def minvar_nls_oracle(sim, isotonic=False, trace=False, upper_bound=True):
"""Oracle eigenvalues for new MinVar nonlinear shrinkage"""
T, N = sim.shape
U, Sigma = sim.U, sim.Sigma
lam = sim.lam
alpha = U.T.dot(np.ones(N))
C = U.T.dot(Sigma).dot(U)
if isotonic:
# Solve non-linear problem with monotonicity constraints
d_min, d_max = lam[-1], lam[0]
_, d_kfold = nls_kfold(sim)
d_isokfold = isotonic_regression(d_kfold)
if trace:
trace = np.sum(d_isokfold)
d = minvar_nls_nlsq_multi_transformed([C], [alpha], trace,
d_isokfold, d_min, d_max,
upper_bound)
else:
trace = None
d = minvar_nls_nlsq_multi([C], [alpha], trace, d_isokfold, d_min,
d_max, upper_bound)
else:
# Solve plain vanilla linear system
A = C * alpha
z = np.linalg.solve(A, alpha)
d = 1. / z
return d
def nls_oracle(sim, isotonic=False, **kwargs):
"""Oracle eigenvalues for LW nonlinear shrinkage"""
T, N = sim.shape
U = sim.U
Sig = sim.Sigma
d = np.zeros(N)
for i in range(N):
u_i = U[:, i]
d[i] = u_i.T.dot(Sig).dot(u_i)
d_min, d_max = sim.lam_N, sim.lam_1
return isotonic_regression(d, y_min=d_min, y_max=d_max) if isotonic else d
def minvar_nls_kfold_oracle(sim,
K=10,
progress=False,
trace=False,
upper_bound=True):
"""
Oracle/K-fold cross-validated eigenvalues for new MinVar nonlinea shrinkage.
"""
T, N = sim.shape
S, Sigma = sim.S, sim.Sigma
X = sim.X
lam = sim.lam
m = int(T / K)
C_list = []
alpha_list = []
if progress:
pbar = tqdm(total=K)
for k in range(K):
k_set = list(range(k * m, (k + 1) * m))
X_k = X[k_set, :]
S_k = (T * S - X_k.T.dot(X_k)) / (T - m)
_, U_k = eig(S_k)
C = U_k.T.dot(Sigma).dot(U_k)
C_list.append(C)
alpha = U_k.T.dot(np.ones(N))
alpha_list.append(alpha)
if progress:
pbar.update()
d_min, d_max = lam[-1], lam[0]
d_kfold = nls_kfold(sim, K)
d_isokfold = isotonic_regression(d_kfold)
if trace:
trace = np.sum(d_isokfold)
d = minvar_nls_nlsq_multi_transformed(C_list, alpha_list, trace,
d_isokfold, d_min, d_max,
upper_bound)
else:
trace = None
d = minvar_nls_nlsq_multi(C_list, alpha_list, trace, d_isokfold, d_min,
d_max, upper_bound)
return d
def minvar_joint_kfold_isotonic(sim,
K,
smoothing='average',
nonnegative=False,
regularization=None):
"""
Base Estimator 4: MinVar $K$-Fold Joint Cross-Validation with Isotonic
Regression
Parameters:
+ K
Variants:
+ Smoothing could be average or median.
+ Nonnegativity constraint
+ Regularization: 'l2' for now, maybe others
"""
T, N = sim.shape
m = int(T / K)
X, S = sim.X, sim.S
P = np.zeros((N, N))
q = np.zeros(N)
for k in range(K):
k_set = list(range(k * m, (k + 1) * m))
_k = np.delete(range(T), k_set)
X_k = X[k_set, :]
S_k = (T * S - X_k.T @ X_k) / (T - m) # 1/(T-m) * X[_k,:].T @ X[_k,:]
_, U_k = eig(S_k)
alpha_k = U_k.T @ np.ones(N)
C_k = U_k.T @ (1 / m * X_k.T @ X_k) @ U_k
A_k = np.diag(alpha_k)
P = P + (A_k @ C_k.T @ C_k @ A_k)
q = q + (A_k @ C_k.T @ alpha_k)
if nonnegative:
z = nnlsq_regularized(P, -q, lmbda=0)
interpolate_zeros(z)
else:
z = np.linalg.solve(P, -q)
d = 1 / z
d = isotonic_regression(d)
return d
def minvar_nls_kfold(sim, K, progress=False, trace=False, upper_bound=True):
"""K-fold cross-validated eigenvalues for new MinVar nonlinear shrinkage"""
T, N = sim.shape
m = int(T / K)
X, S, lam = sim.X, sim.S, sim.lam
C_list = []
alpha_list = []
if progress:
pbar = tqdm(total=K)
for k in range(K):
k_set = list(range(k * m, (k + 1) * m))
X_k = X[k_set, :]
S_k = (T * S - X_k.T.dot(X_k)) / (T - m)
_, U_k = eig(S_k)
# this is a joint version. Lists of C and alpha are collected
# and the system is solved for one z vector. z is not averaged here
# Note that this is also a bona-fide estimator since in C calculation
# sample covariance matrix is used.
C = U_k.T.dot(X_k.T.dot(X_k)).dot(U_k)
C_list.append(C)
alpha = U_k.T.dot(np.ones(N))
alpha_list.append(alpha)
if progress:
pbar.update()
d_min, d_max = lam[-1], lam[0]
d_kfold = nls_kfold(sim, K)
d_isokfold = isotonic_regression(d_kfold)
if trace:
trace = np.sum(d_isokfold)
d = minvar_nls_nlsq_multi_transformed(C_list, alpha_list, trace,
d_isokfold, d_min, d_max,
upper_bound)
else:
trace = None
d = minvar_nls_nlsq_multi(C_list, alpha_list, trace, d_isokfold, d_min,
d_max, upper_bound)
return d
def nls_kfold(sim, K=10, isotonic=False, **kwargs):
"""K-fold cross-validated eigenvalues for LW nonlinear shrinkage"""
d = _nls_cv(sim, K)
d_min, d_max = sim.lam_N, sim.lam_1
return isotonic_regression(d, y_min=d_min, y_max=d_max) if isotonic else d
def nls_asymptotic(sim, isotonic=False):
X = sim.X
S_lw = nlshrink_covariance(X, centered=True)
d = eig(S_lw, return_eigenvectors=False)
return isotonic_regression(d) if isotonic else d