def _embed_solo(t, x, ws, ss):
"""Embed the time series for each window"""
n = len(t)
nw = int(np.floor(float(n - ws) / float(ss)))
tm = np.empty(nw, dtype="object")
m, tau = [np.zeros(nw, dtype="int") for i in range(2)]
xw = np.zeros((nw, ws), dtype="float")
maxlag = 150
maxdim = 10
R = 0.025
pb = _progressbar_start(max_value=nw, pbar_on=args.verbose)
for i in range(nw):
start = i * ss
end = start + ws
x_ = x[start:end]
xw[i] = x_
# get mi
mi, mi_lags = rc.mi(x_, maxlag, pbar_on=False)
mi_filt, _ = utils.boxfilter(mi, filter_width=3, estimate="mean")
try:
tau[i] = rc.first_minimum(mi_filt)
except ValueError:
tau[i] = 1
# FNN
fnn, dims = rc.fnn(x_, tau[i], maxdim=maxdim, r=R, pbar_on=False)
m[i] = dims[rc.first_zero(fnn)]
tm[i] = t[start] + (t[end] - t[start]) / 2
_progressbar_update(pb, i)
_progressbar_finish(pb)
return tm, xw, m, tau
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
def _embed_pair(t, x, y, ws, ss):
"""Determines common embedding parameters for both time series"""
n = len(t)
nw = int(np.floor(float(n - ws) / float(ss)))
tm = np.empty(nw, dtype="object")
m, tau = [np.zeros(nw, dtype="int") for i in range(2)]
xw, yw = [np.zeros((nw, ws), dtype="float") for i in range(2)]
maxlag = 150
maxdim = 10
R = 0.025
pb = _progressbar_start(max_value=nw, pbar_on=args.verbose)
for i in range(nw):
start = i * ss
end = start + ws
x_ = x[start:end]
y_ = y[start:end]
xw[i] = x_
yw[i] = y_
# get mi
mi1, mi_lags1 = rc.mi(x_, maxlag, pbar_on=False)
mi_filt1, _ = utils.boxfilter(mi1, filter_width=3, estimate="mean")
tau1 = rc.first_minimum(mi_filt1)
mi2, mi_lags2 = rc.mi(y_, maxlag, pbar_on=False)
mi_filt2, _ = utils.boxfilter(mi2, filter_width=3, estimate="mean")
tau2 = rc.first_minimum(mi_filt2)
tau[i] = int(max(tau1, tau2))
# FNN
fnn1, dims1 = rc.fnn(x_, tau[i], maxdim=maxdim, r=R, pbar_on=False)
m1 = dims1[rc.first_zero(fnn1)]
fnn2, dims2 = rc.fnn(y_, tau[i], maxdim=maxdim, r=R, pbar_on=False)
m2 = dims2[rc.first_zero(fnn2)]
m[i] = int(max(m1, m2))
tm[i] = t[start] + (t[end] - t[start]) / 2
_progressbar_update(pb, i)
_progressbar_finish(pb)
return tm, xw, yw, m, tau
print("B. Superposed harmonics RP ...")
N = 500
t = np.arange(N)
T1, T2, T3 = 10., 50., 75.
A1, A2, A3 = 1., 1.5, 2.
# T1, T2 = 10., 50.
# A1, A2 = 1., 1.5
twopi = 2. * np.pi
H1 = A1 * np.sin((twopi * t) / T1)
H2 = A2 * np.sin((twopi * t) / T2)
H3 = A3 * np.sin((twopi * t) / T3)
x = H1 + H2 + H3
# get mi
maxlag = 150
mi, mi_lags = rc.mi(x, maxlag)
mi_filt, _ = utils.boxfilter(mi, filter_width=3, estimate="mean")
tau_mi = rc.first_minimum(mi_filt)
# FNN
M = 20
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)]
R = rc.rp(x,
m=m_mi,
tau=tau_mi,
e=0.15,
norm="euclidean",
threshold_by="frr",
normed=True)
ax2.imshow(R,
cmap=pl.cm.gray_r,
def _get_communities():
"""
Identifies the optimal community structure based on modularity.
"""
# load data
utils._printmsg("load data ...", args.verbose)
t, x_enso, x_pdo = _load_indices()
x = {
"enso": x_enso,
"pdo": x_pdo,
}
names = ["enso", "pdo"]
# recurrence plot parameters
EPS = 0.30
thrby = "frr"
# embedding parameters
utils._printmsg("embedding parameters ...", args.verbose)
n = len(t)
m, tau = {}, {}
R = {}
maxlag = 150
maxdim = 20
r_fnn = 0.0010
for name in names:
if args.verbose: print("\t for %s" % name.upper())
# get embedding parameters
## get mi
mi, mi_lags = rc.mi(x[name], maxlag, pbar_on=False)
# mi, mi_lags = rc.acf(x[name], maxlag)
mi_filt, _ = utils.boxfilter(mi, filter_width=3, estimate="mean")
try:
tau[name] = rc.first_minimum(mi_filt)
except ValueError:
tau[name] = 1
## FNN
fnn, dims = rc.fnn(x[name],
tau[name],
maxdim=maxdim,
r=r_fnn,
pbar_on=False)
m[name] = dims[rc.first_zero(fnn)]
# # print out embedding dimensions for documentation in the paper
# print m
# print tau
# sys.exit()
# identify communities using modularity
utils._printmsg("communities based on modularity ...", args.verbose)
COMM = {}
for name in names:
utils._printmsg("\tfor %s" % name.upper(), args.verbose)
A = rc.rn(x[name],
m=m[name],
tau=tau[name],
e=EPS,
norm="euclidean",
threshold_by="frr",
normed=True)
# optimize modularity
utils._printmsg("\toptimize modularity ...", args.verbose)
G = ig.Graph.Adjacency(A.tolist(), mode=ig.ADJ_UNDIRECTED)
dendro = G.community_fastgreedy()
# dendro = G.community_edge_betweenness(directed=False)
clust = dendro.as_clustering()
# clust = G.community_multilevel()
mem = clust.membership
COMM[name] = mem
# get each individual array out of dict to avoid NumPy import error
x_enso = x["enso"]
x_pdo = x["pdo"]
COMM_enso = COMM["enso"]
COMM_pdo = COMM["pdo"]
t = np.array([date.toordinal() for date in t])
# save output
EPS = int(EPS * 100)
FN = DATPATH + "communities_EPS%d" \
% EPS
np.savez(FN,
x_enso=x_enso,
x_pdo=x_pdo,
t=t,
COMM_enso=COMM_enso,
COMM_pdo=COMM_pdo,
m=m,
tau=tau,
e=EPS,
thrby=thrby)
if args.verbose: print("output saved to: %s.npz" % FN)
return None
def _get_rmd():
"""Estimates the RMD between ENSO and PDO"""
# load data
utils._printmsg("load data ...", args.verbose)
t, x_enso, x_pdo = _load_indices()
x = {
"enso": x_enso,
"pdo": x_pdo,
}
names = ["enso", "pdo"]
# recurrence plot parameters
EPS = 0.30
thrby = "frr"
# embedding parameters
utils._printmsg("embedding parameters ...", args.verbose)
n = len(t)
m, tau = {}, {}
R = {}
maxlag = 150
maxdim = 20
r_fnn = 0.0010
for name in names:
if args.verbose: print("\t for %s" % name.upper())
# get embedding parameters
## get mi
mi, mi_lags = rc.mi(x[name], maxlag, pbar_on=False)
# mi, mi_lags = rc.acf(x[name], maxlag)
mi_filt, _ = utils.boxfilter(mi, filter_width=3, estimate="mean")
try:
tau[name] = rc.first_minimum(mi_filt)
except ValueError:
tau[name] = 1
## FNN
fnn, dims = rc.fnn(x[name],
tau[name],
maxdim=maxdim,
r=r_fnn,
pbar_on=False)
m[name] = dims[rc.first_zero(fnn)]
# take the maximum delay and the maximum embedding dimension
tau = np.max([tau["enso"], tau["pdo"]]).astype("int")
m = np.max([m["enso"], m["pdo"]]).astype("int")
# get surrogates
utils._printmsg("surrogates ...", args.verbose)
ns = args.nsurr
SURR = {}
params = {
"m": m,
"tau": tau,
"eps": EPS,
"norm": "euclidean",
"thr_by": thrby,
"tol": 2.
}
for name in names:
utils._printmsg("\t for %s" % name.upper(), args.verbose)
# SURR[name] = rc.surrogates(x[name], ns, "iaaft", verbose=args.verbose)
SURR[name] = rc.surrogates(x[name],
ns,
"twins",
params,
verbose=args.verbose)
# get RMD for original data
utils._printmsg("RMD for original data ...", args.verbose)
ws, ss = args.window_size, args.step_size
nw = int(np.floor(float(n - ws) / float(ss)))
tm = np.empty(nw, dtype="object")
for name in names:
R[name] = rc.rp(
x[name],
m=m,
tau=tau,
e=EPS,
norm="euclidean",
threshold_by=thrby,
)
rmd = np.zeros(nw)
pb = _progressbar_start(max_value=nw, pbar_on=args.verbose)
for i in range(nw):
start = i * ss
end = start + ws
Rw_enso = R["enso"][start:end, start:end]
Rw_pdo = R["pdo"][start:end, start:end]
rmd[i] = rqa.rmd(Rw_enso, Rw_pdo)
tm[i] = t[start] + (t[end] - t[start]) / 2
_progressbar_update(pb, i)
_progressbar_finish(pb)
# get RMD for surrogate data
utils._printmsg("RMD for surrogates ...", args.verbose)
Rs = {}
rmdsurr = np.zeros((ns, nw), dtype="float")
pb = _progressbar_start(max_value=ns, pbar_on=args.verbose)
for k in range(ns):
for name in names:
xs = SURR[name][k]
Rs[name] = rc.rp(
xs,
m=m,
tau=tau,
e=EPS,
norm="euclidean",
threshold_by=thrby,
)
for i in range(nw):
start = i * ss
end = start + ws
Rsw_enso = Rs["enso"][start:end, start:end]
Rsw_pdo = Rs["pdo"][start:end, start:end]
rmdsurr[k, i] = rqa.rmd(Rsw_enso, Rsw_pdo)
_progressbar_update(pb, k)
_progressbar_finish(pb)
# get each individual array out of dict to avoid NumPy import error
SURR_enso = SURR["enso"]
SURR_pdo = SURR["pdo"]
tm = np.array([date.toordinal() for date in tm])
# save output
EPS = int(EPS * 100)
FN = DATPATH + "rmd_WS%d_SS%d_EPS%dpc_NSURR%d" \
% (ws, ss, EPS, ns)
np.savez(
FN,
rmd=rmd,
tm=tm,
rmdsurr=rmdsurr,
SURR_enso=SURR_enso,
SURR_pdo=SURR_pdo,
)
if args.verbose: print("output saved to: %s.npz" % FN)
return None
def _get_spl():
"""
Estimates the average shortest path length SPL for the indices.
"""
# load data
utils._printmsg("load data ...", args.verbose)
t, x_enso, x_pdo = _load_indices()
x = {
"enso": x_enso,
"pdo": x_pdo,
}
names = ["enso", "pdo"]
# get surrogates
utils._printmsg("iAAFT surrogates ...", args.verbose)
ns = args.nsurr
SURR = {}
for name in names:
utils._printmsg("\t for %s" % name.upper(), args.verbose)
SURR[name] = rc.surrogates(x[name], ns, "iaaft", verbose=args.verbose)
# recurrence plot parameters
EPS, LMIN = 0.30, 3
thrby = "frr"
# get SPL for original data
utils._printmsg("SPL for original data ...", args.verbose)
n = len(t)
ws, ss = args.window_size, args.step_size
nw = int(np.floor(float(n - ws) / float(ss)))
tm = np.empty(nw, dtype="object")
m, tau = {}, {}
A = {}
maxlag = 150
maxdim = 20
r_fnn = 0.0010
SPL = {}
for name in names:
if args.verbose: print("\t for %s" % name.upper())
# get embedding parameters
## get mi
mi, mi_lags = rc.mi(x[name], maxlag, pbar_on=False)
# mi, mi_lags = rc.acf(x[name], maxlag)
mi_filt, _ = utils.boxfilter(mi, filter_width=3, estimate="mean")
try:
tau[name] = rc.first_minimum(mi_filt)
except ValueError:
tau[name] = 1
## FNN
fnn, dims = rc.fnn(x[name],
tau[name],
maxdim=maxdim,
r=r_fnn,
pbar_on=False)
m[name] = dims[rc.first_zero(fnn)]
A[name] = rc.rn(
x[name],
m=m[name],
tau=tau[name],
e=EPS,
norm="euclidean",
threshold_by=thrby,
)
A_ = A[name]
G_ = ig.Graph.Adjacency(A_.tolist(), mode=ig.ADJ_UNDIRECTED)
nw = len(tm)
spl = np.zeros(nw)
pb = _progressbar_start(max_value=nw, pbar_on=args.verbose)
for i in range(nw):
start = i * ss
end = start + ws
Gw = G_.subgraph(vertices=G_.vs[start:end])
pl_hist = Gw.path_length_hist(directed=False)
spl[i] = pl_hist.mean
tm[i] = t[start] + (t[end] - t[start]) / 2
_progressbar_update(pb, i)
_progressbar_finish(pb)
SPL[name] = spl
# get SPL for surrogate data
utils._printmsg("SPL for surrogates ...", args.verbose)
SPLSURR = {}
for name in names:
utils._printmsg("\tfor %s" % name.upper(), args.verbose)
xs = SURR[name]
y = np.diff(xs, axis=0)
splsurr = np.zeros((ns, nw), dtype="float")
pb = _progressbar_start(max_value=ns, pbar_on=args.verbose)
for k in range(ns):
As = rc.rp(
xs[k],
m=m[name],
tau=tau[name],
e=EPS,
norm="euclidean",
threshold_by=thrby,
)
Gs = ig.Graph.Adjacency(As.tolist(), mode=ig.ADJ_UNDIRECTED)
for i in range(nw):
start = i * ss
end = start + ws
Gw = Gs.subgraph(vertices=Gs.vs[start:end])
pl_hist = Gw.path_length_hist(directed=False)
splsurr[k, i] = pl_hist.mean
_progressbar_update(pb, k)
_progressbar_finish(pb)
SPLSURR[name] = splsurr
# get each individual array out of dict to avoid NumPy import error
SPL_enso = SPL["enso"]
SPL_pdo = SPL["pdo"]
SPLSURR_enso = SPLSURR["enso"]
SPLSURR_pdo = SPLSURR["pdo"]
SURR_enso = SURR["enso"]
SURR_pdo = SURR["pdo"]
tm = np.array([date.toordinal() for date in tm])
# save output
EPS = int(EPS * 100)
FN = DATPATH + "spl_WS%d_SS%d_EPS%dpc_LMIN%d_NSURR%d" \
% (ws, ss, EPS, LMIN, ns)
np.savez(FN,
SPL_enso=SPL_enso,
SPL_pdo=SPL_pdo,
SPLSURR_enso=SPLSURR_enso,
SPLSURR_pdo=SPLSURR_pdo,
SURR_enso=SURR_enso,
SURR_pdo=SURR_pdo,
tm=tm)
if args.verbose: print("output saved to: %s.npz" % FN)
return None
def _get_det():
"""
Estimates the determinism DET for the indices.
"""
# load data
utils._printmsg("load data ...", args.verbose)
t, x_enso, x_pdo = _load_indices()
x = {
"enso": x_enso,
"pdo": x_pdo,
}
names = ["enso", "pdo"]
# get surrogates
utils._printmsg("iAAFT surrogates ...", args.verbose)
ns = args.nsurr
SURR = {}
for name in names:
utils._printmsg("\t for %s" % name.upper(), args.verbose)
SURR[name] = rc.surrogates(x[name], ns, "iaaft", verbose=args.verbose)
# recurrence plot parameters
EPS, LMIN = 0.30, 3
thrby = "frr"
# get DET for original data
utils._printmsg("DET for original data ...", args.verbose)
n = len(t)
ws, ss = args.window_size, args.step_size
nw = int(np.floor(float(n - ws) / float(ss)))
tm = np.empty(nw, dtype="object")
m, tau = {}, {}
R = {}
maxlag = 150
maxdim = 20
r_fnn = 0.0010
DET = {}
for name in names:
if args.verbose: print("\t for %s" % name.upper())
# get embedding parameters
## get mi
mi, mi_lags = rc.mi(x[name], maxlag, pbar_on=False)
# mi, mi_lags = rc.acf(x[name], maxlag)
mi_filt, _ = utils.boxfilter(mi, filter_width=3, estimate="mean")
try:
tau[name] = rc.first_minimum(mi_filt)
except ValueError:
tau[name] = 1
## FNN
fnn, dims = rc.fnn(x[name],
tau[name],
maxdim=maxdim,
r=r_fnn,
pbar_on=False)
m[name] = dims[rc.first_zero(fnn)]
R[name] = rc.rp(
x[name],
m=m[name],
tau=tau[name],
e=EPS,
norm="euclidean",
threshold_by=thrby,
)
R_ = R[name]
nw = len(tm)
det = np.zeros(nw)
pb = _progressbar_start(max_value=nw, pbar_on=args.verbose)
for i in range(nw):
start = i * ss
end = start + ws
Rw = R_[start:end, start:end]
det[i] = rqa.det(Rw, lmin=LMIN, hist=None, verb=False)
tm[i] = t[start] + (t[end] - t[start]) / 2
_progressbar_update(pb, i)
_progressbar_finish(pb)
DET[name] = det
# get DET for surrogate data
utils._printmsg("DET for surrogates ...", args.verbose)
DETSURR = {}
for name in names:
utils._printmsg("\tfor %s" % name.upper(), args.verbose)
xs = SURR[name]
y = np.diff(xs, axis=0)
detsurr = np.zeros((ns, nw), dtype="float")
pb = _progressbar_start(max_value=ns, pbar_on=args.verbose)
for k in range(ns):
Rs = rc.rp(
xs[k],
m=m[name],
tau=tau[name],
e=EPS,
norm="euclidean",
threshold_by=thrby,
)
for i in range(nw):
start = i * ss
end = start + ws
Rw = Rs[start:end, start:end]
detsurr[k, i] = rqa.det(Rw, lmin=LMIN, hist=None, verb=False)
_progressbar_update(pb, k)
_progressbar_finish(pb)
DETSURR[name] = detsurr
# get each individual array out of dict to avoid NumPy import error
DET_enso = DET["enso"]
DET_pdo = DET["pdo"]
DETSURR_enso = DETSURR["enso"]
DETSURR_pdo = DETSURR["pdo"]
SURR_enso = SURR["enso"]
SURR_pdo = SURR["pdo"]
tm = np.array([date.toordinal() for date in tm])
# save output
EPS = int(EPS * 100)
FN = DATPATH + "det_WS%d_SS%d_EPS%dpc_LMIN%d_NSURR%d" \
% (ws, ss, EPS, LMIN, ns)
np.savez(FN,
DET_enso=DET_enso,
DET_pdo=DET_pdo,
DETSURR_enso=DETSURR_enso,
DETSURR_pdo=DETSURR_pdo,
SURR_enso=SURR_enso,
SURR_pdo=SURR_pdo,
tm=tm)
if args.verbose: print("output saved to: %s.npz" % FN)
return None