def t_stat(data, X_matrix):
"""
Return the estimated betas, t-values, degrees of freedom, and p-values for the glm_multi regression
Parameters
----------
data_4d: numpy array of 4 dimensions
The image data of one subject, one run
X_matrix: numpy array
The design matrix for glm_multi
Note that the fourth dimension of `data_4d` (time or the number
of volumes) must be the same as the number of rows that X has.
Returns
-------
beta: estimated beta values
t: t-values of the betas
df: degrees of freedom
p: p-values corresponding to the t-values and degrees of freedom
"""
beta = glm_beta(data, X_matrix)
# Calculate the parameters - b hat
beta = np.reshape(beta, (-1, beta.shape[-1])).T
fitted = X_matrix.dot(beta)
# Residual error
y = np.reshape(data, (-1, data.shape[-1]))
errors = y.T - fitted
# Residual sum of squares
RSS = (errors**2).sum(axis=0)
df = X_matrix.shape[0] - npl.matrix_rank(X_matrix)
# Mean residual sum of squares
MRSS = RSS / df
# calculate bottom half of t statistic
Cov_beta=npl.pinv(X_matrix.T.dot(X_matrix))
SE =np.zeros(beta.shape)
for i in range(X_matrix.shape[-1]):
c = np.zeros(X_matrix.shape[-1])
c[i]=1
c = np.atleast_2d(c).T
SE[i,:]= np.sqrt(MRSS* c.T.dot(npl.pinv(X_matrix.T.dot(X_matrix)).dot(c)))
zeros = np.where(SE==0)
SE[zeros] = 1
t = beta / SE
t[:,zeros] =0
# Get p value for t value using CDF of t didstribution
ltp = t_dist.cdf(abs(t), df)
p = 1 - ltp # upper tail
return beta.T, t, df, p
def supf(y, x, p):
T = y.shape[0]
range = np.floor(np.array([T * p, T * (1 - p)]))
range = np.arange(range[0], range[1] + 1, dtype=np.int32)
# Demean since intercept doesnt break
x = x - np.mean(x)
y = y - np.mean(y)
b = pinv(x).dot(y)
e = y - x.dot(b)
# Compute full sample R2
R2_r = 1 - e.dot(e) / y.dot(y)
k = x.shape[1]
F_stat = np.zeros(T)
for t in range:
X1 = x[:t]
X2 = x[t:]
# Parameters and errors before the break
b = pinv(X1).dot(y[:t])
e[:t] = y[:t] - X1.dot(b)
# Parameters and errors after the break
b = pinv(X2).dot(y[t:])
e[t:] = y[t:] - X2.dot(b)
# R2 from model with break
R2_u = 1 - e.dot(e) / y.dot(y)
# F stat for break at t
F_stat[t] = ((R2_u - R2_r) / k) / ((1 - R2_u) / (T - 2* k - 1))
# Only return maximum F stat
return F_stat.max()
def td_solver(P, r, X, gm, lm, d=None):
"""Compute the TD solution for an MDP under linear function approximation.
Args:
P : The transition matrix under a given policy.
r : The expected immediate reward for each state under the policy.
X : The feature matrix (one row for each state)
gm : The discount parameter, gamma
lm : The bootstrapping parameter, lambda
d (optional): The stationary distribution to use.
Returns:
theta: the weight vector found by the TD solution.
"""
ns = len(P) # number of states
I = np.eye(ns)
# TODO: Check for validity of P, r, X (size and values)
# TODO: Provide a way to handle terminal states
# account for scalar, vector, or matrix parameters
G = parameter_matrix(gm)
L = parameter_matrix(lm)
# compute the stationary distribution if unspecified
if d is None:
d = stationary(P)
# the stationary distribution as a matrix
D = np.diag(d)
# Solve the equation
A = X.T @ D @ pinv(I - P @ G @ L) @ (I - P @ G) @ X
b = X.T @ D @ pinv(I - P @ G @ L) @ r
return pinv(A) @ b
def predict_all(self):
"""Returns the predictions and the reconstructions for all training / test data."""
outputs, hidden = self.feed_forward(self.pca_transformer.transform(self.X))
hidden_expected = dot(self._inverse_activation(outputs), pinv(self.W_output))[:, :-1]
hidden_reconstruction = self.pca_transformer.inverse_transform(
dot(self._inverse_activation(hidden_expected), pinv(self.W_hidden))[:, :-1])
return outputs.argmax(axis=1), hidden_reconstruction.reshape(self.app.dataset['images'].shape)
def test_entity_array_variable_get_cond_mean_and_var(self):
groups = self.groups
group = groups['u']
variable = [v for v in groups['u'].iter_variables() if v.entity_id
== 'v1' ][0]
related_votes = [
{'review': 'r1', 'author': 'a1', 'voter': 'v1', 'vote': 4},
{'review': 'r2', 'author': 'a1', 'voter': 'v1', 'vote': 5},
{'review': 'r5', 'author': 'a3', 'voter': 'v1', 'vote': 5},
]
v_values = [[v for v in groups['v'].iter_variables() if v.entity_id ==
vote['review']][0].value for vote in related_votes]
v_sum = sum([v.dot(v.T) for v in v_values]) / self.var_H.value
var_matrix = group.var_param.value * identity(const.K)
inv_var = pinv(var_matrix)
true_var = pinv(inv_var + v_sum)
rest_term = sum([variable.get_rest_value(groups, related_votes[i]) *
v_values[i] for i in xrange(len(related_votes))]) / \
group.var_H.value
dot_term = inv_var.dot(group.weight_param.value) \
.dot(variable.features)
true_mean = true_var.dot(rest_term + dot_term)
res_mean, res_var = variable.get_cond_mean_and_var(groups, self.votes)
ntest.assert_allclose(true_var, res_var, rtol=1, atol=1e-7)
ntest.assert_allclose(true_mean, res_mean, rtol=1, atol=1e-7)
def ls(R,W,d):
(n,m) = R.shape
sigma = 0.0001
Id = np.identity(d)
U0 = np.zeros((d,n))
V = np.random.rand(d, m)
for i in range(1000):
U = U0
for g in range(n):
VV = np.zeros(d)
for w in W[g]:
VV = VV+np.dot(V[:,w],V[:,w].T)
X = nlin.pinv(sigma*Id+VV)
#X = sigma*Id + VV
for v in W[g]:
U[:,g] = U[:,g] + R[g,v]*np.dot(V[:,v].T,X)
#U[:,g] = U[:,g] + R[g,v]*slin.solve(X ,V[:,v].T)
Y = np.dot(U,U.T)
Y = nlin.pinv(sigma*Id+Y)
Y = np.dot(U.T,Y)
#Y = np.linalg.solve( U.T, sigma*Id+Y)
#Y = np.linalg.lstsq(U.T, sigma*Id+Y)
for v in range(m):
V[:,v] = np.dot(R[:,v].T,Y)
return (U,V)
def sample_posterior_gibbs(self, X, num_steps=10, Y=None, Z=None):
"""
B{References:}
- Doucet, A. (2010). I{A Note on Efficient Conditional Simulation of
Gaussian Distributions.}
"""
# filter matrix and filter responses
W = pinv(self.A)
WX = dot(W, X)
# nullspace projection matrix
Q = eye(self.num_hiddens) - dot(W, self.A)
# initial hidden state
if Z is None:
Y = WX + dot(Q, Y) if Y is not None else \
WX + dot(Q, self.sample_prior(X.shape[1]))
else:
V = pinv(self.nullspace_basis())
Y = WX + dot(V, Z)
# Gibbs sample between S and Y given X
for step in range(num_steps):
# update scales
S = self.sample_scales(Y)
# update hidden states
Y = self._sample_posterior_cond(Y, X, S, W, WX, Q)
if Distribution.VERBOSITY > 1:
print '{0:6}\t{1:10.2f}'.format(step + 1, mean(self.prior_energy(Y)))
return asarray(Y)
def AND(C, B): dim, col = C.shape tolerance = 1e-14 UC, SC, UtC = svd(C) UB, SB, UtB = svd(B) diag_SC = diag(SC) diag_SB = diag(SB) # sum up how many elements on diagonal # are bigger than tolerance numRankC = (1.0 * (diag_SC > tolerance)).sum() numRankB = (1.0 * (diag_SB > tolerance)).sum() UC0 = matrix(UC[:, numRankC:]) UB0 = matrix(UB[:, numRankB:]) W, Sigma, Wt = svd(UC0 * UC0.transpose() + UB0 * UB0.transpose()) numRankSigma = (1.0 * (diag(Sigma) > tolerance)).sum() Wgk = matrix(W[:, numRankSigma:]) I = matrix(identity(dim)) CandB = \ Wgk * inv(Wgk.transpose() * \ ( pinv(C, tolerance) + pinv(B, tolerance) - \ I) * Wgk) *Wgk.transpose() return CandB
def update_b(i_index, b, alpha, beta, gamma, sigma2, lambda_D,
N_g, uni_id, uni_diet, id_g, p, W, X, Z, y):
i = uni_id[i_index]
for g_search, i_search in id_g.iteritems():
if np.any(i_search == i):
g = g_search
g_index = np.where(uni_diet == g)[0][0]
if np.all(gamma[g_index] == 0): #check if all gamma's are 0
V2 = lambda_D + np.dot(Z[i].T, Z[i])/sigma2
mean2 = np.dot(pinv(V2), np.dot(Z[i].T, y[i]-W[i].dot(alpha)))/sigma2
else:
V2 = lambda_D + np.dot(Z[i].T, Z[i])/sigma2
temp1 = XXsum(g_index, uni_diet, id_g, gamma, X)
temp1 = pinv(temp1)
V2 = V2 + np.dot(np.dot(np.dot(Z[i].T, X[i][:,gamma[g_index]!=0]), temp1), (np.dot(X[i][:,gamma[g_index]!=0].T, Z[i])))/(sigma2*N_g[g])
mean2 = np.dot(Z[i].T, y[i] - W[i].dot(alpha) - np.dot(X[i][:,gamma[g_index]!=0], beta[g_index][gamma[g_index]!=0].reshape(np.sum(gamma[g_index]),1)))
temp2 = np.dot(X[i][:,gamma[g_index]!=0].T, Z[i].dot(b[i_index].reshape(p, 1)))
for j in id_g[g]:
j_index = np.where(uni_id == j)[0][0]
temp2 += np.dot(X[j][:,gamma[g_index]!=0].T, y[j] - W[j].dot(alpha) - Z[j].dot(b[j_index].reshape(p, 1)))
mean2 = mean2 + np.dot(np.dot(Z[i].T.dot(X[i][:,gamma[g_index]!=0]), temp1), temp2)/N_g[g]
mean2 = np.dot(pinv(V2), mean2)/sigma2
#update
b_new = np.random.multivariate_normal(mean2.reshape(p,), pinv(V2)).reshape(p, )
return b_new
def test_ridge_regression_py(X, Y, _lambda):
X = X.astype(np.float32)
Y = Y.astype(np.float32)
t1 = time.time()
n, p = X.shape
if n < p:
tmp = np.dot(X, X.T)
if _lambda:
tmp += _lambda*n*np.eye(n)
tmp = la.pinv(tmp)
beta_out = np.dot(np.dot(X.T, tmp), Y.reshape(-1, 1))
else:
tmp = np.dot(X.T, X)
if _lambda:
tmp += _lambda*n*np.eye(p)
tmp = la.pinv(tmp)
beta_out = np.dot(tmp, np.dot(X.T, Y.reshape(-1, 1)))
t2 = time.time()
dt = t2-t1
print("total time (Python): {}".format(dt))
print(beta_out[:20,0])
def _em_one_pass(self, centered=None, numcmpt=1, thresh=1e-16, out=None):
"""
With numcmpt = 1, computes the first principal component
of the data. Otherwise computes an unnormalized, non-orthogonal
spanning set for the first numcmpt principal components. Assumes
rows are variables, columns are data points.
"""
csize = (self.ndim, numcmpt)
if out != None:
assert out.shape == csize
comp = out
comp[:] = random.normal(size=csize)
else:
comp = random.normal(size=csize)
# Initialize 'old' array to infinity
comp_old = np.empty(csize) + np.inf
if centered == None:
# Center the data with respect to the dataset mean
centered = self._data - self._mean
# Compensate for the shape of the data
if not self._rowvar:
centered = centered.T
while linalg.norm(comp_old - comp, np.inf) > thresh:
pinvc_times_data = np.dot(linalg.pinv(comp), centered)
comp_old[:] = comp
comp[:] = np.dot(centered, linalg.pinv(pinvc_times_data))
# Normalize the eigenvectors we obtained.
comp /= np.apply_along_axis(linalg.norm, 0, comp)[np.newaxis, :]
def t_stat(self):
""" betas, t statistic and significance test given data,
design matix, contrast
This is OLS estimation; we assume the errors to have independent
and identical normal distributions around zero for each $i$ in
$\e_i$ (i.i.d).
"""
if self.design is None:
self.get_design_matrix()
if self.t_values is None:
y = self.data.T
X = self.design
c = [0, 0, 1]
c = np.atleast_2d(c).T
beta = npl.pinv(X).dot(y)
fitted = X.dot(beta)
errors = y - fitted
RSS = (errors**2).sum(axis=0)
df = X.shape[0] - npl.matrix_rank(X)
MRSS = RSS / df
SE = np.sqrt(MRSS * c.T.dot(npl.pinv(X.T.dot(X)).dot(c)))
try:
SE[SE == 0] = np.amin(SE[SE != 0])
except ValueError:
pass
t = c.T.dot(beta) / SE
self.t_values = abs(t[0])
self.t_indices = np.array(self.t_values).argsort(
)[::-1][:self.t_values.size]
return self.t_indices
def fit_to_model(imchunk,model, mode = 'pinv',fit_pix_mask = None,baseline = None):
import numpy as np
#im_array = (imchunk-baseline)#/baseline
if not(baseline is None):
im_array = imchunk-baseline#/baseline
else:
im_array = imchunk
imshape = np.shape(im_array[0])
im_array = im_array.reshape((-1,imshape[0]*imshape[1]))
if mode == 'nnls':
fits = np.empty((np.shape(model)[0],np.shape(im_array)[0]))
for i,im2 in enumerate(im_array):
im = im2.copy()
im[~np.isfinite(im)] = 0
from scipy.optimize import nnls
if not(fit_pix_mask is None):
fits[:,i] = nnls(model[:,fit_pix_mask].T,im[fit_pix_mask])[0]
else:
fits[:,i] = nnls(model.T,im)[0]
else:
im = im_array
print np.shape(im_array)
from numpy.linalg import pinv
if not(fit_pix_mask is None):
fits = np.dot(pinv(model[:,fit_pix_mask]).T,im[:,fit_pix_mask].T)
else:
fits = np.dot(pinv(model).T,im)
return fits
def update(self):
'''Initialize other arrays from fundamental arrays'''
#The sparse matrices are treated a little differently because they are not rectangular
with warnings.catch_warnings():
warnings.filterwarnings("ignore",category=DeprecationWarning)
if self.AtAi.size == 0:
self.AtAi = la.pinv(self.At.dot(self.At.T).todense(),rcond=1e-6).astype(np.float32)#(AtA)^-1
self.BtBi = la.pinv(self.Bt.dot(self.Bt.T).todense(),rcond=1e-6).astype(np.float32)#(BtB)^-1
def noise(L, jump_op, ss, pops, Q, freq):
noise = np.zeros(freq.size, dtype=complex128)
for i in range(len(freq)):
R_plus = np.dot(Q, np.dot(npla.pinv(1.j*freq[i]*np.eye(L.shape[0])-L), Q))
R_minus = np.dot(Q, np.dot(npla.pinv(-1.j*freq[i]*np.eye(L.shape[0])-L), Q))
noise[i] = np.dot(pops, np.dot(jump_op, ss)) \
+ np.dot(pops, np.dot(np.dot(np.dot(jump_op, R_plus), jump_op) \
+ np.dot(np.dot(jump_op, R_minus), jump_op), ss))
return noise
def solve(self, eps=1e-8):
'''
Implement a LCQP solver, with numerical threshold eps.
'''
Cp = npl.pinv(self.C, eps)
xopt = Cp * self.d
P = eye(self.nx * self.N) - Cp * self.C
xopt += npl.pinv(self.A * P, eps) * (self.b - self.A * xopt)
return xopt
def nipals_xy(X, Y, mode="PLS", max_iter=500, tol=1e-06):
"""
NIPALS algorithm; returns the first left and rigth singular
vectors of X'Y.
:param X, Y: data matrix
:type X, Y: :class:`numpy.array`
:param mode: possible values "PLS" (default) or "CCA"
:type mode: string
:param max_iter: maximal number of iterations (default: 500)
:type max_iter: int
:param tol: tolerance parameter; if norm of difference
between two successive left singular vectors is less than tol,
iteration is stopped
:type tol: a not negative float
"""
yScore, uOld, ite = Y[:, [0]], 0, 1
Xpinv = Ypinv = None
# Inner loop of the Wold algo.
while True and ite < max_iter:
# Update u: the X weights
if mode == "CCA":
if Xpinv is None:
Xpinv = linalg.pinv(X) # compute once pinv(X)
u = dot(Xpinv, yScore)
else: # mode PLS
# Mode PLS regress each X column on yScore
u = dot(X.T, yScore) / dot(yScore.T, yScore)
# Normalize u
u /= numpy.sqrt(dot(u.T, u))
# Update xScore: the X latent scores
xScore = dot(X, u)
# Update v: the Y weights
if mode == "CCA":
if Ypinv is None:
Ypinv = linalg.pinv(Y) # compute once pinv(Y)
v = dot(Ypinv, xScore)
else:
# Mode PLS regress each X column on yScore
v = dot(Y.T, xScore) / dot(xScore.T, xScore)
# Normalize v
v /= numpy.sqrt(dot(v.T, v))
# Update yScore: the Y latent scores
yScore = dot(Y, v)
uDiff = u - uOld
if dot(uDiff.T, uDiff) < tol or Y.shape[1] == 1:
break
uOld = u
ite += 1
return u, v
def predict(x): maximum = "null" for param in params: # iterate through all distributions (m, u, v, name) = param mtx = pinv(v)*(np.transpose(np.subtract(x, m)))*(pinv(u))*(np.subtract(x, m)) trace = np.trace(mtx) #butterfy has a small trace l = 1.0e300*np.exp(-0.5*trace)/((la.norm(v)**(n/2.0))*(la.norm(u)**(p/2.0))) # likelihood, excluding the "2pi" term and multiplying by a large positive number (we get overflow otherwise) if maximum == "null": maximum = (l, name) elif l > maximum[0]: maximum = (l, name) print maximum return maximum
def sample_posterior_ais(self, X, num_steps=10, annealing_weights=[]):
"""
Sample posterior distribution over hidden states using annealed importance
sampling with Gibbs sampling transition operator.
"""
if not annealing_weights:
annealing_weights = linspace(0, 1, num_steps + 1)[1:]
# initialize proposal distribution to be Gaussian
model = deepcopy(self)
for gsm in model.subspaces:
gsm.scales[:] = 1.
# filter matrix and filter responses
W = pinv(self.A)
WX = dot(W, X)
# nullspace basis and projection matrix
B = self.nullspace_basis()
Q = dot(B.T, B)
# initialize proposal samples (Z is initially Gaussian and independent of X)
Z = dot(B, randn(self.num_hiddens, X.shape[1]))
Y = WX + dot(pinv(B), Z)
# initialize importance weights (log-determinant of dot(B.T, B) not needed here)
log_is_weights = sum(multiply(Z, dot(inv(dot(B, B.T)), Z)), 0) / 2. \
+ (self.num_hiddens - self.num_visibles) / 2. * log(2. * pi)
log_is_weights.resize(1, X.shape[1])
for step, beta in enumerate(annealing_weights):
# tune proposal distribution
for i in range(len(self.subspaces)):
# adjust standard deviations
model.subspaces[i].scales = (1. - beta) + beta * self.subspaces[i].scales
log_is_weights -= model.prior_energy(Y)
# apply Gibbs sampling transition operator
S = model.sample_scales(Y)
Y = model._sample_posterior_cond(Y, X, S, W, WX, Q)
log_is_weights += model.prior_energy(Y)
if Distribution.VERBOSITY > 1:
print '{0:6}\t{1:10.2f}'.format(step + 1, mean(self.prior_energy(Y)))
log_is_weights += self.prior_loglikelihood(Y) + slogdet(dot(W.T, W))[1] / 2.
return Y, log_is_weights
def _cov_(self, data, params, priors):
'''Calculate covariance matrix of the normal posterior dist.'''
V1 = np.zeros([data.p, data.p])
for gdx in range(data.grp):
g = data.unidiets[gdx]
nzro_gamma = (params.gamma[gdx,:]!=0)
for i in data.grp_uniids[g]:
V1 += np.dot(data.id_W[i].T, data.id_W[i])
if nzro_gamma.any():
V1 += np.dot(self.__nxTw__[g].T,
np.dot(pinv(self.__nxTx__[g]),
self.__nxTw__[g]))
V1 = V1/params.sigma2 + priors.d4
self.cov = pinv(V1)
def get_h(w_matrix, x_matrix):
"""Finds a nonnegative right factor (H) of the perform_net_nmf function
X ~ W.H
Args:
w_matrix: the positive left factor (W) of the perform_net_nmf function
x_matrix: the postive matrix (X) to be decomposed
Returns:
h_matrix: nonnegative right factor (H)
"""
wtw = np.dot(w_matrix.T, w_matrix)
number_of_clusters = wtw.shape[0]
wtx = np.dot(w_matrix.T, x_matrix)
colix = np.arange(0, x_matrix.shape[1])
rowix = np.arange(0, w_matrix.shape[1])
h_matrix = np.dot(LA.pinv(wtw), wtx)
h_pos = h_matrix > 0
h_matrix[~h_pos] = 0
col_log_arr = sum(h_pos == 0) > 0
col_list = colix[col_log_arr]
for cluster in range(0, number_of_clusters):
if col_list.size > 0:
w_ette = wtx[:, col_list]
m_rows = w_ette.shape[0]
n_cols = w_ette.shape[1]
mcode_uniq_col_ix = np.arange(0, n_cols)
h_ette = np.zeros((m_rows, n_cols))
h_pos_ette = h_pos[:, col_list]
mcoding = np.dot(2**(np.arange(0, m_rows)), np.int_(h_pos_ette))
mcode_uniq = np.unique(mcoding)
for u_n in mcode_uniq:
ixidx = mcoding == u_n
c_pat = mcode_uniq_col_ix[ixidx]
if c_pat.size > 0:
r_pat = rowix[h_pos_ette[:, c_pat[0]]]
atmp = wtw[r_pat[:, None], r_pat]
btmp = w_ette[r_pat[:, None], c_pat]
atmptatmp = np.dot(atmp.T, atmp)
atmptatmp = LA.pinv(atmptatmp)
atmptbtmp = np.dot(atmp.T, btmp)
h_ette[r_pat[:, None], c_pat] = np.dot(atmptatmp, atmptbtmp)
h_matrix[:, col_list] = h_ette
h_pos = h_matrix > 0
h_matrix[~h_pos] = 0
col_log_arr = sum(h_pos == 0) > 0
col_list = colix[col_log_arr]
else:
break
return h_matrix
def _Mstep(self,x,L):
"""
VB-M step
Parameters
----------
x: array of shape(nbitem,self.dim)
the data from which the model is estimated
L: array of shape(nbitem,self.k)
the likelihood of the data under each class
"""
from numpy.linalg import pinv
tiny =1.e-15
pop = L.sum(0)
# shrinkage,weights,dof
self.weights = self.prior_weights + pop
pop = pop[0:self.k]
L = L[:,:self.k]
self.shrinkage = self.prior_shrinkage + pop
self.dof = self.prior_dof + pop
#reshape
pop = np.reshape(pop,(self.k,1))
prior_shrinkage = np.reshape(self.prior_shrinkage,(self.k,1))
shrinkage = np.reshape(self.shrinkage,(self.k,1))
# means
means = np.dot(L.T,x)+ self.prior_means*prior_shrinkage
self.means= means/shrinkage
#precisions
empmeans = np.dot(L.T,x)/np.maximum(pop,tiny)
empcov = np.zeros(np.shape(self.prior_scale))
for k in range(self.k):
dx = x-empmeans[k]
empcov[k] = np.dot(dx.T,L[:,k:k+1]*dx)
covariance = np.array([pinv(self.prior_scale[k])
for k in range(self.k)])
covariance += empcov
dx = np.reshape(empmeans-self.prior_means,(self.k,self.dim,1))
addcov = np.array([np.dot(dx[k],dx[k].T) for k in range(self.k)])
apms = np.reshape(prior_shrinkage*pop/shrinkage,(self.k,1,1))
covariance += addcov*apms
self.scale = np.array([pinv(covariance[k]) for k in range(self.k)])
def initialize(self):
S = sum([N.dot(unit.X.T, unit.X) for unit in self.units])
Y = sum([N.dot(unit.X.T, unit.Y) for unit in self.units])
self.a = L.lstsq(S, Y)[0]
D = 0
t = 0
sigmasq = 0
for unit in self.units:
unit.r = unit.Y - N.dot(unit.X, self.a)
if self.q > 1:
unit.b = L.lstsq(unit.Z, unit.r)[0]
else:
Z = unit.Z.reshape((unit.Z.shape[0], 1))
unit.b = L.lstsq(Z, unit.r)[0]
sigmasq += (N.power(unit.Y, 2).sum() -
(self.a * N.dot(unit.X.T, unit.Y)).sum() -
(unit.b * N.dot(unit.Z.T, unit.r)).sum())
D += N.multiply.outer(unit.b, unit.b)
t += L.pinv(N.dot(unit.Z.T, unit.Z))
sigmasq /= (self.N - (self.m - 1) * self.q - self.p)
self.sigma = N.sqrt(sigmasq)
self.D = (D - sigmasq * t) / self.m
def getNormal(xs, Xseen, Yseen):
def cov_matrix(x1s, x2s=None):
if x2s is None:
return covariance(np.asmatrix(x1s).T)
return covariance(np.asmatrix(x1s).T, np.asmatrix(x2s).T)
if len(Xseen) == 0:
mu = np.zeros(xs.shape)
sigma = cov_matrix(xs, xs)
else:
x2s = np.array(Xseen)
o2s = np.array(Yseen)
mu1 = np.zeros(xs.shape)
mu1 = mu1.reshape((mu1.size,))
mu2 = np.zeros(x2s.shape)
a2 = np.matrix(o2s.reshape((len(o2s),1)))
sigma11 = cov_matrix(xs, xs)
sigma12 = cov_matrix(xs, x2s)
sigma21 = cov_matrix(x2s, xs)
sigma22 = cov_matrix(x2s,x2s)
inv22 = la.pinv(sigma22)
plusterm = np.asarray(np.dot(sigma12, np.dot(inv22, (a2 - mu2.reshape(a2.shape))))).squeeze()
# print plusterm.shape
mu = mu1 + plusterm
sigma = sigma11 - np.dot(sigma12,np.dot(inv22,sigma21))
return mu, sigma
def _field_gradient_jac(ref, target):
"""
Given a reference field ref and a target field target
compute the jacobian of the target with respect to ref
Parameters
----------
ref: Field instance that yields the topology of the space
target array of shape(ref.V,dim)
Results
-------
fgj: array of shape (ref.V) that gives the jacobian
implied by the ref.field->target transformation.
"""
import numpy.linalg as nl
n = ref.V
xyz = ref.field
dim = xyz.shape[1]
fgj = []
ln = ref.list_of_neighbors()
for i in range(n):
j = ln[i]
if np.size(j) > dim - 1:
dx = np.squeeze(xyz[j] - xyz[i])
df = np.squeeze(target[j] - target[i])
FG = np.dot(nl.pinv(dx), df)
fgj.append(nl.det(FG))
else:
fgj.append(1)
fgj = np.array(fgj)
return fgj
def reduce(self, assignment):
if all([x is None for x in assignment]):
return MultivariateGaussianDistribution(self.mean, self.cov)
# reordering variables, so that non-reduced variables go before reduced
reduced_idx = [i for i in range(len(assignment)) if assignment[i] is not None]
non_reduced_idx = [i for i in range(len(assignment)) if assignment[i] is None]
x = np.matrix([assignment[idx] for idx in reduced_idx]).T
new_idx = non_reduced_idx + reduced_idx
mean1 = np.matrix([self.mean[idx] for idx in non_reduced_idx]).T
mean2 = np.matrix([self.mean[idx] for idx in reduced_idx]).T
cov11 = self.cov[non_reduced_idx][:, non_reduced_idx]
cov22 = self.cov[reduced_idx][:, reduced_idx]
cov12 = self.cov[non_reduced_idx][:, reduced_idx]
mean = mean1 + cov12 * pinv(cov22) * (x - mean2)
cov = cov11 - cov12 * pinv(cov22) * cov12.T
return MultivariateGaussianDistribution(np.array(mean.T), cov)
def __init__(self,N,xid,covd,modl,rl,sp=1.):
#should take in N data vectors with corresponding models
#xid is list of data vectors; should already be cut to correct scales, given in rl
#covd is full covariance matrix
#modl is list of BAO templates
#rl is the list of r values matching xid and covd
self.xim = [] #these lists will be filled based on the r limits
self.rl = []
self.sp = sp
bs = rl[1]-rl[0] #assumes linear bins
s = 0
nxib = len(xid)
rsw = 0
Bnode = 50.
self.rl = rl
self.xim = []
for j in range(0,N):
for i in range(0,len(xid[0])):
self.xim.append(xid[j][i])
print (self.xim,len(self.xim) )
self.nbin = len(self.rl)
print ('using '+ str(self.nbin)+' xi(s) bins')
#mt = zeros((self.nbin,self.nbin)) #this will be the trimmed covariance matrix
#for i in range(mini,mini+self.nbin):
# for j in range(mini,mini+self.nbin):
# mt[i-mini][j-mini] = covd[i][j]
mt = covd#[self.nbin,mini:mini+self.nbin]
self.invt = linalg.pinv(mt)
self.ximodmin = 10. #minimum of template
self.modl = modl
self.N = N
print (self.wmod(100.))
def smooth_pinv(B, L):
"""Regularized psudo-inverse
Computes a regularized least square inverse of B
Parameters
----------
B : array_like (n, m)
Matrix to be inverted
L : array_like (n,)
Returns
-------
inv : ndarray (m, n)
regularized least square inverse of B
Notes
-----
In the literature this inverse is often written $(B^{T}B+L^{2})^{-1}B^{T}$.
However here this inverse is implemented using the psudo-inverse because it
is more numerically stable than the direct implementation of the matrix
product.
"""
L = diag(L)
inv = pinv(concatenate((B, L)))
return inv[:, : len(B)]
def pdf(self, x): """ Evaluates the pdf of the instantiated Multivariate Normal object """ Z = (2*pi)**(-self.k/2.0)*det(self.sigma)**(0.5) #normalization term pdf = Z*exp(-0.5*(x-self.mu).T.dot(pinv(self.sigma)).dot(x-self.mu)) return pdf
def i_kinematics(self, epsilon, goal, step=1.0):
num_links = self.num_links
ang = self.angles
p_e = vToThree(get_p(ang, self.lengths, num_links-1))
new_ang = ang
while LA.norm(goal - p_e) > epsilon:
if step < 0.0001:
return new_ang
else:
# print(jacobian(ang, self.lengths))
# print(goal)
# print(p_e)
# print("step: " + str(step))
dr = (LA.pinv(jacobian(ang, self.lengths)) * (goal - p_e))
new_ang = []
new = []
for i in range(num_links):
new.append(vectorize(ang)[i] + (step * getvec(dr, i * 3, (i+1) * 3 - 1)))
new_ang = devector(new)
new_p_e = vToThree(get_p(new_ang, self.lengths, num_links-1))
# The new guess is worse
# print("distances are: " + str(LA.norm(goal - p_e)) + " vs " + str(LA.norm(goal - new_p_e)))
if LA.norm(goal - p_e) < LA.norm(goal - new_p_e) + 0.00001:
step = step/2
# The new guess is better
else:
step = 1.0
p_e = new_p_e
ang = new_ang
return new_ang
def find_new_points(imgtf, shapeo, means, covs, k, m, orient):
stop = False
counter = 0
shape = np.copy(shapeo)
normalvectors = gl.get_normals(shape)
normalvectors = np.reshape(normalvectors,(normalvectors.size / 2, 2),'F')
shape = np.reshape(shape,(shape.size / 2, 2),'F')
# For each point
for i in range(shape.size / 2):
# Calculate own sample (slice along normal)
slices = gl.get_slice_indices(shape[i],normalvectors[i],m)
own_gradient_profile = gl.slice_image(slices,imgtf)
dist = np.zeros(2*(m - k) + 1)
for j in range(0,2*(m - k) + 1):
dist[j] = scp.mahalanobis(own_gradient_profile[j:j+(2*k + 1)].flatten(),means[i],LA.pinv(covs[i]))
min_ind = np.argmin(dist)
# Calculate the lower boundary and upper boundary for central 50%
lower = slices[0,:].size / 4.0
upper = lower + slices[0,:].size / 2.0
# Check whether the crown part of upper incisors is not in central 50%
if (orient == 0 and i % 40 >= 10 and i % 40 < 30 and (min_ind + k < lower or min_ind + k > upper)):
counter += 1
# Check whether the crown part of lower incisors is not in central 50%
elif (orient == 1 and (i % 40 < 8 or i % 40 >= 28) and (min_ind + k < lower or min_ind + k > upper)):
counter += 1
new_point = slices[:,min_ind+k]
shape[i] = new_point
# If less than 10% of the newly suggested points are outside of the central 50%, the algorithm has converged
if float(counter) / (shape.size/4) < 0.1:
stop = True
shape = np.reshape(shape,(shape.size, 1),'F')
return shape, stop
def Trajectory_generator(A, B, K_old, x0, noise, alpha, number_of_iteration, control_mode):
''' generate the trajectory and upper-bound based on DGR paper and compare it to DeePC paper '''
# System dimensions
n = np.shape(A)[0]
m = np.shape(B)[1]
mean_w = 0
############# Implementing DeePC for comparison ###################
# Parameters for DeePC algorithm (for comparison)
if control_mode == 0 or control_mode == 2:
T_ini = 1 # 8
N_deepc = 4 # 30
else:
T_ini = 1
N_deepc = 5
# L_deepc = T_ini + N_deepc
T_deepc = (m+1)*(T_ini+N_deepc + n)
# solver parameters
q = 80
r = 0.000000001
lam_sigma = 800000
lam_g = 1
X_deepc = [x0]
U_deepc = []
# offline data
data_u = []
data_x = [x0]
# offline data generation
for t in range(T_deepc):
data_u.append(-K_old@data_x[-1] + np.array(np.random.normal(mean_w, 2, size=(m, 1))))
data_x.append(A @ data_x[-1] + B @ data_u[-1])
print('DeePC started')
# DeePC implementation
for t in range(number_of_iteration):
if noise:
if control_mode == 0 or control_mode == 2:
w = np.array(np.random.normal(mean_w, 0.001, size=(n, 1)))
else:
w = np.array(np.random.normal(mean_w, 0.01, size=(n, 1)))
else:
w = np.zeros((n, 1))
if t <= T_ini:
U_deepc.append(np.array(np.random.normal(mean_w, 0.1, size=(m, 1))))
X_deepc.append(A @ X_deepc[-1] + B @ U_deepc[-1] + w)
else:
if la.norm(X_deepc[-1]) > 1000:
print('DeePC is blown up at')
print(t)
break
try:
u_N_deepc = deepc(U_deepc, X_deepc[:-1], data_u, data_x[:-1], T_ini, N_deepc, q, r, lam_sigma, lam_g)
except:
print('Deepc is stopped at iteration')
print(t)
break
U_deepc.append(u_N_deepc)
X_deepc.append(A@X_deepc[-1] + B @ U_deepc[-1] + w)
print('DeePC ended')
###################### Algorithm 1 implementation ####################
# Parameters for Algorithm 1
X = [x0]
Y = []
K = [np.zeros((m, n))]
u = []
G_alpha = la.pinv(alpha * np.eye(m) + B.T @ B) @ B.T
X_old = [x0]
X_new = [x0]
print('DGR started')
for t in range(number_of_iteration):
if noise:
if control_mode == 0 or control_mode == 2:
w = np.array(np.random.normal(mean_w, 0.001, size=(n, 1)))
else:
w = np.array(np.random.normal(mean_w, 0.01, size=(n, 1)))
else:
w = np.zeros((n, 1))
u.append(-K[-1] @ X[-1])
X.append(A @ X[-1] + B @ u[-1] + w)
Y.append(X[-1] - B @ u[-1] )
K.append(G_alpha @ np.hstack(Y) @ la.pinv(np.hstack(X[:-1])))
if t < T_deepc:
X_new.append(X[-1])
elif t == T_deepc:
xposition = t
Q = np.eye(np.shape(A)[0])
if control_mode == 0 or control_mode ==2:
R = 0.0000001*np.eye(np.shape(B)[1])
else:
R = 0.0000001*np.eye(np.shape(B)[1])
A_hat = np.hstack(Y) @ la.pinv(np.hstack(X[:-1]))
(_,_,K1) = control.dare(A_hat, B, Q, R)
X_new.append((A-B@K1) @ X_new[-1] + w)
else:
X_new.append((A-B@K1) @ X_new[-1] + w)
if la.norm(X_old[-1]) < 200:
X_old.append((A - B @ K_old) @ X_old[-1] + w)
print('DGR ended')
# Find upper-bound based on Lemma 2
z = [X[0]]
z_bar = [X[0] / la.norm(X[0])]
w_bar = [X[0] / la.norm(X[0]) - z_bar[0]]
for t in range(1, number_of_iteration+1):
X_subspace = np.hstack(X[:t])
l = len(X_subspace)
Ux, _ , _ = la.svd(X_subspace, 0)
z.append((np.eye(l) - Ux @ Ux.T) @ X[t])
if la.norm(z[-1]) > 10e-12:
z_bar.append(z[-1] / la.norm(z[-1]))
w_bar.append(X[t]/la.norm(X[t]) - z_bar[-1])
else:
z_bar.append(np.zeros((n, 1)))
w_bar.append(X[t] / la.norm(X[t]))
Ur, _, _ = la.linalg.svd(B, 0)
tilde_A = (np.eye(n) - Ur @ Ur.T) @ A
tilde_B = Ur @ Ur.T @ A
a_t = [la.norm(A)]
for t in range(1, number_of_iteration + 1):
a_t.append(la.norm(la.matrix_power(tilde_A, t) @ A, 2))
L = [0]
UB = [la.norm(X[0])]
L.append(la.norm(A @ z_bar[0]))
UB.append(L[-1] * la.norm(X[0]))
Delta = B @ (la.pinv(B)-G_alpha) @ A
for t in range(2, number_of_iteration + 1):
sum_bl = 0
for r in range(1, t):
sum_bl += np.sqrt( la.norm( la.matrix_power(tilde_A, t-1-r) @ tilde_B @ z_bar[r])**2
+ la.norm( la.matrix_power(tilde_A, t-1-r) @ Delta @ w_bar[r])**2 ) * L[r]
L.append( a_t[t-1] + sum_bl )
UB.append( L[-1] * la.norm(X[0]) )
# ###################### DPC for a system after DGR is done ###############
# online data generation for a system with DGR in the closed loop
data_u = []
data_x = [x0]
# Parameters for DGR
Y = []
u_dgr = []
K_dgr = [np.zeros((m, n))]
G_alpha = la.pinv(alpha * np.eye(m) + B.T @ B) @ B.T
for t in range(T_deepc):
if noise:
if control_mode == 0 or control_mode == 2:
w = np.array(np.random.normal(mean_w, 0.001, size=(n, 1)))
else:
w = np.array(np.random.normal(mean_w, 0.01, size=(n, 1)))
else:
w = np.zeros((n, 1))
data_u.append(np.array(np.random.normal(mean_w, 1, size=(m, 1))))
u_dgr.append(-K_dgr[-1] @ data_x[-1] + data_u[-1])
data_x.append(A @ data_x[-1] + B@u_dgr[-1] + w)
Y.append(data_x[-1] - B @ u_dgr[-1])
K_dgr.append(G_alpha @ np.hstack(Y) @ la.pinv(np.hstack(data_x[:-1])))
print('DGR+DeePC started')
# DeePC implementation
X_deepc_dgr = []
U_deepc_dgr = []
X_deepc_dgr[0:T_deepc] = data_x
U_deepc_dgr[0:T_deepc] = data_u
# solver parameters
if control_mode == 0 or control_mode == 2:
q = 400
r = 0.05
lam_sigma = 10000
lam_g = 1
elif control_mode == 1 or control_mode == 3:
q = 400
r = 0.05
lam_sigma = 10000
lam_g = 1
for t in range(T_deepc, number_of_iteration):
if noise:
if control_mode == 0 or control_mode == 2:
w = np.array(np.random.normal(mean_w, 0.001, size=(n, 1)))
else:
w = np.array(np.random.normal(mean_w, 0.01, size=(n, 1)))
else:
w = np.zeros((n, 1))
if la.norm(X_deepc_dgr[-1]) > 1000:
print('DeePC is blown up at')
print(t)
break
try:
u_N_deepc_dgr = deepc(U_deepc_dgr, X_deepc_dgr[:-1], data_u, data_x[:-1], T_ini, N_deepc, q, r, lam_sigma,
lam_g)
except:
print('DeePC is stopped at iteration')
print(t)
break
U_deepc_dgr.append(u_N_deepc_dgr)
X_deepc_dgr.append(A @ X_deepc_dgr[-1] + B @(-K_dgr[-1]@ X_deepc_dgr[-1] + U_deepc_dgr[-1]) + w)
print('DGR+DeePC ended')
return X, X_old, UB, X_new, xposition, K, X_deepc, X_deepc_dgr
def main():
if len(sys.argv) != 4:
print 'usage: ./lin_reg.py <start time point> <end time point> <laser wavelength>'
sys.exit(1)
hbar = 6.582119514E-16 ## plank's constant in eV s
c_speed = 2.99792458E8 ## speed of light in m/s
t_start = float(sys.argv[1]) ## start time in femtoseconds
t_stop = float(sys.argv[2]) ## stop time in femtoseconds
omega = 1E-6 * 2.0 * np.pi * c_speed / float(
sys.argv[3]) ## ang frequency of sinusoidal component
# read in all data points from file
f = open('fort.400', 'rU') # opens file into the variable f
N_points = 0
time = []
value = []
for l in f:
row = l.split()
time.append(float(row[0]))
value.append(float(row[1]))
N_points = N_points + 1 ## number of time points
# print row[0], row[1]
f.close()
# define data points used in linear regression
t = 0
i = 0
t_arr = []
y_arr = []
while (t < t_stop):
if time[i] > t_start:
t_arr.append(time[i])
y_arr.append(value[i])
t = time[
i] # technically will keep first point AFTER t_stop but that's ok
i = i + 1
m = len(t_arr) ## number of data points
#prepare linear regression variables
s_data_out = open('sample_data.dat', 'w')
n = 4 ## number of linear regression parameters
arr = []
for i in range(m):
s_data_out.write(str(t_arr[i]) + " " + str(y_arr[i]) + '\n')
row = []
for j in range(n):
if j == 0: elem = 1.0
if j == 1: elem = t_arr[i]
if j == 2: elem = np.sin(omega * t_arr[i])
if j == 3: elem = np.cos(omega * t_arr[i])
row.append(elem)
arr.append(row)
x_mat = np.array(arr)
theta_arr = [0.0] * n
# solve via Normal Equation
theta_arr = np.dot(
np.dot(pinv(np.dot(np.transpose(x_mat), x_mat)), np.transpose(x_mat)),
y_arr)
print
print 'theta0: ', theta_arr[0]
print 'theta1: ', theta_arr[1]
print 'theta2: ', theta_arr[2]
print 'theta3: ', theta_arr[3]
print
print 'slope: ', theta_arr[1]
print 'y-int: ', theta_arr[0]
print
print 'rate of electron transfer: ', theta_arr[1], ' electrons per fs'
fit_out_log = open('fit.log', 'w')
fit_out_log.write('theta0: ' + str(theta_arr[0]) + '\n' + 'theta1: ' +
str(theta_arr[1]) + '\n' + 'theta2: ' +
str(theta_arr[2]) + '\n' + 'theta3: ' +
str(theta_arr[3]) + '\n' + '\n')
fit_out_log.write('slope: ' + str(theta_arr[1]) + '\n' + 'y-int: ' +
str(theta_arr[0]) + '\n')
fit_out = open('fit.dat', 'w')
linear_out = open('linear_fit.dat', 'w')
for i in range(N_points):
hyp = theta_arr[0] + theta_arr[1] * time[i] + theta_arr[2] * np.sin(
omega * time[i]) + theta_arr[3] * np.cos(omega * time[i])
hyp_line = theta_arr[0] + theta_arr[1] * time[i]
fit_out.write(str(time[i]) + " " + str(hyp) + '\n')
linear_out.write(str(time[i]) + " " + str(hyp_line) + '\n')
def obs_avoidance_interpolation_moving(x,
xd,
obs=[],
attractor='none',
weightPow=2):
# This function modulates the dynamical system at position x and dynamics xd such that it avoids all obstacles obs. It can furthermore be forced to converge to the attractor.
#
# INPUT
# x [dim]: position at which the modulation is happening
# xd [dim]: initial dynamical system at position x
# obs [list of obstacle_class]: a list of all obstacles and their properties, which present in the local environment
# attractor [list of [dim]]]: list of positions of all attractors
# weightPow [int]: hyperparameter which defines the evaluation of the weight
#
# OUTPUT
# xd [dim]: modulated dynamical system at position x
#
# Initialize Variables
N_obs = len(obs) #number of obstacles
if N_obs == 0:
return xd
d = x.shape[0]
Gamma = np.zeros((N_obs))
if type(attractor) == str:
if attractor == 'default': # Define attractor position
attractor = np.zeros((d))
N_attr = 1
else:
N_attr = 0
else:
N_attr = 1
# Linear and angular roation of velocity
xd_dx_obs = np.zeros((d, N_obs))
xd_w_obs = np.zeros(
(d, N_obs)) #velocity due to the rotation of the obstacle
# Modulation matrices
E = np.zeros((d, d, N_obs))
D = np.zeros((d, d, N_obs))
M = np.zeros((d, d, N_obs))
E_orth = np.zeros((d, d, N_obs))
# Rotation matrix
R = np.zeros((d, d, N_obs))
for n in range(N_obs):
# Move the position into the obstacle frame of reference
if obs[n].th_r: # Nonzero value
R[:, :, n] = compute_R(d, obs[n].th_r)
else:
R[:, :, n] = np.eye(d)
# Move to obstacle centered frame
x_t = R[:, :, n].T @ (x - obs[n].x0)
E[:, :, n], D[:, :,
n], Gamma[n], E_orth[:, :,
n] = compute_modulation_matrix(
x_t, obs[n], R[:, :, n])
if N_attr:
d_a = LA.norm(x - np.array(attractor)) # Distance to attractor
weight = compute_weights(np.hstack((Gamma, [d_a])), N_obs + N_attr)
else:
weight = compute_weights(Gamma, N_obs)
xd_obs = np.zeros((d))
for n in range(N_obs):
if d == 2:
xd_w = np.cross(np.hstack(([0, 0], obs[n].w)),
np.hstack((x - np.array(obs[n].x0), 0)))
xd_w = xd_w[0:2]
elif d == 3:
xd_w = np.cross(obs[n].w, x - obs[n].x0)
else:
warnings.warn('NOT implemented for d={}'.format(d))
#the exponential term is very helpful as it help to avoid the crazy rotation of the robot due to the rotation of the object
exp_weight = np.exp(-1 / obs[n].sigma * (np.max([Gamma[n], 1]) - 1))
xd_obs_n = exp_weight * (np.array(obs[n].xd) + xd_w)
xd_obs_n = E_orth[:, :, n].T @ xd_obs_n
xd_obs_n[0] = np.max(xd_obs_n[0], 0) # Onl use orthogonal part
xd_obs_n = E_orth[:, :, n] @ xd_obs_n
xd_obs = xd_obs + xd_obs_n * weight[n]
xd = xd - xd_obs #computing the relative velocity with respect to the obstacle
# Create orthogonal matrix
xd_norm = LA.norm(xd)
if xd_norm: # nonzero
xd_normalized = xd / xd_norm
else:
xd_normalized = xd
xd_t = np.array([xd_normalized[1], -xd_normalized[0]])
Rf = np.array([xd_normalized, xd_t]).T
xd_hat = np.zeros((d, N_obs))
xd_hat_magnitude = np.zeros((N_obs))
k_ds = np.zeros((d - 1, N_obs))
for n in range(N_obs):
M[:, :, n] = R[:, :, n] @ E[:, :, n] @ D[:, :, n] @ LA.pinv(
E[:, :, n]) @ R[:, :, n].T
xd_hat[:, n] = M[:, :, n] @ xd # velocity modulation
xd_hat_magnitude[n] = np.sqrt(np.sum(xd_hat[:, n]**2))
if xd_hat_magnitude[n]: # Nonzero hat_magnitude
xd_hat_normalized = xd_hat[:, n] / xd_hat_magnitude[
n] # normalized direction
else:
xd_hat_normalized = xd_hat[:, n]
if not d == 2:
warnings.warn('not implemented for d neq 2')
xd_hat_normalized_velocityFrame = Rf @ xd_hat_normalized
# Kappa space - directional space
k_fn = xd_hat_normalized_velocityFrame[1:]
kfn_norm = LA.norm(k_fn) # Normalize
if kfn_norm: # nonzero
k_fn = k_fn / kfn_norm
sumHat = np.sum(xd_hat_normalized * xd_normalized)
if sumHat > 1 or sumHat < -1:
sumHat = max(min(sumHat, 1), -1)
warnings.warn('cosinus out of bound!')
k_ds[:, n] = np.arccos(sumHat) * k_fn.squeeze()
xd_hat_magnitude = np.sqrt(np.sum(xd_hat**2, axis=0))
if N_attr: #nonzero
k_ds = np.hstack((k_ds, np.zeros(
(d - 1, N_attr)))) # points at the origin
xd_hat_magnitude = np.hstack((xd_hat_magnitude, LA.norm(
(xd)) * np.ones(N_attr)))
# Weighted interpolation for several obstacles
weight = weight**weightPow
if not LA.norm(weight, 2):
warnings.warn('trivial weight.')
weight = weight / LA.norm(weight, 2)
xd_magnitude = np.sum(xd_hat_magnitude * weight)
k_d = np.sum(k_ds * np.tile(weight, (d - 1, 1)), axis=1)
norm_kd = LA.norm(k_d)
if norm_kd: # Nonzero
n_xd = Rf.T @ np.hstack(
(np.cos(norm_kd), np.sin(norm_kd) / norm_kd * k_d))
else:
n_xd = Rf.T @ np.hstack((1, k_d))
xd = xd_magnitude * n_xd.squeeze()
# transforming back from object frame of reference to inertial frame of reference
xd = xd + xd_obs
return xd
def get_unseen_tfidf(unseen_event, model, dim):
unseen_event_terms = jieba.analyse.extract_tags(unseen_event)
unseen_event_vector = [0] * dim
for term in unseen_event_terms:
if term in model.vocabulary_:
unseen_event_vector[model.vocabulary_[term]] += model.idf_[
model.vocabulary_[term]]
return unseen_event_vector
if __name__ == "__main__":
VSM_MODEL, TERM_FACTOR_MATRIX = train_mf()
UNSEENID_TO_SENTENCE = load_unseen()
UNSEEN_EMBEDDING_DICT = {}
for id_, (title_string, description) in UNSEENID_TO_SENTENCE.items():
newitem_sentence = title_string + description
unseen_tfidf = np.array([
get_unseen_tfidf(newitem_sentence, VSM_MODEL,
TERM_FACTOR_MATRIX.shape[0])
])
UNSEEN_EMBEDDING_DICT[id_] = np.dot(
unseen_tfidf, pinv(TERM_FACTOR_MATRIX.T)).tolist()[0]
with open('mf.txt', 'wt') as fout:
fout.write("{}\n".format(len(UNSEEN_EMBEDDING_DICT)))
for id_, embedding in UNSEEN_EMBEDDING_DICT.items():
fout.write("{} {}\n".format(
id_, ' '.join(map(lambda x: str(round(x, 6)), embedding))))
def AnDA_analog_forecasting(x, AF):
""" Apply the analog method on catalog of historical data to generate forecasts. """
# initializations
N, n = x.shape
xf = np.zeros([N, n])
xf_mean = np.zeros([N, n])
stop_condition = 0
i_var = np.array([0])
i_var_iter = 0
# local or global analog forecasting
while (stop_condition != 1):
# in case of global approach
if np.all(AF.neighborhood == 1):
i_var_neighboor = np.arange(n, dtype=np.int64)
i_var = np.arange(n, dtype=np.int64)
stop_condition = 1
# in case of local approach
else:
i_var_neighboor = np.where(AF.neighborhood[int(i_var), :] == 1)[0]
# find the indices and distances of the k-nearest neighbors (knn)
#kdt = KDTree(AF.catalog.analogs[:,i_var_neighboor], leaf_size=50, metric='euclidean')
#dist_knn, index_knn = kdt.query(x[:,i_var_neighboor], AF.k)
# find the indices and distances of the k-nearest neighbors (knn)
if not AF.initialized:
if i_var_iter == 0:
AF.kdt = list()
AF.kdt.append(
KDTree(AF.catalog.analogs[:, i_var_neighboor],
leaf_size=50,
metric='euclidean')
) #If we are using the global approach we can compute kdt only once.
kdt = AF.kdt[i_var_iter]
dist_knn, index_knn = kdt.query(x[:, i_var_neighboor], AF.k)
# parameter of normalization for the kernels
if AF.kernel == 'gaussian':
lambdaa = np.median(dist_knn)
# compute weights
if AF.k == 1:
weights = np.ones([N, 1])
else:
weights = mk_stochastic(
np.exp(-np.power(dist_knn, 2) / lambdaa**2))
if AF.kernel == 'tricube':
# compute weights
if AF.k == 1:
weights = np.ones([N, 1])
else:
weights = np.ones((N, AF.k))
for j in range(weights.shape[0]):
lambdaa = max(dist_knn[j, :].squeeze())
weights[j, :] = (
1 - (dist_knn[j, :].squeeze() / lambdaa)**3)**3
weights[j, :] = weights[j, :] / np.sum(weights[j, :])
# for each member/particle
# for each member/particle
for i_N in range(0, N):
# initialization
xf_tmp = np.zeros([AF.k, np.max(i_var) + 1])
# method "locally-constant"
if (AF.regression == 'locally_constant'):
# compute the analog forecasts
xf_tmp[:, i_var] = AF.catalog.successors[np.ix_(
index_knn[i_N, :], i_var)]
# weighted mean and covariance
xf_mean[i_N, i_var] = np.sum(
xf_tmp[:, i_var] *
np.repeat(weights[i_N, :][np.newaxis].T, len(i_var), 1), 0)
E_xf = (xf_tmp[:, i_var] -
np.repeat(xf_mean[i_N, i_var][np.newaxis], AF.k, 0)).T
cov_xf = 1.0 / (
1.0 - np.sum(np.power(weights[i_N, :], 2))) * np.dot(
np.repeat(weights[i_N, :][np.newaxis], len(i_var), 0) *
E_xf, E_xf.T)
# method "locally-incremental"
elif (AF.regression == 'increment'):
# compute the analog forecasts
xf_tmp[:, i_var] = np.repeat(
x[i_N, i_var][np.newaxis],
AF.k, 0) + AF.catalog.successors[np.ix_(
index_knn[i_N, :], i_var)] - AF.catalog.analogs[np.ix_(
index_knn[i_N, :], i_var)]
# weighted mean and covariance
xf_mean[i_N, i_var] = np.sum(
xf_tmp[:, i_var] *
np.repeat(weights[i_N, :][np.newaxis].T, len(i_var), 1), 0)
E_xf = (xf_tmp[:, i_var] -
np.repeat(xf_mean[i_N, i_var][np.newaxis], AF.k, 0)).T
cov_xf = 1.0 / (
1 - np.sum(np.power(weights[i_N, :], 2))) * np.dot(
np.repeat(weights[i_N, :][np.newaxis], len(i_var), 0) *
E_xf, E_xf.T)
# method "locally-linear"
elif (AF.regression == 'local_linear'):
# define analogs, successors and weights
X = AF.catalog.analogs[np.ix_(index_knn[i_N, :],
i_var_neighboor)]
Y = AF.catalog.successors[np.ix_(index_knn[i_N, :], i_var)]
w = weights[i_N, :][np.newaxis]
# compute centered weighted mean and weighted covariance
Xm = np.sum(X * w.T, axis=0)[np.newaxis]
Xc = X - Xm
# regression on principal components
Xr = np.c_[np.ones(X.shape[0]), Xc]
Cxx = np.dot(w * Xr.T, Xr)
Cxx2 = np.dot(w**2 * Xr.T, Xr)
Cxy = np.dot(w * Y.T, Xr)
inv_Cxx = pinv(
Cxx, rcond=0.01
) # in case of error here, increase the number of analogs (AF.k option)
beta = np.dot(inv_Cxx, Cxy.T)
X0 = x[i_N, i_var_neighboor] - Xm
X0r = np.c_[np.ones(X0.shape[0]), X0]
# weighted mean
xf_mean[i_N, i_var] = np.dot(X0r, beta)
pred = np.dot(Xr, beta)
res = Y - pred
xf_tmp[:, i_var] = xf_mean[i_N, i_var] + res
# weigthed covariance
cov_xfc = np.dot(w * res.T,
res) / (1 - np.trace(np.dot(Cxx2, inv_Cxx)))
cov_xf = cov_xfc * (
1 + np.trace(Cxx2 @ inv_Cxx @ X0r.T @ X0r @ inv_Cxx))
# constant weights for local linear
weights[i_N, :] = 1.0 / len(weights[i_N, :])
# error
else:
raise ValueError("""\
Error: choose AF.regression between \
'locally_constant', 'increment', 'local_linear' """)
'''
# method "globally-linear" (to finish)
elif (AF.regression == 'global_linear'):
### REMARK: USE i_var_neighboor IN THE FUTURE! ####
xf_mean[i_N,:] = AF.global_linear.predict(np.array([x[i_N,:]]))
if n==1:
cov_xf = np.cov((AF.catalog.successors - AF.global_linear.predict(AF.catalog.analogs)).T)[np.newaxis][np.newaxis]
else:
cov_xf = np.cov((AF.catalog.successors - AF.global_linear.predict(AF.catalog.analogs)).T)
# method "locally-forest" (to finish)
elif (AF.regression == 'local_forest'):
### REMARK: USE i_var_neighboor IN THE FUTURE! ####
xf_mean[i_N,:] = AF.local_forest.predict(np.array([x[i_N,:]]))
if n==1:
cov_xf = np.cov(((AF.catalog.successors - np.array([AF.local_forest.predict(AF.catalog.analogs)]).T).T))[np.newaxis][np.newaxis]
else:
cov_xf = np.cov((AF.catalog.successors - AF.local_forest.predict(AF.catalog.analogs)).T)
# weighted mean and covariance
#xf_tmp[:,i_var] = AF.local_forest.predict(AF.catalog.analogs[np.ix_(index_knn[i_N,:],i_var)]);
#xf_mean[i_N,i_var] = np.sum(xf_tmp[:,i_var]*np.repeat(weights[i_N,:][np.newaxis].T,len(i_var),1),0)
#E_xf = (xf_tmp[:,i_var]-np.repeat(xf_mean[i_N,i_var][np.newaxis],AF.k,0)).T;
#cov_xf = 1.0/(1.0-np.sum(np.power(weights[i_N,:],2)))*np.dot(np.repeat(weights[i_N,:][np.newaxis],len(i_var),0)*E_xf,E_xf.T);
'''
# Gaussian sampling
if (AF.sampling == 'gaussian'):
# random sampling from the multivariate Gaussian distribution
xf[i_N, i_var] = np.random.multivariate_normal(
xf_mean[i_N, i_var], cov_xf)
# Multinomial sampling
elif (AF.sampling == 'multinomial'):
# random sampling from the multinomial distribution of the weights
i_good = sample_discrete(weights[i_N, :], 1, 1)
xf[i_N, i_var] = xf_tmp[i_good, i_var]
# error
else:
raise ValueError("""\
Error: choose AF.sampling between 'gaussian', 'multinomial'
""")
# stop condition
if (np.array_equal(i_var, np.array([n - 1])) or (len(i_var) == n)):
stop_condition = 1
else:
i_var = i_var + 1
i_var_iter = i_var_iter + 1
AF.initialized = True
return xf, xf_mean
def regression_fit(data, design):
"""Finally uses the design matrix from build_design() and fits a linear regression to each voxel
Parameters
----------
data : fMRI data for a singe sunject
design: matrix returned by build_design()
Returns
-------
numpy array of estimated betas for each voxel
Example
-------
>>> data = get_img(1,1).get_data
>>> behavdata = get_data(1,1)
>>> design = build_design(data,behavdata)
>>> regression_fit(data, design).shape
(64, 64, 34, 4)
>>> regression_fit(data, design)[1,1,...]
array([[ 0., 0., 0., 0.],
[ 0., 0., 0., 0.],
[ 0., 0., 0., 0.],
[ 0., 0., 0., 0.],
[ 0., 0., 0., 0.],
[ 0., 0., 0., 0.],
[ 0., 0., 0., 0.],
[ 0., 0., 0., 0.],
[ 0., 0., 0., 0.],
[ 0., 0., 0., 0.],
[ 0., 0., 0., 0.],
[ 0., 0., 0., 0.],
[ 0., 0., 0., 0.],
[ 0., 0., 0., 0.],
[ 0., 0., 0., 0.],
[ 0., 0., 0., 0.],
[ 0., 0., 0., 0.],
[ 0., 0., 0., 0.],
[ 0., 0., 0., 0.],
[ 0., 0., 0., 0.],
[ 0., 0., 0., 0.],
[ 0., 0., 0., 0.],
[ 0., 0., 0., 0.],
[ 0., 0., 0., 0.],
[ 0., 0., 0., 0.],
[ 0., 0., 0., 0.],
[ 0., 0., 0., 0.],
[ 0., 0., 0., 0.],
[ 0., 0., 0., 0.],
[ 0., 0., 0., 0.],
[ 0., 0., 0., 0.],
[ 0., 0., 0., 0.],
[ 0., 0., 0., 0.],
[ 0., 0., 0., 0.]])
"""
data_2d = np.reshape(data, (-1, data.shape[-1]))
betas = npl.pinv(design).dot(data_2d.T)
betas = np.reshape(betas.T, data.shape[:-1] + (-1,))
return betas
def block_term_tensor_decomposition(tensor,
modes,
n_part,
ranks=None,
n_iter_max=100,
tol=10e-5,
random_state=None):
if ranks == None:
ranks = [tensor.shape[index - 1] for index in modes]
# 随机生成core-tensor，以及factor matrixs
rng = check_random_state(random_state)
core = [np.array(rng.random_sample(ranks)) for i in range(n_part)]
core_2 = [np.array(rng.random_sample(ranks)) for i in range(n_part)]
factors = [
np.array(
rng.random_sample((tensor.shape[index], ranks[index] * n_part)))
for index, mode in enumerate(modes)
]
rec_errors = []
for iteration in range(n_iter_max):
for mode in modes:
index = mode - 1
factors[index] = (blockdiag(core, mode).dot(
pinv(multi_mat_kr(factors, R=n_part,
mode=mode))).dot(unfold(tensor, mode).T)).T
for i in range(n_part):
factors[index][:, i * ranks[index]:(i + 1) *
ranks[index]], _ = QR(
factors[index][:, i * ranks[index]:(i + 1) *
ranks[index]])
# rebuilt core tensor
for i in range(n_part):
factors_rebult = []
for mode in modes:
index = mode - 1
factors_rebult.append(
factors[index][:, i * ranks[index]:(i + 1) * ranks[index]])
core[i] = multi_mode_dot(tensor,
factors_rebult,
modes=modes,
transpose=True)
rebuilt_tensor = rebuilt_block_term_tensor(core, factors, modes)
err = norm(rebuilt_tensor - tensor, 2) / norm(tensor, 2)
print('iteration is:', iteration, 'err is:', err)
vector_core = pinv(multi_mat_kr(factors,
R=n_part)).dot(tensor.reshape(-1, 1))
len_core = vector_core.shape[0] // n_part
for i in range(n_part):
core_2[i] = vector_core[i * len_core:(i + 1) *
len_core].reshape(ranks)
rebuilt_tensor_2 = rebuilt_block_term_tensor(core_2, factors, modes)
err_2 = norm(rebuilt_tensor_2 - tensor, 2) / norm(tensor, 2)
print('iteraton is :', iteration, 'err_2_vector_core is:', err_2)
rec_errors.append(err)
if iteration > 3:
if (np.abs(rec_errors[-1] - rec_errors[-2]) < tol): # 跳出循环的条件
break
return core, factors, rec_errors[-1], iteration
assertNumDiff(contactData.Av, Av_numdiff,
NUMDIFF_MODIFIER * np.sqrt(2 * EPS)) # threshold was 1e-4, is now 2.11e-4 (see assertNumDiff.__doc__)
Aq_numdiff = df_dq(rmodel, lambda _q: returna0(_q, v), q, h=eps)
Av_numdiff = df_dx(lambda _v: returna0(q, _v), v, h=eps)
assert (np.isclose(contactData.Aq, Aq_numdiff, atol=np.sqrt(eps)).all())
assert (np.isclose(contactData.Av, Av_numdiff, atol=np.sqrt(eps)).all())
q = pinocchio.randomConfiguration(rmodel)
v = rand(rmodel.nv) * 2 - 1
pinocchio.computeJointJacobians(rmodel, rdata, q)
J6 = pinocchio.getJointJacobian(rmodel, rdata, rmodel.joints[-1].id, pinocchio.ReferenceFrame.LOCAL).copy()
J = J6[:3, :]
v -= pinv(J) * J * v
x = np.concatenate([m2a(q), m2a(v)])
u = np.random.rand(rmodel.nv - 6) * 2 - 1
actModel = ActuationModelFreeFloating(rmodel)
contactModel3 = ContactModel3D(rmodel,
rmodel.getFrameId('gripper_left_fingertip_2_link'),
ref=np.random.rand(3),
gains=[4., 4.])
rmodel.frames[contactModel3.frame].placement = pinocchio.SE3.Random()
contactModel = ContactModelMultiple(rmodel)
contactModel.addContact(name='fingertip', contact=contactModel3)
model = DifferentialActionModelFloatingInContact(rmodel, actModel, contactModel, CostModelSum(rmodel))
data = model.createData()
import numpy as np
from sys import stdin
from numpy import linalg
from sklearn.preprocessing import PolynomialFeatures
def polynomial(X, order):
poly = PolynomialFeatures(order)
return poly.fit_transform(X)
if __name__ == "__main__":
P = 3
(F, N) = map(int, stdin.readline().split(" "))
train_data = np.array(
[map(float,
stdin.readline().split(" ")) for _ in xrange(N)])
T = int(stdin.readline())
test_data = np.array(
[map(float,
stdin.readline().split(" ")) for _ in xrange(T)])
train_X = polynomial(train_data[:, :-1], P)
train_y = train_data[:, -1]
test_X = polynomial(test_data, P)
w = np.dot(linalg.pinv(train_X), train_y)
test_yhat = np.dot(test_X, w)
print "\n".join([str(v) for v in test_yhat])
def _calc_params(self):
self.W = pinv(self.X_mat.getT() * self.X_mat) * self.X_mat.getT() * self.Y_mat
def bla_periodic(U, Y): #(u, y, nper, fs, fmin, fmax):
"""Calculate the frequency response matrix, and the corresponding noise and
total covariance matrices from the spectra of periodic input/output data.
Note that the term stochastic nonlinear contribution term is a bit
misleading. The NL contribution is deterministic given the same forcing
buy differs between realizations.
G(f) = FRF(f) = Y(f)/U(f) (Y/F in classical notation)
Y and U is the output and input of the system in frequency domain.
Parameters
----------
u : ndarray
Forcing signal
y : ndarray
Response signal (displacements)
fs : float
Sampling frequency
fmin : float
Starting frequency in Hz
fmax : float
Ending frequency in Hz
Returns
-------
G : ndarray
Frequency response matrix(FRM)
covGML : ndarray
Total covariance (= stochastic nonlinear contributions + noise)
covGn : ndarray
Noise covariance
"""
# Number of inputs, realization, periods and frequencies
m, R, P, F = U.shape
p = Y.shape[0] # number of outputs
M = np.floor(R/m).astype(int) # number of block of experiments
if M*m != R:
print('Warning: suboptimal number of experiments: B*m != M')
# Reshape in M blocks of m experiments
U = U[:,:m*M].reshape((m,m,M,P,F))
Y = Y[:,:m*M].reshape((p,m,M,P,F))
if P > 1:
# average input/output spectra over periods
U_mean = np.mean(U,axis=3) # m x m x M x F
Y_mean = np.mean(Y,axis=3)
# Estimate noise spectra
# create new axis. We could have used U_m = np.mean(U,3, keepdims=True)
NU = U - U_mean[:,:,:,None,:] # m x m x M x P x F
NY = Y - Y_mean[:,:,:,None,:]
# Calculate input/output noise (co)variances on averaged(over periods)
# spectra
covU = np.empty((m*m,m*m,M,F), dtype=complex)
covY = np.empty((p*m,p*m,M,F), dtype=complex)
covYU = np.empty((p*m,m*m,M,F), dtype=complex)
for mm in range(M): # Loop over experiment blocks
# noise spectrum of experiment block mm (m x m x P x F)
NU_m = NU[:,:,mm]
NY_m = NY[:,:,mm]
for f in range(F):
# TODO extend this using einsum, so we avoid all loops
# TODO fx: NU_m[...,f].reshape(-1,*NU_m[...,f].shape[2:])
# flatten the m x m dimension and use einsum to take the outer
# product of m*m x m*m and then sum over the p periods.
tmpUU = NU_m[...,f].reshape(-1, P) # create view
tmpYY = NY_m[...,f].reshape(-1, P)
covU[:,:,mm,f] = np.einsum('ij,kj->ik',tmpUU,tmpUU.conj()) / (P-1)/P
covY[:,:,mm,f] = np.einsum('ij,kj->ik',tmpYY,tmpYY.conj()) / (P-1)/P
covYU[:,:,mm,f] = np.einsum('ij,kj->ik',tmpYY,tmpUU.conj()) / (P-1)/P
# Further calculations with averaged spectra
U = U_mean # m x m x M x F
Y = Y_mean
else:
U = U.squeeze(axis=3)
Y = Y.squeeze(axis=3)
# Compute FRM and noise and total covariance on averaged(over experiment
# blocks and periods) FRM
G = np.empty((p,m,F), dtype=complex)
covGML = np.empty((m*p,m*p,F), dtype=complex)
covGn = np.empty((m*p,m*p,F), dtype=complex)
Gm = np.empty((p,m,M), dtype=complex)
U_inv_m = np.empty((m,m,M), dtype=complex)
covGn_m = np.empty((m*p,m*p,M), dtype=complex)
for f in range(F):
# Estimate the frequency response matrix (FRM)
for mm in range(M):
# psudo-inverse by svd. A = usvᴴ, then A⁺ = vs⁺uᴴ where s⁺=1/s
U_inv_m[:,:,mm] = pinv(U[:,:,mm,f])
# FRM of experiment block m at frequency f
Gm[:,:,mm] = Y[:,:,mm,f] @ U_inv_m[:,:,mm]
# Average FRM over experiment blocks
G[:,:,f] = Gm.mean(2)
# Estimate the total covariance on averaged FRM
if M > 1:
NG = G[:,:,f,None] - Gm
tmp = NG.reshape(-1, M)
covGML[:,:,f] = np.einsum('ij,kj->ik',tmp,tmp.conj()) / M/(M-1)
# Estimate noise covariance on averaged FRM (only if P > 1)
if P > 1:
for mm in range(M):
U_invT = U_inv_m[:,:,mm].T
A = np.kron(U_invT, np.eye(p))
B = -np.kron(U_invT, Gm[:,:,mm])
AB = A @ covYU[:,:,mm,f] @ B.conj().T
covGn_m[:,:,mm] = A @ covY[:,:,mm,f] @ A.conj().T + \
B @ covU[:,:,mm,f] @ B.conj().T + \
(AB + AB.conj().T)
covGn[:,:,f] = covGn_m.mean(2)/M
# No total covariance estimate possible if only one experiment block
if M < 2:
covGML = None
# No noise covariance estimate possible if only one period
if P == 1:
covGn = None
return G, covGML, covGn
""" dataproj.py """
from datared import *
from numpy.linalg import svd, pinv
mu21 = (mu2 - mu1).reshape(3,1)
mu31 = (mu3 - mu1).reshape(3,1)
W = np.hstack((mu21, mu31))
U,_,_ = svd(W) # we only need U
P = W @ pinv(W)
R = U.T @ P
RX1 = (R @ X1.T).T
RX2 = (R @ X2.T).T
RX3 = (R @ X3.T).T
# %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
plt.plot(RX1[:,0],RX1[:,1],'b.',alpha=0.5,markersize=2)
plt.plot(RX2[:,0],RX2[:,1],'g.',alpha=0.5,markersize=2)
plt.plot(RX3[:,0],RX3[:,1],'r.',alpha=0.5,markersize=2)
plt.savefig('pcaproj2py.pdf')
plt.show()
# %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
def f1d(mu_SMM1, sig_SMM1, sim_vals):
'''
--------------------------------------------------------------------
This function performs the two-step SMM estimator by using the
estimates from part (c) with two moments to generate an estimator for
the variance covariance matrix omega hat 2step, which then will be
used to get the two-step estimator for the optimal weighting matrix W
hat 2step.
--------------------------------------------------------------------
INPUT: mu_SMM1, sig_SMM1, sim_val
RETURN: none
--------------------------------------------------------------------
'''
pts = np.loadtxt('incomes.txt') # load txt file
S = 300.0
N = 200.0
seed = 1234
err1 = err_vec(pts, sim_vals, mu_SMM1, sig_SMM1, False)
VCV2 = np.dot(err1, err1.T) / pts.shape[0]
W_hat2 = lin.pinv(VCV2)
mu = 11
sigma = 0.2
params_init = np.array([mu_SMM1, sig_SMM1])
args = (pts, W_hat2, N, S, seed)
results_d = opt.minimize(criterion,
params_init,
args=(args),
method='Nelder-Mead',
bounds=((None, None), (1e-10, None)))
mu_SMM2, sig_SMM2 = results_d.x
# Report the estimated parameter values:
print('1d. mu_SMM2=', mu_SMM2, ' sig_SMM2=', sig_SMM2)
# Calculate and report the value of the GMM criterion function at the estimated parameter values:
params_SMM = np.array([mu_SMM2, sig_SMM2])
value = criterion(params_SMM, *args)[0][0]
print(
'The value of the SMM criterion function at the estimated parameter values is',
value)
# Calculate and report and compare my two data moments against my two model moments at the estimated parameter values.
sim_vals = LN_draws(mu_SMM2, sig_SMM2, N, S, seed)
mean_data, std_data = data_moments(pts)
mean_sim, std_sim = data_moments(sim_vals)
mean_model = mean_sim.mean()
std_model = std_sim.mean()
print('Mean of incomes =', mean_data, ', Standard deviation of incomes =',
std_data)
print('Mean of model =', mean_model, ', Standard deviation of model =',
std_model)
# print(results_d)-success
# Plot the estimated lognormal PDF against the histogram from part (a) and the estimated PDF from part (c):
f1a()
#part (c)
dist_pts = np.linspace(0, 150000,
10000) # 150000 is the upper bound of incomes
plt.plot(dist_pts,
LN_pdf(dist_pts, mu_SMM1, sig_SMM1),
linewidth=2,
color='k',
label='$\mu_{b}$: 11.331,$\sigma_{b}$: 0.210')
plt.legend(loc='upper right')
#part (d)
dist_pts = np.linspace(0, 150000,
10000) # 150000 is the upper bound of incomes
plt.plot(dist_pts,
LN_pdf(dist_pts, mu_SMM2, sig_SMM2),
'--',
linewidth=2,
color='r',
label='$\mu_{b}$: 11.330,$\sigma_{b}$: 0.211')
plt.legend(loc='upper right')
central = 1
B_fluxdiv = [[1,-2, 1,4,-2,1,-8, 4,-2], \
[1,-1, 1,1,-1,1,-1, 1,-1], \
[1, 0, 1,0, 0,1,0, 0,0], \
[1, 1, 1,1, 1,1,1, 1,1], \
[1,-2, 0,4, 0,0,-8, 0,0], \
[1,-1, 0,1, 0,0,-1, 0,0], \
[1, 0, 0,0, 0,0,0, 0,0], \
[1, 1, 0,1, 0,0,1, 0,0], \
[1,-2,-1,4, 2,1,-8,-4,-2], \
[1,-1,-1,1, 1,1,-1,-1,-1], \
[1, 0,-1,0, 0,1,0, 0,0], \
[1, 1,-1,1,-1,1,1, -1,1]]
B_fluxdiv_inv = la.pinv(B_fluxdiv)
#print("B_fluxdiv_inv", B_fluxdiv_inv)
B = [[1,-5/2, 1,25/4,-5/2,1,-125/8, 25/4,-5/2], \
[1,-3/2, 1, 9/4,-3/2,1, -27/8, 9/4,-3/2], \
[1,-1/2, 1, 1/4,-1/2,1, -1/8, 1/4,-1/2], \
[1, 1/2, 1, 1/4, 1/2,1, 1/8, 1/4, 1/2], \
[1,-5/2, 0,25/4, 0, 0,-125/8, 0, 0], \
[1,-3/2, 0, 9/4, 0, 0, -27/8, 0, 0], \
central*np.array([1,-1/2, 0, 1/4, 0, 0, -1/8, 0, 0]), \
central*np.array([1, 1/2, 0, 1/4, 0, 0, 1/8, 0, 0]), \
[1,-5/2,-1,25/4, 5/2,1,-125/8,-25/4,-5/2], \
[1,-3/2,-1, 9/4, 3/2,1, -27/8, -9/4,-3/2], \
[1,-1/2,-1, 1/4, 1/2,1, -1/8, -1/4,-1/2], \
[1, 1/2,-1, 1/4,-1/2,1, 1/8, -1/4, 1/2]]
X = X[..., np.newaxis]
X = np.concatenate((np.ones((len(X), 1)), X), axis=1)
return X
standardize = False
np.random.seed(3)
xtrain, ytrain, _, _ = polyDataMake()
if standardize:
xtrain = (xtrain - xtrain.mean(axis=0)) / xtrain.std(axis=0)
N = len(xtrain)
wBatch = np.dot(linalg.pinv(addOnes(xtrain)), ytrain)
ss = {}
w = np.zeros((2, N))
for i in range(N):
w[:, i], ss = linregUpdateSS(ss, xtrain[i], ytrain[i], xtrain, ytrain)
fig, ax1 = plt.subplots(1, 1, figsize=(9, 6))
ax1.plot(np.arange(2, N + 1),
w[0, 1:],
color='#e41a1c',
marker='o',
linestyle='None',
linewidth=2,
label='w0')
ax1.plot(np.arange(2, N + 1),
def test_diff(self, groups, rho=None, weight=None):
"""
test_diff(groups, rho=0)
Test for difference between survival curves
Parameters
----------
groups: A list of the values for exog to test for difference.
tests the null hypothesis that the survival curves for all
values of exog in groups are equal
rho: compute the test statistic with weight S(t)^rho, where
S(t) is the pooled estimate for the Kaplan-Meier survival function.
If rho = 0, this is the logrank test, if rho = 0, this is the
Peto and Peto modification to the Gehan-Wilcoxon test.
weight: User specified function that accepts as its sole arguement
an array of times, and returns an array of weights for each time
to be used in the test
Returns
-------
An array whose zeroth element is the chi-square test statistic for
the global null hypothesis, that all survival curves are equal,
the index one element is degrees of freedom for the test, and the
index two element is the p-value for the test.
Examples
--------
>>> import statsmodels.api as sm
>>> import matplotlib.pyplot as plt
>>> import numpy as np
>>> from statsmodels.sandbox.survival2 import KaplanMeier
>>> dta = sm.datasets.strikes.load()
>>> dta = dta.values()[-1]
>>> censoring = np.ones_like(dta[:,0])
>>> censoring[dta[:,0] > 80] = 0
>>> dta = np.c_[dta,censoring]
>>> km = KaplanMeier(dta,0,exog=1,censoring=2)
>>> km.fit()
Test for difference of survival curves
>>> log_rank = km3.test_diff([0.0645,-0.03957])
The zeroth element of log_rank is the chi-square test statistic
for the difference between the survival curves using the log rank test
for exog = 0.0645 and exog = -0.03957, the index one element
is the degrees of freedom for the test, and the index two element
is the p-value for the test
>>> wilcoxon = km.test_diff([0.0645,-0.03957], rho=1)
wilcoxon is the equivalent information as log_rank, but for the
Peto and Peto modification to the Gehan-Wilcoxon test.
User specified weight functions
>>> log_rank = km3.test_diff([0.0645,-0.03957], weight=np.ones_like)
This is equivalent to the log rank test
More than two groups
>>> log_rank = km.test_diff([0.0645,-0.03957,0.01138])
The test can be performed with arbitrarily many groups, so long as
they are all in the column exog
"""
groups = np.asarray(groups)
if self.exog == None:
raise ValueError("Need an exogenous variable for logrank test")
elif (np.in1d(groups,self.groups)).all():
data = self.data[np.in1d(self.data[:,self.exog],groups)]
t = ((data[:,self.endog]).astype(float)).astype(int)
tind = np.unique(t)
NK = []
N = []
D = []
Z = []
if rho != None and weight != None:
raise ValueError("Must use either rho or weights, not both")
elif rho != None:
s = KaplanMeier(data,self.endog,censoring=self.censoring)
s.fit()
s = (s.results[0][0]) ** (rho)
s = np.r_[1,s[:-1]]
elif weight != None:
s = weight(tind)
else:
s = np.ones_like(tind)
if self.censoring == None:
for g in groups:
dk = np.bincount((t[data[:,self.exog] == g]))
d = np.bincount(t)
if np.max(tind) != len(dk):
dif = np.max(tind) - len(dk) + 1
dk = np.r_[dk,[0]*dif]
dk = dk[:,list(tind)]
d = d[:,list(tind)]
dk = dk.astype(float)
d = d.astype(float)
dkSum = np.cumsum(dk)
dSum = np.cumsum(d)
dkSum = np.r_[0,dkSum]
dSum = np.r_[0,dSum]
nk = len(data[data[:,self.exog] == g]) - dkSum[:-1]
n = len(data) - dSum[:-1]
d = d[n>1]
dk = dk[n>1]
nk = nk[n>1]
n = n[n>1]
s = s[n>1]
ek = (nk * d)/(n)
Z.append(np.sum(s * (dk - ek)))
NK.append(nk)
N.append(n)
D.append(d)
else:
for g in groups:
censoring = ((data[:,self.censoring]).astype(float)).astype(int)
reverseCensoring = -1*(censoring - 1)
censored = np.bincount(t,reverseCensoring)
ck = np.bincount((t[data[:,self.exog] == g]),
reverseCensoring[data[:,self.exog] == g])
dk = np.bincount((t[data[:,self.exog] == g]),
censoring[data[:,self.exog] == g])
d = np.bincount(t,censoring)
if np.max(tind) != len(dk):
dif = np.max(tind) - len(dk) + 1
dk = np.r_[dk,[0]*dif]
ck = np.r_[ck,[0]*dif]
dk = dk[:,list(tind)]
ck = ck[:,list(tind)]
d = d[:,list(tind)]
dk = dk.astype(float)
d = d.astype(float)
ck = ck.astype(float)
dkSum = np.cumsum(dk)
dSum = np.cumsum(d)
ck = np.cumsum(ck)
ck = np.r_[0,ck]
dkSum = np.r_[0,dkSum]
dSum = np.r_[0,dSum]
censored = censored[:,list(tind)]
censored = censored.astype(float)
censoredSum = np.cumsum(censored)
censoredSum = np.r_[0,censoredSum]
nk = (len(data[data[:,self.exog] == g]) - dkSum[:-1]
- ck[:-1])
n = len(data) - dSum[:-1] - censoredSum[:-1]
d = d[n>1]
dk = dk[n>1]
nk = nk[n>1]
n = n[n>1]
s = s[n>1]
ek = (nk * d)/(n)
Z.append(np.sum(s * (dk - ek)))
NK.append(nk)
N.append(n)
D.append(d)
Z = np.array(Z)
N = np.array(N)
D = np.array(D)
NK = np.array(NK)
sigma = -1 * np.dot((NK/N) * ((N - D)/(N - 1)) * D
* np.array([(s ** 2)]*len(D))
,np.transpose(NK/N))
np.fill_diagonal(sigma, np.diagonal(np.dot((NK/N)
* ((N - D)/(N - 1)) * D
* np.array([(s ** 2)]*len(D))
,np.transpose(1 - (NK/N)))))
chisq = np.dot(np.transpose(Z),np.dot(la.pinv(sigma), Z))
df = len(groups) - 1
return np.array([chisq, df, stats.chi2.sf(chisq,df)])
else:
raise ValueError("groups must be in column exog")
def do(self, a, b):
a_ginv = linalg.pinv(a)
assert_almost_equal(dot(a, a_ginv), identity(asarray(a).shape[0]))
assert_(imply(isinstance(a, matrix), isinstance(a_ginv, matrix)))
def main(argv):
seterr(over='raise', divide='raise', invalid='raise')
try:
if int(os.environ['OMP_NUM_THREADS']) > 1 or int(
os.environ['MKL_NUM_THREADS']) > 1:
print 'It seems that parallelization is turned on. This will skew the results. To turn it off:'
print '\texport OMP_NUM_THREADS=1'
print '\texport MKL_NUM_THREADS=1'
except:
print 'Parallelization of BLAS might be turned on. This could skew results.'
experiment = Experiment(seed=42)
if not os.path.exists(EXPERIMENT_PATH):
print 'Could not find file \'{0}\'.'.format(EXPERIMENT_PATH)
return 0
results = Experiment(EXPERIMENT_PATH)
ica = results['model'].model[1].model
# load test data
data = load('data/vanhateren.{0}.0.npz'.format(
results['parameters'][0]))['data']
data = data[:, :100000]
data = preprocess(data)
data = data[:, permutation(data.shape[1] / 2)[:NUM_SAMPLES]]
# transform data
dct = results['model'].transforms[0]
wt = results['model'].model[1].transforms[0]
data = wt(dct(data)[1:])
X = data
for method in sampling_methods:
# disable output and parallelization
Distribution.VERBOSITY = 0
mapp.max_processes = 1
# measure time required by transition operator
start = time()
# initial hidden states
Y = dot(pinv(ica.A), X)
# increase number of steps to reduce overhead
ica.sample_posterior(
X,
method=(method['method'],
dict(method['parameters'],
Y=Y,
num_steps=method['parameters']['num_steps'] *
NUM_STEPS_MULTIPLIER)))
# time required per transition operator application
duration = (time() - start) / NUM_STEPS_MULTIPLIER
# enable output and parallelization
Distribution.VERBOSITY = 2
mapp.max_processes = 2
energies = [mean(ica.prior_energy(Y))]
# Markov chain
for i in range(int(NUM_SECONDS / duration + 1.)):
Y = ica.sample_posterior(X,
method=(method['method'],
dict(method['parameters'], Y=Y)))
energies.append(mean(ica.prior_energy(Y)))
plot(arange(len(energies)) * duration,
energies,
'-',
color=method['color'],
line_width=1.2,
pgf_options=['forget plot'],
comment=str(method['parameters']))
xlabel('time in seconds')
ylabel('average energy')
title('van Hateren')
gca().width = 7
gca().height = 7
gca().xmin = -1
gca().xmax = NUM_SECONDS
savefig('results/vanhateren/vanhateren_trace_.tex')
return 0
def transform(foci, mat):
""" Convert coordinates from one space to another using provided
transformation matrix. """
t = linalg.pinv(mat)
foci = np.hstack((foci, np.ones((foci.shape[0], 1))))
return np.dot(foci, t)[:, 0:3]
np.concatenate(
([data[0]], np.zeros(m - 1))))[m - 1:]
# Unpack data
data = loadmat('audio_data.mat')
noisy_speech = data['reference'].flatten()
noise = data['primary'].flatten()
fs = data['fs'].flatten() # Hz
# Filter order
m = 100
# See http://www.cs.cmu.edu/~aarti/pubs/ANC.pdf page 7 last paragraph
X = regmat(noise, m)
y = noisy_speech[m - 1:]
w = npl.pinv(X).dot(y)
# The error is the signal we seek
speech = y - X.dot(w)
################################################# RECORD
if save_audio:
wavwrite('recovered_ALMS.wav'.format(m), fs, speech)
################################################# VISUALIZATION
if plot_results:
# Compute performance surface for order 2
m2 = 2
dA = np.zeros((n, n))
dA[0:n, 0:n] = np.random.normal(0, .05, (n, n))
tilde_A = (np.eye(n) - Ur @ Ur.T) @ (A + dA)
if np.max(np.abs(la.eigvals(tilde_A))) < 0.94:
break
iterator += 1
A = A + dA
# Generating the trajectory
x0 = 10 * np.random.randn(n, 1)
alpha = 0.0
if control_mode == 0 or control_mode == 2:
alpha = 1e-9
else:
alpha = 5e-7
number_of_iteration = 20*n # or set it to 40 * n
x_a, x_fre, up_a, x_update, xposition, K, x_deepc, x_deepc_dgr = Trajectory_generator(A, B, K1, x0, True, alpha,
number_of_iteration, control_mode)
final_K.append(K)
ideal_K.append(la.pinv(alpha * np.eye(m) + B.T @ B) @ B.T@A)
plots_traj(np.hstack(x_a), np.hstack(x_fre), np.hstack(x_update), np.hstack(x_deepc), np.hstack(x_deepc_dgr), up_a, xposition, K, mode[control_mode], mode_title[control_mode])
plot_controller(final_K, ideal_K)
print('The parameters for DGR+DeePC must be adjusted for each specific instance of perturbation.')
def multi_criteria(lineNo, rootDic, obs_centroid, __iteration_IK__, robot): #
k0 = 1
alpha_vec = np.arange(0, 1.1, 0.1)
q_pareto = []
for alpha in alpha_vec:
q_mc = np.zeros((len(t_vec), 7))
q_mc[0, :] = q_initial
start_IK = time.time()
for i, t in enumerate(t_vec[1:]):
J = jacobian_spatial(q_mc[i, :])
w1_der = manipulability_gradient(
q_mc[i, :], jacobian_spatial) # singularity avoidance
# w2_der = joint_limits_gradient(q_mc[i, :], q_min, q_max)
w3_der = collision(q_input=q_sa[i, :],
centroid=obs_centroid,
robot=robot)
w3_der = w3_der.numpy()
q0 = k0 * (alpha * w1_der + (1 - alpha) * w3_der)
qp_mc = la.pinv(J) @ desired_trajectory_vel[i, :] + (
np.eye(7) - la.pinv(J) @ J) @ q0
q_mc[i + 1, :] = q_mc[i, :] + qp_mc * h
end_IK = time.time()
diff_time_IK = end_IK - start_IK
fig, ax = plt.subplots(1, 1, figsize=(6.472135955, 4), dpi=96)
ax.plot(t_vec, q_mc[:, 0], label=r'$q_1$')
ax.plot(t_vec, q_mc[:, 1], label=r'$q_2$')
ax.plot(t_vec, q_mc[:, 2], label=r'$q_3$')
ax.plot(t_vec, q_mc[:, 3], label=r'$q_4$')
ax.plot(t_vec, q_mc[:, 4], label=r'$q_5$')
ax.plot(t_vec, q_mc[:, 5], label=r'$q_6$')
ax.plot(t_vec, q_mc[:, 6], label=r'$q_7$')
ax.set_xlabel('Time [s]')
ax.set_ylabel('Position [rad]')
ax.set_title('Inverse kinematics with multi-criteria')
ax.legend(loc='ceq_inputnter right', bbox_to_anchor=(1.2, 0.5))
fig.tight_layout()
base_file_name1 = 'Figure_q_mc_multi_criteria'
suffix1 = '.csv'
csv_fileHandle = os.path.join(
rootDic, base_file_name1 + '_' + str(lineNo) + '_' +
str(__iteration_IK__) + '_' + robot + ' ' + str(alpha) + suffix1)
np.savetxt(csv_fileHandle, q_ls, delimiter=",")
suffix2 = '.jpg'
figureHandle = os.path.join(
rootDic, base_file_name1 + '_' + str(lineNo) + '_' +
str(__iteration_IK__) + '_' + robot + ' ' + str(alpha) + suffix2)
fig.savefig(figureHandle)
q_pareto.append(q_mc)
fig, (ax1, ax2) = plt.subplots(1, 2, figsize=(2 * 6.472135955, 4), dpi=96)
for q, alpha in zip(q_pareto, alpha_vec):
manipulability_cost = np.array(
[manipulability(qi, jacobian_spatial) for qi in q])
joint_limit_cost = np.array(
[joint_limits(qi, q_min, q_max) for qi in q])
color = np.random.rand(3, )
if alpha == 0.0 or alpha == 1.0:
lw = 3
else:
lw = 1
ax1.plot(t_vec,
manipulability_cost,
label=(r'$\alpha=$' + '{0:.1f}'.format(alpha)),
color=color,
linewidth=lw)
ax2.plot(t_vec,
joint_limit_cost,
label=(r'$\alpha=$' + '{0:.1f}'.format(alpha)),
color=color,
linewidth=lw)
ax1.legend(ncol=2, prop={'size': 7})
ax1.set_xlabel('Time [s]')
ax2.set_xlabel('Time [s]')
ax1.set_ylabel('Manipulability')
ax2.set_ylabel('Joint Limits')
fig.tight_layout()
base_file_name2 = 'spatial_pareto'
figureHandle = os.path.join(
rootDic,
base_file_name2 + str(__iteration_IK__) + '_' + robot + suffix2)
fig.savefig(figureHandle)
return q_pareto, diff_time_IK
def run(X, y):
#print la.inv(X.T.dot(X)).dot(X.T).dot(y)
return np.dot(la.pinv(X), y)
print('1(b): Mean of incomes =', mean_data, ', Variance of incomes =',
var_data)
print('1(b): Mean of model =', mean_model, ', Variance of model =', var_model)
'''
---------------------------------------
the following code fragment is for 1(c)
---------------------------------------
'''
# calculate the error vector given the GMM estimate using identity matrix as the
# weighting matrix
err1 = err_vec(incomes, mu_GMM1, sig_GMM1, cutoff, False)
# construct an estimator for the variance covariance matrix
VCV2 = np.dot(err1, err1.T) / incomes.shape[0]
# the inverse of the variance covariance matrix is the optimal weighting matrix
W_hat2 = lin.pinv(VCV2)
params_init = np.array([mu_init, sig_init])
gmm_args = (incomes, cutoff, W_hat2)
results = opt.minimize(criterion,
params_init,
args=(gmm_args),
method='TNC',
bounds=((1e-10, None), (1e-10, None)))
mu_GMM2, sig_GMM2 = results.x
print('mu_GMM2=', mu_GMM2, ' sig_GMM2=', sig_GMM2)
# Plot the histogram of the data
plt.title('Histogram of annual incomes')
plt.xlabel(r'Annual income(\$s)')
def lsq_solution_V2(self, X, y):
w = np.dot(la.pinv(X), y)
return w
def _multivariate_ols_fit(endog, exog, method='svd', tolerance=1e-8):
"""
Solve multivariate linear model y = x * params
where y is dependent variables, x is independent variables
Parameters
----------
endog : array_like
each column is a dependent variable
exog : array_like
each column is a independent variable
method : string
'svd' - Singular value decomposition
'pinv' - Moore-Penrose pseudoinverse
tolerance : float, a small positive number
Tolerance for eigenvalue. Values smaller than tolerance is considered
zero.
Returns
-------
a tuple of matrices or values necessary for hypotheses testing
.. [*] https://support.sas.com/documentation/cdl/en/statug/63033/HTML/default/viewer.htm#statug_introreg_sect012.htm
Notes
-----
Status: experimental and incomplete
"""
y = endog
x = exog
nobs, k_endog = y.shape
nobs1, k_exog = x.shape
if nobs != nobs1:
raise ValueError('x(n=%d) and y(n=%d) should have the same number of '
'rows!' % (nobs1, nobs))
# Calculate the matrices necessary for hypotheses testing
df_resid = nobs - k_exog
if method == 'pinv':
# Regression coefficients matrix
pinv_x = pinv(x)
params = pinv_x.dot(y)
# inverse of x'x
inv_cov = pinv_x.dot(pinv_x.T)
if matrix_rank(inv_cov, tol=tolerance) < k_exog:
raise ValueError('Covariance of x singular!')
# Sums of squares and cross-products of residuals
# Y'Y - (X * params)'B * params
t = x.dot(params)
sscpr = np.subtract(y.T.dot(y), t.T.dot(t))
return (params, df_resid, inv_cov, sscpr)
elif method == 'svd':
u, s, v = svd(x, 0)
if (s > tolerance).sum() < len(s):
raise ValueError('Covariance of x singular!')
invs = 1. / s
params = v.T.dot(np.diag(invs)).dot(u.T).dot(y)
inv_cov = v.T.dot(np.diag(np.power(invs, 2))).dot(v)
t = np.diag(s).dot(v).dot(params)
sscpr = np.subtract(y.T.dot(y), t.T.dot(t))
return (params, df_resid, inv_cov, sscpr)
else:
raise ValueError('%s is not a supported method!' % method)
import numpy as np
from numpy import linalg
from sklearn.metrics import mean_squared_error
from math import exp
from sklearn.preprocessing import normalize
A = np.loadtxt("steel_composition_train.csv", delimiter=",", skiprows=1)
numOfRows = A.shape[0]
numOfCols = A.shape[1]
trainingData_temp = np.delete(A, [0, numOfCols-1], 1)
targetVector_train = A[:,-1]
# normalization
trainingData = normalize(trainingData_temp, norm='l2', axis=1)
gramMatrix = np.zeros((numOfRows, numOfRows))
for i in range(0, numOfRows):
for j in range(0, numOfRows):
temp1 = np.dot(trainingData[i].transpose(), trainingData[j])
gramMatrix[i][j] = np.power((temp1 + 1), 3)
eye = np.identity(numOfRows)
temp = np.add(gramMatrix, eye)
a = np.dot(linalg.pinv(temp), targetVector_train)
target_pred = np.zeros(numOfRows)
for i in range(0, numOfRows):
target_pred[i] = np.dot(gramMatrix[:,i], a)
print np.sqrt(mean_squared_error(targetVector_train, target_pred))
def mmmg(
crit_list: List["BaseCrit"],
init: array,
tol: float = 1e-4,
max_iter: int = 500,
) -> Tuple[array, List[float]]:
r"""The Majorize-Minimize Memory Gradient (`3mg`) algorithm.
The `mmmg` (`3mg`) algorithm is a subspace memory-gradient optimization
algorithm with an explicit step formula based on Majorize-Minimize Quadratic
approach [2]_.
Parameters
----------
crit_list : list of `BaseCrit`
A list of :class:`BaseCrit` objects that each represent a `μ ψ(V·x - ω)`.
The criteria are implicitly summed.
init : array
The initial point. The `init` array is updated in place to return the
output. The user must make a copy before calling `mmmg` if this is not
the desired behavior.
tol : float, optional
The stopping tolerance. The algorithm is stopped when the gradient norm
is inferior to `init.size * tol`.
max_iter : int, optional
The maximum number of iterations.
Returns
-------
minimizer : array
The minimizer of the criterion with same shape than `init`.
norm_grad : list of float
The norm of the gradient during iterations.
Notes
-----
The output of :meth:`BaseCrit.operator`, and the `init` value, are
automatically vectorized internally. However, the output is reshaped as the
`init` array.
References
----------
.. [2] E. Chouzenoux, J. Idier, and S. Moussaoui, “A Majorize-Minimize
Strategy for Subspace Optimization Applied to Image Restoration,” IEEE
Trans. on Image Process., vol. 20, no. 6, pp. 1517–1528, Jun. 2011, doi:
10.1109/TIP.2010.2103083.
"""
point = init.reshape((-1, 1))
norm_grad = []
# The first previous moves are initialized with 0 array. Consequently, the
# first iterations implementation can be improved, at the cost of if
# statement.
move = np.zeros_like(point)
op_directions = [
np.tile(_vect(crit.operator, move, init.shape), 2)
for crit in crit_list
]
step = np.ones((2, 1))
for _ in range(max_iter):
# Vectorized gradient
grad = _gradient(crit_list, point, init.shape)
norm_grad.append(la.norm(grad))
# Stopping test
if norm_grad[-1] < point.size * tol:
break
# Memory gradient directions
directions = np.c_[-grad, move]
# Step by Majorize-Minimize
op_directions = [
np.c_[_vect(crit.operator, grad, init.shape), i_op_dir @ step]
for crit, i_op_dir in zip(crit_list, op_directions)
]
step = -la.pinv(
sum(
crit.norm_mat_major(i_op_dir, point.reshape(init.shape))
for crit, i_op_dir in zip(crit_list, op_directions))) @ (
directions.T @ grad)
move = directions @ step
# update
point += move
return np.reshape(point, init.shape), norm_grad
def _calc_params(self):
self.W = pinv(
self.X_mat.getT() * self.X_mat + self.L * np.identity(self.dim + 1)) * self.X_mat.getT() * self.Y_mat