def _get_data():
"""
Runs the analysis and saves the data to be used later for plotting.
"""
# get the Roessler trajectory
print("Roessler trajectory ...")
t = np.linspace(0., 1000., 100000.)
params = (0.432, 2., 4.)
x0 = (1., 3., 5.)
pos = tm.roessler(x0, t, params)
i_eq = 90000
x, y, z = pos[i_eq:, 0], pos[i_eq:, 1], pos[i_eq:, 2]
t = t[i_eq:]
# set the X component as our measured signal
s = x.copy()
# get mi
print("MI ...")
maxlag = np.where(t <= (t[0] + 10.))[0][-1].astype("int")
mi, mi_lags = rc.mi(s, maxlag)
mi_filt, _ = utils.boxfilter(mi, filter_width=25, estimate="mean")
tau_mi = rc.first_minimum(mi_filt)
print("FNN ...")
M = 10
R = 0.50
fnn_mi, dims_mi = rc.fnn(s, tau_mi, maxdim=M, r=R)
m_mi = dims_mi[rc.first_zero(fnn_mi)]
# save data
print("save output ...")
FN = "../data/delay_embedding/results"
np.savez(FN,
x=x,
y=y,
z=z,
t=t,
params=params,
x0=x0,
i_eq=i_eq,
s=s,
maxlag=maxlag,
mi=mi,
mi_lags=mi_lags,
mi_filt=mi_filt,
tau_mi=tau_mi,
M=M,
R=R,
fnn_mi=fnn_mi,
dims_mi=dims_mi,
m_mi=m_mi)
print("saved to: %s.npz" % FN)
return None
nR = len(aR)
mR = 5000
yR_eq = []
kR = 5
x0R = (-1. + 2. * np.random.rand(nR, kR),
-1. + 2. * np.random.rand(nR, kR), 1. * np.random.rand(nR, kR))
count = 0
TOL = 1E-3
pbar_on = True
pb = _progressbar_start(max_value=nR * kR, pbar_on=pbar_on)
for i in range(nR):
params = (aR[i], bR, cR)
yR_eq_ = []
for l in range(kR):
x0R_ = (x0R[0][i, l], x0R[1][i, l], x0R[2][i, l])
XR = tm.roessler(x0R_, TR, params)
xR_ = XR[-mR:, 0]
yR_ = XR[-mR:, 1]
if np.all((yR_[1:] - yR_[:-1]) < TOL):
yR_eq_.extend(yR_)
else:
d2y = np.diff(np.sign(np.diff(yR_)))
iipos = np.where(d2y == -2.)[0] + 1
if len(iipos) > 0:
yR_eq_.extend(yR_[iipos])
iineg = np.where(d2y == 2.)[0] + 1
if len(iineg) > 0:
yR_eq_.extend(yR_[iineg])
_progressbar_update(pb, count)
count += 1
np.random.shuffle(yR_eq_)
for k in range(m):
xe[k] = x[ne:] + E * std * re[k]
xhi = np.percentile(xe, 97.5, axis=0)
xlo = np.percentile(xe, 2.5, axis=0)
# 2) plot
ax1.plot(t, x, "-", c=main_clr, zorder=10, alpha=0.75)
ax1.fill_between(te, xhi, xlo, color=err_clr)
ax1.axvline(te[0], color="IndianRed", linestyle="--")
# nonlinear example
# -----------------
print("nonlinear example ...")
# 1) get primary time series
params = (0.3, 0.4, 7.5)
x0 = (0.0, -1., 1.)
pos = tm.roessler(x0, t, params)
x, y, z = pos[:, 0], pos[:, 1], pos[:, 2]
# 2) get noise limit based on variance of main trajectory
std = np.std(x)
# 3) get upper and lower bounds on trajectory using erroneous IC
ne = int(n / 2)
te = t[ne:]
xe = np.zeros((m, len(te)))
re = np.random.randn(m)
for k in range(m):
xe0 = (x[ne] + E * std * re[k], y[ne], z[ne])
pose = tm.roessler(xe0, t[ne:], params)
xe[k] = pose[:, 0]
xhi = np.percentile(xe, 97.5, axis=0)
xlo = np.percentile(xe, 2.5, axis=0)
# 2) plot
clr3 = "IndianRed"
# time vector
T = np.linspace(0., 600., 600001)
n = len(T)
i = int(n / 2)
t1 = T[:i]
t2 = T[i:]
neq = 50000
# spiral-type chaos
# -----------------
print("spiral-type chaos ...")
params = (0.32, 0.3, 4.5)
x0 = (0.0, -1., 1.)
pos = tm.roessler(x0, t1, params)
x1, y1, z1 = pos[neq:, 0], pos[neq:, 1], pos[neq:, 2]
# screw-type chaos
# ----------------
print("screw-type chaos ...")
params = (0.38, 0.4, 4.820)
x0 = (x1[-1], y1[-1], z1[-1])
pos = tm.roessler(x0, t2, params)
x2, y2, z2 = pos[neq:, 0], pos[neq:, 1], pos[neq:, 2]
# get full time series and the mean and variance filter
print("mean and variance ...")
X = np.r_[x1, x2]
T = T[:-2 * neq]
n = len(T)
ax2.imshow(R,
cmap=pl.cm.gray_r,
interpolation="none",
origin="lower",
rasterized=True)
del R
ax2.set_title(r"$\tau_e = %d, m_e = %d, \varepsilon = 0.15$" %
(tau_mi, m_mi),
fontsize=axlabfs)
# C. Chaotic Roessler
print("C. Chaotic Roessler ...")
t = np.linspace(0., 1000., 100000.)
params = (0.432, 2., 4.)
x0 = (1., 3., 5.)
pos = tm.roessler(x0, t, params)
i_eq = 50000
sample_every = 10
t = t[i_eq::sample_every]
# set the X component as our measured signal
x = pos[i_eq::sample_every, 0]
# get mi
maxlag = np.where(t <= (t[0] + 10.))[0][-1].astype("int")
mi, mi_lags = rc.mi(x, maxlag)
mi_filt, _ = utils.boxfilter(mi, filter_width=25, estimate="mean")
tau_mi = rc.first_minimum(mi_filt)
# FNN
M = 10
R = 0.025
fnn_mi, dims_mi = rc.fnn(x, tau_mi, maxdim=M, r=R)
m_mi = dims_mi[rc.first_zero(fnn_mi)]
clrs = ["MediumTurquoise", "GoldenRod", "IndianRed"]
# Lorenz butterfly
# ----------------
print("Roessler ...")
Tx = np.linspace(0., 1000., 100000.)
params = (0.432, 2., 4.)
x0 = {
1: (1., 3., 5.),
2: (7.5, -4.5, 1.),
3: (-2.5, -4.5, 0.),
}
X = {}
for i in range(1, 4):
print("\tinitial condition #%d" % i)
X[i] = tm.roessler(x0[i], Tx, params)
# Henon map
# ---------
print("Henon map ...")
Ty = np.arange(0, 10000, 1)
params = (1.4, 0.30)
y0 = {
1: (-1., 0.),
2: (1., -0.1),
3: (0.5, 0.1),
}
Y = {}
for i in range(1, 4):
print("\tinitial condition #%d" % i)
Y[i] = tm.henon(y0[i], Ty[-1], params)