distribution = {'norm': (norm(loc=loc, scale=scale),

r'Normal Distribution',

r'$p(x)=\frac{\exp(-x^2/2)}{\sqrt{2\pi}}$' + r'loc = {:.2f}, scale = {:.2f}'.format(loc, scale)),

'expon': (expon(loc=loc, scale=scale),

r'Exponential Distribution',

r'$p(x)=\exp(-x)}$' + r'loc = {:.2f}, scale = {:.2f}'.format(loc, scale)),

'powerlaw': (powerlaw(a=shape+1),

r'Power Law Distribution',

r'$p(x,\alpha)=\alpha x^{\alpha-1}$' + r', $\alpha-1 = {:.2f}$'.format(shape)),

'lognorm': (lognorm(s=shape),

r'Lognormal Distribution',

r'$p(x,s)=\frac{1}{sx\sqrt{2\pi}} exp(-\frac{\log^2(x)}{2s^2})$' +

r', $shape = {:.2f}$'.format(shape))}

shape=1, figsize=(12, 4), log_scale=True):

""" Plot theoretical distribution and a random sample from the same distribution. """

dist = set_distribution(name=name, loc=loc, scale=scale, shape=shape)

y = dist[0].rvs(size=sample_size)

x = np.linspace(dist[0].ppf(0), dist[0].ppf(1), 100)

fig, (ax1, ax2) = plt.subplots(1, 2, figsize=figsize)

plt.suptitle(dist[1] + ', ' + dist[2], y=1.03, fontsize=12)

ax1.hist(y, density=True, bins=bins, alpha=.5,

label='Random Sample (N = {:0d})'.format(sample_size), color=color)

ax1.plot(x, dist[0].pdf(x), 'k-', lw=2, label='Theoretical Distribution') # histtype='stepfilled',

ax1.set_title('PDF')

ax1.set_ylabel(r'$p(x)$')

ax1.set_xlabel(r'$x$')

ax1.legend(loc='best', frameon=False)

ax2.hist(y, density=True, cumulative=True, alpha=.5,

label='Random Sample (N = {:0d})'.format(sample_size), color=color)

ax2.plot(x, dist[0].cdf(x), 'k-', lw=2, label='Theoretical Distribution')

ax2.set_title('CDF')

if log_scale:

ax1.set_xscale('log')

ax1.set_yscale('log')

ax2.set_xscale('log')

ax2.set_yscale('log')

ax1.set_ylabel(r'$log(p(x))$')

ax1.set_xlabel(r'$log(x)$')

ax2.set_ylabel(r'$log(\Phi(x))$')

ax2.set_xlabel(r'$log(x)$')

else:

ax1.set_ylabel(r'$p(x)$')

ax1.set_xlabel(r'$x$')

ax2.set_ylabel(r'$\Phi(x)$')

ax2.set_xlabel(r'$x$')

""" Plot empirical distribution and a random sample from the same distribution. """

fig, (ax1, ax2) = plt.subplots(1, 2, figsize=figsize)

plt.suptitle(title, y=1.03, fontsize=12)

ax1.hist(y, density=True, bins=bins, alpha=.5, color=color)

ax1.set_title('PDF')

ax2.hist(y, density=True, cumulative=True, alpha=.5, color=color)

ax2.set_title('CDF')

if log_scale:

ax1.set_xscale('log')

ax1.set_yscale('log')

ax2.set_xscale('log')

ax2.set_yscale('log')

ax1.set_ylabel(r'$log[p(x)]$')

ax1.set_xlabel(r'$log(x)$')

ax2.set_ylabel(r'$log[\Phi(x)]$')

ax2.set_xlabel(r'$log(x)$')

else:

ax1.set_ylabel(r'$p(x)$')

ax1.set_xlabel(r'$x$')

ax2.set_ylabel(r'$\Phi(x)$')

ax2.set_xlabel(r'$x$')

"""Transform pandas series or list to numpy array."""

data_types = {pd.core.series.Series: (lambda x: x.values),

np.ndarray: (lambda x: x),

list: (lambda x: np.array(x))}

assert type(x) in data_types.keys(), 'invalid data type. Enter numpy array, pandas series , or list of float.'

x = np.sort(numpy_array(x))

n = len(x)

y = np.arange(1, n+1, dtype=float)/n

if plot:

plt.figure(figsize=(6, 4))

plt.step(x, y, c='royalblue', where='pre')

plt.title('Cumulative Density Function (CDF)')

plt.xlabel(r'$x$')

plt.ylabel(r'$\Phi(X \leq x)$')

plt.legend(frameon=False)

plt.yticks(np.arange(0, 1.1, step=.1))

plt.axhline(0.5, c='grey', ls='--', lw=1)

plt.show()

"""Compute the Kolmogorov-Smirnoff Distance between 2 sample distributions: CDF(x) and CDF(y).

Null hypothesis: x and y are samples drawn from the same underlying distribution.

p-value < threshold confidence level e.g., 0.01, 1% ==> reject null hypotheses."""

x = np.sort(numpy_array(x))

y = np.sort(numpy_array(y))

z = np.sort(numpy_array(list(set(x).union(set(y))))) # superset of values

# Compute the CDFs for the superset of values (x and y).

# np.searchsorted() reindexes the original array by inserting superset values

# and maintaining order of original indices. Furthermore, it handles repetitive values.

cdfx = np.searchsorted(x, z, side='right') / len(x)

cdfy = np.searchsorted(y, z, side='right') / len(y)

# Compute the maximum, absolute difference between the CDFs (vertical distance in plot).

max_abs_diff = np.max(np.abs(cdfx - cdfy))

max_loc = np.argmax(np.abs(cdfx - cdfy)) # returns the first occurrence only

vline_bot = np.min([cdfx[max_loc], cdfy[max_loc]])

vline_top = np.max([cdfx[max_loc], cdfy[max_loc]])

# Use scipy implementation to compare distance ("max_abs_diff" should equal "distance")

distance, p_value = ks_2samp(x, y, alternative='two-sided', mode='auto')

critical_value = kstwobign.ppf(confidence_interval) * np.sqrt((len(x)+len(y)) / (len(x)*len(y)))

color1 = 'darkblue'

color2 = 'darkred'

if plot:

fig, (ax1, ax2) = plt.subplots(1, 2, figsize=figsize)

# PDFs

label1 = r'$n_1={:0d}$, $\mu_1={:.2f}$, $\sigma_1={:.2f}$'.format(len(x), np.mean(x), np.std(x))

label2 = r'$n_2={:0d}$, $\mu_2={:.2f}$, $\sigma_2={:.2f}$'.format(len(y), np.mean(y), np.std(y))

ax1.hist(x, bins=bins, density=True, edgecolor=None, facecolor=color1, label=label1, lw=1.5, alpha=0.5)

ax1.hist(y, bins=bins, density=True, edgecolor=None, facecolor=color2, label=label2, lw=1.5, alpha=0.5)

ax1.set_title('PDF')

ax1.set_xlabel(r'$x$')

ax1.set_ylabel(r'$\phi(x)$')

ax1.legend(frameon=False, loc='upper center')

# CDFs

ks_text = r'$d_{KS}=' + r'{:.4f}$, $cv={:.4f}$ ($\alpha={:.2f}$)'.\

format(distance, critical_value, confidence_interval)

ax2.plot(z, cdfx, c=color1, label='$CDF_1$', lw=1.5, alpha=0.5)

ax2.plot(z, cdfy, c=color2, label='$CDF_2$', lw=1.5, alpha=0.5)

ax2.set_title(r'CDF')

ax2.set_xlabel(r'$x$')

ax2.set_ylabel(r'$\Phi(X \leq x)$')

ax2.set_yticks(np.arange(0, 1.2, step=.1))

ax2.set_ylim(0, 1.2)

ax2.vlines(z[max_loc], vline_bot, vline_top, lw=1, ls='--', colors='grey', label=ks_text)

ax2.axhline(vline_top, np.min(z), np.max(z), lw=1, ls='--', c='grey')

ax2.axhline(vline_bot, np.min(z), np.max(z), lw=1, ls='--', c='grey')

ax2.legend(frameon=False, loc='best') # bbox_to_anchor=(0, 1)

plt.show()