def uniform_type_1(a, b, x1, x2):
γ = b
f = 1 / (γ - a)
mean = uniform.mean(a, b - a)
var = uniform.var(a, b - a)
p = uniform.cdf(x2, a, b - a) - uniform.cdf(x1, a, b - a)
return γ, mean, var, p, f, a, b
def test_ns_with_invcdf(self):
"""
Check that results are consistent with uniform sampling
using inverse transform sampling.
"""
ix = (-2, 2)
iy = (1e-30, 2)
xx = np.linspace(ix[0], ix[1], 1000)
xy = np.linspace(iy[0], iy[1], 2000)
cdfx = uniform.cdf(xx, ix[0], ix[1] - ix[0])
cdfy = uniform.cdf(xy, iy[0], iy[1] - iy[0])
icdfx = InvCDF('x', xx, cdfx)
icdfy = InvCDF('y', xy, cdfy)
ns = NestedSampling(seed=42)
lh = partial(lighthouse, data=self.D)
rs = ns.explore(vars=[icdfx, icdfy], initial_samples=100,
maximum_steps=1000,
likelihood=lh, tolZ=1.e-4, tolH=1e30)
ep = rs.getexpt()
ev = rs.getZ()
h = rs.getH()
var = rs.getvar()
m = rs.getmax()
self.assertAlmostEqual(ep[0], 1.233940, 6)
self.assertAlmostEqual(ep[1], 0.981719, 6)
self.assertAlmostEqual(np.sqrt(var[0]), 0.169695, 6)
self.assertAlmostEqual(np.sqrt(var[1]), 0.183914, 6)
self.assertAlmostEqual(m[0], 1.247247, 6)
self.assertAlmostEqual(m[1], 0.908454, 6)
self.assertAlmostEqual(m[2], -156.4192, 4)
self.assertAlmostEqual(ev[0], -160.1016, 4)
self.assertAlmostEqual(ev[1], 0.161775, 6)
self.assertAlmostEqual(h, 2.617102, 6)
def uniform_type_2(a, b, x1, x2):
γ = (a * b - 1) / a
f = a
a = γ
mean = uniform.mean(a, b - a)
var = uniform.var(a, b - a)
p = uniform.cdf(x2, a, b - a) - uniform.cdf(x1, a, b - a)
return γ, mean, var, p, f, a, b
def uniform_type_4(a, b, x1, x2):
γ = 1 / a
f = a
a = (a * b - 1) / (2 * a)
b = a + γ
mean = uniform.mean(a, b - a)
var = uniform.var(a, b - a)
p = uniform.cdf(x2, a, b - a) - uniform.cdf(x1, a, b - a)
return γ, mean, var, p, f, a, b
def cdf(self, x: Tuple[float]):
"""Find the CDF for a certain x value.
Args:
x (float): The value for which the CDF is needed.
Returns:
float: The CDF value at point x.
"""
return uniform.cdf(x[0], loc=self.x_lower_bound, scale=self.x_upper_bound - self.x_lower_bound) \
* uniform.cdf(x[1], loc=self.y_lower_bound, scale=self.y_upper_bound - self.y_lower_bound)
def compute_cdf(self, x_grid):
"""
Returns numpy array of uniform CDF values at the points contained
in x_grid.
"""
return scipy_uniform.cdf(x_grid, self._minimum_value, self._range_size)
def Uniformd():
import numpy as np
from matplotlib import pyplot as plt
from scipy.stats import uniform
print("The amount of time, in minutes, that a person must wait for a bus is uniformly distributed between zero and 15 minutes, inclusive.")
print("What is the probability that a person waits fewer than 12.5 minutes?")
print("On the average, how long must a person wait? Find the mean, μ, and the standard deviation, σ. also plot a graph")
def uniform1(x, a, b):
y = [1 / (b - a) if a <= val and val <= b
else 0 for val in x]
return x, y, np.mean(y), np.std(y)
x = np.arange(-10, 100) # define range of x
for ls in [(0, 15)]:
a, b = ls[0], ls[1]
x, y, u, s = uniform1(x, a, b)
plt.plot(x, y, label=r'$\mu=%.2f,\ \sigma=%.2f$' % (u, s))
plt.legend()
plt.show()
arr=uniform.cdf(np.arange(0,13,0.5),loc=a,scale=b)
res=arr[-1]-arr[0]
print(res)
def punif(q, minimum=0,maximum=1):
"""
Calculates the cumulative of the uniform distribution
"""
from scipy.stats import uniform
result=uniform.cdf(x=q,loc=minimum,scale=maximum-minimum)
return result
def main (M):
k_array=np.array([0.5,1.5])
species=np.linspace(0,M-1, M)
for i in range (np.size(k_array)):
k=k_array[i]
m=math.ceil((M-1)/(3-k)) # maximum number of generous species
print(str('maximum value of generous is %d' %(m)))
prob_f1=np.zeros([m+1])#p(f1)
prob_f1[0]=1#initial condition
q_fav=np.zeros([M])# favorability or probability to be selfish
q_cond=-np.zeros([m+1])#conditional probability to be selfish given f1
for j in range(M-2):
for f1 in range(m+1):
#calculate conditional probability to be selfish
Tf=((M-1)/(3-k)-f1)/(M-j-1)
sd=(1/(12*(M-j-1)))**0.5
q_cond[f1]=1-norm.cdf(Tf,0.5,sd)# conditional prob to be selfish
q_fav[j]+=prob_f1[f1]*q_cond[f1]
prob_new=np.zeros([m+1])
for f1 in range (m+1):
#calculate p(f1) for the next species
if f1==0:
prob_new[f1]=prob_f1[f1]*q_cond[f1]
else:
prob_new[f1]=prob_f1[f1-1]*(1-q_cond[f1-1])+prob_f1[f1]*q_cond[f1]
prob_f1=prob_new#update
#second slowest (M-1)species
for f1 in range(m+1):
Tf=((M-1)/(3-k)-f1)
q_cond[f1]=1-uniform.cdf(Tf,0,1)
q_fav[M-2]+=prob_f1[f1]*q_cond[f1]
#print(q_fav[M-2])
prob_new=np.zeros([m+1])
for f1 in range (m+1):
#calculate p(f1) for the next species
if f1==0:
prob_new[f1]=prob_f1[f1]*q_cond[f1]
else:
prob_new[f1]=prob_f1[f1-1]*(1-q_cond[f1-1])+prob_f1[f1]*q_cond[f1]
prob_f1=prob_new#update
#slowest M species
q_fav[M-1]=prob_f1[m]
#plotthe results
#print(q_fav)
if i==0:
plt.plot(species, q_fav[::-1], color="c",marker="o",markersize=10)
else:
plt.plot(species, q_fav[::-1], color="b",marker="D",markersize=10)
plt.xticks([0,M-1],["slowest","fastest"],fontsize=12)
plt.xlabel("evolutionary rate",fontsize=12)
plt.ylim(0.0,1)
plt.ylabel("favorablity",fontsize=12)
fname=str('favorability_M%d'%(M))
plt.savefig(fname+'.pdf')
plt.show()
def compute_CDF(self, x_grid):
'''
Returns numpy array of uniform CDF values at the points contained
in x_grid
'''
return scipyuniform.cdf(x_grid, self._minimum_value, self._range_size)
def first(data, a, b):
N = len(data)
fig = pyplot.figure()
ax = fig.gca()
ax.set_yticks(arange(0, 1.1, 0.1))
pyplot.grid(True)
x = list([i for i in arange(a, b, 0.01)])
dn = -100
prev_dn = -100
for i in range(len(x) - 1):
y = get_emp_func(data, x[i])
pyplot.plot([x[i], x[i + 1]], [y, y], color='blue', linewidth=0.5)
for i in range(len(x) - 1):
y = (x[i] - a) / (b - a)
pyplot.plot([x[i], x[i + 1]], [y, y], color='red', linewidth=0.5)
pyplot.show()
ans = [0 for i in range(9)]
ans[0], ans[1], ans[2] = a, b, N
for i in range(N):
# y = get_emp_func(data, data[i]) + 1 / N
# y2 = get_emp_func(data, data[i])
y = (i + 1) / N
y2 = i / N
Fx = ((data[i] - a) / (b - a))
dn = max(dn, max(fabs(y - Fx), fabs(y2 - Fx)))
if dn != prev_dn:
ans[5] = data[i]
ans[6] = Fx
ans[7], ans[8] = y, y2
prev_dn = dn
ans[3], ans[4] = dn, dn * sqrt(N)
print(ans)
print(kstest(data, lambda param: uniform.cdf(param, loc=a, scale=b - a)))
print()
def run():
N = 10000 #experiments
x_Values = sorted(genRandos(0, 11, N)) #get rando nums
pdf_list = pdf(x_Values, N) #list returned
cdf_list = cdf(x_Values, N)
fig, axs = plt.subplots(2) #created 2 graphs
fig.suptitle("Part A")
plt.setp(axs, xticks=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11]) #x marks
axs[0].plot(list(x_Values),
pdf_list) #sorted values with appended pdf list
axs[0].set_title("Probability Distribution Function")
axs[1].plot(list(x_Values),
cdf_list) #sorted values with appended cdf list
axs[1].set_title("Cumulative Distribution Function")
#Using stats.uniform for pdf and cdf
x = np.linspace(0, 11, 10000) #spacing for graph points
fig, axs = plt.subplots(2)
fig.suptitle("Part B")
plt.setp(axs, xticks=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11])
axs[0].plot(x, uniform.pdf(x, 1, 9)) #pre built function for pdf
axs[0].set_title("Probability Distribution Function")
axs[1].plot(x, uniform.cdf(x, 1, 9)) #prebuilt function for cdf
axs[1].set_title("Cumulative Distribution Function")
plt.show()
def evaluate(self, group_name):
fig = plt.figure()
Utils.create_folder_if_not_exist(self.output_folder)
save_path = os.path.join(self.output_folder, self.name)
Utils.create_folder_if_not_exist(save_path)
for validation_source in self.method_roc.keys():
method_roc = self.method_roc[validation_source]
methods = []
for method_name, roc in method_roc.items():
pred = roc['pred']
pred = np.nan_to_num(pred, True, 0.0, 1.0, 0.0)
fpr, tpr, _ = roc_curve(roc['y'], pred)
score = roc_auc_score(roc['y'], pred)
tpr = tpr - uniform.cdf(fpr)
plt.plot(fpr, tpr)
plt.xlabel('False Positive Rate', fontsize=16)
plt.ylabel('True Positive Rate', fontsize=16)
methods.append(f"{method_name} AUC {score:.4f}")
name = f"{group_name}_{validation_source}"
path_file = f"{name}.png"
path_method = os.path.join(save_path, path_file)
plt.legend(methods)
plt.savefig(path_method)
plt.title(f"{name}")
plt.clf()
fig.clf()
self.method_roc = {}
self.logger.info("done")
def tirage_uniform(self, x, v_ordered, i):
v_ordered = np.append(v_ordered, np.inf)
if x <= v_ordered[i + 1] and x >= v_ordered[i]:
return uniform.cdf(
(x - v_ordered[i]) / (v_ordered[i + 1] - v_ordered[i]))
else:
return 0
def get_p_in(upper_known, lower_known, attacker, full_width):
"""
Normalizes the upper and lower bounds to compute how likely the attacker is to hit the goal, given the distance
between attacker and last known point of the goal
input:
upper_known: np array or pandas series, holding the presented upper part of the goal (can span several trials)
lower_known: np array or pandas series, holding the lower presented part of the goal (can span several trials)
attacker: np array or pandas series, holding the (estimated) end point of the attacker (must have as many entries as
the two other vectors)
full_width: the span that is covered by the full goal
returns:
P_in: the probability that the attacker is inside the goal (value between 0 and 1)
"""
# part 1: normalize the values to be aligned with the mean of the known goal, and to have positive values
mean_goal = (upper_known + lower_known) / 2
upper_norm = abs(upper_known - mean_goal)
lower_norm = abs(lower_known - mean_goal)
attacker_norm = abs(attacker - mean_goal)
# part 2: compute the covered and the unknown size of the goal
covered_goal = abs(upper_known - lower_known)
free_width = full_width - covered_goal
# part 3: compute the probability that the attacker hits inside the goal
p_in = 1 - uniform.cdf(attacker_norm, loc=upper_norm, scale=free_width)
return p_in
def Uniform_View(self, range_v):
"""
Parameters
----------
range_v :
Returns
-------
"""
NSim, _ = self.MPrior.shape
# get NSim
NViews = self.P_mat.shape[0]
# get NViews
P_ort = Utility.nullspace(self.P_mat)[1].T # compute P_bar
P_bar = r_[self.P_mat, P_ort] # compute P_bar
V = self.MPrior @ P_bar.T
# transform input
W = np.sort(V[:, :NViews], axis=0) # for empirical copula
Cin = np.array(V[:, :NViews].argsort(axis=0),
dtype=np.float64) + 1 # for empirical copula
Grid = np.arange(NSim) + 1.0
C = zeros(Cin.shape)
for k in range(NViews):
f = interpolate.interp1d(Cin[:, k], Grid)
C[:, k] = f(Grid) / (NSim + 1)
F = zeros((NSim, NViews))
F_hat = zeros((NSim, NViews))
F_tilda = zeros((NSim, NViews))
V_tilda = zeros((NSim, NViews))
for k in range(NViews): # determine the posterior margianl per view
F[:, k] = Grid.T / (NSim + 1)
# Uniform view
F_hat[:, k] = uniform.cdf(W[:, k], range_v[k, 0], range_v[k, 1])
F_tilda[:,
k] = (1 - self.Conf[k]) * F[:, k] + self.Conf[k] * F_hat[:,
k]
# weighted distribution
# joint postorior
f = interpolate.interp1d(F_tilda[:, k],
W[:, k],
fill_value='extrapolate')
V_tilda[:, k] = f(C[:, k])
V_tilda = c_[V_tilda, V[:, NViews:]]
# joint posterior distribution
MPost = V_tilda @ inv(P_bar.T)
# new distribution incl. views
return MPost
def cdf(self, x, log=False):
""" Get the value of the cumulative distribution function for a x """
if self.log:
x = np.log(x)
if log:
return uniform.logcdf(x, self._min, self.scale)
else:
return uniform.cdf(x, self._min, self.scale)
def plot_cdf_test(test):
x, y = Utils.ecdf(test)
x = np.append(x, [1.0])
y = np.append(y, [1.0])
y = y - uniform.cdf(x)
plt.plot(x, y)
plt.xlabel('rank', fontsize=16)
plt.ylabel('cdf(r)-r', fontsize=16)
def _Cough_Chen(self):
"""
Distribution fitted from Chen.
:return:
"""
# Small droplets. < 10micron
nparticles_small = 230
k = 3.75
D_small = numpy.arange(0, 20, 0.1)
Fcdf_small = gamma.cdf(D_small, 3.75)
Dsmall_avg = (D_small[:-1] + D_small[1:]) / 2.
F_small = numpy.diff(Fcdf_small)
Volume_small = (4.0 / 3.0) * numpy.pi * (
Dsmall_avg * 1e-6)**3 * F_small * nparticles_small
vol_small_ml = (Volume_small.sum() * m**3).asUnit(ml)
# medium droplets 10micron < x < 225 micron
# upto 100 evaporates in air.
nparticles_medium = 210
params = dict(a=0.2, b=1, loc=53, scale=200)
D_medium = numpy.arange(10, 100, 1)
Fcdf_medium = beta.cdf(D_medium, **params)
Dmedium_avg = (D_medium[:-1] + D_medium[1:]) / 2.
F_medium = numpy.diff(Fcdf_medium)
Volume_small = (4.0 / 3.0) * numpy.pi * (
Dmedium_avg * 1e-6)**3 * F_medium * nparticles_medium
vol_medium_ml = (Volume_small.sum() * m**3).asUnit(ml)
evaporatingDropletsVolume = vol_small_ml + vol_medium_ml
##### == None evaporating
D_medium = numpy.arange(100, 225, 1)
Fcdf_medium = beta.cdf(D_medium, **params)
Dmedium_avg = (D_medium[:-1] + D_medium[1:]) / 2.
F_medium = numpy.diff(Fcdf_medium)
Volume_small = (4.0 / 3.0) * numpy.pi * (
Dmedium_avg * 1e-6)**3 * F_medium * nparticles_medium
vol_medium_ml = (Volume_small.sum() * m**3).asUnit(ml)
nparticles_large = 20
D_large = numpy.arange(225, 800, 1)
Fcdf_large = uniform.cdf(D_large, loc=225, scale=800 - 225)
Dlarge_avg = (D_large[:-1] + D_large[1:]) / 2.
F_large = numpy.diff(Fcdf_large)
Volume_small = (4.0 / 3.0) * numpy.pi * (
Dlarge_avg * 1e-6)**3 * F_large * nparticles_large
vol_large_ml = (Volume_small.sum() * m**3).asUnit(ml)
nonEvaporatingDropletsVolume = vol_large_ml + vol_medium_ml
return evaporatingDropletsVolume, nonEvaporatingDropletsVolume
def cdf(self, x: float):
"""Find the CDF for a certain x value.
Args:
x (float): The value for which the CDF is needed.
"""
return uniform.cdf(x,
loc=self.lower_bound,
scale=self.upper_bound - self.lower_bound)
def uniform_distribution(select_size,
loc=-m.sqrt(3),
scale=2 * m.sqrt(3),
asked=rvs,
x=0):
if asked == rvs:
return uniform.rvs(size=select_size, loc=loc, scale=scale)
elif asked == pdf:
return uniform.pdf(x, loc=loc, scale=scale)
elif asked == cdf:
return uniform.cdf(x, loc=loc, scale=scale)
def compute_uniform_cdf(min_val, max_val, saving_folder):
# range of values for which CDF is computed (100 is arbitrary and should be high enough)
x_range = np.linspace(0, 50, 1000)
# Cumulative distribution function
pdf = uniform.pdf(x_range, loc=min_val, scale=(max_val - min_val))
cdf = uniform.cdf(x_range, loc=min_val, scale=(max_val - min_val))
# Storing cdf, pdf and x_range vectors together
proba_mat = np.vstack([x_range, cdf, pdf]).T
return proba_mat
def distribution_function(x, mu, sigma, distribution):
if distribution == Distribution.NORMAL:
return norm.cdf(x, mu, sigma)
elif distribution == Distribution.CAUCHY:
return cauchy.cdf(x, mu, sigma)
elif distribution == Distribution.LAPLACE:
return laplace.cdf(x, mu, sigma)
elif distribution == Distribution.POISSON:
return poisson.cdf(x, mu, sigma)
elif distribution == Distribution.UNIFORM:
return uniform.cdf(x, mu, sigma)
else:
return None
def update_uniform_data(attrname, old, new):
xmin, xmax = slider_uni_xminmax.value
a, b = slider_uni_ab.value
scale = b - a
x = np.linspace(xmin, xmax, d_N)
y_uni_pdf = uniform.pdf(x, loc=a, scale=scale)
y_uni_cdf = uniform.cdf(x, loc=a, scale=scale)
data_uniform = {
'x': x,
'y_pdf': y_uni_pdf,
'y_cdf': y_uni_cdf
}
source_uniform.data = data_uniform
def univariate_invert_normalization(self, uni_gaussian_data, trans_params):
"""
Inverts the marginal normalization
See the companion, univariate_make_normal.py, for more details
"""
if self.base == "gauss":
uni_uniform_data = norm.cdf(uni_gaussian_data)
elif self.base == "uniform":
uni_uniform_data = uniform.cdf(uni_gaussian_data)
else:
raise ValueError(f"Unrecognized base dist.: {base}.")
uni_data = self.univariate_invert_uniformization(
uni_uniform_data, trans_params)
return uni_data
def analytical(self):
# analytical stress/strain relationship for an infinite number of filaments
# as the actual strain multiplyied by the survival probability and integrated
# over random slack
y_analytical = []
thetas = linspace(self.theta_loc, self.theta_scale, 1000)
CDF = uniform.cdf(thetas, loc = self.theta_loc, scale = self.theta_scale)
for eps in self.e:
integ_term = (eps - thetas)*Heaviside(eps - thetas)*\
(1-weibull_min.cdf((eps - thetas), self.xi_shape, scale = self.xi_scale))
time.clock()
y_analytical.append(trapz(integ_term, CDF))
print time.clock()
self.peaks[2] = max(y_analytical)
return self.e, array(y_analytical)*self.E
def simulate_MDP():
policy = pickle.load(open("./Policies 097 002 001 mean/33.0-17.0-infinite.p", "rb"))
# policy = np.zeros((len(pi.policy), 3))
# policy[np.arange(len(pi.policy)), np.array(pi.policy)] = 1
# policy = pi.policy
policy_unravelled = np.array(policy).reshape((x_max, y_max, z_max))
uniform_cdf = [uniform.cdf(x, gateway_distance_min_q, max(gateway_distance_max_q - gateway_distance_min_q, 0.001))
for x in range(z_max)]
harvesting_rates = getEnergyHarvestingRates()
# 8333.3333
battery = 0
distance = 0
packet = 0
tot_rewards = []
max_time = int(100 / deltaT)
for _ in range(100):
reward = 0
for it in range(max_time):
action = policy_unravelled[battery, packet, distance] # pi.policy[get_state(battery, packet, distance)]
if np.random.random() <= uniform_cdf[distance]:
distance = 0
else:
distance = (distance + 1) % z_max
if action == 0:
battery = int(np.clip(battery + harvesting_rates[battery], 0, energy_quantum_storable_max))
elif action == 1:
battery = int(np.clip(battery + harvesting_rates[battery] - ram_consumption, 0, energy_quantum_storable_max))
else:
battery = int(np.clip(battery + harvesting_rates[battery] - transmit_consumption[packet], 0, energy_quantum_storable_max))
reward += (packet_length[packet] * packets_priorities[packet]) * (action == 2)
if action != 1:
packet = np.random.choice(3, p=gen_prob)
tot_rewards.append(reward)
print(np.mean(tot_rewards))
exit(-1)
print("ok")
def cdf(self, x):
'''
Evaluate the cumulative distribution function (cdf) of the uniform
random variable at points x.
Parameters
----------
x : list or numpy.ndarray
The points at which the cdf is to be evaluated.
Returns
-------
numpy.ndarray
The evaluations of the cdf.
'''
return uniform.cdf(x, self.inf, self.sup - self.inf)
def uniform_stats(a=1, b=10, n=1000000):
x = np.linspace(a-1, b+1, n)
pdf_y = uni.pdf(x, a, b-a)
cdf_y = uni.cdf(x, a, b-a)
# Plot PDF and CDF horizontally
plt.subplot(1,2,1)
plt.title("Uniform Probability Density Function (scipy) For (" + str(a) + ", " + str(b) + ") Over n=" + str(n) + " values")
plt.plot(x, pdf_y, 'r-')
plt.xlabel("Interval Values")
plt.ylabel("f(x)")
plt.subplot(1,2,2)
plt.title("Cumulative Distribution Function (scipy) For (" + str(a) + ", " + str(b) + ") Over n=" + str(n) + " values")
plt.plot(x, cdf_y, 'b-')
plt.xlabel("Interval Values")
plt.ylabel("F(x)")
plt.show()
def compute_asst_loglik(prec_df, param_df, i, precdist):
theta = param_df.loc[i, 'theta']
loc = param_df.loc[i, 'loc']
scale = param_df.loc[i, 'scale']
if precdist == 'uniform':
cdf_val = uniform.cdf(prec_df['midpoint'], loc, scale)
elif precdist == 'norm':
cdf_val = norm.cdf(prec_df['midpoint'], loc, scale)
elif precdist == 'laplace':
cdf_val = laplace.cdf(prec_df['midpoint'], loc, scale)
elif precdist == 'mixnorm':
cdf_val = 0.5 * norm.cdf(prec_df['midpoint'], loc, scale) + \
0.5 * norm.cdf(prec_df['midpoint'], -loc, scale)
else:
print 'Cannot find cdf given precinct distribution, see ' + \
'compute_asst_loglik().'
assert (False)
return binom.logpmf(prec_df['num_votes_0'], prec_df['tot_votes'],
cdf_val) + np.log(theta)
def PDF_CDF():
N = 10000
x_Values = sorted(genRandos(0, 11, N))
pdf_list = pdf(x_Values, N)
cdf_list = cdf(x_Values, N)
fig, axs = plt.subplots(2)
fig.suptitle("Part A")
plt.setp(axs, x_axis=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11])
axs[0].plot(list(x_Values), pdf_list) #sorted values with appended list
axs[0].set_title("Probability Distribution Function")
axs[1].plot(list(x_Values), cdf_list)
axs[1].set_title("Cumulative Distribution Function")
#Using stats.uniform for pdf and cdf
x = np.linspace(0, 11, 10000)
fig, axs = plt.subplots(2)
fig.suptitle("Part B")
plt.setp(axs, x_axis=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11])
axs[0].plot(x, uniform.pdf(x, 1, 9))
axs[0].set_title("Probability Distribution Function")
axs[1].plot(x, uniform.cdf(x, 1, 9))
axs[1].set_title("Cumulative Distribution Function")
plt.show()
def SA_SOBOL(driver):
# Uses the Sobel Method for SA.
# Input:
# inpt : no. of input factors
# N: number of Sobel samples
#
# Output:
# SI[] : sensitivity indices
# STI[] : total effect sensitivity indices
# Other used variables/constants:
# V : total variance
# VI : partial variances
# ---------------------- Setup ---------------------------
methd = 'SOBOL'
method = '7'
mu = [inp.get_I_mu() for inp in driver.inputs]
I_sigma = [inp.get_I_sigma() for inp in driver.inputs]
inpt = len(driver.inputs)
input = driver.inputNames
krig = driver.krig
limstate= driver.limstate
lrflag = driver.lrflag
n_meta = driver.n_meta
nEFAST = driver.nEFAST
nSOBOL = driver.nSOBOL
nMCS = driver.nMCS
nodes = driver.nodes
order = driver.order
otpt = len(driver.outputNames)
output = driver.outputNames
p = driver.p
plotf = 0
r = driver.r
simple = driver.simple
stvars = driver.stvars
# ---------------------- Model ---------------------------
value = asarray(LHS.LHS(2*inpt, nSOBOL))
for j in range(inpt):
if stvars[j].dist == 'NORM':
value[:,j] = norm.ppf(uniform.cdf(value[:,j], 0, 1), stvars[j].param[0], stvars[j].param[1])
value[:,j+inpt] = norm.ppf(uniform.cdf(value[:,j+inpt], 0, 1), stvars[j].param[0], stvars[j].param[1])
elif stvars[j].dist == 'LNORM':
value[:,j] = lognorm.ppf(uniform.cdf(value[:, j], 0, 1), stvars[j].param[1], 0, exp(stvars[j].param[0]))
value[:,j+inpt] = lognorm.ppf(uniform.cdf(value[:, j+inpt], 0, 1), stvars[j].param[1], 0, exp(stvars[j].param[0]))
elif stvars[j].dist == 'BETA':
value[:,j] = beta.ppf(uniform.cdf(value[:, j], 0, 1), stvars[j].param[0], stvars[j].param[1], stvars[j].param[2], stvars[j].param[3] - stvars[j].param[2])
value[:,j+inpt] = beta.ppf(uniform.cdf(value[:, j+inpt], 0, 1), stvars[j].param[0], stvars[j].param[1], stvars[j].param[2], stvars[j].param[3] - stvars[j].param[2])
elif stvars[j].dist == 'UNIF':
value[:,j] = uniform.ppf(uniform.cdf(value[:,j], 0, 1), stvars[j].param[0], stvars[j].param[1])
value[:,j+inpt] = uniform.ppf(uniform.cdf(value[:,j+inpt], 0, 1), stvars[j].param[0], stvars[j].param[1])
values = []
XMA = value[0:nSOBOL, 0:inpt]
XMB = value[0:nSOBOL, inpt:2 * inpt]
YXMA = zeros((nSOBOL, otpt))
YXMB = zeros((nSOBOL, otpt))
if krig == 1:
load("dmodel")
YXMA = predictor(XMA, dmodel)
YXMB = predictor(XMB, dmodel)
else:
values.extend(list(XMA))
values.extend(list(XMB))
YXMC = zeros((inpt, nSOBOL, otpt))
for i in range(inpt):
XMC = deepcopy(XMB)
XMC[:, i] = deepcopy(XMA[:, i])
if krig == 1:
YXMC[i] = predictor(XMC, dmodel)
else:
values.extend(list(XMC))
if krig != 1:
out = iter(run_list(driver, values))
for i in range(nSOBOL):
YXMA[i] = out.next()
for i in range(nSOBOL):
YXMB[i] = out.next()
for i in range(inpt):
for j in range(nSOBOL):
YXMC[i, j] = out.next()
f0 = mean(YXMA,0)
if otpt==1:
V = cov(YXMA,None,0,1)
else: #multiple outputs
V = diag(cov(YXMA,None,0,1))
Vi = zeros((otpt, inpt))
Vci = zeros((otpt, inpt))
for i in range(inpt):
for p in range(otpt):
Vi[p,i] = 1.0/nSOBOL*sum(YXMA[:,p]*YXMC[i,:,p])-f0[p]**2;
Vci[p,i]= 1.0/nSOBOL*sum(YXMB[:,p]*YXMC[i,:,p])-f0[p]**2;
Si = zeros((otpt,inpt));
Sti = zeros((otpt,inpt));
for j in range(inpt):
Si[:, j] = Vi[:, j] / V
Sti[:, j] = 1 - Vci[:, j] / V
if lrflag == 1:
SRC, stat = SRC_regress.SRC_regress(XMA, YXMA, otpt, nSOBOL)
# ---------------------- Analyze ---------------------------
Results = {'FirstOrderSensitivity': Si, 'TotalEffectSensitivity': Sti}
if lrflag == 1:
Results.update({'SRC': SRC, 'R^2': stat})
return Results
def SA_FAST(driver):
# First order indicies for a given model computed with Fourier Amplitude Sensitivity Test (FAST).
# R. I. Cukier, C. M. Fortuin, Kurt E. Shuler, A. G. Petschek and J. H. Schaibly.
# Study of the sensitivity of coupled reaction systems to uncertainties in rate coefficients.
# I-III Theory/Applications/Analysis The Journal of Chemical Physics
#
# Input:
# inpt : no. of input factors
#
# Output:
# SI[] : sensitivity indices
# Other used variables/constants:
# OM[] : frequencies of parameters
# S[] : search curve
# X[] : coordinates of sample points
# Y[] : output of model
# OMAX : maximum frequency
# N : number of sample points
# AC[],BC[]: fourier coefficients
# V : total variance
# VI : partial variances
# ---------------------- Setup ---------------------------
methd = 'FAST'
method = '9'
mu = [inp.get_I_mu() for inp in driver.inputs]
I_sigma = [inp.get_I_sigma() for inp in driver.inputs]
inpt = len(driver.inputs)
input = driver.inputNames
krig = driver.krig
limstate= driver.limstate
lrflag = driver.lrflag
n_meta = driver.n_meta
nEFAST = driver.nEFAST
nSOBOL = driver.nSOBOL
nMCS = driver.nMCS
nodes = driver.nodes
order = driver.order
otpt = len(driver.outputNames)
output = driver.outputNames
p = driver.p
plotf = 0
r = driver.r
simple = driver.simple
stvars = driver.stvars
# ---------------------- Model ---------------------------
#
MI = 4#: maximum number of fourier coefficients that may be retained in
# calculating the partial variances without interferences between the assigned frequencies
#
# Frequency assignment to input factors.
OM = SETFREQ(inpt)
# Computation of the maximum frequency
# OMAX and the no. of sample points N.
OMAX = int(OM[inpt-1])
N = 2 * MI * OMAX + 1
# Setting the relation between the scalar variable S and the coordinates
# {X(1),X(2),...X(inpt)} of each sample point.
S = pi / 2.0 * (2 * arange(1,N+1) - N-1) / N
ANGLE = matrix(OM).T * matrix(S)
X = 0.5 + arcsin(sin(ANGLE.T)) / pi
# Transform distributions from standard uniform to general.
for j in range(inpt):
if stvars[j].dist == 'NORM':
X[:,j] = norm.ppf(uniform.cdf(X[:,j], 0, 1), stvars[j].param[0], stvars[j].param[1])
elif stvars[j].dist == 'LNORM':
X[:,j] = lognorm.ppf(uniform.cdf(X[:, j], 0, 1), stvars[j].param[1], 0, exp(stvars[j].param[0]))
elif stvars[j].dist == 'BETA':
X[:,j] = beta.ppf(uniform.cdf(X[:, j], 0, 1), stvars[j].param[0], stvars[j].param[1], stvars[j].param[2], stvars[j].param[3] - stvars[j].param[2])
elif stvars[j].dist == 'UNIF':
X[:,j] = uniform.ppf(uniform.cdf(X[:,j], 0, 1), stvars[j].param[0], stvars[j].param[1])
# Do the N model evaluations.
Y = zeros((N, otpt))
if krig == 1:
load("dmodel")
Y = predictor(X, dmodel)
else:
values = []
for p in range(N):
# print 'Running simulation on test',p+1,'of',N
# Y[p] = run_model(driver, array(X[p])[0])
values.append(array(X[p])[0])
Y = run_list(driver, values)
# Computation of Fourier coefficients.
AC = zeros((N, otpt))# initially zero
BC = zeros((N, otpt))# initially zero
# q = int(N / 2)-1
q = (N-1)/2
for j in range(2,N+1,2): # j is even
# print "Y[q]",Y[q]
# print "matrix(cos(pi * j * arange(1,q+) / N))",matrix(cos(pi * j * arange(1,q+1) / N))
# print "matrix(Y[q + arange(0,q)] + Y[q - arange(0,q)])",matrix(Y[q + arange(1,q+1)] + Y[q - arange(1,q+1)])
AC[j-1] = 1.0 / N * matrix(Y[q] + matrix(cos(pi * j * arange(1,q+1) / N)) * matrix(Y[q + arange(1,q+1)] + Y[q - arange(1,q+1)]))
for j in range(1,N+1,2): # j is odd
BC[j-1] = 1.0 / N * matrix(sin(pi * j * arange(1,q+1) / N)) * matrix(Y[q + arange(1,q+1)] - Y[q - arange(1,q+1)])
# Computation of the general variance V in the frequency domain.
V = 2 * (matrix(AC).T * matrix(AC) + matrix(BC).T * matrix(BC))
# Computation of the partial variances and sensitivity indices.
# Si=zeros(inpt,otpt);
Si = zeros((otpt,otpt,inpt));
for i in range(inpt):
Vi = zeros((otpt, otpt))
for j in range(1,MI+1):
idx = j * OM[i]-1
Vi = Vi + AC[idx].T * AC[idx] + BC[idx].T * BC[idx]
Vi = 2. * Vi
Si[:, :, i] = Vi / V
if lrflag == 1:
SRC, stat = SRC_regress.SRC_regress(X, Y, otpt, N)
# ---------------------- Analyze ---------------------------
Sti = []# appears right after the call to this method in the original PCC_Computation.m
# if plotf == 1:
# piecharts(inpt, otpt, Si, Sti, method, output)
if simple == 1:
Si_t = zeros((inpt,otpt))
for p in range(inpt):
Si_t[p] = diag(Si[:, :, p])
Si = Si_t.T
Results = {'FirstOrderSensitivity': Si}
if lrflag == 1:
Results.update({'SRC': SRC, 'R^2': stat})
return Results
def UP_MCS(driver):
# Uses the MCS method for UP
mu = [inp.get_I_mu() for inp in driver.inputs]
I_sigma = [inp.get_I_sigma() for inp in driver.inputs]
inpt = len(driver.inputs)
input = driver.inputNames
krig = driver.krig
limstate= driver.limstate
lrflag = driver.lrflag
n_meta = driver.n_meta
nEFAST = driver.nEFAST
nSOBOL = driver.nSOBOL
nMCS = driver.nMCS
nodes = driver.nodes
order = driver.order
otpt = len(driver.outputNames)
output = driver.outputNames
p = driver.p
plotf = 0
r = driver.r
simple = driver.simple
stvars = driver.stvars
#*****************RANDOM DRAWS FROM INPUT DISTRIBUTIONS********************
value = asarray(LHS.LHS(inpt, nMCS))
for j in range(inpt):
if stvars[j].dist == 'NORM':
value[:,j] = norm.ppf(uniform.cdf(value[:,j], 0, 1), stvars[j].param[0], stvars[j].param[1])
elif stvars[j].dist == 'LNORM':
value[:,j] = lognorm.ppf(uniform.cdf(value[:, j], 0, 1), stvars[j].param[1], 0, exp(stvars[j].param[0]))
elif stvars[j].dist == 'BETA':
value[:,j] = beta.ppf(uniform.cdf(value[:, j], 0, 1), stvars[j].param[0], stvars[j].param[1], stvars[j].param[2], stvars[j].param[3] - stvars[j].param[2])
elif stvars[j].dist == 'UNIF':
value[:,j] = uniform.ppf(uniform.cdf(value[:,j], 0, 1), stvars[j].param[0], stvars[j].param[1])
# ---------------------- Model ---------------------------
out = zeros((nMCS, otpt))
if krig == 1:
load("dmodel")
out = predictor(value, dmodel)
else:
# for i in range(nMCS):
# print 'Running simulation',i+1,'of',nMCS,'with inputs',value[i]
# out[i] = run_model(driver, value[i])
out = run_list(driver, value)
limstate = asarray(limstate)
limstate1 = asarray(kron(limstate[:, 0], ones(nMCS))).reshape(otpt,nMCS).transpose()
limstate2 = asarray(kron(limstate[:, 1], ones(nMCS))).reshape(otpt,nMCS).transpose()
B = logical_and(greater_equal(out,limstate1),less_equal(out,limstate2))
PCC = sum(B,0) / nMCS
B_t = B[sum(B,1) == otpt]
if otpt > 1 and not 0 in PCC[0:otpt]:
PCC = append(PCC,len(B_t) / nMCS)
#Moments
CovarianceMatrix = matrix(cov(out,None,0))#.transpose()
Moments = {'Mean': mean(out,0), 'Variance': diag(CovarianceMatrix), 'Skewness': skew(out), 'Kurtosis': kurtosis(out,fisher=False)}
# combine the display of the correlation matrix with setting a var that will be needed below
sigma_mat=matrix(sqrt(diag(CovarianceMatrix)))
CorrelationMatrix= CovarianceMatrix/multiply(sigma_mat,sigma_mat.transpose())
# ---------------------- Analyze ---------------------------
if any(Moments['Variance']==0):
print "Warning: One or more outputs does not vary over given parameter variation."
C_Y = [0]*otpt
for k in range(0,otpt):
if Moments['Variance'][k]!=0:
C_Y[k] = estimate_complexity.with_samples(out[:,k],nMCS)
sigma_mat=matrix(sqrt(diag(CovarianceMatrix)))
seterr(invalid='ignore') #ignore problems with divide-by-zero, just give us 'nan' as usual
CorrelationMatrix= CovarianceMatrix/multiply(sigma_mat,sigma_mat.transpose())
Distribution = {'Complexity': C_Y}
CorrelationMatrix=where(isnan(CorrelationMatrix), None, CorrelationMatrix)
Results = {'Moments': Moments, 'CorrelationMatrix': CorrelationMatrix,
'CovarianceMatrix': CovarianceMatrix, 'Distribution': Distribution, 'PCC': PCC}
return Results
def SA_EFAST(driver):
#[SI,STI] = EFAST(K,WANTEDN)
# First order and total effect indices for a given model computed with
# Extended Fourier Amplitude Sensitivity Test (EFAST).
# Andrea Saltelli, Stefano Tarantola and Karen Chan. 1999
# A quantitative model-independent method for global sensitivity analysis of model output.
# Technometrics 41:39-56
#
# Input:
# inpt : no. of input factors
# WANTEDN : wanted no. of sample points
#
# Output:
# SI[] : first order sensitivity indices
# STI[] : total effect sensitivity indices
# Other used variables/constants:
# OM[] : vector of inpt frequencies
# OMI : frequency for the group of interest
# OMCI[] : set of freq. used for the compl. group
# X[] : parameter combination rank matrix
# AC[],BC[]: fourier coefficients
# FI[] : random phase shift
# V : total output variance (for each curve)
# VI : partial var. of par. i (for each curve)
# VCI : part. var. of the compl. set of par...
# AV : total variance in the time domain
# AVI : partial variance of par. i
# AVCI : part. var. of the compl. set of par.
# Y[] : model output
# N : no. of runs on each curve
# ---------------------- Setup ---------------------------
methd = 'EFAST'
method = '10'
mu = [inp.get_I_mu() for inp in driver.inputs]
I_sigma = [inp.get_I_sigma() for inp in driver.inputs]
inpt = len(driver.inputs)
input = driver.inputNames
krig = driver.krig
limstate= driver.limstate
lrflag = driver.lrflag
n_meta = driver.n_meta
nEFAST = driver.nEFAST
nSOBOL = driver.nSOBOL
nMCS = driver.nMCS
nodes = driver.nodes
order = driver.order
otpt = len(driver.outputNames)
output = driver.outputNames
p = driver.p
plotf = 0
r = driver.r
simple = driver.simple
stvars = driver.stvars
# ---------------------- Model ---------------------------
NR = 1#: no. of search curves
MI = 4#: maximum number of fourier coefficients that may be retained in calculating
# the partial variances without interferences between the assigned frequencies
#
# Computation of the frequency for the group of interest OMi and the no. of sample points N.
OMi = int(floor((nEFAST / NR - 1) / (2 * MI) / inpt))
N = 2 * MI * OMi + 1
total_sims = N*NR*inpt
sim = 0
if (N * NR < 65):
logging.error('sample size must be >= 65 per factor.')
raise ValueError,'sample size must be >= 65 per factor.'
# Algorithm for selecting the set of frequencies. OMci(i), i=1:inpt-1, contains
# the set of frequencies to be used by the complementary group.
OMci = SETFREQ(N - 1, OMi / 2 / MI)
# Loop over the inpt input factors.
Si = zeros((otpt,otpt,inpt));
Sti = zeros((otpt,otpt,inpt));
for i in range(inpt):
# Initialize AV,AVi,AVci to zero.
AV = 0
AVi = 0
AVci = 0
# Loop over the NR search curves.
for L in range(NR):
# Setting the vector of frequencies OM for the inpt factors.
cj = 1
OM = zeros(inpt)
for j in range(inpt):
if (j == i):
# For the factor of interest.
OM[i] = OMi
else:
# For the complementary group.
OM[j] = OMci[cj]
cj = cj + 1
# Setting the relation between the scalar variable S and the coordinates
# {X(1),X(2),...X(inpt)} of each sample point.
FI = zeros(inpt)
for j in range(inpt):
FI[j] = random.random() * 2 * pi # random phase shift
S_VEC = pi * (2 * arange(1,N+1) - N - 1) / N
OM_VEC = OM[range(inpt)]
FI_MAT = transpose(array([FI]*N))
ANGLE = matrix(OM_VEC).T*matrix(S_VEC) + matrix(FI_MAT)
X = 0.5 + arcsin(sin(ANGLE.T)) / pi
# Transform distributions from standard uniform to general.
for j in range(inpt):
if stvars[j].dist == 'NORM':
X[:,j] = norm.ppf(uniform.cdf(X[:,j], 0, 1), stvars[j].param[0], stvars[j].param[1])
elif stvars[j].dist == 'LNORM':
X[:,j] = lognorm.ppf(uniform.cdf(X[:, j], 0, 1), stvars[j].param[1], 0, exp(stvars[j].param[0]))
elif stvars[j].dist == 'BETA':
X[:,j] = beta.ppf(uniform.cdf(X[:, j], 0, 1), stvars[j].param[0], stvars[j].param[1], stvars[j].param[2], stvars[j].param[3] - stvars[j].param[2])
elif stvars[j].dist == 'UNIF':
X[:,j] = uniform.ppf(uniform.cdf(X[:,j], 0, 1), stvars[j].param[0], stvars[j].param[1])
# Do the N model evaluations.
Y = zeros((N, otpt))
if krig == 1:
load("dmodel")
Y = predictor(X, dmodel)
else:
values = []
for p in range(N):
# sim += 1
# print 'Running simulation on test',sim,'of',total_sims
# Y[p] = run_model(driver, array(X[p])[0])
values.append(array(X[p])[0])
Y = run_list(driver, values)
# Subtract the average value.
Y = Y - kron(mean(Y,0), ones((N, 1)))
# Fourier coeff. at [1:OMi/2].
NQ = int(N / 2)-1
N0 = NQ + 1
COMPL = 0
Y_VECP = Y[N0+1:] + Y[NQ::-1]
Y_VECM = Y[N0+1:] - Y[NQ::-1]
# AC = zeros((int(ceil(OMi / 2)), otpt))
# BC = zeros((int(ceil(OMi / 2)), otpt))
AC = zeros((OMi * MI, otpt))
BC = zeros((OMi * MI, otpt))
for j in range(int(ceil(OMi / 2))+1):
ANGLE = (j+1) * 2 * arange(1,NQ+2) * pi / N
C_VEC = cos(ANGLE)
S_VEC = sin(ANGLE)
AC[j] = (Y[N0] +matrix(C_VEC)*matrix(Y_VECP)) / N
BC[j] = matrix(S_VEC) * matrix(Y_VECM) / N
COMPL = COMPL + matrix(AC[j]).T * matrix(AC[j]) + matrix(BC[j]).T * matrix(BC[j])
# Computation of V_{(ci)}.
Vci = 2 * COMPL
AVci = AVci + Vci
# Fourier coeff. at [P*OMi, for P=1:MI].
COMPL = 0
# Do these need to be recomputed at all?
# Y_VECP = Y[N0 + range(NQ)] + Y[N0 - range(NQ)]
# Y_VECM = Y[N0 + range(NQ)] - Y[N0 - range(NQ)]
for j in range(OMi, OMi * MI + 1, OMi):
ANGLE = j * 2 * arange(1,NQ+2) * pi / N
C_VEC = cos(ANGLE)
S_VEC = sin(ANGLE)
AC[j-1] = (Y[N0] + matrix(C_VEC)*matrix(Y_VECP)) / N
BC[j-1] = matrix(S_VEC) * matrix(Y_VECM) / N
COMPL = COMPL + matrix(AC[j-1]).T * matrix(AC[j-1]) + matrix(BC[j-1]).T * matrix(BC[j-1])
# Computation of V_i.
Vi = 2 * COMPL
AVi = AVi + Vi
# Computation of the total variance in the time domain.
AV = AV + matrix(Y).T * matrix(Y) / N
# Computation of sensitivity indicies.
AV = AV / NR
AVi = AVi / NR
AVci = AVci / NR
Si[:, :, i] = AVi / AV
Sti[:, :, i] = 1 - AVci / AV
if lrflag == 1:
SRC, stat = SRC_regress.SRC_regress(X, Y, otpt, N)
# ---------------------- Analyze ---------------------------
# if plotf == 1:
# piecharts(inpt, otpt, Si, Sti, methd, output)
if simple == 1:
Si_t = zeros((inpt,otpt))
for p in range(inpt):
Si_t[p] = diag(Si[:, :, p])
Si = Si_t.T
if simple == 1:
Sti_t = zeros((inpt,otpt))
for p in range(inpt):
Sti_t[p] = diag(Sti[:, :, p])
Sti = Sti_t.T
Results = {'FirstOrderSensitivity': Si, 'TotalEffectSensitivity': Sti}
if lrflag == 1:
Results.update({'SRC': SRC, 'R^2': stat})
return Results
def cdf(self, dist, v):
return uniform.cdf(v, *self._get_params(dist))