/
mtm_functions.py
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/
mtm_functions.py
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# Functions for MultiTaper Method-Singular Value Decomposition (MTM-SVD) in python
#
# ------------------------------------------------------------------
#
# This script is a direct adaptation of the Matlab toolbox developed by
# Marco Correa-Ramirez and Samuel Hormazabal at
# Pontificia Universidad Catolica de Valparaiso
# Escuela de Ciencias del Mar, Valparaiso, Chile
# and is available through
# http://www.meteo.psu.edu/holocene/public_html/Mann/tools/tools.php
#
# This script was adapted by Mathilde Jutras at McGill University, Canada
# Copyright (C) 2020, Mathilde Jutras
# and is available under the GNU General Public License v3.0
#
# The script may be used, copied, or redistributed as long as it is cited as follow:
#
#
# This software may be used, copied, or redistributed as long as it is not
# sold and that this copyright notice is reproduced on each copy made.
# This routine is provided as is without any express or implied warranties.
#
# Questions or comments to:
# M. Jutras, mathilde.jutras@mail.mcgill.ca
#
# Last update:
# July 2020
#
# ------------------------------------------------------------------
#
# This file contains the functions called in the file
# mmt-svd-python.py
#
# ------------------------------------------------------------------
#
from scipy.signal.windows import dpss
from scipy import signal
import numpy as np
from numpy.matlib import repmat
# Function 1) Determine the local fractional variance spectrum LFV
def mtm_svd_lfv(ts2d,nw,kk,dt) :
# Compute spectrum at each grid point
p, n = ts2d.shape
# Remove the mean and divide by std
vm = np.nanmean(ts2d, axis=0) # mean
vmrep = repmat(vm,ts2d.shape[0],1)
ts2d = ts2d - vmrep
vs = np.nanstd(ts2d, axis=0) # standard deviation
vsrep = repmat(vs,ts2d.shape[0],1)
ts2d = np.divide(ts2d,vsrep)
ts2d = np.nan_to_num(ts2d)
# Slepian tapers
psi = dpss(p,nw,kk)
npad = 2**int(np.ceil(np.log2(abs(p)))+2)
nf = int(npad/2)
ddf = 1./(npad*dt)
fr = np.arange(0,nf)*ddf
# Get the matrix of spectrums
psimats = []
for k in range(kk):
psimat2 = np.transpose( repmat(psi[k,:],ts2d.shape[1],1) )
psimat2 = np.multiply(psimat2,ts2d)
psimats.append(psimat2)
psimats=np.array(psimats)
nev = np.fft.fft(psimats,n=npad,axis=1)
nev = np.fft.fftshift(nev,axes=(1))
nev = nev[:,nf:,:]
# Calculate svd for each frequency
lfvs = np.zeros(nf)*np.nan
for j in range(nf) :
U,S,V = np.linalg.svd(nev[:,j,:], full_matrices=False)
lfvs[j] = S[0]**2/(np.nansum(S[1:])**2)
return fr, lfvs
# Function 2) Calculate the confidence interval of the LFV calculations
def mtm_svd_conf(ts2d,nw,kk,dt,niter,sl) :
# Compute spectrum at each grid point
p, n = ts2d.shape
npad = 2**int(np.ceil(np.log2(abs(p)))+2)
nf = int(npad/2)
ddf = 1./(npad*dt)
fr = np.arange(0,nf)*ddf
# range of frequencies
fran = [0, 0.5/dt]
nfmin = (np.abs(fr-fran[0])).argmin()
nfmax = (np.abs(fr-fran[1])).argmin()
fr = fr[nfmin:nfmax]
q = [int(niter*each) for each in sl]
# Remove the mean and divide by std
vm = np.nanmean(ts2d, axis=0) # mean
vmrep = repmat(vm,ts2d.shape[0],1)
ts2d = ts2d - vmrep
vs = np.nanstd(ts2d, axis=0) # standard deviation
vsrep = repmat(vs,ts2d.shape[0],1)
ts2d = np.divide(ts2d,vsrep)
ts2d = np.nan_to_num(ts2d)
# Slepian tapers
psi = dpss(p,nw,kk)
partvar = np.ones((niter, len(fr)+1))*np.nan
for it in range(niter):
print('Iter %i'%it)
shr = np.random.permutation(ts2d) # random permutation of each time series
# Spectral estimation
psimats = []
for k in range(kk):
psimat2 = np.transpose( repmat(psi[k,:],shr.shape[1],1) )
psimat2 = np.multiply(psimat2,shr)
psimats.append(psimat2)
psimats=np.array(psimats)
nevconf = np.fft.fft(psimats,n=npad,axis=1)
nevconf = np.fft.fftshift(nevconf,axes=(1))
nevconf = nevconf[:,nf:,:]
# Calculate svd for each frequency
for j in range(nfmin,nfmax) :
U,S,V = np.linalg.svd(nevconf[:,j,:], full_matrices=False)
partvar[it,j] = S[0]**2/(np.nansum(S[1:])**2)
np.sort(partvar, axis=1)
freq_sec = nw/(p*dt)
ibs = (np.abs(fr-freq_sec)).argmin() ; ibs=range(ibs)
ibns = range(ibs[-1]+1, len(fr))
fray = 1./(p*dt)
fbw = 2*nw*fray
ifbw = int(round(fbw/(fr[2]-fr[1])))
evalper = np.zeros((len(q),len(fr)))
for i in range(len(q)):
y2 = np.zeros(len(fr))
y = partvar[q[i],:] ### right order indices?
y1 = signal.medfilt(y,ifbw)
y2[ibs] = np.mean(y[ibs])
a = np.polyfit(fr[ibns],y1[ibns],10)
y2[ibns] = np.polyval(a, fr[ibns])
evalper[i,:] = y2
return fr, evalper
def envel(ff0, iif, fr, dt, ddf, p, kk, psi, V) :
ex = np.ones(p)
df1 = 0
c0=1; s0=0;
c=[c0]
s=[s0]
cs=np.cos(2.*np.pi*df1*dt)
sn=np.sin(2.*np.pi*df1*dt)
for i in range(1,p) :
c.append( c[i-1]*cs-s[i-1]*sn )
s.append( c[i-1]*sn+s[i-1]*cs )
cl = np.ones(p) ## REMOVE?
sl = np.zeros(p) ##
d = V[0,:]
d = np.conj(d)*2
if iif == 1 :
d = V[0,:] ; d = np.conj(d)
g = []
for i0 in range(kk) :
cn = [complex( psi[i0,i]*c[i], -psi[i0,i]*s[i] ) for i in range(len(s))]
g.append( ex*cn )
g=np.array(g).T
za = np.conj(sum(g))
[g1,qrsave1] = np.linalg.qr(g)
dum1 = np.linalg.lstsq( np.conj(qrsave1), np.linalg.lstsq( np.conj(qrsave1.T), d )[0] )[0].T
amp0=sum(np.conj(za)*dum1)
dum2 = np.linalg.lstsq( np.conj(qrsave1), np.linalg.lstsq( np.conj(qrsave1.T), za )[0] )[0].T
amp1=sum(np.conj(za)*dum2)
amp0=amp0/amp1
sum1=sum(abs(d)**2)
d=d-za*amp0
sum2=sum(abs(d)**2)
env0= np.linalg.lstsq( np.conj((qrsave1.T)), d.T )[0].T
env = np.matmul(g1, env0.T)
env = env + amp0*np.ones(len(c))
return env
# Function 3) Reconstruct the spatial patterns associated with peaks in the spectrum
def mtm_svd_recon(ts2d, nw, kk, dt, fo) :
imode = 0
lan = 0
vw = 0
# Compute spectrum at each grid point
p, n = ts2d.shape
# Remove the mean and divide by std
vm = np.nanmean(ts2d, axis=0) # mean
vmrep = repmat(vm,ts2d.shape[0],1)
ts2d = ts2d - vmrep
vs = np.nanstd(ts2d, axis=0) # standard deviation
vsrep = repmat(vs,ts2d.shape[0],1)
ts2d = np.divide(ts2d,vsrep)
ts2d = np.nan_to_num(ts2d)
# Slepian tapers
psi = dpss(p,nw,kk)
npad = 2**int(np.ceil(np.log2(abs(p)))+2)
nf = int(npad/2)
ddf = 1./(npad*dt)
fr = np.arange(0,nf)*ddf
# Get the matrix of spectrums
psimats = []
for k in range(kk):
psimat2 = np.transpose( repmat(psi[k,:],ts2d.shape[1],1) )
psimat2 = np.multiply(psimat2,ts2d)
psimats.append(psimat2)
psimats=np.array(psimats)
nev = np.fft.fft(psimats,n=npad,axis=1)
nev = np.fft.fftshift(nev,axes=(1))
nev = nev[:,nf:,:]
# Initialiser les matrices de sorties
S = np.ones((kk, len(fo)))*np.nan
vexp = [] ; totvarexp = [] ; iis = []
D = vsrep
envmax = np.zeros(len(fo))*np.nan
for i1 in range(len(fo)):
# closest frequency
iif = (np.abs(fr - fo[i1])).argmin()
iis.append(iif)
ff0 = fr[iif]
print('( %i ) %.2f cyclesyr | %.2f yr'%(iif,ff0,1/ff0))
U,S0,Vh = np.linalg.svd(nev[:,iif,:].T,full_matrices=False)
##V = Vh.T.conj()
V = Vh
S[:,i1] = S0
env1 = envel(ff0, iif, fr, dt, ddf, p, kk, psi, V) # condition 1
cs=[1]
sn=[0]
c=np.cos(2*np.pi*ff0*dt)
s=np.sin(2*np.pi*ff0*dt)
for i2 in range(1,p):
cs.append( cs[i2-1]*c-sn[i2-1]*s )
sn.append( cs[i2-1]*s+sn[i2-1]*c )
CS = [complex(cs[i], sn[i]) for i in range(len(cs))]
CS=np.conj(CS)
# Reconstructions
R = np.real( D * S[imode, i1] * np.outer(U[:,imode], CS*env1).T )
vsr=np.var(R,axis=0)
vexp.append( vsr/(vs**2)*100 )
totvarexp.append( np.nansum(vsr)/np.nansum(vs**2)*100 )
return vexp, totvarexp, iis