Interacting with the Kernel: IPython widgets

Note that if you're viewing this notebook statically (e.g. on nbviewer) the examples below will not work. They require connection to a running Python kernel



In [1]:

    
%run talktools
%matplotlib inline
import numpy as np
import matplotlib.pyplot as plt

Scientific Visualization

In practice, it's a long process of trial and error.

For example, let's say we have some data and want to draw a scatter plot:



In [2]:

    
np.random.seed(0)
x, y = np.random.normal(size=(2, 100))
s, c = np.random.random(size=(2, 100))
    
def draw_scatter(size=100, cmap='jet', alpha=1.0):
    fig, ax = plt.subplots(figsize=(8, 6))
    points = ax.scatter(x, y, s=size*s, c=c, alpha=alpha, cmap=cmap)
    fig.colorbar(points, ax=ax)
    return fig



In [3]:

    
draw_scatter(size=100, cmap='jet', alpha=1.0);

Let's make this interactive...

IPython 2.0 (released at the beginning of April 2014) adds an interactive widget interface:



In [4]:

    
from IPython.html.widgets import interact

colormaps = sorted(m for m in plt.cm.datad if not m.endswith("_r"))
interact(draw_scatter, size=[0, 2000], alpha=[0.0, 1.0], cmap=colormaps);

You don't even realize how cool this is...

These are Javascript events calling back to the Python Kernel which leads to display events in the browser.

We can use NetworkX to explore some types of random graphs:



In [5]:

    
import networkx as nx

def random_lobster(n, m, k, p):
    return nx.random_lobster(n, p, p / m)

def powerlaw_cluster(n, m, k, p):
    return nx.powerlaw_cluster_graph(n, m, p)

def erdos_renyi(n, m, k, p):
    return nx.erdos_renyi_graph(n, p)

def newman_watts_strogatz(n, m, k, p):
    return nx.newman_watts_strogatz_graph(n, k, p)

def plot_random_graph(n, m, k, p, generator):
    g = generator(n, m, k, p)
    nx.draw(g)
    plt.show()



In [6]:

    
interact(plot_random_graph, n=(2,30), m=(1,10), k=(1,10), p=(0.0, 1.0, 0.001),
        generator={'lobster': random_lobster,
                   'power law': powerlaw_cluster,
                   'Newman-Watts-Strogatz': newman_watts_strogatz,
                   u'Erdős-Rényi': erdos_renyi,
                   });

Exploring the Lorenz System

We can also call-back to arbitrarily complicated Python scripts.

One well-known set of differential equations is the Lorenz system:

$$ \begin{aligned} \dot{x} & = \sigma(y-x) \\ \dot{y} & = \rho x - y - xz \\ \dot{z} & = -\beta z + xy \end{aligned} $$

It exhibits a range of different behaviors as the parameters ($\sigma$, $\beta$, $\rho$) are varied.

Let's write a quick Lorenz solver...



In [7]:

    
import numpy as np
from scipy import integrate

from matplotlib import pyplot as plt
from mpl_toolkits.mplot3d import Axes3D
from matplotlib.colors import cnames
from matplotlib import animation



In [8]:

    
def solve_lorenz(N=10, angle=0.0, max_time=4.0, sigma=10.0, beta=8./3, rho=28.0):
    
    def lorenz_deriv((x, y, z), t0, sigma=sigma, beta=beta, rho=rho):
        """Compute the time-derivative of a Lorentz system."""
        return [sigma * (y - x), x * (rho - z) - y, x * y - beta * z]

    # Choose random starting points, uniformly distributed from -15 to 15
    np.random.seed(1)
    x0 = -15 + 30 * np.random.random((N, 3))

    # Solve for the trajectories
    t = np.linspace(0, max_time, int(250*max_time))
    x_t = np.asarray([integrate.odeint(lorenz_deriv, x0i, t)
                      for x0i in x0])
    
    # choose a different color for each trajectory
    colors = plt.cm.jet(np.linspace(0, 1, N))

    # plot the results
    fig = plt.figure()
    ax = fig.add_axes([0, 0, 1, 1], projection='3d')
    ax.axis('off')

    # prepare the axes limits
    ax.set_xlim((-25, 25))
    ax.set_ylim((-35, 35))
    ax.set_zlim((5, 55))

    for i in range(N):
        x, y, z = x_t[i,:,:].T
        lines = ax.plot(x, y, z, '-', c=colors[i])
        plt.setp(lines, linewidth=2)

    ax.view_init(30, angle)
    plt.show()

    return t, x_t



In [9]:

    
t, x_t = solve_lorenz()



In [10]:

    
from IPython.html.widgets import interactive

w = interactive(solve_lorenz, angle=(0.,360.), N=(0,50), sigma=(0.0,50.0), rho=(0.0,50.0))
display(w)



In [11]:

    
w.result









    Out[11]:





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         3.4034034 ,  3.40740741,  3.41141141,  3.41541542,  3.41941942,
         3.42342342,  3.42742743,  3.43143143,  3.43543544,  3.43943944,
         3.44344344,  3.44744745,  3.45145145,  3.45545546,  3.45945946,
         3.46346346,  3.46746747,  3.47147147,  3.47547548,  3.47947948,
         3.48348348,  3.48748749,  3.49149149,  3.4954955 ,  3.4994995 ,
         3.5035035 ,  3.50750751,  3.51151151,  3.51551552,  3.51951952,
         3.52352352,  3.52752753,  3.53153153,  3.53553554,  3.53953954,
         3.54354354,  3.54754755,  3.55155155,  3.55555556,  3.55955956,
         3.56356356,  3.56756757,  3.57157157,  3.57557558,  3.57957958,
         3.58358358,  3.58758759,  3.59159159,  3.5955956 ,  3.5995996 ,
         3.6036036 ,  3.60760761,  3.61161161,  3.61561562,  3.61961962,
         3.62362362,  3.62762763,  3.63163163,  3.63563564,  3.63963964,
         3.64364364,  3.64764765,  3.65165165,  3.65565566,  3.65965966,
         3.66366366,  3.66766767,  3.67167167,  3.67567568,  3.67967968,
         3.68368368,  3.68768769,  3.69169169,  3.6956957 ,  3.6996997 ,
         3.7037037 ,  3.70770771,  3.71171171,  3.71571572,  3.71971972,
         3.72372372,  3.72772773,  3.73173173,  3.73573574,  3.73973974,
         3.74374374,  3.74774775,  3.75175175,  3.75575576,  3.75975976,
         3.76376376,  3.76776777,  3.77177177,  3.77577578,  3.77977978,
         3.78378378,  3.78778779,  3.79179179,  3.7957958 ,  3.7997998 ,
         3.8038038 ,  3.80780781,  3.81181181,  3.81581582,  3.81981982,
         3.82382382,  3.82782783,  3.83183183,  3.83583584,  3.83983984,
         3.84384384,  3.84784785,  3.85185185,  3.85585586,  3.85985986,
         3.86386386,  3.86786787,  3.87187187,  3.87587588,  3.87987988,
         3.88388388,  3.88788789,  3.89189189,  3.8958959 ,  3.8998999 ,
         3.9039039 ,  3.90790791,  3.91191191,  3.91591592,  3.91991992,
         3.92392392,  3.92792793,  3.93193193,  3.93593594,  3.93993994,
         3.94394394,  3.94794795,  3.95195195,  3.95595596,  3.95995996,
         3.96396396,  3.96796797,  3.97197197,  3.97597598,  3.97997998,
         3.98398398,  3.98798799,  3.99199199,  3.995996  ,  4.        ]),
 array([[[ -2.48933986e+00,   6.60973480e+00,  -1.49965688e+01],
         [ -2.14077645e+00,   6.18646806e+00,  -1.48962127e+01],
         [ -1.82130748e+00,   5.82299967e+00,  -1.47853208e+01],
         ..., 
         [  6.87667416e+00,   1.07734499e+01,   1.78286316e+01],
         [  7.03431517e+00,   1.10115546e+01,   1.79410297e+01],
         [  7.19515729e+00,   1.12517722e+01,   1.80659207e+01]],
 
        [[ -5.93002282e+00,  -1.05973233e+01,  -1.22298422e+01],
         [ -6.13136261e+00,  -1.15193465e+01,  -1.18344052e+01],
         [ -6.36138540e+00,  -1.24619735e+01,  -1.14104771e+01],
         ..., 
         [ -1.11085203e+01,  -1.62261151e+01,   2.32341935e+01],
         [ -1.13121656e+01,  -1.63640151e+01,   2.37150767e+01],
         [ -1.15128567e+01,  -1.64827701e+01,   2.42098140e+01]],
 
        [[ -9.41219366e+00,  -4.63317819e+00,  -3.09697577e+00],
         [ -9.24717733e+00,  -5.76936651e+00,  -2.87083263e+00],
         [ -9.13260659e+00,  -6.87477643e+00,  -2.60897637e+00],
         ..., 
         [  8.94933599e+00,   1.00204877e+01,   2.61339939e+01],
         [  8.99188828e+00,   1.00458891e+01,   2.62149083e+01],
         [  9.03371359e+00,   1.00685518e+01,   2.62975123e+01]],
 
        ..., 
        [[  1.40478473e+01,  -5.59727466e+00,   5.76967847e+00],
         [  1.33015844e+01,  -4.35137818e+00,   5.43762841e+00],
         [  1.26324566e+01,  -3.15868945e+00,   5.18607122e+00],
         ..., 
         [ -5.62183444e+00,  -9.08227752e+00,   1.57576353e+01],
         [ -5.76241216e+00,  -9.32397964e+00,   1.57989149e+01],
         [ -5.90705639e+00,  -9.57059699e+00,   1.58506576e+01]],
 
        [[  1.12916746e+01,   1.18381999e+01,  -1.24486737e+01],
         [  1.13481065e+01,   1.36039497e+01,  -1.17430178e+01],
         [  1.14710812e+01,   1.53431579e+01,  -1.09606825e+01],
         ..., 
         [ -1.47677316e-02,  -1.23193351e-02,   1.01977111e+01],
         [ -1.46914304e-02,  -1.33208729e-02,   1.00894066e+01],
         [ -1.46573919e-02,  -1.43208005e-02,   9.98225246e+00]],
 
        [[ -1.38283565e+01,  -9.90508741e+00,   1.13442751e+01],
         [ -1.36915606e+01,  -1.07690154e+01,   1.17903119e+01],
         [ -1.35933180e+01,  -1.15964765e+01,   1.22727276e+01],
         ..., 
         [  1.61306741e+01,   1.93584734e+01,   3.43088188e+01],
         [  1.62473985e+01,   1.88446346e+01,   3.51764615e+01],
         [  1.63382575e+01,   1.82747191e+01,   3.60076113e+01]]]))



In [11]: