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multiple_spacing_tests.py
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multiple_spacing_tests.py
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#!/usr/bin/env python3
# -*- coding: utf-8 -*-
"""
Created on Fri Apr 19 16:35:42 2019
@author: Y. DE CASTRO, yohann.decastro "at" gmail
This py file contains functions for running multiple spacing testing on the LAR's knots
"""
# %%
import numpy as np
import matplotlib.pyplot as plt
import time
import scipy as sc
import random
from scipy.special import erfc
from scipy.stats import norm, binom
# %% Generate generate data
def generate_data(n, p, k, eta, A, X=None, theta0=None, S=None):
'''
We consider the linear model: y = X @ theta0 + W where
- X is a (n x p) matrix drawn from N(0, Sigma) with Sigma_ij = eta^{|i-j|} with eta \in (0,1). Columns are normalized
- theta0 has k non-zero coefficients. The non-zero coeffs are set uniformly at random to -A or +A
- W is a standard gaussian vector
'''
if X is None:
sig = np.ones(p)
for i in range(1,p):
sig[i] = eta * sig[i-1]
Sigma = sc.linalg.circulant(sig)
Sigma = np.tril(Sigma) + np.tril(Sigma, -1).T
X = np.random.multivariate_normal(np.zeros(p), Sigma, size=n)
for j in range(p):
X[:,j] /= np.linalg.norm(X[:,j])
if theta0 is None:
S = random.sample(range(p), k)
theta0 = np.zeros(p)
theta0[S] = 2*(np.random.normal(0,1,k)>0)-1
theta0 *= A
y = X @ theta0 + np.random.normal(0,1,n)
return theta0, X, S, y
# %% Generate variables
def generate_variables(predictors_number,
sample_size,
sparsity=0,
sigma=1,
noise_correlation=0,
normalization=False,
covariance_design=0):
"""Generate a linear model
"""
p = predictors_number # % vector size
n = sample_size # % sample size
""" design
"""
if np.all(covariance_design) == 0:
X = norm.rvs(0, 1 / np.sqrt(n), [n, p])
else:
for k in range(0, n):
X[k, ] = np.random.multivariate_normal(np.zeros(p), covariance_design)
if normalization:
X = np.dot(X, np.diag(np.dot(np.ones((1, np.shape(X)[0])), X * X)[0] ** (-0.5)))
""" target vector
"""
signs_beta_0 = 2 * binom.rvs(1, 0.5, size=sparsity) - 1 # % signs of sparse target vector
beta_0 = np.concatenate([signs_beta_0, np.zeros(p - sparsity)]) # % sparse vector
""" noise vector
"""
if np.all(noise_correlation) == 0:
noise_correlation = np.identity(n)
epsilon = np.random.multivariate_normal(np.zeros(n),
(sigma ** 2) * noise_correlation) # % noise vector
""" observation
"""
Y = np.dot(X, beta_0) + epsilon
""" whithening the observation
"""
A = np.dot(np.matrix.transpose(X),
np.linalg.inv(noise_correlation))
R_bar = np.dot(A, X)
Z = np.dot(A, Y)
Z = np.concatenate([Z, -Z])
R = np.block(
[[R_bar, -R_bar],
[-R_bar, R_bar]
])
return [Z, R, sigma, X, noise_correlation, sparsity, beta_0, Y]
# %% Empirical Laws
def empirical_cdf(x):
return (1 + np.argsort(np.argsort(x))) / len(x)
def empirical_law(predictors_number,
sample_size,
start,
end,
middle,
iterations=int(1e2),
display=0,
sparsity=0,
sigma=1,
noise_correlation=0,
normalization=False,
covariance_design=0,
eval_points=9973): # 99991
rep = 1
result = []
for i in range(0, iterations):
Z, R, __, __, __, __, __, __ = generate_variables(predictors_number,
sample_size,
sparsity,
sigma,
noise_correlation,
normalization,
covariance_design)
T = np.zeros(np.size(Z))
lambdas = np.zeros(sample_size)
indexes = np.zeros(sample_size)
correls = np.zeros(sample_size)
# irrepresentable_check = np.zeros(sample_size)
var_R = R
var_Z = Z
for k in range(min(sample_size, predictors_number, np.linalg.matrix_rank(R, hermitian=True))):
var_R, var_Z, T, lambdas[k], indexes[k], correls[k] = rec(var_R, var_Z, T)
# irrepresentable_check[k] = bool(np.amax(np.abs(T))<1)
# order_irrep = sum(np.cumprod(irrepresentable_check))
lars = [lambdas, indexes, correls]
alpha, __, __ = observed_significance_CBC(lars,
sigma,
start,
end,
middle,
eval_points, # 99991,
rep)
result += [alpha]
if display:
print("CBC: Significance of knot %s between knot %s and knot %s over %s iterations" % (
middle, start, end, iterations))
lispace_size = 2000
x = np.linspace(0, 1, lispace_size)
a = np.asarray(result)
z = np.arange(0, 1, 1. / len(a))
t = np.sort(a)
# sns.set_style("white")
plt.ylim(0, 1)
plt.xlim(0, 1)
plt.step(t, z, color="lightcoral", lw=3)
plt.plot(x, x, color="yellowgreen", linestyle="--")
plt.legend([r'CDF of $\widehat\alpha_{%s,%s,%s}$' % (start, middle, end)], loc=2)
plt.title("$p=$%s, $n=$%s over %s iterations" % (predictors_number, sample_size, iterations))
plt.tight_layout()
plt.savefig('law.png', format='png', dpi=600)
plt.show()
return result
def empirical_law_H0_H1(predictors_number,
sample_size,
start,
end,
middle,
iterations=int(1e2),
display=1,
sparsity=1,
sigma=1,
noise_correlation=0,
normalization=True,
covariance_design=0,
eval_points=9973): # 99991,
result0 = []
result1 = []
rep = 1
for i in range(0, iterations):
Z, R, __, __, __, __, __, __ = generate_variables(predictors_number,
sample_size,
sparsity=0, # sparsity = 0
sigma=sigma,
noise_correlation=noise_correlation,
normalization=normalization,
covariance_design=covariance_design)
T = np.zeros(np.size(Z))
lambdas = np.zeros(sample_size)
indexes = np.zeros(sample_size)
correls = np.zeros(sample_size)
# irrepresentable_check = np.zeros(sample_size)
var_R = R
var_Z = Z
for k in range(min(sample_size, predictors_number, np.linalg.matrix_rank(R, hermitian=True))):
var_R, var_Z, T, lambdas[k], indexes[k], correls[k] = rec(var_R, var_Z, T)
# irrepresentable_check[k] = bool(np.amax(np.abs(T))<1)
# order_irrep = sum(np.cumprod(irrepresentable_check))
lars = [lambdas, indexes, correls]
alpha0, __, __ = observed_significance_CBC(lars,
sigma,
start,
end,
middle,
eval_points, # 99991,
rep)
result0 += [alpha0]
for i in range(0, iterations):
Z, R, __, __, __, __, __, __ = generate_variables(predictors_number,
sample_size,
sparsity=sparsity, # sparsity = 0
sigma=sigma,
noise_correlation=noise_correlation,
normalization=normalization,
covariance_design=covariance_design)
T = np.zeros(np.size(Z))
lambdas = np.zeros(sample_size)
indexes = np.zeros(sample_size)
correls = np.zeros(sample_size)
# irrepresentable_check = np.zeros(sample_size)
var_R = R
var_Z = Z
for k in range(min(sample_size, predictors_number, np.linalg.matrix_rank(R, hermitian=True))):
var_R, var_Z, T, lambdas[k], indexes[k], correls[k] = rec(var_R, var_Z, T)
# irrepresentable_check[k] = bool(np.amax(np.abs(T))<1)
# order_irrep = sum(np.cumprod(irrepresentable_check))
lars = [lambdas, indexes, correls]
alpha1, __, __ = observed_significance_CBC(lars,
sigma,
start,
end,
middle,
eval_points, # 99991,
rep)
result1 += [alpha1]
if display:
print("CBC: Significance of knot %s between knot %s and knot %s over %s iterations" % (
middle, start, end, iterations))
lispace_size = 2000
x = np.linspace(0, 1, lispace_size)
a1 = np.asarray(result0)
z1 = np.arange(0, 1, 1. / len(a1))
t1 = np.sort(a1)
a2 = np.asarray(result1)
z2 = np.arange(0, 1, 1. / len(a2))
t2 = np.sort(a2)
# sns.set_style("white")
plt.ylim(0, 1)
plt.xlim(0, 1)
plt.step(t1, z1, color="royalblue", lw=3)
plt.step(t2, z2, color="lightcoral", lw=3)
plt.plot(x, x, color="yellowgreen", linestyle="--")
plt.legend([r'CDF of $\widehat\alpha_{%s,%s,%s}$ under $H_0$' % (start, middle, end),
r'CDF of $\widehat\alpha_{%s,%s,%s}$ under $H_1$' % (start, middle, end)],
loc=4)
# plt.title ("$p=$%s, $n=$%s over %s iterations" %(predictors_number,sample_size, iterations))
plt.tight_layout()
plt.savefig('law_a-%s_b-%s_c-%s_sparisty-%s_CBC-%s_on_%s_at_%s.png' % (
start, middle, end, sparsity, eval_points, time.strftime("%d_%m_%Y"), time.strftime("%H_%M_%S")),
format='png',
dpi=600)
plt.show()
return [result0, result1]
# %% Figure 1 and 2
def empirical_law_first_knots(predictors_number: int,
sample_size: int,
end: int,
start=0,
iterations=int(5e2),
display=0,
sparsity=0,
sigma=1,
noise_correlation=0,
normalization=False,
covariance_design=0,
eval_points=9973):
spacing = end - start
result = []
linspace_size = 2000
x = np.linspace(0, 1, linspace_size)
if display:
plt.close('all')
plt.figure(figsize=(8, 6))
for l1 in range(start, end - 1):
for l2 in range(l1 + 1, end):
values = empirical_law(predictors_number,
sample_size,
start=l1,
end=end,
middle=l2,
iterations=int(iterations),
display=0,
sparsity=int(sparsity),
sigma=sigma,
noise_correlation=noise_correlation,
normalization=normalization,
covariance_design=covariance_design,
eval_points=int(eval_points))
result += [values, "$\lambda_{%s}$, $\lambda_{%s}$, $\lambda_{%s}$" % (l1, l2, end)]
if display:
plt.subplot(spacing - 1, spacing - 1, (l1 - start) * (spacing - 1) + (l2 - start))
a = np.asarray(values)
z = np.arange(0, 1, 1. / len(a))
t = np.sort(a)
plt.ylim(0, 1)
plt.xlim(0, 1)
plt.xticks([0, 0.5, 1], fontsize='x-small')
plt.yticks([0, 1], fontsize='x-small')
plt.step(t, z, color="lightcoral", lw=2)
plt.plot(x, x, color="yellowgreen", linestyle="--")
plt.legend([r'CDF of $\widehat\alpha_{%s,%s,%s}$' % (l1, l2, end)], fontsize='small', loc=2)
if display:
plt.tight_layout()
plt.savefig('laws.png', format='png', dpi=600)
plt.show()
return result
# %%
# % Recursive formulation of the LAR + computation of the correlations rho
def rec(R, Z, T):
regressed_process = (T < 1) * (Z / (1 - (T != 1) * T))
val_lambda = np.amax(regressed_process)
signed_index = np.argmax(regressed_process)
p = int(np.size(Z) / 2)
val_index = int(signed_index % p)
# % safe division
def safe_div(x, y):
if y == 0:
return 0
return x / y
# % parameters
val_rho = safe_div(np.sqrt(R[signed_index, signed_index]),
1 - T[signed_index])
x_return = R[signed_index, ] / R[signed_index, signed_index]
R_return = R - np.dot(x_return.reshape(np.size(x_return), 1),
np.asarray(R[signed_index]).reshape(1, np.size(x_return)))
# % set to 0 the variance-covariance of the chosen index
R_return[val_index, ] = np.zeros(2 * p)
R_return[:, val_index] = np.zeros(2 * p)
R_return[val_index + p, ] = np.zeros(2 * p)
R_return[:, val_index + p] = np.zeros(2 * p)
# % residuals
Z_return = Z - Z[signed_index] * x_return
Z_return[val_index] = 0
Z_return[val_index + p] = 0
# % parameter theta: regressed coefficient (<1 implies Irrepresentability)
T_return = T + (1 - T[signed_index]) * x_return
T_return[val_index] = 0
T_return[val_index + p] = 0
return [R_return, Z_return, T_return, val_lambda, val_index, val_rho]
def lar_rec(X, Y,
kmax=0,
normalization=False,
noise_correlation=0):
sample_size, predictors_number = np.shape(X)
if normalization:
X = np.dot(X, np.diag(np.dot(np.ones((1, np.shape(X)[0])), X * X)[0] ** (-0.5)))
if np.all(noise_correlation) == 0:
noise_correlation = np.identity(sample_size)
A = np.dot(np.matrix.transpose(X),
np.linalg.inv(noise_correlation))
R_bar = np.dot(A, X)
Z = np.dot(A, Y)
Z = np.concatenate([Z, -Z])
R = np.block(
[[R_bar, -R_bar],
[-R_bar, R_bar]
])
if kmax == 0:
kmax = min(sample_size, predictors_number, np.linalg.matrix_rank(R, hermitian=True))
T = np.zeros(np.size(Z))
lambdas = np.zeros(kmax)
indexes = np.zeros(kmax)
correls = np.zeros(kmax)
var_R = R
var_Z = Z
for k in range(kmax):
var_R, var_Z, T, lambdas[k], indexes[k], correls[k] = rec(var_R, var_Z, T)
return [lambdas, indexes, correls, R, Z]
# %%
# % Compute the Empirical Irrepresentable Check
def get_order(var_Z, var_R):
T = np.zeros(np.size(var_Z))
irrepresentable = True
order_irrep = 0
while irrepresentable:
var_R, var_Z, T, __, __, __ = rec(var_R, var_Z, T)
irrepresentable = bool(np.amax(np.abs(T)) < 1)
order_irrep += 1
return order_irrep
# % Compute the Empirical Irrepresentable Check
def get_residual(var_R, var_Z, k):
T = np.zeros(np.size(var_Z))
for n in range(k):
var_R, var_Z, T, __, __, __ = rec(var_R, var_Z, T)
return [var_R, var_Z]
# % Compute the esitmation of the variance of the residuals
def get_variance(R, Z, t):
global R2
R1, Z1 = get_residual(R, Z, t)
w, v = np.linalg.eig(R1)
w = np.real(w)
# We take R2 = R1^(-1/2)
w2 = np.zeros(np.shape(w))
for k in range(np.size(w)):
if abs(w[k]) > 1e-8:
w2[k] = abs(w[k]) ** (-0.5)
R2 = np.real(np.dot(v, np.dot(np.diag(w2), np.transpose(v))))
Z2 = np.dot(R2, Z1)
# We compute the variance estimation
d = np.linalg.matrix_rank(R2, hermitian=True)
V2 = v[:, 0:d]
Y2 = np.real(np.dot(np.transpose(V2), Z2))
var = (np.sum(Y2 ** 2) - d * ((np.sum(Y2) / d) ** 2)) / (d - 1)
return var
# %%
def observed_significance_spacing(lars, sigma, start):
lambdas, indexes, correls = lars
middle = start + 1
end = middle + 1
if start != 0:
lambda_a = lambdas[start - 1]
else:
lambda_a = np.inf
lambda_b = lambdas[middle - 1]
rho_b = correls[middle - 1]
lambda_c = lambdas[end - 1]
num2 = (erfc(lambda_b / (np.sqrt(2) * sigma * rho_b)) - erfc(lambda_a / (np.sqrt(2) * sigma * rho_b)))
den2 = (erfc(lambda_c / (np.sqrt(2) * sigma * rho_b)) - erfc(lambda_a / (np.sqrt(2) * sigma * rho_b)))
hat_alpha = num2 / den2
return hat_alpha
# %% MCQMC Fast-CBC method
# % closest prime less than n
def prime(n):
test = n
while not isPrime(test):
test -= 1
return int(test)
# % prime test
def isPrime(n):
for i in range(2, int(n ** 0.5) + 1):
if n % i == 0:
return False
return True
# % vector generating the lattice for cubature
def fast_rank_1(n,
s_max,
omega=lambda x: x ** 2 - x + 1 / 6,
gamma=0.9,
print_CBC=False):
# % n has to be prime
if isPrime(n):
n_prime = n
else:
n_prime = int(prime(n))
if print_CBC:
print("Warning: number of points is not a prime number, changed to n=%s points" % n_prime)
# % generating vector
z = np.zeros(s_max)
# e2 = np.zeros(s_max)
m = int((n_prime - 1) / 2)
q = np.ones(m)
q0 = 1
gamma_vec = np.cumprod(gamma * np.ones(s_max))
beta_vec = 1 + gamma_vec / 3
# cumbeta = np.cumprod(beta_vec)
g = np.random.randint(2, n_prime)
perm = np.zeros(m)
temp = 1
for j in range(m):
perm[j] = int(temp)
temp = temp * g % n_prime
perm = np.minimum(perm, n_prime - perm)
psi = omega(perm / n_prime)
psi0 = omega(0)
fft_psi = np.fft.fft(psi)
for s in range(s_max):
E2 = np.fft.ifft(fft_psi * np.fft.fft(q))
E2 = np.real(E2)
w = np.argmin(E2)
# min_E2 = E2[w]
if s == 0:
w = 1
# noise=np.abs(E2[0]-min_E2)
z[s] = perm[w]
# e2[s] = -cumbeta[s]+(beta_vec[s]*(q0+2*np.sum(q))+gamma_vec[s]*(psi0*q0+2*min_E2))/n_prime
ppsi = np.concatenate((psi[w::-1], psi[:w:-1]), axis=0)
q = (beta_vec[s] + gamma_vec[s] * ppsi) * q
q0 = (beta_vec[s] + gamma_vec[s] * psi0) * q0
if print_CBC:
print("s=%s, z=%s, w=%s" % (s, z[s], w))
return [z, n_prime]
def observed_significance_CBC(lars,
sigma,
start,
end,
middle=-1,
eval_points=9973, # 99991,
rep=20): # rep is used to repeat the computations rep times to estimate the precision
# error
# % calculus is different if middle = start+1 (int_the_middle=False in this case)
lambdas, indexes, correls = lars
lambdas = lambdas / sigma
sigma = 1
restarts = 1
normal_cut = 4
in_the_middle = True
if middle == -1:
middle = start + 1
if middle == start + 1:
in_the_middle = False
lambda_a = lambdas[int(start - 1)]
if start == 0:
lambda_a = np.max([normal_cut * np.log(len(lambdas)) * sigma * correls[0],
2 * normal_cut * lambdas[0]]) # should be np.inf but this technique only works on hypercubes
lambda_b = lambdas[middle - 1]
lambda_c = lambdas[end - 1]
variables = []
for k in range(start, end - 1):
var = "%s" % k
variables += [var]
s_max = len(variables)
def stat_pdf(*args):
args_ordered = np.asarray(sorted(args, reverse=True))
temp = 1
for k in range(s_max):
temp *= np.exp((lambda_c ** 2 - (args_ordered[k] ** 2)) / (2 * (sigma * correls[int(variables[k])] ** 2)))
return temp
def stat_pdf_with_middle(*args):
args_ordered = np.asarray(sorted(args, reverse=True))
temp = 1
for k in range(s_max):
if (k == middle - start - 1) & (args_ordered[k] > lambda_b):
return 0
temp *= np.exp((lambda_c ** 2 - (args_ordered[k] ** 2)) / (2 * (sigma * correls[int(variables[k])] ** 2)))
return temp
alpha = []
for t in range(restarts):
[z, n_prime] = fast_rank_1(eval_points, s_max)
if not in_the_middle:
for r in range(rep):
# z_current = z
# if r!=0:
z_current = z + np.random.uniform(0, 1, size=s_max) % 1
def stat_cdf(t):
values = []
for k in range(n_prime):
current_point = np.asarray(lambda_c + (t - lambda_c) * (k * z_current / n_prime % 1))
values += [stat_pdf(*current_point)]
return ((t - lambda_c) ** s_max) * np.mean(np.asarray(values))
stat1 = stat_cdf(lambda_b)
stat2 = stat_cdf(lambda_a)
alpha += [1 - stat1 / stat2]
else:
for r in range(rep):
z_current = z
if r != 0:
z_current = z + np.sqrt(n_prime) * np.random.uniform(-1, 1, size=s_max)
values = []
values_normaliszation = []
for k in range(n_prime):
current_point = np.asarray(lambda_c + (lambda_a - lambda_c) * (k * z_current / n_prime % 1))
values += [stat_pdf_with_middle(*current_point)]
values_normaliszation += [stat_pdf(*current_point)]
stat1 = ((lambda_a - lambda_c) ** s_max) * np.mean(np.asarray(values))
stat2 = ((lambda_a - lambda_c) ** s_max) * np.mean(np.asarray(values_normaliszation))
alpha += [1 - stat1 / stat2]
return np.median(alpha), 4 * np.std(alpha), np.mean(alpha)
# %%
def stacked_bar(data, series_labels, category_labels=None,
show_values=False, value_format="{}", y_label=None,
grid=True, reverse=False):
"""Plots a stacked bar chart with the data and labels provided.
Keyword arguments:
data -- 2-dimensional numpy array or nested list
containing data for each series in rows
series_labels -- list of series labels (these appear in
the legend)
category_labels -- list of category labels (these appear
on the x-axis)
show_values -- If True then numeric value labels will
be shown on each bar
value_format -- Format string for numeric value labels
(default is "{}")
y_label -- Label for y-axis (str)
grid -- If True display grid
reverse -- If True reverse the order that the
series are displayed (left-to-right
or right-to-left)
"""
ny = len(data[0])
ind = list(range(ny))
axes = []
cum_size = np.zeros(ny)
data = np.array(data)
if reverse:
data = np.flip(data, axis=1)
category_labels = reversed(category_labels)
for i, row_data in enumerate(data):
axes.append(plt.bar(ind, row_data, bottom=cum_size,
label=series_labels[i]))
cum_size += row_data
if category_labels:
plt.xticks(ind, category_labels)
if y_label:
plt.ylabel(y_label)
plt.legend()
if grid:
plt.grid()
if show_values:
for axis in axes:
for bar in axis:
w, h = bar.get_width(), bar.get_height()
plt.text(bar.get_x() + w / 2, bar.get_y() + h / 2,
value_format.format(h), ha="center",
va="center")