import ipywidgets as widgets
from ipywidgets import interact
import matplotlib.pyplot as plt
import matplotlib.ticker as ticker
from mpl_toolkits.mplot3d.axes3d import Axes3D
import numpy as np
import scipy.stats as stats
from sympy import lambdify, symbols
[docs]def Plot_continuous(x_limits, func, *args, **kwargs):
"""
Plots a continuous distribution
Parameters
----------
x_limits : iterable
Array, list, or tuple of size 2, containing the lower and upper bound
to be plotted.
func : Sympy expression, function, or str
Univariate density function or a String for the scipy dist name.
args : floats (optional)
Arguments of the distribution if func was a String.
Optional Named Parameters
-------------------------
tails : boolean (optional)
If True, left_end will shade from the lower bound up to left_end, and
right_end will shade from right_end up to the upper bound. If False,
left_end will shade to right_end. (Default: False)
left_end : float (optional)
Left side of event to be shaded. (Default: None)
right_end : float (optional)
Right side of event to be shaded. (Default: None)
cdf : boolean (optional)
If True and func was string, the cdf will be plotted (Default: False)
All pyplot named arguments (such as color) should work as well. See
http://matplotlib.org/api/pyplot_api.html#matplotlib.pyplot.plot
Returns
-------
None
"""
interval = x_limits
tails = kwargs.pop('tails', False)
if isinstance(func, str):
func = func.lower()
rv = getattr(stats, func)
cdf = kwargs.pop('cdf', False)
if cdf:
f = rv(*args).cdf
else:
f = rv(*args).pdf
elif callable(func):
f = func
else:
assert len(func.free_symbols) <= 1, 'Must have exactly 1 variable'
if len(func.free_symbols) == 0:
new_symbol = symbols('x')
replace_variables = [new_symbol]
else:
replace_variables = list(func.free_symbols)
f = lambdify(replace_variables, func)
lower = interval[0]
upper = interval[1]
x = np.linspace(lower, upper, 100)
y = [f(xi) for xi in x]
left = kwargs.pop('left_end', None)
right = kwargs.pop('right_end', None)
options = {'lw': 2, 'color': 'darkblue'}
options.update(kwargs)
plt.plot(x, y, **options)
plt.ylim(0, max(y) * 1.1)
if tails:
if left and left >= lower:
x2 = np.linspace(lower, left, 100)
plt.fill_between(x2, f(x2), alpha=0.7, color='gold')
if right and right <= upper:
x2 = np.linspace(right, upper, 100)
plt.fill_between(x2, f(x2), alpha=0.7, color='gold')
else:
lb = left if left else lower
rb = right if right else upper
if left or right:
x2 = np.linspace(lb, rb, 100)
plt.fill_between(x2, f(x2), alpha=0.7, color='gold')
# Multiple all y-ticks by 100 to get percent per unit.
ax = plt.gca()
ticks = ticker.FuncFormatter(lambda x, pos: '{0:g}'.format(x * 100.))
ax.yaxis.set_major_formatter(ticks)
plt.ylabel('Percent per unit')
[docs]def Plot_3d(x_limits, y_limits, f, interactive=False, figsize=(12, 8), **kwargs):
"""
Plots a 3d graph.
Parameters
----------
x_limits : iterable
Array, list, or tuple of size 2, containing the lower and upper bound
of the x-axis.
y_limits : iterable
Array, list, or tuple of size 2, containing the lower and upper bound
of the x-axis.
f : bivariate function
Joint density
interactive : boolean (optional)
If True, creates a widget to adjust elevation and azimuth.
(default: False)
kwargs
Optional named arguments for `plot_surface`.
Returns
-------
None
"""
def plot(elev, azim):
fig = plt.figure(figsize=figsize)
ax = fig.add_subplot(111, projection='3d')
x = np.linspace(*x_limits, 100)
y = np.linspace(*y_limits, 100)
X, Y = np.meshgrid(x, y)
v = np.vectorize(f)
Z = v(X, Y)
ax.plot_surface(X, Y, Z, **kwargs)
ax.set_xlabel('x')
ax.set_ylabel('y')
ax.set_zlabel('f(x, y)')
ax.view_init(elev, azim)
if interactive:
elevation_slider = widgets.FloatSlider(
value=20,
min=0,
max=90,
step=1,
description='elevation'
)
azimuth_slider = widgets.FloatSlider(
value=-100,
min=-180,
max=180,
step=1,
description='azimuth'
)
@interact(elev=elevation_slider, azim=azimuth_slider)
def wrapper(elev, azim):
plot(elev, azim)
else:
plot(20, -100)
[docs]def Plot_bivariate_normal(mu, cov, **kwargs):
"""
Plots the density of a bivariate normal distribution.
Parameters
----------
mu : array
Array of length 2 for mean.
cov : array
Covariance matrix of dimension 2x2.
"""
def normal_density(x, y):
return stats.multivariate_normal.pdf([x, y], mean=mu, cov=cov)
sd = np.sqrt(np.diag(cov))
lower = mu - sd * 4
upper = mu + sd * 4
options = {
'x_limits': (lower[0], upper[0]),
'y_limits': (lower[1], upper[1]),
'f': normal_density,
}
options.update(kwargs)
Plot_3d(**options)
[docs]def Scatter_multivariate_normal(mu, cov, n):
"""
Draws scatterplot for a trivariate normal distribution.
Parameters
----------
mu : array
Array of length 3 corresponding to the means.
cov : array
3x3 Covariance Matrix.
n : int
Number of points to plot.
"""
points = stats.multivariate_normal.rvs(mu, cov, n)
fig = plt.figure()
ax = fig.add_subplot(111, projection='3d')
ax.scatter(points[:, 0], points[:, 1], points[:, 2], alpha=0.5, s=10,
color='darkblue')
ax.set_xlabel('Variable 1')
ax.set_ylabel('Variable 2')
ax.set_zlabel('Variable 3')
ax.dist = 12
[docs]def multivariate_normal_regression(mu, cov, n=100, figsize=(8, 6)):
"""
Draws a scatter plot of points drawn from a trivariate normal distribution
and the corresponding regresson plane.
Random vectors should be of the form [Y X1 X2]^T.
Parameters
----------
mu : array
[mu_Y, mu_X1, mu_X2].
cov : array
3x3 covariance matrix.
n : int (optional)
Number of points to plot.
figsize : tuple (optional)
Size of figure.
"""
y, x1, x2 = stats.multivariate_normal.rvs(mu, cov, n).T
sigma_X = cov[1:, 1:]
sigma_Y = cov[1:, 0:1]
A = sigma_Y.T.dot(np.linalg.inv(sigma_X)).flatten()
d = max(np.sqrt(np.diag(cov))[1:]) * 4
X1 = np.linspace(mu[1] - d, mu[1] + d, 100)
X2 = np.linspace(mu[2] - d, mu[2] + d, 100)
X1, X2 = np.meshgrid(X1, X2)
Y = A[0] * (X1 - mu[1]) + A[1] * (X2 - mu[2]) + mu[0]
fig = plt.figure(figsize=figsize)
ax = fig.add_subplot(111, projection='3d')
ax.scatter(x1, x2, y, s=10)
ax.set_xlabel('$x1$')
ax.set_ylabel('$x2$')
ax.set_zlabel('$y$')
ax.plot_surface(X1, X2, Y, alpha=0.3, color='gold')
[docs]def Plot_expon(x_limits, lamb, **kwargs):
"""
Plots an exponential distribution
Parameters
----------
x_limits : iterable
Array, list, or tuple of size 2, containing the lower and upper bound
to be plotted.
lamb : float
Rate.
Optional Named Parameters
-------------------------
tails : boolean (optional)
If True, left_end will shade from the lower bound up to left_end, and
right_end will shade from right_end up to the upper bound. If False,
left_end will shade to right_end. (Default: False)
left_end : float (optional)
Left side of event to be shaded. (Default: None)
right_end : float (optional)
Right side of event to be shaded. (Default: None)
cdf : boolean (optional)
If True and func was string, the cdf will be plotted (Default: False)
All pyplot named arguments (such as color) should work as well. See
http://matplotlib.org/api/pyplot_api.html#matplotlib.pyplot.plot
Returns
-------
None
"""
Plot_continuous(x_limits, 'expon', 0, 1/lamb, **kwargs)
[docs]def Plot_norm(x_limits, mu, sigma, **kwargs):
"""
Plots a gaussian distribution.
Parameters
----------
x_limits : iterable
Array, list, or tuple of size 2, containing the lower and upper bound
to be plotted.
mu : float
Mean.
sigma : float
Standard Deviation.
Optional Named Parameters
-------------------------
tails : boolean (optional)
If True, left_end will shade from the lower bound up to left_end, and
right_end will shade from right_end up to the upper bound. If False,
left_end will shade to right_end. (Default: False)
left_end : float (optional)
Left side of event to be shaded. (Default: None)
right_end : float (optional)
Right side of event to be shaded. (Default: None)
cdf : boolean (optional)
If True and func was string, the cdf will be plotted (Default: False)
All pyplot named arguments (such as color) should work as well. See
http://matplotlib.org/api/pyplot_api.html#matplotlib.pyplot.plot
Returns
-------
None
"""
Plot_continuous(x_limits, 'norm', mu, sigma, **kwargs)
[docs]def Plot_arcsine(x_limits, **kwargs):
"""
Plots an arcsine distribution.
Parameters
----------
x_limits : iterable
Array, list, or tuple of size 2, containing the lower and upper bound
to be plotted.
Optional Named Parameters
-------------------------
tails : boolean (optional)
If True, left_end will shade from the lower bound up to left_end, and
right_end will shade from right_end up to the upper bound. If False,
left_end will shade to right_end. (Default: False)
left_end : float (optional)
Left side of event to be shaded. (Default: None)
right_end : float (optional)
Right side of event to be shaded. (Default: None)
cdf : boolean (optional)
If True and func was string, the cdf will be plotted (Default: False)
All pyplot named arguments (such as color) should work as well. See
http://matplotlib.org/api/pyplot_api.html#matplotlib.pyplot.plot
Returns
-------
None
"""
Plot_continuous(x_limits, 'arcsine', **kwargs)
[docs]def Plot_beta(x_limits, a, b, **kwargs):
"""
Plots a beta distribution.
Parameters
----------
x_limits : iterable
Array, list, or tuple of size 2, containing the lower and upper bound
to be plotted.
a : float
Shape.
b : float
Shape.
Optional Named Parameters
-------------------------
tails : boolean (optional)
If True, left_end will shade from the lower bound up to left_end, and
right_end will shade from right_end up to the upper bound. If False,
left_end will shade to right_end. (Default: False)
left_end : float (optional)
Left side of event to be shaded. (Default: None)
right_end : float (optional)
Right side of event to be shaded. (Default: None)
cdf : boolean (optional)
If True and func was string, the cdf will be plotted (Default: False)
All pyplot named arguments (such as color) should work as well. See
http://matplotlib.org/api/pyplot_api.html#matplotlib.pyplot.plot
Returns
-------
None
"""
Plot_continuous(x_limits, 'beta', a, b, **kwargs)
[docs]def Plot_cauchy(x_limits, loc=0, scale=1, **kwargs):
"""
Plots a cauchy distribution.
Parameters
----------
x_limits : iterable
Array, list, or tuple of size 2, containing the lower and upper bound
to be plotted.
Optional Named Parameters
-------------------------
tails : boolean (optional)
If True, left_end will shade from the lower bound up to left_end, and
right_end will shade from right_end up to the upper bound. If False,
left_end will shade to right_end. (Default: False)
left_end : float (optional)
Left side of event to be shaded. (Default: None)
right_end : float (optional)
Right side of event to be shaded. (Default: None)
cdf : boolean (optional)
If True and func was string, the cdf will be plotted (Default: False)
All pyplot named arguments (such as color) should work as well. See
http://matplotlib.org/api/pyplot_api.html#matplotlib.pyplot.plot
Returns
-------
None
"""
Plot_continuous(x_limits, 'cauchy', loc, scale, **kwargs)
[docs]def Plot_chi2(x_limits, df, **kwargs):
"""
Plots a chi-squared distribution.
Parameters
----------
x_limits : iterable
Array, list, or tuple of size 2, containing the lower and upper bound
to be plotted.
df : Integer
Number of degrees of freedom.
Optional Named Parameters
-------------------------
tails : boolean (optional)
If True, left_end will shade from the lower bound up to left_end, and
right_end will shade from right_end up to the upper bound. If False,
left_end will shade to right_end. (Default: False)
left_end : float (optional)
Left side of event to be shaded. (Default: None)
right_end : float (optional)
Right side of event to be shaded. (Default: None)
cdf : boolean (optional)
If True and func was string, the cdf will be plotted (Default: False)
All pyplot named arguments (such as color) should work as well. See
http://matplotlib.org/api/pyplot_api.html#matplotlib.pyplot.plot
Returns
-------
None
"""
Plot_continuous(x_limits, 'chi2', df, **kwargs)
[docs]def Plot_erlang(x_limits, r, lamb, **kwargs):
"""
Plots an erlang distribution.
Parameters
----------
x_limits : iterable
Array, list, or tuple of size 2, containing the lower and upper bound
to be plotted.
r : int
Shape.
lamb : float
Rate.
Optional Named Parameters
-------------------------
tails : boolean (optional)
If True, left_end will shade from the lower bound up to left_end, and
right_end will shade from right_end up to the upper bound. If False,
left_end will shade to right_end. (Default: False)
left_end : float (optional)
Left side of event to be shaded. (Default: None)
right_end : float (optional)
Right side of event to be shaded. (Default: None)
cdf : boolean (optional)
If True and func was string, the cdf will be plotted (Default: False)
All pyplot named arguments (such as color) should work as well. See
http://matplotlib.org/api/pyplot_api.html#matplotlib.pyplot.plot
Returns
-------
None
"""
Plot_continuous(x_limits, 'erlang', r, 0, 1 / lamb, **kwargs)
[docs]def Plot_f(x_limits, dfn, dfd, **kwargs):
"""
Plots an F distribution.
Parameters
----------
x_limits : iterable
Array, list, or tuple of size 2, containing the lower and upper bound
to be plotted.
dfn : int
Degree of freedom 1.
dfd : int
Degree of freedom 2.
Optional Named Parameters
-------------------------
tails : boolean (optional)
If True, left_end will shade from the lower bound up to left_end, and
right_end will shade from right_end up to the upper bound. If False,
left_end will shade to right_end. (Default: False)
left_end : float (optional)
Left side of event to be shaded. (Default: None)
right_end : float (optional)
Right side of event to be shaded. (Default: None)
cdf : boolean (optional)
If True and func was string, the cdf will be plotted (Default: False)
All pyplot named arguments (such as color) should work as well. See
http://matplotlib.org/api/pyplot_api.html#matplotlib.pyplot.plot
Returns
-------
None
"""
Plot_continuous(x_limits, 'f', dfn, dfd, **kwargs)
[docs]def Plot_gamma(x_limits, r, lamb, **kwargs):
"""
Plots a gamma distribution.
Parameters
----------
x_limits : iterable
Array, list, or tuple of size 2, containing the lower and upper bound
to be plotted.
r : int
Shape.
lamb : float
Rate.
Optional Named Parameters
-------------------------
tails : boolean (optional)
If True, left_end will shade from the lower bound up to left_end, and
right_end will shade from right_end up to the upper bound. If False,
left_end will shade to right_end. (Default: False)
left_end : float (optional)
Left side of event to be shaded. (Default: None)
right_end : float (optional)
Right side of event to be shaded. (Default: None)
cdf : boolean (optional)
If True and func was string, the cdf will be plotted (Default: False)
All pyplot named arguments (such as color) should work as well. See
http://matplotlib.org/api/pyplot_api.html#matplotlib.pyplot.plot
Returns
-------
None
"""
Plot_continuous(x_limits, 'gamma', r, 0, 1 / lamb, **kwargs)
[docs]def Plot_lognorm(x_limits, mu, sigma, **kwargs):
"""
Plots a log-normal distribution.
Parameters
----------
x_limits : iterable
Array, list, or tuple of size 2, containing the lower and upper bound
to be plotted.
mu : float
Mean.
sigma : float
Standard Deviation.
Optional Named Parameters
-------------------------
tails : boolean (optional)
If True, left_end will shade from the lower bound up to left_end, and
right_end will shade from right_end up to the upper bound. If False,
left_end will shade to right_end. (Default: False)
left_end : float (optional)
Left side of event to be shaded. (Default: None)
right_end : float (optional)
Right side of event to be shaded. (Default: None)
cdf : boolean (optional)
If True and func was string, the cdf will be plotted (Default: False)
All pyplot named arguments (such as color) should work as well. See
http://matplotlib.org/api/pyplot_api.html#matplotlib.pyplot.plot
Returns
-------
None
"""
Plot_continuous(x_limits, 'lognorm', sigma, 0, np.exp(mu), **kwargs)
[docs]def Plot_pareto(x_limits, alpha, **kwargs):
"""
Plots an alpha distribution.
Parameters
----------
x_limits : iterable
Array, list, or tuple of size 2, containing the lower and upper bound
to be plotted.
a : float
Shape.
Optional Named Parameters
-------------------------
tails : boolean (optional)
If True, left_end will shade from the lower bound up to left_end, and
right_end will shade from right_end up to the upper bound. If False,
left_end will shade to right_end. (Default: False)
left_end : float (optional)
Left side of event to be shaded. (Default: None)
right_end : float (optional)
Right side of event to be shaded. (Default: None)
cdf : boolean (optional)
If True and func was string, the cdf will be plotted (Default: False)
All pyplot named arguments (such as color) should work as well. See
http://matplotlib.org/api/pyplot_api.html#matplotlib.pyplot.plot
Returns
-------
None
"""
Plot_continuous(x_limits, 'pareto', alpha, **kwargs)
[docs]def Plot_powerlaw(x_limits, a, **kwargs):
"""
Plots a powerlaw distribution.
Parameters
----------
x_limits : iterable
Array, list, or tuple of size 2, containing the lower and upper bound
to be plotted.
a : float
Shape.
Optional Named Parameters
-------------------------
tails : boolean (optional)
If True, left_end will shade from the lower bound up to left_end, and
right_end will shade from right_end up to the upper bound. If False,
left_end will shade to right_end. (Default: False)
left_end : float (optional)
Left side of event to be shaded. (Default: None)
right_end : float (optional)
Right side of event to be shaded. (Default: None)
cdf : boolean (optional)
If True and func was string, the cdf will be plotted (Default: False)
All pyplot named arguments (such as color) should work as well. See
http://matplotlib.org/api/pyplot_api.html#matplotlib.pyplot.plot
Returns
-------
None
"""
Plot_continuous(x_limits, 'powerlaw', a, **kwargs)
[docs]def Plot_rayleigh(x_limits, sigma, **kwargs):
"""
Plots a rayleigh distribution.
Parameters
----------
x_limits : iterable
Array, list, or tuple of size 2, containing the lower and upper bound
to be plotted.
sigma : float
Scale.
Optional Named Parameters
-------------------------
tails : boolean (optional)
If True, left_end will shade from the lower bound up to left_end, and
right_end will shade from right_end up to the upper bound. If False,
left_end will shade to right_end. (Default: False)
left_end : float (optional)
Left side of event to be shaded. (Default: None)
right_end : float (optional)
Right side of event to be shaded. (Default: None)
cdf : boolean (optional)
If True and func was string, the cdf will be plotted (Default: False)
All pyplot named arguments (such as color) should work as well. See
http://matplotlib.org/api/pyplot_api.html#matplotlib.pyplot.plot
Returns
-------
None
"""
Plot_continuous(x_limits, 'rayleigh', 0, sigma, **kwargs)
[docs]def Plot_t(x_limits, df, **kwargs):
"""
Plots a t distribution.
Parameters
----------
x_limits : iterable
Array, list, or tuple of size 2, containing the lower and upper bound
to be plotted.
df : int
Degree of freedom
Optional Named Parameters
-------------------------
tails : boolean (optional)
If True, left_end will shade from the lower bound up to left_end, and
right_end will shade from right_end up to the upper bound. If False,
left_end will shade to right_end. (Default: False)
left_end : float (optional)
Left side of event to be shaded. (Default: None)
right_end : float (optional)
Right side of event to be shaded. (Default: None)
cdf : boolean (optional)
If True and func was string, the cdf will be plotted (Default: False)
All pyplot named arguments (such as color) should work as well. See
http://matplotlib.org/api/pyplot_api.html#matplotlib.pyplot.plot
Returns
-------
None
"""
Plot_continuous(x_limits, 't', df, **kwargs)
[docs]def Plot_triang(x_limits, a, b, c, **kwargs):
"""
Plots a triangular distribution.
Parameters
----------
x_limits : iterable
Array, list, or tuple of size 2, containing the lower and upper bound
to be plotted.
a : float
Minimum value.
b : float
Maximum value.
c : float
Intermediate value
Optional Named Parameters
-------------------------
tails : boolean (optional)
If True, left_end will shade from the lower bound up to left_end, and
right_end will shade from right_end up to the upper bound. If False,
left_end will shade to right_end. (Default: False)
left_end : float (optional)
Left side of event to be shaded. (Default: None)
right_end : float (optional)
Right side of event to be shaded. (Default: None)
cdf : boolean (optional)
If True and func was string, the cdf will be plotted (Default: False)
All pyplot named arguments (such as color) should work as well. See
http://matplotlib.org/api/pyplot_api.html#matplotlib.pyplot.plot
Returns
-------
None
"""
Plot_continuous(x_limits, 'triang', (c - a) / (b - a), a, b - a, **kwargs)
