def constraints(X, pos):
n, p = X.shape
while True:
Y = np.random.standard_normal(n)
con, _, idx, sign = covtest(X, Y, sigma=1)
if idx == pos and sign == +1:
initial = Y.copy()
break
return con, initial
def test_covtest():
n, p = 30, 50
X = np.random.standard_normal((n,p)) + np.random.standard_normal(n)[:,None]
X /= X.std(0)[None,:]
Y = np.random.standard_normal(n) * 1.5
for exact, covariance in itertools.product([True, False],
[None, np.identity(n)]):
con, pval, idx, sign = covtest(X, Y, sigma=1.5, exact=exact,
covariance=covariance)
for covariance in [None, np.identity(n)]:
con, pval, idx, sign = reduced_covtest(X, Y, sigma=1.5,
covariance=covariance)
return pval
def test_covtest():
n, p = 30, 50
X = np.random.standard_normal(
(n, p)) + np.random.standard_normal(n)[:, None]
X /= X.std(0)[None, :]
Y = np.random.standard_normal(n) * 1.5
for exact, covariance in itertools.product([True, False],
[None, np.identity(n)]):
con, pval, idx, sign = covtest(X,
Y,
sigma=1.5,
exact=exact,
covariance=covariance)
for covariance in [None, np.identity(n)]:
con, pval, idx, sign = reduced_covtest(X,
Y,
sigma=1.5,
covariance=covariance)
return pval
def sample_split(X,
Y,
sigma=None,
nstep=10,
burnin=1000,
ndraw=5000,
reduced=True):
n, p = X.shape
half_n = int(n / 2)
X1, Y1 = X[:half_n, :] * 1., Y[:half_n] * 1.
X1 -= X1.mean(0)[None, :]
Y1 -= Y1.mean()
X2, Y2 = X[half_n:], Y[half_n:]
X2 -= X2.mean(0)[None, :]
Y2 -= Y2.mean()
FS_half = forward_stepwise(X1, Y1) # sample splitting model
FS_full = forward_stepwise(X.copy(), Y.copy()) # full data model
spacings_P = []
split_P = []
reduced_Pknown = []
reduced_Punknown = []
covtest_P = []
for i in range(nstep):
FS_half.next()
if FS_half.P[i] is not None:
RX = FS_half.X - FS_half.P[i](FS_half.X)
RY = FS_half.Y - FS_half.P[i](FS_half.Y)
covariance = centering(FS_half.Y.shape[0]) - np.dot(
FS_half.P[i].U, FS_half.P[i].U.T)
else:
RX = FS_half.X
RY = FS_half.Y
covariance = centering(FS_half.Y.shape[0])
RX -= RX.mean(0)[None, :]
RX /= (RX.std(0)[None, :] * np.sqrt(RX.shape[0]))
# covtest on half -- not saved
con, pval, idx, sign = covtest(RX,
RY,
sigma=sigma,
covariance=covariance,
exact=True)
# spacings on half -- not saved
eta1 = RX[:, idx] * sign
Acon = constraints(FS_half.A,
np.zeros(FS_half.A.shape[0]),
covariance=centering(FS_half.Y.shape[0]))
Acon.covariance *= sigma**2
Acon.pivot(eta1, FS_half.Y)
# sample split
eta2 = np.linalg.pinv(X2[:, FS_half.variables])[-1]
eta_sigma = np.linalg.norm(eta2) * sigma
split_P.append(2 * ndist.sf(np.fabs((eta2 * Y2).sum() / eta_sigma)))
# inference on full mu using split model, this \beta^+_s.
zero_block = np.zeros((Acon.linear_part.shape[0], (n - half_n)))
linear_part = np.hstack([Acon.linear_part, zero_block])
Fcon = constraints(linear_part, Acon.offset, covariance=centering(n))
Fcon.covariance *= sigma**2
if i > 0:
U = np.linalg.pinv(X[:, FS_half.variables[:-1]])
Uy = np.dot(U, Y)
Fcon = Fcon.conditional(U, Uy)
else:
Fcon = Fcon
eta_full = np.linalg.pinv(X[:, FS_half.variables])[-1]
if reduced:
reduced_pval = gibbs_test(Fcon,
Y,
eta_full,
ndraw=ndraw,
burnin=burnin,
sigma_known=sigma is not None,
alternative='twosided')[0]
reduced_Pknown.append(reduced_pval)
reduced_pval = gibbs_test(Fcon,
Y,
eta_full,
ndraw=ndraw,
burnin=burnin,
sigma_known=False,
alternative='twosided')[0]
reduced_Punknown.append(reduced_pval)
# now use all the data
FS_full.next()
if FS_full.P[i] is not None:
RX = X - FS_full.P[i](X)
RY = Y - FS_full.P[i](Y)
covariance = centering(RY.shape[0]) - np.dot(
FS_full.P[i].U, FS_full.P[i].U.T)
else:
RX = X
RY = Y.copy()
covariance = centering(RY.shape[0])
RX -= RX.mean(0)[None, :]
RX /= RX.std(0)[None, :]
con, pval, idx, sign = covtest(RX,
RY,
sigma=sigma,
covariance=covariance,
exact=False)
covtest_P.append(pval)
# spacings on full data
eta1 = RX[:, idx] * sign
Acon = constraints(FS_full.A, np.zeros(FS_full.A.shape[0]),
centering(RY.shape[0]))
Acon.covariance *= sigma**2
spacings_P.append(Acon.pivot(eta1, Y))
return split_P, reduced_Pknown, reduced_Punknown, spacings_P, covtest_P, FS_half.variables
def sample_split(X, Y, sigma=None,
nstep=10,
burnin=1000,
ndraw=5000,
reduced=True):
n, p = X.shape
half_n = int(n/2)
X1, Y1 = X[:half_n,:]*1., Y[:half_n]*1.
X1 -= X1.mean(0)[None,:]
Y1 -= Y1.mean()
X2, Y2 = X[half_n:], Y[half_n:]
X2 -= X2.mean(0)[None,:]
Y2 -= Y2.mean()
FS_half = forward_stepwise(X1, Y1) # sample splitting model
FS_full = forward_stepwise(X.copy(), Y.copy()) # full data model
spacings_P = []
split_P = []
reduced_Pknown = []
reduced_Punknown = []
covtest_P = []
for i in range(nstep):
FS_half.next()
if FS_half.P[i] is not None:
RX = FS_half.X - FS_half.P[i](FS_half.X)
RY = FS_half.Y - FS_half.P[i](FS_half.Y)
covariance = centering(FS_half.Y.shape[0]) - np.dot(FS_half.P[i].U, FS_half.P[i].U.T)
else:
RX = FS_half.X
RY = FS_half.Y
covariance = centering(FS_half.Y.shape[0])
RX -= RX.mean(0)[None,:]
RX /= (RX.std(0)[None,:] * np.sqrt(RX.shape[0]))
# covtest on half -- not saved
con, pval, idx, sign = covtest(RX, RY, sigma=sigma,
covariance=covariance,
exact=True)
# spacings on half -- not saved
eta1 = RX[:,idx] * sign
Acon = constraints(FS_half.A, np.zeros(FS_half.A.shape[0]),
covariance=centering(FS_half.Y.shape[0]))
Acon.covariance *= sigma**2
Acon.pivot(eta1, FS_half.Y)
# sample split
eta2 = np.linalg.pinv(X2[:,FS_half.variables])[-1]
eta_sigma = np.linalg.norm(eta2) * sigma
split_P.append(2*ndist.sf(np.fabs((eta2*Y2).sum() / eta_sigma)))
# inference on full mu using split model, this \beta^+_s.
zero_block = np.zeros((Acon.linear_part.shape[0], (n-half_n)))
linear_part = np.hstack([Acon.linear_part, zero_block])
Fcon = constraints(linear_part, Acon.offset,
covariance=centering(n))
Fcon.covariance *= sigma**2
if i > 0:
U = np.linalg.pinv(X[:,FS_half.variables[:-1]])
Uy = np.dot(U, Y)
Fcon = Fcon.conditional(U, Uy)
else:
Fcon = Fcon
eta_full = np.linalg.pinv(X[:,FS_half.variables])[-1]
if reduced:
reduced_pval = gibbs_test(Fcon, Y, eta_full,
ndraw=ndraw,
burnin=burnin,
sigma_known=sigma is not None,
alternative='twosided')[0]
reduced_Pknown.append(reduced_pval)
reduced_pval = gibbs_test(Fcon, Y, eta_full,
ndraw=ndraw,
burnin=burnin,
sigma_known=False,
alternative='twosided')[0]
reduced_Punknown.append(reduced_pval)
# now use all the data
FS_full.next()
if FS_full.P[i] is not None:
RX = X - FS_full.P[i](X)
RY = Y - FS_full.P[i](Y)
covariance = centering(RY.shape[0]) - np.dot(FS_full.P[i].U, FS_full.P[i].U.T)
else:
RX = X
RY = Y.copy()
covariance = centering(RY.shape[0])
RX -= RX.mean(0)[None,:]
RX /= RX.std(0)[None,:]
con, pval, idx, sign = covtest(RX, RY, sigma=sigma,
covariance=covariance,
exact=False)
covtest_P.append(pval)
# spacings on full data
eta1 = RX[:,idx] * sign
Acon = constraints(FS_full.A, np.zeros(FS_full.A.shape[0]),
centering(RY.shape[0]))
Acon.covariance *= sigma**2
spacings_P.append(Acon.pivot(eta1, Y))
return split_P, reduced_Pknown, reduced_Punknown, spacings_P, covtest_P, FS_half.variables
def marginal(n,
snr,
pos,
rho=0.25,
ndraw=5000,
burnin=1000,
nsim=5000,
sigma=1.):
X, mu, beta = parameters(n, rho, pos)
Psplit = []
Pselect = []
hypotheses = []
for _ in range(nsim):
Y_select = (snr * mu / np.sqrt(2) +
np.random.standard_normal(n)) * sigma
con, _, select_pos, sign = covtest(X,
Y_select,
sigma=sigma,
exact=True)
cond_ncp = snr * np.dot(X.T[select_pos], mu) / np.sqrt(2) * sign
correct = (sign == +1) and (pos == select_pos)
hypotheses.append(correct)
Y_null = sample_from_constraints(con,
Y_select,
ndraw=ndraw,
burnin=burnin)
Z_null = (np.dot(X.T[select_pos], Y_null.T) +
sigma * np.random.standard_normal(ndraw)) / np.sqrt(2)
Z_inference = sigma * (cond_ncp + np.random.standard_normal())
Z_observed = (np.dot(X.T[select_pos], Y_select) * sign +
Z_inference) / np.sqrt(2)
dfam = discrete_family(Z_null, np.ones(Z_null.shape))
Pselect.append(dfam.ccdf(0, Z_observed))
if sign == +1:
Psplit.append(ndist.sf(Z_inference / sigma))
else:
Psplit.append(ndist.cdf(Z_inference / sigma))
Ugrid = np.linspace(0, 1, 101)
Psplit = np.array(Psplit)
Pselect = np.array(Pselect)
hypotheses = np.array(hypotheses, np.bool)
# plot of marginal distribution of p-values
fig1 = plt.figure(figsize=(8, 8))
ax1 = fig1.gca()
ax1.plot(Ugrid,
ECDF(Psplit)(Ugrid),
label='Sample splitting',
c='red',
linewidth=5,
alpha=0.5)
ax1.plot(Ugrid,
ECDF(Pselect)(Ugrid),
label='Selected using $i^*(Z_S)$',
c='blue',
linewidth=5,
alpha=0.5)
ax1.set_xlabel('P-value, $p$', fontsize=20)
ax1.set_ylabel('ECDF($p$)', fontsize=20)
ax1.plot([0.05, 0.05], [0, 1], 'k--')
ax1.legend(loc='lower right')
# conditional distribution of p-values
# conditioned on selection choosing correct position and sign
fig2 = plt.figure(figsize=(8, 8))
ax2 = fig2.gca()
ax2.plot(Ugrid,
ECDF(Psplit[hypotheses])(Ugrid),
label='Sample splitting',
c='red',
linewidth=5,
alpha=0.5)
ax2.plot(Ugrid,
ECDF(Pselect[hypotheses])(Ugrid),
label='Selected using $i^*(Z_S)$',
c='blue',
linewidth=5,
alpha=0.5)
ax2.set_xlabel('P-value, $p$', fontsize=20)
ax2.set_ylabel('ECDF($p$)', fontsize=20)
ax2.plot([0.05, 0.05], [0, 1], 'k--')
ax2.legend(loc='lower right')
dbn1 = {}
dbn1['split'] = Psplit
dbn1['select'] = Pselect
dbn1['hypotheses'] = hypotheses
return fig1, fig2, dbn1
def marginal(n, snr, pos, rho=0.25, ndraw=5000,
burnin=1000, nsim=5000, sigma=1.):
X, mu, beta = parameters(n, rho, pos)
Psplit = []
Pselect = []
hypotheses = []
for _ in range(nsim):
Y_select = (snr * mu / np.sqrt(2) + np.random.standard_normal(n)) * sigma
con, _, select_pos, sign = covtest(X, Y_select, sigma=sigma, exact=True)
cond_ncp = snr * np.dot(X.T[select_pos], mu) / np.sqrt(2) * sign
correct = (sign == +1) and (pos == select_pos)
hypotheses.append(correct)
Y_null = sample_from_constraints(con, Y_select, ndraw=ndraw, burnin=burnin)
Z_null = (np.dot(X.T[select_pos], Y_null.T) + sigma * np.random.standard_normal(ndraw)) / np.sqrt(2)
Z_inference = sigma * (cond_ncp + np.random.standard_normal())
Z_observed = (np.dot(X.T[select_pos], Y_select) * sign + Z_inference) / np.sqrt(2)
dfam = discrete_family(Z_null, np.ones(Z_null.shape))
Pselect.append(dfam.ccdf(0, Z_observed))
if sign == +1:
Psplit.append(ndist.sf(Z_inference / sigma))
else:
Psplit.append(ndist.cdf(Z_inference / sigma))
Ugrid = np.linspace(0,1,101)
Psplit = np.array(Psplit)
Pselect = np.array(Pselect)
hypotheses = np.array(hypotheses, np.bool)
# plot of marginal distribution of p-values
fig1 = plt.figure(figsize=(8,8))
ax1 = fig1.gca()
ax1.plot(Ugrid, ECDF(Psplit)(Ugrid), label='Sample splitting', c='red', linewidth=5, alpha=0.5)
ax1.plot(Ugrid, ECDF(Pselect)(Ugrid), label='Selected using $i^*(Z_S)$', c='blue', linewidth=5, alpha=0.5)
ax1.set_xlabel('P-value, $p$', fontsize=20)
ax1.set_ylabel('ECDF($p$)', fontsize=20)
ax1.plot([0.05,0.05],[0,1], 'k--')
ax1.legend(loc='lower right')
# conditional distribution of p-values
# conditioned on selection choosing correct position and sign
fig2 = plt.figure(figsize=(8,8))
ax2 = fig2.gca()
ax2.plot(Ugrid, ECDF(Psplit[hypotheses])(Ugrid), label='Sample splitting', c='red', linewidth=5, alpha=0.5)
ax2.plot(Ugrid, ECDF(Pselect[hypotheses])(Ugrid), label='Selected using $i^*(Z_S)$', c='blue', linewidth=5, alpha=0.5)
ax2.set_xlabel('P-value, $p$', fontsize=20)
ax2.set_ylabel('ECDF($p$)', fontsize=20)
ax2.plot([0.05,0.05],[0,1], 'k--')
ax2.legend(loc='lower right')
dbn1 = {}
dbn1['split'] = Psplit
dbn1['select'] = Pselect
dbn1['hypotheses'] = hypotheses
return fig1, fig2, dbn1
def forward_step(X, Y, sigma=None,
nstep=5,
exact=False,
burnin=1000,
ndraw=5000):
"""
A simple implementation of forward stepwise
that uses the `reduced_covtest` iteratively
after adjusting fully for the selected variable.
This implementation is not efficient, in
that it computes more SVDs than it really has to.
Parameters
----------
X : np.float((n,p))
Y : np.float(n)
sigma : float (optional)
Noise level (not needed for reduced).
nstep : int
How many steps of forward stepwise?
exact : bool
Which version of covtest should we use?
burnin : int
How many iterations until we start
recording samples?
ndraw : int
How many samples should we return?
tests : ['reduced_known', 'covtest', 'reduced_unknown']
Which test to use? A subset of the above sequence.
"""
n, p = X.shape
FS = forward_stepwise(X, Y)
spacings_P = []
covtest_P = []
reduced_Pknown = []
reduced_Punknown = []
for i in range(nstep):
FS.next()
# covtest
if FS.P[i] is not None:
RX = X - FS.P[i](X)
RY = Y - FS.P[i](Y)
covariance = np.identity(n) - np.dot(FS.P[i].U, FS.P[i].U.T)
else:
RX = X
RY = Y
covariance = None
RX -= RX.mean(0)[None,:]
RX /= RX.std(0)[None,:]
con, pval, idx, sign = covtest(RX, RY, sigma=sigma,
covariance=covariance,
exact=exact)
covtest_P.append(pval)
# reduced
eta = RX[:,idx] * sign
Acon = constraints(FS.A, np.zeros(FS.A.shape[0]))
Acon.covariance *= sigma**2
if i > 0:
U = FS.P[-2].U.T
Uy = np.dot(U, Y)
Bcon = Acon.conditional(U, Uy)
else:
Bcon = Acon
spacings_P.append(Acon.pivot(eta, Y))
reduced_pval, _, _ = gibbs_test(Bcon, Y, eta,
ndraw=ndraw,
burnin=burnin,
sigma_known=sigma is not None,
alternative='greater')
reduced_Pknown.append(reduced_pval)
reduced_pval, _, _ = gibbs_test(Bcon, Y, eta,
ndraw=ndraw,
burnin=burnin,
sigma_known=False,
alternative='greater')
reduced_Punknown.append(reduced_pval)
return covtest_P, reduced_Pknown, reduced_Punknown, spacings_P, FS.variables
def simulation(n, snr, pos, rho=0.25, nsim=5000, sigma=1.5):
# Design, mean vector and parameter vector
X, mu, beta = parameters(n, rho, pos)
Pcov = []
Pexact = []
Pu = []
Pr = []
Pfixed = []
Pmax = []
hypotheses = []
# Set seed
np.random.seed(0)
# Max test
max_stat = np.fabs(np.dot(X.T, np.random.standard_normal(
(n, 10000)))).max(0) * sigma
max_fam = discrete_family(max_stat, np.ones(max_stat.shape))
max_fam.theta = 0
for i in range(nsim):
Y = (snr * mu + np.random.standard_normal(n)) * sigma
Z = np.dot(X.T, Y)
# did this find the correct position and sign?
correct = np.all(np.less_equal(np.fabs(Z), Z[pos]))
hypotheses.append(correct)
Pcov.append(covtest(X, Y, sigma=sigma, exact=False)[1])
Pexact.append(covtest(X, Y, sigma=sigma, exact=True)[1])
Pfixed.append(2 * ndist.sf(np.fabs(np.dot(X.T, Y))[pos] / sigma))
Pu.append(reduced_covtest(X, Y, burnin=500, ndraw=5000)[1])
Pr.append(
reduced_covtest(X, Y, burnin=500, ndraw=5000, sigma=sigma)[1])
p = max_fam.ccdf(0, np.fabs(np.dot(X.T, Y)).max())
Pmax.append(p)
Ugrid = np.linspace(0, 1, 101)
Pcov = np.array(Pcov)
Pexact = np.array(Pexact)
Pu = np.array(Pu)
Pr = np.array(Pr)
Pfixed = np.array(Pfixed)
Pmax = np.array(Pmax)
# plot of marginal distribution of p-values
fig1 = plt.figure(figsize=(8, 8))
ax1 = fig1.gca()
ax1.plot(Ugrid,
ECDF(Pcov)(Ugrid),
label='Full (exact)',
c='red',
linewidth=5,
alpha=0.5)
ax1.plot(Ugrid,
ECDF(Pexact)(Ugrid),
label='Full (asymptotic)',
c='k',
linewidth=5,
alpha=0.5)
ax1.plot(Ugrid,
ECDF(Pmax)(Ugrid),
label='Max test',
c='cyan',
linewidth=5,
alpha=0.5)
ax1.plot(Ugrid,
ECDF(Pu)(Ugrid),
label=r'Selected 1-sparse, $\sigma$ unknown',
c='blue',
linewidth=5,
alpha=0.5)
ax1.plot(Ugrid,
ECDF(Pr)(Ugrid),
label=r'Selected 1-sparse, $\sigma$ known',
c='green',
linewidth=5,
alpha=0.5)
ax1.plot(Ugrid,
ECDF(Pfixed)(Ugrid),
label=r'Fixed 1-sparse, $\sigma$ known',
c='yellow',
linewidth=5,
alpha=0.5)
ax1.set_xlabel('P-value, $p$', fontsize=20)
ax1.set_ylabel('ECDF($p$)', fontsize=20)
ax1.plot([0.05, 0.05], [0, 1], 'k--')
ax1.legend(loc='lower right')
# conditional distribution of p-values
# conditioned on selection choosing correct position and sign
fig2 = plt.figure(figsize=(8, 8))
hypotheses = np.array(hypotheses, np.bool)
ax2 = fig2.gca()
ax2.plot(Ugrid,
ECDF(Pcov[hypotheses])(Ugrid),
label='Full (exact)',
c='red',
linewidth=5,
alpha=0.5)
ax2.plot(Ugrid,
ECDF(Pexact[hypotheses])(Ugrid),
label='Full (asymptotic)',
c='k',
linewidth=5,
alpha=0.5)
ax2.plot(Ugrid,
ECDF(Pu[hypotheses])(Ugrid),
label=r'Selected 1-sparse, $\sigma$ unknown',
c='blue',
linewidth=5,
alpha=0.5)
ax2.plot(Ugrid,
ECDF(Pr[hypotheses])(Ugrid),
label=r'Selected 1-sparse, $\sigma$ known',
c='green',
linewidth=5,
alpha=0.5)
ax2.set_xlabel('P-value, $p$', fontsize=20)
ax2.set_ylabel('ECDF($p$)', fontsize=20)
ax2.plot([0.05, 0.05], [0, 1], 'k--')
ax2.legend(loc='lower right')
dbn1 = {}
dbn1['exact'] = Pexact
dbn1['covtest'] = Pcov
dbn1['unknown'] = Pu
dbn1['known'] = Pr
dbn1['fixed'] = Pfixed
dbn1['max'] = Pmax
dbn1['hypotheses'] = hypotheses
return fig1, fig2, dbn1
def simulation(n, snr, pos, rho=0.25, nsim=5000, sigma=1.5):
# Design, mean vector and parameter vector
X, mu, beta = parameters(n, rho, pos)
Pcov = []
Pexact = []
Pu = []
Pr = []
Pfixed = []
Pmax = []
hypotheses = []
# Set seed
np.random.seed(0)
# Max test
max_stat = np.fabs(np.dot(X.T, np.random.standard_normal((n, 10000)))).max(0) * sigma
max_fam = discrete_family(max_stat, np.ones(max_stat.shape))
max_fam.theta = 0
for i in range(nsim):
Y = (snr * mu + np.random.standard_normal(n)) * sigma
Z = np.dot(X.T, Y)
# did this find the correct position and sign?
correct = np.all(np.less_equal(np.fabs(Z), Z[pos]))
hypotheses.append(correct)
Pcov.append(covtest(X, Y, sigma=sigma, exact=False)[1])
Pexact.append(covtest(X, Y, sigma=sigma, exact=True)[1])
Pfixed.append(2 * ndist.sf(np.fabs(np.dot(X.T, Y))[pos] / sigma))
Pu.append(reduced_covtest(X, Y, burnin=500, ndraw=5000)[1])
Pr.append(reduced_covtest(X, Y, burnin=500, ndraw=5000, sigma=sigma)[1])
p = max_fam.ccdf(0, np.fabs(np.dot(X.T, Y)).max())
Pmax.append(p)
Ugrid = np.linspace(0,1,101)
Pcov = np.array(Pcov)
Pexact = np.array(Pexact)
Pu = np.array(Pu)
Pr = np.array(Pr)
Pfixed = np.array(Pfixed)
Pmax = np.array(Pmax)
# plot of marginal distribution of p-values
fig1 = plt.figure(figsize=(8,8))
ax1 = fig1.gca()
ax1.plot(Ugrid, ECDF(Pcov)(Ugrid), label='Full (exact)', c='red', linewidth=5, alpha=0.5)
ax1.plot(Ugrid, ECDF(Pexact)(Ugrid), label='Full (asymptotic)', c='k', linewidth=5, alpha=0.5)
ax1.plot(Ugrid, ECDF(Pmax)(Ugrid), label='Max test', c='cyan', linewidth=5, alpha=0.5)
ax1.plot(Ugrid, ECDF(Pu)(Ugrid), label=r'Selected 1-sparse, $\sigma$ unknown', c='blue', linewidth=5, alpha=0.5)
ax1.plot(Ugrid, ECDF(Pr)(Ugrid), label=r'Selected 1-sparse, $\sigma$ known', c='green', linewidth=5, alpha=0.5)
ax1.plot(Ugrid, ECDF(Pfixed)(Ugrid), label=r'Fixed 1-sparse, $\sigma$ known', c='yellow', linewidth=5, alpha=0.5)
ax1.set_xlabel('P-value, $p$', fontsize=20)
ax1.set_ylabel('ECDF($p$)', fontsize=20)
ax1.plot([0.05,0.05],[0,1], 'k--')
ax1.legend(loc='lower right')
# conditional distribution of p-values
# conditioned on selection choosing correct position and sign
fig2 = plt.figure(figsize=(8,8))
hypotheses = np.array(hypotheses, np.bool)
ax2 = fig2.gca()
ax2.plot(Ugrid, ECDF(Pcov[hypotheses])(Ugrid), label='Full (exact)', c='red', linewidth=5, alpha=0.5)
ax2.plot(Ugrid, ECDF(Pexact[hypotheses])(Ugrid), label='Full (asymptotic)', c='k', linewidth=5, alpha=0.5)
ax2.plot(Ugrid, ECDF(Pu[hypotheses])(Ugrid), label=r'Selected 1-sparse, $\sigma$ unknown', c='blue', linewidth=5, alpha=0.5)
ax2.plot(Ugrid, ECDF(Pr[hypotheses])(Ugrid), label=r'Selected 1-sparse, $\sigma$ known', c='green', linewidth=5, alpha=0.5)
ax2.set_xlabel('P-value, $p$', fontsize=20)
ax2.set_ylabel('ECDF($p$)', fontsize=20)
ax2.plot([0.05,0.05],[0,1], 'k--')
ax2.legend(loc='lower right')
dbn1 = {}
dbn1['exact'] = Pexact
dbn1['covtest'] = Pcov
dbn1['unknown'] = Pu
dbn1['known'] = Pr
dbn1['fixed'] = Pfixed
dbn1['max'] = Pmax
dbn1['hypotheses'] = hypotheses
return fig1, fig2, dbn1
def forward_step(X,
Y,
sigma=None,
nstep=5,
exact=False,
burnin=1000,
ndraw=5000):
"""
A simple implementation of forward stepwise
that uses the `reduced_covtest` iteratively
after adjusting fully for the selected variable.
This implementation is not efficient, in
that it computes more SVDs than it really has to.
Parameters
----------
X : np.float((n,p))
Y : np.float(n)
sigma : float (optional)
Noise level (not needed for reduced).
nstep : int
How many steps of forward stepwise?
exact : bool
Which version of covtest should we use?
burnin : int
How many iterations until we start
recording samples?
ndraw : int
How many samples should we return?
tests : ['reduced_known', 'covtest', 'reduced_unknown']
Which test to use? A subset of the above sequence.
"""
n, p = X.shape
FS = forward_stepwise(X, Y)
spacings_P = []
covtest_P = []
reduced_Pknown = []
reduced_Punknown = []
for i in range(nstep):
FS.next()
# covtest
if FS.P[i] is not None:
RX = X - FS.P[i](X)
RY = Y - FS.P[i](Y)
covariance = np.identity(n) - np.dot(FS.P[i].U, FS.P[i].U.T)
else:
RX = X
RY = Y
covariance = None
RX -= RX.mean(0)[None, :]
RX /= RX.std(0)[None, :]
con, pval, idx, sign = covtest(RX,
RY,
sigma=sigma,
covariance=covariance,
exact=exact)
covtest_P.append(pval)
# reduced
eta = RX[:, idx] * sign
Acon = constraints(FS.A, np.zeros(FS.A.shape[0]))
Acon.covariance *= sigma**2
if i > 0:
U = FS.P[-2].U.T
Uy = np.dot(U, Y)
Bcon = Acon.conditional(U, Uy)
else:
Bcon = Acon
spacings_P.append(Acon.pivot(eta, Y))
reduced_pval, _, _ = gibbs_test(Bcon,
Y,
eta,
ndraw=ndraw,
burnin=burnin,
sigma_known=sigma is not None,
alternative='greater')
reduced_Pknown.append(reduced_pval)
reduced_pval, _, _ = gibbs_test(Bcon,
Y,
eta,
ndraw=ndraw,
burnin=burnin,
sigma_known=False,
alternative='greater')
reduced_Punknown.append(reduced_pval)
return covtest_P, reduced_Pknown, reduced_Punknown, spacings_P, FS.variables
