maxPrice=10; #gamma Parameter shape=2; scale=1.5 nsamples=300; #np.random.seed(852952) tau = np.random.gamma(shape,scale,nsamples) actions= np.random.rand(nsamples)*(maxPrice-minPrice) + minPrice; nbins=10; actionsBins = np.linspace(minPrice, maxPrice, nbins+1); print(gamma.mean(shape,scale=scale)); print(actionsBins) discreteActions=np.digitize(actions,actionsBins); print(actions[:10]); print(discreteActions[:10]) rewardsW=np.zeros(actions.size); rewardsDiscrete=np.zeros(actions.size); rewardsW[actions<=tau]=actions[actions<=tau]; rewardsDiscrete[discreteActions<=tau]=discreteActions[discreteActions<=tau]; print(rewardsW); #plt.scatter(actions,rewardsW) #plt.show()
def mean(self, dist):
return gamma.mean(*self._get_params(dist))
def mean(self, n, p):
mu = gamma.mean(self, n, p)
return mu
plt.hist(mean_val, density=True, alpha=0.5, label='hist mean n ' + str(n))
# мат. ожидание sigma нормального распределения
sigma = np.math.sqrt(Dx / n)
print('мат.ожидание=', Ex)
print('sigma=', sigma)
# зададим нормальное распределенние
norm_rv = sts.norm(loc=Ex, scale=sigma)
x = np.linspace(6, 14, 100)
pdf = norm_rv.pdf(x)
plt.plot(x, pdf, 'r-', lw=3, alpha=0.7, label='erlang pdf n ' + str(n))
plt.ylabel('samples')
plt.xlabel('$x$')
plt.legend(loc='upper right')
plt.show()
# Вычисление теоритических EX, std, DX распределения
EX = gamma.mean(k)
std = gamma.std(k)
DX = std**2
print('Ex=', EX, ' STD=', std, ' DX=', DX)
gamma_func(5, EX, DX)
gamma_func(10, EX, DX)
gamma_func(50, EX, DX)
# ## Вывод:
# Распределение выборочных средних для функции gamma хорошо описывается нормальным распределением.
# С ростом n точность аппроксимации увеличивается.
def mean(self) -> np.ndarray:
return np.atleast_1d(
gamma.mean(self._shape, scale=self._scale)
)
def analytical_MPVS(
infection_ts: pd.DataFrame,
smoothing: Callable,
alpha: float = 3.0, # shape
beta: float = 2.0, # rate
CI: float = 0.95, # confidence interval
infectious_period: int = 5*days, # inf period = 1/gamma,
variance_shift: float = 0.99, # how much to scale variance parameters by when anomaly detected
totals: bool = True # are these case totals or daily new cases?
):
"""Estimates Rt ~ Gamma(alpha, 1/beta), and implements an analytical expression for a mean-preserving variance increase whenever case counts fall outside the CI defined by a negative binomial distribution"""
# infection_ts = infection_ts.copy(deep = True)
dates = infection_ts.index
if totals:
# daily_cases = np.diff(infection_ts.clip(lower = 0)).clip(min = 0) # infection_ts clipped because COVID19India API does weird stuff
daily_cases = infection_ts.clip(lower = 0).diff().clip(lower = 0).iloc[1:]
else:
daily_cases = infection_ts
total_cases = np.cumsum(smoothing(np.squeeze(daily_cases)))
v_alpha, v_beta = [], []
RR_pred, RR_CI_upper, RR_CI_lower = [], [], []
T_pred, T_CI_upper, T_CI_lower = [], [], []
new_cases_ts = []
anomalies = []
anomaly_dates = []
for i in range(2, len(total_cases)):
new_cases = max(0, total_cases[i] - total_cases[i-1])
old_new_cases = max(0, total_cases[i-1] - total_cases[i-2])
alpha += new_cases
beta += old_new_cases
v_alpha.append(alpha)
v_beta.append(beta)
RR_est = max(0, 1 + infectious_period*np.log(Gamma.mean( a = alpha, scale = 1/beta)))
RR_upper = max(0, 1 + infectious_period*np.log(Gamma.ppf(CI, a = alpha, scale = 1/beta)))
RR_lower = max(0, 1 + infectious_period*np.log(Gamma.ppf(1-CI, a = alpha, scale = 1/beta)))
RR_pred.append(RR_est)
RR_CI_upper.append(RR_upper)
RR_CI_lower.append(RR_lower)
if (new_cases == 0 or old_new_cases == 0):
if new_cases == 0:
logger.debug("new_cases at time %s: 0", i)
if old_new_cases == 0:
logger.debug("old_new_cases at time %s: 0", i)
T_pred.append(0)
T_CI_upper.append(10) # <- where does this come from?
T_CI_lower.append(0)
new_cases_ts.append(0)
if (new_cases > 0 and old_new_cases > 0):
new_cases_ts.append(new_cases)
r, p = alpha, beta/(old_new_cases + beta)
T_pred.append(nbinom.mean(r, p))
T_upper = nbinom.ppf(CI, r, p)
T_lower = nbinom.ppf(1-CI, r, p)
T_CI_upper.append(T_upper)
T_CI_lower.append(T_lower)
_np = p
_nr = r
anomaly_noted = False
counter = 0
while not (T_lower < new_cases < T_upper):
if not anomaly_noted:
anomalies.append(new_cases)
anomaly_dates.append(dates[i])
# logger.debug("anomaly identified at time %s: %s < %s < %s, r: %s, p: %s, annealing iteration: %s", i, T_lower, new_cases, T_upper, _nr, _np, counter+1)
# nnp = 0.95 *_np # <- where does this come from
_nr = variance_shift * _nr * ((1-_np)/(1-variance_shift*_np) )
_np = variance_shift * _np
T_upper = nbinom.ppf(CI, _nr, _np)
T_lower = nbinom.ppf(1-CI, _nr, _np)
T_lower, T_upper = sorted((T_lower, T_upper))
if T_lower == T_upper == 0:
T_upper = 1
logger.debug("CI collapse, setting T_upper -> 1")
anomaly_noted = True
counter += 1
if counter >= 10000:
raise ValueError("Number of iterations exceeded")
else:
if anomaly_noted:
alpha = _nr # update distribution on R with new parameters that enclose the anomaly
beta = _np/(1-_np) * old_new_cases
T_pred[-1] = nbinom.mean(_nr, _np)
T_CI_lower[-1] = nbinom.ppf(CI, _nr, _np)
T_CI_upper[-1] = nbinom.ppf(1-CI, _nr, _np)
# annealing leaves the RR mean unchanged, but we need to adjust its widened CI
RR_upper = max(0, 1 + infectious_period * np.log(Gamma.ppf(CI , a = alpha, scale = 1/beta)))
RR_lower = max(0, 1 + infectious_period * np.log(Gamma.ppf(1 - CI, a = alpha, scale = 1/beta)))
# replace latest CI time series entries with adjusted CI
RR_CI_upper[-1] = RR_upper
RR_CI_lower[-1] = RR_lower
return (
dates[2:],
RR_pred, RR_CI_upper, RR_CI_lower,
T_pred, T_CI_upper, T_CI_lower,
total_cases, new_cases_ts,
anomalies, anomaly_dates
)
def update_lambda(self, data):
self.occurrence_shape += sum(data)
self.occurrence_scale += len(data) * self.interval
self._gamma_map = self._gamma_mode(self.occurrence_shape, self.occurrence_scale)
self._gamma_mean = gamma.mean(self.occurrence_shape, scale=1/float(self.occurrence_scale))
d0 = d0[0]/d0[0]
return dm2, dm1, d0, d1, d2, d3
#dm2, dm1, d0, d1, d2, d3 = getProbabilities(5)
# Example from:
# http://docs.scipy.org/doc/scipy-0.14.0/reference/generated/scipy.stats.gamma.html
# http://docs.scipy.org/doc/scipy-0.14.0/reference/tutorial/integrate.html
# gamma function
a = 5
rv = gamma(a, loc=0., scale = 1.) # scale = 1.0 / lambda.
median = gamma.median(a)
mean = gamma.mean(a)
x0 = scipy.optimize.fsolve(lambda x: rv.pdf(x), 0.1) # 0.3 is the starting point
# find the top
increase = True
df0 = 0
delta = 0.01
x = 0.01
while increase == True:
h = rv.pdf(x+delta) - rv.pdf(x)
if h < 0:
increase = False
else:
x = x + delta
def reset(self):
self.occurrence_scale = 1.0
self.occurrence_shape = 1.1
self._gamma_map = self._gamma_mode(self.occurrence_shape, self.occurrence_scale)
self._gamma_mean = gamma.mean(self.occurrence_shape, scale=1/float(self.occurrence_scale))
def update_rate(self, data):
self.alpha += sum(data)
self.beta += len(data) * self.interval
self.mode = self._mode(self.alpha, self.beta)
self.mean = gamma.mean(self.alpha, scale=1/float(self.beta))
def reset(self):
self.beta = 1.1
self.alpha = 1.1
self.mode = self._mode(self.alpha, self.beta)
self.mean = gamma.mean(self.alpha, scale=1/float(self.beta))
def mean(self):
return gamma.mean(self.shape, self.loc, self.scale)
return dm2, dm1, d0, d1, d2, d3
# dm2, dm1, d0, d1, d2, d3 = getProbabilities(5)
# Example from:
# http://docs.scipy.org/doc/scipy-0.14.0/reference/generated/scipy.stats.gamma.html
# http://docs.scipy.org/doc/scipy-0.14.0/reference/tutorial/integrate.html
# gamma function
a = 5
rv = gamma(a, loc=0.0, scale=1.0) # scale = 1.0 / lambda.
median = gamma.median(a)
mean = gamma.mean(a)
x0 = scipy.optimize.fsolve(lambda x: rv.pdf(x), 0.1) # 0.3 is the starting point
# find the top
increase = True
df0 = 0
delta = 0.01
x = 0.01
while increase == True:
h = rv.pdf(x + delta) - rv.pdf(x)
if h < 0:
increase = False
else:
x = x + delta
