def roots_of_matrix(M, x=sympy.abc.x):
'''
Find roots of det( M(x) ) = 0 equation, where M(x) = A + B*x.
Parameters
----------
M : sympy.Matrix
Square matrix depends linearly on only one variable
x : sympy.Symbol, optional
Symbol which m depends on
Returns
-------
numpy.ndarray
1-D array of roots
'''
M_wo_x = M.replace(x, 0.)
A = ml.empty(M.shape, dtype=np.complex)
for (i, j), element in np.ndenumerate(M_wo_x):
A[i, j] = complex(element)
M_coeffs_x = (M - M_wo_x).replace(x, 1.)
B = ml.empty(M.shape, dtype=np.complex)
for (i, j), element in np.ndenumerate(M_coeffs_x):
B[i, j] = complex(element)
C = -A.I * B
evs = np.linalg.eigvals(C)
roots = 1 / evs[np.nonzero(evs)]
return roots
def getOrthColumns(m):
'''
Constructs the orthogonally complementing columns of the input.
Input of the form pxr is assumed to have r<=p,
and have either full column rank r or rank 0 (scalar or matrix)
Output is of the form px(p-r), except:
a) if M square and full rank p, returns scalar 0
b) if rank(M)=0 (zero matrix), returns I_p
(Note you cannot pass scalar zero, because dimension info would be
missing.)
Return type is as input type.
'''
if type(m) == type(asarray(m)):
m = mat(m)
output = 'array'
else: output = 'matrix'
p, r = m.shape
# first catch the stupid input case
if p < r: raise ValueError, 'need at least as many rows as columns'
# we use lstsq(M, ones) just to exploit its rank-finding algorithm,
rk = lstsq(m, ones(p).T)[2]
# first the square and full rank case:
if rk == p: result = zeros((p,0)) # note the shape! hopefully octave-like
# then the zero-matrix case (within machine precision):
elif rk == 0: result = eye(p)
# now the rank-deficient case:
elif rk < r:
raise ValueError, 'sorry, matrix does not have full column rank'
# (what's left should be ok)
else:
# we have to watch out for zero rows in M,
# if they are in the first p-r positions!
# so the (probably inefficient) algorithm:
# 1. check the rank of each row
# 2. if zero, then also put a zero row in c
# 3. if not, put the next unit vector in c-row
idr = eye(r)
idpr = eye(p-r)
c = empty([0,r]) # starting point
co = empty([0, p-r]) # will hold orth-compl.
idrcount = 0
for row in range(p):
# (must be ones() instead of 1 because of 2d-requirement
if lstsq( m[row,:], ones(1) )[2] == 0 or idrcount >= r:
c = r_[ c, zeros(r) ]
co = r_[ co, idpr[row-idrcount, :] ]
else: # row is non-zero, and we haven't used all unit vecs
c = r_[ c, idr[idrcount, :] ]
co = r_[ co, zeros(p-r) ]
idrcount += 1
# earlier non-general (=bug) line: c = mat(r_[eye(r), zeros((p-r, r))])
# and: co = mat( r_[zeros((r, p-r)), eye(p-r)] )
# old:
# result = ( eye(p) - c * (M.T * c).I * M.T ) * co
result = co - c * solve(m.T * c, m.T * co)
if output == 'array': return result.A
else: return result
def simulate_LDS_with_LPF(A, B, K, x0, DT, N, ndt, fc, plot):
'''Simulate a Linear Dynamical System (LDS) forward in time assuming
the control inputs are low-pass filtered
A: transition matrix
B: control matrix
K: feedback gain matrix
x0: initial state
DT: time step
N: number of time steps to simulate
ndt: number of sub time steps (to make simulation more accurate)
'''
n = A.shape[0]
m = B.shape[1]
x = matlib.empty((n, N))
u = matlib.empty((m, N - 1))
u_filt = matlib.empty((m, N - 1))
x[:, 0] = x0
u[:, 0] = -K * x0
u_filt[:, 0] = -K * x0
dt = DT / ndt
lpf = FirstOrderLowPassFilter(dt, fc, u[:, 0].A1)
for i in range(N - 1):
u[:, i] = -K * x[:, i]
x_pre = x[:, i]
for ii in range(ndt):
if (fc > 0):
u_filt[:, i] = np.matrix(lpf.filter_data(u[:, i].A1)).T
else:
u_filt[:, i] = u[:, i]
x_post = x_pre + dt * (A * x_pre + B * u_filt[:, i])
x_pre = x_post
x[:, i + 1] = x_post
if plot:
max_rows = 4
n_cols = 1 + (n + m + 1) / max_rows
n_rows = int(np.ceil(float(n + m) / n_cols))
f, ax = plt.subplots(n_rows, n_cols, sharex=True)
ax = ax.reshape(n_cols * n_rows)
time = np.arange(N * DT, step=DT)
for i in range(n):
ax[i].plot(time, x[i, :].A1)
ax[i].set_title('x ' + str(i))
for i in range(m):
ax[n + i].plot(time[:-1], u[i, :].A1, label='u')
ax[n + i].plot(time[:-1],
u_filt[i, :].A1,
'--',
label='u filtered')
ax[n + i].set_title('u ' + str(i))
ax[n + 1].legend(loc='best')
return (x, u)
def record_activity(self, inputs):
"""
Returns states and outputs through time after beeing fed inputs
"""
nb_inputs = len(inputs)
states = npmat.empty((self.state_size, nb_inputs))
outputs = npmat.empty((self.output_size, nb_inputs))
for i in range(nb_inputs):
self.next_output(inputs[i])
states[:, i] = self.state
outputs[:, i] = self.output_values
return states, outputs
def simulate_ALDS(H, x0, dt, N, plot=False, show_plot=None):
'''Simulate an Autonomous Linear Dynamical System (ALDS) forward in time
H: transition matrix
x0: initial state
dt: time step
N: number of time steps to simulate
plot: if True it plots the time evolution of the state
'''
n = H.shape[0]
x = matlib.empty((n, N))
x[:, 0] = x0
e_dtH = expm(dt * H)
for i in range(N - 1):
x[:, i + 1] = e_dtH * x[:, i]
if plot:
max_rows = 4
n_cols = 1 + (n + 1) / max_rows
n_rows = int(np.ceil(float(n) / n_cols))
f, ax = plt.subplots(n_rows, n_cols, sharex=True)
ax = ax.reshape(n_cols * n_rows)
time = np.arange(N * dt, step=dt)
for i in range(n):
ax[i].plot(time, x[i, :].A1)
ax[i].set_title(str(i))
if (show_plot is None):
plt.show()
return x
def FluidStationaryDistr (mass0, ini, K, clo, x):
"""
Returns the stationary distribution of a Markovian
fluid model at the given points.
Parameters
----------
mass0 : matrix, shape (1,Np+Nm)
The stationary probability vector of zero level
ini : matrix, shape (1,Np)
The initial vector of the stationary density
K : matrix, shape (Np,Np)
The matrix parameter of the stationary density
clo : matrix, shape (Np,Np+Nm)
The closing matrix of the stationary density
x : vector, length (K)
The distribution function is computed at these
points.
Returns
-------
pi : matrix, shape (K,Nm+Np)
The ith row of pi is the probability that the fluid
level is less than or equal to x(i), while being in
different states of the background process.
"""
m = clo.shape[1]
y = ml.empty((len(x),m))
closing = -K.I*clo
for i in range(len(x)):
y[i,:] = mass0 + ini*(ml.eye(K.shape[0])-la.expm(K*x[i]))*closing
return y
def QBDStationaryDistr (pi0, R, K):
"""
Returns the stationary distribution of a QBD up to a
given level K.
Parameters
----------
pi0 : matrix, shape (1,N)
The stationary probability vector of level zero
R : matrix, shape (N,N)
The matrix parameter of the matrix geometrical
distribution of the QBD
K : integer
The stationary distribution is returned up to
this level.
Returns
-------
pi : array, length (K+1)*N
The stationary probability vector up to level K
"""
m = R.shape[0]
qld = ml.empty((1,(K+1)*m))
qld[0,0:m] = pi0
pix = pi0
for k in range(1,K+1):
pix = pix*R
qld[0,k*m:(k+1)*m] = pix
return qld
def compute_closed_loop_eigenvales(robot, gains_array):
''' Compute eigenvalues of linear part of closed-loop system:
d3f = -Kd_bar*d2f - (K*Upsilon+Kp_bar)*df - Kp_bar*K*A*Kf*e_f + ...
'''
ny=3
ei_cls_f = matlib.empty((3*nf,T), dtype=complex)*np.nan
ei_cls = matlib.empty((3*nf+2*ny,T), dtype=complex)*np.nan
for t in range(T):
H = compute_closed_loop_transition_matrix(gains_array, robot, q[:,t], v[:,t])
H_f = H[2*ny:, 2*ny:]
ei_cls_f[:,t] = np.sort_complex(eigvals(H_f)).reshape((3*nf,1))
ei_cls[:,t] = np.sort_complex(eigvals(H)).reshape((3*nf+2*ny,1))
plot_stats_eigenvalues(ei_cls_f, name='Closed-loop force tracking')
plot_stats_eigenvalues(ei_cls, name='Closed-loop momentum tracking')
return ei_cls
def __init__(self,g,d,sigma,L,U,Q):
"""
defines a generative model for diffusion of disease
Arguments
=========
g : function
describes the individual contribution of a location to the dynamics
d : function
defines the distance between two locations
L : list of ntuples
the spatial locations (in R^n) over which the model is defined
sigma : scalar
controls the scale of the diffusion
U : matrix
input matrix
Q : matrix
disturbance covariance matrix
"""
# number of states
self.n = len(L)
# store state locations
self.L = L
# diffusion matrix
F = mb.empty((self.n,self.n))
for i,li in enumerate(L):
for j,lj in enumerate(L):
F[i,j] = (1./np.sqrt(2*np.pi*sigma**2))*np.exp(-d(li,lj)/(2*sigma**2))
# normalise columns
for j,col in enumerate(F.T):
F[:,j] = F[:,j]/np.sum(col)
# initialise A matrix
self.A = mb.empty((self.n,self.n))
# populate A matrix
for i,li in enumerate(L):
for j,lj in enumerate(L):
self.A[i,j] = F[i,j] * g(lj)
# store U matrix
self.U = U
# store disturbance covariance matrix
self.Q = Q
# store distance function
self.d = d
def predict(self, X, *args, **kwargs):
m = X.shape[0]
RBF_result = matlib.empty((m, self._n_RBF + 1))
for x_index in range(m):
x_output = matlib.empty((1, self._n_RBF + 1))
# RBF process
for RBF_index in range(self._n_RBF):
x_output[0, RBF_index + 1] = Euler_distance(X[x_index], self._Theta1[RBF_index])
sigmoid_output = sigmoid(x_output)
sigmoid_output[0] = 1
RBF_result[x_index] = x_output
# Logistic Regression Process
raw_output = RBF_result * self._Theta2
return sigmoid(raw_output) > 0.5
def simulate_LDS(A, B, K, x0, dt, N):
'''Simulate a Linear Dynamical System (LDS) forward in time
A: transition matrix
B: control matrix
K: feedback gain matrix
x0: initial state
dt: time step
N: number of time steps to simulate
'''
n = A.shape[0]
m = B.shape[1]
x = matlib.empty((n, N))
u = matlib.empty((m, N - 1))
x[:, 0] = x0
H = A - B * K
e_dtH = expm(dt * H)
for i in range(N - 1):
x[:, i + 1] = e_dtH * x[:, i]
u[:, i] = -K * x[:, i]
return (x, u)
def make_C_matrix(self, r_t, alpha_t, o_t):
c = len(o_t)
# initialise C matrix
C = mb.empty((c, self.n))
# populate C matrix
for i in range(c):
for j in range(self.n):
rti2 = r_t[i]**2
const = alpha_t[i] / np.sqrt(2*np.pi*rti2)
C[i,j] = const * np.exp(-self.d(o_t[i], self.L[j]) / (2*rti2))
return C
def trouverParenthesageOptimalAvecStockage(dim, frontiere, i, j):
"""Calcule récursivement le meilleure parenthésage pour une
série de multiplications de matrices. Il effectue le calcul
de façon récursive
frontiere: matrice de position des parentheses pour deux matrices
dim: tableau de dimensions des matrices à multiplier
m: résultats déjà calculés
"""
m = matlib.empty((j, j), dtype=int)
m.fill(-1)
return __trouverParenthesageOptimalAvecStockage(frontiere, dim, m, i, j)
def colliding_boxes(self):
nb_solids = self.nb_solids
points = npmat.hstack([s.center_position for s in self.solids])
center = npmat.asmatrix(npmat.average(points, axis = 1)).T
centered_points = points - center
correlation_matrix = (centered_points * centered_points.T) / nb_solids
u = iterate(correlation_matrix, self.NB_ITERATIONS)
if self.projected_bounds is None:
self.projected_bounds = []
for i in range(nb_solids):
self.projected_bounds.append((i, 0, 0.))
self.projected_bounds.append((i, 1, 0.))
min_max_projections = npmat.empty((nb_solids, 2))
for solid_id in range(nb_solids):
solid_AABB_corners = self.solids[solid_id].AABB_corners()
corners_projections = u.T * solid_AABB_corners
min_max_projections[solid_id, 0] = np.min(corners_projections)
min_max_projections[solid_id, 1] = np.max(corners_projections)
for i in range(2 * nb_solids):
solid_id, begin_or_end_id, value = self.projected_bounds[i]
new_value = min_max_projections[solid_id, begin_or_end_id]
self.projected_bounds[i] = (solid_id, begin_or_end_id, new_value)
# TODO: linear sorting
self.projected_bounds.sort(key = lambda x : x[2])
output = []
active_solids = []
for i in range(2 * nb_solids):
solid_id, begin_or_end_id, value = self.projected_bounds[i]
if begin_or_end_id == 0:
for active_solid_id in active_solids:
if self.solids[active_solid_id].AABB_intersect_with(self.solids[solid_id]):
output.append((active_solid_id, solid_id))
active_solids.append(solid_id)
else:
active_solids.remove(solid_id)
return output
def getMfeature(self):
mata = npmatrix.empty((3, 4)) # random data
mata = npmatrix.zeros((3, 4)) # zeros
mata = npmatrix.ones((3, 4)) # one
mata = npmatrix.eye(3) # one along diagonal
mata = npmatrix.eye(3, 5) # one along diagonal
mata = npmatrix.identity(3) # identity square matrix
mata = npmatrix.rand(3, 7) # rand data
mata = npmatrix.ones((3, 1)) # one
print(mata)
print(mata.shape)
print(mata.dtype)
def compute_costs(x, u, dt, P, Q_x, Q_u):
N = x.shape[1]
T = N * dt
xu_proj = matlib.empty((P.shape[0], N)) # com state
state_cost, control_cost = 0.0, 0.0
for t in range(N):
if (t < N - 1):
xup = P * np.vstack([x[:, t], u[:, t]])
else:
xup = P * np.vstack([x[:, t], matlib.zeros_like(u[:, 0])])
xu_proj[:, t] = xup
state_cost += dt * (xup.T * Q_x * xup)[0, 0]
control_cost += dt * (xup.T * Q_u * xup)[0, 0]
return xu_proj, np.sqrt(state_cost / T), np.sqrt(control_cost / T)
def AABB_corners(self):
AABB = self.AABB
output = npmat.empty((3, 8))
i = 0
for x_i in range(2):
for y_i in range(2):
for z_i in range(2):
output[0, i] = AABB[0, x_i]
output[1, i] = AABB[1, y_i]
output[2, i] = AABB[2, z_i]
i += 1
return output
def getImpulseDummies(sampledateslist, periodslist):
'''
Returns a (numpy-)matrix of impulse dummies for the specified periods.
sampledateslist must consist of 1999.25 -style dates (quarterly or monthly).
However, because periodslist is probably human-made, it expects strings
such as '1999q3' or '1999M12'.
Variables in columns.
So far only for quarterly and monthly data.
'''
nobs = len(sampledateslist)
result = empty([nobs, 0])
for periodstring in periodslist:
period = dateString2dateFloat(periodstring)
result = c_[result, getDeterministics(nobs, 'i', \
sampledateslist.index(period))]
return result
def getImpulseDummies(sampledateslist, periodslist):
'''
Returns a (numpy-)matrix of impulse dummies for the specified periods.
sampledateslist must consist of 1999.25 -style dates (quarterly or monthly).
However, because periodslist is probably human-made, it expects strings
such as '1999q3' or '1999M12'.
Variables in columns.
So far only for quarterly and monthly data.
'''
nobs = len(sampledateslist)
result = empty([nobs,0])
for periodstring in periodslist:
period = dateString2dateFloat(periodstring)
result = c_[result, getDeterministics(nobs, 'i', \
sampledateslist.index(period))]
return result
def multiplierMatrice(a, b):
"""Multiplie les matrices a et b, et retourne
le résultat dans une nouvelle matrice
"""
#Vérifie si on peut bien multiplier les matrices
dimA = a.shape
dimB = b.shape
#On multiplie -- vérifier le type de données
c = matlib.empty((dimA[0], dimB[1]), dtype=a.dtype)
for i in range(0, dimA[0]):
for j in range(0, dimB[1]):
total = 0
for k in range(0, dimA[1]):
total += a[i,k] * b[k,j]
c[i,j] = total
return c
def betas(prices_m):
"""
Count beta parameters for all stocks, described in 'prices_m' matrix,
according to 'index' benchmark.
:param prices_m: matrix of prices. Each column represents a stock.
Each row represents price at successive time stamp
:return: matrix with betas. Each column represents a stock. Each row
represents beta at successive time period
"""
returns_m = ml.divide(ml.subtract(prices_m[1:], prices_m[:-1]),
prices_m[:-1])
index_m = ml.matrix(index).T
index_returns_m = ml.divide(ml.subtract(index_m[1:], index_m[:-1]),
index_m[:-1])
result = ml.empty((k, prices_m.shape[1]))
for i in range(k):
for j in range(stock_amount):
x = returns_m[:, j]
y = index_returns_m[:, i]
result[i, j] = np.cov(x, y, rowvar=0)[0][1]/np.var(y)
return result
def generate(self, x0, V, O, R, Alpha):
"""
Arguments
========
x0 : nx1 matrix
initial state
V : T-length list of nx1 matrices
inputs
O : T-length list of lists
location of each clinic reporting at time t
R : T-length list of lists
radius proportional to the catchment of clinic i at time t
Alpha : T-length list of lists
some number proportional to the daily throughput of clinic i at time t
"""
# check that x0 is the right size and type
assert type(x0) is np.matrix
assert x0.shape == (self.n,1)
x = x0
# assign the multivariate normal function to a local function (faster and prettier!)
# we can do a bit of wrapping to make life easier too
def N(mean,sigma):
mean = np.array(mean).flatten()
x = np.random.multivariate_normal(mean, sigma)
return np.matrix(x).T
# yield each new state variable by looping through the input
for t, v in enumerate(V):
# draw next state
x = N(self.A*x + self.U*v, self.Q)
# build observation matrix
C = self.make_C_matrix(R[t], Alpha[t], O[t])
# calculate rate parameters at each clinic
rates = np.array(C*x).flatten()**2
# initialise next y
y = mb.empty((C.shape[0],1))
for i,rate in enumerate(rates):
# draw from the poisson
y[i]=np.random.poisson(rate)
yield x,y
def _record_state_matrices(self, inputs, target_outputs, nb_washing_steps = 0):
training_size = len(inputs)
state_recording_matrix = npmat.empty((self.state_size, training_size - nb_washing_steps))
for i in range(training_size):
self.next_output(inputs[i])
if self.use_feedback:
#teacher forcing : output_values are changed for the next self.next_output() call
self.output_values = target_outputs[i]
if i < nb_washing_steps:
continue
state_recording_matrix[:, i - nb_washing_steps] = self.state
output_target_matrix = np.hstack(target_outputs[nb_washing_steps : training_size])
return state_recording_matrix, output_target_matrix
def dare(F, G1, G2, H):
"""Solves the discrete-time algebraic Riccati equation
0 = F ^ T * X * F
- X - F ^ T * X * G1 * (G2 + G1 ^ T * X * G1) ^ -1 * G1 ^ T * X * F + H
Under the assumption that X ^ -1 exists, this equation is equivalent to
0 = F ^ T * (X ^ -1 + G1 * G2 ^ -1 * G1 ^ T) ^ -1 * F - X + H
Parameters
==========
Inputs are real matrices:
F : n x n
G1 : n x m
G2 : m x m, symmetric, positive definite
H : n x n, symmetric, positive semi-definite
Assumptions
===========
(F, G1) is a stabilizable pair
(C, F) is a detectable pair (where C is full rank factorization of H, i.e.,
C ^ T * C = H and rank(C) = rank(H).
F is invertible
Returns
=======
Unique nonnegative definite solution of discrete Algrebraic Ricatti
equation.
Notes
=====
This is an implementation of the Schur method for solving algebraic Riccati
eqautions as described in dx.doi.org/10.1109/TAC.1979.1102178
"""
# Verify that F is non-singular
u, s, v = la.svd(F)
assert(np.all(s > 0.0))
# Verify that (F, G1) controllable
C = ctrb(F, G1)
u, s, v = la.svd(C)
assert(np.all(s > 0.0))
# Verify that (H**.5, F) is observable
O = obsv(H**.5, F)
u, s, v = la.svd(O)
assert(np.all(s > 0.0))
n = F.shape[0]
m = G2.shape[0]
G = np.dot(G1, np.dot(inv(G2), G1.T))
Finv = inv(F)
Finvt = Finv.T
# Form symplectic matrix
Z = empty((2*n, 2*n))
Z[:n, :n] = F + np.dot(G, np.dot(Finvt, H))
Z[:n, n:] = -np.dot(G, Finvt)
Z[n:, :n] = -np.dot(Finvt, H)
Z[n:, n:] = Finvt
S, U, sdim = schur(Z, sort='iuc')
# Verify that the n eigenvalues of the upper left block stable
assert(sdim == n)
U11 = U[:n, :n]
U21 = U[n:, :n]
return solve(U[:n, :n].T, U[n:, :n].T).T
#!/usr/bin/python # Generate absorbance data and save testing file for A->B import random from numpy import matrix, matlib import csv from frange import frange t_0 = matrix(frange(-1e-6, 0, 5e-11)).transpose() t_1 = matrix(frange(0, 1.9999500e-6, 5e-11)).transpose() k = [2.2e7, 3.124e7] # rate constant a_0 = 1e-3 # initial concentration of A a_1 = 2e-3 # initial concentration of C c = matlib.empty([t_1.size, 2]) c[:,0] = a_0 * matlib.exp(-k[0] * t_1) c[:,1] = a_1 * matlib.exp(-k[1] * t_1) # molar absorption of species A a = matlib.empty([2, 1]) a[0,0] = 1e3 a[1,0] = 1e3 y_1 = matlib.dot(c, a) y_1 = y_1.transpose().tolist()[0] y_1 = map(lambda y: y + (0.04 * random.random() - 0.02), y_1) t_0 = t_0.transpose().tolist()[0] t_1 = t_1.transpose().tolist()[0] fullLightVoltage = -0.0951192897786 y_1 = map(lambda y:fullLightVoltage*(10**-y), y_1)
#!/usr/bin/python
# Generate absorbance data and save testing file for A->B
import random
from numpy import matrix, matlib
import csv
from frange import frange
t_0 = matrix(frange(-1e-6, 0, 5e-11)).transpose()
t_1 = matrix(frange(0, 1.9999500e-6, 5e-11)).transpose()
k = 2.2e7 # rate constant
a_0 = 1e-3 # initial concentration of A
c = a_0 * matlib.exp(-k * t_1)
# molar absorption of species A
a = matlib.empty([1, 1])
a[0, 0] = 1e3
y_1 = matlib.dot(c, a)
y_1 = y_1.transpose().tolist()[0]
y_1 = map(lambda y: y + (0.04 * random.random() - 0.02), y_1)
t_0 = t_0.transpose().tolist()[0]
t_1 = t_1.transpose().tolist()[0]
fullLightVoltage = -0.0951192897786
y_1 = map(lambda y: fullLightVoltage * (10**-y), y_1)
y_0 = []
for i in range(0, len(t_0)):
y_0.append(fullLightVoltage + 0.01 * random.random() - 0.005)
# All seeds the same
#
#===============================================================================
mainPath = 'C:/DataSets/Results/Fraust'
os.chdir(mainPath)
# Parameters List (of Dictionaries)
param = {'alpha' : 0.996,
'rr' : 4,
'sci' : 1}
# Results list (of dictionaris)
# results = []
e_qq_mat = npm.empty((10000,numRuns))
for i in range(numRuns):
# Generate artificial data streams
streams = genCosSignals(i, -3)
# Run Fast row householder subspace tracker
Q_t, S_t, rr, E_t, E_dash_t, hid_var, z_dash, RSRE, no_inp_count, \
no_inp_marker = FRHH(streams, param['rr'], param['alpha'], param['sci'])
# Calculate deviations from orthogonality and subspace
e_qq, f_qq = plotEqqFqq(streams, Q_t, param['alpha'],0, 0)
# Store results in Dictionary
dic_name = 'res_' + str(i) # string of the name of the Dictionary
vars()[dic_name] = {'param' : param,
#!/usr/bin/python
# Generate absorbance data and save testing file for A->B
import random
from numpy import matrix, matlib
import csv
from frange import frange
t_0 = matrix(frange(-1e-6, 0, 5e-11)).transpose()
t_1 = matrix(frange(0, 1.9999500e-6, 5e-11)).transpose()
k = 2.2e7 # rate constant
a_0 = 1e-3 # initial concentration of A
c = a_0 * matlib.exp(-k * t_1)
# molar absorption of species A
a = matlib.empty([1, 1])
a[0,0] = 1e3
y_1 = matlib.dot(c, a)
y_1 = y_1.transpose().tolist()[0]
y_1 = map(lambda y: y + (0.04 * random.random() - 0.02), y_1)
t_0 = t_0.transpose().tolist()[0]
t_1 = t_1.transpose().tolist()[0]
fullLightVoltage = -0.0951192897786
y_1 = map(lambda y:fullLightVoltage*(10**-y), y_1)
y_0 = []
for i in range(0, len(t_0)):
y_0.append(fullLightVoltage + 0.01 * random.random() - 0.005)
# -*- encoding: utf-8 -*-
"""
4.6.1 创建矩阵
"""
import numpy as np
import numpy.matlib as mat
print(np.mat([[1, 2, 3], [4, 5, 6]], dtype=np.int)) # 使用列表创建矩阵
print(np.mat(np.arange(6).reshape((2, 3)))) # 使用数组创建矩阵
print(np.mat('1 4 7; 2 5 8; 3 6 9')) # 使用Matlab风格的字符串创建矩阵
print(mat.zeros((2, 3))) # 全0矩阵
print(mat.ones((2, 3))) # 全1矩阵
print(mat.eye(3)) # 单位矩阵
print(mat.empty((2, 3))) # 空矩阵
print(mat.rand((2, 3))) # [0,1)区间随机数矩阵
print(mat.randn((2, 3))) # 均值0方差1的高斯（正态）分布矩阵
def MG1FundamentalMatrix (A, precision=1e-14, maxNumIt=50, method="ShiftPWCR", maxNumRoot=2048, shiftType="one"):
"""
Returns matrix G corresponding to the M/G/1 type Markov
chain defined by matrices A.
Matrix G is the minimal non-negative solution of the
following matrix equation:
.. math::
G = A_0 + A_1 G + A_2 G^2 + A_3 G^3 + \dots.
The implementation is based on [1]_, please cite it if
you use this method.
Parameters
----------
A : length(M) list of matrices of shape (N,N)
Matrix blocks of the M/G/1 type generator from
0 to M-1.
precision : double, optional
Matrix G is computed iteratively up to this
precision. The default value is 1e-14
maxNumIt : int, optional
The maximal number of iterations. The default value
is 50.
method : {"CR", "RR", "NI", "FI", "IS"}, optional
The method used to solve the matrix-quadratic
equation (CR: cyclic reduction, RR: Ramaswami
reduction, NI: Newton iteration, FI: functional
iteration, IS: invariant subspace method). The
default is "CR".
Returns
-------
G : matrix, shape (N,N)
The G matrix of the M/G/1 type Markov chain.
(G is stochastic.)
References
----------
.. [1] Bini, D. A., Meini, B., Steffé, S., Van Houdt,
B. (2006, October). Structured Markov chains
solver: software tools. In Proceeding from the
2006 workshop on Tools for solving structured
Markov chains (p. 14). ACM.
"""
if not isinstance(A,np.ndarray):
D = np.hstack(A)
else:
D = ml.matrix(A)
if method=="ShiftPWCR":
if butools.verbose:
Dold = ml.matrix(D)
D, drift, tau, v = MG1TypeShifts (D, shiftType)
m = D.shape[0]
M = D.shape[1]
I = ml.eye(m)
D = D.T
D = np.vstack((D, ml.zeros(((2**(1+math.floor(math.log(M/m-1,2)))+1)*m-M,m))))
# Step 0
G = ml.zeros((m,m))
Aeven = D[np.remainder(np.kron(np.arange(D.shape[0]/m),np.ones(m)),2)==0,:]
Aodd = D[np.remainder(np.kron(np.arange(D.shape[0]/m),np.ones(m)),2)==1,:]
Ahatodd = np.vstack((Aeven[m:,:], D[-m:,:]))
Ahateven = Aodd
Rj = D[m:2*m,:]
for i in range(3,M//m+1):
Rj = Rj + D[(i-1)*m:i*m,:]
Rj = la.inv(I-Rj)
Rj = D[:m,:] * Rj
numit = 0
while numit < maxNumIt:
numit+=1
nj=Aodd.shape[0]//m
if nj > 0:
# Evaluate the 4 functions in the nj+1 roots using FFT
# prepare for FFTs (such that they can be performed in 4 calls)
temp1=np.reshape(Aodd[:nj*m,:].T,(m*m,nj),order='F').T
temp2=np.reshape(Aeven[:nj*m,:].T,(m*m,nj),order='F').T
temp3=np.reshape(Ahatodd[:nj*m,:].T,(m*m,nj),order='F').T
temp4=np.reshape(Ahateven[:nj*m,:].T,(m*m,nj),order='F').T
# FFTs
temp1=fft(temp1,nj,0)
temp2=fft(temp2,nj,0)
temp3=fft(temp3,nj,0)
temp4=fft(temp4,nj,0)
# reform the 4*nj matrices
temp1=np.reshape(temp1.T,(m,m*nj),order='F').T
temp2=np.reshape(temp2.T,(m,m*nj),order='F').T
temp3=np.reshape(temp3.T,(m,m*nj),order='F').T
temp4=np.reshape(temp4.T,(m,m*nj),order='F').T
# Next, we perform a point-wise evaluation of (6.20) - Thesis Meini
Ahatnew = ml.empty((nj*m,m), dtype=complex)
Anew = ml.empty((nj*m,m), dtype=complex)
for cnt in range(1,nj+1):
Ahatnew[(cnt-1)*m:cnt*m,:] = temp4[(cnt-1)*m:cnt*m,:] + temp2[(cnt-1)*m:cnt*m,:] * la.inv(I-temp1[(cnt-1)*m:cnt*m,:]) * temp3[(cnt-1)*m:cnt*m,:]
Anew[(cnt-1)*m:cnt*m,:] = cmath.exp(-(cnt-1)*2.0j*math.pi/nj) * temp1[(cnt-1)*m:cnt*m,:] + temp2[(cnt-1)*m:cnt*m,:] * la.inv(I-temp1[(cnt-1)*m:cnt*m,:]) * temp2[(cnt-1)*m:cnt*m,:]
# We now invert the FFTs to get Pz and Phatz
# prepare for IFFTs (in 2 calls)
Ahatnew = np.reshape(Ahatnew[:nj*m,:].T,(m*m,nj),order='F').T
Anew = np.reshape(Anew[:nj*m,:].T,(m*m,nj),order='F').T
# IFFTs
Ahatnew = np.real(ifft(Ahatnew,nj,0))
Anew = np.real(ifft(Anew,nj,0))
# reform matrices Pi and Phati
Ahatnew = np.reshape(Ahatnew.T,(m,m*nj),order='F').T
Anew = np.reshape(Anew.T,(m,m*nj),order='F').T
else: # series Aeven, Aodd, Ahateven and Ahatodd are constant
temp = Aeven * la.inv(I-Aodd)
Ahatnew = Ahateven + temp*Ahatodd
Anew = np.vstack((temp*Aeven, Aodd))
nAnew = 0
deg = Anew.shape[0]//m
for i in range(deg//2,deg):
nAnew = max(nAnew, la.norm(Anew[i*m:(i+1)*m,:],np.inf))
nAhatnew = 0
deghat = Ahatnew.shape[0]//m
for i in range(deghat//2,deghat):
nAhatnew = max(nAhatnew, la.norm(Ahatnew[i*m:(i+1)*m,:],np.inf))
# c) the test
while (nAnew > nj*precision or nAhatnew > nj*precision) and nj < maxNumRoot:
nj *= 2
stopv = min(nj, Aodd.shape[0]/m)
# prepare for FFTs
temp1=np.reshape(Aodd[:stopv*m,:].T,(m*m,stopv),order='F').T
temp2=np.reshape(Aeven[:stopv*m,:].T,(m*m,stopv),order='F').T
temp3=np.reshape(Ahatodd[:stopv*m,:].T,(m*m,stopv),order='F').T
temp4=np.reshape(Ahateven[:stopv*m,:].T,(m*m,stopv),order='F').T
# FFTs
temp1=fft(temp1,nj,0)
temp2=fft(temp2,nj,0)
temp3=fft(temp3,nj,0)
temp4=fft(temp4,nj,0)
# reform the 4*(nj+1) matrices
temp1=np.reshape(temp1.T,(m,m*nj),order='F').T
temp2=np.reshape(temp2.T,(m,m*nj),order='F').T
temp3=np.reshape(temp3.T,(m,m*nj),order='F').T
temp4=np.reshape(temp4.T,(m,m*nj),order='F').T
# Next, we perform a point-wise evaluation of (6.20) - Thesis Meini
Ahatnew = ml.empty((nj*m,m), dtype=complex)
Anew = ml.empty((nj*m,m), dtype=complex)
for cnt in range(1,nj+1):
Ahatnew[(cnt-1)*m:cnt*m,:] = temp4[(cnt-1)*m:cnt*m,:] + temp2[(cnt-1)*m:cnt*m,:] * la.inv(I-temp1[(cnt-1)*m:cnt*m,:]) * temp3[(cnt-1)*m:cnt*m,:]
Anew[(cnt-1)*m:cnt*m,:] = cmath.exp(-(cnt-1)*2j*math.pi/nj) * temp1[(cnt-1)*m:cnt*m,:] + temp2[(cnt-1)*m:cnt*m,:] * la.inv(I-temp1[(cnt-1)*m:cnt*m,:]) * temp2[(cnt-1)*m:cnt*m,:]
# We now invert the FFTs to get Pz and Phatz
# prepare for IFFTs
Ahatnew = np.reshape(Ahatnew[:nj*m,:].T,(m*m,nj),order='F').T
Anew = np.reshape(Anew[:nj*m,:].T,(m*m,nj),order='F').T
# IFFTs
Ahatnew = ml.matrix(np.real(ifft(Ahatnew,nj,0)))
Anew = ml.matrix(np.real(ifft(Anew,nj,0)))
# reform matrices Pi and Phati
Ahatnew = np.reshape(Ahatnew.T,(m,m*nj),order='F').T
Anew = np.reshape(Anew.T,(m,m*nj),order='F').T
vec1 = ml.zeros((1,m))
vec2 = ml.zeros((1,m))
for i in range(1,Anew.shape[0]//m):
vec1 += i*np.sum(Anew[i*m:(i+1)*m,:],0)
vec2 += i*np.sum(Ahatnew[i*m:(i+1)*m,:],0)
nAnew = 0
deg = Anew.shape[0]//m
for i in range(deg//2,deg):
nAnew = max(nAnew, la.norm(Anew[i*m:(i+1)*m,:],np.inf))
nAhatnew = 0
deghat = Ahatnew.shape[0]//m
for i in range(deghat//2,deghat):
nAhatnew = max(nAhatnew, la.norm(Ahatnew[i*m:(i+1)*m,:],np.inf))
if (nAnew > nj*precision or nAhatnew > nj*precision) and nj >= maxNumRoot:
print("MaxNumRoot reached, accuracy might be affected!")
if nj > 2:
Anew = Anew[:m*nj/2,:]
Ahatnew = Ahatnew[:m*nj/2,:]
# compute Aodd, Aeven, ...
Aeven = Anew[np.remainder(np.kron(np.arange(Anew.shape[0]/m),np.ones(m)),2)==0,:]
Aodd = Anew[np.remainder(np.kron(np.arange(Anew.shape[0]/m),np.ones(m)),2)==1,:]
Ahateven = Ahatnew[np.remainder(np.kron(np.arange(Ahatnew.shape[0]/m),np.ones(m)),2)==0,:]
Ahatodd = Ahatnew[np.remainder(np.kron(np.arange(Ahatnew.shape[0]/m),np.ones(m)),2)==1,:]
if butools.verbose==True:
if method == "PWCR":
print("The Point-wise evaluation of Iteration ", numit, " required ", nj, " roots")
else:
print("The Shifted PWCR evaluation of Iteration ", numit, " required ", nj, " roots")
# test stopcriteria
if method == "PWCR":
Rnewj = Anew[m:2*m,:]
for i in range (3, Anew.shape[0]/m+1):
Rnewj = Rnewj + Anew[(i-1)*m:i*m,:]
Rnewj = la.inv(I-Rnewj)
Rnewj = Anew[:m,:]*Rnewj
if np.max(np.abs(Rj-Rnewj)) < precision or np.max(np.sum(I-Anew[:m,:]*la.inv(I-Anew[m:2*m,:]),0)) < precision:
G = Ahatnew[:m,:]
for i in range (2,Ahatnew.shape[0]/m+1):
G = G + Rnewj * Ahatnew[(i-1)*m:i*m,:]
G = D[:m,:]*la.inv(I-G)
break
Rj = Rnewj
# second condition tests whether Ahatnew is degree 0 (numerically)
if la.norm(Anew[:m,:m]) < precision or np.sum(Ahatnew[m:,:]) < precision or np.max(np.sum(I-D[:m,:]*la.inv(I-Ahatnew[:m,:]),0)) < precision:
G = D[0:m,:] * la.inv(I-Ahatnew[:m,:])
break
else:
Gold = G
G = D[:m,:]*la.inv(I-Ahatnew[:m,:])
if la.norm(G-Gold,np.inf) < precision or la.norm(Ahatnew[m:,:],np.inf) < precision:
break
if numit == maxNumIt and G==ml.zeros((m,m)):
print("Maximum Number of Iterations reached!")
G = D[:m,:] * la.inv(I-Ahatnew[:m,:])
G=G.T
if method=="ShiftPWCR":
if shiftType=="one":
G = G + (drift<1)*ml.ones((m,m))/m
elif shiftType=="tau":
G = G + (drift>1)*tau*v*ml.ones((1,m))
elif shiftType=="dbl":
G = G + (drift<1)*ml.ones((m,m))/m+(drift>1)*tau*v*ml.ones((1,m))
if butools.verbose:
if method=="PWCR":
D = D.T
else:
D = Dold
temp = D[:,-m:]
for i in range (D.shape[1]//m-1,0,-1):
temp = D[:,(i-1)*m:i*m] + temp*G
res_norm = la.norm(G-temp,np.inf)
print("Final Residual Error for G: ", res_norm)
return G
def getDeterministics(nobs, which = 'c', date = 0.5):
'''
Returns various useful deterministic terms for a given sample length T.
Return object is a numpy-matrix-type of dimension Tx(len(which));
(early periods first, where relevant).
In the 'which' argument pass a string composed of the following letters,
in arbitrary order:
c - constant (=1) term
t - trend (starting with 0)
q - centered quarterly seasonal dummies (starting with 0.75, -0.25...)
m - centered monthly seasonal dummies (starting with 11/12, -1/12, ...)
l - level shift (date applies)
s - slope shift (date applies)
i - impulse dummy (date applies)
If the date argument is a floating point number (between 0 and 1),
it is treated as the fraction of the sample where the break occurs.
If instead it is an integer between 0 and T, then that observation is
treated as the shift date.
'''
# some input checks (as well as assignment of shiftperiod):
if type(nobs) != type(4): # is not an integer
raise TypeError, 'need integer for sample length'
if nobs <=0: raise ValueError, 'need positive sample length'
if type(date) == type(0.5): #is a float, treat as break fraction
if date < 0 or date > 1:
raise ValueError, 'need break fraction between 0 and 1'
shiftperiod = int(date * nobs)
elif type(date) == type(4): # is integer, treat as period number
if date not in range(1, nobs+1):
raise ValueError, 'need period within sample range'
shiftperiod = date
else: raise TypeError, 'need float or integer input for date'
if type(which) != type('a string'):
raise TypeError, 'need string for case spec'
# end input checks
out = empty([nobs,0]) # create starting point
if 'c' in which: out = c_[ out, ones(nobs).T ]
if 't' in which: out = c_[ out, r_['c', :nobs] ]
if 'l' in which:
shift = r_[ zeros(shiftperiod).T, ones(nobs-shiftperiod).T ]
out = c_[ out, shift ]
if 's' in which:
slopeshift = r_[ zeros(shiftperiod).T, r_['c', 1:(nobs - shiftperiod + 1)] ]
out = c_[ out, slopeshift ]
if 'i' in which:
impulse = r_[ zeros(shiftperiod).T, ones(1), zeros(nobs-shiftperiod-1).T ]
out = c_[ out, impulse ]
if 'q' in which or 'Q' in which:
# to end of next full year, thus need to slice at T below:
q1 = [0.75, -0.25, -0.25, -0.25] * (1 + nobs/4)
q2 = [-0.25, 0.75, -0.25, -0.25] * (1 + nobs/4)
q3 = [-0.25, -0.25, 0.75, -0.25] * (1 + nobs/4)
out = c_[ out, mat(q1[:nobs]).T, mat(q2[:nobs]).T, mat(q3[:nobs]).T ]
if 'm' in which or 'M' in which:
temp = [-1./12] * 11
for month in range(11):
temp.insert(month, 1-temp[0])
# again, to end of next full year, thus need to slice at T below:
monthly = temp * (1 + nobs/12) # temp is still a list here!
out = c_[ out, mat(monthly[:nobs]).T ]
return out
def kernal(x, gamma, n):
k = ml.empty((n, n))
for i in range(n):
for j in range(n):
k[i, j] = np.exp(-gamma * la.norm(x[i, :] - x[j, :]) ** 2)
return k
#-*-coding:utf-8-*-
"""
NumPy - 矩阵库
NumPy 包包含一个 Matrix库numpy.matlib。此模块的函数返回矩阵而不是返回ndarray对象。
"""
import numpy.matlib as mt
import numpy as np
if __name__ == '__main__':
#numpy.matlib.empty(shape, dtype, order)函数返回一个新的矩阵，而不初始化元素
print mt.empty((2, 2))
#numpy.matlib.zeros()返回以零填充的矩阵。
print mt.zeros((3, 3))
#numpy.matlib.ones()返回以一填充的矩阵。
print mt.ones((4, 4))
#numpy.matlib.eye(n, M,k, dtype)返回一个矩阵，对角线元素为 1，其他位置为零。
print mt.eye(n=5, M=5, k=0, dtype=float)
#numpy.matlib.identity()函数返回给定大小的单位矩阵。单位矩阵是主对角线元素都为 1 的方阵。
print mt.identity(5)
#numpy.matlib.rand()函数返回给定大小的填充随机值的矩阵。
print mt.rand(3, 3)
#矩阵总是二维的，而ndarray是一个 n 维数组。 两个对象都是可互换的。
m = np.matrix('1,2;3,4')
print m
print type(m)
print np.asarray(m)
def signal(prices, index):
"""
Signals to buy/sell stocks.
Returns tuple (X1, X2, ..., Xn), where n is amount of stocks,
Xi is one of the ('sell', 'buy', None).
:param prices: list of price lists of stocks, in time ascending order
:param index: list of k(!) price lists of benchmark values for k last
periods, in time ascending order, with same time frame as stocks.
"""
def betas(prices_m):
"""
Count beta parameters for all stocks, described in 'prices_m' matrix,
according to 'index' benchmark.
:param prices_m: matrix of prices. Each column represents a stock.
Each row represents price at successive time stamp
:return: matrix with betas. Each column represents a stock. Each row
represents beta at successive time period
"""
returns_m = ml.divide(ml.subtract(prices_m[1:], prices_m[:-1]),
prices_m[:-1])
index_m = ml.matrix(index).T
index_returns_m = ml.divide(ml.subtract(index_m[1:], index_m[:-1]),
index_m[:-1])
result = ml.empty((k, prices_m.shape[1]))
for i in range(k):
for j in range(stock_amount):
x = returns_m[:, j]
y = index_returns_m[:, i]
result[i, j] = np.cov(x, y, rowvar=0)[0][1]/np.var(y)
return result
def regime(reduced_returns_m):
"""
Make a regime switch based on PCA standard deviation acceleration.
:param reduced_returns_m: matrix of PCA. Each column represents a
stock. Each row represents price at successive time stamp
:return: one of the strings, 'momentum' - if trend is ment to
continue its movement, 'mean_reversion' - otherwise
"""
cross_sect_vol = np.std(reduced_returns_m, axis=1)
changes = cross_sect_vol[1:] - cross_sect_vol[:-1]
squared_changes = np.square(changes)
distance_times = reduced_returns_m.shape[0] - 1 # because there is
# T - 1 changes
distance = np.zeros(distance_times)
for t in range(distance_times):
sum_amount = min(t + 1, H)
for i in range(sum_amount):
distance[t] += squared_changes[t - i, 0]
distance[t] = np.sqrt(distance[t])
signal = distance[1:] - distance[:-1]
if np.max(signal) > 0:
return 'momentum'
else:
return 'mean_reversion'
prices_m = ml.matrix(prices).T
# Preparing main matrices for further computations
try:
log_returns_m = np.log(ml.divide(prices_m[1:], prices_m[:-1]))
except TypeError as e:
raise WrongPricesError(prices_m)
time_period, stock_amount = log_returns_m.shape
mean_log_returns_m = ml.average(log_returns_m, axis=0)
demeaned_log_returns_m = log_returns_m - mean_log_returns_m
covariation_m = demeaned_log_returns_m.T * demeaned_log_returns_m
# Count eigenvectors of covariation matrix and compose PCA matrix from them
e_values, e_vectors = eig(covariation_m)
abs_e_values = np.absolute(e_values)
# TODO: np.absolute(e_vectors) or something like that
indexed_abs_e_values = [(i, v) for i, v in enumerate(abs_e_values)]
w = sorted(indexed_abs_e_values, reverse=False,
key=lambda x: x[1])
e_vectors_m = ml.empty((stock_amount, k))
for j in range(k):
e_vectors_m[:, j] = e_vectors[:, w[j][0]]
# Main part: project returns on PCA universe
reduced_returns_m = (e_vectors_m.T * demeaned_log_returns_m.T).T
# Count beta parameters with respect to given benchmark index
betas_m = betas(prices_m)
time = time_period - time_shift
if time < H:
raise WrongParameterException("time_shift should be less than H")
# Collect data from returns in one vector
accumulated_reduced_returns = ml.zeros((1, k))
for i in range(H):
accumulated_reduced_returns += reduced_returns_m[time - 1 - i]
# Make a prediction about further returns behaviour
estimation = accumulated_reduced_returns * betas_m + \
mean_log_returns_m
if regime_switcher:
regime = regime(reduced_returns_m)
else:
regime = 'mean_reversion'
# Finally, decide for each stock, whether we need to sell it as
# overvalued or buy as undervalued. Other way around for momentum switch
max_recent_log_returns = log_returns_m[-H:].max(0)
result = []
for i in range(stock_amount):
if max_recent_log_returns[0, i] > estimation[0, i] + epsilon:
if regime == 'mean_reversion':
result.append('sell')
else:
result.append('buy')
elif max_recent_log_returns[0, i] < estimation[0, i] - epsilon:
if regime == 'mean_reversion':
result.append('buy')
else:
result.append('sell')
else:
result.append(None)
return result
#!/usr/bin/python # Generate absorbance data and save testing file for A->B import random from numpy import matrix, matlib import csv from frange import frange t_0 = matrix(frange(-1e-6, 0, 5e-11)).transpose() t_1 = matrix(frange(0, 1.9999500e-6, 5e-11)).transpose() #t_1 = matrix(frange(0, 4000, 25)).transpose() # the following constants work well k = [2.2e7, 3.17e6] # rate constant #k = [0.003, 0.0015] a_0 = 1e-3 # initial concentration of A c = matlib.empty([t_1.size, 3]) c[:,0] = a_0 * matlib.exp(-k[0] * t_1) # concentration of A c[:,1] = a_0 * (k[0] / (k[1] - k[0])) * (matlib.exp(-k[0] * t_1) - matlib.exp(-k[1] * t_1)) # concentration of B c[:,2] = a_0 - c[:,0] - c[:,1] # concentration of C a = matlib.empty([3, 1]) a[0,0] = 1e3 # molar absorption of species A a[1,0] = 4e2 # molar absorption of species B a[2,0] = 7e2 # molar absorption of species C y_1 = matlib.dot(c, a) y_1 = y_1.transpose().tolist()[0] y_1 = map(lambda y: y + (0.04 * random.random() - 0.02), y_1) t_0 = t_0.transpose().tolist()[0] t_1 = t_1.transpose().tolist()[0]
def mesh(self, lsid: int, lpid: int):
from pygeom.geom3d import Vector
from numpy.matlib import empty
nums = len(self.sects)
self.shts = []
for i in range(nums - 1):
a, b = i, i + 1
secta = self.sects[a]
sectb = self.sects[b]
self.shts.append(LatticeSheet(secta, sectb))
self.strps = []
for sht in self.shts:
lsid = sht.mesh_strips(lsid)
self.strps += sht.strps
pnts = [strp.pnt1 for strp in self.strps]
pnts.append(self.strps[-1].pnt2)
crds = [strp.crd1 for strp in self.strps]
crds.append(self.strps[-1].crd2)
lenb = len(pnts)
lenc = len(self.cspc)
self.pnts = empty((lenb, lenc + 1), dtype=Vector)
for i in range(lenb):
minx = pnts[i].x
y = pnts[i].y
z = pnts[i].z
c = crds[i]
cd = self.cspc[0][0]
x = minx + cd * c
self.pnts[i, 0] = Vector(x, y, z)
for j in range(1, lenc + 1):
cd = self.cspc[j - 1][-1]
x = minx + cd * c
self.pnts[i, j] = Vector(x, y, z)
self.pnls = empty((lenb - 1, lenc), dtype=LatticePanel)
for i, strp in enumerate(self.strps):
for j in range(lenc):
pnts = [
self.pnts[i, j], self.pnts[i + 1, j], self.pnts[i, j + 1],
self.pnts[i + 1, j + 1]
]
cspc = self.cspc[j]
pnl = LatticePanel(lpid, pnts, cspc, strp)
self.pnls[i, j] = pnl
lpid += 1
if self.mirror:
self.sgrp = [[], []]
numstrp = len(self.strps)
hlfstrp = int(numstrp / 2)
for i in range(hlfstrp):
lstrp = self.strps[numstrp - 1 - i]
mstrp = self.strps[i]
self.sgrp[0].append(lstrp.lsid)
self.sgrp[1].append(mstrp.lsid)
else:
self.sgrp = [[]]
numstrp = len(self.strps)
for i in range(numstrp):
lstrp = self.strps[numstrp - 1 - i]
self.sgrp[0].append(lstrp.lsid)
bpos = [0.0]
for sht in self.shts:
sht.inherit_panels()
sht.set_control_panels()
bpos.append(bpos[-1] + sht.width)
if self.mirror:
numsht = len(self.shts)
wmir = bpos[int(numsht / 2)]
for i in range(len(bpos)):
bpos[i] = bpos[i] - wmir
for i, sect in enumerate(self.sects):
sect.bpos = bpos[i]
for sht in self.shts:
sht.set_strip_bpos()
bmax = max(bpos)
for func in self.funcs:
func.set_spline(bmax)
var = func.var
if var == 'twist':
var = '_ang'
if self.mirror:
for i in range(hlfstrp):
strp = self.strps[numstrp - 1 - i]
mstrp = self.strps[i]
bpos = strp.bpos
val = func.interpolate(bpos)
strp.__dict__[var] = val
mstrp.__dict__[var] = val
else:
for strp in self.strps:
bpos = strp.bpos
val = func.interpolate(bpos)
strp.__dict__[var] = val
self.area = 0.0
for sht in self.shts:
if not sht.noload:
self.area += sht.area
return lsid, lpid
#!/usr/bin/env python
# -*- coding: utf-8 -*-
# @Time : 2019/9/21 14:23
# @Author : ganliang
# @File : npmatlib.py
# @Desc : 矩阵
import numpy as np
import numpy.matlib as ml
print ("empty")
print(ml.empty((3, 3), dtype=np.int, order='F'))
print(ml.empty((3, 3), dtype=np.int, order='C'))
print ("\nzeros")
print(ml.zeros((3, 3), dtype=np.int, order='C'))
print ("\nones")
print(ml.ones((3, 3), dtype=np.int, order='C'))
print ("\neye")
print(ml.eye(3, dtype=np.int, order='C'))
print ("\nidentity")
print(ml.identity(3, dtype=np.int))
print ("\nrand")
print(ml.rand(2, 3))
print ("\nmatrix")
a = np.arange(12).reshape(3, 4)
mr = ml.matrix(a)
finalState = oldState
else:
finalState = newState
t = sum(finalState)
#print(finalState)
return finalState, t
def print20x20(
array): # prints the 20x20 row by row rather than as a single line
for i in range(0, 20):
print(array[i])
z = 0
m = empty((20, 20), int)
mplot = []
# creates the 20x20 canvas
for i in range(0, 20):
# m.append([])
for j in range(0, 20):
# # m[i].append([])
x = random.rand()
if x > 0.5:
m[i, j] = -1
else:
m[i, j] = 1
z = sum(m[i, 0]) # Counts to check net magnetization
#print20x20(m)
print(m)
print(z)
#numpy.matlib is used for matrices instead of numpy ndarray objects
#matrices functions
#importing the numpy module
import numpy as np
#importing the numpy.matlib module
import numpy.matlib as nm
#matlib empty function
print(
"\nprinting the matrix with uninitializing the values using empty function:"
)
print(nm.empty((3, 3)))
#matlib zeros function
print(
"\n\nprinting the matrix with initializing the values with value \"0\" using zero function:"
)
print(nm.zeros((3, 3), dtype=int))
#matlib ones function
print(
"\n\nprinting the matrix with initializing the values with value \"1\" using ones function:"
)
print(nm.ones((3, 4), dtype=int))
#matlib eye function
print("\n\nprinting the matrix with diagonal elements values equals to 1:")
print(nm.eye(n=4, k=0, dtype=int, M=5))
def graph2featureVect(training_files,featureNames):
graphs=[]
for training_file in training_files:
try:
graphs.append( graph_analysis.input_graph(training_file)) # open graph data and copy contents to memory
print ("Succefully read in %s\n" %training_file)
except:
print("Error reading in %s\n" %training_file)
#### initialize the feature_vector to be zeros(#datapoints=sum(len(g.edge_list()) w label!=0 , len(featureNames))
## initialize the target_vector to be zeros(#datapoints=sum(len(g.edge_list()) w label!=0 , 1)
MRI_labels=[]
if 'mri_label_dist' in featureNames:
MRI_labels=[l[0] for l in graphs[0].edge_property(graphs[0].edge_list()[0],'mri_label_dist')]
feature_vector = Matlib.empty((0,len(featureNames)+len(MRI_labels)-1),float)
else:
feature_vector = Matlib.empty((0,len(featureNames)),float)
target_vector = [] #Matlib.empty((0,1),float)
labelNumerics = []
edge_w_indx = {}
g_cnt=-1
e_cnt = 0
for g in graphs:
g_cnt =g_cnt+1
for e in g.edge_list():
if 'label' not in g.edge_properties(e).keys():
print ("ERROR! The edge (%d,%d) in graph %s doesn't have label property.\nAborted!\n" %(e[0],e[1],training_files[g_cnt]))
exit (0)
#else:
if g.edge_property(e,'label')>-1: #including the edges with label 0
#target_vector= numpy.vstack([target_vector, array(g.edge_property(e,'label'))])
target_vector.append(int(g.edge_property(e,'label')))
if g.edge_property(e,'label') not in labelNumerics and g.edge_property(e,'label')>0:
labelNumerics.append(int(g.edge_property(e,'label')))
features=[]
for f in featureNames:
if f not in g.edge_properties(e).keys():
print ("ERROR! The edge (%d,%d) in graph %s doesn't have feature %s property.\nAborted!\n" %(e[0],e[1],training_files[g_cnt],f))
exit (0)
if not f== 'mri_label_dist' and not f=='rel_dir' and not f=='rel_diameter':
features.append(g.edge_property(e,f))
elif f== 'mri_label_dist':
for mr_l in MRI_labels:
#if type(g.edge_property(e,'mri_label_dist'))==list:
#print mr_l,":",e, " list ", training_files[g_cnt]
#ind=[l[0] for l in g.edge_property(e,'mri_label_dist')].index(mr_l)
#else:
#print mr_l,":",e, ", ", training_files[g_cnt]
#ind=[l[0] for l in g.edge_property(e,'mri_label_dist').tolist()].index(mr_l)
mri_label_dist_list = [int(l[0]) for l in g.edge_property(e,'mri_label_dist')]
if mr_l in mri_label_dist_list:
ind = mri_label_dist_list.index(mr_l)
features.append(g.edge_property(e,'mri_label_dist')[ind][1])
elif f=='rel_dir' or f=='rel_diameter':
rel_f = 1.0
for rel_i in range(min(len(g.edge_property(e,f)),4)): #we only consider up to 4 adjacent edges, if there are less 1.0 will be used
rel_f = rel_f * g.edge_property(e,f)[rel_i]
features.append(rel_f)
feature_vector = numpy.vstack([feature_vector, array(features)]) #hstack #a = matrix([[10,20,30]]); a=append(a,[[1,2,3]],axis=0); a=append(a,[[15],[15]],axis=1)
edge_w_indx[e_cnt]= e
e_cnt = e_cnt+1
#### !!find neighbouring edges and in iterations find their
adjacency_matrix = (1.0/len(labelNumerics))*Matlib.ones((len(labelNumerics),len(labelNumerics)+1),float) #### smoothing (for 0s in the adj_matrix)! to be minimum of 1 occurance for the nieghbourhood!
edge_neighboring_indx = {}
g_cnt=-1
e_cnt = 0
for g in graphs:
g_cnt =g_cnt+1
for e in g.edge_list():
#if 'label' not in g.edge_properties(e).keys():
#print ("ERROR! The edge (%d,%d) in graph %s doesn't have label property.\nAborted!\n" %(e[0],e[1],training_files[g_cnt]))
#exit (0)
#else:
if g.edge_property(e,'label')>0:
indx0= labelNumerics.index(g.edge_property(e,'label'))
e1_neighbours= g.vertices[e[0]].edges
#e1_neighbours.remove(e2)
for v in e1_neighbours:
if v!=e[1]:
neighbor_edge=tuple((min(e[0],v),max(e[0],v)))
if g.edge_property(neighbor_edge,'label')>0:
indx1= labelNumerics.index(g.edge_property(neighbor_edge,'label'))
adjacency_matrix[indx0,indx1]= adjacency_matrix[indx0,indx1]+1
e2_neighbours= g.vertices[e[1]].edges
#e2_neighbours.remove(e1)
for v in e2_neighbours:
if v!=e[0]:
neighbor_edge=tuple((min(e[1],v),max(e[1],v)))
if g.edge_property(neighbor_edge,'label')>0:
indx1= labelNumerics.index(g.edge_property(neighbor_edge,'label'))
adjacency_matrix[indx0,indx1]= adjacency_matrix[indx0,indx1]+1
if len(g.vertices[e[0]].edges)==1: #end point
adjacency_matrix[indx0,len(labelNumerics)] +=1
if len(g.vertices[e[1]].edges)==1: #end point
adjacency_matrix[indx0,len(labelNumerics)] +=1
#find edge neighbouring indeces
neighbor_indx=[]
e1_neighbours= g.vertices[e[0]].edges
for v in e1_neighbours:
if v!=e[1]:
neighbor_edge=tuple((min(e[0],v),max(e[0],v)))
neighbor_indx.append(find_key(edge_w_indx, neighbor_edge, e_cnt))
e2_neighbours= g.vertices[e[1]].edges
for v in e2_neighbours:
if v!=e[0]:
neighbor_edge=tuple((min(e[1],v),max(e[1],v)))
neighbor_indx.append(find_key(edge_w_indx, neighbor_edge, e_cnt))
#if (len(neighbor_indx)!=2 and len(neighbor_indx)!=4):
#print ("\n\nERROR! edge (%d,%d) has %d neighbouring vertices!" %(e[0],e[1],len(neighbor_indx)))
#if (len(neighbor_indx)!=0):
#print "Aborted!\n"
#exit(0)
edge_neighboring_indx[e_cnt]=neighbor_indx
e_cnt = e_cnt+1
#### adjacency matrix normalization
for i in range(adjacency_matrix.shape[0]):
adjacency_matrix[i,:]= adjacency_matrix[i,:]/sum(adjacency_matrix[i,:])
print_adj_mat (adjacency_matrix,labelNumerics)
return labelNumerics, feature_vector, target_vector, adjacency_matrix, edge_w_indx, edge_neighboring_indx #featureVect
def MG1FundamentalMatrix(A,
precision=1e-14,
maxNumIt=50,
method="ShiftPWCR",
maxNumRoot=2048,
shiftType="one"):
"""
Returns matrix G corresponding to the M/G/1 type Markov
chain defined by matrices A.
Matrix G is the minimal non-negative solution of the
following matrix equation:
.. math::
G = A_0 + A_1 G + A_2 G^2 + A_3 G^3 + \dots.
The implementation is based on [1]_, please cite it if
you use this method.
Parameters
----------
A : length(M) list of matrices of shape (N,N)
Matrix blocks of the M/G/1 type generator from
0 to M-1.
precision : double, optional
Matrix G is computed iteratively up to this
precision. The default value is 1e-14
maxNumIt : int, optional
The maximal number of iterations. The default value
is 50.
method : {"CR", "RR", "NI", "FI", "IS"}, optional
The method used to solve the matrix-quadratic
equation (CR: cyclic reduction, RR: Ramaswami
reduction, NI: Newton iteration, FI: functional
iteration, IS: invariant subspace method). The
default is "CR".
Returns
-------
G : matrix, shape (N,N)
The G matrix of the M/G/1 type Markov chain.
(G is stochastic.)
References
----------
.. [1] Bini, D. A., Meini, B., Steffé, S., Van Houdt,
B. (2006, October). Structured Markov chains
solver: software tools. In Proceeding from the
2006 workshop on Tools for solving structured
Markov chains (p. 14). ACM.
"""
if not isinstance(A, np.ndarray):
D = np.hstack(A)
else:
D = ml.matrix(A)
if method == "ShiftPWCR":
if butools.verbose:
Dold = ml.matrix(D)
D, drift, tau, v = MG1TypeShifts(D, shiftType)
m = D.shape[0]
M = D.shape[1]
I = ml.eye(m)
D = D.T
D = np.vstack(
(D,
ml.zeros(
((2**(1 + math.floor(math.log(M / m - 1, 2))) + 1) * m - M, m))))
# Step 0
G = ml.zeros((m, m))
Aeven = D[np.remainder(np.kron(np.arange(D.shape[0] /
m), np.ones(m)), 2) == 0, :]
Aodd = D[np.remainder(np.kron(np.arange(D.shape[0] /
m), np.ones(m)), 2) == 1, :]
Ahatodd = np.vstack((Aeven[m:, :], D[-m:, :]))
Ahateven = Aodd
Rj = D[m:2 * m, :]
for i in range(3, M // m + 1):
Rj = Rj + D[(i - 1) * m:i * m, :]
Rj = la.inv(I - Rj)
Rj = D[:m, :] * Rj
numit = 0
while numit < maxNumIt:
numit += 1
nj = Aodd.shape[0] // m
if nj > 0:
# Evaluate the 4 functions in the nj+1 roots using FFT
# prepare for FFTs (such that they can be performed in 4 calls)
temp1 = np.reshape(Aodd[:nj * m, :].T, (m * m, nj), order='F').T
temp2 = np.reshape(Aeven[:nj * m, :].T, (m * m, nj), order='F').T
temp3 = np.reshape(Ahatodd[:nj * m, :].T, (m * m, nj), order='F').T
temp4 = np.reshape(Ahateven[:nj * m, :].T, (m * m, nj),
order='F').T
# FFTs
temp1 = fft(temp1, nj, 0)
temp2 = fft(temp2, nj, 0)
temp3 = fft(temp3, nj, 0)
temp4 = fft(temp4, nj, 0)
# reform the 4*nj matrices
temp1 = np.reshape(temp1.T, (m, m * nj), order='F').T
temp2 = np.reshape(temp2.T, (m, m * nj), order='F').T
temp3 = np.reshape(temp3.T, (m, m * nj), order='F').T
temp4 = np.reshape(temp4.T, (m, m * nj), order='F').T
# Next, we perform a point-wise evaluation of (6.20) - Thesis Meini
Ahatnew = ml.empty((nj * m, m), dtype=complex)
Anew = ml.empty((nj * m, m), dtype=complex)
for cnt in range(1, nj + 1):
Ahatnew[(cnt - 1) * m:cnt *
m, :] = temp4[(cnt - 1) * m:cnt * m, :] + temp2[
(cnt - 1) * m:cnt * m, :] * la.inv(I - temp1[
(cnt - 1) * m:cnt * m, :]) * temp3[
(cnt - 1) * m:cnt * m, :]
Anew[(cnt - 1) * m:cnt * m, :] = cmath.exp(
-(cnt - 1) * 2.0j * math.pi / nj) * temp1[
(cnt - 1) * m:cnt * m, :] + temp2[
(cnt - 1) * m:cnt * m, :] * la.inv(I - temp1[
(cnt - 1) * m:cnt * m, :]) * temp2[
(cnt - 1) * m:cnt * m, :]
# We now invert the FFTs to get Pz and Phatz
# prepare for IFFTs (in 2 calls)
Ahatnew = np.reshape(Ahatnew[:nj * m, :].T, (m * m, nj),
order='F').T
Anew = np.reshape(Anew[:nj * m, :].T, (m * m, nj), order='F').T
# IFFTs
Ahatnew = np.real(ifft(Ahatnew, nj, 0))
Anew = np.real(ifft(Anew, nj, 0))
# reform matrices Pi and Phati
Ahatnew = np.reshape(Ahatnew.T, (m, m * nj), order='F').T
Anew = np.reshape(Anew.T, (m, m * nj), order='F').T
else: # series Aeven, Aodd, Ahateven and Ahatodd are constant
temp = Aeven * la.inv(I - Aodd)
Ahatnew = Ahateven + temp * Ahatodd
Anew = np.vstack((temp * Aeven, Aodd))
nAnew = 0
deg = Anew.shape[0] // m
for i in range(deg // 2, deg):
nAnew = max(nAnew, la.norm(Anew[i * m:(i + 1) * m, :], np.inf))
nAhatnew = 0
deghat = Ahatnew.shape[0] // m
for i in range(deghat // 2, deghat):
nAhatnew = max(nAhatnew,
la.norm(Ahatnew[i * m:(i + 1) * m, :], np.inf))
# c) the test
while (nAnew > nj * precision
or nAhatnew > nj * precision) and nj < maxNumRoot:
nj *= 2
stopv = min(nj, Aodd.shape[0] / m)
# prepare for FFTs
temp1 = np.reshape(Aodd[:stopv * m, :].T, (m * m, stopv),
order='F').T
temp2 = np.reshape(Aeven[:stopv * m, :].T, (m * m, stopv),
order='F').T
temp3 = np.reshape(Ahatodd[:stopv * m, :].T, (m * m, stopv),
order='F').T
temp4 = np.reshape(Ahateven[:stopv * m, :].T, (m * m, stopv),
order='F').T
# FFTs
temp1 = fft(temp1, nj, 0)
temp2 = fft(temp2, nj, 0)
temp3 = fft(temp3, nj, 0)
temp4 = fft(temp4, nj, 0)
# reform the 4*(nj+1) matrices
temp1 = np.reshape(temp1.T, (m, m * nj), order='F').T
temp2 = np.reshape(temp2.T, (m, m * nj), order='F').T
temp3 = np.reshape(temp3.T, (m, m * nj), order='F').T
temp4 = np.reshape(temp4.T, (m, m * nj), order='F').T
# Next, we perform a point-wise evaluation of (6.20) - Thesis Meini
Ahatnew = ml.empty((nj * m, m), dtype=complex)
Anew = ml.empty((nj * m, m), dtype=complex)
for cnt in range(1, nj + 1):
Ahatnew[(cnt - 1) * m:cnt *
m, :] = temp4[(cnt - 1) * m:cnt * m, :] + temp2[
(cnt - 1) * m:cnt * m, :] * la.inv(I - temp1[
(cnt - 1) * m:cnt * m, :]) * temp3[
(cnt - 1) * m:cnt * m, :]
Anew[(cnt - 1) * m:cnt *
m, :] = cmath.exp(-(cnt - 1) * 2j * math.pi / nj) * temp1[
(cnt - 1) * m:cnt * m, :] + temp2[
(cnt - 1) * m:cnt * m, :] * la.inv(I - temp1[
(cnt - 1) * m:cnt * m, :]) * temp2[
(cnt - 1) * m:cnt * m, :]
# We now invert the FFTs to get Pz and Phatz
# prepare for IFFTs
Ahatnew = np.reshape(Ahatnew[:nj * m, :].T, (m * m, nj),
order='F').T
Anew = np.reshape(Anew[:nj * m, :].T, (m * m, nj), order='F').T
# IFFTs
Ahatnew = ml.matrix(np.real(ifft(Ahatnew, nj, 0)))
Anew = ml.matrix(np.real(ifft(Anew, nj, 0)))
# reform matrices Pi and Phati
Ahatnew = np.reshape(Ahatnew.T, (m, m * nj), order='F').T
Anew = np.reshape(Anew.T, (m, m * nj), order='F').T
vec1 = ml.zeros((1, m))
vec2 = ml.zeros((1, m))
for i in range(1, Anew.shape[0] // m):
vec1 += i * np.sum(Anew[i * m:(i + 1) * m, :], 0)
vec2 += i * np.sum(Ahatnew[i * m:(i + 1) * m, :], 0)
nAnew = 0
deg = Anew.shape[0] // m
for i in range(deg // 2, deg):
nAnew = max(nAnew, la.norm(Anew[i * m:(i + 1) * m, :], np.inf))
nAhatnew = 0
deghat = Ahatnew.shape[0] // m
for i in range(deghat // 2, deghat):
nAhatnew = max(nAhatnew,
la.norm(Ahatnew[i * m:(i + 1) * m, :], np.inf))
if (nAnew > nj * precision
or nAhatnew > nj * precision) and nj >= maxNumRoot:
print("MaxNumRoot reached, accuracy might be affected!")
if nj > 2:
Anew = Anew[:m * nj / 2, :]
Ahatnew = Ahatnew[:m * nj / 2, :]
# compute Aodd, Aeven, ...
Aeven = Anew[
np.remainder(np.kron(np.arange(Anew.shape[0] /
m), np.ones(m)), 2) == 0, :]
Aodd = Anew[
np.remainder(np.kron(np.arange(Anew.shape[0] /
m), np.ones(m)), 2) == 1, :]
Ahateven = Ahatnew[
np.remainder(np.kron(np.arange(Ahatnew.shape[0] /
m), np.ones(m)), 2) == 0, :]
Ahatodd = Ahatnew[
np.remainder(np.kron(np.arange(Ahatnew.shape[0] /
m), np.ones(m)), 2) == 1, :]
if butools.verbose == True:
if method == "PWCR":
print("The Point-wise evaluation of Iteration ", numit,
" required ", nj, " roots")
else:
print("The Shifted PWCR evaluation of Iteration ", numit,
" required ", nj, " roots")
# test stopcriteria
if method == "PWCR":
Rnewj = Anew[m:2 * m, :]
for i in range(3, Anew.shape[0] / m + 1):
Rnewj = Rnewj + Anew[(i - 1) * m:i * m, :]
Rnewj = la.inv(I - Rnewj)
Rnewj = Anew[:m, :] * Rnewj
if np.max(np.abs(Rj - Rnewj)) < precision or np.max(
np.sum(I - Anew[:m, :] * la.inv(I - Anew[m:2 * m, :]),
0)) < precision:
G = Ahatnew[:m, :]
for i in range(2, Ahatnew.shape[0] / m + 1):
G = G + Rnewj * Ahatnew[(i - 1) * m:i * m, :]
G = D[:m, :] * la.inv(I - G)
break
Rj = Rnewj
# second condition tests whether Ahatnew is degree 0 (numerically)
if la.norm(Anew[:m, :m]) < precision or np.sum(
Ahatnew[m:, :]) < precision or np.max(
np.sum(I - D[:m, :] * la.inv(I - Ahatnew[:m, :]),
0)) < precision:
G = D[0:m, :] * la.inv(I - Ahatnew[:m, :])
break
else:
Gold = G
G = D[:m, :] * la.inv(I - Ahatnew[:m, :])
if la.norm(G - Gold, np.inf) < precision or la.norm(
Ahatnew[m:, :], np.inf) < precision:
break
if numit == maxNumIt and G == ml.zeros((m, m)):
print("Maximum Number of Iterations reached!")
G = D[:m, :] * la.inv(I - Ahatnew[:m, :])
G = G.T
if method == "ShiftPWCR":
if shiftType == "one":
G = G + (drift < 1) * ml.ones((m, m)) / m
elif shiftType == "tau":
G = G + (drift > 1) * tau * v * ml.ones((1, m))
elif shiftType == "dbl":
G = G + (drift < 1) * ml.ones(
(m, m)) / m + (drift > 1) * tau * v * ml.ones((1, m))
if butools.verbose:
if method == "PWCR":
D = D.T
else:
D = Dold
temp = D[:, -m:]
for i in range(D.shape[1] // m - 1, 0, -1):
temp = D[:, (i - 1) * m:i * m] + temp * G
res_norm = la.norm(G - temp, np.inf)
print("Final Residual Error for G: ", res_norm)
return G
def Kent_calculation (Lpos): #receiving Lpos= feature_vector[indx,axialfeatureidx], Li=[lx,ly,lz]) of type numpy.matrix
print "Kent_calculation:"
for i in range(Lpos.shape[0]):
Lpos[i] = Lpos[i]/numpy.linalg.norm(Lpos[i])
Lneg=-Lpos
L=bmat('Lpos; Lneg') #?matrix(numpy.negative(array(Lpos))) #[a b; c d] {MATLAB}= vstack([hstack([a,b]),hstack([c,d])]) {numpy.array}= bmat('a b; c d') {numpy.matrix}
#f(x) = 1/c(k,beta) exp(k*gamma1*x + beta*((gamma2*x)^2 - (gamma3*x)^2))
#g(x) = exp(k*gamma1*x + beta*((gamma2*x)^2 - (gamma3*x)^2))
#LL = log likelihood = k*gamma1*x + beta*((gamma2*x)^2 - (gamma3*x)^2)
#dLL/dk = gamma1*x
#dLL/dbeta = (gamma2*x)^2 - (gamma3*x)^2
T= (L.transpose())*L
D,V = numpy.linalg.eig(T) # a: array_like shape (M, M), D:eigenvalues ndarray shape (M,) NOT ordered, V: eignevectors ndarray shape (M, M) The normalized (unit “length”) eigenvectors, column V[:,i] is the eigenvector corresponding to the eigenvalue D[i]
#i = numpy.argsort(D) #Returns the indices that would sort an array, ascending
i = numpy.argsort(D[::-1]) #Returns the indices that would sort an array, descending
Tau = D[i] #sorted eigenvals
t = V[:,i] #sorted a=eigenvects
Tau_bar = Tau/float(L.shape[0])
gamma1 = t[:,0]
gamma2 = t[:,1]
gamma3 = t[:,2]
noOutlierL = Matlib.empty((0,L.shape[1]),float)
for i in range(L.shape[0]):
if ( arccos((dot(L[i,:],gamma1))/(linalg.norm(L[i,:])*linalg.norm(gamma1))) < (pi/4.0)):
noOutlierL = numpy.vstack([noOutlierL, array(L[i,:])])
#### run EM to find k and beta
#### 1.initialize
k0 = 10
beta0 = 0
v0 = [k0,beta0]
v = scipy.optimize.fmin(fp,v0, args=(noOutlierL,gamma1,gamma2,gamma3),maxiter=10000,maxfun=10000)
k=v[0]
beta=v[1]
if (k==0):
k=1
elif (k<0):
k=-k
if (beta<0):
beta=-beta
#we should have 2*abs(b) < abs(k)
if (2*beta >= k):
beta = (k-1)/2
if (beta <0):
beta = 0
ck = sqrt((k-2*beta)*(k+2*beta))/(2*pi*exp(k))
if math.isnan(ck) or math.isnan(k) or math.isnan(beta) or math.isinf(ck) or math.isinf(k) or math.isinf(beta) or (k <0) or (beta< 0):
print "Error!\n"
print "ck :", ck , "\nbeta : ", beta, "\nk :", k, "\ngamma :", gamma1, gamma2, gamma3
#print "\nAborted!"
#exit (0)
#put unit kent distribution there!
k = 0.1
beta = 0
#print type(gamma1), " ", gamma1
return [k,beta,ck,gamma1,gamma2,gamma3]
from numpy import nan
from numpy.linalg import norm as norm
import matplotlib.pyplot as plt
import plot_utils as plut
import time
import romeo_conf as conf
from tsid_biped import TsidBiped
print "".center(conf.LINE_WIDTH, '#')
print " Test Task Space Inverse Dynamics ".center(conf.LINE_WIDTH, '#')
print "".center(conf.LINE_WIDTH, '#'), '\n'
tsid = TsidBiped(conf)
N = conf.N_SIMULATION
com_pos = matlib.empty((3, N)) * nan
com_vel = matlib.empty((3, N)) * nan
com_acc = matlib.empty((3, N)) * nan
com_pos_ref = matlib.empty((3, N)) * nan
com_vel_ref = matlib.empty((3, N)) * nan
com_acc_ref = matlib.empty((3, N)) * nan
com_acc_des = matlib.empty(
(3, N)) * nan # acc_des = acc_ref - Kp*pos_err - Kd*vel_err
offset = tsid.robot.com(tsid.formulation.data())
amp = np.matrix([0.0, 0.05, 0.0]).T
two_pi_f = 2 * np.pi * np.matrix([0.0, 0.5, 0.0]).T
two_pi_f_amp = np.multiply(two_pi_f, amp)
two_pi_f_squared_amp = np.multiply(two_pi_f, two_pi_f_amp)
if variable == 0:
filepath = './Run' + str(N) + '.dat'
with open(filepath, 'r') as myfile:
result = pickle.load(myfile)
else:
filepath = './Run' + str(N) + '.dat'
with open(filepath, 'r') as myfile:
temp = pickle.load(myfile)
result = temp[variable]
return result
if __name__ == '__main__':
numRuns = 10
e_qq_mat = npm.empty((10000, 10))
f_qq_mat = npm.empty((10001, 10))
for i in range(numRuns):
result = viewResult(i)
figure(1)
plot(result['e_qq'])
e_qq_mat[:, i] = result['e_qq']
figure(2)
plot(result['f_qq'])
f_qq_mat[:, i] = result['f_qq']
figure(1)
title('Deviation of true tracked subspace')
figure(2)
转置矩阵
NumPy 中除了可以使用 numpy.transpose 函数来对换数组的维度，还可以使用 T 属性。。
例如有个 m 行 n 列的矩阵，使用 t() 函数就能转换为 n 行 m 列的矩阵。
'''
a = np.arange(12).reshape(3, 4)
print('原数组：')
print(a)
print('转置数组：')
print(a.T)
print('\n')
'''
matlib.empty()
matlib.empty() 函数返回一个新的矩阵，语法格式为：
numpy.matlib.empty(shape, dtype, order)
参数说明：
shape: 定义新矩阵形状的整数或整数元组
Dtype: 可选，数据类型
order: C（行序优先） 或者 F（列序优先）
numpy.matlib.zeros()
numpy.matlib.zeros() 函数创建一个以 0 填充的矩阵。
numpy.matlib.ones()
numpy.matlib.ones()函数创建一个以 1 填充的矩阵。
def runBatch(numRuns,BatchName,param):
"""
Function to run Batch of 'numRuns' initial conditions for FRHH using the
'param' parameters
"""
# Directory in which to create new data directory
mainPath = 'C:/DataSets/Results/Fraust/'
os.chdir(mainPath)
# Results matrix
e_qq_mat = npm.empty((10000,numRuns))
f_qq_mat = npm.empty((10001,numRuns))
g_qq_mat = npm.empty((10000,numRuns))
for i in range(numRuns):
# Generate artificial data streams
streams = genSimSignalsA(i, -3.0)
# Run Basic Fast row householder subspace tracker
Q_t, S_t = frhh_min(streams, param['alpha'], param['rr'])
#Q_t, S_t, rr, E_t, E_dash_t, hid_var, z_dash, RSRE, no_inp_count, \
#no_inp_marker = FRHH32(streams, param['rr'], param['alpha'], param['sci'])
# Calculate deviations from orthogonality and subspace
e_qq, f_qq, g_qq = plotEqqFqqA(streams, Q_t, param['alpha'])
# Store results in Dictionary
dic_name = 'res_' + str(i) # string of the name of the Dictionary
vars()[dic_name] = {'param' : param,
'Q_t' : Q_t,
'S_t': S_t,
'e_qq' : e_qq,
'f_qq' : f_qq,
'g_qq' : g_qq}
# 'rr' : rr,
# 'E_t' : E_t,
# 'E_dash_t' : E_dash_t,
# 'hid_var' : hid_var,
# 'z_dash' : z_dash,
# 'RSRE' : RSRE,
# 'no_inp_count' : no_inp_count,
# 'no_inp_marker' : no_inp_marker,
myDic = vars()[dic_name]
e_qq_mat[:,i] = e_qq
f_qq_mat[:,i] = f_qq
g_qq_mat[:,i] = g_qq
# Save data files
mypath = mainPath + '\\' + BatchName
if not os.path.exists(mypath):
os.makedirs(mypath)
runfile = mypath + '/Run' + str(i) + '.dat'
with open(runfile, 'w') as outfile:
pickle.dump(myDic, outfile)
streamfile = mypath + '/stream_inputs' + str(i) + '.dat'
streams.tofile(streamfile)
print 'finished ', i, 'th batch'
del myDic, vars()[dic_name] # saves memory
# Save summary Matrixs
e_qq_path = mypath + '/e_qq_mat.dat'
e_qq_mat.tofile(e_qq_path)
f_qq_path = mypath + '/f_qq_mat.dat'
f_qq_mat.tofile(f_qq_path)
g_qq_path = mypath + '/g_qq_mat.dat'
g_qq_mat.tofile(g_qq_path)
return e_qq_mat, f_qq_mat, g_qq_mat