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cvxEDA.py
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"""
______________________________________________________________________________
File: cvxEDA.py
Last revised: 07 Nov 2015 r69
______________________________________________________________________________
Copyright (C) 2014-2015 Luca Citi, Alberto Greco
This program is free software; you can redistribute it and/or modify it under
the terms of the GNU General Public License as published by the Free Software
Foundation; either version 3 of the License, or (at your option) any later
version.
This program is distributed in the hope that it will be useful, but WITHOUT
ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS
FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details.
You may contact the author by e-mail ([email protected]).
______________________________________________________________________________
This method was first proposed in:
A Greco, G Valenza, A Lanata, EP Scilingo, and L Citi
"cvxEDA: a Convex Optimization Approach to Electrodermal Activity Processing"
IEEE Transactions on Biomedical Engineering, 2015
DOI: 10.1109/TBME.2015.2474131
If you use this program in support of published research, please include a
citation of the reference above. If you use this code in a software package,
please explicitly inform the end users of this copyright notice and ask them
to cite the reference above in their published research.
______________________________________________________________________________
"""
import numpy as np
import cvxopt as cv
import cvxopt.solvers
def cvxEDA(y, delta, tau0=2., tau1=0.7, delta_knot=10., alpha=8e-4, gamma=1e-2,
solver=None, options={'reltol':1e-9}):
"""CVXEDA Convex optimization approach to electrodermal activity processing
This function implements the cvxEDA algorithm described in "cvxEDA: a
Convex Optimization Approach to Electrodermal Activity Processing"
(http://dx.doi.org/10.1109/TBME.2015.2474131, also available from the
authors' homepages).
Arguments:
y: observed EDA signal (we recommend normalizing it: y = zscore(y))
delta: sampling interval (in seconds) of y
tau0: slow time constant of the Bateman function
tau1: fast time constant of the Bateman function
delta_knot: time between knots of the tonic spline function
alpha: penalization for the sparse SMNA driver
gamma: penalization for the tonic spline coefficients
solver: sparse QP solver to be used, see cvxopt.solvers.qp
options: solver options, see:
http://cvxopt.org/userguide/coneprog.html#algorithm-parameters
Returns (see paper for details):
r: phasic component
p: sparse SMNA driver of phasic component
t: tonic component
l: coefficients of tonic spline
d: offset and slope of the linear drift term
e: model residuals
obj: value of objective function being minimized (eq 15 of paper)
"""
n = len(y)
y = cv.matrix(y)
# bateman ARMA model
a1 = 1./min(tau1, tau0) # a1 > a0
a0 = 1./max(tau1, tau0)
ar = np.array([(a1*delta + 2.) * (a0*delta + 2.), 2.*a1*a0*delta**2 - 8.,
(a1*delta - 2.) * (a0*delta - 2.)]) / ((a1 - a0) * delta**2)
ma = np.array([1., 2., 1.])
# matrices for ARMA model
i = np.arange(2, n)
A = cv.spmatrix(np.tile(ar, (n-2,1)), np.c_[i,i,i], np.c_[i,i-1,i-2], (n,n))
M = cv.spmatrix(np.tile(ma, (n-2,1)), np.c_[i,i,i], np.c_[i,i-1,i-2], (n,n))
# spline
delta_knot_s = int(round(delta_knot / delta))
spl = np.r_[np.arange(1.,delta_knot_s), np.arange(delta_knot_s, 0., -1.)] # order 1
spl = np.convolve(spl, spl, 'full')
spl /= max(spl)
# matrix of spline regressors
i = np.c_[np.arange(-(len(spl)//2), (len(spl)+1)//2)] + np.r_[np.arange(0, n, delta_knot_s)]
nB = i.shape[1]
j = np.tile(np.arange(nB), (len(spl),1))
p = np.tile(spl, (nB,1)).T
valid = (i >= 0) & (i < n)
B = cv.spmatrix(p[valid], i[valid], j[valid])
# trend
C = cv.matrix(np.c_[np.ones(n), np.arange(1., n+1.)/n])
nC = C.size[1]
# Solve the problem:
# .5*(M*q + B*l + C*d - y)^2 + alpha*sum(A,1)*p + .5*gamma*l'*l
# s.t. A*q >= 0
old_options = cv.solvers.options.copy()
cv.solvers.options.clear()
cv.solvers.options.update(options)
if solver == 'conelp':
# Use conelp
z = lambda m,n: cv.spmatrix([],[],[],(m,n))
G = cv.sparse([[-A,z(2,n),M,z(nB+2,n)],[z(n+2,nC),C,z(nB+2,nC)],
[z(n,1),-1,1,z(n+nB+2,1)],[z(2*n+2,1),-1,1,z(nB,1)],
[z(n+2,nB),B,z(2,nB),cv.spmatrix(1.0, range(nB), range(nB))]])
h = cv.matrix([z(n,1),.5,.5,y,.5,.5,z(nB,1)])
c = cv.matrix([(cv.matrix(alpha, (1,n)) * A).T,z(nC,1),1,gamma,z(nB,1)])
res = cv.solvers.conelp(c, G, h, dims={'l':n,'q':[n+2,nB+2],'s':[]})
obj = res['primal objective']
else:
# Use qp
Mt, Ct, Bt = M.T, C.T, B.T
H = cv.sparse([[Mt*M, Ct*M, Bt*M], [Mt*C, Ct*C, Bt*C],
[Mt*B, Ct*B, Bt*B+gamma*cv.spmatrix(1.0, range(nB), range(nB))]])
f = cv.matrix([(cv.matrix(alpha, (1,n)) * A).T - Mt*y, -(Ct*y), -(Bt*y)])
res = cv.solvers.qp(H, f, cv.spmatrix(-A.V, A.I, A.J, (n,len(f))),
cv.matrix(0., (n,1)), solver=solver)
obj = res['primal objective'] + .5 * (y.T * y)
cv.solvers.options.clear()
cv.solvers.options.update(old_options)
l = res['x'][-nB:]
d = res['x'][n:n+nC]
t = B*l + C*d
q = res['x'][:n]
p = A * q
r = M * q
e = y - r - t
return (np.array(a).ravel() for a in (r, p, t, l, d, e, obj))