def _init_params(self, X):
init = self.init
n_samples, n_features = X.shape
n_components = self.n_components
if (init == 'kmeans'):
km = Kmeans(n_components)
clusters, mean, cov = km.cluster(X)
coef = sp.array([c.shape[0] / n_samples for c in clusters])
comps = [multivariate_normal(mean[i], cov[i], allow_singular=True)
for i in range(n_components)]
elif (init == 'rand'):
coef = sp.absolute(sprand.randn(n_components))
coef = coef / coef.sum()
means = X[sprand.permutation(n_samples)[0: n_components]]
clusters = [[] for i in range(n_components)]
for x in X:
idx = sp.argmin([spla.norm(x - mean) for mean in means])
clusters[idx].append(x)
comps = []
for k in range(n_components):
mean = means[k]
cov = sp.cov(clusters[k], rowvar=0, ddof=0)
comps.append(multivariate_normal(mean, cov, allow_singular=True))
self.coef = coef
self.comps = comps
def main():
N = 10000
D = 10
mean, covar = moments_from_samples(random.rand(N), random.randn(D, N))
print("residual norm^2:",
linalg.norm(mean)**2,
linalg.norm(covar - eye(D))**2)
def generate_lin_system(n, d, filepath=None):
X = random.randn(n, d)
A = 1. / (d * n) * X.T.dot(X)
y = random.uniform(low=-1, high=1, size=d)
b = A.dot(y)
# -10 and 10 are an arbitrary choice here
mask_A = random.uniform(low=-10, high=10, size=(d, d))
mask_b = random.uniform(low=-10, high=10, size=d)
if filepath:
write_system(A, b, y, filepath)
return (A, mask_A, b, mask_b, y)
def generate_lin_regression(n, d, sigma):
"""
See cgd.pdf
"""
X = random.randn(n, d)
for i in xrange(d):
X[:, i] /= numpy.max(numpy.abs(X[:, i]))
beta = random.uniform(low=0, high=1, size=d)
e = numpy.array(random.normal(0, sigma, n))
y = X.dot(beta) + e.T
return (X, y, beta, e)
def calloptionMC(spot,strike,vol,T,Rf,M,alpha):
'''
call option using Monte Carlo simulation
returns option price and lower and upper bounds
'''
S = np.zeros(M)
S = spot * np.exp( (Rf - 0.5*vol**2)*T + vol * math.sqrt(T) * random.randn(M))
Carray = np.exp(-Rf * T) * np.maximum(S - strike, 0)
c = np.average(Carray)
c_std = np.std(Carray) / math.sqrt(M)
bounds = c + np.array([-1,1]) * stats.norm.ppf(0.5 + alpha/2) * c_std
return c, bounds
def draw(self):
"""
If an arm has been drawn less than 2 times, select that arm
Otherwise return:
argmax([ ... random.normal(mean=expected_return[i], sd=std[i]) ...])
:return: The numerical index of the selected arm
"""
mu = array(self.expected_payouts)
sd = array([float('inf') if n < 2 else sqrt(s / (n - 1)) for s, n in zip(self.M2, self.draws)])
return argmax(random.randn(self.n_arms) * sd + mu)
def _em(self, X):
# Constants
n_samples, n_features = X.shape
n_components = self.n_components
max_iter = self.max_iter
# tol = self.tol
mu = X.mean(axis=0)
X_centered = X - sp.atleast_2d(mu)
# Initialize parameters
latent_mean = 0
sigma2 = 1
weight = sprd.randn(n_features, n_components)
# Main loop of EM algorithm
for i in range(max_iter):
# E step
M = sp.dot(weight.T, weight) + sigma2 * sp.eye(n_components)
inv_M = spla.inv(M)
latent_mean = sp.dot(inv_M, sp.dot(weight.T, X_centered.T)).T
# M step
expectation_zzT = n_samples * sigma2 * inv_M + sp.dot(latent_mean.T, latent_mean)
# Re-estimate W
weight = sp.dot(sp.dot(X_centered.T, latent_mean), spla.inv(expectation_zzT))
weight2 = sp.dot(weight.T, weight)
# Re-estimate \sigma^2
sigma2 = ((spla.norm(X_centered)**2 -
2 * sp.dot(latent_mean.ravel(), sp.dot(X_centered, weight).ravel()) +
sp.trace(sp.dot(expectation_zzT, weight2))) /
(n_samples * n_features))
self.predict_mean = mu
self.predict_cov = sp.dot(weight, weight.T) + sigma2 * sp.eye(n_features)
self.latent_mean = latent_mean
self.latent_cov = sigma2 * inv_M
self.sigma2 = sigma2
self.weight = weight
self.inv_M = inv_M
return self.latent_mean
def add_awgn(x, SNRdB=100):
'''
add additive withe gaussian noise with a given SNR (dB)
INPUT
x: ndarray (TODO check sulle istanze per righe o per colonne)
OUTPUT
xn: copy of x corrupted by awgn
'''
if SNRdB < np.inf:
ps = linalg.norm(x, axis=-1)
noise = random.randn(*x.shape)
pn0 = linalg.norm(noise, axis=-1)
pn = ps / pn0 / (10**(SNRdB / 20))
if len(x.shape) > 1:
pn = np.tile(pn, (x.shape[1], 1)).T
xn = x + noise * pn
else:
xn = x
return xn
def Inference_2D(Data, Data_pred, confidence=0.95):
## Objective: To fit the hierarchical approach on univariate functions
## Input:
## 1. Data: The dataset to be modeled
## 2. Data_pred: Prediction locations
## 3. confidence: % confidence level for the model
## Output:
## 1. results: The inference results for the prediction
## 2. Sp: The sparse representation at each scale
## 3. qoptx: Optimal penalty at each considered scale (either 1 or 2) in x direction
## 4. qopty: Optimal penalty at each considered scale (either 1 or 2) in y direction
## 5. fun_vec: The cost of fitting at each scale
## Initializations
points = Data.shape[0]
Gaussian = np.empty([points, points])
## Computing T
dist = squareform(pdist(Data[:, 0:2], 'euclidean'))
max_d = np.amax(dist)
k1 = 2 * (max_d / 2)**2
T = k1
l_s = 0
s = 0
n = Data.shape[0]
tol = 1.0e9
fun_vec = []
results = {}
qoptx = []
qopty = []
Sp = {}
while l_s <= n:
epsilon = T / (2**s)
Gaussian = np.exp(-((dist**2) / epsilon))
# Finding the rank of the Gaussian Kernel
l_s = LA.matrix_rank(Gaussian)
k = l_s + 8
#print('epsilon,T,s for this iteration is '+str(epsilon)+' '+str(T)+' '+str(s))
# Calculating the W matrix
points = Data.shape[0]
A = random.randn(k, points)
W = A.dot(Gaussian)
W = np.matrix(W)
# Applying Pivoted QR on W
Q, R, P = linalg.qr(W, mode='full', pivoting=True)
Perm = np.zeros([len(P), len(P)], dtype=int)
for w in range(0, len(P)):
Perm[P[w], w] = 1
sparse = np.zeros([l_s, 3])
for i in range(l_s):
sparse[i, 0] = Data[P[i], 0]
sparse[i, 1] = Data[P[i], 1]
sparse[i, 2] = Data[P[i], 3]
# Selecting Relevant columns of Kernel into B Matrix
Mat = Gaussian.dot(Perm)
B = Mat[:, 0:l_s]
## Getting the Permutation matrix for the coordinate vector
ttx = sparse[:, 0]
ttx_sort = np.argsort(ttx)
Permmx = np.zeros([len(ttx_sort), len(ttx_sort)], dtype=int)
for w in range(0, len(ttx_sort)):
Permmx[w, ttx_sort[w]] = 1
tty = sparse[:, 1]
tty_sort = np.argsort(tty)
Permmy = np.zeros([len(tty_sort), len(tty_sort)], dtype=int)
for w in range(0, len(tty_sort)):
Permmy[w, tty_sort[w]] = 1
### COmputing the Penalty and Fitting the Regularization network
q1x, q1y = 1, 1
Pex_temp = Penalty_p(q1x, l_s)
Pe_x = Permmx.T.dot(Pex_temp).dot(Permmx)
Pey_temp = Penalty_p(q1y, l_s)
Pe_y = Permmy.T.dot(Pey_temp).dot(Permmy)
args = (Data, B, Pe_x, Pe_y)
bnds = [(1.0e-12, None), (1.0e-12, None)]
par = [0.1, 0.1]
l = minimize(Cost_func_2D, par, args, bounds=bnds, method='SLSQP')
lam1_x_1 = l.x[0]
lam1_y_1 = l.x[1]
fun1_1 = l.fun
q1x, q1y = 1, 2
Pex_temp = Penalty_p(q1x, l_s)
Pe_x = Permmx.T.dot(Pex_temp).dot(Permmx)
Pey_temp = Penalty_p(q1y, l_s)
Pe_y = Permmy.T.dot(Pey_temp).dot(Permmy)
args = (Data, B, Pe_x, Pe_y)
bnds = [(1.0e-12, None), (1.0e-12, None)]
par = [0.1, 0.1]
l = minimize(Cost_func_2D, par, args, bounds=bnds, method='SLSQP')
lam1_x_2 = l.x[0]
lam1_y_2 = l.x[1]
fun1_2 = l.fun
if fun1_1 <= fun1_2:
fun1 = fun1_1
lam_x1 = lam1_x_1
lam_y1 = lam1_y_1
optq_x1 = 1
optq_y1 = 1
else:
fun1 = fun1_2
lam_x1 = lam1_x_2
lam_y1 = lam1_y_2
optq_x1 = 1
optq_y1 = 2
q2x, q2y = 2, 1
Pex_temp = Penalty_p(q2x, l_s)
Pe_x = Permmx.T.dot(Pex_temp).dot(Permmx)
Pey_temp = Penalty_p(q2y, l_s)
Pe_y = Permmy.T.dot(Pey_temp).dot(Permmy)
args = (Data, B, Pe_x, Pe_y)
bnds = [(1.0e-12, None), (1.0e-12, None)]
par = [0.1, 0.1]
l = minimize(Cost_func_2D, par, args, bounds=bnds, method='SLSQP')
lam2_x_1 = l.x[0]
lam2_y_1 = l.x[1]
fun2_1 = l.fun
q2x, q2y = 2, 2
Pex_temp = Penalty_p(q2x, l_s)
Pe_x = Permmx.T.dot(Pex_temp).dot(Permmx)
Pey_temp = Penalty_p(q2y, l_s)
Pe_y = Permmy.T.dot(Pey_temp).dot(Permmy)
args = (Data, B, Pe_x, Pe_y)
bnds = [(1.0e-12, None), (1.0e-12, None)]
par = [0.1, 0.1]
l = minimize(Cost_func_2D, par, args, bounds=bnds, method='SLSQP')
lam2_x_2 = l.x[0]
lam2_y_2 = l.x[1]
fun2_2 = l.fun
if fun2_1 <= fun2_2:
fun2 = fun2_1
lam_x2 = lam2_x_1
lam_y2 = lam2_y_1
optq_x2 = 2
optq_y2 = 1
else:
fun2 = fun2_2
lam_x2 = lam2_x_2
lam_y2 = lam2_y_2
optq_x2 = 2
optq_y2 = 2
if fun1 <= fun2:
fun_vec.append(fun1)
lam_x = lam_x1
lam_y = lam_y1
optq_x = optq_x1
optq_y = optq_y1
else:
fun_vec.append(fun2)
lam_x = lam_x2
lam_y = lam_y2
optq_x = optq_x2
optq_y = optq_y2
#8. Returning the mean Prediction
Bpred = np.zeros([Data_pred.shape[0], l_s])
for i in range(Data_pred.shape[0]):
for j in range(l_s):
temp = distance.euclidean(
Data_pred[i, :], np.array([sparse[j, 0], sparse[j, 1]]))
Bpred[i, j] = np.exp(-((temp**2) / epsilon))
Pex_temp = Penalty_p(optq_x, l_s)
Pe_x = Permmx.T.dot(Pex_temp).dot(Permmx)
Pey_temp = Penalty_p(optq_y, l_s)
Pe_y = Permmy.T.dot(Pey_temp).dot(Permmy)
inver = np.linalg.inv(B.T.dot(B) + n * lam_x * Pe_x + n * lam_y * Pe_y)
theta = inver.dot(B.T.dot(Data[:, 3].reshape(-1, 1)))
pred = Bpred.dot(theta)
#9. Standard Error bounds
nr = (Data[:, 3].reshape(-1, 1) - B.dot(theta)).reshape(-1, 1)
term = B.dot(inver.dot(B.T))
df_res = n - 2 * np.trace(term) + np.trace(term.dot(term.T))
sigmasq = (nr.T.dot(nr)) / (df_res)
sigmasq = sigmasq[0][0]
std = np.sqrt(np.diag(sigmasq * Bpred.dot(inver).dot(Bpred.T)))
stdev_t = sp.stats.t._ppf((1 + confidence) / 2., df_res) * std
results[s] = [pred, stdev_t]
qoptx.append(optq_x)
qopty.append(optq_y)
Sp[s] = sparse
print(s)
if l_s == n:
break
s = s + 1
return [results, Sp, qoptx, qopty, fun_vec]
def Inference_1D(Data_temp, Data_pred, confidence=0.95):
## Objective: To fit the hierarchical approach on univariate functions
## Input:
## 1. Data_temp: The dataset to be modeled
## 2. Data_pred: Prediction locations
## 3. confidence: % confidence level for the model
## Output:
## 1. results: The inference results for the prediction
## 2. Sparse: The sparse representation at each scale
## 3. qopt: Optimal penalty at each considered scale (either 1 or 2)
## 4. fun_vec: The cost of fitting at each scale
## Formatting the Data matrix
if Data_temp.shape[1] == 3:
Data = zeros([Data_temp.shape[0], 4])
Data[:, 0] = Data_temp[:, 0]
Data[:, 2] = Data_temp[:, 1]
Data[:, 3] = Data_temp[:, 2]
if Data_temp.shape[1] == 4:
Data = Data_temp
## Initializations
points = Data.shape[0]
Gaussian = np.empty([points, points])
## Computing T
dist = squareform(pdist(Data[:, 0:2], 'euclidean'))
max_d = np.amax(dist)
k1 = 2 * (max_d / 2)**2
T = k1
l_s = 0
s = 0
n = Data.shape[0]
tol = 1.0e9
fun_vec = []
results = {}
qopt = []
Sp = {}
while l_s <= n:
epsilon = T / (2**s)
Gaussian = np.exp(-((dist**2) / epsilon))
# Finding the rank of the Gaussian Kernel
l_s = LA.matrix_rank(Gaussian)
k = l_s + 8
# Getting Permutation ordering in terms of importance
points = Data.shape[0]
A = random.randn(k, points)
W = A.dot(Gaussian)
W = np.matrix(W)
Q, R, P = linalg.qr(W, mode='full', pivoting=True)
# Getting Perm: the permutation matrix
Perm = np.zeros([len(P), len(P)], dtype=int)
for w in range(0, len(P)):
Perm[P[w], w] = 1
# Getting sparse representation ordered by importance (Most important point at index 0)
sparse = np.zeros([l_s, 3])
for i in range(l_s):
sparse[i, 0] = Data[P[i], 0]
sparse[i, 1] = Data[P[i], 1]
sparse[i, 2] = Data[P[i], 3]
# Selecting Relevant columns of Kernel into B Matrix
Mat = Gaussian.dot(Perm)
B = Mat[:, 0:l_s]
# Getting the Permutation operator for the coordinate vector which
# changes ordering of theta from decreasing order of importance to
# increasing sequentially (Relation <=)
tt = sparse[:, 0]
tt_sort = np.argsort(tt)
Permm = np.zeros([len(tt_sort), len(tt_sort)], dtype=int)
for w in range(0, len(tt_sort)):
Permm[w, tt_sort[w]] = 1
# Computing the Penalty and Fitting the Regularization network
q1 = 1
Pe_temp = Penalty_p(q1, l_s)
Pe = Permm.T.dot(Pe_temp).dot(Permm)
args = (Data, B, Pe)
bnds = [(1.0e-12, None)]
par = [0.01]
l = minimize(Cost_func_1D, par, args, bounds=bnds, method='SLSQP')
lam1 = l.x[0]
fun1 = l.fun
q2 = 2
Pe_temp = Penalty_p(q2, l_s)
Pe = Permm.T.dot(Pe_temp).dot(Permm)
args = (Data, B, Pe)
bnds = [(1.0e-12, None)]
par = [0.01]
l = minimize(Cost_func_1D, par, args, bounds=bnds, method='SLSQP')
lam2 = l.x[0]
fun2 = l.fun
if fun1 <= fun2:
fun_vec.append(fun1)
lam = lam1
optq = q1
else:
fun_vec.append(fun2)
lam = lam2
optq = q2
#8. Returning the mean Prediction
Bpred = np.zeros([Data_pred.shape[0], l_s])
for i in range(Data_pred.shape[0]):
for j in range(l_s):
temp = distance.euclidean(
Data_pred[i, :], np.array([sparse[j, 0], sparse[j, 1]]))
Bpred[i, j] = np.exp(-((temp**2) / epsilon))
Pe_temp = Penalty_p(optq, l_s)
Pe = Permm.T.dot(Pe_temp).dot(Permm)
inver = np.linalg.inv(B.T.dot(B) + n * lam * Pe)
theta = inver.dot(B.T.dot(Data[:, 3].reshape(-1, 1)))
pred = Bpred.dot(theta)
#9. Error bounds
nr = (Data[:, 3].reshape(-1, 1) - B.dot(theta)).reshape(-1, 1)
term = B.dot(inver.dot(B.T))
df_res = n - 2 * np.trace(term) + np.trace(term.dot(term.T))
sigmasq = (nr.T.dot(nr)) / (df_res)
sigmasq = sigmasq[0][0]
std = np.sqrt(np.diag(sigmasq * Bpred.dot(inver).dot(Bpred.T)))
stdev_t = sp.stats.t._ppf((1 + confidence) / 2., df_res) * std
results[s] = [pred, stdev_t]
qopt.append(optq)
Sp[s] = sparse
print(s)
if l_s == n:
break
s = s + 1
return [results, Sp, qopt, fun_vec]
k = db['k_1d_from_2d']['center']
mask = np.isfinite(P_sim)
P_sim = P_sim[mask]
C_sim = C_sim[mask][:,mask]
k = k[mask]
n = len(k)
# Generate model for Covariance.
C_mod, f = get_C_model(C_sim, 10)
# Make a fake power spectrum.
P_fake = 1.3 * P_sim + 3.1 * np.mean(P_sim) * (k / np.mean(k))**1.4
e, v = linalg.eigh(C_sim)
noise = np.sqrt(e) * random.randn(n)
noise = np.dot(v, noise)
P_fake += noise
# Choose PS and covariance to process.
P = P_fake
C = C_sim
C_inv = linalg.inv(C)
# Process and fit.
fg_residuals = P - P_sim
def PL_model(pars):
A = pars[0]
beta = pars[1]
return A * (k/k[0])**beta
nx = ny = 20
nz = 6
model.generate_uniform_mesh(nx, ny, nz, xmin=0, xmax=L, ymin=0, ymax=L,
generate_pbcs=True)
model.set_parameters(src.physical_constants.IceParameters())
model.initialize_variables()
F = src.solvers.SteadySolver(model,config)
F.solve()
model.eps_reg = 1e-5
config['adjoint']['control_variable'] = [model.beta2]
config['adjoint']['bounds'] = [(0.0,5000.0)]
dolfin.File('results/beta2_obs.xml') << model.beta2
A = src.solvers.AdjointSolver(model,config)
u_o = model.u.vector().get_local()
v_o = model.v.vector().get_local()
U_e = 10.0
from scipy import random
for i in range(50):
config['output_path'] = 'results/run_'+str(i)+'/'
model.beta2.vector()[:] = 1000.
u_error = U_e*random.randn(len(u_o))
v_error = U_e*random.randn(len(v_o))
model.u_o.vector().set_local(u_o+u_error)
model.v_o.vector().set_local(v_o+v_error)
A.solve()
hankel1_ratio_2d(m.astype(float64), x, br)
return br
def hankel_ratio_scipy(m, x):
br = x*hankel1(m-1,x)/hankel1(m,x) - m
return br
def hankel_ratio_scipy_p(m, x):
br = x*hankel1p(m,x)/hankel1(m,x)
return br
t = timer.timer()
Nt = 100
mr = random.random_integers(-500,500,(100,2))
xr = random.randn(Nt)*300 + random.randn(Nt)*30j
#"Exact" defined by scipy
exact_y = [hankel_ratio_amos(mr, x) for x in xr]
t.start("asym")
y = zeros(mr.shape, complex_)
for ii in range(Nt):
z=xr[ii]
m=mr
d = (e*z/2/(m+1))**2*((m-1)/(m+1))**(m-0.5)
b1 = m*(1-2*d)
t.stop("asym")
nans_found = error = 0
t.start("scipy")
def f(x):
return exp(-x[0]**2 - x[1]**2) * sin(x[1]**2) + cos(x[0])
def neg_f(x):
return -f(x)
minXY = [-1.5, -2] # punkt startowy
local_X_Y = minXY
local_fmin = fmin(neg_f, local_X_Y, disp=False)
minVAL = -1000
for i in range(500):
lel = random.randn(2)
xd = fmin(neg_f, lel, disp=False)
if minVAL < f(xd):
minXY = xd
minVAL = f(minXY)
delta = 4.5
x_knots = linspace(minXY[0] - delta, minXY[0] + delta, 41)
y_knots = linspace(minXY[1] - delta, minXY[1] + delta, 41)
X, Y = meshgrid(x_knots, y_knots)
Z = zeros(X.shape)
for i in range(Z.shape[0]):
for j in range(Z.shape[1]):
Z[i][j] = f([X[i, j], Y[i, j]])
print(50 * "#")
mom = MomentumHybrid(model) mom.solve(annotate=False) u = Function(model.Q) v = Function(model.Q) assign(u, model.u.sub(0)) assign(v, model.u.sub(1)) u_o = u.vector().array() v_o = v.vector().array() n = len(u_o) #U_e = model.get_norm(as_vector([model.u, model.v]), 'linf') / 500 U_e = 0.18 print_min_max(U_e, 'U_e') u_error = U_e * random.randn(n) v_error = U_e * random.randn(n) u_ob = u_o + u_error v_ob = v_o + v_error model.init_U_ob(u_ob, v_ob) model.save_xdmf(model.u, 'U_true') model.save_xdmf(model.U_ob, 'U_ob') model.save_xdmf(model.beta, 'beta_true') model.init_beta(30.0**2) #model.init_beta_SIA() #model.save_xdmf(model.beta, 'beta_SIA') model.set_out_dir(out_dir + 'inversion/')
count_list = []
for x in list1:
if x not in unique_list:
unique_list.append(x)
count_list.append(1)
else:
count_list[unique_list.index(x)] += 1
return unique_list, count_list
resultList = []
currentTime = time.perf_counter()
while time.perf_counter() - currentTime < 5:
x0 = random.randn(2)
x_min = fmin(f, x0) #minimum
resultList.append([round(x_min[0], 2), round(x_min[1], 2)])
val_list, count_list = unique(resultList)
best_probability = 0
for i in val_list:
print("Point: {} | Value: {} | Probability: {}".format(
i, f(i), count_list[val_list.index(i)] / sum(count_list) * 100))
if best_probability <= count_list[val_list.index(i)] / sum(
count_list) * 100:
best_point = i
best_probability = count_list[val_list.index(i)] / sum(
count_list) * 100
def mcmc(a, b, phi, sst_dict, n, ld_blk, blk_size, n_iter, n_burnin, thin,
chrom, out_dir, beta_std):
print('... MCMC ...')
# derived stats
beta_mrg = sp.array(sst_dict['BETA'], ndmin=2).T
maf = sp.array(sst_dict['MAF'], ndmin=2).T
n_pst = (n_iter - n_burnin) / thin
p = len(sst_dict['SNP'])
n_blk = len(ld_blk)
# initialization
beta = sp.zeros((p, 1))
psi = sp.ones((p, 1))
sigma = 1.0
if phi == None:
phi = 1.0
phi_updt = True
else:
phi_updt = False
beta_est = sp.zeros((p, 1))
psi_est = sp.zeros((p, 1))
sigma_est = 0.0
phi_est = 0.0
# MCMC
for itr in range(1, n_iter + 1):
if itr % 100 == 0:
print('--- iter-' + str(itr) + ' ---')
mm = 0
quad = 0.0
for kk in range(n_blk):
if blk_size[kk] == 0:
continue
else:
idx_blk = range(mm, mm + blk_size[kk])
dinvt = ld_blk[kk] + sp.diag(1.0 / psi[idx_blk].T[0])
dinvt_chol = linalg.cholesky(dinvt)
beta_tmp = linalg.solve_triangular(
dinvt_chol, beta_mrg[idx_blk], trans='T') + sp.sqrt(
sigma / n) * random.randn(len(idx_blk), 1)
beta[idx_blk] = linalg.solve_triangular(dinvt_chol,
beta_tmp,
trans='N')
quad += sp.dot(sp.dot(beta[idx_blk].T, dinvt), beta[idx_blk])
mm += blk_size[kk]
err = max(n / 2.0 * (1.0 - 2.0 * sum(beta * beta_mrg) + quad),
n / 2.0 * sum(beta**2 / psi))
sigma = 1.0 / random.gamma((n + p) / 2.0, 1.0 / err)
delta = random.gamma(a + b, 1.0 / (psi + phi))
for jj in range(p):
psi[jj] = gigrnd.gigrnd(a - 0.5, 2.0 * delta[jj],
n * beta[jj]**2 / sigma)
psi[psi > 1] = 1.0
if phi_updt == True:
w = random.gamma(1.0, 1.0 / (phi + 1.0))
phi = random.gamma(p * b + 0.5, 1.0 / (sum(delta) + w))
# posterior
if (itr > n_burnin) and (itr % thin == 0):
beta_est += beta / n_pst
psi_est += psi / n_pst
sigma_est += sigma / n_pst
phi_est += phi / n_pst
# convert standardized beta to per-allele beta
if beta_std == False:
beta_est /= sp.sqrt(2.0 * maf * (1.0 - maf))
# write posterior effect sizes
if phi_updt == True:
eff_file = out_dir + '_pst_eff_a%d_b%.1f_phiauto_chr%d.txt' % (a, b,
chrom)
else:
eff_file = out_dir + '_pst_eff_a%d_b%.1f_phi%1.0e_chr%d.txt' % (
a, b, phi, chrom)
with open(eff_file, 'w') as ff:
for snp, bp, a1, a2, beta in zip(sst_dict['SNP'], sst_dict['BP'],
sst_dict['A1'], sst_dict['A2'],
beta_est):
ff.write('%d\t%s\t%d\t%s\t%s\t%.6e\n' %
(chrom, snp, bp, a1, a2, beta))
print('... Done ...')
from scipy import linspace, cos, exp, random, meshgrid, zeros
from scipy.optimize import fmin
from matplotlib.pyplot import plot, show, legend, figure, cm, contour, clabel
from mpl_toolkits.mplot3d import Axes3D
def f(x):
return exp(-x[0]**2 - x[1]**2)
def neg_f(x):
return -f(x)
x0 = random.randn(2) * 2 + 3
x_min = fmin(neg_f, x0)
delta = 3
x_knots = linspace(x_min[0] - delta, x_min[0] + delta, 41)
y_knots = linspace(x_min[1] - delta, x_min[1] + delta, 41)
X, Y = meshgrid(x_knots, y_knots)
Z = zeros(X.shape)
for i in range(Z.shape[0]):
for j in range(Z.shape[1]):
Z[i][j] = f([X[i, j], Y[i, j]])
ax = Axes3D(figure(figsize=(8, 5)))
ax.plot_surface(X, Y, Z, rstride=1, cstride=1, cmap=cm.coolwarm, linewidth=0.4)
ax.plot([x0[0]], [x0[1]], [f(x0)],
mom.solve(annotate=False) # add noise with a signal-to-noise ratio of 100 : snr = 100.0 u = Function(model.Q) v = Function(model.Q) assign(u, model.u.sub(0)) assign(v, model.u.sub(1)) u_o = u.vector().array() v_o = v.vector().array() n = len(u_o) sig = model.get_norm(as_vector([u, v]), 'linf') / snr print_min_max(snr, 'SNR') print_min_max(sig, 'sigma') u_error = sig * random.randn(n) v_error = sig * random.randn(n) u_ob = u_o + u_error v_ob = v_o + v_error # init the 'observed' velocity : model.init_U_ob(u_ob, v_ob) u_ob_ex = model.vert_extrude(model.u_ob, 'down') v_ob_ex = model.vert_extrude(model.v_ob, 'down') model.init_U_ob(u_ob_ex, v_ob_ex) # init the traction to the SIA approximation : model.init_beta_SIA() # solving the incomplete adjoint is more efficient : mom.linearize_viscosity()
'animate': False,
'bounds': None,
'control_variable': None,
'regularization_type': 'Tikhonov'
}
}
F = SteadySolver(model, config)
F.solve()
model.eps_reg = 1e-5
config['adjoint']['control_variable'] = [model.beta2]
config['adjoint']['bounds'] = [(0.0, 5000.0)]
File('results/beta2_obs.xml') << model.beta2
File('results/beta2_obs.pvd') << model.beta2
A = AdjointSolver(model, config)
u_o = model.u.vector().get_local()
v_o = model.v.vector().get_local()
U_e = 10.0
for i in range(50):
model.beta2.vector()[:] = 1000.
u_error = U_e * random.randn(len(u_o))
v_error = U_e * random.randn(len(v_o))
model.u_o.vector().set_local(u_o + u_error)
model.v_o.vector().set_local(v_o + v_error)
model.u_o.vector().apply('insert')
model.v_o.vector().apply('insert')
A.solve()
Count_list = []
for x in list:
if x not in Value_list:
Value_list.append(x)
Count_list.append(1)
else:
Count_list[Value_list.index(x)] += 1
return Value_list, Count_list
Xmin_list = []
currenttime = time.perf_counter()
while time.perf_counter() - currenttime < 30:
x_0 = random.randn(2) * 20 - 10
x_min = fmin(neg_f, x_0)
Xmin_list.append([round(x_min[0], 2), round(x_min[1], 2)])
Val_list, count_list = value_and_count(Xmin_list)
probab = 0
for i in Val_list:
print("point:", i, "value:", round(f(i), 2), "probability:",
round(count_list[Val_list.index(i)] / sum(count_list) * 100, 4))
if probab <= count_list[Val_list.index(i)] / sum(count_list) * 100:
best_point = i
probab = count_list[Val_list.index(i)] / sum(count_list) * 100
print("best point:", best_point, probab)
k = db["k_1d_from_2d"]["center"]
mask = np.isfinite(P_sim)
P_sim = P_sim[mask]
C_sim = C_sim[mask][:, mask]
k = k[mask]
n = len(k)
# Generate model for Covariance.
C_mod, f = get_C_model(C_sim, 10)
# Make a fake power spectrum.
P_fake = 1.3 * P_sim + 3.1 * np.mean(P_sim) * (k / np.mean(k)) ** 1.4
e, v = linalg.eigh(C_sim)
noise = np.sqrt(e) * random.randn(n)
noise = np.dot(v, noise)
P_fake += noise
# Choose PS and covariance to process.
P = P_fake
C = C_sim
C_inv = linalg.inv(C)
# Process and fit.
fg_residuals = P - P_sim
def PL_model(pars):
A = pars[0]
beta = pars[1]
def mcmc(a, b, phi, snp_dict, beta_mrg, frq_dict, idx_dict, n, ld_blk, blk_size, n_iter, n_burnin, thin, pop, chrom, out_dir, out_name, meta, seed):
print('... MCMC ...')
# seed
if seed != None:
random.seed(seed)
# derived stats
n_pst = (n_iter-n_burnin)/thin
n_pop = len(pop)
p_tot = len(snp_dict['SNP'])
p = {}
n_blk = {}
het = {}
for pp in range(n_pop):
p[pp] = len(beta_mrg[pp])
n_blk[pp] = len(ld_blk[pp])
het[pp] = sp.sqrt(2.0*frq_dict[pp]*(1.0-frq_dict[pp]))
n_grp = sp.zeros((p_tot,1))
for jj in range(p_tot):
for pp in range(n_pop):
if jj in idx_dict[pp]:
n_grp[jj] += 1
# initialization
beta = {}
sigma = {}
for pp in range(n_pop):
beta[pp] = sp.zeros((p[pp],1))
sigma[pp] = 1.0
psi = sp.ones((p_tot,1))
if phi == None:
phi = 1.0; phi_updt = True
else:
phi_updt = False
# space allocation
beta_est = {}
sigma_est = {}
for pp in range(n_pop):
beta_est[pp] = sp.zeros((p[pp],1))
sigma_est[pp] = 0.0
psi_est = sp.zeros((p_tot,1))
phi_est = 0.0
# MCMC
for itr in range(1,n_iter+1):
if itr % 100 == 0:
print('--- iter-' + str(itr) + ' ---')
for pp in range(n_pop):
mm = 0; quad = 0.0
psi_pp = psi[idx_dict[pp]]
for kk in range(n_blk[pp]):
if blk_size[pp][kk] == 0:
continue
else:
idx_blk = range(mm,mm+blk_size[pp][kk])
dinvt = ld_blk[pp][kk]+sp.diag(1.0/psi_pp[idx_blk].T[0])
dinvt_chol = linalg.cholesky(dinvt)
beta_tmp = linalg.solve_triangular(dinvt_chol, beta_mrg[pp][idx_blk], trans='T') \
+ sp.sqrt(sigma[pp]/n[pp])*random.randn(len(idx_blk),1)
beta[pp][idx_blk] = linalg.solve_triangular(dinvt_chol, beta_tmp, trans='N')
quad += sp.dot(sp.dot(beta[pp][idx_blk].T, dinvt), beta[pp][idx_blk])
mm += blk_size[pp][kk]
err = max(n[pp]/2.0*(1.0-2.0*sum(beta[pp]*beta_mrg[pp])+quad), n[pp]/2.0*sum(beta[pp]**2/psi_pp))
sigma[pp] = 1.0/random.gamma((n[pp]+p[pp])/2.0, 1.0/err)
delta = random.gamma(a+b, 1.0/(psi+phi))
xx = sp.zeros((p_tot,1))
for pp in range(n_pop):
xx[idx_dict[pp]] += n[pp]*beta[pp]**2/sigma[pp]
for jj in range(p_tot):
while True:
try:
psi[jj] = gigrnd.gigrnd(a-0.5*n_grp[jj], 2.0*delta[jj], xx[jj])
except:
continue
else:
break
psi[psi>1] = 1.0
if phi_updt == True:
w = random.gamma(1.0, 1.0/(phi+1.0))
phi = random.gamma(p_tot*b+0.5, 1.0/(sum(delta)+w))
# posterior
if (itr > n_burnin) and (itr % thin == 0):
for pp in range(n_pop):
beta_est[pp] = beta_est[pp] + beta[pp]/n_pst
sigma_est[pp] = sigma_est[pp] + sigma[pp]/n_pst
psi_est = psi_est + psi/n_pst
phi_est = phi_est + phi/n_pst
# convert standardized beta to per-allele beta
for pp in range(n_pop):
beta_est[pp] /= het[pp]
# meta
if meta == 'TRUE':
vv = sp.zeros((p_tot,1))
zz = sp.zeros((p_tot,1))
for pp in range(n_pop):
vv[idx_dict[pp]] += n[pp]/sigma_est[pp]/psi_est[idx_dict[pp]]*het[pp]**2
zz[idx_dict[pp]] += n[pp]/sigma_est[pp]/psi_est[idx_dict[pp]]*het[pp]**2*beta_est[pp]
mu = zz/vv
# write posterior effect sizes
for pp in range(n_pop):
if phi_updt == True:
eff_file = out_dir + '/' + '%s_%s_pst_eff_a%d_b%.1f_phiauto_chr%d.txt' % (out_name, pop[pp], a, b, chrom)
else:
eff_file = out_dir + '/' + '%s_%s_pst_eff_a%d_b%.1f_phi%1.0e_chr%d.txt' % (out_name, pop[pp], a, b, phi, chrom)
snp_pp = [snp_dict['SNP'][ii] for ii in idx_dict[pp]]
bp_pp = [snp_dict['BP'][ii] for ii in idx_dict[pp]]
a1_pp = [snp_dict['A1'][ii] for ii in idx_dict[pp]]
a2_pp = [snp_dict['A2'][ii] for ii in idx_dict[pp]]
with open(eff_file, 'w') as ff:
for snp, bp, a1, a2, beta in zip(snp_pp, bp_pp, a1_pp, a2_pp, beta_est[pp]):
ff.write('%d\t%s\t%d\t%s\t%s\t%.6e\n' % (chrom, snp, bp, a1, a2, beta))
if meta == 'TRUE':
if phi_updt == True:
eff_file = out_dir + '/' + '%s_META_pst_eff_a%d_b%.1f_phiauto_chr%d.txt' % (out_name, a, b, chrom)
else:
eff_file = out_dir + '/' + '%s_META_pst_eff_a%d_b%.1f_phi%1.0e_chr%d.txt' % (out_name, a, b, phi, chrom)
with open(eff_file, 'w') as ff:
for snp, bp, a1, a2, beta in zip(snp_dict['SNP'], snp_dict['BP'], snp_dict['A1'], snp_dict['A2'], mu):
ff.write('%d\t%s\t%d\t%s\t%s\t%.6e\n' % (chrom, snp, bp, a1, a2, beta))
# print estimated phi
if phi_updt == True:
print('... Estimated global shrinkage parameter: %1.2e ...' % phi_est )
print('... Done ...')
def hankel_ratio_scipy(m, x):
br = x * hankel1(m - 1, x) / hankel1(m, x) - m
return br
def hankel_ratio_scipy_p(m, x):
br = x * hankel1p(m, x) / hankel1(m, x)
return br
t = timer.timer()
Nt = 100
mr = random.random_integers(-500, 500, (100, 2))
xr = random.randn(Nt) * 300 + random.randn(Nt) * 30j
#"Exact" defined by scipy
exact_y = [hankel_ratio_amos(mr, x) for x in xr]
t.start("asym")
y = zeros(mr.shape, complex_)
for ii in range(Nt):
z = xr[ii]
m = mr
d = (e * z / 2 / (m + 1))**2 * ((m - 1) / (m + 1))**(m - 0.5)
b1 = m * (1 - 2 * d)
t.stop("asym")
nans_found = error = 0
t.start("scipy")
def assimilate(h, H, g):
"""
Run the full assimilation process for a cell size <h>, ice thickness <H>,
and surface slope <g>.
"""
out_dir = 'dump/stokes/full/h_%i/H_%i/g_%.2f/' % (h, H, g)
n = 25
L = n * h
alpha = g * pi / 180
p1 = Point(0.0, 0.0, 0.0)
p2 = Point(L, L, 1)
mesh = BoxMesh(p1, p2, n, n, 10)
model = D3Model(mesh, out_dir + 'initial/', use_periodic=True)
surface = Expression('- x[0] * tan(alpha)',
alpha=alpha,
element=model.Q.ufl_element())
bed = Expression('- x[0] * tan(alpha) - H',
alpha=alpha,
H=H,
element=model.Q.ufl_element())
beta = Expression('H - H * sin(2*pi*x[0]/L) * sin(2*pi*x[1]/L)',
H=H,
alpha=alpha,
L=L,
element=model.Q.ufl_element())
model.calculate_boundaries()
model.deform_mesh_to_geometry(surface, bed)
model.init_beta(beta)
model.init_b(model.A0(0)**(-1 / model.n(0)))
model.init_E(1.0)
nparams = {
'newton_solver': {
'linear_solver': 'mumps',
'relative_tolerance': 1e-8,
'relaxation_parameter': 1.0,
'maximum_iterations': 25,
'error_on_nonconvergence': False
}
}
m_params = {'solver': nparams}
mom = MomentumDukowiczStokes(model, m_params, isothermal=True)
mom.solve(annotate=False)
u, v, w = model.U3.split(True)
u_s = model.vert_extrude(u, d='down')
v_s = model.vert_extrude(v, d='down')
sigma = 100.0
U_mag = model.get_norm(as_vector([u_s, v_s]), 'linf')[1]
n = len(U_mag)
U_avg = sum(U_mag) / n
U_e = U_avg / sigma
print_min_max(U_e, 'U_error')
u_o = u.vector().array()
v_o = v.vector().array()
n = len(u_o)
u_error = U_e * random.randn(n)
v_error = U_e * random.randn(n)
u_ob = u_o + u_error
v_ob = v_o + v_error
model.assign_variable(u, u_ob)
model.assign_variable(v, v_ob)
print_min_max(u_error, 'u_error')
print_min_max(v_error, 'v_error')
model.init_U_ob(u, v)
#model.save_pvd(model.U3, 'U_true')
#model.save_pvd(model.U_ob, 'U_ob')
#model.save_pvd(model.beta, 'beta_true')
#model.set_out_dir(model.out_dir + 'xml/')
model.save_xml(interpolate(model.u, model.Q), 'u_true')
model.save_xml(interpolate(model.v, model.Q), 'v_true')
model.save_xml(interpolate(model.w, model.Q), 'w_true')
model.save_xml(model.U_ob, 'U_ob')
model.save_xml(model.beta, 'beta_true')
# save the true beta for MSE calculation :
beta_true = model.beta.copy(True)
m_params['solver']['newton_solver']['maximum_iterations'] = 3
mom = MomentumDukowiczStokes(model, m_params, linear=True, isothermal=True)
#model.init_beta_SIA()
#model.save_pvd(model.beta, 'beta_SIA')
#alphas = [1e-6, 1e-5, 1e-4, 1e-3, 1e-2, 1e-1, 1e0,
# 1e1, 1e2, 1e3, 1e4, 1e5, 1e6]
#alphas = [1e-2, 5e-2, 1e-3, 5e-3, 1e-2, 5e-2, 1e-1,
# 5e-1, 1e0, 5e0, 1e1, 5e1, 1e2]
alphas = [1e-2, 5e-2, 1e-1, 5e-1, 1e0, 5e0, 1e1]
Js = []
Rs = []
MSEs = []
REs = []
nits = []
for alpha in alphas:
model.init_beta(30.0**2)
mom.solve(annotate=True)
model.set_out_dir(out_dir=out_dir + 'assimilated/alpha_%.1E/' % alpha)
J = mom.form_obj_ftn(integral=model.dSrf,
kind='log_lin_hybrid',
g1=0.01,
g2=1000)
R = mom.form_reg_ftn(model.beta,
integral=model.dBed,
kind='Tikhonov',
alpha=alpha)
I = J + R
#controls = File(model.out_dir + "control_viz/beta_control.pvd")
#beta_viz = Function(model.Q, name="beta_control")
def eval_cb(I, beta):
# commented out because the model variables are not updated by
# dolfin-adjoint (yet) :
#mom.print_eval_ftns()
#print_min_max(mom.U, 'U')
print_min_max(I, 'I')
print_min_max(beta, 'beta')
def deriv_cb(I, dI, beta):
#print_min_max(I, 'I')
print_min_max(dI, 'dI/dbeta')
#print_min_max(beta, 'beta')
#beta_viz.assign(beta)
#controls << beta_viz
def hessian_cb(I, ddI, beta):
print_min_max(ddI, 'd/db dI/db')
m = FunctionControl('beta')
F = ReducedFunctional(Functional(I),
m,
eval_cb_post=eval_cb,
derivative_cb_post=deriv_cb,
hessian_cb=hessian_cb)
b_opt, res = minimize(F,
method="L-BFGS-B",
tol=1e-9,
bounds=(0, 4000),
options={
"disp": True,
"maxiter": 1000,
"gtol": 1e-5,
"factr": 1e7
})
#problem = MinimizationProblem(F, bounds=(0, 4000))
#parameters = {"tol" : 1e8,
# "acceptable_tol" : 1000.0,
# "maximum_iterations" : 1000,
# "linear_solver" : "ma57"}
#
#solver = IPOPTSolver(problem, parameters=parameters)
#b_opt = solver.solve()
print_min_max(b_opt, 'b_opt')
model.assign_variable(model.beta, b_opt)
mom.solve(annotate=False)
mom.calc_eval_ftns()
beta_true_v = beta_true.vector()
beta_opt_v = b_opt.vector()
mse = norm(beta_opt_v - beta_true_v)**2 / len(beta_true_v)
re = norm(beta_opt_v - beta_true_v) / norm(beta_true_v)
print_min_max(mse, 'MSE')
print_min_max(re, 'RE')
Rs.append(assemble(mom.Rp))
Js.append(assemble(mom.J))
MSEs.append(mse)
REs.append(re)
nits.append(res['nit'])
#model.save_pvd(model.beta, 'beta_opt')
#model.save_pvd(model.U3, 'U_opt')
#model.set_out_dir(model.out_dir + 'xml/')
model.save_xml(model.beta, 'beta_opt')
model.save_xml(interpolate(model.u, model.Q), 'u_opt')
model.save_xml(interpolate(model.v, model.Q), 'v_opt')
model.save_xml(interpolate(model.w, model.Q), 'w_opt')
# reset entire dolfin-adjoint state :
adj_reset()
from numpy import savetxt, array
import os
if model.MPI_rank == 0:
d = out_dir + 'plot/'
if not os.path.exists(d):
os.makedirs(d)
savetxt(d + 'Rs.txt', array(Rs))
savetxt(d + 'Js.txt', array(Js))
savetxt(d + 'as.txt', array(alphas))
savetxt(d + 'MSEs.txt', array(MSEs))
savetxt(d + 'REs.txt', array(REs))
savetxt(d + 'nits.txt', array(nits))
model.init_mask(0.0) # all grounded model.init_beta(beta) model.init_b(model.A0(0)**(-1 / model.n(0))) mom = MomentumHybrid(model, isothermal=True) mom.solve(annotate=True) u, v, w = model.U3.split(True) u_o = u.vector().array() v_o = v.vector().array() n = len(u_o) #U_e = model.get_norm(as_vector([model.u, model.v]), 'linf')[1] / 500 U_e = 0.18 print_min_max(U_e, 'U_e') u_error = U_e * random.randn(n) v_error = U_e * random.randn(n) u_ob = u_o + u_error v_ob = v_o + v_error model.init_U_ob(u_ob, v_ob) model.save_pvd(model.U3, 'U_true') model.save_pvd(model.U_ob, 'U_ob') model.save_pvd(model.beta, 'beta_true') model.init_beta(30.0**2) #model.init_beta_SIA() #model.save_pvd(model.beta, 'beta_SIA') model.set_out_dir(out_dir=out_dir + 'inverted/')
u_i = Expression('sqrt(pow(x[0],2) + pow(x[1],2))')
v_i = Expression('exp(-pow(x[0],2)/2 - pow(x[1],2)/2)')
z_i = Expression('10 + 10 * sin(2*pi*x[0]) * sin(2*pi*x[1])')
x_i = Expression('x[0]')
y_i = Expression('x[1]')
u = interpolate(u_i, V)
v = interpolate(v_i, V)
z = interpolate(z_i, V)
x = interpolate(x_i, V)
y = interpolate(y_i, V)
w = project(u * v * z, V)
# apply some noise
w_v = w.vector().array()
w_v += 0.05 * random.randn(len(w_v))
w.vector().set_local(w_v)
w.vector().apply('insert')
bmesh = BoundaryMesh(mesh, "exterior") # surface boundary mesh
cellmap = bmesh.entity_map(2)
pb = CellFunction("size_t", bmesh, 0)
for c in cells(bmesh):
if Facet(mesh, cellmap[c.index()]).normal().z() < 0:
pb[c] = 1
submesh = SubMesh(bmesh, pb, 1) # subset of surface mesh
Vs = FunctionSpace(submesh, "CG", 1) # submesh function space
us = Function(Vs) # desired function