]/( pi[x]*Q[x, y] )\n", " return np.min([1., A])\n", "\n", "def sample_next_state(graph,x):\n", " '''Return next random state y according to MH transition matrix P(x -> y).'''\n", " y = sample_proposal(graph, x)\n", " return y if rng.random() < acceptance_probability(graph, x, y) else x" ] }, { "cell_type": "code", "execution_count": 9, "id": "058d3728", "metadata": { "deletable": false, "editable": false, "nbgrader": { "cell_type": "code", "checksum": "6bf922f8787f02c80681390ce8a8f84c", "grade": true, "grade_id": "cell-fbfc2607999d0c99", "locked": true, "points": 10, "schema_version": 3, "solution": false, "task": false } }, "outputs": [], "source": [ "assert_almost_equal(acceptance_probability(example_graph,3,4),0.6,delta=1e-9)\n", "assert_almost_equal(acceptance_probability(example_graph,8,7),0.5,delta=1e-9)\n", "assert_almost_equal(acceptance_probability(example_graph,7,8),1.0,delta=1e-9)\n", "assert_almost_equal(acceptance_probability(nx.Graph([(0,1)]),0,1),1,delta=1e-9)" ] }, { "cell_type": "markdown", "id": "cbf205bf", "metadata": { "deletable": false, "editable": false, "nbgrader": { "cell_type": "markdown", "checksum": "da69b710a3b1986ac9f9997235c1a472", "grade": false, "grade_id": "cell-97f7d289177acf39", "locked": true, "schema_version": 3, "solution": false, "task": false } }, "source": [ "__(d)__ Do the same as in part (b) but now for the Markov chain corresponding to $P$. Verify that the histogram of the Markov chain approaches a flat distribution and corroborate this by calculating the explicit matrix $P$ and applying `stationary_distributions` to it. _Hint_: for determining the explicit matrix $P(x\\to y)$, remember that the formula $P(x\\to y) = Q(x\\to y)A(x\\to y)$ only holds for $x\\neq y$. What is $P(x\\to x)$? **(15 pts)**" ] }, { "cell_type": "code", "execution_count": 10, "id": "5de68351", "metadata": { "deletable": false, "nbgrader": { "cell_type": "code", "checksum": "bd25c9f6b39133973fd2d1594e210b30", "grade": false, "grade_id": "cell-3b7fde395331916d", "locked": false, "schema_version": 3, "solution": true, "task": false } }, "outputs": [], "source": [ "def chain_P_histogram(graph,start,n):\n", " '''Produce a histogram of the states visited (excluding initial state) \n", " by the P Markov chain in the first n steps when started at start.'''\n", " n = graph.number_of_nodes()\n", " number_of_visits = np.zeros(n)\n", " x = start\n", " \n", " for _ in range(k):\n", " x = sample_next_state(graph, x)\n", " number_of_visits[x] += 1\n", " \n", " return number_of_visits\n", "\n", "def transition_matrix_P(graph):\n", " '''Construct transition matrix Q from graph as numpy array.'''\n", " n = graph.number_of_nodes()\n", " P = np.zeros((n, n))\n", " Q = transition_matrix_Q(graph)\n", " \n", " for x in range(n):\n", " for y in range(n):\n", " if x == y:\n", " P[y, x] = 1 - np.sum([Q[x, z]*acceptance_probability(graph, x, z) if z != x else 0 for z in range(n)])\n", " else:\n", " P[y, x] = Q[x, y]*acceptance_probability(graph, x, y)\n", " \n", " return P\n", "\n", "# plotting\n", "# YOUR CODE HERE\n", "#raise NotImplementedError()" ] }, { "cell_type": "code", "execution_count": 11, "id": "4b5cc504", "metadata": { "deletable": false, "editable": false, "nbgrader": { "cell_type": "code", "checksum": "1b756f8dff589048e791db63f89d8624", "grade": true, "grade_id": "cell-8c4b6c60ee96f037", "locked": true, "points": 10, "schema_version": 3, "solution": false, "task": false } }, "outputs": [], "source": [ "assert_almost_equal(transition_matrix_P(example_graph)[3,4],1/5,delta=1e-9)\n", "assert_almost_equal(transition_matrix_P(example_graph)[3,7],0.0,delta=1e-9)\n", "assert_almost_equal(transition_matrix_P(example_graph)[2,2],0.41666666,delta=1e-5)\n", "assert_almost_equal(np.sum(transition_matrix_P(example_graph)[7]),1.0,delta=1e-9)" ] }, { "cell_type": "code", "execution_count": 12, "id": "f4e9d8aa", "metadata": { "deletable": false, "editable": false, "nbgrader": { "cell_type": "code", "checksum": "22ff259ba7ad6066040f3417cc8d4483", "grade": true, "grade_id": "cell-a63274314d1b2a2d", "locked": true, "points": 5, "schema_version": 3, "solution": false, "task": false } }, "outputs": [ { "ename": "AssertionError", "evalue": "11133.0 != 2222 within 180 delta (8911.0 difference)", "output_type": "error", "traceback": [ "\u001b[0;31m---------------------------------------------------------------------------\u001b[0m", "\u001b[0;31mAssertionError\u001b[0m Traceback (most recent call last)", "Input \u001b[0;32mIn [12]\u001b[0m, in \u001b[0;36m\u001b[0;34m()\u001b[0m\n\u001b[1;32m 1\u001b[0m \u001b[38;5;28;01massert\u001b[39;00m \u001b[38;5;28mlen\u001b[39m(chain_P_histogram(example_graph,\u001b[38;5;241m0\u001b[39m,\u001b[38;5;241m100\u001b[39m)) \u001b[38;5;241m==\u001b[39m example_graph\u001b[38;5;241m.\u001b[39mnumber_of_nodes()\n\u001b[0;32m----> 2\u001b[0m \u001b[43massert_almost_equal\u001b[49m\u001b[43m(\u001b[49m\u001b[43mchain_P_histogram\u001b[49m\u001b[43m(\u001b[49m\u001b[43mexample_graph\u001b[49m\u001b[43m,\u001b[49m\u001b[38;5;241;43m0\u001b[39;49m\u001b[43m,\u001b[49m\u001b[38;5;241;43m20000\u001b[39;49m\u001b[43m)\u001b[49m\u001b[43m[\u001b[49m\u001b[38;5;241;43m8\u001b[39;49m\u001b[43m]\u001b[49m\u001b[43m,\u001b[49m\u001b[38;5;241;43m2222\u001b[39;49m\u001b[43m,\u001b[49m\u001b[43mdelta\u001b[49m\u001b[38;5;241;43m=\u001b[39;49m\u001b[38;5;241;43m180\u001b[39;49m\u001b[43m)\u001b[49m\n", "File \u001b[0;32m/opt/jupyter-conda/lib/python3.9/unittest/case.py:891\u001b[0m, in \u001b[0;36mTestCase.assertAlmostEqual\u001b[0;34m(self, first, second, places, msg, delta)\u001b[0m\n\u001b[1;32m 885\u001b[0m standardMsg \u001b[38;5;241m=\u001b[39m \u001b[38;5;124m'\u001b[39m\u001b[38;5;132;01m%s\u001b[39;00m\u001b[38;5;124m != \u001b[39m\u001b[38;5;132;01m%s\u001b[39;00m\u001b[38;5;124m within \u001b[39m\u001b[38;5;132;01m%r\u001b[39;00m\u001b[38;5;124m places (\u001b[39m\u001b[38;5;132;01m%s\u001b[39;00m\u001b[38;5;124m difference)\u001b[39m\u001b[38;5;124m'\u001b[39m \u001b[38;5;241m%\u001b[39m (\n\u001b[1;32m 886\u001b[0m safe_repr(first),\n\u001b[1;32m 887\u001b[0m safe_repr(second),\n\u001b[1;32m 888\u001b[0m places,\n\u001b[1;32m 889\u001b[0m safe_repr(diff))\n\u001b[1;32m 890\u001b[0m msg \u001b[38;5;241m=\u001b[39m \u001b[38;5;28mself\u001b[39m\u001b[38;5;241m.\u001b[39m_formatMessage(msg, standardMsg)\n\u001b[0;32m--> 891\u001b[0m \u001b[38;5;28;01mraise\u001b[39;00m \u001b[38;5;28mself\u001b[39m\u001b[38;5;241m.\u001b[39mfailureException(msg)\n", "\u001b[0;31mAssertionError\u001b[0m: 11133.0 != 2222 within 180 delta (8911.0 difference)" ] } ], "source": [ "assert len(chain_P_histogram(example_graph,0,100)) == example_graph.number_of_nodes()\n", "assert_almost_equal(chain_P_histogram(example_graph,0,20000)[8],2222,delta=180)" ] }, { "cell_type": "markdown", "id": "89f65139", "metadata": { "deletable": false, "editable": false, "nbgrader": { "cell_type": "markdown", "checksum": "9b277d17c7b55bb15e5708e6ca90c4f9", "grade": false, "grade_id": "cell-da50533d4d46c293", "locked": true, "schema_version": 3, "solution": false, "task": false } }, "source": [ "## MCMC simulation of disk model\n", "\n", "**(50 points)**\n", "\n", "Recall that in the disk model with we would like to sample the positions $x = (x_1,y_1,\\ldots,x_N,y_N)\\in [0,L)^{2N}$ of $N$ disks of radius $1$ in the torus $[0,L)^2$ with uniform density $\\pi(x) = \\mathbf{1}_{\\{\\text{all pairwise distance }\\geq 2\\}}(x) / Z$, where $Z$ is the unknown partition function of the model. We will assume $L > 2$ and $N\\geq 1$. For the purposes of this simulation we will store the state $x$ in a `np.array` of dimension $(N,2)$ with values in $[0,L)$. Such a configuration can be conveniently plotted using the following function:" ] }, { "cell_type": "code", "execution_count": null, "id": "683d2de3", "metadata": { "deletable": false, "editable": false, "nbgrader": { "cell_type": "code", "checksum": "4c7cb2d90103cd790145cae15d46e99e", "grade": false, "grade_id": "cell-d661f575ab9f80ea", "locked": true, "schema_version": 3, "solution": false, "task": false } }, "outputs": [], "source": [ "def plot_disk_configuration(positions,L):\n", " fig,ax = plt.subplots()\n", " ax.set_aspect('equal')\n", " ax.set_ylim(0,L)\n", " ax.set_xlim(0,L)\n", " ax.set_yticklabels([])\n", " ax.set_xticklabels([])\n", " ax.set_yticks([])\n", " ax.set_xticks([])\n", " for x,y in positions:\n", " # consider all horizontal and vertical copies that may be visible\n", " for x_shift in [z for z in x + [-L,0,L] if -10$ is a parameter and $\\mathcal{N}_i$ are independent normal random variables (make sure to keep positions within $[0,L)^2$ by taking the new position modulo $L$). Test run your simulation for $L=11.3$ and $N=20$ and $\\delta = 0.3$ and about $10000$ Markov chain steps and plot the final state. **(15 pts)**" ] }, { "cell_type": "code", "execution_count": null, "id": "ccaa8f9f", "metadata": { "deletable": false, "nbgrader": { "cell_type": "code", "checksum": "24f09b7137c0184ddb872ef558f756a9", "grade": true, "grade_id": "cell-e180b99ca699c610", "locked": false, "points": 15, "schema_version": 3, "solution": true, "task": false } }, "outputs": [], "source": [ "def MH_disk_move(x,L,delta):\n", " '''Perform random MH move on configuration x, thus changing the array x (if accepted). \n", " Return True if move was accepted, False otherwise.'''\n", " # YOUR CODE HERE\n", " raise NotImplementedError()\n", "\n", "# Test run and plot resulting configuration\n", "# YOUR CODE HERE\n", "raise NotImplementedError()" ] } ], "metadata": { "kernelspec": { "display_name": "Python 3", "language": "python", "name": "python3" }, "language_info": { "codemirror_mode": { "name": "ipython", "version": 3 }, "file_extension": ".py", "mimetype": "text/x-python", "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", "version": "3.9.12" } }, "nbformat": 4, "nbformat_minor": 5 }