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04-distribution-weibull.py
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04-distribution-weibull.py
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from pathlib import Path
import pandas as pd
import numpy as np
from scipy import stats
import matplotlib.pylab as plt
import matplotlib.ticker as ticker
import math
import random
X_RANGE = 1000 # up to 1000 hours
Y_RANGE = 1000 # up to 20 events per hour
LAMBDA_1 = 1 # mean of successes, 1 success in average per the given time range
LAMBDA_2 = 2 # mean of successes, 2 successes in average per the given time range
LAMBDA_3 = 3 # mean of successes, 3 successes in average per the given time range
P1 = LAMBDA_1 / Y_RANGE # (ex. 1% out of 100, 0.1% out of 1000)
P2 = LAMBDA_2 / Y_RANGE # (ex. 2% out of 100, 0.2% out of 1000)
P3 = LAMBDA_3 / Y_RANGE # (ex. 3% out of 100, 0.3% out of 1000)
THETA_1 = (Y_RANGE - LAMBDA_1)/LAMBDA_1 # mean of time interval between the successes
# (ex. 1 successes in 1000 means ~ 999 time interval in average)
THETA_2 = (Y_RANGE - LAMBDA_2)/LAMBDA_2 # mean of time interval between the successes
# (ex. 2 successes in 1000 means ~ 499 time interval in average)
THETA_3 = (Y_RANGE - LAMBDA_3)/LAMBDA_3 # mean of time interval between the successes
# (ex. 3 successes in 1000 means ~ 332 time interval in average)
BETA_1 = 3.0
BETA_2 = 1.5
BETA_3 = 0.9
K_1 = 1/BETA_1
K_2 = 1/BETA_2
K_3 = 1/BETA_3
YLIM1 = 3 * (Y_RANGE / LAMBDA_1)**K_1 # up to 3 zero samples in a row and 1 success per sample in average - time in K_1 power
YLIM2 = 4 * (Y_RANGE / LAMBDA_2)**K_2 # up to 4 zero samples in a row and 2 successes per sample in average - time in K_2 power
YLIM3 = 5 * (Y_RANGE / LAMBDA_3)**K_3 # up to 5 zero samples in a row and 3 successes per sample in average - time in K_3 power
distr_1 = pd.DataFrame(columns = ['time'])
distr_2 = pd.DataFrame(columns = ['time'])
distr_3 = pd.DataFrame(columns = ['time'])
# https://stackoverflow.com/questions/53978121/how-can-i-plot-four-subplots-with-different-colspans
ax1 = plt.subplot2grid((10, 6), (0, 0), rowspan=5, colspan=2)
ax2 = plt.subplot2grid((10, 6), (0, 2), rowspan=5, colspan=2)
ax3 = plt.subplot2grid((10, 6), (0, 4), rowspan=5, colspan=2)
ax1.grid(axis='both', linestyle='--', color='0.95')
ax1.set_xlim(0, X_RANGE * LAMBDA_1)
ax1.set_ylim(0, YLIM1)
ax1.set_xlabel('number of measures')
ax1.set_ylabel(f'time between successes of power k={K_1:.2f}')
ax1.set_title(f'Time between the successes of power k={K_1:.2f} (p = {P1})')
ax1.title.set_size(10)
ax2.grid(axis='both', linestyle='--', color='0.95')
ax2.set_xlim(0, X_RANGE * LAMBDA_2)
ax2.set_ylim(0, YLIM2)
ax2.set_xlabel('number of measures')
ax2.set_ylabel(f'time between successes of power k={K_2:.2f}')
ax2.set_title(f'Time between the successes of power k={K_2:.2f} (p = {P2})')
ax2.title.set_size(10)
ax3.grid(axis='both', linestyle='--', color='0.95')
ax3.set_xlim(0, X_RANGE * LAMBDA_3)
ax3.set_ylim(0, YLIM3)
ax3.set_xlabel('number of measures')
ax3.set_ylabel(f'time between successes of power k={K_3:.2f}')
ax3.set_title(f'Time between the successes of power k={K_3:.2f} (p = {P3})')
ax3.title.set_size(10)
# https://stackoverflow.com/questions/42435446/how-to-put-text-outside-of-plots
text_1 = ax1.text(50, YLIM1 * 0.9, '', color='r', fontweight='bold') # , transform=plt.gcf().transFigure
text_2 = ax2.text(50, YLIM2 * 0.9, '', color='g', fontweight='bold') # , transform=plt.gcf().transFigure
text_3 = ax3.text(50, YLIM3 * 0.9, '', color='b', fontweight='bold') # , transform=plt.gcf().transFigure
line_1, = ax1.plot([], color='r', label=f'p = {P1}')
line_2, = ax2.plot([], color='g', label=f'p = {P2}')
line_3, = ax3.plot([], color='b', label=f'p = {P3}')
# https://stackoverflow.com/questions/53978121/how-can-i-plot-four-subplots-with-different-colspans
ax4 = plt.subplot2grid((10, 6), (5, 0), rowspan=5, colspan=2)
ax5 = plt.subplot2grid((10, 6), (5, 2), rowspan=5, colspan=2)
ax6 = plt.subplot2grid((10, 6), (5, 4), rowspan=5, colspan=2)
ax1.legend(loc="upper right")
ax2.legend(loc="upper right")
ax3.legend(loc="upper right")
X_1 = np.linspace(0, YLIM1, 1000)
X_2 = np.linspace(0, YLIM2, 1000)
X_3 = np.linspace(0, YLIM3, 1000)
# https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.weibull_min.html
# Suppose X is an exponentially distributed random variable with scale s.
# Then Y = X**k is weibull_min distributed with shape c = 1/k and scale s**k.
PDF_1 = stats.weibull_min.pdf(X_1, c=BETA_1, scale = THETA_1**K_1)
PDF_2 = stats.weibull_min.pdf(X_2, c=BETA_2, scale = THETA_2**K_2)
PDF_3 = stats.weibull_min.pdf(X_3, c=BETA_3, scale = THETA_3**K_3)
# 1 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 1
# Number of successes: 4
# Times between successes: [3, 8, 4]
def calc_times(sample: list, df: pd.DataFrame, remainder: int):
time = remainder
for event in sample:
if event == 1:
df.loc[len(df), 'time'] = time
time = 0
elif event == 0:
time += 1
return df, time
remainder1 = 0
remainder2 = 0
remainder3 = 0
for i in range(X_RANGE):
# https://numpy.org/doc/stable/reference/random/generated/numpy.random.poisson.html
# The Poisson distribution is the limit of the binomial distribution for large N.
sample_1 = [1 if r < P1 else 0 for r in [random.random() for i in range(Y_RANGE)]]
sample_2 = [1 if r < P2 else 0 for r in [random.random() for i in range(Y_RANGE)]]
sample_3 = [1 if r < P3 else 0 for r in [random.random() for i in range(Y_RANGE)]]
distr_1, remainder1 = calc_times(sample_1, distr_1, remainder1)
distr_2, remainder2 = calc_times(sample_2, distr_2, remainder2)
distr_3, remainder3 = calc_times(sample_3, distr_3, remainder3)
# https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.weibull_min.html
# Suppose X is an exponentially distributed random variable with scale s.
# Then Y = X**k is weibull_min distributed with shape c = 1/k and scale s**k.
distr_1_pow_k = np.power(distr_1.values, K_1)
distr_2_pow_k = np.power(distr_2.values, K_2)
distr_3_pow_k = np.power(distr_3.values, K_3)
if (i < 100) or (i == X_RANGE - 1):
mean_1 = distr_1["time"].mean()
mean_2 = distr_2["time"].mean()
mean_3 = distr_3["time"].mean()
text_1.set_text(f'{i}: {mean_1}')
text_2.set_text(f'{i}: {mean_2}')
text_3.set_text(f'{i}: {mean_3}')
# https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.weibull_min.html
# Y = X**k is weibull_min distributed with shape c = 1/k and scale s**k.
line_1.set_data(distr_1.index.values, distr_1_pow_k)
line_2.set_data(distr_2.index.values, distr_2_pow_k)
line_3.set_data(distr_3.index.values, distr_3_pow_k)
bins_1 = 20 # int(distr_1_pow_k.max() - distr_1_pow_k.min()) if len(distr_1_pow_k) > 0 else 1
bins_2 = 20 # int(distr_2_pow_k.max() - distr_2_pow_k.min()) if len(distr_2_pow_k) > 0 else 1
bins_3 = 20 # int(distr_3_pow_k.max() - distr_3_pow_k.min()) if len(distr_3_pow_k) > 0 else 1
ax4.cla()
ax5.cla()
ax6.cla()
# https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.weibull_min.html
# Y = X**k is weibull_min distributed with shape c = 1/k and scale s**k.
ax4.hist(distr_1_pow_k, bins = bins_1 if bins_1 > 0 else 1, density=True, rwidth=0.8, alpha=0.4, color='r', label=f'λ={LAMBDA_1:.2f} β={BETA_1:.2f} n={Y_RANGE} p={P1}')
ax5.hist(distr_2_pow_k, bins = bins_2 if bins_2 > 0 else 1, density=True, rwidth=0.8, alpha=0.4, color='g', label=f'λ={LAMBDA_2:.2f} β={BETA_2:.2f} n={Y_RANGE} p={P2}')
ax6.hist(distr_3_pow_k, bins = bins_3 if bins_3 > 0 else 1, density=True, rwidth=0.8, alpha=0.4, color='b', label=f'λ={LAMBDA_3:.2f} β={BETA_3:.2f} n={Y_RANGE} p={P3}')
ax4.plot(X_1, PDF_1, alpha=1.0, color='r', linewidth=2.0)
ax5.plot(X_2, PDF_2, alpha=1.0, color='g', linewidth=2.0)
ax6.plot(X_3, PDF_3, alpha=1.0, color='b', linewidth=2.0)
ax4.grid(axis='both', linestyle='--', color='0.95')
# ax4.xaxis.set_major_locator(ticker.MultipleLocator(int(YLIM1 / 10)))
# ax4.set_xlim(0, YLIM1 / 2)
# ax3.set_ylim(0, 1)
# ax3.set_xlabel('')
# ax3.set_ylabel('')
# ax3.set_title('')
ax4.legend(loc="upper right")
ax5.grid(axis='both', linestyle='--', color='0.95')
# ax5.xaxis.set_major_locator(ticker.MultipleLocator(int(YLIM2 / 10)))
# ax5.set_xlim(0, YLIM2 / 2)
ax5.legend(loc="upper right")
ax6.grid(axis='both', linestyle='--', color='0.95')
# ax6.xaxis.set_major_locator(ticker.MultipleLocator(int(YLIM3 / 10)))
# ax6.set_xlim(0, YLIM3 / 2)
ax6.legend(loc="upper right")
ax4.text(15, 0.06, f'W(λ={LAMBDA_1},β={BETA_1:.2f})')
ax5.text(100, 0.006, f'W(λ={LAMBDA_2},β={BETA_2:.2f})')
ax6.text(1000, 0.0005, f'W(λ={LAMBDA_3},β={BETA_3:.2f})')
(i < 100) and (i % 20 == 0) and plt.tight_layout()
# pause the plot for 0.01s before next point is shown
# plt.pause(0.5 if i < 100 else 0.0001)
(i < 100) and plt.pause(0.05)
plt.show()