def test_glmmexpfam_qs_none():
nsamples = 10
random = RandomState(0)
X = random.randn(nsamples, 5)
K = linear_eye_cov().value()
z = random.multivariate_normal(0.2 * ones(nsamples), K)
ntri = random.randint(1, 30, nsamples)
nsuc = zeros(nsamples, dtype=int)
for (i, ni) in enumerate(ntri):
nsuc[i] += sum(z[i] + 0.2 * random.randn(ni) > 0)
ntri = ascontiguousarray(ntri)
glmm = GLMMExpFam(nsuc, ("binomial", ntri), X, None)
assert_allclose(glmm.lml(), -38.30173374439622, atol=ATOL, rtol=RTOL)
glmm.fix("beta")
glmm.fix("scale")
glmm.fit(verbose=False)
assert_allclose(glmm.lml(), -32.03927471370041, atol=ATOL, rtol=RTOL)
glmm.unfix("beta")
glmm.unfix("scale")
glmm.fit(verbose=False)
assert_allclose(glmm.lml(), -19.575736561760586, atol=ATOL, rtol=RTOL)
def cmc(g, xdists, u_to_x, T, seed, maxitr):
"""
Crude Monte Carlo simulation.
"""
# Seed the random number generator if required
if seed == -1:
prng = RandomState()
else:
prng = RandomState(seed)
# Generate standard normal samples centered at the origin
u0 = zeros(len(xdists))
covmat = eye(len(xdists))
u = prng.multivariate_normal(u0, covmat, size=maxitr).T
g_mc = g(u_to_x(u, xdists, T))
# Convert g-function output to pass/fail indicator function and estimate pf
g_mc[g_mc>0] = 0
g_mc[g_mc<0] = 1
mu_pf = g_mc.mean()
beta = -norm.ppf(mu_pf) if mu_pf < 0.5 else norm.ppf(mu_pf)
# Convergence metrics (standard deviation, standard error, CoV of s.e.)
std_pf = g_mc.std(ddof=1) # Calculate sample standard deviation
se_pf = std_pf/sqrt(maxitr)
cv_pf = se_pf/mu_pf
return {'vars': xdists, 'beta': beta, 'Pf': mu_pf, 'stderr': se_pf,
'stdcv': cv_pf}
def test_glmmexpfam_optimize():
nsamples = 10
random = RandomState(0)
X = random.randn(nsamples, 5)
K = linear_eye_cov().value()
z = random.multivariate_normal(0.2 * ones(nsamples), K)
QS = economic_qs(K)
ntri = random.randint(1, 30, nsamples)
nsuc = zeros(nsamples, dtype=int)
for (i, ni) in enumerate(ntri):
nsuc[i] += sum(z[i] + 0.2 * random.randn(ni) > 0)
ntri = ascontiguousarray(ntri)
glmm = GLMMExpFam(nsuc, ("binomial", ntri), X, QS)
assert_allclose(glmm.lml(), -29.102168129099287, atol=ATOL, rtol=RTOL)
glmm.fix("beta")
glmm.fix("scale")
glmm.fit(verbose=False)
assert_allclose(glmm.lml(), -27.635788105778012, atol=ATOL, rtol=RTOL)
glmm.unfix("beta")
glmm.unfix("scale")
glmm.fit(verbose=False)
assert_allclose(glmm.lml(), -19.68486269551159, atol=ATOL, rtol=RTOL)
def test_glmmexpfam_predict():
random = RandomState(4)
n = 100
p = n + 1
X = ones((n, 2))
X[:, 1] = random.randn(n)
G = random.randn(n, p)
G /= G.std(0)
G -= G.mean(0)
G /= sqrt(p)
K = dot(G, G.T)
i = asarray(arange(0, n), int)
si = random.choice(i, n, replace=False)
ntest = int(n // 5)
itrain = si[:-ntest]
itest = si[-ntest:]
Xtrain = X[itrain, :]
Ktrain = K[itrain, :][:, itrain]
Xtest = X[itest, :]
beta = random.randn(2)
z = random.multivariate_normal(dot(X, beta), 0.9 * K + 0.1 * eye(n))
ntri = random.randint(1, 100, n)
nsuc = zeros(n, dtype=int)
for (i, ni) in enumerate(ntri):
nsuc[i] += sum(z[i] + 0.2 * random.randn(ni) > 0)
ntri = ascontiguousarray(ntri)
QStrain = economic_qs(Ktrain)
nsuc_train = ascontiguousarray(nsuc[itrain])
ntri_train = ascontiguousarray(ntri[itrain])
nsuc_test = ascontiguousarray(nsuc[itest])
ntri_test = ascontiguousarray(ntri[itest])
glmm = GLMMExpFam(nsuc_train, ("binomial", ntri_train), Xtrain, QStrain)
glmm.fit(verbose=False)
ks = K[itest, :][:, itrain]
kss = asarray([K[i, i] for i in itest])
pm = glmm.predictive_mean(Xtest, ks, kss)
pk = glmm.predictive_covariance(Xtest, ks, kss)
r = nsuc_test / ntri_test
assert_(corrcoef([pm, r])[0, 1] > 0.8)
assert_allclose(pk[0], 54.263705682514846)
def test_glmmexpfam_poisson():
from numpy import ones, stack, exp, zeros
from numpy.random import RandomState
from numpy_sugar.linalg import economic_qs
from pandas import DataFrame
random = RandomState(1)
# sample size
n = 30
# covariates
offset = ones(n) * random.randn()
age = random.randint(16, 75, n)
M = stack((offset, age), axis=1)
M = DataFrame(stack([offset, age], axis=1), columns=["offset", "age"])
M["sample"] = [f"sample{i}" for i in range(n)]
M = M.set_index("sample")
# genetic variants
G = random.randn(n, 4)
# sampling the phenotype
alpha = random.randn(2)
beta = random.randn(4)
eps = random.randn(n)
y = M @ alpha + G @ beta + eps
# Whole genotype of each sample.
X = random.randn(n, 50)
# Estimate a kinship relationship between samples.
X_ = (X - X.mean(0)) / X.std(0) / sqrt(X.shape[1])
K = X_ @ X_.T + eye(n) * 0.1
# Update the phenotype
y += random.multivariate_normal(zeros(n), K)
y = (y - y.mean()) / y.std()
z = y.copy()
y = random.poisson(exp(z))
M = M - M.mean(0)
QS = economic_qs(K)
glmm = GLMMExpFam(y, "poisson", M, QS)
assert_allclose(glmm.lml(), -52.479557279193585)
glmm.fit(verbose=False)
assert_allclose(glmm.lml(), -34.09720756737648)
def mutate_transformed(
self, sample_transformed: list, *,
rng: RandomState,
relscale: t.Union[float, t.Iterable[float]],
) -> list:
if not isinstance(relscale, t.Iterable):
relscale = [relscale] * self.n_dims
cov = np.diag(relscale)
retries = int(20 * np.sqrt(self.n_dims))
for _ in range(retries):
mut_sample = rng.multivariate_normal(sample_transformed, cov)
if self.is_valid_transformed(mut_sample):
return mut_sample
cov *= 0.9 # make feasibility more likely
raise RuntimeError(
f"mutation failed to produce values within bounds"
f"\n last mut_sample = {mut_sample}"
f"\n input sample = {sample_transformed}")
def test_glmmexpfam_poisson():
random = RandomState(1)
# sample size
n = 30
# covariates
offset = ones(n) * random.randn()
age = random.randint(16, 75, n)
M = stack((offset, age), axis=1)
# genetic variants
G = random.randn(n, 4)
# sampling the phenotype
alpha = random.randn(2)
beta = random.randn(4)
eps = random.randn(n)
y = M @ alpha + G @ beta + eps
# Whole genotype of each sample.
X = random.randn(n, 50)
# Estimate a kinship relationship between samples.
X_ = (X - X.mean(0)) / X.std(0) / sqrt(X.shape[1])
K = X_ @ X_.T + eye(n) * 0.1
# Update the phenotype
y += random.multivariate_normal(zeros(n), K)
y = (y - y.mean()) / y.std()
z = y.copy()
y = random.poisson(exp(z))
M = M - M.mean(0)
QS = economic_qs(K)
glmm = GLMMExpFam(y, "poisson", M, QS)
assert_allclose(glmm.lml(), -52.479557279193585)
glmm.fit(verbose=False)
assert_allclose(glmm.lml(), -34.09720756737648)
def ismc(g, xdists, u_to_x, T, seed, maxitr, tol, ftol, eps):
"""
Importance sampling Monte Carlo.
"""
# Seed the random number generator if required
if seed == -1:
prng = RandomState()
else:
prng = RandomState(seed)
# Use FORM to get estimate of u*
u_beta = slsqp(g, xdists, u_to_x, T, maxitr, tol, ftol, eps)['u_beta']
# Generate standard normal samples centred at u*
covmat = eye(len(xdists))
v = prng.multivariate_normal(u_beta, covmat, size=maxitr).T
g_mc = g(u_to_x(v, xdists, T))
# Define importance sampling functions weighting functions
fv = multivariate_normal(zeros(len(xdists)), covmat).pdf
hv = multivariate_normal(u_beta, covmat).pdf
# Convert g-function output to pass/fail indicator function and estimate pf
g_mc[g_mc>0] = 0
g_mc[g_mc<0] = 1
indfunc = g_mc * fv(v.T)/hv(v.T)
mu_pf = indfunc.mean()
beta = -norm.ppf(mu_pf) if mu_pf < 0.5 else norm.ppf(mu_pf)
# Convergence metrics (standard deviation, standard error, CoV of s.e.)
std_pf = indfunc.std(ddof=1) # Calculate sample standard deviation
se_pf = std_pf/sqrt(maxitr)
cv_pf = se_pf/mu_pf
return {'vars': xdists, 'beta': beta, 'Pf': mu_pf, 'stderr': se_pf,
'stdcv': cv_pf}
def test_glmmexpfam_copy():
nsamples = 10
random = RandomState(0)
X = random.randn(nsamples, 5)
K = linear_eye_cov().value()
z = random.multivariate_normal(0.2 * ones(nsamples), K)
QS = economic_qs(K)
ntri = random.randint(1, 30, nsamples)
nsuc = zeros(nsamples, dtype=int)
for (i, ni) in enumerate(ntri):
nsuc[i] += sum(z[i] + 0.2 * random.randn(ni) > 0)
ntri = ascontiguousarray(ntri)
glmm0 = GLMMExpFam(nsuc, ("binomial", ntri), X, QS)
assert_allclose(glmm0.lml(), -29.10216812909928, atol=ATOL, rtol=RTOL)
glmm0.fit(verbose=False)
v = -19.575736562427252
assert_allclose(glmm0.lml(), v)
glmm1 = glmm0.copy()
assert_allclose(glmm1.lml(), v)
glmm1.scale = 0.92
assert_allclose(glmm0.lml(), v, atol=ATOL, rtol=RTOL)
assert_allclose(glmm1.lml(), -30.832831740038056, atol=ATOL, rtol=RTOL)
glmm0.fit(verbose=False)
glmm1.fit(verbose=False)
v = -19.575736562378573
assert_allclose(glmm0.lml(), v)
assert_allclose(glmm1.lml(), v)
from sklearn.tree import DecisionTreeClassifier
from sklearn.ensemble import AdaBoostClassifier
from skTMVA import convert_bdt_sklearn_tmva
import cPickle
import numpy as np
from numpy.random import RandomState
RNG = RandomState(21)
# Construct an example dataset for binary classification
n_vars = 2
n_events = 10000
signal = RNG.multivariate_normal(
np.ones(n_vars), np.diag(np.ones(n_vars)), n_events)
background = RNG.multivariate_normal(
np.ones(n_vars) * -1, np.diag(np.ones(n_vars)), n_events)
X = np.concatenate([signal, background])
y = np.ones(X.shape[0])
w = RNG.randint(1, 10, n_events * 2)
y[signal.shape[0]:] *= -1
permute = RNG.permutation(y.shape[0])
X = X[permute]
y = y[permute]
# Use all dataset for training
X_train, y_train, w_train = X, y, w
# Declare BDT - we are going to use AdaBoost Decision Tree
dt = DecisionTreeClassifier(max_depth=3,
def _test_linear_mixed_model_low_rank(self):
seed = 0
n_populations = 8
fst = n_populations * [.9]
n_samples = 500
n_variants = 200
n_orig_markers = 100
n_culprits = 10
n_covariates = 3
sigma_sq = 1
tau_sq = 1
from numpy.random import RandomState
prng = RandomState(seed)
x = np.hstack((np.ones(shape=(n_samples, 1)),
prng.normal(size=(n_samples, n_covariates - 1))))
mt = hl.balding_nichols_model(n_populations=n_populations,
n_samples=n_samples,
n_variants=n_variants,
fst=fst,
af_dist=hl.rand_unif(0.1, 0.9, seed=seed),
seed=seed)
pa_t_path = utils.new_temp_file(suffix='bm')
a_t_path = utils.new_temp_file(suffix='bm')
BlockMatrix.write_from_entry_expr(mt.GT.n_alt_alleles(), a_t_path)
a = BlockMatrix.read(a_t_path).T.to_numpy()
g = a[:, -n_orig_markers:]
g_std = self._filter_and_standardize_cols(g)
n_markers = g_std.shape[1]
k = (g_std @ g_std.T) * n_samples / n_markers
beta = np.arange(n_covariates)
beta_stars = np.array([1] * n_culprits)
y = prng.multivariate_normal(
np.hstack((a[:, 0:n_culprits], x)) @ np.hstack((beta_stars, beta)),
sigma_sq * k + tau_sq * np.eye(n_samples))
# low rank computation of S, P
l = g_std.T @ g_std
sl, v = np.linalg.eigh(l)
n_eigenvectors = int(np.sum(sl > 1e-10))
sl = sl[-n_eigenvectors:]
v = v[:, -n_eigenvectors:]
s = sl * (n_samples / n_markers)
p = (g_std @ (v / np.sqrt(sl))).T
# compare with full rank S, P
sk0, uk = np.linalg.eigh(k)
sk = sk0[-n_eigenvectors:]
pk = uk[:, -n_eigenvectors:].T
assert np.allclose(sk, s)
assert np.allclose(np.abs(pk), np.abs(p))
# build and fit model
py = p @ y
px = p @ x
pa = p @ a
model = LinearMixedModel(py, px, s, y, x)
assert model.n == n_samples
assert model.f == n_covariates
assert model.r == n_eigenvectors
assert model.low_rank
model.fit()
# check effect sizes tend to be near 1 for first n_marker alternative models
BlockMatrix.from_numpy(pa).T.write(pa_t_path, force_row_major=True)
df_lmm = model.fit_alternatives(pa_t_path, a_t_path).to_pandas()
assert 0.9 < np.mean(df_lmm['beta'][:n_culprits]) < 1.1
# compare NumPy and Hail LMM per alternative
df_numpy = model.fit_alternatives_numpy(pa, a).to_pandas()
assert np.min(df_numpy['chi_sq']) > 0
na_numpy = df_numpy.isna().any(axis=1)
na_lmm = df_lmm.isna().any(axis=1)
assert na_numpy.sum() <= 10
assert na_lmm.sum() <= 10
assert np.logical_xor(na_numpy, na_lmm).sum() <= 5
mask = ~(na_numpy | na_lmm)
lmm_vs_numpy_p_value = np.sort(np.abs(df_lmm['p_value'][mask] - df_numpy['p_value'][mask]))
assert lmm_vs_numpy_p_value[10] < 1e-12 # 10 least p-values differences
assert lmm_vs_numpy_p_value[-1] < 1e-8 # all p-values
# <codecell>
# initialize the MR dataframe to store the simulated process
MR = pd.DataFrame(0.1, columns=["MR1", "MR2"], index=arange(T))
# intial value of the process
MR.ix[0] = [mu1, mu2]
# <codecell>
# set random seed
from numpy.random import RandomState
prng = RandomState(12345)
# simulate the process
for t in xrange(T - 1):
dW = prng.multivariate_normal([0, 0], [[1, rho], [rho, 1]])
next1 = (
exp(-lam1) * MR["MR1"][t] + (1 - exp(-lam1)) * mu1 + sigma1 * sqrt((1 - exp(-2 * lam1)) / (2 * lam1)) * dW[0]
)
next2 = (
exp(-lam2) * MR["MR2"][t] + (1 - exp(-lam2)) * mu2 + sigma2 * sqrt((1 - exp(-2 * lam2)) / (2 * lam2)) * dW[1]
)
MR.ix[t + 1] = [next1, next2]
# <codecell>
MR.index = pd.date_range("1/1/2000", periods=T) # add an arbitrary date index
# <codecell>
MR.tail()
def test_linear_mixed_model_full_rank(self):
seed = 0
n_populations = 8
fst = n_populations * [.9]
n_samples = 200
n_variants = 500
n_orig_markers = 500
n_culprits = 20
n_covariates = 3
sigma_sq = 1
tau_sq = 1
from numpy.random import RandomState
prng = RandomState(seed)
x = np.hstack((np.ones(shape=(n_samples, 1)),
prng.normal(size=(n_samples, n_covariates - 1))))
mt = hl.balding_nichols_model(n_populations=n_populations,
n_samples=n_samples,
n_variants=n_variants,
fst=fst,
seed=seed)
pa_t_path = utils.new_temp_file(suffix='bm')
a = BlockMatrix.from_entry_expr(mt.GT.n_alt_alleles()).T.to_numpy()
g = a[:, -n_orig_markers:]
g_std = self._filter_and_standardize_cols(g)
n_markers = g_std.shape[1]
k = (g_std @ g_std.T) * n_samples / n_markers
beta = np.arange(n_covariates)
beta_stars = np.array([1] * n_culprits)
y = prng.multivariate_normal(
np.hstack((a[:, 0:n_culprits], x)) @ np.hstack((beta_stars, beta)),
sigma_sq * k + tau_sq * np.eye(n_samples))
s, u = np.linalg.eigh(k)
p = u.T
# build and fit model
py = p @ y
px = p @ x
pa = p @ a
model = LinearMixedModel(py, px, s)
assert model.n == n_samples
assert model.f == n_covariates
assert model.r == n_samples
assert (not model.low_rank)
model.fit()
# check effect sizes tend to be near 1 for first n_marker alternative models
BlockMatrix.from_numpy(pa).T.write(pa_t_path, force_row_major=True)
df_lmm = model.fit_alternatives(pa_t_path).to_pandas()
assert 0.9 < np.mean(df_lmm['beta'][:n_culprits]) < 1.1
# compare NumPy and Hail LMM per alternative
df_numpy = model.fit_alternatives_numpy(pa, a).to_pandas()
na_numpy = df_numpy.isna().any(axis=1)
na_lmm = df_lmm.isna().any(axis=1)
assert na_numpy.sum() <= 20
assert na_lmm.sum() <= 20
assert np.logical_xor(na_numpy, na_lmm).sum() <= 10
mask = ~(na_numpy | na_lmm)
lmm_vs_numpy_p_value = np.sort(
np.abs(df_lmm['p_value'][mask] - df_numpy['p_value'][mask]))
assert lmm_vs_numpy_p_value[10] < 1e-12 # 10 least p-values differences
assert lmm_vs_numpy_p_value[-1] < 1e-8 # all p-values
=============================================
"""
from array import array
import numpy as np
from numpy.random import RandomState
from root_numpy.tmva import add_classification_events, evaluate_reader
from root_numpy import ROOT_VERSION
import matplotlib.pyplot as plt
from ROOT import TMVA, TFile, TCut
plt.style.use('ggplot')
RNG = RandomState(42)
# Construct an example multiclass dataset
n_events = 1000
class_0 = RNG.multivariate_normal(
[-2, -2], np.diag([1, 1]), n_events)
class_1 = RNG.multivariate_normal(
[0, 2], np.diag([1, 1]), n_events)
class_2 = RNG.multivariate_normal(
[2, -2], np.diag([1, 1]), n_events)
X = np.concatenate([class_0, class_1, class_2])
y = np.ones(X.shape[0])
w = RNG.randint(1, 10, n_events * 3)
y[:class_0.shape[0]] *= 0
y[-class_2.shape[0]:] *= 2
permute = RNG.permutation(y.shape[0])
X = X[permute]
y = y[permute]
# Split into training and test datasets
X_train, y_train, w_train = X[:n_events], y[:n_events], w[:n_events]
class GPPairwise:
"""
Gaussian process with a probit likelihood for pairwise comparisons.
"""
def __init__(self,
num_objectives,
std_noise=0.01,
kernel_width=0.15,
prior_mean_type='zero',
seed=None):
"""
:param num_objectives: number of objectives of input for utility function we want to approximate
:param std_noise: the standard deviation of the normal distributed noise we assume for the utility function
:param prior_mean_type: prior mean function type (zero/linear), default is zero
:param kernel_width: parameter for kernel width, deafult is 0.2
:param seed: seed for random state
"""
self.num_objectives = num_objectives
self.std_noise = std_noise
self.kernel_width = kernel_width
self.prior_mean_type = prior_mean_type
self.random_state = RandomState(seed)
# variables for the observed data
self.datapoints = None
self.comparisons = None
# approximate utility values of the datapoints
self.utility_vals = None
# covariance matrix of datapoints
self.cov_mat = None
self.cov_mat_inv = None
# hessian (second deriv) of the pairwise likelihoods for observed data
self.hess_likelihood = None
self.hess_likelihood_inv = None
# needed for predictive distribution (cov - hess_likelihood_inv)^(-1)
self.pred_cov_factor = None
def sample(self, sample_points):
"""
Get a sample from the current GP at the given points.
:param sample_points: the points at which we want to take the sample
:return: the values of the GP sample at the input points
"""
# bring sample points in right shape
sample_points = utils_data.format_data(sample_points,
self.num_objectives)
# get the mean and the variance of the predictive (multivariate gaussian) distribution at the sample points
mean, var = self.get_predictive_params(sample_points, pointwise=False)
# sample from the multivariate gaussian with the given parameters
f_sample = self.random_state.multivariate_normal(mean, var, 1)[0]
return f_sample
def get_predictive_params(self, x_new, pointwise):
"""
Returns the predictive parameters (mean, variance) of the Gaussian distribution
at the given datapoints
:param x_new: the points for which we want the predictive params
:param pointwise: whether we want pointwise variance or the entire covariance matrix
:return:
"""
# bring input points into right shape
x_new = utils_data.format_data(x_new, self.num_objectives)
# if we don't have any data yet, use prior GP to make predictions
if self.datapoints is None or self.utility_vals is None:
pred_mean, pred_var = self._evaluate_prior(x_new)
# otherwise compute predictive mean and covariance
else:
cov_xnew_x = self._cov_mat(x_new, self.datapoints, noise=False)
cov_x_xnew = self._cov_mat(self.datapoints, x_new, noise=False)
cov_xnew = self._cov_mat(x_new, noise=False)
pred_mean = self.prior_mean(x_new) + np.dot(
np.dot(cov_xnew_x, self.cov_mat_inv),
(self.utility_vals - self.prior_mean(self.datapoints)))
pred_var = cov_xnew - np.dot(
np.dot(cov_xnew_x, self.pred_cov_factor), cov_x_xnew)
if pointwise:
pred_var = pred_var.diagonal()
return pred_mean, pred_var
def update(self, dataset):
"""
Update the Gaussian process using the given data
:param dataset:
:return: """
self.datapoints = dataset.datapoints
self.comparisons = dataset.comparisons
# compute the covariance matrix given the new datapoints
self.cov_mat = self._cov_mat(self.datapoints)
self.cov_mat_inv = np.linalg.inv(self.cov_mat)
# compute the map estimate of f
self.utility_vals = self._compute_posterior()
# compute the hessian of the likelihood given f_MAP
self.hess_likelihood = self._compute_hess_likelihood()
try:
self.hess_likelihood_inv = np.linalg.inv(self.hess_likelihood)
except:
self.hess_likelihood_inv = np.linalg.pinv(self.hess_likelihood)
self.pred_cov_factor = np.linalg.inv(self.cov_mat -
self.hess_likelihood_inv)
def _evaluate_prior(self, input_points):
"""
Given some datapoints, evaluate the prior
:param input_points: input datapoints at which to evaluate prior
:return: predictive mean and covariance at the given inputs
"""
pred_mean = self.prior_mean(input_points)
num_inputs = input_points.shape[0]
pred_cov = self._kernel(np.repeat(input_points, num_inputs, axis=0),
np.tile(input_points,
(num_inputs, 1))).reshape(
(num_inputs, num_inputs))
return pred_mean, pred_cov
def _cov_mat(self, x1, x2=None, noise=True):
"""
Covariance matrix for preference data using the kernel function.
:param x1: datapoints for which to compute covariance matrix
:param x2: if None, covariance matrix will be square for the input x1
if not None, covariance will be between x1 (rows) and x2 (cols_
:param noise: whether to add noise to the diagonal of the covariance matrix
:return:
"""
if x2 is None:
x2 = x1
else: # if x1 != x2 we don't add noise!
noise = False
x1 = utils_data.format_data(x1, self.num_objectives)
x2 = utils_data.format_data(x2, self.num_objectives)
cov_mat = self._kernel(np.repeat(x1, x2.shape[0], axis=0),
np.tile(x2, (x1.shape[0], 1)))
cov_mat = cov_mat.reshape((x1.shape[0], x2.shape[0]))
if noise:
cov_mat += self.std_noise**2 * np.eye(cov_mat.shape[0])
return cov_mat
def prior_mean(self, x):
"""
Prior mean function
:param x: num_datapoints * num_objectives
:return:
"""
x = utils_data.format_data(x, self.num_objectives)
m = np.zeros(x.shape[0])
if self.prior_mean_type == 'linear':
m += np.sum(x, axis=1) / self.num_objectives
else:
TypeError('Prior mean type not understood.')
return m
def _kernel(self, x1, x2):
x1 = utils_data.format_data(x1, self.num_objectives)
x2 = utils_data.format_data(x2, self.num_objectives)
k = 0.8**2 * np.exp(-(1. / (2. * (self.kernel_width**2))) *
np.linalg.norm(x1 - x2, axis=1)**2)
return k
def _compute_hess_likelihood(self, z=None):
"""
Compute the hessian of the likelihood given utility values f
:return:
"""
if z is None:
# compute z
f_winner = np.array([
self.utility_vals[self.comparisons[i, 0]]
for i in range(self.comparisons.shape[0])
])
f_loser = np.array([
self.utility_vals[self.comparisons[i, 1]]
for i in range(self.comparisons.shape[0])
])
z = (f_winner - f_loser) / (np.sqrt(2) * self.std_noise)
z_logpdf = norm.logpdf(z)
z_logcdf = norm.logcdf(z)
# initialise with zeros
lambda_mat = np.zeros(
(self.datapoints.shape[0], self.datapoints.shape[0]))
# build up diagonal for pairs (xi, xi)
diag_arr = np.array([
self._compute_hess_likelihood_entry(m, m, z, z_logpdf, z_logcdf)
for m in range(self.datapoints.shape[0])
])
np.fill_diagonal(lambda_mat, diag_arr) # happens in-place
# go through the list of comparisons collected so far and update lambda
for k in range(self.comparisons.shape[0]):
m = self.comparisons[k, 0] # winner
n = self.comparisons[k, 1] # loser
lambda_mat[m, n] = self._compute_hess_likelihood_entry(
m, n, z, z_logpdf, z_logcdf)
lambda_mat[n, m] = self._compute_hess_likelihood_entry(
n, m, z, z_logpdf, z_logcdf)
# add jitter term to make lambda positive definite for computational stability
lambda_mat += np.eye(self.datapoints.shape[0]) * 0.01
return lambda_mat
def _compute_hess_likelihood_entry(self, m, n, z, z_logpdf, z_logcdf):
"""
Get a single entry for the Hessian matrix at indices (m,n)
:param m:
:param n:
:param f:
:return:
"""
h_x_m = np.array(self.comparisons[:, 0] == m, dtype=int) - np.array(
self.comparisons[:, 1] == m, dtype=int)
h_x_n = np.array(self.comparisons[:, 0] == n, dtype=int) - np.array(
self.comparisons[:, 1] == n, dtype=int)
p = h_x_m * h_x_n * (np.exp(2. * z_logpdf - 2. * z_logcdf) +
z * np.exp(z_logpdf - z_logcdf))
c = -np.sum(p) / (2 * self.std_noise**2)
return c
def _compute_posterior(self):
""" Approximate the posterior distribution """
converged = False
try_no = 0
f_map = None
# using Newton-Raphson, approximate f_MAP
while not converged and try_no < 1:
# randomly initialise f_map
f_map = self.random_state.uniform(0., 1., self.datapoints.shape[0])
# f_map = np.zeros(gp.datapoints.shape[0])
for m in range(100):
# compute z
f_winner = np.array([
f_map[self.comparisons[i, 0]]
for i in range(self.comparisons.shape[0])
])
f_loser = np.array([
f_map[self.comparisons[i, 1]]
for i in range(self.comparisons.shape[0])
])
z = (f_winner - f_loser) / (np.sqrt(2) * self.std_noise)
z_logpdf = norm.logpdf(z)
z_logcdf = norm.logcdf(z)
# compute b
h_j = np.array([
np.array(self.comparisons[:, 0] == j, dtype=int) -
np.array(self.comparisons[:, 1] == j, dtype=int)
for j in range(self.datapoints.shape[0])
])
b = np.sum(h_j * np.exp(z_logpdf - z_logcdf),
axis=1) / (np.sqrt(2) * self.std_noise)
# compute gradient g
g = -np.dot(self.cov_mat_inv,
(f_map - self.prior_mean(self.datapoints))) + b
# compute approximation of the hessian of the posterior
hess_likelihood = self._compute_hess_likelihood(z)
hess_posterior = -self.cov_mat_inv + hess_likelihood
try:
hess_posterior_inv = np.linalg.inv(hess_posterior)
except:
hess_posterior_inv = np.linalg.pinv(hess_posterior)
# perform update
update = np.dot(hess_posterior_inv, g)
f_map -= update
# stop criterion
if np.linalg.norm(update) < 0.0001:
converged = True
break
if not converged:
print("Did not converge.")
try_no += 1
return f_map
from sklearn.tree import DecisionTreeClassifier
from sklearn.ensemble import AdaBoostClassifier
from sklearn.metrics import roc_curve
from sklearn import tree
import cPickle
import numpy as np
from numpy.random import RandomState
RNG = RandomState(45)
# Construct an example dataset for binary classification
n_vars = 2
n_events = 300
signal = RNG.multivariate_normal(
np.ones(n_vars), np.diag(np.ones(n_vars)), n_events)
background = RNG.multivariate_normal(
np.ones(n_vars) * -1, np.diag(np.ones(n_vars)), n_events)
X = np.concatenate([signal, background])
y = np.ones(X.shape[0])
w = RNG.randint(1, 10, n_events * 2)
y[signal.shape[0]:] *= -1
permute = RNG.permutation(y.shape[0])
X = X[permute]
y = y[permute]
# Some print-out
print "Event numbers total:", 2 * n_events
# Plot the testing points
c1 = ROOT.TCanvas("c1","Testing Dataset",200,10,700,500)
bpy.ops.wm.open_mainfile(filepath=“/Users/bacheletlab/Downloads/stereo_equirectangular_base.blend")
import bpy
from numpy.random import RandomState
import numpy as np
count = 200
rs = RandomState(1234)
min_pt = 0
max_pt = 10
cube_size = 0.1
# Generate some data from a multivariate normal distribution
# (http://stackoverflow.com/questions/16024677/generate-correlated-data-in-python-3-3)
mu = np.array([0.0, 5.0, 5.0])
r = np.array([[5.5, 0.3, 4.3],
[0.4, 1., 0.5],
[4.8, 0.5, 5.0]])
data = rs.multivariate_normal(mu, r, size=count)
print(data)
for x, y, z in data:
print("xyz", x, y, z)
bpy.ops.mesh.primitive_cube_add(location=(x, y, z))
bpy.ops.transform.resize(value=(cube_size, cube_size, cube_size))
bpy.data.scenes['Scene'].render.filepath = '/Users/bacheletlab/Downloads/output.png'
bpy.ops.render.render(write_still=True)
class GP_pairwise:
"""Gaussian process with a discrete-choice probit model (latent utility function).
Attributes:
sigma: Hyperparameter for std of the normal distributed noise for the utility function.
theta: Hyperparameter for kernel width.
seed: Seed for random state.
datapoints: Matrix for the observed data.
comparisons: Matrix for comparisons of the observed data.
f_map: Approximate utility values of the datapoints.
K: Covariance matrix of datapoints.
K_inv: Inverse of the covariance matrix of datapoints.
C: The matrix C for observed data.
C_inv: The inverse of matrix C for observed data.
"""
def __init__(self, sigma=0.01, theta=50, seed=None):
"""Inits GP class with all attributes."""
self.sigma = sigma
self.theta = theta
self.random_state = RandomState(seed)
self.datapoints = None
self.comparisons = None
self.f_map = None
self.K = None
self.K_inv = None
self.C = None
self.C_inv = None
def update(self, dataset):
"""Update the Gaussian process using the given data.
Args:
dataset: Dataset consists of datapoints and comparison matrix data
"""
self.datapoints = dataset.datapoints
self.comparisons = dataset.comparisons
# compute the covariance matrix and its inverse
self.K = self._get_K(self.datapoints)
self.K_inv = self._get_inv(self.K)
# compute the MAP estimate of f
self.f_map = self._get_f_map()
# compute C matrix given f_MAP and its inverse (psudo-inverse)
self.C = self._get_C()
self.C_inv = self._get_inv(self.C)
return True
def sample(self, sample_points):
"""Get a sample from the current GP at the given points.
Args:
sample_points: The points at which we want to take the sample.
Returns:
The values of the GP sample at the input points.
"""
# get the mean and the variance of the predictive (multivariate gaussian) distribution at the sample points
mean, var = self.get_Gaussian_params(sample_points, pointwise=False)
# sample from the multivariate gaussian with the given parameters
f_sample = self.random_state.multivariate_normal(mean, var, 1)[0]
return f_sample
def predict(self, sample_point):
"""Predicts value function for a single datapoint"""
mean, var = self.get_Gaussian_params(np.array([sample_point]),
pointwise=False)
f_sample = self.random_state.multivariate_normal(mean, var, 1)[0]
return mean
def get_Gaussian_params(self, x_new, pointwise):
"""Gets the Gaussian parameters.
Args:
x_new: the points for which we want the predictive parameters.
pointwise: whether we want pointwise variance or the entire covariance matrix.
Returns:
The predictive parameters of the Gaussian distribution at the given datapoints.
"""
# if we don't have any data yet, use prior GP to make predictions
if self.datapoints is None or self.f_map is None:
pred_mean, pred_var = self._evaluate_prior(x_new)
# otherwise compute predictive mean and covariance
else:
k_T = self._get_K(x_new, self.datapoints, noise=False)
k = self._get_K(self.datapoints, x_new, noise=False)
k_plus = self._get_K(x_new, noise=False)
pred_mean = self._prior_mean(k_plus) + np.dot(
np.dot(k_T, self.K_inv),
(self.f_map - self._prior_mean(self.datapoints)))
pred_var = k_plus - np.dot(
np.dot(k_T, self._get_inv(self.K + self.C_inv)), k)
if pointwise:
pred_var = pred_var.diagonal()
return pred_mean, pred_var
def _get_K(self, x1, x2=None, noise=True):
"""Computes covariance matrix for preference data using the kernel function.
Args:
x1: The datapoints for which to compute covariance matrix.
x2: If None, covariance matrix will be square for the input x1,
If not None, covariance will be between x1 (rows) and x2 (cols)
noise: Whether to add noise to the diagonal of the covariance matrix.
Returns:
The covariance matrix K.
"""
if x2 is None:
x2 = x1
else:
noise = False
K = self._k(np.repeat(x1, x2.shape[0], axis=0),
np.tile(x2, (x1.shape[0], 1)))
K = K.reshape((x1.shape[0], x2.shape[0]))
if noise:
K += self.sigma**2 * np.eye(K.shape[0])
return K
def _k(self, x1, x2):
"""Exponentiated quadratic kernel function"""
k = 0.8**2 * np.exp(-(1. / (2. * (self.theta**2))) *
np.linalg.norm(x1 - x2, axis=1)**2)
return k
def _get_f_map(self):
"""Computes maximum a posterior (MAP) evaluation of f given the data using Newton's method
Returns:
MAP of the Gassian processes values at current datapoints
"""
converged = False
try_no = 0
f_map = None
# Newton's method to approximate f_MAP
while not converged and try_no < 1:
# randomly initialise f_map
f_map = self.random_state.uniform(0., 1., self.datapoints.shape[0])
for m in range(100):
# compute Z
f_sup = np.array([
f_map[self.comparisons[i, 0]]
for i in range(self.comparisons.shape[0])
])
f_inf = np.array([
f_map[self.comparisons[i, 1]]
for i in range(self.comparisons.shape[0])
])
Z = self._get_Z(f_sup, f_inf)
Z_logpdf = norm.logpdf(Z)
Z_logcdf = norm.logcdf(Z)
# compute b
b = self._get_b(Z_logpdf, Z_logcdf)
# compute gradient g
g = self._get_g(f_map, b)
# compute hessian H
C = self._get_C(Z)
H = -self.K_inv + C
H_inv = self._get_inv(H)
# perform update
update = np.dot(H_inv, g)
f_map -= update
# stop criterion
if np.linalg.norm(update) < 0.0001:
converged = True
break
if not converged:
print("Did not converge.")
try_no += 1
return f_map
def _get_Z(self, f_sup, f_inf):
"""Gets the random variable Z based on given sup and inf pair."""
return (f_sup - f_inf) / (np.sqrt(2) * self.sigma)
def _get_b(self, Z_logpdf, Z_logcdf):
"""Gets the N-dimensional vector b"""
h_j = np.array([
np.array(self.comparisons[:, 0] == j, dtype=int) -
np.array(self.comparisons[:, 1] == j, dtype=int)
for j in range(self.datapoints.shape[0])
])
b = np.sum(h_j * np.exp(Z_logpdf - Z_logcdf),
axis=1) / (np.sqrt(2) * self.sigma)
return b
def _get_g(self, f_map, b):
"""Gets the gradient g"""
return -np.dot(self.K_inv,
(f_map - self._prior_mean(self.datapoints))) + b
def _get_C(self, Z=None):
"""Gets the matrix C"""
if Z is None:
# compute z
f_sup = np.array([
self.f_map[self.comparisons[i, 0]]
for i in range(self.comparisons.shape[0])
])
f_inf = np.array([
self.f_map[self.comparisons[i, 1]]
for i in range(self.comparisons.shape[0])
])
Z = (f_sup - f_inf) / (np.sqrt(2) * self.sigma)
Z_logpdf = norm.logpdf(Z)
Z_logcdf = norm.logcdf(Z)
# init with zeros
C = np.zeros((self.datapoints.shape[0], self.datapoints.shape[0]))
# build up diagonal for pairs (x, x)
diag_arr = np.array([
self._get_C_entry(m, m, Z, Z_logpdf, Z_logcdf)
for m in range(self.datapoints.shape[0])
])
np.fill_diagonal(C, diag_arr) # happens in-place
# go through the existing list of comparisons and update C
for k in range(self.comparisons.shape[0]):
m = self.comparisons[k, 0] # superior
n = self.comparisons[k, 1] # inferior
C[m, n] = self._get_C_entry(m, n, Z, Z_logpdf, Z_logcdf)
C[n, m] = self._get_C_entry(n, m, Z, Z_logpdf, Z_logcdf)
# add jitter terms to make matrix C positive semidefinite for stable computation
C += np.eye(self.datapoints.shape[0]) * 0.01
return C
def _get_C_entry(self, m, n, Z, Z_logpdf, Z_logcdf):
"""Gets a single entry for the Hessian matrix at indices (m,n)"""
h_x_m = np.array(self.comparisons[:, 0] == m, dtype=int) - np.array(
self.comparisons[:, 1] == m, dtype=int)
h_x_n = np.array(self.comparisons[:, 0] == n, dtype=int) - np.array(
self.comparisons[:, 1] == n, dtype=int)
p = h_x_m * h_x_n * (np.exp(2. * Z_logpdf - 2. * Z_logcdf) +
Z * np.exp(Z_logpdf - Z_logcdf))
c = -np.sum(p) / (2 * self.sigma**2)
return c
def _evaluate_prior(self, input_points):
"""Evaluates the prior distribution given some datapoints.
Args:
input_points: input datapoints at which to evaluate prior distribution.
Returns:
The predictive mean and covariance at the given inputs.
"""
pred_mean = self._prior_mean(input_points)
num_inputs = input_points.shape[0]
pred_cov = self._k(np.repeat(input_points, num_inputs, axis=0),
np.tile(input_points, (num_inputs, 1))).reshape(
(num_inputs, num_inputs))
return pred_mean, pred_cov
def _get_inv(self, M):
"""Computes the inverse of the given matrix or the psudoinverse."""
try:
M_inv = np.linalg.inv(M)
except:
M_inv = np.linalg.pinv(M)
return M_inv
def _prior_mean(self, x):
"""Returns the prior mean of zeros"""
m = np.zeros(x.shape[0])
return m