{
 "cells": [
  {
   "cell_type": "markdown",
   "id": "7711bd97",
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   "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,
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     "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,
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     "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": {
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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "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": {
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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "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": [
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "def chain_P_histogram(graph,start,k):\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",
    "    return P\n",
    "\n",
    "# plotting\n",
    "x_start = 1\n",
    "k = 40000\n",
    "\n",
    "plt.figure()\n",
    "x_list = list(range(example_graph.number_of_nodes()))\n",
    "plt.bar(x_list, chain_P_histogram(example_graph, x_start, k)/k, color=\"lightblue\", label=\"sampled\")\n",
    "plt.axhline(y = 1/example_graph.number_of_nodes(), marker=\"_\", linestyle = \"dashed\", 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": 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": [],
   "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": 13,
   "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": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "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 -1<z<L+1]:\n",
    "            for y_shift in [z for z in y + [-L,0,L] if -1<z<L+1]:\n",
    "                ax.add_patch(plt.Circle((x_shift,y_shift),1))\n",
    "    plt.show()\n",
    "    \n",
    "# Example with N=3 and L=5\n",
    "positions = np.array([[0.1,0.5],[2.1,1.5],[3.2,3.4]])\n",
    "plot_disk_configuration(positions,5)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "efa2f1c9",
   "metadata": {
    "deletable": false,
    "editable": false,
    "nbgrader": {
     "cell_type": "markdown",
     "checksum": "d29e40f9d0e8d304d266f016150aeab7",
     "grade": false,
     "grade_id": "cell-527056f298dedf7c",
     "locked": true,
     "schema_version": 3,
     "solution": false,
     "task": false
    }
   },
   "source": [
    "__(a)__ Write a function `two_disks_overlap` that tests whether disks at position $\\mathbf{x}_1 \\in [0,L)^{2}$ and position $\\mathbf{x}_2 \\in [0,L)^{2}$ overlap and a function `disk_config_valid` that checks whether a full configuration is valid (non-overlapping and non-touching). _Hint:_ The minimal separation in the $x$-direction can be expressed as a function of `x1[0]-x2[0]` and the minimal separation in the y-direction as a function of `x1[1]-x2[1]`. Then use pythagoras. **(15 pts)**"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "id": "19c010eb",
   "metadata": {
    "deletable": false,
    "nbgrader": {
     "cell_type": "code",
     "checksum": "bbf8ae650d9d4a66a3e4ac4722a3f60e",
     "grade": false,
     "grade_id": "cell-1b2a61bf719003e0",
     "locked": false,
     "schema_version": 3,
     "solution": true,
     "task": false
    }
   },
   "outputs": [],
   "source": [
    "def two_disks_overlap(x1,x2,L):\n",
    "    '''Return True if the disks centered at x1 and x2 (represented as 2-element arrays) overlap in [0,L)^2.'''\n",
    "    # To take into account all overlap with the boundaries, we will also\n",
    "    # test shifted disks. As d(x1 + d, x2) = d(x1, x2 - d), we do not have\n",
    "    # to shift both disks explicitly.\n",
    "    for dx in [0, -L, L]:\n",
    "        for dy in [0, -L, L]:\n",
    "            if (x1[0] - x2[0] + dx)**2 + (x1[1] - x2[1] + dy)**2 <= 4:\n",
    "                return True\n",
    "    return False\n",
    "    \n",
    "def disk_config_valid(x,L):\n",
    "    '''Return True if the configuration x (as two-dimensional array) is non-overlapping in [0,L)^2.'''\n",
    "    n = len(x)\n",
    "    for idx_x1 in range(n):\n",
    "        # We should not compare x1 with itself, and the previous disks have\n",
    "        # already been compared.\n",
    "        for idx_x2 in range(idx_x1 + 1, n):\n",
    "            if two_disks_overlap(x[idx_x1], x[idx_x2], L):\n",
    "                return False\n",
    "    return True"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "id": "e0f2024a",
   "metadata": {
    "deletable": false,
    "editable": false,
    "nbgrader": {
     "cell_type": "code",
     "checksum": "05fc821fed0690aabc13fd66da800694",
     "grade": true,
     "grade_id": "cell-9f544deda0526691",
     "locked": true,
     "points": 10,
     "schema_version": 3,
     "solution": false,
     "task": false
    }
   },
   "outputs": [],
   "source": [
    "assert two_disks_overlap(np.array([1,1]),np.array([1,1]),5)\n",
    "assert two_disks_overlap(np.array([0.6,0.6]),np.array([4.1,0.5]),5)\n",
    "assert two_disks_overlap(np.array([0.3,0.3]),np.array([4.6,4.6]),5)\n",
    "assert not two_disks_overlap(np.array([1,1]),np.array([3.1,1]),7)\n",
    "assert not two_disks_overlap(np.array([1,1]),np.array([1,3.1]),7)\n",
    "assert not two_disks_overlap(np.array([1,1]),np.array([1.01+np.sqrt(2),1.01+np.sqrt(2)]),6)\n",
    "assert two_disks_overlap(np.array([1,1]),np.array([0.99+np.sqrt(2),0.99+np.sqrt(2)]),6)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "id": "68e57e57",
   "metadata": {
    "deletable": false,
    "editable": false,
    "nbgrader": {
     "cell_type": "code",
     "checksum": "8705cb3a08fc636963fe61806348d093",
     "grade": true,
     "grade_id": "cell-699454de327d56d5",
     "locked": true,
     "points": 5,
     "schema_version": 3,
     "solution": false,
     "task": false
    }
   },
   "outputs": [],
   "source": [
    "assert disk_config_valid(np.array([[0.1,0.5],[2.1,1.5],[3.2,3.4]]),5)\n",
    "assert not disk_config_valid(np.array([[0.1,0.5],[2.1,1.5],[3.2,3.4],[4.1,2.3]]),5)\n",
    "assert disk_config_valid(np.array([[1,1],[3.1,1],[1,3.1]]),6)\n",
    "assert not disk_config_valid(np.array([[1,1],[3.1,1],[1,3.1],[2.5,2.5]]),6)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "fe7c5736",
   "metadata": {
    "deletable": false,
    "editable": false,
    "nbgrader": {
     "cell_type": "markdown",
     "checksum": "db023e65fc2613d624f29977a82bf1f1",
     "grade": false,
     "grade_id": "cell-b0e00f11b0f85525",
     "locked": true,
     "schema_version": 3,
     "solution": false,
     "task": false
    }
   },
   "source": [
    "__(b)__ Assuming $N \\leq \\lceil \\frac12 L -1 \\rceil^2$ where $\\lceil r\\rceil$ is the smallest integer larger or equal to $r$, write a function `generate_initial_positions` that produces an arbitrary non-overlapping (and non-touching) initial condition given $N$ and $L$. The layout need not be random, any deterministic layout is ok (e.g. grid). **(10 pts)**"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "f86a04ed",
   "metadata": {
    "deletable": false,
    "nbgrader": {
     "cell_type": "code",
     "checksum": "b17579d74c8ff8572d81f18997b49c90",
     "grade": false,
     "grade_id": "cell-53c1bd894d6fe27d",
     "locked": false,
     "schema_version": 3,
     "solution": true,
     "task": false
    }
   },
   "outputs": [],
   "source": [
    "def generate_initial_positions(N,L):\n",
    "    '''Return array of positions of N disks in non-overlapping positions.'''\n",
    "    # YOUR CODE HERE\n",
    "    raise NotImplementedError()\n",
    "    \n",
    "plot_disk_configuration(generate_initial_positions(33,14.5),14.5)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "3d3dc3db",
   "metadata": {
    "deletable": false,
    "editable": false,
    "nbgrader": {
     "cell_type": "code",
     "checksum": "fa73c21133aa2b913b931b266e7782f0",
     "grade": true,
     "grade_id": "cell-e1c80d8b59301b93",
     "locked": true,
     "points": 10,
     "schema_version": 3,
     "solution": false,
     "task": false
    }
   },
   "outputs": [],
   "source": [
    "for l in [4,9.2,14.5]:\n",
    "    for n in range(1,int(np.ceil(l/2-1)**2)+1):\n",
    "        assert disk_config_valid(generate_initial_positions(n,l),l), \"Failed for n = {}, l = {}\".format(n,l)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "eb5dbe07",
   "metadata": {
    "deletable": false,
    "editable": false,
    "nbgrader": {
     "cell_type": "markdown",
     "checksum": "b4bb37c92263c95c89055362336f894a",
     "grade": false,
     "grade_id": "cell-1bd76964ec9620ce",
     "locked": true,
     "schema_version": 3,
     "solution": false,
     "task": false
    }
   },
   "source": [
    "__(c)__ Write a function `remains_valid_after_move` that determines whether in a non-overlapping configuration $x$ moving the $i$th disk to `next_position` results in a valid non-overlapping configuration. **(10 pts)**"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "56252046",
   "metadata": {
    "deletable": false,
    "nbgrader": {
     "cell_type": "code",
     "checksum": "04ff82e1b0e3386f2cb15412695c1864",
     "grade": false,
     "grade_id": "cell-d54b4fa9b2f8eb92",
     "locked": false,
     "schema_version": 3,
     "solution": true,
     "task": false
    }
   },
   "outputs": [],
   "source": [
    "def remains_valid_after_move(x,i,next_position,L):\n",
    "    '''Returns True if replacing x[i] by next_position would yield a valid configuration,\n",
    "    otherwise False.'''\n",
    "    # YOUR CODE HERE\n",
    "    raise NotImplementedError()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "167bdc3e",
   "metadata": {
    "deletable": false,
    "editable": false,
    "nbgrader": {
     "cell_type": "code",
     "checksum": "37fe75b415d4e31369d6b6bb1041609b",
     "grade": true,
     "grade_id": "cell-e4902309b9869f3d",
     "locked": true,
     "points": 10,
     "schema_version": 3,
     "solution": false,
     "task": false
    }
   },
   "outputs": [],
   "source": [
    "assert remains_valid_after_move([[0.1,0.5],[2.1,1.5],[3.2,3.4]],0,[4.5,0.5],5)\n",
    "assert not remains_valid_after_move([[0.1,0.5],[2.1,1.5],[3.2,3.4]],1,[4.5,0.5],5)\n",
    "assert not remains_valid_after_move([[0.1,0.5],[2.1,1.5],[3.2,3.4]],2,[3.2,2.5],5)\n",
    "assert remains_valid_after_move([[0.1,0.5],[2.1,1.5],[3.2,3.4]],2,[3.2,3.8],5)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "a8cd0c44",
   "metadata": {
    "deletable": false,
    "editable": false,
    "nbgrader": {
     "cell_type": "markdown",
     "checksum": "8e0cfe7c5bce12cc34e939c5c151ed65",
     "grade": false,
     "grade_id": "cell-582ca3e158908e6c",
     "locked": true,
     "schema_version": 3,
     "solution": false,
     "task": false
    }
   },
   "source": [
    "__(d)__ Implement the Metropolis-Hastings transition by selecting a uniformly chosen disk and displacing it by $ (\\delta\\,\\mathcal{N}_1,\\delta\\,\\mathcal{N}_2)$ where $\\delta>0$ 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()"
   ]
  }
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