Random Walks & Cellular Automata

Transcription

1 Random Walks & Cellular Automata 1 Particle Movements basic version of the simulation vectorized implementation 2 Cellular Automata pictures of matrices an animation of matrix plots the game of life of John Conway 3 Vectorizing the Game of Life performance issues vectorizing the neighbor count vectorizing the rules MCS 507 Lecture 30 Mathematical, Statistical and Scientific Software Jan Verschelde, 5 November 2012 Scientific Software (MCS 507) random walks and cellular automata 5 November / 39

2 Random Walks & Cellular Automata 1 Particle Movements basic version of the simulation vectorized implementation 2 Cellular Automata pictures of matrices an animation of matrix plots the game of life of John Conway 3 Vectorizing the Game of Life performance issues vectorizing the neighbor count vectorizing the rules Scientific Software (MCS 507) random walks and cellular automata 5 November / 39

3 particle movements Scientific Software (MCS 507) random walks and cellular automata 5 November / 39

4 designing the simulation All particles originate at (0, 0). For every particle, for every time step do: 1 generate a random integer d {1, 2, 3, 4}, 2 move according to the value of d: 1 if d = 1: move particle north 2 if d = 2: move particle south 3 if d = 3: move particle east 4 if d = 4: move particle west Scientific Software (MCS 507) random walks and cellular automata 5 November / 39

5 the script walk2d.py import numpy as np import random from scitools.std import plot def particles(npa,nst,pls): """ Shows random particle movement with npa : number of particles, nst : number of time steps, pls : how many steps for next plot. """ Scientific Software (MCS 507) random walks and cellular automata 5 November / 39

6 the script continued x = np.zeros(npa); y = np.zeros(npa) xymax = 3*np.sqrt(nst); xymin = -xymax for step in range(nst): for i in range(npa): d = random.randint(1,4) if d == 1: y[i] += 1 # north elif d == 2: y[i] -= 1 # south elif d == 3: x[i] += 1 # east elif d == 4: x[i] -= 1 # west if (step+1) % pls == 0: plot(x,y, bo, \ axis=[xymin, xymax, xymin, xymax], \ title= %d particles after %d steps \ % (npa, step+1)) Scientific Software (MCS 507) random walks and cellular automata 5 November / 39

7 the main function def main(): """ Fixes the seed for the random numbers and starts the particle simulation. """ random.seed(10) particles(3000,4000,20) will run a simulation of 3000 particles over 4000 stages plotted every 20 time steps. Scientific Software (MCS 507) random walks and cellular automata 5 November / 39

8 Random Walks & Cellular Automata 1 Particle Movements basic version of the simulation vectorized implementation 2 Cellular Automata pictures of matrices an animation of matrix plots the game of life of John Conway 3 Vectorizing the Game of Life performance issues vectorizing the neighbor count vectorizing the rules Scientific Software (MCS 507) random walks and cellular automata 5 November / 39

9 vectorization Vectorization: replace the Python for loops by NumPy operations on arrays. To speed up the simulation: 1 generate all random directions at once, 2 use where to update coordinates. The built-in function where has the syntax numpy.where(condition, [x, y]) and returns elements either from x or y (optional) depending on condition. Scientific Software (MCS 507) random walks and cellular automata 5 November / 39

10 generating all moves def particles(npa,nst,pls): """ Shows random particle movement with npa : number of particles, nst : number of time steps, pls : how many steps for next plot. """ x = np.zeros(npa); y = np.zeros(npa) xymax = 3*np.sqrt(nst); xymin = -xymax moves = np.random.random_integers(1,4,nst*npa) moves.shape = (nst,npa) Scientific Software (MCS 507) random walks and cellular automata 5 November / 39

11 script walk2dv.py continued for step in range(nst): this_move = moves[step,:] y += np.where(this_move == 1,1,0) y -= np.where(this_move == 2,1,0) x += np.where(this_move == 3,1,0) x -= np.where(this_move == 4,1,0) if (step+1) % pls == 0: plot(x,y, bo, \ axis=[xymin, xymax, xymin, xymax], \ title= %d particles after %d steps \ % (npa, step+1)) Scientific Software (MCS 507) random walks and cellular automata 5 November / 39

12 Random Walks & Cellular Automata 1 Particle Movements basic version of the simulation vectorized implementation 2 Cellular Automata pictures of matrices an animation of matrix plots the game of life of John Conway 3 Vectorizing the Game of Life performance issues vectorizing the neighbor count vectorizing the rules Scientific Software (MCS 507) random walks and cellular automata 5 November / 39

13 visualizing a matrix Scientific Software (MCS 507) random walks and cellular automata 5 November / 39

14 running the code $ python spy_plot.py [[ , 0 0 0] [ , 0 1 1] [ , 0 0 0]..., [ , 0 0 0] [ , 0 1 0] [ , 0 0 0]] number of nonzeros : 964 quit? (y/n) Scientific Software (MCS 507) random walks and cellular automata 5 November / 39

15 a plot of a matrix Every nonzero element of the matrix is represented by a red disk in a plot. Extract rows and columns of nonzeroes of A: 1 from scipy import sparse 2 convert the matrix A into a sparse matrix, in coordinate format: S = sparse.coo_matrix(a) 3 x = S.row; y = S.col give in x and y the rows and columns of nonzeroes. Scientific Software (MCS 507) random walks and cellular automata 5 November / 39

16 the script spy_plot.py import numpy as np from scitools.std import plot r = 0.1 # ratio of nonzeroes n = 100 # dimension of the matrix A = np.random.rand(n,n) A = np.matrix(a < r,int) print A from scipy import sparse S = sparse.coo_matrix(a) print number of nonzeros :, S.nnz x = S.row; y = S.col plot(x,y, ro,axis=[-1, n, -1, n]) ans = raw_input( quit? (y/n) ) Scientific Software (MCS 507) random walks and cellular automata 5 November / 39

17 Random Walks & Cellular Automata 1 Particle Movements basic version of the simulation vectorized implementation 2 Cellular Automata pictures of matrices an animation of matrix plots the game of life of John Conway 3 Vectorizing the Game of Life performance issues vectorizing the neighbor count vectorizing the rules Scientific Software (MCS 507) random walks and cellular automata 5 November / 39

18 a movie of matrices We generate a sequence of random matrices of the same dimension n = 100 and ratio of nonzeroes r = 0.1. For each generated random matrix: 1 compute rows and columns of nonzeroes, 2 plot rows and columns as red disks. Scientific Software (MCS 507) random walks and cellular automata 5 November / 39

19 spy_plot_movie.py import numpy as np from scitools.std import plot from scipy import sparse r = 0.1 # ratio of nonzeroes n = 100 # dimension of the matrix for i in xrange(n): A = np.random.rand(n,n) A = np.matrix(a < r,int) S = sparse.coo_matrix(a) x = S.row; y = S.col plot(x,y, ro,axis=[-1, n, -1, n], \ title= matrix %d has %d nonzeroes \ % (i, S.nnz)) Scientific Software (MCS 507) random walks and cellular automata 5 November / 39

20 Random Walks & Cellular Automata 1 Particle Movements basic version of the simulation vectorized implementation 2 Cellular Automata pictures of matrices an animation of matrix plots the game of life of John Conway 3 Vectorizing the Game of Life performance issues vectorizing the neighbor count vectorizing the rules Scientific Software (MCS 507) random walks and cellular automata 5 November / 39

21 visualizing living cells Scientific Software (MCS 507) random walks and cellular automata 5 November / 39

22 simulating cellular growth The game of life is a discovery of John Conway. Consider a rectangular grid of cells with rules: 1 An empty cell is born when it has 3 neighbors. 2 A living cell can either die or survive, as follows: 1 die by loneliness, if the cell has one or no neighbors; 2 die by overpopulation, if the cell has 4 neighbors; 3 survive, if the cell has two or three neighbors. Scientific Software (MCS 507) random walks and cellular automata 5 November / 39

23 design of the code Three ingredients: 1 The rectangular grid is represented by a NumPy matrix A of integers: Ai,j {0, 1}, Ai,j = 0: cell (i, j) is dead, Ai,j = 1: cell (i, j) is alive. 2 We update the matrix applying the rules, running over all pairs of indices (i, j). 3 We use plot of scitools.std to the rows and columns of the nonzero entries. Scientific Software (MCS 507) random walks and cellular automata 5 November / 39

24 the main function def main(): """ Generates a random matrix and applies the rules for Conway s game of life. """ r = 0.2 # ratio of nonzeroes n = 100 # dimension of the matrix A = np.random.rand(n,n) A = np.matrix(a < r,int) for i in xrange(10*n): S = sparse.coo_matrix(a) x = S.row; y = S.col plot(x,y, ro,axis=[-1, n, -1, n], \ title= stage %d % i) A = update(a) Scientific Software (MCS 507) random walks and cellular automata 5 November / 39

25 applying the rules def update(a): """ Applies the rules of Conway s game of life. """ B = np.zeros((a.shape[0],a.shape[1]),int) for i in range(0,a.shape[0]): for j in range(0,a.shape[1]): nb = neighbors(a,i,j) if A[i,j] == 1: if nb < 2 or nb > 3: B[i,j] = 0 else: B[i,j] = 1 else: if nb == 3: B[i,j] = 1 return B Scientific Software (MCS 507) random walks and cellular automata 5 November / 39

26 counting live neighbors def neighbors(a,i,j): cnt = 0 if i > 0: if A[i-1,j]: cnt = cnt + 1 if j > 0: if A[i-1,j-1]: cnt = cnt + 1 if j < A.shape[1]-1: if A[i-1,j+1]: cnt = cnt + 1 if i < A.shape[0]-1: if A[i+1,j]: cnt = cnt + 1 if j > 0: if A[i+1,j-1]: cnt = cnt + 1 if j < A.shape[1]-1: if A[i+1,j+1]: cnt = cnt + 1 if j > 0: if A[i,j-1]: cnt = cnt + 1 if j < A.shape[1]-1: if A[i,j+1]: cnt = cnt + 1 return cnt Scientific Software (MCS 507) random walks and cellular automata 5 November / 39

27 Random Walks & Cellular Automata 1 Particle Movements basic version of the simulation vectorized implementation 2 Cellular Automata pictures of matrices an animation of matrix plots the game of life of John Conway 3 Vectorizing the Game of Life performance issues vectorizing the neighbor count vectorizing the rules Scientific Software (MCS 507) random walks and cellular automata 5 November / 39

28 performance issues The straightforward code does not work well on a MacBook when the dimension gets at n = 300. Reasons for the problems with performance: Traversals through the matrix with double for loops in Python: incrementing loop counters is expensive. The loop counts i and j are Python objects. In counting the neighbors we access not only the (i, j)-th data element, but also (i 1, j), (i + 1, j), (i, j 1), and (i, j + 1). In the row or column oriented storing of the matrix, getting access to respectively (i 1, j),(i + 1, j) or (i, j 1),(i, j + 1) means accessing n elements before or after the (i, j)-th element. Vectorization reorganizes the counting of the live neighbors and the application of the rules. Scientific Software (MCS 507) random walks and cellular automata 5 November / 39

29 Random Walks & Cellular Automata 1 Particle Movements basic version of the simulation vectorized implementation 2 Cellular Automata pictures of matrices an animation of matrix plots the game of life of John Conway 3 Vectorizing the Game of Life performance issues vectorizing the neighbor count vectorizing the rules Scientific Software (MCS 507) random walks and cellular automata 5 November / 39

30 vectorizing the neighbor count For every cell (i, j) in the matrix we need to count the number of neighbors that are alive. We need to avoid to access at the same time the left, right, upper and lower neighbors because of the way matrices are stored. Idea: to count the right neighbors of live cells add to the matrix the same matrix shifted one column. Scientific Software (MCS 507) random walks and cellular automata 5 November / 39

31 counting the right live neighbors >>> import numpy as np >>> A = np.random.rand(5,5) >>> A = np.matrix(a < 0.3,int); A matrix([[1, 1, 1, 1, 1], [1, 1, 0, 0, 1], [0, 0, 0, 0, 0], [0, 0, 1, 1, 0], [0, 0, 0, 0, 0]]) >>> B = np.copy(a[:,+1:]) >>> B array([[1, 1, 1, 1], [1, 0, 0, 1], [0, 0, 0, 0], [0, 1, 1, 0], [0, 0, 0, 0]]) >>> A[:,:-1] = np.copy(a[:,:-1]) + B >>> A matrix([[2, 2, 2, 2, 1], [2, 1, 0, 1, 1], [0, 0, 0, 0, 0], [0, 1, 2, 1, 0], [0, 0, 0, 0, 0]]) Scientific Software (MCS 507) random walks and cellular automata 5 November / 39

32 the script game_neighbors.py def neighbor_count_matrix(a): """ Returns a matrix counting the number of live neighbors. """ AA = np.copy(a); R = np.copy(a) B = np.copy(r[:,:-1]) # omit last column C = np.copy(r[:,+1:]) # omit first column R[:,+1:] = np.copy(r[:,+1:]) + B R[:,:-1] = np.copy(r[:,:-1]) + C D = np.copy(r[:-1,:]) # omit last row E = np.copy(r[+1:,:]) # omit first row R[+1:,:] = np.copy(r[+1:,:]) + D R[:-1,:] = np.copy(r[:-1,:]) + E R = R - AA return R Scientific Software (MCS 507) random walks and cellular automata 5 November / 39

33 the main test def main(): """ Test on neighbor count. """ n = 8; r = 0.2 A = np.random.rand(n,n) A = np.matrix(a < r,int) B = neighbor_count_matrix(a) print cells alive print A print vectorized neighbor count print B C = neighbor_count(a) print original neighbor count print C print equality check : print np.equal(b,c) if name == " main ": main() Scientific Software (MCS 507) random walks and cellular automata 5 November / 39

34 Random Walks & Cellular Automata 1 Particle Movements basic version of the simulation vectorized implementation 2 Cellular Automata pictures of matrices an animation of matrix plots the game of life of John Conway 3 Vectorizing the Game of Life performance issues vectorizing the neighbor count vectorizing the rules Scientific Software (MCS 507) random walks and cellular automata 5 November / 39

35 recall the where method Converting a 0/1 matrix into a Boolean matrix: >>> A = np.random.rand(4,4) >>> A = np.matrix(a < 0.5, int) >>> A matrix([[0, 1, 1, 1], [0, 1, 0, 1], [1, 1, 1, 0], [0, 0, 1, 1]]) >>> B = np.where(a > 0, True, False) >>> B matrix([[false, True, True, True], [False, True, False, True], [ True, True, True, False], [False, False, True, True]], dtype=bool) Scientific Software (MCS 507) random walks and cellular automata 5 November / 39

36 die of loneliness >>> import numpy as np >>> A = np.random.rand(5,5) >>> A = np.matrix(a < 0.4,int) >>> A matrix([[0, 0, 0, 0, 0], [1, 1, 0, 0, 1], [1, 0, 0, 1, 0], [0, 0, 0, 1, 0], [0, 0, 0, 0, 1]]) >>> from game_neighbors import neighbor_count_matrix >>> B = neighbor_count_matrix(a) >>> B array([[2, 2, 1, 1, 1], [2, 2, 2, 2, 1], [2, 3, 3, 2, 3], [1, 1, 2, 2, 3], [0, 0, 1, 2, 1]]) >>> lonely = np.where(b < 2,0,A); lonely array([[0, 0, 0, 0, 0], [1, 1, 0, 0, 0], [1, 0, 0, 1, 0], [0, 0, 0, 1, 0], [0, 0, 0, 0, 0]]) Scientific Software (MCS 507) random walks and cellular automata 5 November / 39

37 the script game_rules.py import numpy as np from game_neighbors import neighbor_count_matrix def update_matrix(a,b): """ The vectorized version of update. """ starve = np.where(b > 3,0,A) lonely = np.where(b < 2,0,starve) alive = np.where(b == 3,1,lonely) return alive To test, we compare the update methods. Scientific Software (MCS 507) random walks and cellular automata 5 November / 39

38 the vectorized game The script game_vector.py has a different update: import numpy as np from scitools.std import plot from scipy import sparse from game_neighbors import neighbor_count_matrix from game_rules import update_matrix def update(a): """ Applies the rules of Conway s game of life. """ B = neighbor_count_matrix(a) C = update_matrix(a,b) return C Scientific Software (MCS 507) random walks and cellular automata 5 November / 39

39 Summary + Exercises Random walks and cellular automata simulate and model complicated dynamical systems. Vectorization is often critical for good performance. 1 Use the canvas of Tkinter for a GUI to make the game of life. Allow the user to define configurations of living cells via mouse clicks on the canvas. 2 Apply vectorization to your GUI. Compare the performance of your versions with and without factorization. Compare the performance of your vectorized GUI with the version that uses scitools.std. Homework due Friday 16 November at 10AM: exercises 1 and 3 of lecture 21; exercise 1 of lecture 22; exercise 2 of lecture 23; exercises 2 and 4 of lecture 24; exercises 1 and 2 of lecture 25; exercises 1 and 3 of lecture 26. Scientific Software (MCS 507) random walks and cellular automata 5 November / 39