Python implementation Tarvainen et.al. "An Advanced Detrending Method With Application to HRV Analysis"

tarvainen_filter.py

def tarvainen_filter(sig, lam):
    """ Returns detrended signal
    
    Keyword arguments:
    sig - input signal (1d array)
    lam - smoothing parameter(lambda)
    
    The implementation of
    "An Advanced Detrending Method With Application to HRV Analysis"
    Mika P. Tarvainen, Perttu O. Ranta-aho, and Pasi A. Karjalainen  
    IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, VOL. 49, NO. 2, FEBRUARY 2002
    
    It is the same method as method:
    W. Gersch, Smoothness priors , in New Directions in Time Series , Analysis, Part II, pp. 113-146.
    Springer-Verlag, 1991.

    Matlab implementation
    N=100; % Number of points
    x=randn(N,1); % Random data
    T = length(x);
    lambda = 100;
    I = speye(T);
    D2 = spdiags(ones(T-2,1)[1 -2 1], [0:2], T-2, T);
    xhat = (I-inv(I+lambda^2D2'D2))x
    figure()
    plot(1:N,x,'b',1:N,xhat,'k')
    
    Code written by Konstantin Purtov <k.s.purtov@gmail.com> 2015
    
    example of using:
    signal_with_noise = np.random.rand(200)
    detrended_signal = tarvainen_filter(signal_with_noise, 2000)
    """
    import numpy as np
    import scipy.sparse
    sig = np.array(sig)

    T = len(sig)
    I = np.identity(T)
    filt = [1, -2, 1] * np.ones((1, T), dtype=np.int).T
    D2 = scipy.sparse.spdiags(filt.T, (range(0, 3)), T - 2, T).toarray()
    xhat = np.dot((I - np.linalg.inv(I + (lam ** 2) * np.dot(D2.T, D2))), sig) # filtered signal
    return xhat