def create(m, n):
mu = 1
rho = 1
sigma = 0.1
A = problem_util.normalized_data_matrix(m, n, mu)
x0 = sp.rand(n, 1, rho)
x0.data = np.random.randn(x0.nnz)
x0 = x0.toarray().ravel()
b = np.sign(A.dot(x0) + sigma*np.random.randn(m))
A[b>0,:] += 0.7*np.tile([x0], (np.sum(b>0),1))
A[b<0,:] -= 0.7*np.tile([x0], (np.sum(b<0),1))
P = la.block_diag(np.random.randn(n-1,n-1), 0)
lam = 1
x = cp.Variable(A.shape[1])
# Straightforward formulation w/ no constraints
# TODO(mwytock): Fix compiler so this works
z0 = 1 - sp.diags([b],[0])*A*x + cp.norm1(P.T*x)
f_eval = lambda: (lam*cp.sum_squares(x) + cp.sum_entries(cp.max_elemwise(z0, 0))).value
# Explicit epigraph constraint
t = cp.Variable(1)
z = 1 - sp.diags([b],[0])*A*x + t
f = lam*cp.sum_squares(x) + cp.sum_entries(cp.max_elemwise(z, 0))
C = [cp.norm1(P.T*x) <= t]
return cp.Problem(cp.Minimize(f), C), f_eval
def solve_sparse(self, y, w, x_mask=None):
assert y.ndim == 1 and w.ndim == 1 and y.shape == w.shape
assert w.shape[0] == self.n
x = cp.Variable(np.count_nonzero(x_mask))
inv_mask = np.logical_not(x_mask)
term1 = cp.square(cp.norm(x-y[x_mask]))
term2 = self.c * cp.norm1(cp.diag(w[x_mask]) * x)
objective = cp.Minimize(term1 + term2)
constraints = [cp.quad_form(x, self.B[np.ix_(x_mask, x_mask)]) <= 1]
problem = cp.Problem(objective, constraints)
result = problem.solve(solver=cp.SCS)
if problem.status != cp.OPTIMAL:
warnings.warn(problem.status)
if problem.status not in (cp.OPTIMAL, cp.OPTIMAL_INACCURATE,
cp.UNBOUNDED_INACCURATE):
raise ValueError(problem.status)
out = np.zeros(self.n)
x = np.asarray(x.value).flatten()
out[x_mask] = x
return out
def create(m, n):
# Generate data
X = np.hstack([np.random.randn(m,n), np.ones((m,1))])
theta0 = np.random.randn(n+1)
y = np.sign(X.dot(theta0) + 0.1*np.random.randn(m))
X[y>0,:] += np.tile([theta0], (np.sum(y>0),1))
X[y<0,:] -= np.tile([theta0], (np.sum(y<0),1))
# Generate uncertainty envelope
P = la.block_diag(np.random.randn(n,n), 0)
lam = 1e-8
theta = cp.Variable(n+1)
# TODO(mwytock): write this as:
# f = (lam/2*cp.sum_squares(theta) +
# problem_util.hinge(1 - y[:,np.newaxis]*X*theta+cp.norm1(P.T*theta)))
# already in prox form
t1 = cp.Variable(m)
t2 = cp.Variable(1)
z = cp.Variable(n+1)
f = lam/2*cp.sum_squares(theta) + problem_util.hinge(1-t1)
C = [t1 == y[:,np.newaxis]*X*theta - t2,
cp.norm1(z) <= t2,
P.T*theta == z]
return cp.Problem(cp.Minimize(f), C)
def create(**kwargs):
A, B = problem_util.create_regression(**kwargs)
lambda_max = np.abs(A.T.dot(B)).max()
lam = 0.5*lambda_max
X = cp.Variable(A.shape[1], B.shape[1] if len(B.shape) > 1 else 1)
f = cp.sum_squares(A*X - B) + lam*cp.norm1(X)
return cp.Problem(cp.Minimize(f))
def create(m, n):
np.random.seed(0)
A = np.random.randn(m,n)
x0 = sp.rand(n, 1, 0.1)
b = A*x0
x = cp.Variable(n)
return cp.Problem(cp.Minimize(cp.norm1(x)), [A*x == b])
def lasso_dense(n):
m = 2*n
A = np.random.randn(m, n)
A /= np.sqrt(np.sum(A**2, 0))
b = A*sp.rand(n, 1, 0.1) + 1e-2*np.random.randn(m,1)
lam = 0.2*np.max(np.abs(A.T.dot(b)))
x = cvx.Variable(n)
f = cvx.sum_squares(A*x - b) + lam*cvx.norm1(x)
return cvx.Problem(cvx.Minimize(f))
def create(**kwargs):
A, b = problem_util.create_classification(**kwargs)
m = kwargs["m"]
n = kwargs["n"]
sigma = 0.05
mu = kwargs.get("mu", 1)
lam = 0.5*sigma*np.sqrt(m*np.log(mu*n))
x = cp.Variable(A.shape[1])
f = ep.hinge_loss(x, A, b) + lam*cp.norm1(x)
return cp.Problem(cp.Minimize(f))
def create(**kwargs):
A, b = problem_util.create_classification(**kwargs)
ratio = float(np.sum(b==1)) / len(b)
lambda_max = np.abs((1-ratio)*A[b==1,:].sum(axis=0) +
ratio*A[b==-1,:].sum(axis=0)).max()
lam = 0.5*lambda_max
x = cp.Variable(A.shape[1])
f = ep.logistic_loss(x, A, b) + lam*cp.norm1(x)
return cp.Problem(cp.Minimize(f))
def create(m, n):
np.random.seed(0)
A = np.random.randn(m,n);
A = A*sp.diags([1 / np.sqrt(np.sum(A**2, 0))], [0])
b = A.dot(10*np.random.randn(n))
k = max(m/50, 1)
idx = np.random.randint(0, m, k)
b[idx] += 100*np.random.randn(k)
x = cp.Variable(n)
return cp.Problem(cp.Minimize(cp.norm1(A*x - b)))
def lasso_conv(n):
sigma = n/10
c = np.exp(-np.arange(-n/2., n/2.)**2./(2*sigma**2))/np.sqrt(2*sigma**2*np.pi)
c[c < 1e-4] = 0
x0 = np.array(sp.rand(n, 1, 0.1).todense()).ravel()
b = np.convolve(c, x0) + 1e-2*np.random.randn(2*n-1)
lam = 0.2*np.max(np.abs(np.convolve(b, c, "valid")))
print lam
x = cvx.Variable(n)
f = cvx.sum_squares(cvx.conv(c, x) - b) + lam*cvx.norm1(x)
return cvx.Problem(cvx.Minimize(f))
def lasso_sparse(n):
m = 2*n
A = sp.rand(m, n, 0.1)
A.data = np.random.randn(A.nnz)
N = A.copy()
N.data = N.data**2
A = A*sp.diags([1 / np.sqrt(np.ravel(N.sum(axis=0)))], [0])
b = A*sp.rand(n, 1, 0.1) + 1e-2*np.random.randn(m,1)
lam = 0.2*np.max(np.abs(A.T*b))
x = cvx.Variable(n)
f = cvx.sum_squares(A*x - b) + lam*cvx.norm1(x)
return cvx.Problem(cvx.Minimize(f))
def create(n, data):
np.random.seed(0)
X, y = load_data(data)
X = random_features(X, n)
Y = one_hot_encoding(y)
n = X.shape[1]
k = Y.shape[1]
lam = 0.1
# TODO(mwytock): Use softmax here
Theta = cp.Variable(n, k)
f = cp.sum_squares(X*Theta - Y) + lam*cp.norm1(Theta)
return cp.Problem(cp.Minimize(f))
def create(n):
np.random.seed(0)
k = max(int(np.sqrt(n)/2), 1)
x0 = np.ones((n,1))
idxs = np.random.randint(0, n, (k,2))
idxs.sort()
for a, b in idxs:
x0[a:b] += 10*(np.random.rand()-0.5)
b = x0 + np.random.randn(n, 1)
lam = np.sqrt(n)
x = cp.Variable(n)
f = 0.5*cp.sum_squares(x-b) + lam*cp.norm1(x[1:]-x[:-1])
return cp.Problem(cp.Minimize(f))
def solve(self, y, w):
assert y.ndim == 1 and w.ndim == 1 and y.shape == w.shape
assert w.shape[0] == self.n
x = cp.Variable(self.n)
term1 = x.T*y - self.c*cp.norm1(cp.diag(w) * x)
constraints = [cp.quad_form(x, self.B) <= 1]
problem = cp.Problem(cp.Maximize(term1), constraints)
result = problem.solve(solver=cp.SCS)
if problem.status != cp.OPTIMAL:
warnings.warn(problem.status)
if problem.status not in (cp.OPTIMAL, cp.OPTIMAL_INACCURATE):
raise ValueError(problem.status)
return np.asarray(x.value).flatten()
def create(n, r=10, density=0.1):
np.random.seed(0)
L1 = np.random.randn(n, r)
L2 = np.random.randn(r, n)
L0 = L1.dot(L2)
S0 = sp.rand(n, n, density)
S0.data = 10 * np.random.randn(len(S0.data))
M = L0 + S0
lam = 0.1
L = cp.Variable(n, n)
S = cp.Variable(n, n)
f = cp.norm(L, "nuc") + lam * cp.norm1(S)
C = [L + S == M]
return cp.Problem(cp.Minimize(f), C)
def create(m, n, lam):
np.random.seed(0)
m = int(n)
n = int(n)
lam = float(lam)
A = sp.rand(n,n, 0.01)
A = np.asarray(A.T.dot(A).todense() + 0.1*np.eye(n))
L = np.linalg.cholesky(np.linalg.inv(A))
X = np.random.randn(m,n).dot(L.T)
S = X.T.dot(X)/m
W = np.ones((n,n)) - np.eye(n)
Theta = cp.Variable(n,n)
return cp.Problem(cp.Minimize(
lam*cp.norm1(cp.mul_elemwise(W,Theta)) +
cp.sum_entries(cp.mul_elemwise(S,Theta)) -
cp.log_det(Theta)))
def test_norm1(self):
"""Test L1 norm prox fn.
"""
# No modifiers.
tmp = Variable(10)
fn = norm1(tmp)
rho = 1
v = np.arange(10) * 1.0 - 5.0
x = fn.prox(rho, v.copy())
self.assertItemsAlmostEqual(x, np.sign(v) * np.maximum(np.abs(v) - 1.0 / rho, 0))
rho = 2
x = fn.prox(rho, v.copy())
self.assertItemsAlmostEqual(x, np.sign(v) * np.maximum(np.abs(v) - 1.0 / rho, 0))
# With modifiers.
mod_fn = norm1(tmp, alpha=0.1, beta=5,
c=np.ones(10) * 1.0, b=np.ones(10) * -1.0, gamma=4)
rho = 2
v = np.arange(10) * 1.0
x = mod_fn.prox(rho, v.copy())
# vhat = mod_fn.beta*(v - mod_fn.c/rho)*rho/(rho+2*mod_fn.gamma) - mod_fn.b
# rho_hat = rho/(mod_fn.alpha*mod_fn.beta**2)
# xhat = fn.prox(rho_hat, vhat)
x_var = cvx.Variable(10)
cost = 0.1 * cvx.norm1(5 * x_var + np.ones(10)) + np.ones(10).T * x_var + \
4 * cvx.sum_squares(x_var) + (rho / 2) * cvx.sum_squares(x_var - v)
prob = cvx.Problem(cvx.Minimize(cost))
prob.solve()
self.assertItemsAlmostEqual(x, x_var.value, places=3)
# With weights.
tmp = Variable(10)
v = np.arange(10) * 1.0 - 5.0
fn = weighted_norm1(tmp, -v + 1)
rho = 2
x = fn.prox(rho, v)
self.assertItemsAlmostEqual(x, np.sign(v) *
np.maximum(np.abs(v) - np.abs(-v + 1) / rho, 0))
def solve_sparse(self, y, w, x_mask):
assert y.ndim == 1 and w.ndim == 1 and y.shape == w.shape
assert w.shape[0] == self.n
x = cp.Variable(np.count_nonzero(x_mask))
term1 = x.T*y[x_mask] - self.c*cp.norm1(cp.diag(w[x_mask]) * x)
constraints = [cp.quad_form(x, self.B[np.ix_(x_mask, x_mask)]) <= 1]
problem = cp.Problem(cp.Maximize(term1), constraints)
result = problem.solve(solver=cp.SCS)
if problem.status != cp.OPTIMAL:
warnings.warn(problem.status)
if problem.status not in (cp.OPTIMAL, cp.OPTIMAL_INACCURATE):
raise ValueError(problem.status)
out = np.zeros(self.n)
x = np.asarray(x.value).flatten()
out[x_mask] = x
return out
def glrd_sparse(V, G, F, r, err_V, err_F):
# sparsity threshold is num_nodes / num_roles
for k in xrange(r):
G_copy = np.copy(G) # create local copies for excluding the k^th col and row of G and F resp.
F_copy = np.copy(F)
G_copy[:, k] = 0.0
F_copy[k, :] = 0.0
R = V - np.dot(G_copy, F_copy) # compute residual
# Solve for optimal G(.)(k) with sparsity constraints
F_k = F[k, :]
x_star_G = linalg.lstsq(R.T, F_k.T)[0].T
x_G = cvx.Variable(x_star_G.shape[0])
objective_G = cvx.Minimize(cvx.norm2(x_star_G - x_G))
constraints_G = [x_G >= 0]
constraints_G += [cvx.norm1(x_G) <= err_V]
prob_G = cvx.Problem(objective_G, constraints_G)
result = prob_G.solve(solver='SCS')
if not np.isinf(result):
G_k_min = np.asarray(x_G.value)
G[:, k] = G_k_min[:, 0]
else:
print result
# Solve for optimal F(k)(.) with sparsity constraints
G_k = G[:, k]
x_star_F = linalg.lstsq(R, G_k)[0]
x_F = cvx.Variable(x_star_F.shape[0])
objective_F = cvx.Minimize(cvx.norm2(x_star_F - x_F))
constraints_F = [x_F >= 0]
constraints_F += [cvx.sum_entries(x_F) <= err_F]
prob_F = cvx.Problem(objective_F, constraints_F)
result = prob_F.solve(solver='SCS')
if not np.isinf(result):
F_k_min = np.asarray(x_F.value)
F[k, :] = F_k_min[0, :]
else:
print result
return G, F
A = sps.rand(n,n, 0.01)
A = np.asarray(A.T.dot(A).todense() + 0.1*np.eye(n))
L = np.linalg.cholesky(np.linalg.inv(A))
X = np.random.randn(m,n).dot(L.T)
S = X.T.dot(X)/m
W = np.ones((n,n)) - np.eye(n)
# Problem construction
Theta = cp.Variable(n,n)
prob = cp.Problem(cp.Minimize(
lam*cp.norm1(cp.mul_elemwise(W,Theta)) +
cp.sum_entries(cp.mul_elemwise(S,Theta)) -
cp.log_det(Theta)))
# Problem collection
# Single problem collection
problemDict = {
"problemID" : problemID,
"problem" : prob,
"opt_val" : opt_val
}
problems = [problemDict]
def fit(self, x, y, loss = 'l2', interactions = False, m = 1, constraints = None, \
verbose = False):
n_samples, n_features = x.shape
n_coeffs = n_features * self.deg + 1
designmatrix = self.build_designmatrix(x, interactions=interactions)
n_coeffs = designmatrix.shape[1]
column_norms_designmatrix = self.column_norms(designmatrix)
designmatrix = designmatrix / column_norms_designmatrix
coeffs = cv.Variable(n_coeffs)
#calculate residuals
residuals = designmatrix @ coeffs - y
#define loss function
loss_options = {
'l2': cv.sum_squares(residuals),
'l1': cv.norm1(residuals),
'huber': cv.sum(cv.huber(residuals, m))
}
data_term = loss_options[loss]
if self.regularization is not None:
regularization_options = {
'l2': cv.pnorm(coeffs, 2, axis=0)**2,
'l1': cv.norm1(coeffs)
}
regularization_term = regularization_options[self.regularization]
objective = cv.Minimize(data_term + self.lam * regularization_term)
else:
objective = cv.Minimize(data_term)
#build constraints
constraint_list = []
if constraints is not None:
#loop over all features
for feature_index in constraints:
Feature_constraints = constraints[feature_index]
xvals_feature = x[:, feature_index]
coefficient_index = feature_index * self.deg + 1
feature_coefficients = coeffs[
coefficient_index:coefficient_index + self.deg]
if Feature_constraints.sign == 'positive':
constraint_list.append(feature_coefficients >= 0)
elif Feature_constraints.sign == 'negative':
constraint_list.append(feature_coefficients <= 0)
monotonic = Feature_constraints.monotonicity is not None
strict_curvature = Feature_constraints.curvature is not None
if monotonic or strict_curvature or ybound:
if Feature_constraints.constraint_range is None:
constraint_min = np.amin(xvals_feature)
constraint_max = np.amax(xvals_feature)
Feature_constraints.constraint_range = [
constraint_min, constraint_max
]
constraints_grid = np.linspace(Feature_constraints.constraint_range[0], \
Feature_constraints.constraint_range[1], num=Feature_constraints.gridpoints)
if monotonic:
vander_grad = self.vander_grad(constraints_grid)[:, 1:]
norms = column_norms_designmatrix[
coefficient_index:coefficient_index + self.deg]
vander_grad = vander_grad / norms
if Feature_constraints.monotonicity == 'inc':
constraint_list.append(
vander_grad @ feature_coefficients >= 0)
elif Feature_constraints.monotonicity == 'dec':
constraint_list.append(
vander_grad @ feature_coefficients <= 0)
if strict_curvature:
vander_hesse = self.vander_hesse(constraints_grid)[:, 1:]
norms = column_norms_designmatrix[
coefficient_index:coefficient_index + self.deg]
vander_hesse = vander_hesse / norms
if Feature_constraints.curvature == 'convex':
constraint_list.append(
vander_hesse @ feature_coefficients >= 0)
elif Feature_constraints.curvature == 'concave':
constraint_list.append(
vander_hesse @ feature_coefficients <= 0)
#set up cvxpy problem
problem = cv.Problem(objective, constraints=constraint_list)
try:
if loss == 'l1' or self.regularization == 'l2':
#l1 loss solved by ECOS. Lower its tolerances for convergence
problem.solve(abstol=1e-8, reltol=1e-8, max_iters=1000000, \
feastol=1e-8, abstol_inacc = 1e-7, \
reltol_inacc=1e-7, verbose = verbose)
else:
#l2 and huber losses solved by OSQP. Lower its tolerances for convergence
problem.solve(eps_abs=1e-8, eps_rel=1e-8, max_iter=10000000, \
eps_prim_inf = 1e-9, eps_dual_inf = 1e-9, verbose = verbose, \
adaptive_rho = True)
#in case OSQP or ECOS fail, use SCS
except cv.SolverError:
try:
problem.solve(solver=cv.SCS,
max_iters=100000,
eps=1e-4,
verbose=verbose)
except cv.SolverError:
print("cvxpy optimization failed!")
#if optimal solution found, set parameters
if problem.status == 'optimal':
coefficients = coeffs.value / column_norms_designmatrix
self.coeffs_ = coefficients
#if not try SCS optimization
else:
try:
problem.solve(solver=cv.SCS,
max_iters=100000,
eps=1e-6,
verbose=verbose)
except cv.SolverError:
pass
if problem.status == 'optimal':
coefficients = coeffs.value / column_norms_designmatrix
self.coeffs_ = coefficients
else:
print("CVXPY optimization failed")
return self
def lasso_sure_cvxpy(X, y, alpha, sigma, random_state=42):
# lambda_alpha = [alpha, alpha]
n_samples, n_features = X.shape
epsilon = 2 * sigma / n_samples**0.3
rng = check_random_state(random_state)
delta = rng.randn(n_samples)
y2 = y + epsilon * delta
Xth, yth, y2th, deltath = map(torch.from_numpy, [X, y, y2, delta])
# set up variables and parameters
beta_cp = cp.Variable(n_features)
lambda_cp = cp.Parameter(nonneg=True)
# set up objective
loss = ((1 / (2 * n_samples)) * cp.sum(cp.square(Xth @ beta_cp - yth)))
reg = lambda_cp * cp.norm1(beta_cp)
objective = loss + reg
# define problem
problem1 = cp.Problem(cp.Minimize(objective))
assert problem1.is_dpp()
# solve problem1
layer = CvxpyLayer(problem1, [lambda_cp], [beta_cp])
alpha_th1 = torch.tensor(alpha, requires_grad=True)
beta1, = layer(alpha_th1)
# get test loss and it's gradient
test_loss1 = (Xth @ beta1 - yth).pow(2).sum()
test_loss1 -= 2 * sigma**2 / epsilon * (Xth @ beta1) @ deltath
test_loss1.backward()
val1 = test_loss1.detach().numpy()
grad1 = np.array(alpha_th1.grad)
# set up variables and parameters
beta_cp = cp.Variable(n_features)
lambda_cp = cp.Parameter(nonneg=True)
# set up objective
loss = ((1 / (2 * n_samples)) * cp.sum(cp.square(Xth @ beta_cp - y2th)))
reg = lambda_cp * cp.norm1(beta_cp)
objective = loss + reg
# define problem
problem2 = cp.Problem(cp.Minimize(objective))
assert problem2.is_dpp()
# solve problem2
layer = CvxpyLayer(problem2, [lambda_cp], [beta_cp])
alpha_th2 = torch.tensor(alpha, requires_grad=True)
beta2, = layer(alpha_th2)
# get test loss and it's gradient
test_loss2 = 2 * sigma**2 / epsilon * (Xth @ beta2) @ deltath
test_loss2.backward()
val2 = test_loss2.detach().numpy()
grad2 = np.array(alpha_th2.grad)
val = val1 + val2 - len(y) * sigma**2
grad = grad1 + grad2
return val, grad
lam = float(0.1)
import scipy.sparse as sps
A = sps.rand(n, n, 0.01)
A = np.asarray(A.T.dot(A).todense() + 0.1 * np.eye(n))
L = np.linalg.cholesky(np.linalg.inv(A))
X = np.random.randn(m, n).dot(L.T)
S = X.T.dot(X) / m
W = np.ones((n, n)) - np.eye(n)
# Problem construction
Theta = cp.Variable(n, n)
prob = cp.Problem(
cp.Minimize(lam * cp.norm1(cp.mul_elemwise(W, Theta)) +
cp.sum_entries(cp.mul_elemwise(S, Theta)) - cp.log_det(Theta)))
# Problem collection
# Single problem collection
problemDict = {"problemID": problemID, "problem": prob, "opt_val": opt_val}
problems = [problemDict]
# For debugging individual problems:
if __name__ == "__main__":
def printResults(problemID="", problem=None, opt_val=None):
print(problemID)
problem.solve()
print("\tstatus: {}".format(problem.status))
import cvxpy as cp import pandas as pd #%% u = ut.get_cvx_utility() #%% t = get_data() r = get_target_return(t, months=3) r = r.fillna(0) # A voir si c'est necessaire? #%% n, p = r.shape w_cp = cp.Variable(p) λ = 1 reg = λ * cp.norm1(w_cp) objective = cp.Maximize(1 / n * cp.sum(u(r.values @ w_cp)) - reg) constraints = [w_cp >= 0, cp.sum(w_cp) == 1] prob = cp.Problem(objective, constraints) result = prob.solve(solver="ECOS", verbose=True) # %% w = pd.Series(w_cp.value, index=t.columns) w = w[w >= 0.01] # On ne garde que les titres avec un poids >= 1% w.sort_values(ascending=False).plot.bar() # %% w.index = w.index.map(fund_name) w.sort_values(ascending=False)
def total_variation_plus_seasonal_quantile_filter(signal,
use_ixs=None,
tau=0.995,
c1=1e3,
c2=1e2,
c3=1e2,
solver='ECOS',
residual_weights=None,
tv_weights=None):
'''
This performs total variation filtering with the addition of a seasonal baseline fit. This introduces a new
signal to the model that is smooth and periodic on a yearly time frame. This does a better job of describing real,
multi-year solar PV power data sets, and therefore does an improved job of estimating the discretely changing
signal.
:param signal: A 1d numpy array (must support boolean indexing) containing the signal of interest
:param c1: The regularization parameter to control the total variation in the final output signal
:param c2: The regularization parameter to control the smoothness of the seasonal signal
:return: A 1d numpy array containing the filtered signal
'''
n = len(signal)
if residual_weights is None:
residual_weights = np.ones_like(signal)
if tv_weights is None:
tv_weights = np.ones(len(signal) - 1)
if use_ixs is None:
use_ixs = np.ones(n, dtype=np.bool)
# selected_days = np.arange(n)[index_set]
# np.random.shuffle(selected_days)
# ix = 2 * n // 3
# train = selected_days[:ix]
# validate = selected_days[ix:]
# train.sort()
# validate.sort()
s_hat = cvx.Variable(n)
s_seas = cvx.Variable(max(n, 366))
s_error = cvx.Variable(n)
s_linear = cvx.Variable(n)
c1 = cvx.Parameter(value=c1, nonneg=True)
c2 = cvx.Parameter(value=c2, nonneg=True)
c3 = cvx.Parameter(value=c3, nonneg=True)
tau = cvx.Parameter(value=tau)
# w = len(signal) / np.sum(index_set)
beta = cvx.Variable()
objective = cvx.Minimize(
# (365 * 3 / len(signal)) * w * cvx.sum(0.5 * cvx.abs(s_error) + (tau - 0.5) * s_error)
2 * cvx.sum(0.5 * cvx.abs(cvx.multiply(residual_weights, s_error)) +
(tau - 0.5) * cvx.multiply(residual_weights, s_error)) +
c1 * cvx.norm1(cvx.multiply(tv_weights, cvx.diff(s_hat, k=1))) +
c2 * cvx.norm(cvx.diff(s_seas, k=2)) + c3 * beta**2)
constraints = [
signal[use_ixs] == s_hat[use_ixs] + s_seas[:n][use_ixs] +
s_error[use_ixs],
cvx.sum(s_seas[:365]) == 0
]
if True:
constraints.append(s_seas[365:] - s_seas[:-365] == beta)
constraints.extend([beta <= 0.01, beta >= -0.1])
problem = cvx.Problem(objective=objective, constraints=constraints)
problem.solve(solver='MOSEK')
return s_hat.value, s_seas.value[:n]
def solve(self, data):
is_labeled_train = data.is_train & data.is_labeled
x = data.x[is_labeled_train, :]
#self.transform.with_mean = False
#self.transform.with_std = False
x = self.transform.fit_transform(x)
y = data.y[is_labeled_train]
n = x.shape[0]
p = x.shape[1]
C = self.C
C2 = self.C2
C3 = self.C3
#C = .001
#C2 = 0
use_nonneg_ridge = self.method == MixedFeatureGuidanceMethod.METHOD_RIDGE and self.use_nonneg
if self.method == MixedFeatureGuidanceMethod.METHOD_ORACLE:
assert False, 'Update this'
#Refit with standardized data to clear transform
#Is there a better way of doing this?
self.transform.fit_transform(x)
self.w = data.metadata['true_w']
self.b = 0
return
elif self.method in {MixedFeatureGuidanceMethod.METHOD_RELATIVE, MixedFeatureGuidanceMethod.METHOD_HARD_CONSTRAINT}\
or use_nonneg_ridge:
opt_data = optimize_data(x, y, C, C2, C3)
'''
opt_data.pairs = [
(0, 9),
(1, 8),
(2, 7),
(3, 6)
]
'''
opt_data.pairs = list()
constraints = list()
pairs = self.pairs
feats_to_constraint = self.feats_to_constrain
true_w = data.metadata['true_w']
if self.method == MixedFeatureGuidanceMethod.METHOD_HARD_CONSTRAINT:
assert not self.use_corr
constraints = list()
for j, k in pairs:
constraints.append({
'fun': lambda w, j=j, k=k: w[j] - w[k],
'type': 'ineq'
})
#for i in range(num_signs):
# j = np.random.choice(p)
for j in feats_to_constraint:
fun = lambda w, j=j: w[j]*np.sign(true_w[j])
constraints.append({
'fun': fun,
'type': 'ineq',
'idx': j
})
else:
opt_data.pairs = pairs
if self.method == MixedFeatureGuidanceMethod.METHOD_RELATIVE or use_nonneg_ridge:
assert len(feats_to_constraint) == 0 or len(pairs) == 0
w = cvx.Variable(p)
b = cvx.Variable(1)
z = cvx.Variable(len(feats_to_constraint) + len(pairs))
loss = cvx.sum_entries(
cvx.power(
x*w + b - y,
2
)
)
loss /= n
constraints = list()
idx = 0
corr = data.metadata['corr']
if self.method != MixedFeatureGuidanceMethod.METHOD_RIDGE:
for j, k in pairs:
if self.use_corr:
if corr[j] > corr[k]:
constraints.append(w[j] - w[k] + z[idx] >= 0)
else:
constraints.append(w[k] - w[j] + z[idx] >= 0)
else:
constraints.append(w[j] - w[k] + z[idx] >= 0)
idx += 1
for j in feats_to_constraint:
assert not self.use_nonneg
if self.use_corr:
constraints.append(w[j] * np.sign(corr[j]) + z[idx] >= 0)
else:
constraints.append(w[j]*np.sign(true_w[j]) + z[idx] >= 0)
idx += 1
reg = cvx.norm2(w) ** 2
if self.use_l1:
reg_guidance = cvx.norm1(z)
else:
reg_guidance = cvx.norm2(z) ** 2
if np.isinf(C2):
constraints = []
C2 = 0
constraints.append(z >= 0)
if self.use_nonneg:
constraints.append(w >= 0)
obj = cvx.Minimize(loss + C*reg + C2*reg_guidance)
prob = cvx.Problem(obj, constraints)
try:
#prob.solve(solver='SCS')
prob.solve(solver=self.cvx_method)
assert w.value is not None
self.w = np.squeeze(np.asarray(w.value))
except:
self.w = np.zeros(p)
'''
if not self.running_cv:
assert False, 'Failed to converge when done with CV!'
'''
'''
if b.value is not None:
assert abs(b.value - y.mean())/abs(b.value) <= 1e-3
'''
else:
assert not self.use_corr
eval_func = lambda a: MixedFeatureGuidanceMethod.eval(opt_data, a)
w0 = np.zeros(p)
options = dict()
options['maxiter'] = 1000
options['disp'] = False
bounds = [(None, None)] * p
'''
w1 = optimize.minimize(
eval_func,
a0,
method=self.configs.scipy_opt_method,
jac=None,
options=options,
bounds = bounds,
constraints=constraints
).x
'''
if self.method == MixedFeatureGuidanceMethod.METHOD_ORACLE_SPARSITY:
assert False
#with Capturing() as output:
results = optimize.minimize(
eval_func,
w0,
method=self.configs.scipy_opt_method,
jac=None,
options=options,
bounds = bounds,
constraints=constraints
)
w2 = results.x
self.w = np.asarray(results.x)
else:
#assert not self.use_corr
self.w = self.solve_w(x, y, C)
self.b = y.mean()
if not self.running_cv:
try:
print prob.status
except:
pass
if not self.running_cv and self.method != MixedFeatureGuidanceMethod.METHOD_RIDGE:
w2 = self.solve_w(x,y,C)
true_w = data.metadata['true_w']
err1 = array_functions.normalized_error(self.w, true_w)
err2 = array_functions.normalized_error(w2, true_w)
print str(err1 - err2)
c2 = deepcopy(self.configs)
c2.method = MixedFeatureGuidanceMethod.METHOD_RIDGE
t2 = MixedFeatureGuidanceMethod(c2)
t2.quiet = True
t2.train_and_test(data)
w = self.w
w2 = t2.w
pass
#print self.w
self.true_w = data.metadata['true_w']
pass
def compile_lrtop(data):
"""
Create the local rendezvous trajectory optimization problem (LRTOP).
Parameters
----------
data : dict
Problem data.
"""
csm = data['csm']
N = data['N']
p_traj = data['state_traj_init'][:, :3]
v_traj = data['state_traj_init'][:, 3:6]
w_traj = data['state_traj_init'][:, 10:13]
state_init = np.concatenate(
[data['x0']['p'], data['x0']['v'], data['x0']['q'], data['x0']['w']])
state_final = np.concatenate(
[data['xf']['p'], data['xf']['v'], data['xf']['q'], data['xf']['w']])
# Compute scaling terms
# state (=P_x*state_scaled+p_x)
p_center = np.mean(p_traj, axis=0)
v_center = np.mean(v_traj, axis=0)
q_center = np.zeros(4)
w_center = np.mean(w_traj, axis=0)
p_box = tools.bounding_box(p_traj - p_center)
v_box = tools.bounding_box(v_traj - v_center)
q_box = np.ones(4)
w_box = tools.bounding_box(w_traj - w_center)
p_x = np.concatenate([p_center, v_center, q_center, w_center])
P_x = np.diag(np.concatenate([p_box, v_box, q_box, w_box]))
P_x_inv = la.inv(P_x)
data['scale_x'] = lambda x: P_x_inv.dot(x - p_x)
# virtual control (=P_v*vc_scaled)
p_box = tools.bounding_box(p_traj)
v_box = tools.bounding_box(v_traj)
q_box = np.ones(4)
w_box = tools.bounding_box(w_traj)
P_v = np.diag(np.concatenate([p_box, v_box, q_box, w_box]))
# input (=P_u*input_scaled+p_u)
p_u = np.array([0.5 * data['t_pulse_max'] for i in range(csm.M)])
P_u = np.diag(p_u)
# fuel (impulse) cost (=P_xi*xi_scaled+p_xi)
# Heuristic: compute as having a quarter of the thrusters active at minimum
# pulse width, the whole time
mean_active_thruster_count = 0.25 * csm.M
mean_pulse_width = csm.t_pulse_min
p_xi = 0.
P_xi = mean_active_thruster_count * mean_pulse_width * N
# General quantites
data['t_grid'] = np.linspace(0., data['t_f'],
N + 1) # discretization time grid
n_x = data['state_traj_init'].shape[1]
n_u = csm.M
e = -tools.rotate(np.array([1., 0., 0.]),
csm.q_dock) # dock axis in LM frame
I_e = tools.rotate(e, data['lm']['q']) # dock axis in inertial frame
data['dock_axis'] = I_e
# Optimization variables
# non-dimensionalized
x_hat = [cvx.Variable(n_x) for k in range(N + 1)]
v_hat = [cvx.Variable(n_x) for k in range(N + 1)]
xi_hat = cvx.Variable(N + 1)
u_hat = [cvx.Variable(n_u) for k in range(N)]
eta_hat = cvx.Variable(N) # quadratic trust region size
# dimensionalized (physical units)
x = [P_x * x_hat[k] + p_x for k in range(N + 1)] # unscaled state
xi = P_xi * xi_hat + p_xi
u = [P_u * u_hat[k] + p_u for k in range(N)] # unscaled control
v = [P_v * v_hat[k] for k in range(N)] # virtual control
data['lrtop_var'] = dict(x=x, u=u, xi=xi, v=v)
# Optimization parameters
A = [cvx.Parameter((n_x, n_x)) for k in range(N)]
B = [cvx.Parameter((n_x, n_u)) for k in range(N)]
r = [cvx.Parameter(n_x) for k in range(N)]
u_lb = [cvx.Parameter(n_u, value=np.zeros(n_u)) for k in range(N)]
stc_lb = [cvx.Parameter(n_u) for k in range(N)]
stc_q = cvx.Parameter(N + 1)
x_prev_hat = [cvx.Parameter(n_x) for k in range(N + 1)]
data['lrtop_par'] = dict(A=A,
B=B,
r=r,
u_lb=u_lb,
stc_lb=stc_lb,
stc_q=stc_q,
x_prev_hat=x_prev_hat)
# Cost
# minimum-impulse
J = xi_hat[-1]
# trust region penalty
J_tr = data['w_tr'] * sum(eta_hat)
# virtual control penalty
J_vc = data['w_vc'] * sum([cvx.norm1(v_hat[k]) for k in range(N)])
data['lrtop_cost'] = dict(J=J, J_tr=J_tr, J_vc=J_vc)
cost = cvx.Minimize(J + J_tr + J_vc)
# constraints
constraints = []
constraints += [
x[k + 1] == A[k] * x[k] + B[k] * u[k] + r[k] + v[k] for k in range(N)
]
constraints += [xi[k + 1] == xi[k] + sum(u[k]) for k in range(N)]
constraints += [x[0] == state_init, x[-1] == state_final, xi[0] == 0]
constraints += [(x[k][:3] - data['lm']['p']).T * I_e >=
cvx.norm(x[k][:3] - data['lm']['p']) *
np.cos(np.deg2rad(data['gamma'])) for k in range(N + 1)]
constraints += [u[k] <= data['t_pulse_max'] for k in range(N)]
constraints += [u[k] >= u_lb[k] for k in range(N)]
constraints += [
u[k][i] == 0. for k in range(N) for i in range(n_u)
if i in data['thrusters_off']
]
constraints += [
cvx.quad_form(x_hat[k + 1] - x_prev_hat[k + 1], np.eye(n_x)) <=
eta_hat[k] for k in range(N)
]
constraints += [
stc_lb[k][i] * u[k][i] == 0 for k in range(N) for i in range(n_u)
]
constraints += [
stc_q[k] * u[k][i] == 0 for k in range(N) for i in range(n_u)
if 'p_f' in csm.i2thruster[i]
]
constraints += [
stc_q[k + 1] *
(tools.rqpmat(data['xf']['q'])[0].dot(np.diag([1, -1, -1, -1])) *
x[k][6:10] - np.cos(0.5 * np.deg2rad(data['ang_app']))) <= 0
for k in range(N)
]
data['lrtop'] = cvx.Problem(cost, constraints)
def modalityRegularizer(beta):
return cp.norm1(beta)
import matplotlib.pyplot as plt
# generate a problem instance
n = 50
d = 1000
np.random.seed(42)
A = np.random.randn(n, d)
b = np.random.randn(n)
R = 5
L = np.sqrt(d) # Lipschtz
DELTA = 1e-12
x1 = np.random.rand(d) * 2 - 1 # starting point
# compute optimal value by solving a problem using cvxpy
x = cp.Variable(d)
prob = cp.Problem(objective=cp.Minimize(cp.norm1(x)),
constraints=[cp.norm(x - x1) <= R])
prob.solve()
f_opt = prob.value
x_opt = x.value
# run projected subgradient method with decreasing step size
x = x1.copy()
MAX_ITERS = 3000
xs = [x]
for k in range(1, MAX_ITERS):
g = np.zeros(x.shape[0])
g[np.where(x > +DELTA)] = +1
g[np.where(x < -DELTA)] = -1
x = x - (R / (L * np.sqrt(k))) * g
if np.linalg.norm(x - x1) > R:
def loss_fn_l1(X, Y, beta):
return cp.norm1(cp.matmul(X, beta) - Y)
def regularizer(beta):
return cp.norm1(beta)
def conv_sensing(measurement_ratios,
filter_size,
solver="AMP",
sparse_in_dct=False,
width=50,
depth=20,
delta=1e-2,
sparsity=0.5,
n_rep=10):
N = width * depth
signal = GaussBernouilliPrior(size=(N, ), rho=sparsity)
def sample_trnsf():
if (sparse_in_dct):
D = scipy.linalg.block_diag(
*[dctmtx(width).T for _ in range(depth)])
else:
D = np.eye(N)
return D
recovery_per_alpha = []
for alpha in measurement_ratios:
recoveries = []
for rep in range(n_rep):
out_channels = int(np.rint(alpha * depth))
ensemble = ChannelConvEnsemble(width=width,
in_channels=depth,
out_channels=out_channels,
k=filter_size)
A = ensemble.generate()
C = sample_trnsf()
teacher = conv_model(
A, C, signal) @ GaussianChannel(var=delta) @ O(id="y")
teacher = teacher.to_model()
sample = teacher.sample()
if (solver == "AMP"):
max_iter = 20
damping = 0.1
student = conv_model(A, C, signal) @ GaussianLikelihood(
y=sample['y'], var=delta)
student = student.to_model_dag()
student = student.to_model()
ep = ExpectationPropagation(student)
ep.iterate(max_iter=max_iter,
damping=damping,
callback=EarlyStopping(tol=1e-8))
data_ep = ep.get_variables_data((['x']))
mse = np.mean((data_ep['x']['r'] - sample['x'])**2)
recoveries.append(mse)
elif (solver == "CVX"):
reg_param = 0.001
x = cp.Variable(shape=(N, ), name="x")
lmbda = cp.Parameter(nonneg=True)
objective = cp.norm2(A @ C @ x -
sample['y'])**2 + lmbda * cp.norm1(x)
problem = cp.Problem(cp.Minimize(objective))
lmbda.value = reg_param
problem.solve(abstol=1e-6)
mse = np.mean((x.value - sample['x'])**2)
recoveries.append(mse)
else:
raise ValueError("Solver must be 'AMP' or 'CVX'")
recovery_per_alpha.append(recoveries)
return recovery_per_alpha
def cvx_inequality_time_graphical_lasso(S, K_init, max_iter, loss, C, theta,
psi, gamma, tol):
"""Inequality constrained time-varying graphical LASSO solver.
Solves the following problem via ADMM:
min sum_{i=1}^T ||K_i||_{od,1} + beta sum_{i=2}^T Psi(K_i - K_{i-1})
s.t. objective =< c_i for i = 1, ..., T
where S_i = (1/n_i) X_i^T X_i is the empirical covariance of data
matrix X (training observations by features).
Parameters
----------
emp_cov : ndarray, shape (n_features, n_features)
Empirical covariance of data.
alpha, beta : float, optional
Regularisation parameter.
rho : float, optional
Augmented Lagrangian parameter.
max_iter : int, optional
Maximum number of iterations.
n_samples : ndarray
Number of samples available for each time point.
gamma: float, optional
Kernel parameter when psi is chosen to be 'kernel'.
tol : float, optional
Absolute tolerance for convergence.
rtol : float, optional
Relative tolerance for convergence.
return_history : bool, optional
Return the history of computed values.
return_n_iter : bool, optional
Return the number of iteration before convergence.
verbose : bool, default False
Print info at each iteration.
update_rho_options : dict, optional
Arguments for the rho update.
See regain.update_rules.update_rho function for more information.
compute_objective : bool, default True
Choose to compute the objective value.
init : {'empirical', 'zero', ndarray}
Choose how to initialize the precision matrix, with the inverse
empirical covariance, zero matrix or precomputed.
Returns
-------
K : numpy.array, 3-dimensional (T x d x d)
Solution to the problem for each time t=1...T .
history : list
If return_history, then also a structure that contains the
objective value, the primal and dual residual norms, and tolerances
for the primal and dual residual norms at each iteration.
"""
if loss == 'LL':
loss_function = neg_logl
else:
loss_function = dtrace
T, p, _ = S.shape
K = [cp.Variable(shape=(p, p), PSD=True) for t in range(T)]
# Z_1 = [cp.Variable(shape=(p, p), PSD=True) for t in range(T-1)]
# Z_2 = [cp.Variable(shape=(p, p), PSD=True) for t in range(T-1)]
if psi == 'laplacian':
objective = cp.Minimize(
theta * cp.sum(
[cp.norm(K[t] - cp.diag(cp.diag(K[t])), 1)
for t in range(T)]) + (1 - theta) *
cp.sum([cp.norm(K[t] - K[t - 1], 'fro') for t in range(1, T)]))
elif psi == 'l1':
objective = cp.Minimize(theta * cp.sum([
cp.norm(K[t] - cp.diag(cp.diag(K[t])), 1) for t in range(T)
]) + (1 - theta) * cp.sum(
[cp.sum(cp.norm1(K[t] - K[t - 1], axis=1)) for t in range(1, T)]))
elif psi == 'l2':
objective = cp.Minimize(theta * cp.sum(
[cp.norm(K[t] - cp.diag(cp.diag(K[t])), 1)
for t in range(T)]) + (1 - theta) * cp.sum([
cp.sum(cp.norm(K[t] - K[t - 1], p=2, axis=1))
for t in range(1, T)
]))
elif psi == 'linf':
objective = cp.Minimize(theta * cp.sum(
[cp.norm(K[t] - cp.diag(cp.diag(K[t])), 1)
for t in range(T)]) + (1 - theta) * cp.sum([
cp.sum(cp.norm_inf(K[t] - K[t - 1], axis=1))
for t in range(1, T)
]))
# if loss_function == neg_logl:
constraints = [(cp.sum(cp.multiply(K[t], S[t])) - cp.log_det(K[t]) <= C[t])
for t in range(T)]
# [(cp.trace(K[t] @ S[t]) - cp.log_det(K[t]) <= C[t]) for t in range(T)] # + \
# [(Z_1[t] == K[t]) for t in range(T-1)] + \
# [(Z_2[t] == K[t+1]) for t in range(T-1)]
# else:
# constraints = [(cp.trace(K[t] @ K[t] @ S[t]) - cp.trace(K[t]) <= C[t]) for t in range(T)] # + \
# # [(Z_1[t] == K[t]) for t in range(T-1)] + \
# # [(Z_2[t] == K[t+1]) for t in range(T-1)]
prob = cp.Problem(objective, constraints)
# prob.solve(solver=cp.SCS, max_iters=np.int(max_iter), eps=tol, verbose=True)
prob.solve(solver=cp.MOSEK, verbose=True)
print(prob.status)
print(prob.value)
K = np.array([k.value for k in K])
covariance_ = np.array([linalg.pinvh(k) for k in K])
return_list = [K, covariance_]
return return_list
def total_variation(C, p):
return _solve_convex(C, p, lambda p, q: 0.5 * cvxpy.norm1(p - q))
#!/usr/bin/env python
import numpy as np
import cvxpy as cp
from epopt import cvxpy_expr
from epopt import expression_vis
from epopt.compiler import canonicalize
if __name__ == "__main__":
n = 5
x = cp.Variable(n)
# Lasso expression tree
m = 10
A = np.random.randn(m,n)
b = np.random.randn(m)
lam = 1
f = cp.sum_squares(A*x - b) + lam*cp.norm1(x)
prob0 = cvxpy_expr.convert_problem(cp.Problem(cp.Minimize(f)))[0]
expression_vis.graph(prob0.objective).write("expr_lasso.dot")
# Canonicalization of a more complicated example
c = np.random.randn(n)
f = cp.exp(cp.norm(x) + c.T*x) + cp.norm1(x)
prob0 = cvxpy_expr.convert_problem(cp.Problem(cp.Minimize(f)))[0]
expression_vis.graph(prob0.objective).write("expr_epigraph.dot")
prob1 = canonicalize.transform(prob0)
expression_vis.graph(prob1.objective).write("expr_epigraph_canon.dot")
def reg(self, X): return self.nu*cp.norm1(X) def __str__(self): return "linear reg"
def recovery(measurement_ratios,
filter_size,
solver="AMP",
prior="conv",
sparse_in_dct=False,
N=1000,
delta=1e-2,
sparsity=0.5,
n_rep=10):
signal = GaussBernouilliPrior(size=(N, ), rho=sparsity)
prior_conv_ens = ConvEnsemble(N, filter_size)
def sample_trnsf():
if (sparse_in_dct):
D = dctmtx(N).T
else:
D = np.eye(N)
if (prior == "conv"):
C = prior_conv_ens.generate()
elif (prior == "sparse"):
C = np.eye(N)
else:
raise ValueError("Prior must be 'conv' or 'sparse'")
return D @ C
recovery_per_alpha = []
for alpha in measurement_ratios:
recoveries = []
for rep in range(n_rep):
M = int(alpha * N)
ensemble = GaussianEnsemble(M, N)
A = ensemble.generate()
C = sample_trnsf()
teacher = conv_model(
A, C, signal) @ GaussianChannel(var=delta) @ O(id="y")
teacher = teacher.to_model()
sample = teacher.sample()
if (solver == "AMP"):
max_iter = 20
damping = 0.1
student = conv_model(A, C, signal) @ GaussianLikelihood(
y=sample['y'], var=delta)
student = student.to_model_dag()
student = student.to_model()
ep = ExpectationPropagation(student)
ep.iterate(max_iter=max_iter,
damping=damping,
callback=EarlyStopping(tol=1e-8))
data_ep = ep.get_variables_data((['x']))
mse = np.mean((data_ep['x']['r'] - sample['x'])**2)
recoveries.append(mse)
elif (solver == "CVX"):
reg_param = 0.001
x = cp.Variable(shape=(N, ), name="x")
lmbda = cp.Parameter(nonneg=True)
objective = cp.norm2(A @ C @ x -
sample['y'])**2 + lmbda * cp.norm1(x)
problem = cp.Problem(cp.Minimize(objective))
lmbda.value = reg_param
problem.solve(abstol=1e-6)
mse = np.mean((x.value - sample['x'])**2)
recoveries.append(mse)
else:
raise ValueError("Solver must be 'AMP' or 'CVX'")
recovery_per_alpha.append(recoveries)
return recovery_per_alpha
def total_variation_plus_seasonal_filter(signal,
c1=10,
c2=500,
residual_weights=None,
tv_weights=None,
use_ixs=None,
periodic_detector=False,
transition_locs=None,
seas_max=None):
'''
This performs total variation filtering with the addition of a seasonal baseline fit. This introduces a new
signal to the model that is smooth and periodic on a yearly time frame. This does a better job of describing real,
multi-year solar PV power data sets, and therefore does an improved job of estimating the discretely changing
signal.
:param signal: A 1d numpy array (must support boolean indexing) containing the signal of interest
:param c1: The regularization parameter to control the total variation in the final output signal
:param c2: The regularization parameter to control the smoothness of the seasonal signal
:return: A 1d numpy array containing the filtered signal
'''
if residual_weights is None:
residual_weights = np.ones_like(signal)
if tv_weights is None:
tv_weights = np.ones(len(signal) - 1)
if use_ixs is None:
index_set = ~np.isnan(signal)
else:
index_set = np.logical_and(use_ixs, ~np.isnan(signal))
s_hat = cvx.Variable(len(signal))
s_seas = cvx.Variable(len(signal))
s_error = cvx.Variable(len(signal))
c1 = cvx.Constant(value=c1)
c2 = cvx.Constant(value=c2)
#w = len(signal) / np.sum(index_set)
if transition_locs is None:
objective = cvx.Minimize(
# (365 * 3 / len(signal)) * w *
# cvx.sum(cvx.huber(cvx.multiply(residual_weights, s_error)))
10 * cvx.norm(cvx.multiply(residual_weights, s_error)) +
c1 * cvx.norm1(cvx.multiply(tv_weights, cvx.diff(s_hat, k=1))) +
c2 * cvx.norm(cvx.diff(s_seas, k=2))
# + c2 * .1 * cvx.norm(cvx.diff(s_seas, k=1))
)
else:
objective = cvx.Minimize(
10 * cvx.norm(cvx.multiply(residual_weights, s_error)) +
c2 * cvx.norm(cvx.diff(s_seas, k=2)))
constraints = [
signal[index_set] == s_hat[index_set] + s_seas[index_set] +
s_error[index_set],
cvx.sum(s_seas[:365]) == 0
]
if len(signal) > 365:
constraints.append(s_seas[365:] - s_seas[:-365] == 0)
if periodic_detector:
constraints.append(s_hat[365:] - s_hat[:-365] == 0)
if transition_locs is not None:
loc_mask = np.ones(len(signal) - 1, dtype=bool)
loc_mask[transition_locs] = False
# loc_mask[transition_locs + 1] = False
constraints.append(cvx.diff(s_hat, k=1)[loc_mask] == 0)
if seas_max is not None:
constraints.append(s_seas <= seas_max)
problem = cvx.Problem(objective=objective, constraints=constraints)
problem.solve()
return s_hat.value, s_seas.value
def fit_ellipse_stack2(dx, dy, dz, di, norm_type="l2"):
"""
fit ellipoid using squared loss
idea to learn all stacks together including smoothness
"""
#TODO create flag for norm1 vs norm2
assert norm_type in ["l1", "l2", "huber"]
# sanity check
assert len(dx) == len(dy)
assert len(dx) == len(dz)
assert len(dx) == len(di)
# unique zs
dat = defaultdict(list)
# resort data
for idx in range(len(dx)):
dat[dz[idx]].append( [dx[idx], dy[idx], di[idx]] )
# init ret
ellipse_stack = []
for idx in range(max(dz)):
ellipse_stack.append(Ellipse(0, 0, idx, 1, 1, 0))
total_N = len(dx)
M = len(dat.keys())
#D = 5
D = 4
X_matrix = []
thetas = []
slacks = []
eps_slacks = []
mean_di = float(numpy.mean(di))
for z in dat.keys():
x = numpy.array(dat[z])[:,0]
y = numpy.array(dat[z])[:,1]
# intensities
i = numpy.array(dat[z])[:,2]
ity = numpy.diag(i) / mean_di
# dimensionality
N = len(x)
d = numpy.zeros((N, D))
d[:,0] = x*x
d[:,1] = y*y
#d[:,2] = x*y
d[:,2] = x
d[:,3] = y
#d[:,4] = numpy.ones(N)
#d[:,0] = x*x
#d[:,1] = y*y
#d[:,2] = x*y
#d[:,3] = x
#d[:,4] = y
#d[:,5] = numpy.ones(N)
# consider intensities
old_shape = d.shape
#d = numpy.dot(ity, d)
assert d.shape == old_shape
print d.shape
d = cvxpy.matrix(d)
#### parameters
# da
X = cvxpy.parameter(N, D, name="X" + str(z))
X.value = d
X_matrix.append(X)
#### varibales
# parameter vector
theta = cvxpy.variable(D, name="theta" + str(z))
thetas.append(theta)
# construct obj
objective = 0
print "norm type", norm_type
for i in xrange(M):
if norm_type == "l1":
objective += cvxpy.norm1(X_matrix[i] * thetas[i] + 1.0)
if norm_type == "l2":
objective += cvxpy.norm2(X_matrix[i] * thetas[i] + 1.0)
#TODO these need to be summed
#objective += cvxpy.huber(X_matrix[i] * thetas[i], 1)
#objective += cvxpy.deadzone(X_matrix[i] * thetas[i], 1)
# add smoothness regularization
reg_const = float(total_N) / float(M-1)
for i in xrange(M-1):
objective += reg_const * cvxpy.norm2(thetas[i] - thetas[i+1])
# create problem
p = cvxpy.program(cvxpy.minimize(objective))
prob = p
import ipdb
ipdb.set_trace()
# add constraints
#for i in xrange(M):
# #p.constraints.append(cvxpy.eq(thetas[i][0,:] + thetas[i][1,:], 1))
# p.constraints.append(cvxpy.eq(thetas[i][4,:], 1))
# set solver settings
p.options['reltol'] = 1e-1
p.options['abstol'] = 1e-1
#p.options['feastol'] = 1e-1
# invoke solver
p.solve()
# wrap up result
ellipse_stack = {}
active_layers = dat.keys()
assert len(active_layers) == M
for i in xrange(M):
w = numpy.array(thetas[i].value)
## For clarity, fill in the quadratic form variables
#A = numpy.zeros((2,2))
#A[0,0] = w[0]
#A.ravel()[1:3] = w[2]
#A[1,1] = w[1]
#bv = w[3:5]
#c = w[5]
A = numpy.zeros((2,2))
A[0,0] = w[0]
A.ravel()[1:3] = 0 #w[2]
A[1,1] = w[1]
#bv = w[2:4]
bv = w[2:]
#c = w[4]
c = 1.0
## find parameters
z, a, b, alpha = util.conic2parametric(A, bv, c)
print "layer (i,z,a,b,alpha):", i, z, a, b, alpha
layer = active_layers[i]
ellipse_stack[layer] = Ellipse(z[0], z[1], layer, a, b, alpha)
return ellipse_stack
def sparse_kmeans(AllDataMatrix,s,niter,group):
w = [1/np.sqrt(AllDataMatrix.shape[1])]*AllDataMatrix.shape[1]
wx = np.zeros((len(AllDataMatrix),AllDataMatrix.shape[1]))
for j in range(AllDataMatrix.shape[1]):
wx[:,j] = AllDataMatrix[:,j]*(np.sqrt(w)[j])
alpha_group = s
s_orig = s
nclust = 6
#kmt = KMeans(n_clusters=nclust, init='random', n_init=100,verbose=False, tol=0.0000000001)
#kmt.fit(wx)
#print kmt.labels_
#print adjusted_rand_score(labels, kmt.labels_)
kmt = rkmeans(x=numpy2ri(wx),centers=nclust,nstart=100)
kmlabels = np.array(kmt[0])
print adjusted_rand_score(labels, np.array(kmt[0]))
#overall iterations
for i in range(niter):
#print i
#1.get bcssj
aj_list = []
for j in range(AllDataMatrix.shape[1]):
dat_j = AllDataMatrix[:,j].reshape((len(AllDataMatrix),1))
djall = euclidean_distances(dat_j, dat_j)
sumd_all = np.sum(djall**2)/len(AllDataMatrix)
nk_list = [];sumd_k_list = []
for k in range(nclust):
dat_j = AllDataMatrix[kmlabels==k,j]
dat_j = dat_j.reshape((len(dat_j),1))
if(len(dat_j)<1):
d = 0
else:
d = euclidean_distances(dat_j, dat_j)
nk = len(dat_j)
sumd_k = np.sum(d**2)
nk_list.append(nk)
sumd_k_list.append(sumd_k)
nk_list = np.array(nk_list)
sumd_k_list = np.array(sumd_k_list)
#compute within-sum of squares over feature j
nk_list[nk_list==0] = -1
one_nk_list = 1./nk_list
one_nk_list[np.sign(one_nk_list)== -1 ] = 0
withinssj = np.sum(one_nk_list*sumd_k_list)
#aj = totalss/n - wss/nk
aj = sumd_all - withinssj
aj_list.append(aj)
#2. get w
a = np.array(aj_list)
lenseq = np.array([256,128,64,32,16,8,4,2,1,1])
lenseq = np.array([256,128,64,32,16,8,8])
nlevels = len(lenseq)
sqrtlenseq = np.sqrt(lenseq)
indseq = np.cumsum(lenseq)
wvar = cvx.Variable(len(a))
t = cvx.Variable(nlevels)
## Form objective.
if group:
####GROUP SPARSE
#obj = cvx.Minimize(sum(-1*a*wvar) + alpha_group*sum(t))
obj = cvx.Minimize(sum(-1*a*wvar))
group0 = [cvx.norm(wvar[0:(indseq[0]-1)],2)<=t[0]]
group1 = [cvx.norm(wvar[indseq[0]:(indseq[1])],2)<=t[1]]
group2 = [cvx.norm(wvar[indseq[1]:(indseq[2])],2)<=t[2]]
group3 = [cvx.norm(wvar[indseq[2]:(indseq[3])],2)<=t[3]]
group4 = [cvx.norm(wvar[indseq[3]:(indseq[4])],2)<=t[4]]
group5 = [cvx.norm(wvar[indseq[4]:(indseq[5])],2)<=t[5]]
group6 = [cvx.norm(wvar[indseq[5]:(indseq[6])],2)<=t[6]]
# group0 = [cvx.norm(wvar[0:indseq[0]],2)<=t[0]]
# group1 = [cvx.norm(wvar[indseq[0]:indseq[1]],2)<=t[1]]
# group0 = [sqrtlenseq[0]*cvx.norm(wvar[0:(indseq[0]-1)],2)<=t[0]]
# group1 = [sqrtlenseq[1]*cvx.norm(wvar[indseq[0]:(indseq[1])],2)<=t[1]]
# group2 = [sqrtlenseq[2]*cvx.norm(wvar[indseq[1]:(indseq[2])],2)<=t[2]]
# group3 = [sqrtlenseq[3]*cvx.norm(wvar[indseq[2]:(indseq[3])],2)<=t[3]]
# group4 = [sqrtlenseq[4]*cvx.norm(wvar[indseq[3]:(indseq[4])],2)<=t[4]]
# group5 = [sqrtlenseq[5]*cvx.norm(wvar[indseq[4]:(indseq[5])],2)<=t[5]]
# group6 = [sqrtlenseq[6]*cvx.norm(wvar[indseq[5]:(indseq[6])],2)<=t[6]]
#
#group7 = [cvx.norm(wvar[indseq[6]:(indseq[7])],2)<=t[7]]
# group8 = [cvx.norm(wvar[indseq[7]:(indseq[8])],2)<=t[8]]
# group9 = [cvx.norm(wvar[indseq[8]:(indseq[9])],2)<=t[9]]
###"correct" constraints
#constr = [wvar>=0,sum(wvar)==1] + group0 + group1 + group2 + group3 + group4 + group5 + group6
##l2 constraints
#constr = [cvx.square(cvx.norm2(wvar))<=1,wvar>=0] + group0 + group1 + group2 + group3 + group4 + group5 + group6 + group7 + group8 + group9
#constr = [cvx.square(cvx.norm2(wvar))<=1,wvar>=0] + group0 + group1 + group2 + group3 + group4 + group5 + group6 + group7
constr = [cvx.square(cvx.norm2(wvar))<=1,wvar>=0] + group0 + group1 + group2 + group3 + group4 + group5 + group6
constr = constr + [sum(t)<=alpha_group]#cvx.norm1(wvar)<=s_orig
####GROUP NORM AS IN LASSO
# groupnormvec = [cvx.norm(wvar[0:(indseq[0]-1)],2),cvx.norm(wvar[indseq[0]:(indseq[1])],2),
# cvx.norm(wvar[indseq[1]:(indseq[2])],2),cvx.norm(wvar[indseq[2]:(indseq[3])],2),
# cvx.norm(wvar[indseq[3]:(indseq[4])],2),cvx.norm(wvar[indseq[4]:(indseq[5])],2),
# cvx.norm(wvar[indseq[5]:(indseq[6])],2)]
# obj = cvx.Minimize(sum(-1*a*wvar) + alpha_group*sum(groupnormvec))
# constr = [cvx.square(cvx.norm2(wvar))<=1,wvar>=0]
else:
####ORIGINAL SPARSE KMEANS PROBLEM
#obj = cvx.Minimize(cvx.sum(cvx.mul_elemwise(-1*a,wvar)))
obj = cvx.Minimize(sum(-1*a*wvar))
constr = [cvx.square(cvx.norm2(wvar))<=1,cvx.norm1(wvar)<=s_orig, wvar>=0]
#constr = [cvx.square(cvx.norm2(wvar))<=1, wvar>=0]
prob = cvx.Problem(obj, constr)
#prob.solve()
try:
prob.solve()
#print "default solver"
except:
#print "SCS SOLVER"
#prob.solve(solver =cvx.CVXOPT)
prob.solve(solver = cvx.SCS,verbose=False)#use_indirect=True
#print prob.value
w = wvar.value
#3. update kmeans
wx = np.zeros((len(AllDataMatrix),AllDataMatrix.shape[1]))
for j in range(AllDataMatrix.shape[1]):
wj = np.sqrt(w[j][0,0])
#wj = w[j][0,0]
if np.isnan(wj):
#print "bad"
wj = 10**-20
# else:
# #print "yes"
# #print wj
wx[:,j] = AllDataMatrix[:,j]*wj
# kmt = KMeans(n_clusters=nclust, init='random', n_init=100,verbose=False,tol=0.0000000001)
# kmt.fit(wx)
kmt = rkmeans(x=numpy2ri(wx),centers=nclust,nstart=100)
kmlabels = np.array(kmt[0])
return prob.value,kmlabels
constr = [cvx.square(cvx.norm2(wvar))<=1,wvar>=0] + group0 + group1 + group2 + group3 + group4 + group5 + group6
constr = constr + [sum(t)<=alpha_group]#cvx.norm1(wvar)<=s_orig
####GROUP NORM AS IN LASSO
# groupnormvec = [cvx.norm(wvar[0:(indseq[0]-1)],2),cvx.norm(wvar[indseq[0]:(indseq[1])],2),
# cvx.norm(wvar[indseq[1]:(indseq[2])],2),cvx.norm(wvar[indseq[2]:(indseq[3])],2),
# cvx.norm(wvar[indseq[3]:(indseq[4])],2),cvx.norm(wvar[indseq[4]:(indseq[5])],2),
# cvx.norm(wvar[indseq[5]:(indseq[6])],2)]
# obj = cvx.Minimize(sum(-1*a*wvar) + alpha_group*sum(groupnormvec))
# constr = [cvx.square(cvx.norm2(wvar))<=1,wvar>=0]
else:
####ORIGINAL SPARSE KMEANS PROBLEM
#obj = cvx.Minimize(cvx.sum(cvx.mul_elemwise(-1*a,wvar)))
obj = cvx.Minimize(sum(-1*a*wvar))
constr = [cvx.square(cvx.norm2(wvar))<=1,cvx.norm1(wvar)<=s_orig, wvar>=0]
#constr = [cvx.square(cvx.norm2(wvar))<=1, wvar>=0]
prob = cvx.Problem(obj, constr)
#prob.solve()
try:
prob.solve()
print "default solver"
except:
print "SCS SOLVER"
#prob.solve(solver =cvx.CVXOPT)
prob.solve(solver = cvx.SCS,verbose=False)#use_indirect=True
print prob.value
w = wvar.value
# Proximal operators
PROX_TESTS = [
#prox("MATRIX_FRAC", lambda: cp.matrix_frac(p, X)),
#prox("SIGMA_MAX", lambda: cp.sigma_max(X)),
prox("AFFINE", lambda: randn(n).T*x),
prox("CONSTANT", lambda: 0),
prox("LAMBDA_MAX", lambda: cp.lambda_max(X)),
prox("LOG_SUM_EXP", lambda: cp.log_sum_exp(x)),
prox("MAX", lambda: cp.max_entries(x)),
prox("NEG_LOG_DET", lambda: -cp.log_det(X)),
prox("NON_NEGATIVE", None, C_non_negative_scaled),
prox("NON_NEGATIVE", None, C_non_negative_scaled_elemwise),
prox("NON_NEGATIVE", None, lambda: [x >= 0]),
prox("NORM_1", f_norm1_weighted),
prox("NORM_1", lambda: cp.norm1(x)),
prox("NORM_2", lambda: cp.norm(X, "fro")),
prox("NORM_2", lambda: cp.norm2(x)),
prox("NORM_NUCLEAR", lambda: cp.norm(X, "nuc")),
prox("SECOND_ORDER_CONE", None, C_soc_scaled),
prox("SECOND_ORDER_CONE", None, C_soc_scaled_translated),
prox("SECOND_ORDER_CONE", None, C_soc_translated),
prox("SECOND_ORDER_CONE", None, lambda: [cp.norm(X, "fro") <= t]),
prox("SECOND_ORDER_CONE", None, lambda: [cp.norm2(x) <= t]),
prox("SEMIDEFINITE", None, lambda: [X >> 0]),
prox("SUM_DEADZONE", f_dead_zone),
prox("SUM_EXP", lambda: cp.sum_entries(cp.exp(x))),
prox("SUM_HINGE", f_hinge),
prox("SUM_HINGE", lambda: cp.sum_entries(cp.max_elemwise(1-x, 0))),
prox("SUM_HINGE", lambda: cp.sum_entries(cp.max_elemwise(1-x, 0))),
prox("SUM_INV_POS", lambda: cp.sum_entries(cp.inv_pos(x))),
def SLS_synthesis_p1(Pssd, T, regularization=-1, test_signal='step',
Mp=1.01, m=0.5, b=10, k=80,
freqs_bnd_yn=[1e-2, 255], mag_bnd_yn=[-10, -10],
freqs_bnd_T=[1e-2, 357], mag_bnd_T=[6, 6], T_delay=11):
"""Synthesize a controller using SLS.
Procedure p1
Constraints
- 20c, 20a, 20b (achievability constraints),
- lower bound on impulse response to prevent negative response,
- steady-state displacement given constant acting force;
- noise attenuation: upper bound on the mapping from noise to displacement;
- robust stability: upper bound on the complementary sensitivity transfer function;
Objectives
- regularization using scaled l1 norm on the impulse responses L, MB2
- distance to a desired mass/spring/damper model (m,b,k)
"""
print("-- Starting SLS_synthesis_p1")
# parameters
nu = 1 # 1 dof controller
ny = 1
# form response matrices
R, N, M, L, H, constraints = Ss.SLS.form_SLS_response_matrices(
Pssd, nu, ny, T)
# # constants
nx = Pssd.states
A, B1, B2, C1, C2, D11, D12, D21, D22 = Ss.get_partitioned_mats(
Pssd, nu, ny)
# select component of H that correspond to the mapping from acting
# force (and noise) to actual robot displacement (H_yn) and the
# mapping that corresponds to the complementary transfer function T.
H_yn = H[0::2, :]
H_T = H[1::2, :]
# objective: match the impulse response of a given system
sys_model = co.c2d(co.tf([1], [m, b, k]), dT)
_, imp_model = co.impulse_response(sys_model, np.arange(T) * dT)
# NOTE: have to use norm, sum_of_squares does not work. The reason
# is the magnitude of the objective function must not be too
# small. The optimizer seems to get confuse and simply stop
# working.
imp_diff = H_yn[T_delay:, 0] - imp_model[0, :T - T_delay]
weight = np.diag(1 + 2 * (1.0 / T) * np.arange(T - T_delay))
objective = 1e6 * cvx.norm(weight * imp_diff)
# try some regularization
if regularization > 0:
reg = regularization * (cvx.norm1(H_yn))
else:
reg = cvx.abs(cvx.Variable())
# constraint in frequency-domain, if specified
W_dft = Ss.dft_matrix(T)
Hz_yn = W_dft * H_yn
Hz_T = W_dft * H_T
omegas = np.arange(int(T / 2)) * 2 * np.pi / T / dT
omegas[0] = 1e-2
# upper bound for noise attenuation
wN_inv = np.ones((T, 1)) * 100 # infinity
wN_inv[:int(T / 2), 0] = np.power(10, np.interp(np.log10(omegas),
np.log10(freqs_bnd_yn), mag_bnd_yn) / 20)
# upper bound for complementary sensitivity transfer function
wT_inv = np.ones((T, 1)) * 100 # infinity
wT_inv[:int(T / 2), 0] = np.power(
10, np.interp(np.log10(omegas), np.log10(freqs_bnd_T), mag_bnd_T) / 20)
# add both frequency-domian constraints
constraints.append(cvx.abs(Hz_yn) <= wN_inv)
constraints.append(cvx.abs(Hz_T) <= wT_inv)
# optimize
obj = cvx.Minimize(objective + reg)
prob = cvx.Problem(obj, constraints)
print("-- [SLS_synthesis_p1] Preparing problem with cvxpy!")
prob.solve(verbose=True, solver=cvx.MOSEK)
print("-- [SLS_synthesis_p1] optimization status: {:}".format(prob.status))
print(
"-- [SLS_synthesis_p1] obj = {:}, reg = {:}".format(objective.value, reg.value))
if prob.status != "optimal":
return None, None
print("-- [SLS_synthesis_p1] Forming controllers!")
# form controllers (Structure 1, Figure 4b, Wang 2018)
L_value = np.array(L.value).reshape(ny, nu, -1)
MB2_value = np.array((M * B2).value).reshape(nu, nu, -1)
# since ny=nu=1, we have
fir_den = [1] + [0 for n in range(T - 1)]
MB2_tf = co.tf(MB2_value[0, 0], fir_den, dT)
L_tf = co.tf(L_value[0, 0], fir_den, dT)
K = co.feedback(1, MB2_tf, sign=-1) * L_tf
K = Ss.tf2ss(K, minreal=True)
# response mapping
Rval = np.array(
[R[n * nx: (n + 1) * nx, :].value for n in range(T)]).reshape(-1, nx, nx)
Nval = np.array(
[N[n * nx: (n + 1) * nx, :].value for n in range(T)]).reshape(-1, nx, ny)
Mval = np.array(
[M[n * nu: (n + 1) * nu, :].value for n in range(T)]).reshape(-1, nu, nx)
Lval = np.array(
[L[n * nu: (n + 1) * nu, :].value for n in range(T)]).reshape(-1, nu, ny)
Hval_yn = np.array(H_yn.value)
Hval_T = np.array(H_T.value)
return K, {'internal responses': (Rval, Nval, Mval, Lval),
'output impulse': (Hval_yn, Hval_T),
'L': L_value, 'MB2': MB2_value}
import cvxpy as cp
import matplotlib.pyplot as plt
# generate a problem instance
n = 1000 # number of variables
m = 50 # number of equality constraints
# randn('state',1); # set state so problem is reproducable
A = np.random.randn(m, n)
b = np.random.randn(m)
# threshold value below which we consider an element to be zero
DELTA = 1e-8
# compute an optimal point by solving an LP (use CVX)
x = cp.Variable(n)
objective = cp.Minimize(cp.norm1(x))
constraint = [A @ x == b]
prob = cp.Problem(objective, constraint)
result = prob.solve()
f_min = sum(np.abs(x.value))
print(f_min)
print(f'Optimal value is {f_min:0.4f}')
nnz = len(np.where(abs(x.value) > DELTA))
print(f'Found a feasible x in R^{n} that hs {nnz}')
# initial point needs to satisfy A*x1 = b (can use least-norm solution)
x1 = np.linalg.pinv(A).dot(b)
# ********************************************************************
# subgradient method computation
# Variable declarations
import scipy.sparse as sps
n = 500
m = 300
np.random.seed(0)
A = np.random.rand(m, n)
x0 = sps.rand(n, 1, 0.1)
b = A*x0
# Problem construction
x = cp.Variable(n)
prob = cp.Problem(cp.Minimize(cp.norm1(x)), [A*x == b])
# Problem collection
# Single problem collection
problemDict = {
"problemID" : problemID,
"problem" : prob,
"opt_val" : opt_val
}
problems = [problemDict]
# For debugging individual problems:
# A: the complete connection matrix; A_hat: the selected connection matrix;
# b: the complete threshold; b_hat: the selected threshold
# x: the encoding bits; y: the merged test cube to be encoded; y_hat: the decoded/re-constructed test cube
y = np.random.choice(2, (num_sc), p=[0.8, 0.2])
x = cp.Variable(num_ctrl, boolean=True)
# A = state_dict_np['decoder.0.linear.weight'] * np.expand_dims(thred_sign, axis=1)
# b = thred_decoder * thred_sign
A = np.load('checkpoint/decoder.linear.weight.npy')
b = np.load('checkpoint/encoder.bn.thred.npy')
A_hat = A[y.astype(bool)]
b_hat = b[y.astype(bool)]
# print(A_hat)
# print(b_hat)
# exit()
# constraints
# cost = cp.norm1(A @ x - b) # This cost is too computational complex to solve.
cost = cp.norm1(x)
objective = cp.Minimize(cost)
constraint = [A_hat @ x >= b_hat]
prob = cp.Problem(objective, constraint)
prob.solve()
print("Status: ", prob.status)
print("The optimal value is", prob.value)
print("A solution x is")
print(x.value)
y_recover = np.matmul(A, x.value) >= b
# print(y_recover)
print('OnePercent: ', np.sum(y_recover) / num_sc)
def linear_builder(er: np.ndarray,
lbound: Union[np.ndarray, float],
ubound: Union[np.ndarray, float],
risk_constraints: np.ndarray,
risk_target: Tuple[np.ndarray, np.ndarray],
turn_over_target: float = None,
current_position: np.ndarray = None,
method: str = 'ecos') -> Tuple[str, np.ndarray, np.ndarray]:
er = er.flatten()
n, m = risk_constraints.shape
if not risk_target:
risk_lbound = -np.inf * np.ones((m, 1))
risk_ubound = np.inf * np.ones((m, 1))
else:
risk_lbound = risk_target[0].reshape((-1, 1))
risk_ubound = risk_target[1].reshape((-1, 1))
if isinstance(lbound, float):
lbound = np.ones(n) * lbound
if isinstance(ubound, float):
ubound = np.ones(n) * ubound
if not turn_over_target or current_position is None:
cons_matrix = np.concatenate(
(risk_constraints.T, risk_lbound, risk_ubound), axis=1)
prob = LPOptimizer(cons_matrix, lbound, ubound, -er, method)
if prob.status() == 0:
return 'optimal', prob.feval(), prob.x_value()
else:
raise PortfolioBuilderException(prob.status())
else:
if method in ("simplex", "interior"):
# we need to expand bounded condition and constraint matrix to handle L1 bound
w_u_bound = np.minimum(
np.maximum(np.abs(current_position - lbound),
np.abs(current_position - ubound)),
turn_over_target).reshape((-1, 1))
current_position = current_position.reshape((-1, 1))
lbound = np.concatenate((lbound, np.zeros(n)), axis=0)
ubound = np.concatenate((ubound, w_u_bound.flatten()), axis=0)
risk_lbound = np.concatenate((risk_lbound, [[0.]]), axis=0)
risk_ubound = np.concatenate((risk_ubound, [[turn_over_target]]),
axis=0)
risk_constraints = np.concatenate(
(risk_constraints.T, np.zeros((m, n))), axis=1)
er = np.concatenate((er, np.zeros(n)), axis=0)
turn_over_row = np.zeros(2 * n)
turn_over_row[n:] = 1.
risk_constraints = np.concatenate(
(risk_constraints, [turn_over_row]), axis=0)
turn_over_matrix = np.zeros((2 * n, 2 * n))
for i in range(n):
turn_over_matrix[i, i] = 1.
turn_over_matrix[i, i + n] = -1.
turn_over_matrix[i + n, i] = 1.
turn_over_matrix[i + n, i + n] = 1.
risk_constraints = np.concatenate(
(risk_constraints, turn_over_matrix), axis=0)
risk_lbound = np.concatenate((risk_lbound, -np.inf * np.ones(
(n, 1))),
axis=0)
risk_lbound = np.concatenate((risk_lbound, current_position),
axis=0)
risk_ubound = np.concatenate((risk_ubound, current_position),
axis=0)
risk_ubound = np.concatenate((risk_ubound, np.inf * np.ones(
(n, 1))),
axis=0)
cons_matrix = np.concatenate(
(risk_constraints, risk_lbound, risk_ubound), axis=1)
prob = LPOptimizer(cons_matrix, lbound, ubound, -er, method)
if prob.status() == 0:
return 'optimal', prob.feval(), prob.x_value()[:n]
else:
raise PortfolioBuilderException(prob.status())
elif method.lower() == 'ecos':
from cvxpy import Problem
from cvxpy import Variable
from cvxpy import norm1
from cvxpy import Minimize
w = Variable(n)
current_risk_exposure = risk_constraints.T @ w
constraints = [
w >= lbound, w <= ubound,
current_risk_exposure >= risk_lbound.flatten(),
current_risk_exposure <= risk_ubound.flatten(),
norm1(w - current_position) <= turn_over_target
]
objective = Minimize(-w.T * er)
prob = Problem(objective, constraints)
prob.solve(solver='ECOS',
feastol=1e-10,
abstol=1e-10,
reltol=1e-10)
if prob.status == 'optimal' or prob.status == 'optimal_inaccurate':
return prob.status, prob.value, w.value.flatten()
else:
raise PortfolioBuilderException(prob.status)
else:
raise ValueError("{0} is not recognized".format(method))
def sparse_kmeans(AllDataMatrix, nclust, s, niter, group=False, tree=False, multi=False, groups_vector=None):
w = [1 / np.sqrt(AllDataMatrix.shape[1])] * AllDataMatrix.shape[1]
wx = np.zeros((len(AllDataMatrix), AllDataMatrix.shape[1]))
for j in range(AllDataMatrix.shape[1]):
wx[:, j] = AllDataMatrix[:, j] * (np.sqrt(w)[j])
alpha_group = s
s_orig = s
print nclust
kmt = KMeans(n_clusters=nclust, init="k-means++", max_iter=200, n_init=100)
kmt.fit(wx)
kmlabels = np.array(kmt.labels_)
for i in range(niter):
aj_list = []
for j in range(AllDataMatrix.shape[1]):
dat_j = AllDataMatrix[:, j].reshape((len(AllDataMatrix), 1))
djall = euclidean_distances(dat_j, dat_j)
sumd_all = np.sum(djall ** 2) / len(AllDataMatrix)
nk_list = []
sumd_k_list = []
for k in range(nclust):
dat_j = AllDataMatrix[kmlabels == k, j]
dat_j = dat_j.reshape((len(dat_j), 1))
if len(dat_j) < 1:
d = 0
else:
d = euclidean_distances(dat_j, dat_j)
nk = len(dat_j)
sumd_k = np.sum(d ** 2)
nk_list.append(nk)
sumd_k_list.append(sumd_k)
nk_list = np.array(nk_list)
sumd_k_list = np.array(sumd_k_list)
# compute within-sum of squares over feature j
nk_list[nk_list == 0] = -1
one_nk_list = 1.0 / nk_list
one_nk_list[np.sign(one_nk_list) == -1] = 0
withinssj = np.sum(one_nk_list * sumd_k_list)
aj = sumd_all - withinssj
aj_list.append(aj)
# 2. get w
a = np.array(aj_list)
wvar = cvx.Variable(len(a))
if tree:
print "tree structure not supported"
return -1
else:
if group:
obj = cvx.Minimize(sum(-1 * a * wvar))
if groups_vector is None:
lenseq = np.hstack((np.power(2, np.arange(np.log2(AllDataMatrix.shape[1])))[::-1], [1]))
else:
lenseq = np.array(groups_vector)
lenseq = lenseq.astype(int)
nlevels = len(lenseq)
sqrtlenseq = np.sqrt(lenseq)
indseq = np.cumsum(lenseq)
t = cvx.Variable(nlevels)
group0 = [sqrtlenseq[0] * cvx.norm(wvar[0 : (indseq[0])], 2) <= t[0]]
group_constraints = group0
for level in range(1, nlevels):
# print level
group_const = [
sqrtlenseq[level] * cvx.norm(wvar[indseq[(level - 1)] : (indseq[level])], 2) <= t[level]
]
group_constraints = group_constraints + group_const
constr = [cvx.square(cvx.norm2(wvar)) <= 1, wvar >= 0] + group_constraints
constr = constr + [sum(t) <= alpha_group]
if multi:
T = AllDataMatrix.shape[1] / 3
t = cvx.Variable(T - 1)
constr_list = []
for coeff in range((T - 1)):
penalty = cvx.norm(wvar[coeff : (T * 3) : T], 2) <= t[coeff]
constr_list.append(penalty)
constr = [cvx.square(cvx.norm2(wvar)) <= 1, wvar >= 0] + constr_list
constr = constr + [sum(t) <= alpha_group]
else:
####ORIGINAL SPARSE KMEANS PROBLEM
print "ORIGINAL"
obj = cvx.Minimize(sum(-1 * a * wvar))
constr = [cvx.square(cvx.norm2(wvar)) <= 1, cvx.norm1(wvar) <= s_orig, wvar >= 0]
prob = cvx.Problem(obj, constr)
try:
prob.solve()
except:
prob.solve(solver=cvx.SCS, verbose=False) # use_indirect=True
w = wvar.value
# 3. update kmeans
wx = np.zeros((len(AllDataMatrix), AllDataMatrix.shape[1]))
for j in range(AllDataMatrix.shape[1]):
wj = np.sqrt(w[j][0, 0])
if np.isnan(wj):
wj = 10 ** -30
wx[:, j] = AllDataMatrix[:, j] * wj
kmt = KMeans(n_clusters=nclust, init="k-means++", max_iter=200, n_init=100)
kmt.fit(wx)
kmlabels = np.array(kmt.labels_)
return prob.value, kmlabels, w
def nonlinearmodel(x,
library,
dt,
params,
options={
'smooth': True,
'solver': 'MOSEK'
}):
'''
Use the integral form of SINDy to find a sparse dynamical system model for the output, x, given a library of features.
Then take the derivative of that model to estimate the derivative.
Inputs
------
x : (np.array of floats, 1xN) time series to differentiate
library : (list of 1D arrays) list of features to use for building the model
dt : (float) time step
Parameters
----------
params : (list) [gamma, : (int) sparsity knob (higher = more sparse model)
window_size], : (int) if option smooth, this determines the smoothing window
options : (dict) {'smooth', : (bool) if True, apply gaussian smoothing to the result with the same window size
'solver'} : (str) solver to use with cvxpy, MOSEK is default
Outputs
-------
x_hat : estimated (smoothed) x
dxdt_hat : estimated derivative of x
'''
# Features
int_library = integrate_library(library, dt)
w_int_library, w_int_library_func, dw_int_library_func = whiten_library(
int_library)
# Whitened states
w_state, w_state_func, dw_state_func = whiten_library([x])
w_x_hat, = w_state
# dewhiten integral library coefficients
integrated_library_std = []
integrated_library_mean = []
for d in dw_int_library_func:
integrated_library_std.append(d.std)
integrated_library_mean.append(d.mean)
integrated_library_std = np.array(integrated_library_std)
integrated_library_mean = np.array(integrated_library_mean)
# dewhiten state coefficients
state_std = []
state_mean = []
for d in dw_state_func:
state_std.append(d.std)
state_mean.append(d.mean)
state_std = np.array(state_std)
state_mean = np.array(state_mean)
# Define loss function
var = cvxpy.Variable((1, len(library)))
sum_squared_error_x = cvxpy.sum_squares(w_x_hat[1:-1] -
(w_int_library * var[0, :])[1:-1])
sum_squared_error = cvxpy.sum([sum_squared_error_x])
# Solve convex optimization problem
gamma = params[0]
solver = options['solver']
L = cvxpy.sum(sum_squared_error + gamma * cvxpy.norm1(var))
obj = cvxpy.Minimize(L)
prob = cvxpy.Problem(obj)
r = prob.solve(solver=solver)
sindy_coefficients = var.value
integrated_library_offset = np.matrix(
sindy_coefficients[0, :] /
integrated_library_std) * np.matrix(integrated_library_mean).T
estimated_coefficients = sindy_coefficients[
0, :] / integrated_library_std * np.tile(state_std[0],
[len(int_library), 1]).T
offset = -1 * (state_std[0] *
np.ravel(integrated_library_offset)) + state_mean
# estimate derivative
dxdt_hat = np.ravel(np.matrix(estimated_coefficients) * np.matrix(library))
if options['smooth']:
window_size = params[1]
kernel = __gaussian_kernel__(window_size)
dxdt_hat = pynumdiff.smooth_finite_difference.__convolutional_smoother__(
dxdt_hat, kernel, 1)
x_hat = utility.integrate_dxdt_hat(dxdt_hat, dt)
x0 = utility.estimate_initial_condition(x, x_hat)
x_hat = x_hat + x0
return x_hat, dxdt_hat
A[b<0,:] -= 0.7*np.tile([x0], (np.sum(b<0),1))
P = la.block_diag(np.random.randn(n-1,n-1), 0)
lam = 1
# Problem construction
# Explicit epigraph constraint
x = cp.Variable(n)
t = cp.Variable(1)
z = 1 - sps.diags([b],[0])*A*x + t
f = lam*cp.sum_squares(x) + cp.sum_entries(cp.max_elemwise(z, 0))
C = [cp.norm1(P.T*x) <= t]
prob = cp.Problem(cp.Minimize(f), C)
# Problem collection
# Single problem collection
problemDict = {
"problemID" : problemID,
"problem" : prob,
"opt_val" : opt_val
}
problems = [problemDict]
def f_norm1_weighted():
w = np.random.randn(n)
w[0] = 0
return cp.norm1(cp.mul_elemwise(w, x))
# Choose regularization parameter
# lambda > lambda_max -> zero solution
lambda_max = 2*norm(dot(nA.T, rSig.T), np.inf)
lamb = 1.0e-8*lambda_max
print('Solving L1 penalized system with cvxpy...')
coefs = cvx.variable(n_qpnts,1)
A = cvx.matrix(nA)
rhs = cvx.matrix(rSig).T
objective = cvx.minimize(cvx.norm2(A*coefs - rhs) +
lamb*cvx.norm1(coefs) )
constraints = [cvx.geq(coefs,0.0)]
prob = cvx.program(objective, constraints)
# Call the solver
prob.solve(quiet=True) #Use quiet=True to suppress output
# Convert the cvxmod objects to plain numpy arrays for further processing
nd_coefs_l1 = np.array(coefs.value).squeeze()
# Cutoff those coefficients that are less than cutoff
cutoff = nd_coefs_l1.mean() + 2.0*nd_coefs_l1.std(ddof=1)
nd_coefs_l1_trim = np.where(nd_coefs_l1 > cutoff, nd_coefs_l1, 0)
# Get indices needed for sorting coefs, in reverse order.
def backtest(returns,
Z_returns,
benchmark,
means,
covs,
lev_lim,
bottom_sec_limit,
upper_sec_limit,
shorting_cost,
tcost,
MAXRISK,
kappa=None,
bid_ask=None):
_, num_assets = returns.shape
T, K = Z_returns.shape
Zs_time = np.zeros((T - 1, K))
value_strat, value_benchmark = 1, 1
vals_strat, vals_benchmark = [value_strat], [value_benchmark]
benchmark_returns = benchmark.loc[Z_returns.index].copy().values.flatten()
"""
On the fly computing for stratified model policy
"""
W = [np.zeros(18)]
W[0][8] = 1 #vti
for date in range(1, T):
# if date % 50 == 0: print(date)
dt = Z_returns.iloc[date].name.strftime("%Y-%m-%d")
node = tuple(Z_returns.iloc[date])
Zs_time[date - 1, :] = [*node]
if date == 1:
w_prev = W[0].copy()
else:
w_prev = W[-1].flatten().copy()
w = cp.Variable(num_assets)
#adding cash asset into returns and covariances
SIGMA = covs[node]
MU = means[node]
roll = 15
#get last 5 days tcs, lagged by one! so this doesnt include today's date
tau = np.maximum(bid_ask.loc[:dt].iloc[-(roll + 1):-1].mean().values,
0) / 2
obj = -w @ (MU + 1) + shorting_cost * (kappa @ cp.neg(w)) + tcost * (
cp.abs(w - w_prev)) @ tau
cons = [
cp.quad_form(w, SIGMA) <= MAXRISK * 100 * 100,
sum(w) == 1,
cp.norm1(w) <= lev_lim,
bottom_sec_limit * np.ones(num_assets) <= w,
w <= upper_sec_limit * np.ones(num_assets),
]
prob_sm = cp.Problem(cp.Minimize(obj), cons)
prob_sm.solve(verbose=False)
returns_date = 1 + returns[date, :]
#get TODAY's tc
tau_sim = bid_ask.loc[dt].values.flatten() / 2
value_strat *= returns_date @ w.value - (kappa @ cp.neg(w)).value - (
cp.abs(w - w_prev) @ tau_sim).value
vals_strat += [value_strat]
value_benchmark *= 1 + benchmark_returns[date]
vals_benchmark += [value_benchmark]
w_prev = w.value.copy() * returns_date
W += [w_prev.reshape(-1, 1)]
vals = pd.DataFrame(data=np.vstack([vals_strat, vals_benchmark]).T,
columns=["policy", "benchmark"],
index=Z_returns.index.rename("Date"))
Zs_time = pd.DataFrame(data=Zs_time,
index=Z_returns.index[1:],
columns=Z_returns.columns)
#calculate sharpe
rr = vals.pct_change()
sharpes = np.sqrt(250) * rr.mean() / rr.std()
returns = 250 * rr.mean()
return vals, Zs_time, W, sharpes, returns
# Proximal operators
PROX_TESTS = [
#prox("MATRIX_FRAC", lambda: cp.matrix_frac(p, X)),
#prox("SIGMA_MAX", lambda: cp.sigma_max(X)),
prox("AFFINE", lambda: randn(n).T*x),
prox("CONSTANT", lambda: 0),
prox("LAMBDA_MAX", lambda: cp.lambda_max(X)),
prox("LOG_SUM_EXP", lambda: cp.log_sum_exp(x)),
prox("MAX", lambda: cp.max_entries(x)),
prox("NEG_LOG_DET", lambda: -cp.log_det(X)),
prox("NON_NEGATIVE", None, C_non_negative_scaled),
prox("NON_NEGATIVE", None, C_non_negative_scaled_elemwise),
prox("NON_NEGATIVE", None, lambda: [x >= 0]),
prox("NORM_1", f_norm1_weighted),
prox("NORM_1", lambda: cp.norm1(x)),
prox("NORM_2", lambda: cp.norm(X, "fro")),
prox("NORM_2", lambda: cp.norm2(x)),
prox("NORM_NUCLEAR", lambda: cp.norm(X, "nuc")),
#prox("QUAD_OVER_LIN", lambda: cp.quad_over_lin(p, q1)),
prox("SECOND_ORDER_CONE", None, C_soc_scaled),
prox("SECOND_ORDER_CONE", None, C_soc_scaled_translated),
prox("SECOND_ORDER_CONE", None, C_soc_translated),
prox("SECOND_ORDER_CONE", None, lambda: [cp.norm(X, "fro") <= t]),
prox("SECOND_ORDER_CONE", None, lambda: [cp.norm2(x) <= t]),
prox("SEMIDEFINITE", None, lambda: [X >> 0]),
prox("SUM_DEADZONE", f_dead_zone),
prox("SUM_EXP", lambda: cp.sum_entries(cp.exp(x))),
prox("SUM_HINGE", f_hinge),
prox("SUM_HINGE", lambda: cp.sum_entries(cp.max_elemwise(1-x, 0))),
prox("SUM_HINGE", lambda: cp.sum_entries(cp.max_elemwise(1-x, 0))),
def test_web(self):
"""
Note:
Alternatively, here is a simpler (Gaussian) model for the z variables:
def get_z_data(self, p, p_pos, q):
z = []
for m in range(M):
mu = None
if m < p_pos:
mu = 0.5*numpy.ones(q)
else:
mu = -0.5*numpy.ones(q)
Sigma = numpy.eye(q)
rv = RandomVariableFactory().create_normal_rv(mu, Sigma)
z += [rv]
return z
"""
if self.run_test_web is False:
self.assertAlmostEqual(1, 1)
return
# Create problem data.
n = 3 # Num dimensions of x
m = 20 # Num (x,y) train points
m_pos = math.floor(m / 2)
m_neg = m - m_pos
q = 4 # Num dimensions of z
p = 30 # Num (z,w) train points
p_pos = math.floor(p / 2)
p_neg = p - p_pos
l = math.floor((n + q) / 3)
u = math.floor(2 * (n + q) / 3)
C = 1 # L2 regularization trade-off parameter
ns = 10 # Num samples
# Create (x,y) data.
mu_pos = 0.5 * numpy.ones(n)
mu_neg = -0.5 * numpy.ones(n)
Sigma = numpy.eye(n)
x = numpy.matrix(
numpy.vstack((
numpy.random.randn(m_pos, n) + mu_pos,
numpy.random.randn(m_neg, n) + mu_neg,
)))
y = numpy.hstack((numpy.ones(m_pos), -1 * numpy.ones(m_neg)))
# Set up probabilistic model for (z,w) data.
z = self.get_z_data(p, p_pos, q)
w = numpy.hstack((numpy.ones(p_pos), -1 * numpy.ones(p_neg)))
# Create optimization variables.
a, b = cp.Variable(n), cp.Variable()
c, d = cp.Variable(q), cp.Variable()
# Create second stage problem.
obj2 = [
cp.log_sum_exp(vstack(0, -w[i] * (c.T * z[i] + d)))
for i in range(p)
]
budget = cp.norm1(a) + cp.norm1(c)
p2 = cp.Problem(cp.Minimize(sum(obj2) + C * cp.norm(c, 2)),
[budget <= u])
Q = partial_optimize(p2, [c, d], [a, b])
# Create and solve first stage problem.
obj1 = [
cp.log_sum_exp(cp.vstack([0, -y[i] * (x[i] * a + b)]))
for i in range(m)
]
p1 = cp.Problem(
cp.Minimize(
cp.sum(obj1) + C * cp.norm(a, 2) + expectation(Q(a, b), ns)),
[])
p1.solve()
self.assert_feas(p1)
def f_norm1_weighted():
w = np.random.randn(n)
w[0] = 0
return cp.norm1(cp.mul_elemwise(w, x))
def analysis_z(extracted_data, cmd, keys, gam=0):
"""Analyze data and identify model in discrete time.
Note: This script analyse interaction between two scalar signal
only. MIMO is possible, but that is for a future project.
cmd = "z tmin,tmax order idx1,idx2"
Params:
z: Indicator that analysis_z is to be ran.
tmin, tmax: Two ends of the analysis interval.
order: Desired Order.
idx1: Index of the input signal.
idx2: Index of the output signal.
"""
tmin, tmax = map(float, cmd.split(" ")[1].split(","))
Norder = int(cmd.split(" ")[2])
idx1, idx2 = cmd.split(" ")[3].split(",")
print(
"-- Analysis interval: [{:.3f}, {:.3f}]\n"
"-- Desired Norder: {:d}\n"
"-- Input/Output indices: {:}, {:}".format(tmin, tmax, Norder, idx1,
idx2))
data1 = get_data(extracted_data, idx1, keys)
data2 = get_data(extracted_data, idx2, keys)
Nstart1 = np.argmin(np.abs(data1['t'] - tmin))
Nend1 = np.argmin(np.abs(data1['t'] - tmax))
Nstart2 = np.argmin(np.abs(data2['t'] - data1['t'][Nstart1]))
Nend2 = Nstart2 + (Nend1 - Nstart1)
# notation: Let a=[a0..a[T-1]] and b=[b[0]...b[T-1]] be the
# coefficients. By definition we have:
# a[0] x[T] + a[1] x[T-1] + ... + a[T-1] x[0] = b[0] y[T] + b[1] y[T-1] + ... + b[T-1] y[0]
# rearrange the equations to obtain:
# X a = Y b,
# where X and Y are not the vectors of xs and ys, but a matrix
# formed by rearranging the vectors appropriately. Note that T is
# the degree (Norder + 1) of the relevant polynomial coefficients.
# x[0] is substracted from X as do y[0] is substracted from Y.
xoffset = 1.0
X = []
Y = []
for i in range(Norder - 1, Nend1 - Nstart1):
X.append(data1['data'][Nstart1 + i:Nstart1 + i + Norder][::-1] -
xoffset)
Y.append(data2['data'][Nstart2 + i:Nstart2 + i + Norder][::-1])
X = np.array(X)
Y = np.array(Y)
N = Nend1 - Nstart1
# optimization
a = cvx.Variable(Norder)
b = cvx.Variable(Norder)
obj = cvx.sum_squares(X * a - Y * b) / N
reg = gam * cvx.norm1(a) + gam * cvx.norm1(b)
constraints = [b[0] == 1]
prob = cvx.Problem(cvx.Minimize(obj + reg), constraints)
prob.solve(solver='MOSEK')
print("obj: {:f}\nreg: {:f}\na={:}\nb={:}".format(obj.value, reg.value,
a.value, b.value))
# validation by running the filter obtained on data
xin = data1['data'][Nstart1:Nend1] - xoffset
yact = data2['data'][Nstart2:Nend2]
bval = np.array(b.value).reshape(-1)
aval = np.array(a.value).reshape(-1)
ypred = signal.lfilter(bval, aval, xin)
plt.plot(-160 * xin, label='xin')
plt.plot(yact, label='yactual')
# plt.plot(ypred, '--', label='ypredict')
plt.legend()
plt.show()
scipy.io.savemat("11_1_J3_human.mat", {'x': xin, 'y': yact})
def reg(self, X): return self.nu*cp.norm1(X) def __str__(self): return "linear reg"
def fit_ellipse(x, y):
"""
fit ellipoid using squared loss and abs loss
"""
#TODO introduce flag for switching between losses
assert len(x) == len(y)
N = len(x)
D = 5
dat = numpy.zeros((N, D))
dat[:,0] = x*x
dat[:,1] = y*y
#dat[:,2] = x*y
dat[:,2] = x
dat[:,3] = y
dat[:,4] = numpy.ones(N)
print dat.shape
dat = cvxpy.matrix(dat)
#### parameters
# data
X = cvxpy.parameter(N, D, name="X")
#### varibales
# parameter vector
theta = cvxpy.variable(D, name="theta")
# simple objective
objective = cvxpy.norm1(X*theta)
# create problem
p = cvxpy.program(cvxpy.minimize(objective))
p.constraints.append(cvxpy.eq(theta[0,:] + theta[1,:], 1))
###### set values
X.value = dat
p.solve()
w = numpy.array(theta.value)
#print weights
## For clarity, fill in the quadratic form variables
A = numpy.zeros((2,2))
A[0,0] = w[0]
A.ravel()[1:3] = 0 #w[2]
A[1,1] = w[1]
bv = w[2:4]
c = w[4]
## find parameters
z, a, b, alpha = util.conic2parametric(A, bv, c)
print "XXX", z, a, b, alpha
return z, a, b, alpha