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exercise_sheet_04 (Score: 100.0 / 100.0)

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  1. Test cell (Score: 10.0 / 10.0)
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  11. Test cell (Score: 10.0 / 10.0)
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Exercise sheet

Some general remarks about the exercises:

+
    +
  • For your convenience functions from the lecture are included below. Feel free to reuse them without copying to the exercise solution box.
    • +
  • 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().
    • +
  • 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 for the widely adopted coding conventions or this guide for explanation.
    • +
  • 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.
    • +
  • 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.
    • +
  • 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.
    • +
+

Please fill in your name here:

+ +
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+ +
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+
In [1]:
+
+ 
NAME = "Kees van Kempen"
+NAMES_OF_COLLABORATORS = ""
+
+ +
+
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+ +
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+ +
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+ +
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+

Exercise sheet 4

+

Code from the lectures:

+ +
+
+ +
+
+
In [2]:
+
+ 
import numpy as np
+import matplotlib.pylab as plt
+import networkx as nx
+
+rng = np.random.default_rng()
+%matplotlib inline
+
+def draw_transition_graph(P):
+    # construct a directed graph directly from the matrix
+    graph = nx.DiGraph(P)        
+    # draw it in such a way that edges in both directions are visible and have appropriate width
+    nx.draw_networkx(graph,connectionstyle='arc3, rad = 0.15',width=[6*P[u,v] for u,v in graph.edges()])
+
+def sample_next(P,current):
+    return rng.choice(len(P),p=P[current])
+
+def sample_chain(P,start,n):
+    chain = [start]
+    for _ in range(n):
+        chain.append(sample_next(P,chain[-1]))
+    return chain
+
+def stationary_distributions(P):
+    eigenvalues, eigenvectors = np.linalg.eig(np.transpose(P))
+    # make list of normalized eigenvectors for which the eigenvalue is very close to 1
+    return [eigenvectors[:,i]/np.sum(eigenvectors[:,i]) for i in range(len(eigenvalues)) 
+              if np.abs(eigenvalues[i]-1) < 1e-10]
+
+def markov_sample_mean(P,start,function,n):
+    total = 0
+    state = start
+    for _ in range(n):
+        state = sample_next(P,state)
+        total += function[state]
+    return total/n
+
+ +
+
+
+ +
+
+
+
+
+

Markov Chain on a graph

(50 points)

+

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:

+ +
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+ +
+
+
In [3]:
+
+ 
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)]
+example_graph = nx.Graph(edges)
+nx.draw(example_graph,with_labels=True)
+
+ +
+
+
+ +
+
+ + +
+ +
+ + + + +
+ +
+ +
+ +
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+ +
+
+
+
+
+

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.

+

(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. (10 pts)

+ +
+
+ +
+
+
In [4]:
+
Student's answer(Top)
+
+
+ 
def sample_proposal(graph,x):
+    '''Pick a random node y from the neighbors of x in graph with uniform
+    probability, according to Q.'''
+    y_list = np.fromiter(graph.neighbors(x), dtype=int)
+    return rng.choice(y_list)
+
+ +
+
+ +
+
+ +
+
+
+
In [5]:
+
Grade cell: cell-c70fec7da167b7be + + Score: 10.0 / 10.0 (Top) +
+
+
+ 
from nose.tools import assert_almost_equal
+assert sample_proposal(nx.Graph([(0,1)]),0)==1
+assert_almost_equal([sample_proposal(example_graph,3) for _ in range(1000)].count(4),333,delta=50)
+assert_almost_equal([sample_proposal(example_graph,8) for _ in range(1000)].count(5),500,delta=60)
+
+ +
+
+ +
+
+ +
+
+
+
+
+

(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. (15 pts)

+ +
+
+ +
+
+
In [6]:
+
Student's answer(Top)
+
+
+ 
def chain_Q_histogram(graph,start,k):
+    '''Produce a histogram (a Numpy array of length equal to the number of 
+    nodes of graph) of the states visited (excluding initial state) by the 
+    Q Markov chain in the first k steps when started at start.'''
+    n = graph.number_of_nodes()
+    number_of_visits = np.zeros(n)
+    x = start
+    
+    for _ in range(k):
+        x = sample_proposal(graph, x)
+        number_of_visits[x] += 1
+    
+    return number_of_visits
+
+def transition_matrix_Q(graph):
+    '''Construct transition matrix Q from graph as two-dimensional Numpy array.'''
+    n = example_graph.number_of_nodes()
+    Q = np.zeros((n, n))
+    for x in range(n): 
+        for k in example_graph.neighbors(x):
+            Q[x, k] = 1/example_graph.degree(x)
+    return Q
+
+# Compare histogram and stationary distribution in a plot
+x_start = 1
+k = 100000
+
+plt.figure()
+plt.title("Markov chain visiting density for uniform transition matrix $Q(x \\to y)$")
+x_list = list(range(example_graph.number_of_nodes()))
+plt.bar(x_list, chain_Q_histogram(example_graph, x_start, k)/k, color="lightblue", label="sampled")
+plt.scatter(x_list, stationary_distributions(transition_matrix_Q(example_graph)), s=200, marker="_", color="black", label="theoretical")
+plt.ylabel("visiting density")
+plt.xlabel("node $x$")
+plt.legend()
+plt.show()
+
+ +
+
+
+
+ Comments:

Nice plot

+ +
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+ + +
+ +
+ + + + +
+ +
+ +
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+ +
+
+
+
In [7]:
+
Grade cell: cell-bdea40714e10d0e8 + + Score: 15.0 / 15.0 (Top) +
+
+
+ 
assert_almost_equal(transition_matrix_Q(example_graph)[3,4],1/3,delta=1e-9)
+assert_almost_equal(transition_matrix_Q(example_graph)[3,7],0.0,delta=1e-9)
+assert_almost_equal(transition_matrix_Q(example_graph)[2,2],0.0,delta=1e-9)
+assert_almost_equal(np.sum(transition_matrix_Q(example_graph)[7]),1.0,delta=1e-9)
+assert chain_Q_histogram(nx.Graph([(0,1)]),0,100)[1] == 50
+assert len(chain_Q_histogram(example_graph,0,100)) == example_graph.number_of_nodes()
+
+ +
+
+ +
+
+ +
+
+
+
+
+

(c) Determine the appropriate Metropolis-Hastings acceptance probability $A(x \to y)$ for $x\neq y$ +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)

+ +
+
+ +
+
+
In [8]:
+
Student's answer(Top)
+
+
+ 
def acceptance_probability(graph,x,y):
+    '''Compute A(x -> y) for the supplied graph (assuming x!=y).'''
+    assert x != y
+    
+    Q = transition_matrix_Q(graph)
+    n = graph.number_of_nodes()
+    # We want to sample a uniform mass distribution pi:
+    pi = np.ones(n)/n
+    
+    if Q[x, y] == 0:
+        return 0
+    else:
+        A = pi[y]*Q[y, x]/( pi[x]*Q[x, y] )
+        return np.min([1., A])
+
+def sample_next_state(graph,x):
+    '''Return next random state y according to MH transition matrix P(x -> y).'''
+    y = sample_proposal(graph, x)
+    return y if rng.random() < acceptance_probability(graph, x, y) else x
+
+ +
+
+ +
+
+ +
+
+
+
In [9]:
+
Grade cell: cell-fbfc2607999d0c99 + + Score: 10.0 / 10.0 (Top) +
+
+
+ 
assert_almost_equal(acceptance_probability(example_graph,3,4),0.6,delta=1e-9)
+assert_almost_equal(acceptance_probability(example_graph,8,7),0.5,delta=1e-9)
+assert_almost_equal(acceptance_probability(example_graph,7,8),1.0,delta=1e-9)
+assert_almost_equal(acceptance_probability(nx.Graph([(0,1)]),0,1),1,delta=1e-9)
+
+ +
+
+ +
+
+ +
+
+
+
+
+

(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)

+ +
+
+ +
+
+
In [10]:
+
Student's answer(Top)
+
+
+ 
def chain_P_histogram(graph,start,k):
+    '''Produce a histogram of the states visited (excluding initial state) 
+    by the P Markov chain in the first n steps when started at start.'''
+    n = graph.number_of_nodes()
+    number_of_visits = np.zeros(n)
+    x = start
+    
+    for _ in range(k):
+        x = sample_next_state(graph, x)
+        number_of_visits[x] += 1
+    
+    return number_of_visits
+
+def transition_matrix_P(graph):
+    '''Construct transition matrix Q from graph as numpy array.'''
+    n = graph.number_of_nodes()
+    P = np.zeros((n, n))
+    Q = transition_matrix_Q(graph)
+    
+    for x in range(n):
+        for y in range(n):
+            if x != y:
+                P[y, x] = Q[x, y]*acceptance_probability(graph, x, y)
+        # Finally, we have to calculate P(x -> x):
+        P[x, x] = 1 - np.sum(P[:, x])
+    return P
+
+# plotting
+x_start = 1
+k = 40000
+
+plt.figure()
+plt.title("Markov chain visiting density for uniform Metropolis-Hastings transition matrix $P(x \\to y)$")
+x_list = list(range(example_graph.number_of_nodes()))
+plt.bar(x_list, chain_P_histogram(example_graph, x_start, k)/k, color="lightblue", label="sampled")
+plt.axhline(y = 1/example_graph.number_of_nodes(), marker="_", linestyle = "dashed", color="black", label="theoretical")
+plt.ylabel("visiting density")
+plt.xlabel("node $x$")
+plt.legend()
+plt.show()
+
+ +
+
+ +
+
+ +
+
+ + +
+ +
+ + + + +
+ +
+ +
+ +
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+ +
+
+
+
In [11]:
+
Grade cell: cell-8c4b6c60ee96f037 + + Score: 10.0 / 10.0 (Top) +
+
+
+ 
assert_almost_equal(transition_matrix_P(example_graph)[3,4],1/5,delta=1e-9)
+assert_almost_equal(transition_matrix_P(example_graph)[3,7],0.0,delta=1e-9)
+assert_almost_equal(transition_matrix_P(example_graph)[2,2],0.41666666,delta=1e-5)
+assert_almost_equal(np.sum(transition_matrix_P(example_graph)[7]),1.0,delta=1e-9)
+
+ +
+
+ +
+
+ +
+
+
+
In [12]:
+
Grade cell: cell-a63274314d1b2a2d + + Score: 5.0 / 5.0 (Top) +
+
+
+ 
assert len(chain_P_histogram(example_graph,0,100)) == example_graph.number_of_nodes()
+assert_almost_equal(chain_P_histogram(example_graph,0,20000)[8],2222,delta=180)
+
+ +
+
+ +
+
+ +
+
+
+
+
+

MCMC simulation of disk model

(50 points)

+

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:

+ +
+
+ +
+
+
In [13]:
+
+ 
def plot_disk_configuration(positions,L):
+    fig,ax = plt.subplots()
+    ax.set_aspect('equal')
+    ax.set_ylim(0,L)
+    ax.set_xlim(0,L)
+    ax.set_yticklabels([])
+    ax.set_xticklabels([])
+    ax.set_yticks([])
+    ax.set_xticks([])
+    for x,y in positions:
+        # consider all horizontal and vertical copies that may be visible
+        for x_shift in [z for z in x + [-L,0,L] if -1<z<L+1]:
+            for y_shift in [z for z in y + [-L,0,L] if -1<z<L+1]:
+                ax.add_patch(plt.Circle((x_shift,y_shift),1))
+    plt.show()
+    
+# Example with N=3 and L=5
+positions = np.array([[0.1,0.5],[2.1,1.5],[3.2,3.4]])
+plot_disk_configuration(positions,5)
+
+ +
+
+
+ +
+
+ + +
+ +
+ + + + +
+ +
+ +
+ +
+
+ +
+
+
+
+
+

(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)

+ +
+
+ +
+
+
In [14]:
+
Student's answer(Top)
+
+
+ 
def two_disks_overlap(x1,x2,L):
+    '''Return True if the disks centered at x1 and x2 (represented as 2-element arrays) overlap in [0,L)^2.'''
+    # To take into account all overlap with the boundaries, we will also
+    # test shifted disks. As d(x1 + d, x2) = d(x1, x2 - d), we do not have
+    # to shift both disks explicitly.
+    for dx in [0, -L, L]:
+        for dy in [0, -L, L]:
+            if (x1[0] - x2[0] + dx)**2 + (x1[1] - x2[1] + dy)**2 <= 4:
+                return True
+    return False
+    
+def disk_config_valid(x,L):
+    '''Return True if the configuration x (as two-dimensional array) is non-overlapping in [0,L)^2.'''
+    n = len(x)
+    for idx_x1 in range(n):
+        # We should not compare x1 with itself, and the previous disks have
+        # already been compared.
+        for idx_x2 in range(idx_x1 + 1, n):
+            if two_disks_overlap(x[idx_x1], x[idx_x2], L):
+                return False
+    return True
+
+ +
+
+ +
+
+ +
+
+
+
In [15]:
+
Grade cell: cell-9f544deda0526691 + + Score: 10.0 / 10.0 (Top) +
+
+
+ 
assert two_disks_overlap(np.array([1,1]),np.array([1,1]),5)
+assert two_disks_overlap(np.array([0.6,0.6]),np.array([4.1,0.5]),5)
+assert two_disks_overlap(np.array([0.3,0.3]),np.array([4.6,4.6]),5)
+assert not two_disks_overlap(np.array([1,1]),np.array([3.1,1]),7)
+assert not two_disks_overlap(np.array([1,1]),np.array([1,3.1]),7)
+assert not two_disks_overlap(np.array([1,1]),np.array([1.01+np.sqrt(2),1.01+np.sqrt(2)]),6)
+assert two_disks_overlap(np.array([1,1]),np.array([0.99+np.sqrt(2),0.99+np.sqrt(2)]),6)
+
+ +
+
+ +
+
+ +
+
+
+
In [16]:
+
Grade cell: cell-699454de327d56d5 + + Score: 5.0 / 5.0 (Top) +
+
+
+ 
assert disk_config_valid(np.array([[0.1,0.5],[2.1,1.5],[3.2,3.4]]),5)
+assert not disk_config_valid(np.array([[0.1,0.5],[2.1,1.5],[3.2,3.4],[4.1,2.3]]),5)
+assert disk_config_valid(np.array([[1,1],[3.1,1],[1,3.1]]),6)
+assert not disk_config_valid(np.array([[1,1],[3.1,1],[1,3.1],[2.5,2.5]]),6)
+
+ +
+
+ +
+
+ +
+
+
+
+
+

(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)

+ +
+
+ +
+
+
In [17]:
+
Student's answer(Top)
+
+
+ 
def generate_initial_positions(N,L):
+    '''Return array of positions of N disks in non-overlapping positions.'''
+    assert N <= np.ceil( (.5*L**2 - 1)**2 )
+    
+    disks = np.zeros((N, 2))
+    i = 0
+    # To have the disks non-touchting, we have them 2.0001 apart instead of 2.
+    for x in np.arange(1, L - 1, 2.0001):
+        for y in np.arange(1, L - 1, 2.0001):
+            if i + 1 > N: return disks
+            disks[i] = x, y
+            i += 1
+    return disks
+    
+plot_disk_configuration(generate_initial_positions(33,14.5),14.5)
+
+ +
+
+
+
+ Comments:

Be careful with .0001

+ +
+
+
+
+ +
+
+ + +
+ +
+ + + + +
+ +
+ +
+ +
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+ +
+
+
+
In [18]:
+
Grade cell: cell-e1c80d8b59301b93 + + Score: 10.0 / 10.0 (Top) +
+
+
+ 
for l in [4,9.2,14.5]:
+    for n in range(1,int(np.ceil(l/2-1)**2)+1):
+        assert disk_config_valid(generate_initial_positions(n,l),l), "Failed for n = {}, l = {}".format(n,l)
+
+ +
+
+ +
+
+ +
+
+
+
+
+

(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)

+ +
+
+ +
+
+
In [19]:
+
Student's answer(Top)
+
+
+ 
def remains_valid_after_move(x,i,next_position,L):
+    '''Returns True if replacing x[i] by next_position would yield a valid configuration,
+    otherwise False.'''
+    # We need to create a copy as not to alter the original x
+    # outside this function.
+    #
+    # NOTE: The copying in this function is the heaviest operation.
+    # Reducing this overhead would be great, but I am not familiar
+    # enough with references/pointers in Python.
+    copy_x = np.copy(x)
+    copy_x[i] = next_position
+    return disk_config_valid(copy_x, L)
+
+ +
+
+ +
+
+ +
+
+
+
In [20]:
+
Grade cell: cell-e4902309b9869f3d + + Score: 10.0 / 10.0 (Top) +
+
+
+ 
assert remains_valid_after_move([[0.1,0.5],[2.1,1.5],[3.2,3.4]],0,[4.5,0.5],5)
+assert not remains_valid_after_move([[0.1,0.5],[2.1,1.5],[3.2,3.4]],1,[4.5,0.5],5)
+assert not remains_valid_after_move([[0.1,0.5],[2.1,1.5],[3.2,3.4]],2,[3.2,2.5],5)
+assert remains_valid_after_move([[0.1,0.5],[2.1,1.5],[3.2,3.4]],2,[3.2,3.8],5)
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+

(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)

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In [21]:
+
Student's answer + Score: 15.0 / 15.0 (Top) +
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+ 
def MH_disk_move(x,L,delta):
+    '''Perform random MH move on configuration x, thus changing the array x (if accepted). 
+    Return True if move was accepted, False otherwise.'''
+    
+    # Although it is tempting to use the wacky method of pulling two independent
+    # normal RVs using the method developed two weeks ago, let's just use Numpy.
+    N1, N2 = rng.normal(), rng.normal()
+    n = len(x)
+    i = np.floor(rng.random()*n).astype(int)
+    new_position = x[i] + delta*np.array([N1, N2])
+    new_position %= L
+    
+    if remains_valid_after_move(x, i, new_position, L):
+        x[i] = new_position
+        return True
+    return False
+    
+# Test run and plot resulting configuration
+steps = 10000
+num_moves = 0
+
+L = 11.3
+N = 20
+delta = .3
+
+x = generate_initial_positions(N, L)
+plot_disk_configuration(x, L)
+
+for _ in range(steps):
+    num_moves += MH_disk_move(x, L, delta)
+
+plot_disk_configuration(x, L)
+print("Of {} proposed moves, {} were taken ({:.1f}%).".format(steps, num_moves, num_moves/steps*100))
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Of 10000 proposed moves, 5219 were taken (52.2%).
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