def periodogramfft(d, time, data):
from scipy.signal import hann, periodogram
import numpy.fft as fft
print "d=", d
pidx, pv, tidx, tv = peaks.turning_points(data, minidx=0)
tts = time[pidx]
rspl = UnivariateSpline(time, data, k=4, s=0)
t0 = tts[1]
tf = tts[d + 1]
dt = time[1] - time[0]
twin = np.arange(t0, tf, dt)
N = len(twin)
fs = 1 / dt
#twin1 = np.arange(t0, tts[3], dt)
#Nfft = len(twin1)
#fs = N/(tf-t0)
Nfft = 100000
rwin = rspl(twin)
rwin = rwin - np.mean(rwin)
hwin = hann(len(twin))
rwin = rwin * hwin
#f, pgram = periodogram(hwin, fs=fs, nfft=Nfft)
fftout = fft.rfft(rwin, 50 * len(rwin))
fftfreq = fft.rfftfreq(50 * len(rwin), dt)
print len(fftout)
print len(fftfreq)
return 2 * np.pi * fftfreq, np.abs(fftout)
def fit_sinusoids(time, data, width=2):
#plot the data
#plt.plot(time,data,color="red",label="data")
#get the amplitude and mean of the data (approx.constant)
amp, mean = get_amp_mean(data, time)
#take peak-to-peak ranges of data
ptimes, pvals, ttimes, tvals = pks.turning_points(data)
rngs = zip(turns[:-width:width], turns[width::width])
#number of cycles each rng covers
ncycles = width / 2.0
#set this to 2*pi if you want an angular frequency
fscale = 1.0
freqs = []
for a, b in rngs:
#the frequency in radians per second
period = (time[b] - time[a]) / ncycles
freqs.append(fscale / period)
#set the phase to ensure the fitted curve will pass through the data at the
#initial point
phases = []
for rng, freq in zip(rngs, freqs):
t = time[rng[0]]
th = data[rng[0]]
phases.append(sin_phase(amp, freq, mean, th, t))
return amp, mean, rngs, freqs, phases
def trajpassage(N, traj):
"""Compute from the trajectory the index of the resonance (i.e. the halfway
index) and those of N full passages in either direction from it.
"""
residx = len(traj)/2
pidx, pv = pk.turning_points(traj, joined=True)
pidx = np.array(pidx, dtype=int)
try:
right_ex = np.argmax(pidx>residx)
except ValueError:
return [-1, residx, -1]
right_done = right_ex + N
if right_done >= len(pidx):
right_idx = len(traj)-1
else:
right_idx = pidx[right_done]
left_ex = right_ex - 1
left_done = left_ex - N
if left_done < 0:
left_idx = 0
else:
left_idx = pidx[left_done]
# plt.plot(traj)
# plt.plot(residx, traj[residx], marker="o", markersize=10)
# plt.plot(left_idx, traj[left_idx], marker="o", markersize=10)
# plt.plot(right_idx, traj[right_idx], marker="o", markersize=10)
# plt.show()
return [left_idx, residx, right_idx]
def divvy_coef(time, data, coef=0.6):
"""Finds the indices d_k at which data is closest to minimal, with the 0th and last
indices being 0 and len(data). Computes M_k = (d_k + d_{k+1} / 2),
W_k = d_{k+1} - d_k, and returns an array such that each row is an index range:
[[ 0 , M_0 + coef * W_0],
[M_1 - coef * W_1 , M_1 + coef * W_1],
.
.
.
[M_N - coef * W_N , -1 ]]
The terms 'M_0 -/+ coef * W_0' are rounded to the nearest integer.
(they must be good indices)
"""
# Get a sorted list of all extremal indices.
pk_idx, pk_v, tr_idx, tr_v = peaks.turning_points(data)
both_idx = [0] + tr_idx + [len(time)]
both_idx.sort()
M_k = [(d0 + d1)/2. for d0, d1 in zip(both_idx[:-1], both_idx[1:])]
fW_k = [coef*(d1 - d0) for d0, d1 in zip(both_idx[:-1], both_idx[1:])]
left = [M - f for M, f in zip(M_k, fW_k)]
right = [M + f for M, f in zip(M_k, fW_k)]
index_ranges = np.column_stack((left, right))
index_ranges[0, 0] = 0
index_ranges[-1, 1] = len(time)
return np.around(index_ranges, decimals=0).astype(int)
def get_amp_mean(data, time):
#to get the amplitude, we first get the peaks.
#then at first approximation we consider the amplitude to be the average
#distance between peak and mean.
ptimes, pvals, ttimes, tvals = pks.turning_points(data)
vals = pvals + tvals
mean = np.mean(data)
distances = mean - vals
amp = np.mean(distances)
return amp, mean
def get_amp_mean(data, time):
mean = np.mean(data)
#to get the amplitude, we first get the peaks.
#then at first approximation we consider the amplitude to be the average
#distance between peak and mean.
turns, vals = pks.turning_points(data)
distances = np.fabs(mean - vals)
amp = np.mean(distances)
return amp, mean
def savgol_omega(path, N=7, polorder=3, delta=0.5):
t = get_coord("t", path)
xA = np.column_stack((t, get_coord("xA", path)))
xB = np.column_stack((t, get_coord("xB", path)))
Omega, OmegaMag = savgol_omegafunc(xA,
xB,
N,
polorder=polorder,
delta=delta)
peaklist, smooth = peaks.turning_points(OmegaMag[:, 1], return_smooth=True)
OmegaMag[:, 1] = smooth
#smoothed = smooth_savgol(OmegaMag)
#OmegaMag_smooth = smooth_omegamag(OmegaMag)
return Omega, OmegaMag
def periodogram(d, time, data):
from scipy.signal import hann, lombscargle
print "d=", d
#datamean = np.mean(data)
datashift = data - np.mean(data)
pidx, pv, tidx, tv = peaks.turning_points(datashift, minidx=0)
tts = time[pidx]
rspl = UnivariateSpline(time, datashift, k=4, s=0)
twin = np.linspace(tts[1], tts[d + 1], 5000)
rwin = rspl(twin)
hwin = hann(len(twin))
rwin = rwin * hwin
f = np.linspace(0.005, 0.1, 3000)
pgram = lombscargle(twin, rwin, f)
pgram = np.sqrt(4 * (pgram / rwin.shape))
return f, pgram
def divvy(time, data, N_overlap=20):
""" Breaks 'time' into pieces, each containing about N cycles of
data, such that each piece overlaps its successor by N_overlap points.
Returns: a 2D numpy array 'pieces' such that pieces[i, 0] is the start
and pieces[i, 1] the end index of each piece.
"""
# Get a sorted list of all extremal indices.
pk_idx, pk_v, tr_idx, tr_v = peaks.turning_points(data)
both_idx = pk_idx + tr_idx
both_idx.sort()
assert both_idx[-2]+N_overlap < len(time), "failed size assumptions!"
assert both_idx[-1] < len(time), "failed size assumptions!"
pieces = np.zeros((len(both_idx)+1, 2))
pieces[1:, 0] = both_idx[:]
pieces[:-1, 1] = np.copy(both_idx[:]) + N_overlap
pieces[-1, 1] = len(time)
return pieces.astype(int)
def flip(data, junk_inds=0, shift=True):
"""Reflect descending parts of an oscillating function such that they increase.
"""
pk_idx, pk_v, tr_idx, tr_v = peaks.turning_points(data, minidx=junk_inds)
for a in [pk_idx, pk_v, tr_idx, tr_v]:
if len(a) <= 0:
return data
#assert len(a) > 0, "No peaks found!"
#we always want to flip between peaks and troughs
initially_descending = True
if pk_idx[0] == 0:
initially_descending = True
elif tr_idx[0] == 0:
initially_descending = False
elif pk_idx[0] < tr_idx[0]:
initially_descending = False
elif pk_idx[0] > tr_idx[0]:
initially_descending = True
pk_idx = [0] + pk_idx
pk_v = [data[0]] + pk_v
else:
raise ValueError("Identified same point as peak and trough")
last = len(data)
if pk_idx[-1] != last and tr_idx[-1] != last:
if pk_idx[-1] > tr_idx[
-1]: #last extremum is a peak so we end descending
tr_idx = tr_idx + [last]
if tr_idx[0] == 0:
tr_idx = tr_idx[1:]
ranges = zip(pk_idx, tr_idx, pk_v)
output = np.copy(data)
for l, r, this_max in ranges:
output[l:r] = 2 * this_max - data[l:r]
##account for the initial phase
if initially_descending and shift:
output -= 2 * output[0]
return output
def unwrap(data, thresh=0.2, initial_fraction=1., scale=1., junk_inds=200):
"""Produce a continuous 'unwrapped phase' which increases monotonically and accretes
by 2Pi with every period in the data.
Step 1: make monotonic by reflecting around the y=ymax line
at every point where the derivative changes sign.
2: This produces a set of discontinous curves, one per cycle. Each
should represent an increase by scale. Thus, multiply every range
by scale*(max - min).
3: Make the curve continuous by
shifting each cycle up such that it joins with its predecessor.
Params: x'thresh' is a minimum discontinuity required to identify a cycle.
Useful for ignoring junk.
x'starts_ascending' should be set to True if the data is initially
ascending (in principle this could be detected automatically).
If your results seem to be correct up to a y-reflection for each
period, try flipping this value.
x'intial_fraction' is the fraction of the relevant phase the orbit
begins at. For r-data, for example, we typically begin at
apastron and this should be set to 0.5. If your initial cycle
seems distorted compared to the others in a systematic (i.e.
non-junk like way) you probably need to adjust this.
"""
flipped = flip(data, junk_inds=junk_inds)
pk_idx, pk_v, tr_idx, tr_v = peaks.turning_points(flipped,
minidx=junk_inds)
for a in [pk_idx, pk_v, tr_idx, tr_v]:
if len(a) <= 0:
return flipped, flipped
output = np.copy(flipped)
if pk_idx[0] == 0:
zipvals = zip(pk_idx[1:], pk_v[1:])
else:
zipvals = zip(pk_idx, pk_v)
for idx, val in zipvals:
output[idx + 1:] += val #+ central
return output, flipped
def monotonic_acos(time, data, tnew=None, unwrap=True, minidx=200, Niter=0):
"""Identifies increasing and decreasing branches of 'data' by finding
extrema
and comparing their heights (e.g. the range between a minimum [maximum]
and a maximum [minimum] is increasing [decreasing]). Then, applies
the following piecewise function:
y = acos(data) {decreasing branch}
y = 2Pi - acos(data) {increasing branch}
This produces a monotonically increasing phase up to jump discontinuities
at the increasing-decreasing transition points (i.e. at the branch cuts
of acos). If unwrap is True, this function will add factors of 2Pi to
the jump discontinuities in order to produce a smooth curve.
If tnew is not None, the above function is evaluated at the points
data(tnew) only, which are interpolated from the original time series
'time' using a third order spline. The arguments 'time' and 'tnew'
must both be supplied if either one is.
Typically tnew will be the radial extremum times found from omega.
"""
pidx, pv, tidx, tv = peaks.turning_points(data, minidx=minidx)
amp, off = ampoffset(time, data)
dscale = (data - off) / amp
et, ev = peaks.joined_peaks(time, data)
nrngs = len(et) + 1
npi = np.zeros(nrngs)
if tidx[0] < pidx[0]: #initially descending
#ascending = [0, 1, 0, 1, 0, 1...]
#npi = [0, 0, 1, 1, 2, 2, 3, 3...]
ascending = truefalse(False, nrngs)
npi[::2] = np.arange(0, len(npi[::2]))
npi[1::2] = np.arange(0, len(npi[1::2]))
elif pidx[0] < tidx[0]: #initially ascending
#ascending = [1, 0, 1, 0, 1...]
#npi = [0, 1, 1, 2, 2, 3, 3...]
ascending = truefalse(True, nrngs)
npi[1::2] = np.arange(1, len(npi[1::2]) + 1)
npi[2::2] = np.arange(1, len(npi[2::2]) + 1)
else:
raise ValueError("First extremum identified as both trough and peak.")
pzip = zip(et, ascending, npi)
if tnew is not None:
spl = UnivariateSpline(time, dscale, s=0, k=4)
dnew = spl(tnew)
else:
dnew = dscale
tnew = time
chi = np.zeros(len(tnew))
lidx = 0
breaknow = False
for t_r, ascend, thisnpi in pzip:
if t_r > tnew[-1]:
ridx = len(tnew)
breaknow = True
else:
ridx = np.argmin(tnew <= t_r)
chi[lidx:ridx] = piecewise_acos(dnew[lidx:ridx], ascend)
if unwrap:
chi[lidx:ridx] += 2.0 * np.pi * thisnpi
if breaknow:
break
lidx = ridx
return chi
def freq_from_splines(
data,
offset_order=1, #
amp_order=3,
amp_smoothness=0,
signal_order=5,
signal_smoothness=1):
t = data[:, 0]
y = data[:, 1]
#Enforce correct types and assumptions
if not isinstance(data, np.ndarray):
raise TypeError("data should be a numpy array")
if data.ndim != 2:
raise ValueError("data should be 2 dimensional; first axis time")
if not isinstance(offset_order, int):
raise TypeError("offset_order must be of integer type")
if (offset_order < 0):
raise ValueError("offset_order must be nonnegative.")
if not isinstance(amp_order, int):
raise TypeError("amp_order must be of integer type")
if (amp_order < 0):
raise ValueError("amp_order must be nonnegative.")
if not isinstance(amp_smoothness, int):
raise TypeError("amp_smoothness must be of integer type")
if (amp_smoothness < 0):
raise ValueError("amp_smoothness must be nonnegative.")
if not isinstance(signal_order, int):
raise TypeError("signal_order must be of integer type")
if (signal_order < 3):
raise ValueError("signal_order must be at least 3.")
if not isinstance(signal_smoothness, int):
raise TypeError("signal_smoothness must be of integer type")
if (signal_smoothness < 0):
raise ValueError("signal_smoothness must be nonnegative.")
#Compute the offset M
M = do_fit(t, y, offset_order)
#Get peaks and troughs
#The indices and values of the peaks.
turns, turn_vals = pks.turning_points(y, 0)
#Distinguish the peaks from the troughs
peaks, peak_vals, troughs, trough_vals = pks.peaks_and_troughs(
turns, turn_vals)
#Spline interpolations of both (this seems to work better than polyfits)
pspl = interpolate.UnivariateSpline(t[peaks],
peak_vals,
k=amp_order,
s=amp_smoothness)
trspl = interpolate.UnivariateSpline(t[troughs],
trough_vals,
k=amp_order,
s=amp_smoothness)
P = np.zeros(len(t))
P = pspl(t)
Tr = np.zeros(len(t))
Tr = trspl(t)
#The amplitude is half the distanc between the peaks and trough
#at each point.
A = np.zeros(len(t))
A[:] = 0.5 * (P - Tr)
#Now we can form Q
Q = np.zeros(len(t))
Q[:] = (y[:] - M[:]) / A[:]
hsig = signal.hilbert(Q)
#print hsig
#print np.absolute(hsig)
#plt.plot(t,hsig.real,color="green",label="real")
#print hsig.imag
#plt.plot(t,hsig.imag,color="grey",label="imag")
#We need Qdot
#interpolate and take derivative
Qspl = interpolate.UnivariateSpline(t,
Q,
k=signal_order,
s=signal_smoothness)
Qdot_spl = Qspl.derivative()
Qdot = Qdot_spl(t)
#plt.plot(t,Qdot/hsig.imag)
#plt.plot(t,Qspl(t),label="Qspl")
#plt.plot(t,Qdot_spl(t),label="Qdotspl")
#We also need Q^2
#restrict to [-1,1] and square
np.clip(Q, -1.0, 1.0, out=Q)
Qsq = np.square(Q)
denom = np.sqrt(1.0 - Qsq)
freq = np.zeros(len(t))
freq[:] = (1. / (2. * np.pi)) * (Qdot / denom)
#spline interpolate the arcin to extract its derivative
asQ = np.arcsin(Q)
asQspl = interpolate.UnivariateSpline(t,
asQ,
k=signal_order,
s=signal_smoothness)
asQdot_spl = asQspl.derivative()
#freq = np.zeros( len(t) )
#freq[:] = (1./(2.*np.pi))*Qdot_spl( t )
#plt.plot(t,freq,label="freq")
#plt.plot(t,y,color = "blue", label="data")
#plt.plot(t, asQ, color="violet", label="asQ")
#plt.plot( t, Q, color="blue", label="Q" )
#plt.plot( t, M, color = "black", label="offset" )
#plt.plot( t, P, color = "red", label="peaks" )
#plt.plot( t, Tr, color = "red", label="troughs" )
plt.legend()
plt.show()
return 0
def t_resonance(phase): turns, turn_vals = pk.turning_points(phase[:,1]) i = np.argmax(turn_vals) return phase[turns[i],0]
def unwrapped_ang(ang):
peaktimes, peakvals, troughtimes, troughvals = peaks.turning_points(ang)
unwrapped = np.zeros(ang.shape)
grad = np.gradient(ang, 0.5)
unwrapped = ang[0]+np.cumsum(np.fabs(grad))
return unwrapped