. Author manuscript; available in PMC: 2023 Dec 13.

Published in final edited form as: IEEE Trans Biomed Circuits Syst. 2014 Apr 17;8(2):228–239. doi: 10.1109/TBCAS.2014.2306732

TABLE I.

The LPR-ICI Method for Time-varying Covariance Estimation

Step 1.	Calculate the inner product of two locally stationary processes as $m (n, l) = X_{n, N}^{(1)} X_{n + l, N}^{(2)}$
Step 2.	At each time instant $τ$ , estimate ${\hat{β}}_{k} (τ, l; h_{γ})$ as in (14) by a $p$ th order LPR with each bandwidth $h_{γ}$ in the set $\tilde{H}$ .
Step 3.	Approximate the residual variance $σ^{2} (τ, l)$ and the covariance $V (\hat{β} (τ, l))$ as in (21) and (22).
Step 4.	Calculate the optimal bandwidth ${\hat{h}}_{k}^{opt} (τ, l)$ using the ICI method as in (18 – 20).
Step 5.	Smooth ${\hat{h}}_{k}^{opt} (τ, l)$ slightly to reduce the variance, resulting in a smoothed variable bandwidth ${\tilde{h}}_{k}^{opt} (τ, l)$ .
Step 6.	Perform a $p$ th order LPR with bandwidth ${\tilde{h}}_{k}^{opt} (τ, l)$ to obtain ${\hat{β}}_{k} (τ, l; {\tilde{h}}_{k}^{opt} (τ, l))$ . ${\hat{β}}_{0} (τ, l; {\tilde{h}}_{0}^{opt} (τ, l))$ is the final estimate of time-varying covariance $\hat{c} (τ, l)$ .