def poisson_mode_and_alpha(expected, alpha):
"""
Returns mode number of expected substitutions and upper & lower limits
within which falls `alpha` fraction of the occurrences for a poisson
distribution with lambda=`expected` value.
Parameters
----------
expected : float
Expected substitutions.
alpha : float
Probability interval containing alpha fraction of the Poisson
distribution.
Returns
-------
mode : int
Typical number of substitutions expected.
lower_limit, upper_limit : int
Lower number of substitutions between which alpha fraction of the
Poisson distribution is contained.
upper_limit : int
Lower number of substitutions between which alpha fraction of the
Poisson distribution is contained.
"""
mean, var = poisson.stats(expected, moments='mv')
mode = math.floor(mean) # round down to closest integer
lower_limit, upper_limit = poisson.interval(alpha, expected, loc=0)
return mode, lower_limit, upper_limit
def test_poisson(self):
mu = 0.6
mean, var, skew, kurt = poisson.stats(mu, moments='mvsk')
self.assertEqual(mean, 0.6)
self.assertEqual(var, 0.6)
self.assertEqual(skew, 1.2909944487358056)
self.assertEqual(kurt, 1.6666666666666667)
n = np.array([0., 1., 2.])
x = np.arange(poisson.ppf(0.01, mu), poisson.ppf(0.99, mu))
self.assertTrue(np.array_equal(n, x))
def create_poisson_distribution():
fig, ax = plt.subplots(1, 1)
mu = 0.6
mean, var, skew, kurt = poisson.stats(mu, moments='mvsk')
x = np.arange(poisson.ppf(0.01, mu), poisson.ppf(0.99, mu))
ax.plot(x, poisson.pmf(x, mu), 'bo', ms=8, label='poisson pmf')
ax.vlines(x, 0, poisson.pmf(x, mu), colors='b', lw=5, alpha=0.5)
rv = poisson(mu)
ax.vlines(x,
0,
rv.pmf(x),
colors='k',
linestyles='-',
lw=1,
label='frozen pmf')
ax.legend(loc='best', frameon=False)
interactive(False)
plt.show()
def get_mean_kurt_variance(self):
"""
Function to get the mean of the data set.
Args:
None
Returns:
float: the mean value,
float: the kurtosis value.
"""
mean, variance, skew, kurt = poisson.stats(mu, moments='mvsk')
self.mean = mean
self.variance = variance
self.kurt = kurt
return mean, kurt, variance
def plot_poison_pmf(mean_calculated, title):
#mu - calculate (of counts) + sum and divide by length.
fig, ax = plt.subplots(1, 1)
mu = mean_calculated
mean, var, skew, kurt = poisson.stats(mu, moments='mvsk')
x = np.arange(poisson.ppf(0.01, mu), poisson.ppf(0.99, mu))
ax.plot(x, poisson.pmf(x, mu), 'bo', ms=8, label='poisson pmf')
ax.vlines(x, 0, poisson.pmf(x, mu), colors='b', lw=5, alpha=0.5)
rv = poisson(mu)
ax.vlines(x,
0,
rv.pmf(x),
colors='k',
linestyles='-',
lw=1,
label='frozen pmf')
ax.legend(loc='best', frameon=False)
plt.title(title)
interactive(False)
plt.show()
def testPoisson(): # {{{
"""
Poisson Distribution (泊松分布)
泊松分布的例子：已知某路口发生事故的比率是每天2次，那么在此处一天内发生n次事故的概率是多少？
"""
# 准备数据: 已知lam:单位时间内发生某件事的平均次数
# X轴: 单位时间内该事件发生的次数
# Y轴: 次数对应的概率
lam = 10 # p = 1/lam
xs = np.arange(poisson.ppf(0.01, mu=lam),
poisson.ppf(0.99, mu=lam),
step=1)
# E(X) = lam, D(X) = lam
mean, var, skew, kurt = poisson.stats(mu=lam, loc=0, moments='mvsk')
print("mean: %.2f, var: %.2f, skew: %.2f, kurt: %.2f" %
(mean, var, skew, kurt))
fig, axs = plt.subplots(1, 3)
# 显示pmf
ys = poisson.pmf(xs, mu=lam)
axs[0].plot(xs, ys, 'bo', markersize=5, label='poisson pmf')
axs[0].legend()
# 显示cdf
ys = poisson.cdf(xs, mu=lam)
axs[1].plot(xs, ys, 'bo', markersize=5, label='poisson cdf')
axs[1].legend()
# 随机变量RVS
data = poisson.rvs(mu=lam, size=1000)
import sys
sys.path.append("../../thinkstats")
import Pmf
pmf = Pmf.MakePmfFromList(data)
xs, ys = pmf.Render()
axs[2].plot(xs, ys, 'bo', markersize=5, label='rvs pmf')
axs[2].legend()
plt.show()
def test_poisson(self):
from scipy.stats import poisson
import matplotlib.pyplot as plt
fig, ax = plt.subplots(1, 1)
mu = 0.6
mean, var, skew, kurt = poisson.stats(mu, moments='mvsk')
x = np.arange(poisson.ppf(0.01, mu), poisson.ppf(0.99, mu))
ax.plot(x, poisson.pmf(x, mu), 'bo', ms=8, label='poisson pmf')
ax.vlines(x, 0, poisson.pmf(x, mu), colors='b', lw=5, alpha=0.5)
rv = poisson(mu)
ax.vlines(x,
0,
rv.pmf(x),
colors='k',
linestyles='-',
lw=1,
label='frozen pmf')
ax.legend(loc='best', frameon=False)
self.assertEqual(str(ax), "AxesSubplot(0.125,0.11;0.775x0.77)")
def poisson_distribution(*args):
try:
a = float(args[2])
except (ValueError, TypeError):
return {'is_valid': False}
if not 0 < a < 1:
return {'is_valid': False}
t = Symbol('t')
mean, var = poisson.stats(a)
cf = exp(a * (exp(I * t) - 1))
d_cf = diff(cf, t)
dd_cf = diff(d_cf, t)
mean2 = round(float(lambdify(t, -dd_cf)(0)), 2)
return {
'a': a,
'mean': mean,
'mean2': mean2,
'var': var,
'g': latex(nsimplify(cf, tolerance=0.1)),
'g1': latex(nsimplify(d_cf, tolerance=0.1)),
'g2': latex(nsimplify(dd_cf, tolerance=0.1)),
'type': type,
'is_valid': True,
}
#from https://docs.scipy.org/doc/scipy-0.18.1/reference/generated/scipy.stats.poisson.html#scipy.stats.poisson from scipy.stats import poisson import matplotlib.pyplot as plt import numpy as np fig, ax = plt.subplots(1, 1) mu = 0.6 mean, var, skew, kurt = poisson.stats(mu, moments='mvsk') x = np.arange(poisson.ppf(0.01, mu), poisson.ppf(0.99, mu)) ax.plot(x, poisson.pmf(x, mu), 'bo', ms=8, label='poisson pmf') ax.vlines(x, 0, poisson.pmf(x, mu), colors='b', lw=5, alpha=0.5) rv = poisson(mu) ax.vlines(x, 0, rv.pmf(x), colors='k', linestyles='-', lw=1,label='frozen pmf') ax.legend(loc='best', frameon=False) plt.show()
from scipy.stats import poisson import numpy as np import matplotlib.pyplot as plt fig, ax = plt.subplots(1, 1) mu = 5.0 mean, var, skew, kurt = poisson.stats(mu, moments='mvsk') x = np.arange(poisson.ppf(0.01, mu), poisson.ppf(0.99, mu)) ax.plot(x, poisson.pmf(x, mu), 'bo', ms=8, label='poisson pmf') ax.vlines(x, 0, poisson.pmf(x, mu), colors='b', lw=5, alpha=0.5) plt.show()
data = [pkt_segs, time_segs]
#print(poisson.pmf(test, mean))
#print((mean**test)*np.exp(-mean))#/np.math.factorial(int(np.round(test))))
#calculate mean of samples - max likelyhood estimator for poisson distribution
pp = PdfPages(
'/home/francesco/Documents/Thesis_project/Results/Trace_packet_dist.pdf')
ks_res = np.zeros((len(pkt_segs), 2))
for i, metric in enumerate(data):
for j, seg in enumerate(metric):
fig, ax = plt.subplots(1, 1)
psn_param = np.mean(seg)
mean, var, skew, kurt = poisson.stats(psn_param, moments='mvsk')
#fig, ax = plt.subplots(1, 1)
ax.plot(np.sort(seg),
poisson.pmf(np.sort(seg), psn_param),
ms=8,
label='trace seg: ' + str(j + 1) + ', ' + r'$\lambda$: ' +
str(round(psn_param, 3)))
n, x, _ = ax.hist(seg,
histtype='step',
density=True,
label='Emperical Distribution')
density = stats.gaussian_kde(np.sort(seg))
ax.plot(x, density(x), 'r--', label='Gaussian Kernel Smoothing ')
if i == 0:
fig.suptitle("Distribución Binomial Negativa")
plt.show()
# DISTRIBUCIÓN POISSON
from scipy.stats import poisson
poisson.pmf(0, mu=3)
poisson.cdf(0, mu=3)
sum(poisson.pmf(range(0, 10), mu=3))
poisson.cdf(10, mu=3)
poisson.stats(mu=3, moments='mv') # Tonteria
mu = 10 # mu = lambda
x = np.arange(poisson.ppf(0.01, mu), poisson.ppf(0.99, mu))
fig = plt.figure(figsize=(5, 2.7))
ax = fig.add_subplot(1, 2, 1)
ax.plot(x, poisson.pmf(x, mu), 'bo', ms=5, label='poisson pmf')
ax.vlines(x, 0, poisson.pmf(x, mu), colors='b', lw=2, alpha=0.5)
for tick in ax.xaxis.get_major_ticks():
tick.label.set_fontsize(5)
for tick in ax.yaxis.get_major_ticks():
tick.label.set_fontsize(5)
ax = fig.add_subplot(1, 2, 2)
ax.plot(x, poisson.cdf(x, mu), 'bo', ms=5, label='poisson cdf')
ax.vlines(x, 0, poisson.cdf(x, mu), colors='b', lw=2, alpha=0.5)
for tick in ax.xaxis.get_major_ticks():
# ちょうど7人来る確率を計算したいので、k=7です。
k = 7
# 確率質量関数をつかって確率を計算します。
prob = (lamb**k) * exp(-lamb) / factorial(k)
print(' 昼のピーク時にお客さんが7人である確率は、{:0.2f}%です。'.format(100 * prob))
"""
確率質量関数を手作りできました。scipyを使うともう少し楽です。
"""
# 平均は10です。
mu = 10
# 平均と分散を計算できます。
mean, var = poisson.stats(mu)
# 確率質量関数を使って、特定の確率を計算することも可能です。
odds_seven = poisson.pmf(7, mu)
print('ピーク時に7人の確率は{:0.2f}％'.format(odds_seven * 100))
print('平均={}'.format(mean))
"""
ピーク時に7人の確率は9.01％
平均=10.0
分布の全体を見ておくことにします。これは、10人よりお客が多い確率を求めるのに必要です。
"""
# 確率質量関数をプロットしてみましょう。
from scipy.stats import binom, geom, hypergeom, poisson, expon
import numpy as np
#Utilizamos una distribución de poisson
mu = 2
mean, var = poisson.stats(mu, moments = 'mv')
print("La esperanza es de: %f"%mean)
print("La varianza es de: %f"%var)
prob_0 = poisson.pmf(0,mu)
print("La probabilidad de que haya 0 accidentes es de un %f %s"%((round(prob_0*100,2),"%")))
plt.legend(loc='upper left', shadow=True)
plt.show()
# ### Poisson Distribution
# In[5]:
#Poisson Distribution
from scipy.stats import poisson
loc, mu = 0, 10 # Mu is basically Lambda
x = np.arange(poisson.ppf(0.01, mu, loc), poisson.ppf(
0.99, mu, loc)) #Percent Point Function (inverse of cdf — percentiles)
print("Mean : ", poisson.stats(mu, loc, moments='m'))
print("Variance : ", poisson.stats(mu, loc, moments='v'))
print("Prob. Dens. Func. : ", poisson.pmf(x, mu, loc))
print("Cum. Density Func.: ", poisson.cdf(x, mu, loc))
CDF = poisson.cdf(x, mu, loc)
fig = plt.figure(figsize=(20, 10))
plt.subplot(221)
plt.plot(x, poisson.pmf(x, mu, loc), 'go', ms=8, label='PMF')
plt.vlines(x, 0, poisson.pmf(x, mu, loc), colors='g', lw=5, alpha=0.5)
plt.xlabel("Sample Space of Poisson Distribution", fontsize=14)
plt.ylabel("PMF", fontsize=14)
plt.title("Probability Distribution of Poisson(λ=10) Distribution",
fontsize=16)
plt.xticks(np.arange(0, 20, 1))
def histogram(data, x_axis, y, name):
# Create the regular raw data figure
plt.clf()
plt.bar(x_axis, y)
plt.xlabel("Count")
plt.ylabel("Frequency")
plt.savefig(name + "_count_histo.png")
plt.clf()
# Graphing expected values from the poisson probability mass function
fig, ax = plt.subplots(1, 1)
# Caluclating first moments below
mu = statistics.mean(data) # Get the average from all the data
print("average is ", mu) # Print it
mean, var, skew, kurt = poisson.stats(mu, moments='mvsk')
ax.plot(data, poisson.pmf(data, mu), 'bo', ms=8, label='poisson pmf')
ax.vlines(data, 0, poisson.pmf(data, mu), colors='b', lw=5, alpha=0.5)
plt.xlabel("Count")
plt.ylabel("Probability")
plt.savefig(name + "_poisson.png")
plt.clf()
poisson_pts = poisson.pmf(data, mu)
print("Chi Square is for poisson before multiplication ",
chisquare(poisson_pts))
poisson_pts = [i * len(data) for i in poisson_pts]
print("Chi Square is for poisson after multiplication ",
chisquare(poisson_pts))
#print("Poisson stuff is ", poisson_pts) # Printing poisson points we see that they match hand calculated numbers
# Graph with histogram and poisson fitted values
#plt.bar(x_axis,y, yerr = statistics.stdev(data))
plt.hist(data)
plt.plot(
data,
poisson_pts,
marker='o',
linestyle='',
label='Fit result',
)
plt.legend()
plt.xlabel("Count")
plt.ylabel("Frequency")
plt.savefig(name + "_poisson2.png")
plt.clf()
# Fit a normal distribution to the data:
mu, std = norm.fit(data) #Gets mean and standard_dev with scipy statistics
plt.hist(data) # create histogram from our data
xmin, xmax = plt.xlim(
) # Set limits on x axis for gaussian distribution to be over
x = np.linspace(
xmin, xmax,
100) #Return evenly spaced numbers over a specified interval.
p = norm.pdf(x, mu, std)
print("Chi Square of the normal distribution before multiplication is ",
chisquare(p))
list = p
list = [i * len(data) for i in p]
print(
"Chi Square of the normal distribution after multiplication by len(data) ",
chisquare(list))
#print("p is ", list) Output the gaussian distribution numbers
plt.plot(x, list, 'k', linewidth=2)
title = "Fit results: mu = %.2f, std = %.2f" % (mu, std)
plt.title(title)
plt.xlabel("Count")
plt.ylabel("Probability")
plt.savefig(name + "_normal.png")
plt.clf()
# Now normal with poisson and bar graph
#print(x)
#print(y)
plt.hist(data)
plt.plot(data, poisson_pts, marker='o', linestyle='', label='Fit result')
plt.plot(x, list, 'k', linewidth=2)
plt.xlabel("Count")
plt.ylabel("Frequency")
plt.savefig(name + "normal_poisson.png")
#Generate four histogram plots, one for each data set, and display them all
#simultaneously
#The first plot will be in the top left
plt.figure(1)
plt.subplot(2,2,1) #(number of rows, number of colums, figure number)
count1, bins1, ignored1 = plt.hist(samples1, 12, facecolor = 'blue',
normed=False)
#Label the axes and title the plot
plt.xlabel('Value')
plt.ylabel('Counts')
plt.title('N=10')
#Calculate the mean, mode, variance, skew, and kurtosis, then print them
mean1, var1, skew1, kurt1 = poisson.stats(5, moments='mvsk')
mode1 = np.bincount(samples1)
print'For N=10, the average value is ', samples1.mean(), ', the mode is ',\
np.argmax(mode1), ', the variance is ', samples1.var(), ', the skew is ', skew1, \
', and the kurtosis is ', kurt1
print
#The second plot will be on the top right
plt.subplot(2,2,2)
count2, bins2, ignored2 = plt.hist(samples2, 12, facecolor = 'green',
normed = False)
#Label the axes and title the plot
plt.xlabel('Value')
plt.ylabel('Counts')
plt.title('N=100')
