Systems and Control Theory Lecture Notes. Laura Giarré

Transcription

1 Systems and Control Theory Lecture Notes Laura Giarré L. Giarré

2 Lesson 23: Regularized LMS methods for baseline wandering removal in wearable ECG devices Regularized LMS method Baseline wandering removal Wearable ECG devices Detrend of Economic Data L. Giarré- Systems and Control Theory

3 Outline & Goal Introduction L. Giarré- Systems and Control Theory

4 Outline & Goal Introduction Quadratic Regularization l 1 and l 2 methods L. Giarré- Systems and Control Theory

5 Outline & Goal Introduction Quadratic Regularization l 1 and l 2 methods LMS methods L. Giarré- Systems and Control Theory

6 Outline & Goal Introduction Quadratic Regularization l 1 and l 2 methods LMS methods Numerical and experimental results L. Giarré- Systems and Control Theory

7 Introduction Wearable electrocardiogram (ECG) devices are light-weight, low-consumption systems used to acquire and transmit (with wireless connection) physiological signals. Baseline wandering (BW): patient movements and respiration produce a low frequency (up to 0.8 Hz) random variation of the ECG signal trend. Removing this artifact is not simple since its spectrum is partly overlapped to the informative signal. L. Giarré- Systems and Control Theory

8 State of the art on baseline removal Several methods and tools for solving the baseline wandering problem 1. based on notch filters and time-varying filters [AS85], [Sorn1993] L. Giarré- Systems and Control Theory

9 State of the art on baseline removal Several methods and tools for solving the baseline wandering problem 1. based on notch filters and time-varying filters [AS85], [Sorn1993] 2. linear spline and cubic approximations [Meyer1977],[Papa2001] L. Giarré- Systems and Control Theory

10 State of the art on baseline removal Several methods and tools for solving the baseline wandering problem 1. based on notch filters and time-varying filters [AS85], [Sorn1993] 2. linear spline and cubic approximations [Meyer1977],[Papa2001] 3. adaptive filters [Lagu1992], L. Giarré- Systems and Control Theory

11 State of the art on baseline removal Several methods and tools for solving the baseline wandering problem 1. based on notch filters and time-varying filters [AS85], [Sorn1993] 2. linear spline and cubic approximations [Meyer1977],[Papa2001] 3. adaptive filters [Lagu1992], 4. discrete wavelet transform (DWT) [Park1998] L. Giarré- Systems and Control Theory

12 State of the art on baseline removal Several methods and tools for solving the baseline wandering problem 1. based on notch filters and time-varying filters [AS85], [Sorn1993] 2. linear spline and cubic approximations [Meyer1977],[Papa2001] 3. adaptive filters [Lagu1992], 4. discrete wavelet transform (DWT) [Park1998] 5. empirical mode decomposition (EMD) [Blan2008] L. Giarré- Systems and Control Theory

13 State of the art on baseline removal Several methods and tools for solving the baseline wandering problem 1. based on notch filters and time-varying filters [AS85], [Sorn1993] 2. linear spline and cubic approximations [Meyer1977],[Papa2001] 3. adaptive filters [Lagu1992], 4. discrete wavelet transform (DWT) [Park1998] 5. empirical mode decomposition (EMD) [Blan2008] 6. quadratic variation reduction (QVR) and a linear time invariant (LTI) implementation approximating the QVR method [Fasa2013] L. Giarré- Systems and Control Theory

14 Our contribution: Novelty Online implementations: a new baseline sample is estimated after the acquisition of a new ECG sample. Generalized cost function to be optimized, including an either l 1 or l 2 penalty term. L. Giarré- Systems and Control Theory

15 State of the art on regularization and Detrending Regularization using the l 1 -norm has attracted a lot of interest 1. in statistics [Tib1985] 2. signal processing [Chen01] 3. machine learning [Boyd04] L. Giarré- Systems and Control Theory

16 Our target:regularized mean square error Let y[k], k = 1, 2,...,n, be the acquired ECG signal affected by a baseline q[k], k = 1, 2,...,n. q is a lowpass signal that introduces slow variations (or trend) into the ECG. The objective of a BW removal algorithm is that of estimating q from y and remove it, so that y q has the same shape of y and a constant trend. L. Giarré- Systems and Control Theory

17 Regularized mean square error Consider the following penalized mean square error problem to estimate the baseline: ˆq = arg min q = arg min q J(q) y q λp(q), (1) where y and q are n-length column vectors, 2 is the l 2 norm of a vector, and λ is a given positive constant. L. Giarré- Systems and Control Theory

18 Regularized mean square error The first term is a fidelity term between the acquired ECG signal and the unknown baseline. The penalty term P(q) must be chosen in order to induce smoothness on the signal q: P l2 (q) = Δq 2 2 (2) P l1 (q) = Δ 2 q 1, (3) where Δ=1 z 1 is the derivative operator (with z 1 denoting the unitary delay). L. Giarré- Systems and Control Theory

19 ARMA modeling of BW We obtain the baseline q from the observed ECG signal y as an ARMA model: Q(z) =F (z)y (z) = B(z) A(z) Y (z) B(z) = M b k z k, k=0 A(z) =1 + N a k z k, k=1 with b k, k = 0, 1,...,M, anda k, k = 1,...,N, the MA and AR parameters. L. Giarré- Systems and Control Theory

20 ARMA modeling of BW Thus, the baseline is given by M q[n] = b k x[n k] k=0 = ϕ T 1 [n]θ, N a k q[n k] k=1 where ϕ 1 [n] = [ y[n]... y[n M] q[n 1]... q[n N] ] T θ = [ b 0... b M a 1... a N ] T, L. Giarré- Systems and Control Theory

21 Cost function J l2 (q) = y ϕ T 1 θ λ ϕ T 2 θ 2 2 J l1 (q) = y ϕ T 1 θ λ ϕ T 2 θ 1 L. Giarré- Systems and Control Theory

22 Penalty P l2 (q) Let h =[1 1] T, so that [ ] Δq = h T q[n] q[n 1] [ ] = h T ϕ T 1 [n] ϕ T θ 1 [n 1] = ϕ T 2 [n]θ where ϕ 2 [n] = [ ϕ 1 [n] ϕ 1 [n 1] ] h. L. Giarré- Systems and Control Theory

23 Penalty P l1 (q) q[n] Δ 2 q = h T q[n 1] q[n 2] ϕ T = h T 1 [n] ϕ T 1 [n 1] θ [n 2] ϕ T 1 = ϕ T 2 [n]θ, ϕ 2 [n] = [ ϕ 1 [n] ϕ 1 [n 1] ϕ 1 [n 2] ] h. L. Giarré- Systems and Control Theory

24 LMS Algorithms for l 2 -penalty Defining x[k] = [ y[k] 0 ] T ; ϕ[k] = [ ϕ 1 [k] λϕ 2 [k] ] The LMS solution is given by e[k] =x[k] ϕ T [k]θ. ˆθ[n] =ˆθ[n 1] μ 2 e[n] 2 μ is the updating gain The LMS update is ˆθ[n] =ˆθ[n 1]+μϕ[n]e[n] L. Giarré- Systems and Control Theory

25 LMS Algorithms for l 1 -penalty Approximating the subdifferential of the l 1 -norm at ϕ T 2 θ as ϕ T 2 θ 1 ϕ T 2 sign(ϕ T 2 θ) The LMS solution is ˆθ[n] =ˆθ[n 1] μ 2 J l 1 [n] where μ is the updating gain The LMS update is ˆθ[n] =ˆθ[n 1]+μ [ ϕ 1 [n]e[n] 1 2 ϕt 2 [n]sign(ϕ T 2 [n]ˆθ[n 1]) ] L. Giarré- Systems and Control Theory

26 Numerical Results: Triangular wave 5 4 y=x+q q y,q samples L. Giarré- Systems and Control Theory

27 Numerical Results: Triangular wave 3 2 q LMS2 LMS1 1 q samples L. Giarré- Systems and Control Theory

28 MSE values Since the trend is known, the methods can be compared in terms of mean square error (MSE): MSE = 1 (q[n] ˆq[n]) 2 N q n Table: MSE values (averaged over 50 realizations of the trend). LMS-L2 LMS-L L. Giarré- Systems and Control Theory

29 Numerical Results: Synthetic ECG The algorithm to generate synthetic baseline-free ECG signalsis is a Matlab implementation of the one in PhysioNet. We set the heart rate to 60 bpm, with a sampling frequency f s = 256 Hz and an additive Gaussian noise with standard deviation σ n = The output is an ECG-like signal normalized between -0.4 and 1.2 mv. A synthetic pseudo-random baseline was added to the ECG signal. The baseline is a filtered white Gaussian process with a fourth-order Butterworth filter with a 3-dB cutoff frequency set to f t. L. Giarré- Systems and Control Theory

30 Numerical Results: Synthetic ECG L. Giarré- Systems and Control Theory

31 MSE values Table: MSE values (averaged over 50 realizations of the trend). Method f t = 0.2 Hz f t = 0.4 Hz f t = 0.6 Hz QVR-LTI LMS-L LMS-L L. Giarré- Systems and Control Theory

32 Experimental Results: REAL ECG data L. Giarré- Systems and Control Theory

33 Prototype A prototype of ECG acquisition device was developed at the UNIFI laboratory. Features: acquisition of 3 ECG bipolar derivations (DI, DII, DIII) and 1 pre-cordial derivation (V1), by using 5 standard electrodes; analog front-end and ADC at 24 bit (Texas Instruments ADS1293), sampling frequency up to 25.6 ksps; micro-controller ARM STM32F411; storage onto microsd; transmission of ECG signals in real time by means of wireless Bluetooth 4.0 Low Energy (Nordic Semiconductor nrf8001) or by means of USB connection (developed dedicated APP using HL7 FHIR standard; PCB dimension of 44x60 mm; long duration battery with capacity of 1300 mah; Standard ECG connectors DIN, diameter 1.5mm. L. Giarré- Systems and Control Theory

34 Experimental Results: REAL ECG data Figure: Prototype of ECG acquisition device. L. Giarré- Systems and Control Theory

35 Economic data Results We apply the developed methods also to financial time series trend estimation Here, the trend is just the information we would like to extract from the observed data for economic analysis purposes. Data are taken daily on a 10 years interval (from September 29th, 2006 to October 3rd, 2016). L. Giarré- Systems and Control Theory

36 Results obtained from the SP500 dataset We plot the estimated trends obtained by using the RLS-12, LMS-12 and HP algorithms as well as the real data. These results were obtained by setting the order of the derivatives d 1 = 1andd 2 = 2, λ 1 = 50, λ 2 = 100, μ = 10 8 The ARMA model was identified with M = 3andN = 1. L. Giarré- Systems and Control Theory

37 Economic real data (Standard&Poor) SP500 RLS-12 LMS-12 HP Figure: Real SP500 data and estimated trends with different methods. L. Giarré- Systems and Control Theory

38 More work New mixed norm cost (Penalty cost with both l 1 and l 2 ) RLS Solution L. Giarré- Systems and Control Theory

39 Publications Adaptive quadratic regularization for baseline wandering removal in wearable ECG devices, Eusipco 2016 Regularized LMS methods for baseline wandering removal in wearable ECG devices, CDC 2016 Mixed l 2 and l 1 -norm regularization for adaptive detrending with ARMA modeling, Journal of Franklin Institute, 2017 L. Giarré- Systems and Control Theory

40 References on baseline removal AS85 Sorn1993 Meyer1977 Papa2001 Lagu1992 Park1998 Blan2008 Fasa2013b J. V. Alste and T. Schilder, Removal of base-line wander and power-line interference from the ECG by an efficient FIR filter with a reduced number of taps, IEEE Transactions on Biomedical Engineering, vol. BME-32, no. 12, pp , Dec 1985 L. Sornmo, Time-varying digital filtering of ECG baseline wander, Medical and Biological Engineering and Computing, vol. 31, no. 5, pp , C. R. Meyer and H. N. Keiser, Electrocardiogram baseline noise estimation and removal using cubic splines and state-space computation techniques, Computers and Biomedical Research, vol. 10, no. 5, pp , C. Papaloukas, D. I. Fotiadis, A. P. Liavas, A. Likas, and L. K. Michalis, A knowledge-based technique for automated detection of ischaemic episodes in long duration electrocardiograms, Medical and Biological Engineering and Computing, vol. 39, no. 1, pp , P. Laguna, R. Jané, and P. Caminal, Adaptive filtering of ECG baseline wander, in 14th Annual International Conference of the IEEE Engineering in Medicine and Biology Society, vol. 2, Oct 1992, pp K. L. Park, K. J. Lee, and H. R. Yoon, Application of a wavelet adaptive filter to minimise distortion of the ST-segment, Medical and Biological Engineering and Computing, vol. 36, no. 5, pp , M. Blanco-Velasco, B. Weng, and K. E. Barner, ECG signal denoising and baseline wander correction based on the empirical mode decomposition, Computers in Biology and Medicine, vol. 38, no. 1, pp. 1 13, A. Fasano and V. Villani, Baseline wander removal in ECG and AHA recommendations, in Computing in Cardiology Conference (CinC), 2013, Sept 2013, pp L. Giarré- Systems and Control Theory

41 References on regularized penalty and de-trending tib1985 R. Tibshirani, Regression shrinkage and selection via the lasso, Journal of the Royal Statistical Society, vol. 58, no. 1, p. 267âĂŞ288, chen01 S. Chen, D. Donoho, and M. Saunders, Atomic decomposition by basis pursuit, SIAM Review, vol. 43, no. 1, p. 129âĂŞ159, boyd04 S. Boyd and L. Vandenberghe, Convex optimization Cambridge Univ. Press, L. Giarré- Systems and Control Theory

42 Thanks DIEF- Tel: giarre.wordpress.com