def continuous(save = False, name = 'img/parameters/incubation.png'):
"""Plot incubation period distributions.
After call, use `plt.show()` to show the figure.
Args:
save (bool, optional): Whether to save the figure, defaultly not.
name (str, optional): Path to save the plot to.
"""
def _pars(a,b): return '%.3f,%.3f' % (a,b)
# grid
xgrid = np.linspace(0, 14, 1000)
# distributions
distr = continuous()
lognorm_pdf = lognorm.pdf(xgrid, *distr['lognorm'])
gamma_pdf = gamma.pdf(xgrid, *distr['gamma'])
weibull_pdf = weibull.pdf(xgrid, *distr['weibull'])
erlang_pdf = erlang.pdf(xgrid, *distr['erlang'])
# plot
fig1, ax1 = plt.subplots()
ax1.plot(xgrid, lognorm_pdf,
label=f"LN({_pars(np.log(distr['lognorm'][2]),distr['lognorm'][0])}^2)")#'LN(1.621,0.418)')
ax1.plot(xgrid, gamma_pdf,
label=f"Gamma({_pars(distr['gamma'][0],1/distr['gamma'][2])})")
ax1.plot(xgrid, weibull_pdf,
label=f"W({_pars(*distr['weibull'])})")
ax1.plot(xgrid, erlang_pdf,
label=f"E({_pars(distr['erlang'][0],distr['erlang'][2])})")
ax1.set_xlabel('Incubation period')
ax1.set_ylabel('Density')
ax1.legend()
if save: fig1.savefig(name)
def _pdf(self, value: float):
"""
Defines the K-Erlang distribution
:param value:
:return: Function value at point x
"""
if self._research_mode:
return erlang.pdf(value, a=self._k, scale=1 / self.rate)
else:
if value >= 0:
return math.pow(self.rate * value, self.k - 1) / (math.factorial(self.k - 1)) \
* self.rate * math.exp(-self.rate * value)
else:
return 0
def _pdf(self, value: float):
"""
Defines the K-Erlang distribution
:param value:
:return: Function value at point x
"""
if self._research_mode:
return erlang.pdf(value, a=self._k, scale=1 / self.rate)
else:
if value >= 0:
return math.pow(self.rate * value, self.k - 1) / (math.factorial(self.k - 1)) \
* self.rate * math.exp(-self.rate * value)
else:
return 0
def update_custom_data(attrname, old, new):
xmin, xmax = slider_cust_xminmax.value
k = slider_cust_k.value
teta = slider_cust_teta.value
x = np.linspace(xmin, xmax, d_N)
y_cust_pdf = erlang.pdf(x, a=k, scale=teta)
y_cust_cdf = erlang.cdf(x, a=k, scale=teta)
data_custom = {
'x': x,
'y_pdf': y_cust_pdf,
'y_cdf': y_cust_cdf
}
source_custom.data = data_custom
def exp_erl(rate, rate_exp, k, y0=1, neval=35):
tspan = [0, 16.]
t_eval = np.linspace(*tspan, neval)
# Exp
pdf_exp = expon.pdf(t_eval, scale=1. / rate_exp)
survival_exp = y0 * expon.sf(t_eval, scale=1. / rate_exp)
pdf_erlang = erlang.pdf(t_eval, a=k, scale=1. / rate)
survival_erlang = y0 * erlang.sf(t_eval, a=k, scale=1. / rate)
fig, axs = plt.subplots(2, 1, figsize=(15, 15), sharex=True)
pdf_idx = 0
survival_idx = 1
exp_label = 'Exp(x; $r$={:5.3f})'.format(rate_exp)
erlang_label = 'Erlang(x; $r$={:5.3f},$k$={})'.format(rate, k)
axs[pdf_idx].plot(t_eval, pdf_exp, 'k-', label='PDF {}'.format(exp_label))
axs[pdf_idx].plot(t_eval,
pdf_erlang,
'k--',
label='PDF {}'.format(erlang_label))
axs[survival_idx].plot(t_eval,
survival_exp,
'k-',
label='{}'.format(exp_label))
axs[survival_idx].plot(t_eval,
survival_erlang,
'k--',
label='{}'.format(erlang_label))
# Styling
font_size = 16
for ax in axs:
ax.legend(fontsize=font_size)
ax.tick_params(axis='x', labelsize=font_size)
ax.tick_params(axis='y', labelsize=font_size)
axs[-1].set_xlabel('x', fontsize=font_size)
plt.show()
def calc_mdelays_hist(d, ich=0, m=10, period=(0, -1), bins_s=(0, 10, 0.02),
ph_sel=Ph_sel('all'), bursts=False, bg_fit=True,
bg_F=0.8):
"""Compute histogram of m-photons delays (or waiting times).
Arguments:
dx (Data object): contains the burst data to process.
ich (int): the channel number. Default 0.
m (int): number of photons used to compute each delay.
period (int or 2-element tuple): index of the period to use. If
tuple, the period range between period[0] and period[1]
(included) is used.
bins_s (3-element tuple): start, stop and step for the bins
ph_sel (Ph_sel object): photon selection to use.
Returns:
Tuple of values:
* bin_x (array): array of bins centers
* histograms_y (array): arrays of histograms, contains 1 or 2
histograms (when `bursts` is False or True)
* bg_dist (random distribution): erlang distribution with same
rate as background (kcps)
* a, rate_kcps (floats, optional): amplitude and rate for an
Erlang distribution fitted to the histogram for
bin_x > bg_mean*bg_F. Returned only if `bg_fit` is True.
"""
assert ph_sel in [Ph_sel('all'), Ph_sel(Dex='Dem'), Ph_sel(Dex='Aem')]
if np.size(period) == 1: period = (period, period)
periods = slice(d.Lim[ich][period[0]][0], d.Lim[ich][period[1]][1] + 1)
bins = np.arange(*bins_s)
if ph_sel == Ph_sel('all'):
ph = d.ph_times_m[ich][periods]
if bursts:
phb = ph[d.ph_in_burst[ich][periods]]
elif ph_sel == Ph_sel(Dex='Dem'):
donor_ph_period = -d.A_em[ich][periods]
ph = d.ph_times_m[ich][periods][donor_ph_period]
if bursts:
phb = ph[d.ph_in_burst[ich][periods][donor_ph_period]]
elif ph_sel == Ph_sel(Dex='Aem'):
accept_ph_period = d.A_em[ich][periods]
ph = d.ph_times_m[ich][periods][accept_ph_period]
if bursts:
phb = ph[d.ph_in_burst[ich][periods][accept_ph_period]]
ph_mdelays = np.diff(ph[::m])*d.clk_p*1e3 # millisec
if bursts:
phb_mdelays = np.diff(phb[::m])*d.clk_p*1e3 # millisec
phb_mdelays = phb_mdelays[phb_mdelays < 5]
# Compute the PDF through histograming
hist_kwargs = dict(bins=bins, normed=True)
mdelays_hist_y, _ = np.histogram(ph_mdelays, **hist_kwargs)
bin_x = bins[:-1] + 0.5*(bins[1] - bins[0])
if bursts:
mdelays_b_hist_y, _ = np.histogram(phb_mdelays, **hist_kwargs)
mdelays_b_hist_y *= phb_mdelays.size/ph_mdelays.size
if bursts:
histograms_y = np.vstack([mdelays_hist_y, mdelays_b_hist_y])
else:
histograms_y = mdelays_hist_y
results = [bin_x, histograms_y]
# Compute the BG distribution
bg_dist = _get_bg_distrib_erlang(d, ich=ich, m=m, period=period,
ph_sel=ph_sel)
bg_mean = bg_dist.mean()
results.append(bg_dist)
if bg_fit:
## Fitting the BG portion of the PDF to an Erlang
_x = bin_x[bin_x > bg_mean*bg_F]
_y = mdelays_hist_y[bin_x > bg_mean*bg_F]
fit_func = lambda x, a, rate_kcps: a*erlang.pdf(x, a=m,
scale=1./rate_kcps)
err_func = lambda p, x, y: fit_func(x, p[0], p[1]) - y
p, flag = leastsq(err_func, x0=[0.9, 3.], args=(_x, _y))
print(p, flag)
a, rate_kcps = p
results.extend([a, rate_kcps])
return results
def Erlang(x, k, lam):
""" 爱尔兰分布概率密度函数.
"""
return erlang.pdf(x, k) * np.exp(-k * lam * x) * pow(k * lam,
k) / np.exp(-x)
def example(rate, rate_exp, k, neval=35):
tspan = [0, 16.]
t_eval = np.linspace(*tspan, neval)
y0 = [
1,
]
# k = 5
# scale to same mean
# scale = 1.5
# r = 1./scale
# Exp
pdf_exp = expon.pdf(t_eval, scale=1. / rate_exp)
survival_exp = y0[0] * expon.sf(t_eval, scale=1. / rate_exp)
y0_lct = np.zeros(k)
y0_lct[0] = y0[0]
sol_lct = solve_ivp(dydt_lct,
tspan,
y0_lct,
t_eval=t_eval,
args=(rate, k),
method='Radau')
# print('mean exp: ', 1/rate_exp)
# print('rate erlang: ', rate)
# print('scale erlang: ', 1./rate)
# print('mean erlang: ', k/rate)
pdf_erlang = erlang.pdf(t_eval, a=k, scale=1. / rate)
survival_erlang = y0[0] * erlang.sf(t_eval, a=k, scale=1. / rate)
fig, axs = plt.subplots(2, 1, figsize=(15, 15), sharex=True)
pdf_idx = 0
survival_idx = 1
exp_label = 'Exp(x; $r$={:5.3f})'.format(rate_exp)
erlang_label = 'Erlang(x; $r$={:5.3f},$k$={})'.format(rate, k)
axs[pdf_idx].plot(t_eval, pdf_exp, 'k-', label='PDF {}'.format(exp_label))
axs[pdf_idx].plot(t_eval,
pdf_erlang,
'k--',
label='PDF {}'.format(erlang_label))
axs[survival_idx].plot(t_eval,
survival_exp,
'k-',
label='{}'.format(exp_label))
axs[survival_idx].plot(t_eval,
survival_erlang,
'k--',
label='{}'.format(erlang_label))
axs[survival_idx].plot(t_eval,
np.sum(sol_lct.y, axis=0),
marker='o',
linewidth=0,
markeredgecolor='k',
markerfacecolor='none',
label=r'$I = I_1 + I_2 + I_3 + I_4 + I_5$')
for istage in range(k):
axs[survival_idx].plot(t_eval,
sol_lct.y[istage],
':',
label='$I_{}$'.format(istage + 1))
# Styling
font_size = 16
for ax in axs:
ax.legend(fontsize=font_size)
ax.tick_params(axis='x', labelsize=font_size)
ax.tick_params(axis='y', labelsize=font_size)
axs[-1].set_xlabel('x', fontsize=font_size)
plt.show()
from scipy.stats import erlang
import numpy as np
import matplotlib.pyplot as plt
k = 47
laba = 1
x = np.arange(1, 100)
print(x)
temp1 = erlang.cdf(x, k, laba)
temp2 = erlang.pdf(x, k, laba)
print(temp2)
plt.plot(x, temp1, 'o-', color='orange')
plt.plot(x, temp2, 'o-', color='green')
plt.xlabel('$x$')
plt.ylabel('$y$')
plt.grid()
plt.show()
def fit_func(x, a, rate_kcps):
return a * erlang.pdf(x, a=m, scale=1./rate_kcps)
def calc_mdelays_hist(d, ich=0, m=10, period=(0, -1), bins_s=(0, 10, 0.02),
ph_sel=Ph_sel('all'), bursts=False, bg_fit=True,
bg_F=0.8):
"""Compute histogram of m-photons delays (or waiting times).
Arguments:
dx (Data object): contains the burst data to process.
ich (int): the channel number. Default 0.
m (int): number of photons used to compute each delay.
period (int or 2-element tuple): index of the period to use. If
tuple, the period range between period[0] and period[1]
(included) is used.
bins_s (3-element tuple): start, stop and step for the bins
ph_sel (Ph_sel object): photon selection to use.
Returns:
Tuple of values:
* bin_x (array): array of bins centers
* histograms_y (array): arrays of histograms, contains 1 or 2
histograms (when `bursts` is False or True)
* bg_dist (random distribution): erlang distribution with same
rate as background (kcps)
* a, rate_kcps (floats, optional): amplitude and rate for an
Erlang distribution fitted to the histogram for
bin_x > bg_mean*bg_F. Returned only if `bg_fit` is True.
"""
assert ph_sel in [Ph_sel('all'), Ph_sel(Dex='Dem'), Ph_sel(Dex='Aem')]
if np.size(period) == 1: period = (period, period)
periods = slice(d.Lim[ich][period[0]][0], d.Lim[ich][period[1]][1] + 1)
bins = np.arange(*bins_s)
if ph_sel == Ph_sel('all'):
ph = d.ph_times_m[ich][periods]
if bursts:
phb = ph[d.ph_in_burst[ich][periods]]
elif ph_sel == Ph_sel(Dex='Dem'):
donor_ph_period = -d.A_em[ich][periods]
ph = d.ph_times_m[ich][periods][donor_ph_period]
if bursts:
phb = ph[d.ph_in_burst[ich][periods][donor_ph_period]]
elif ph_sel == Ph_sel(Dex='Aem'):
accept_ph_period = d.A_em[ich][periods]
ph = d.ph_times_m[ich][periods][accept_ph_period]
if bursts:
phb = ph[d.ph_in_burst[ich][periods][accept_ph_period]]
ph_mdelays = np.diff(ph[::m])*d.clk_p*1e3 # millisec
if bursts:
phb_mdelays = np.diff(phb[::m])*d.clk_p*1e3 # millisec
phb_mdelays = phb_mdelays[phb_mdelays < 5]
# Compute the PDF through histograming
hist_kwargs = dict(bins=bins, normed=True)
mdelays_hist_y, _ = np.histogram(ph_mdelays, **hist_kwargs)
bin_x = bins[:-1] + 0.5*(bins[1] - bins[0])
if bursts:
mdelays_b_hist_y, _ = np.histogram(phb_mdelays, **hist_kwargs)
mdelays_b_hist_y *= phb_mdelays.size/ph_mdelays.size
if bursts:
histograms_y = np.vstack([mdelays_hist_y, mdelays_b_hist_y])
else:
histograms_y = mdelays_hist_y
results = [bin_x, histograms_y]
# Compute the BG distribution
bg_dist = _get_bg_distrib_erlang(d, ich=ich, m=m, period=period,
ph_sel=ph_sel)
bg_mean = bg_dist.mean()
results.append(bg_dist)
if bg_fit:
## Fitting the BG portion of the PDF to an Erlang
_x = bin_x[bin_x > bg_mean*bg_F]
_y = mdelays_hist_y[bin_x > bg_mean*bg_F]
fit_func = lambda x, a, rate_kcps: a*erlang.pdf(x, a=m,
scale=1./rate_kcps)
err_func = lambda p, x, y: fit_func(x, p[0], p[1]) - y
p, flag = leastsq(err_func, x0=[0.9, 3.], args=(_x, _y))
print(p, flag)
a, rate_kcps = p
results.extend([a, rate_kcps])
return results
###-----------------------------------------------------------------------###
###------------------DATA SOURCES AND INITIALIZATION----------------------###
### This section defines the data sources which will be used in the Bokeh ###
### plots. To update a Bokeh plot in the server, each of the sources will ###
### be modified in the CALLBACKS section. ("Model") ###
###-----------------------------------------------------------------------###
# x data
d_x = np.linspace(d_xmin, d_xmax, d_N)
# uniform distribution
d_y_uni_pdf = uniform.pdf(d_x, loc=d_uni_a, scale=(d_uni_b - d_uni_a))
d_y_uni_cdf = uniform.cdf(d_x, loc=d_uni_a, scale=(d_uni_b - d_uni_a))
# custom distribution
d_y_cust_pdf = erlang.pdf(d_x, a=d_cust_k, scale=d_cust_teta)
d_y_cust_cdf = erlang.cdf(d_x, a=d_cust_k, scale=d_cust_teta)
d_data_uniform = {
'x': d_x,
'y_pdf': d_y_uni_pdf,
'y_cdf': d_y_uni_cdf
}
source_uniform = ColumnDataSource(data=d_data_uniform)
d_data_custom = {
'x': d_x,
'y_pdf': d_y_cust_pdf,
'y_cdf': d_y_cust_cdf
}
source_custom = ColumnDataSource(data=d_data_custom)
def f_z_smaller__x(aa, n, A, a, UL=30):
return erlang.pdf(aa, n, loc=0, scale=1) * quad(
f_z_smaller_x_, aa, UL, args=(aa, n, A, a))[0]
