def Reff(data, si_mean, si_sd, tau=7, conf=0.95, mu=5):
""" effective reproduction number
assuming exponential distribution for prior Reff
input:
data = daily number of incidence
si_mean = mean of serial interval
si_sd = standard deviation of serial interval
tau = length of time window (integer in days)
conf = confidence level of estimated Reff
mu = mean of prior ditribution of Reff
return:
R = daily Reff of shape (3,len(data))
R[0:3] = median, min, max
reference:
A. Cori et al
American Journal of Epidemiology 178 (2013) 1505
Web Appendix 1
"""
N = len(data)
w = si_distr(N, si_mean, si_sd)
L = np.convolve(data, w)[:N]
u = np.ones(tau)
a = 1 + np.convolve(data, u)[:N]
b = mu / (1 + mu * np.convolve(L, u)[:N])
return np.vstack([gamma.median(a, 0, b), gamma.interval(conf, a, 0, b)])
def calibration_test(self, x, y_norm):
mean, var, shape, rate, mixture_var = self(x)
y = y_norm * self.y_std + self.y_mean
y_norm = y_norm.detach().numpy()
y = y.detach().numpy()
confidence_values = np.expand_dims(np.arange(0.1, 1, 0.1), axis=1)
norm_lower, norm_upper = norm.interval(
confidence_values,
loc=mean.detach().numpy(),
scale=np.sqrt(var.detach().numpy()),
)
gamma_lower, gamma_upper = gamma.interval(confidence_values,
shape.detach().numpy(),
scale=1 /
rate.detach().numpy())
output = torch.zeros_like(norm_upper)
normal_check = np.logical_and(norm_lower < y_norm[np.newaxis, :],
y_norm[np.newaxis, :] < norm_upper)
gamma_check = np.logical_and(gamma_lower < y[np.newaxis, :],
y[np.newaxis, :] < gamma_upper)
output[mixture_var < 0.5] = normal_check
output[mixture_var > 0.5] = gamma_check
return output
def calibration_test(y, shape, rate):
confidence_values = np.expand_dims(np.arange(0.1, 1, 0.1), axis=1)
lower_bounds, upper_bounds = gamma.interval(confidence_values, shape, scale=1 / rate)
lower_check = lower_bounds < y[np.newaxis, :]
upper_check = y[np.newaxis, :] < upper_bounds
return np.logical_and(lower_check, upper_check).T
def dist_pdf(z, loc, scale, a, tail=0, tail_len=0):
"""
Separation probability density function for confinement cell modelling.
Used for modelling neutron reflectometry data collected using the
confinment cell of Prescott et al. 2016-2020.
It is produced by the normalised summation of
a typical gamma distribution (parameterised by shape and scale varaiables)
and a custom tail distribution (linear decay).
Parameters
----------
z : np.array
Seperations at which the function is evaluated.
loc : float
The separation at which the PDF starts (i.e. shits PDF to lower /
higher seperations).
scale : float
The width of the gamma-component of the distribution.
a : float, optional
The shape of the gamma-component of the distribution. Lower a
result in 'exponential' PDFs, whilst higher a results in more
'normal' PDFs.
tail : float, optional
The weighting of the tail-component. Higher values result in a
higher tail. Values of zero result in no tail. The default is 0.
The default is 0.
tail_len : float, optional
The length of the tail-component. The default is 0.
Returns
-------
np.array
Probability density at separations provided.
"""
pdf1 = gamma.pdf(z, loc=loc, scale=scale, a=a)
tpeak = loc + (a - 1) * scale
tcut = tail_len + loc + (a - 1) * scale
tstart = gamma.interval(0.999, loc=loc, scale=scale,
a=a)[0] #start at 1% of cdf
pdf2 = np.ones_like(z)
pdf2[z < tpeak] = (z[z < tpeak] - tstart) / (tpeak - tstart)
pdf2[z > tpeak] = (tcut - z[z > tpeak]) / (tcut - tpeak)
pdf2[z > tcut] = 0
pdf2[z < tstart] = 0
pdf = pdf1 + pdf2 * tail
pdf[z < 100] = 0
return pdf / np.trapz(pdf, z)
def poisson_interval(mu, cl=0.683):
return (mu - gamma.interval(cl, mu)[0] if mu > 0. else 0.,
gamma.interval(cl, mu + 1.)[1] - mu)
