%pylab inline
# Pylab is a module that provides a Matlab like namespace by importing functions from the modules Numpy and Matplotlib.
#                      %matplotlib inline turns on “inline plotting”, where plot graphics will appear in your notebook.
from itertools import cycle  # This module implements a number of iterator building blocks inspired by constructs from APL, Haskell, and SML.
#                           see <https://docs.python.org/3/library/itertools.html#itertools.cycle>
from mpl_toolkits.mplot3d import Axes3D  # see <https://matplotlib.org/stable/api/toolkits/mplot3d.html>
colors = cycle('bgrcmykbgrcmykbgrcmykbgrcmyk')  # see cycle description above.
import numpy as np

# Define parameters for the walk
dims = 1          # in this case the walk is either forward or backward
step_n = 10000    # There will be 10,000 steps
step_set = [-1, 0, 1] # create an array of one step forward or one step backward, or no step

origin = np.zeros((1,dims)) # create an array of zeros
print(origin.shape)

# Simulate steps in 1D
step_shape = (step_n,dims)     # step-shape array created with dimensions of (10,000, 1)
print(type(step_shape))
steps = np.random.choice(a=step_set, size=step_shape) # a random sample is chosen from [step_set] and step_n *

path = np.concatenate([origin, steps]).cumsum(0)
start = path[:1]
stop = path[-1:]

# Plot the path
fig = plt.figure(figsize=(8,4),dpi=200) # create the plot frame
ax = fig.add_subplot(111)               # create one subplot
ax.scatter(np.arange(step_n+1), path, c='blue',alpha=0.25,s=0.05); # plot the scatter diagram of the path data and the array index
ax.plot(path,c='blue',alpha=0.5,lw=0.5,ls='-',); # plot the lines between the scatter points
ax.plot(0, start, c='red', marker='+')           # plot the start point
ax.plot(step_n, stop, c='black', marker='o')     # plot the stop point
plt.title('1D Random Walk')                     # label the plot
plt.tight_layout(pad=0)
#plt.savefig('plots/random_walk_1d.png',dpi=250);

# Define parameters for the walk for a two dimensional walk the same as the one dimensional
dims = 2
step_n = 10000
step_set = [-1, 0, 1]
origin = np.zeros((1,dims))

# Simulate steps in 2D
step_shape = (step_n,dims)
steps = np.random.choice(a=step_set, size=step_shape) # calculate the next step
#print(steps)
path = np.concatenate([origin, steps]).cumsum(0)     # calculate the next position
#print(path)
start = path[:1]
stop = path[-1:]

# Plot the path
fig = plt.figure(figsize=(8,8),dpi=200)
ax = fig.add_subplot(111)
ax.scatter(path[:,0], path[:,1],c='blue',alpha=0.25,s=0.05);  # here the scatter diagram shows the two dimensional position.
ax.plot(path[:,0], path[:,1],c='blue',alpha=0.5,lw=0.25,ls='-'); # and here the dots are connected
ax.plot(start[:,0], start[:,1],c='red', marker='+')
ax.plot(stop[:,0], stop[:,1],c='black', marker='o')
plt.title('2D Random Walk')
plt.tight_layout(pad=0)
#plt.savefig(‘plots/random_walk_2d.png’,dpi=250);

# Define parameters for the walk in three dimenions
dims = 3
step_n = 1000
step_set = [-1, 0, 1]
origin = np.zeros((1,dims))

# Simulate steps in 3D
step_shape = (step_n,dims)
steps = np.random.choice(a=step_set, size=step_shape)
path = np.concatenate([origin, steps]).cumsum(0)
start = path[:1]
stop = path[-1:]

# Plot the path
fig = plt.figure(figsize=(10,10),dpi=200)
ax = fig.add_subplot(111, projection='3d')
ax.grid(False)
ax.xaxis.pane.fill = ax.yaxis.pane.fill = ax.zaxis.pane.fill = False
ax.set_xlabel('X')
ax.set_ylabel('Y')
ax.set_zlabel('Z')
ax.scatter3D(path[:,0], path[:,1], path[:,2], c='blue', alpha=0.25,s=1)
ax.plot3D(path[:,0], path[:,1], path[:,2], c='blue', alpha=0.5, lw=0.5)
ax.plot3D(start[:,0], start[:,1], start[:,2], c='red', marker='+')
ax.plot3D(stop[:,0], stop[:,1], stop[:,2], c='black', marker='o')
plt.title('3D Random Walk')
#plt.savefig(‘plots/random_walk_3d.png’,dpi=250);

# Define parameters for the walk for ten walks of 1000 steps in 3 dimensions
dims = 3
n_runs = 10
step_n = 1000
step_set = [-1, 0 ,1]
runs = np.arange(n_runs)
step_shape = (step_n,dims)

# Plot
fig = plt.figure(figsize=(10,10),dpi=250)
ax = fig.add_subplot(111, projection='3d')
ax.grid(False)
ax.xaxis.pane.fill = ax.yaxis.pane.fill = ax.zaxis.pane.fill = False
ax.set_xlabel('X')
ax.set_ylabel('Y')
ax.set_zlabel('Y')

for i, col in zip(runs, colors):    
# Simulate steps in 3D
    origin = np.random.randint(low=-10,high=10,size=(1,dims))
    steps = np.random.choice(a=step_set, size=step_shape)
    path = np.concatenate([origin, steps]).cumsum(0)
    start = path[:1]
    stop = path[-1:]    
# Plot the path
    ax.scatter3D(path[:,0], path[:,1], path[:,2],c=col,alpha=0.15,s=1);
    ax.plot3D(path[:,0], path[:,1], path[:,2], c=col, alpha=0.25,lw=0.25)
    ax.plot3D(start[:,0], start[:,1], start[:,2],c=col, marker='+')
    ax.plot3D(stop[:,0], stop[:,1], stop[:,2],c=col, marker='o');

plt.title('3D Random Walk - Multiple runs')
#plt.savefig(‘plots/random_walk_3d_multiple_runs.png’,dpi=250);