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teaching:exos:simulations_random_walks_codes

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.

Génération de nombres aléatoires

01_random.py
#!/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 * 
 
facepiece = ['pile','face']
valeurpiece = [0.01,0.02,0.05,0.1,0.2,0.5,1.,2.]
 
for i in range(1):
    # choice : random choice of an element from a list
    print(choice(facepiece), choice(valeurpiece))
    # randint : return a random integer number between 2 values (including limits)
    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
 
 
# seed(ANY_DATA) : seeding of the random number generator with any (constant) data
# in order to generate reproducible random sequences.
# seed() - without data - uses internal clock value to "randomly" initiate the generator !
 
for j in range(3):
    #seed('ma chaîne personnielle')  # reproducible initialization
    seed()   # to randomly initiate the generator
    for i in range(10):
        print(randint(1000,9999))
    print(" ")

Histogrammes de nombres aléatoires

02_random_histogram.py
#!/usr/bin/env python
# -*- coding: utf-8 -*-
 
from random import *    # cf. documentation cf 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
 
#seed('ma chaîne personnelle')  # reproducible initialization
seed()
 
rval = []
for j in range(100000):
    rval.append(randint(0,99))   # append to the list a random (integer) number between 0 and 99
 
# print rval  # uncomment just to see the list of random numbers
 
# analysis - histogram  -  see http://matplotlib.sourceforge.net/examples/api/histogram_demo.html
# 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
# print(xh)
 
fig = plt.figure()
ax = fig.add_subplot(111)
 
n, bins, patches = ax.hist(xh, 50, facecolor='green', alpha=0.75)
print(n)  # les nombres d'occurences par classe
print(bins)  # les classes, de largeur identique
 
# modifier le nombre de nombres générés, les nombres de classes-bins, 
 
plt.show()

Représenter le déplacement d'un objet

03_tkinter_simple_move.py
#!/usr/bin/python
# -*- coding: utf-8 -*-
 
from tkinter import *
import time
 
window = Tk()
sizex = 400
sizey = 200
canvas = Canvas(window, width = sizex, height = sizey)
canvas.pack()
x = 100		# initial left-most edge of first ball
y = 30		# initial top-most edge of first ball
r = 20                  # ball diameter 
depx = 2            # displacement at each move in x direction
depy = 1            # displacement at each move in y direction
 
ball=canvas.create_oval(x,y,x+r,y+r,fill="blue")
 
#moves
no_moves = 140
for j in range(no_moves):
    canvas.move(ball, depx, depy)
    canvas.after(20)         # time delay in milliseconds
    canvas.update()
 
time.sleep(5) # on attend quelques secondes
window.destroy()

Représenter le déplacement de nombreux points

04_tkinter_many_moves.py
#!/usr/bin/python
# -*- coding: utf-8 -*-
 
from tkinter import *
import time
from random import * 
 
window = Tk()
sizex = 400
sizey = 600
canvas = Canvas(window, width = sizex, height = sizey)
canvas.pack()
x = 100		# initial left-most edge of first ball
y = 30		# initial top-most edge of first ball
r = 16                  # ball diameter 
depx = 2            # displacement at each move in x direction
depy = 0            # displacement at each move in y direction
 
# create balls:
no_particles = 20
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="blue")
    y = y+dy
    ball_list.append(ball)
 
#moves
no_moves = 100
for j in range(no_moves):
    for ball in ball_list:
        canvas.move(ball, depx, choice([-2, 2]) )
#        canvas.move(ball, depx, depy)
    canvas.after(10)
    canvas.update()
 
time.sleep(5) # on attend quelques secondes
window.destroy()

Marche aléatoire d'un petit nombre de pas

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()

Marche aléatoire d'un grand nombre de pas

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()

Avec analyse de la distribution :

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()
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teaching/exos/simulations_random_walks_codes.txt · Dernière modification: 2018/11/05 12:09 par villersd