def coerce_gene(X, X_star):
"""Coerces vector X to have same summary statistics as X_star.
Using the process outlined in 'Generating Data with Identical Statistics but
Dissimilar Graphics' by Sangit Chatterjee and Aykut Firat, this function
takes a randomly drawn vector X and a vector X_star and produces a new
vector from X that has the same mean, std, correlation and possibly other
statistics as X_star. The reference is:
Generating Data with Identical Statistics but Dissimilar Graphics. Sangit
Chatterjee and Aykut Firat 2007. The American Statistician 61:3, 248-254.
Inputs:
X, X_star - nX2 arrays.
Reshaping is required because the way numpy handles matrix multiplication
elementwise unless it recognizes the array as 2D.
"""
# step ii: set mean value of columns of X to 0 using X=X-e_nx1*X_mean
n = X.shape[0]
X_new = X - dot(ones((n,1)),X.mean(0).reshape(1,2)) #a long col axis
# step iii: orthonormalize cols of X_new with Gram-Schmidt process
x, y = X_new[:,0], X_new[:,1]
u1 = x
u2 = y-(dot(x,y)/dot(x,x))*x
e1, e2 = u1/norm(u1), u2/norm(u2)
X_on = array([e1,e2]).T #orthnormalized X_new, .T to keep nX2
# step iv: transform X_on to ensure summary stat agreement
tmp_a = ((n-1.)**.5)*X_on
tmp_b = sqrtm(cov(X_star.T)).astype(float)
tmp_c = dot(ones((n,1)),X_star.mean(0).reshape(1,2))
return dot(tmp_a, tmp_b)+tmp_c
def fBM_nd(dims, H, return_mat=False, use_eig_ev=True):
"""
creates fractional Brownian motion
parameters: dims is a tuple of the shape of the sample path (nxd);
H: Hurst exponent
this is the slow version of fBM. It might, however, be more precise than
fBM, however - sometimes, the matrix square root has a problem, which might
induce inaccuracy
use_eig_ev: use eigenvalue decomposition for matrix square root computation
(faster)
"""
n = dims[0]
d = dims[1]
Gamma = zeros((n, n))
print('building ...\n')
for t in arange(n):
for s in arange(n):
Gamma[t, s] = .5 * ((s + 1)**(2. * H) +
(t + 1)**(2. * H) - abs(t - s)**(2. * H))
print('rooting ...\n')
if use_eig_ev:
ev, ew = eig(Gamma.real)
Sigma = dot(ew, dot(diag(sqrt(ev)), ew.T))
else:
Sigma = sqrtm(Gamma)
if return_mat:
return Sigma
v = randn(n, d)
return dot(Sigma, v)
def coerce_gene(X, X_star):
"""Coerces vector X to have same summary statistics as X_star.
Using the process outlined in 'Generating Data with Identical Statistics but
Dissimilar Graphics' by Sangit Chatterjee and Aykut Firat, this function
takes a randomly drawn vector X and a vector X_star and produces a new
vector from X that has the same mean, std, correlation and possibly other
statistics as X_star. The reference is:
Generating Data with Identical Statistics but Dissimilar Graphics. Sangit
Chatterjee and Aykut Firat 2007. The American Statistician 61:3, 248-254.
Inputs:
X, X_star - nX2 arrays.
Reshaping is required because the way numpy handles matrix multiplication
elementwise unless it recognizes the array as 2D.
"""
# step ii: set mean value of columns of X to 0 using X=X-e_nx1*X_mean
n = X.shape[0]
X_new = X - dot(ones((n, 1)), X.mean(0).reshape(1, 2)) #a long col axis
# step iii: orthonormalize cols of X_new with Gram-Schmidt process
x, y = X_new[:, 0], X_new[:, 1]
u1 = x
u2 = y - (dot(x, y) / dot(x, x)) * x
e1, e2 = u1 / norm(u1), u2 / norm(u2)
X_on = array([e1, e2]).T #orthnormalized X_new, .T to keep nX2
# step iv: transform X_on to ensure summary stat agreement
tmp_a = ((n - 1.)**.5) * X_on
tmp_b = sqrtm(cov(X_star.T)).astype(float)
tmp_c = dot(ones((n, 1)), X_star.mean(0).reshape(1, 2))
return dot(tmp_a, tmp_b) + tmp_c
def fBM_nd(dims, H, return_mat = False, use_eig_ev = True):
"""
creates fractional Brownian motion
parameters: dims is a tuple of the shape of the sample path (nxd);
H: Hurst exponent
this is the slow version of fBM. It might, however, be more precise than
fBM, however - sometimes, the matrix square root has a problem, which might
induce inaccuracy
use_eig_ev: use eigenvalue decomposition for matrix square root computation
(faster)
"""
n = dims[0]
d = dims[1]
Gamma = zeros((n,n))
print ('building ...\n')
for t in arange(n):
for s in arange(n):
Gamma[t,s] = .5*((s+1)**(2.*H) + (t+1)**(2.*H) - abs(t-s)**(2.*H))
print('rooting ...\n')
if use_eig_ev:
ev,ew = eig(Gamma.real)
Sigma = dot(ew, dot(diag(sqrt(ev)),ew.T) )
else:
Sigma = sqrtm(Gamma)
if return_mat:
return Sigma
v = randn(n,d)
return dot(Sigma,v)
def gp_gen(num_point, num_dim, domain, noise_level, mix_list=[[0, 1], [2]]):
"""Generate matrix variate normally distributed data"""
reg_param = 1e-8
num_class = len(flatten_list(mix_list))
X = domain*rand(num_dim, num_point)
Kx = GaussKernel(1.0)
Kx.compute(X)
Ky = list2matrix(mix_list, neg_corr=True)
K = JointKernel(Kx, Ky)
L = real(sqrtm(0.5*(K.kmat)+reg_param*eye(num_point*num_class)).T)
mu = zeros((num_class, num_point))
Y = L*matrix(randn(num_point*num_class, 1))
Y.shape = (num_point, num_class)
Y = real(Y.T)
Y += mu + noise_level*randn(num_class, num_point)
Y = array(Y)
return (X, Y)
def gp_gen(num_point, num_dim, domain, noise_level, mix_list=[[0, 1], [2]]):
"""Generate matrix variate normally distributed data"""
reg_param = 1e-8
num_class = len(flatten_list(mix_list))
X = domain * rand(num_dim, num_point)
Kx = GaussKernel(1.0)
Kx.compute(X)
Ky = list2matrix(mix_list, neg_corr=True)
K = JointKernel(Kx, Ky)
L = real(sqrtm(0.5 * (K.kmat) + reg_param * eye(num_point * num_class)).T)
mu = zeros((num_class, num_point))
Y = L * matrix(randn(num_point * num_class, 1))
Y.shape = (num_point, num_class)
Y = real(Y.T)
Y += mu + noise_level * randn(num_class, num_point)
Y = array(Y)
return (X, Y)
def mydeftensor (v): j = v.reshape([N,N]) s = sqrtm(j.transpose()*j) return S.reshape([1,N*N])
def mydeftensor(v):
j = v.reshape([N, N])
s = sqrtm(j.transpose() * j)
return S.reshape([1, N * N])
#http://en.wikipedia.org/wiki/Square_root_of_a_matrix #Dirac oscillator http://iopscience.iop.org/0305-4470/22/17/002/pdf/0305-4470_22_17_002.pdf from pylab import * from scipy.linalg.matfuncs import sqrtm N = 400 s = 1 # 1/sqrt(m\omega_0) n = arange(1,N) m = sqrt(n) x = matrix(s /sqrt(2) * (diag(m,-1) + diag(m,1))) p = matrix(1j/(s*sqrt(2)) * (diag(m,-1) - diag(m,1) )) #H = (p*p + x*x) /2. #H = (p*p + x**4) /2. #Klein-Gordon Hamiltonian mass = 1000 H_KG = sqrtm( p*p + x*x + mass) - mass H = H_KG def eigval_KG_analytic(n): return sqrt(mass**2 + 2 * (n + .5) * mass) - mass def eigval_classical(n): return (n+.5) value, vector = eig(H) value = sort(value)[1:] plot(value) plot(*array([(i,eigval_KG_analytic(i)) for i in range(len(value))]).T) plot(*array([(i,eigval_classical(i)) for i in range(len(value))]).T) legend(['numerical','analytical','nonrelativistic']) show()