Systems and Control Theory Lecture Notes

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Systems and Control Theory Lecture Notes

Laura Giarr´ e

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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. Giarr´e- Systems and Control Theory 2017-2018 - 2

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Outline & Goal

Introduction

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Outline & Goal

Introduction

Quadratic Regularization ¹ and ² methods

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Outline & Goal

Introduction

Quadratic Regularization ¹ and ² methods

LMS methods

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Outline & Goal

Introduction

Quadratic Regularization ¹ and ² methods

LMS methods

Numerical and experimental results

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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.

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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]

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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]

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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],

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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]

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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]

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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]

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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

¹ or ² penalty term.

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State of the art on regularization and Detrending

Regularization using the ¹ -norm has attracted a lot of interest 1. in statistics [Tib1985]

2. signal processing [Chen01]

3. machine learning [Boyd04]

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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.

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Regularized mean square error

Consider the following penalized mean square error problem to estimate the baseline:

ˆq = arg min

q J (q)

= arg min

q y − q ² 2 + λP(q), (1)

where y and q are n-length column vectors, · ² is the ² norm of

a vector, and λ is a given positive constant.

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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

2

(q) = Δq ² 2 (2)

P

1

(q) = Δ ² q ¹ , (3)

where Δ = 1 − z ⁻¹ is the derivative operator (with z ⁻¹ denoting the unitary delay).

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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 k =0

b _k z ^−k ,

A (z) = 1 +

N k =1

a _k z ^−k ,

with b _k , k = 0, 1, . . . , M, and a ^k , k = 1, . . . , N, the MA and

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ARMA modeling of BW

Thus, the baseline is given by

q [n] =

M k =0

b _k x [n − k] −

N k =1

a _k q [n − k]

= ϕ ^T 1 [n]θ,

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

,

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Cost function

J

₂

(q) = y − ϕ ^T 1 θ ² 2 + λϕ ^T 2 θ ² 2

J

₁

(q) = y − ϕ ^T 1 θ ² 2 + λϕ ^T 2 θ ¹

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Penalty P

²

(q)

Let h = [1 − 1] ^T , so that

Δq = h ^T

q [n]

q [n − 1]

= h ^T

ϕ ^T 1 [n]

ϕ ^T ₁ [n − 1]

θ

= ϕ ^T 2 [n]θ where

ϕ ² [n] =

ϕ ¹ [n] ϕ ¹ [n − 1] h .

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Penalty P

¹

(q)

Δ ² q = h ^T

⎡

⎣ q [n]

q [n − 1]

q [n − 2]

⎤

⎦

= h ^T

⎡

⎣ ϕ ^T 1 [n]

ϕ ^T 1 [n − 1]

ϕ ^T 1 [n − 2]

⎤

⎦ θ

= ϕ ^T 2 [n]θ, ϕ ² [n] =

ϕ ¹ [n] ϕ ¹ [n − 1] ϕ ¹ [n − 2]

h .

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LMS Algorithms for ² -penalty

Deﬁning

x [k] =

y [k] 0 T

; ϕ[k] =

ϕ ¹ [k] √

λϕ ² [k] e [k] = x[k] − ϕ ^T [k]θ.

The LMS solution is given by

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]

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LMS Algorithms for ¹ -penalty

Approximating the subdiﬀerential of the 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

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 ϕ ^T 2 [n]sign(ϕ ^T 2 [n]ˆθ[n − 1])

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Numerical Results: Triangular wave

3000 3500 4000 4500 5000 5500 6000

samples -5

-4 -3 -2 -1 0 1 2 3 4 5

y,q

y=x+q q

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Numerical Results: Triangular wave

3000 3500 4000 4500 5000 5500 6000

samples -3

-2 -1 0 1 2 3

q

q LMS2 LMS1

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MSE values

Since the trend is known, the methods can be compared in terms of mean square error (MSE):

MSE = 1 N _q

n

(q[n] − ˆq[n]) ²

Table: MSE values (averaged over 50 realizations of the trend).

LMS-L2 LMS-L1 0.0045 0.0037

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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 = 0.01.

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 .

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Numerical Results: Synthetic ECG

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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 0.0105 0.0293 0.0567

LMS-L2 0.0162 0.0268 0.0372

LMS-L1 0.0220 0.0310 0.0360

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Experimental Results: REAL ECG data

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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

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Experimental Results: REAL ECG data

Figure: Prototype of ECG acquisition device.

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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).

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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 = 1 and d ² = 2, λ ¹ = 50, λ ² = 100, μ = 10 ⁻⁸

The ARMA model was identiﬁed with M = 3 and N = 1.

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Economic real data (Standard&Poor)

2011 2012 2013 2014

1000 1100 1200 1300 1400 1500 1600 1700 1800

SP500 RLS-12 LMS-12 HP

Figure: Real SP500 data and estimated trends with diﬀerent methods.

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More work

New mixed norm cost (Penalty cost with both ¹ and ² )

RLS Solution

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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 ² and ¹ -norm regularization for adaptive detrending

with ARMA modeling, Journal of Franklin Institute, 2017

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References on baseline removal

AS85 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. 1052–1060, Dec 1985

Sorn1993 L. Sornmo, “Time-varying digital ﬁltering of ECG baseline wander,” Medical and Biological Engineering and Computing, vol. 31, no. 5, pp. 503–508, 1993.

Meyer1977 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.

459–470, 1977.

Papa2001 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. 105–112, 2001.

Lagu1992 P. Laguna, R. Jan´e, 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.

508–509.

Park1998 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. 581–586, 1998.

Blan2008 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, 2008.

Fasa2013b A. Fasano and V. Villani, “Baseline wander removal in ECG and AHA recommendations,” in Computing in Cardiology Conference (CinC), 2013, Sept 2013, pp. 1171–1174.

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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, 1996.

chen01 S. Chen, D. Donoho, and M. Saunders, “Atomic decomposition by basis pursuit,” SIAM Review, vol. 43, no. 1, p. 129âĂŞ159, 2001.

boyd04 S. Boyd and L. Vandenberghe, Convex optimization Cambridge Univ. Press, 2004.

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Systems and Control Theory Lecture Notes