teaching:exos:simulations_random_walks_codes

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 ====== Simulations numériques de marches aléatoires : programmes en Python ====== ====== Simulations numériques de marches aléatoires : programmes en Python ======
  
-Pour une bonne compréhension, ces programmes doivent être étudiés successivement. Il est important d'exécuter le code Python et même de tester des petites modifications. +<note tip>Pour une bonne compréhension, ces programmes doivent être étudiés successivement. Il est important d'exécuter le code Python et même de tester des petites modifications. 
 +</note>
 ===== Génération de nombres aléatoires ===== ===== Génération de nombres aléatoires =====
  
-<sxh python; title : 01_Random.py> +<code python 01_random.py>
-#!/usr/bin/env python+
 #!/usr/bin/python #!/usr/bin/python
 +# -*- coding: utf-8 -*-
 +"""
 +cf. documentation cf http://docs.python.org/library/random.html 
 +random number generation - génération de nombres aléatoires
 +functions of interest : choice, randint, seed
 +"""
  
 from random import *  from random import * 
-# cf. documentation cf http://docs.python.org/library/random.html  
-# random number generation - génération de nombres aléatoires 
-# functions of interest : choice, randint, seed 
  
-facepiece=['pile','face'] +facepiece = ['pile','face'] 
-valeurpiece=[0.01,0.02,0.05,0.1,0.2,0.5,1.,2.]+valeurpiece = [0.01,0.02,0.05,0.1,0.2,0.5,1.,2.]
  
-#for i in range(1):+for i in range(1):
     # choice : random choice of an element from a list     # choice : random choice of an element from a list
-    #print choice(facepiece), choice(valeurpiece)+    print(choice(facepiece), choice(valeurpiece))
     # randint : return a random integer number between 2 values (including limits)     # randint : return a random integer number between 2 values (including limits)
-    #print randint(0,10)       # imprime un nombre aléatoire entre 0 et 10 +    print(randint(0,10))       # imprime un nombre aléatoire entre 0 et 10 
-    #print choice(range(0,11,1))  # same function, using choice and range to create the list+    print(choice(range(0,11,1)))  # same function, using choice and range to create the list 
 +    
  
 # seed(ANY_DATA) : seeding of the random number generator with any (constant) data # seed(ANY_DATA) : seeding of the random number generator with any (constant) data
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     seed()   # to randomly initiate the generator     seed()   # to randomly initiate the generator
     for i in range(10):     for i in range(10):
-        print randint(1000,9999) +        print(randint(1000,9999)) 
-    print " " +    print(" ") 
-</sxh>+</code>
  
 ===== Histogrammes de nombres aléatoires  ===== ===== Histogrammes de nombres aléatoires  =====
  
-<sxh python; title : 02_random_histogram.py>+<code python 02_random_histogram.py>
 #!/usr/bin/env python #!/usr/bin/env python
 # -*- coding: utf-8 -*- # -*- coding: utf-8 -*-
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 seed() seed()
  
-rval=[] +rval = [] 
-for j in range(10000):+for j in range(100000):
     rval.append(randint(0,99))   # append to the list a random (integer) number between 0 and 99     rval.append(randint(0,99))   # append to the list a random (integer) number between 0 and 99
  
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 # analysis - histogram  -  see http://matplotlib.sourceforge.net/examples/api/histogram_demo.html # analysis - histogram  -  see http://matplotlib.sourceforge.net/examples/api/histogram_demo.html
 # http://fr.wikipedia.org/wiki/Histogramme # http://fr.wikipedia.org/wiki/Histogramme
-xh=np.array(rval)  # see http://www.scipy.org/Cookbook/BuildingArrays  transforme une liste en un tableau numérique de Numpy +xh = np.array(rval)  # see http://www.scipy.org/Cookbook/BuildingArrays  transforme une liste en un tableau numérique de Numpy 
-# print xh+# print(xh)
  
 fig = plt.figure() fig = plt.figure()
 ax = fig.add_subplot(111) ax = fig.add_subplot(111)
  
-n, bins, patches = ax.hist(xh, 10, facecolor='green', alpha=0.75) +n, bins, patches = ax.hist(xh, 50, facecolor='green', alpha=0.75) 
-print n  # les nombres d'occurences par classe +print(n)  # les nombres d'occurences par classe 
-print bins  # les classes, de largeur identique+print(bins)  # les classes, de largeur identique
  
 # modifier le nombre de nombres générés, les nombres de classes-bins,  # modifier le nombre de nombres générés, les nombres de classes-bins, 
  
 plt.show() plt.show()
-</sxh>+ 
 +</code>
  
 ===== Représenter le déplacement d'un objet  ===== ===== Représenter le déplacement d'un objet  =====
  
-<sxh python; title : 03_tkinter_simple_move.py>+<code python 03_tkinter_simple_move.py>
 #!/usr/bin/python #!/usr/bin/python
 # -*- coding: utf-8 -*- # -*- coding: utf-8 -*-
  
-from Tkinter import *+from tkinter import *
 import time import time
  
 window = Tk() window = Tk()
-sizex=400 +sizex = 400 
-sizey=100+sizey = 200
 canvas = Canvas(window, width = sizex, height = sizey) canvas = Canvas(window, width = sizex, height = sizey)
 canvas.pack() canvas.pack()
 x = 100 # initial left-most edge of first ball x = 100 # initial left-most edge of first ball
 y = 30 # initial top-most edge of first ball y = 30 # initial top-most edge of first ball
-r=20                  # ball diameter  +r = 20                  # ball diameter  
-depx=2            # displacement at each move in x direction +depx = 2            # displacement at each move in x direction 
-depy=0            # displacement at each move in y direction+depy = 1            # displacement at each move in y direction
  
 ball=canvas.create_oval(x,y,x+r,y+r,fill="blue") ball=canvas.create_oval(x,y,x+r,y+r,fill="blue")
  
 #moves #moves
-no_moves=10+no_moves = 140
 for j in range(no_moves): for j in range(no_moves):
     canvas.move(ball, depx, depy)     canvas.move(ball, depx, depy)
-    canvas.after(10)         # time delay in milliseconds+    canvas.after(20)         # time delay in milliseconds
     canvas.update()     canvas.update()
  
 time.sleep(5) # on attend quelques secondes time.sleep(5) # on attend quelques secondes
 window.destroy() window.destroy()
- +</code>
-</sxh>+
  
 ===== Représenter le déplacement de nombreux points ===== ===== Représenter le déplacement de nombreux points =====
-<sxh python; title : 04_tkinter_many_moves.py>+<code python 04_tkinter_many_moves.py>
 #!/usr/bin/python #!/usr/bin/python
 # -*- coding: utf-8 -*- # -*- coding: utf-8 -*-
  
-from Tkinter import *+from tkinter import *
 import time import time
 +from random import * 
  
 window = Tk() window = Tk()
-sizex=400 +sizex = 400 
-sizey=600+sizey = 600
 canvas = Canvas(window, width = sizex, height = sizey) canvas = Canvas(window, width = sizex, height = sizey)
 canvas.pack() canvas.pack()
 x = 100 # initial left-most edge of first ball x = 100 # initial left-most edge of first ball
 y = 30 # initial top-most edge of first ball y = 30 # initial top-most edge of first ball
-r=20                  # ball diameter  +r = 16                  # ball diameter  
-depx=2            # displacement at each move in x direction +depx = 2            # displacement at each move in x direction 
-depy=0            # displacement at each move in y direction+depy = 0            # displacement at each move in y direction
  
 # create balls: # create balls:
-no_particles= 20 +no_particles = 20 
-dy = (sizey-2.)/(no_particles+1)       # y initial separation between balls +dy = (sizey-2.*y)/(no_particles+1)       # y initial separation between balls 
-print dy +print(dy) 
-ball_list=[]+ball_list = []
 for i in range(no_particles): for i in range(no_particles):
-    ball=canvas.create_oval(x,y,x+r,y+r,fill="blue")+    ball = canvas.create_oval(x,y,x+r,y+r,fill="blue")
     y = y+dy     y = y+dy
     ball_list.append(ball)     ball_list.append(ball)
  
 #moves #moves
-no_moves=100+no_moves = 100
 for j in range(no_moves): for j in range(no_moves):
     for ball in ball_list:     for ball in ball_list:
-        canvas.move(ball, depx, depy)+        canvas.move(ball, depx, choice([-2, 2]) ) 
 +#        canvas.move(ball, depx, depy)
     canvas.after(10)     canvas.after(10)
     canvas.update()     canvas.update()
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 time.sleep(5) # on attend quelques secondes time.sleep(5) # on attend quelques secondes
 window.destroy() window.destroy()
-</sxh>+</code>
  
 +===== Marche aléatoire d'un petit nombre de pas =====
 +
 +<code python 05_tkinter_random_walk_few_steps_1D.py>
 +#!/usr/bin/env python
 +# -*- coding: utf-8 -*-
 +
 +from tkinter import *
 +from random import choice     # http://docs.python.org/library/random.html
 +import numpy as np
 +import matplotlib.pyplot as plt     # http://matplotlib.sourceforge.net/api/pyplot_api.html#module-matplotlib.pyplot
 +import matplotlib.mlab as mlab    # http://matplotlib.sourceforge.net/api/mlab_api.html#module-matplotlib.mlab
 +
 +window = Tk()
 +sizex = 200
 +sizey = 600
 +canvas = Canvas(window, width = sizex, height = sizey)
 +canvas.pack()
 +x = 100 # initial left-most edge of first ball
 +y = 1 # initial top-most edge of first ball
 +r = 4                  # ball diameter 
 +depx = 10            # displacement at each move in x direction
 +depy = 0
 +
 +# create balls:
 +no_particles = 6400
 +dy = (sizey-2.*y)/(no_particles+1)        # y initial separation between balls
 +print(dy)
 +ball_list = []
 +for i in range(no_particles):
 +    ball = canvas.create_oval(x,y,x+r,y+r,fill="red")
 +    y = y+dy
 +    ball_list.append(ball)
 +
 +#moves  
 +no_moves = 6  # number of moves
 +for j in range(no_moves):
 +    for ball in ball_list:
 +        canvas.move(ball, choice([-1,1])*depx, depy)
 +    canvas.after(1)
 +    canvas.update()
 +
 +#analysis - histogram
 +# see http://matplotlib.sourceforge.net/examples/api/histogram_demo.html
 +xpos=[]
 +for ball in ball_list:
 +    posi = canvas.coords(ball)
 +    xpos.append(((no_moves+1.)/no_moves)*(posi[0]-x)/depx)
 +    # le facteur (no_moves+1.)/no_moves) permet de gérer la largeur des barres de l'histogramme
 +xh = np.array(xpos)  # see http://www.scipy.org/Cookbook/BuildingArrays
 +#print(xh) 
 +
 +fig = plt.figure()
 +ax = fig.add_subplot(111)
 +n, bins, patches = ax.hist(xh, (no_moves)+1, facecolor='green', alpha=0.75)
 +print(n,bins, patches)
 +
 +plt.show()
 +
 +#window.mainloop()
 +
 +</code>
 +
 +===== Marche aléatoire d'un grand nombre de pas =====
 +
 +<code python 06_tkinter_random_walk_many_steps_1D.py>
 +#!/usr/bin/env python
 +# -*- coding: utf-8 -*-
 +
 +from tkinter import *
 +from random import choice     # http://docs.python.org/library/random.html
 +import numpy as np
 +import matplotlib.pyplot as plt     # http://matplotlib.sourceforge.net/api/pyplot_api.html#module-matplotlib.pyplot
 +import matplotlib.mlab as mlab    # http://matplotlib.sourceforge.net/api/mlab_api.html#module-matplotlib.mlab
 +
 +window = Tk()
 +sizex = 400
 +sizey = 400
 +canvas = Canvas(window, width = sizex, height = sizey)
 +canvas.pack()
 +x = 200 # initial left-most edge of first ball
 +y = 1 # initial top-most edge of first ball
 +r = 4                  # ball diameter 
 +depx = 1            # displacement at each move in x direction
 +depy = 0
 +
 +# create balls:
 +no_particles = 1600
 +dy = (sizey-2.)/(no_particles+1)         # y initial separation between balls
 +print(dy)
 +ball_list = []
 +for i in range(no_particles):
 +    ball = canvas.create_oval(x,y,x+r,y+r,fill="blue")
 +    y = y+dy
 +    ball_list.append(ball)
 +
 +#moves
 +no_moves = 200
 +for j in range(no_moves):
 +    for ball in ball_list:
 +        canvas.move(ball, choice([-1,1])*depx, depy)
 +    canvas.after(1)
 +    canvas.update()
 +
 +#analysis - histogram
 +# see http://matplotlib.sourceforge.net/examples/api/histogram_demo.html
 +xpos = []
 +for ball in ball_list:
 +    posi = canvas.coords(ball)
 +    xpos.append((posi[0]-x)/depx)
 +xh = np.array(xpos)  # see http://www.scipy.org/Cookbook/BuildingArrays
 +#  compute the mean mu and sigma  from xh (and/or theoretical value from random walk result)
 +mu = np.mean(xh)
 +sigma = np.std(xh)
 +fig = plt.figure()
 +ax = fig.add_subplot(111)
 +# print xh 
 +n, bins, patches = ax.hist(xh, 10, facecolor='green', alpha=0.75)
 +print(n,bins, patches)
 +# hist uses np.histogram to create 'n' and 'bins'. cf. http://docs.scipy.org/doc/numpy/reference/generated/numpy.histogram.html
 +
 +ax.set_xlabel('X positions')
 +ax.set_ylabel('Occurences')
 +
 +ax.grid(True)
 +
 +plt.show()
 +
 +#window.mainloop()
 +
 +</code>
 +
 +==== Avec analyse de la distribution : ====
 +<code python 07_tkinter_random_walk_many_steps_1D-analysis.py>
 +# -*- coding: utf-8 -*-
 +
 +from tkinter import *
 +from random import choice     # http://docs.python.org/library/random.html
 +import numpy as np
 +import matplotlib.pyplot as plt     # http://matplotlib.sourceforge.net/api/pyplot_api.html#module-matplotlib.pyplot
 +import matplotlib.mlab as mlab    # http://matplotlib.sourceforge.net/api/mlab_api.html#module-matplotlib.mlab
 +
 +window = Tk()
 +sizex = 400
 +sizey = 400
 +canvas = Canvas(window, width = sizex, height = sizey)
 +canvas.pack()
 +x = 200 # initial left-most edge of first ball
 +y = 1 # initial top-most edge of first ball
 +r = 4                  # ball diameter 
 +depx = 1            # displacement at each move in x direction
 +depy = 0
 +
 +# create balls:
 +no_particles = 1000
 +dy = (sizey-2.)/(no_particles+1)         # y initial separation between balls
 +#print dy
 +ball_list=[]
 +for i in range(no_particles):
 +    ball = canvas.create_oval(x,y,x+r,y+r,fill="blue")
 +    y = y+dy
 +    ball_list.append(ball)
 +
 +#moves
 +no_moves = 400
 +for j in range(no_moves):
 +    for ball in ball_list:
 +        canvas.move(ball, choice([-1,-1,-1,-1,1,1,1,1,1,1])*depx, depy) #drift
 +    canvas.after(1)
 +    canvas.update()
 +
 +#analysis - histogram
 +# see http://matplotlib.sourceforge.net/examples/api/histogram_demo.html
 +xpos = []
 +for ball in ball_list:
 +    posi = canvas.coords(ball)
 +    xpos.append(posi[0]-x)
 +xh = np.array(xpos)  # see http://www.scipy.org/Cookbook/BuildingArrays
 +#  compute the mean mu and sigma  from xh (and/or theoretical value from random walk result)
 +mu = np.mean(xh)
 +sigma = np.std(xh)
 +fig = plt.figure()
 +ax = fig.add_subplot(111)
 +# print xh 
 +n, bins, patches = ax.hist(xh, 20, facecolor='green', alpha=0.75)
 +print(mu, sigma)
 +print(n,bins, patches)
 +# hist uses np.histogram to create 'n' and 'bins'.
 +# np.histogram returns the bin edges, so there will be ii probability
 +# density values in n, ii+1 bin edges in bins and ii patches.  To get
 +# everything lined up, we'll compute the bin centers
 +bincenters = 0.5*(bins[1:]+bins[:-1])
 +# add a 'best fit' line for the normal PDF
 +yh = (bins[1]-bins[0])*no_particles*mlab.normpdf( bincenters, mu, sigma)  # http://matplotlib.sourceforge.net/api/mlab_api.html#matplotlib.mlab.normpdf
 +l = ax.plot(bincenters, yh, 'r--', linewidth=1)
 +#print n
 +ax.set_xlabel('X positions')
 +ax.set_ylabel('Occurences')
 +
 +ax.grid(True)
 +
 +plt.show()
  
 +#window.mainloop()
 +</code>
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  • Dernière modification : 2013/11/13 16:57
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