Heart rate variability detection - Revision history

Jureslak: /* Fine local search */

2019-09-08T16:26:30Z

‎Fine local search

Jureslak: /* Detection of heart rate variabilityfrom a wearable differential ECG device */

2018-05-31T11:39:01Z

‎Detection of heart rate variabilityfrom a wearable differential ECG device

Gkosec at 14:39, 24 February 2018

2018-02-24T14:39:25Z

Anja at 18:15, 7 November 2016

2016-11-07T18:15:59Z

Anja at 17:50, 7 November 2016

2016-11-07T17:50:37Z

Anja at 17:36, 7 November 2016

2016-11-07T17:36:38Z

Anja: Created page with "We used MLS and WLS aproximation to extract heart rate variability from a wearable ECG sensor. [http://www-e6.ijs.si/ParallelAndDistributedSystems/MeshlessMachine/technical_do..."

2016-11-07T16:47:52Z

Created page with "We used MLS and WLS aproximation to extract heart rate variability from a wearable ECG sensor. [http://www-e6.ijs.si/ParallelAndDistributedSystems/MeshlessMachine/technical_do..."

New page

We used MLS and WLS aproximation to extract heart rate variability from a wearable ECG sensor.
[http://www-e6.ijs.si/ParallelAndDistributedSystems/MeshlessMachine/technical_docs/html/heartratevar.pdf Full paper available for download here.]
[http://www-e6.ijs.si/ParallelAndDistributedSystems/MeshlessMachine/technical_docs/html/heartratevar_pres.pdf Presentation available for download here.]

The code can be found in <syntaxhighlight lang="bash" inline> /EKG/detect.cpp </syntaxhighlight>.
<xr id="fig:real"/> shows how we detected beat to beat times from an actual heartbeat.
<figure id="fig:real">
[[File:real.png|600px|thumb|upright=2|alt=actual heartbeat detected beat to beat|<caption> We detected beat to beat times from an actual heartbeat in this way. </caption>]]
</figure>

A slightly abridged version of the paper is presented below.

=Detection of heart rate variability<br>from a wearable differential ECG device=

[mailto:jure.slak@student.fmf.uni-lj.si Jure Slak], [mailto:gregor.kosec@ijs.si Gregor Kosec],
Jožef Stefan Insitute, Department of Communication Systems, Ljubljana.

==Abstract==
The precise heart rate variability is extracted from an ECG signal
measured by a wearable sensor that constantly records the heart activity of an
active subject for several days. Due to the limited resources of the wearable
ECG device the signal can only be sampled at relatively low, approximately $100$
Hz, frequency. Besides low sampling rate the signal from a wearable sensor is
also burdened with much more noise than the standard $12$-channel ambulatory
ECG, mostly due to the design of the device, i.e. the electrodes are
positioned relatively close to each other, and the fact that the subject is
active during the measurements. To extract heart rate variability with $1$ ms
precision, i.e. $10$ times more accurate than the sample rate of the measured
signal, a two-step algorithm is proposed. In first step an approximate global
search is performed, roughly determining the point of interest, followed by a
local search based on the Moving Least Squares approximation to refine the
result. The methodology is evaluated in terms of accuracy, noise sensitivity,
and computational complexity. All tests are performed on simulated as well as
measured data. It is demonstrated that the proposed algorithm provides
accurate results at a low computational cost and it is robust enough for
practical application.

<h2>Introduction</h2>

<figure id="fig:beat">
[[File:beat.png|600px|thumb|upright=2|alt= Beat graph|<caption>Beat to beat time between two characteristic points.</caption>]]
</figure>

It is well known that the morphology of ECG signals changes from beat to beat as
a consequence of physical activity, sensations, emotions, breathing, etc. of the
subject. The most straightforward measure of these changes is
the heart rate variability (HRV), i.e. small variations of beat duration. HRV
characterizes the timings of hearth cells repolarization and depolarization
processes. The HRV is typically determined by measuring the intervals between
two consecutive R-waves (RRI) or intervals between R and T waves (RTI). Several
vital signals can be identified from the HRV and therefore it is often used as
a health status indicator in different fields of medicine, e.g. neurology,
cardiac surgery, heart transplantation and many more. Typical HRV values of
healthy subjects are approximately $40$ ms for RRI and $2$ ms for RTI (see
<xr id="fig:beat"/>). Therefore it is important to detect considered waves with at least $1$
ms accuracy for practical use. This paper deals with the detection of HRV in
ECG signal provided by a Wearable ECG Device (WECGD) that is paired with a
personal digital assistant (PDA) via Bluetooth Smart protocol. The WECGD, due
to the hardware limitations, only measures the signal, while the PDA takes care
of data visualization, basic analysis and transmission of the data to a more
powerful server for further analyses. In contrast to a standard ambulatory
$12$-channel ECG measurement, where trained personnel prepare and supervise the
measurement of subject at rest, the WECGD works on a single channel, the
subject is active and since the WECGD is often placed by an untrained user its
orientation might be random, resulting in additional decrease of signal
quality. In order to maintain several days of battery autonomy The WECGD also
records the heart rate with significantly lower frequencies and resolution in
comparison to ambulatory measurements. All these factors render the standard
ECG analysis algorithms ineffective. In this paper we analyse a possible local,
i.e. only short history of measurement data is required, algorithm for detection
of heart rate variability with $1$ ms precision of a signal recorded with $120$ Hz.

=Detection method=

In order to evaluate the HRV, the ''characteristic point'' of each heart
beat has to be detected in the signal, which is provided as values of electric
potential sampled with a frequency $120$ Hz. Since the HRV is computed from
differences of consequent characteristic points the choice of the
characteristic point does not play any role, as long as it is the same in every
beat. In this work we choose to characterise the beat with a minimal first
derivative, in another words, we seek the points in the signal with the most
violent drop in electric potential (<xr id="fig:beat"/>) that occurs between R and S
peaks.

The detection method is separated in two stages, namely global and local. The
goal of the global method is to approximately detect the characteristic point,
while the local method serves as a fine precision detection, enabling us to
detect HRV with much higher accuracy.

==Coarse global search==
In the first step the algorithm finds a minimal first derivative of a given signal
on a sample rate accuracy, i.e. $\frac{1}{\nu}$. The global search method is next to
trivial. The algorithm simply travels along the signal, calculating the
discrete derivative and storing the position of minimal values found so far.
Since the points are sampled equidistantly, minimizing $\frac{\Delta y}{\Delta
t}$ is equal to minimizing $\Delta y$. The middle of the interval where the
largest drop was detected is taken as the global guess $t_G$. The results
of the global search are presented in <xr id="fig:global"/>.

<figure id="fig:global">
[[File:global.png|600px|thumb|upright=2|alt= Beat graph|<caption>Global search detection of two beats.</caption>]]
</figure>

==Fine local search==
The global search provides only coarse, limited to sample points, positions of
the characteristic points. To push accuracy beyond $1/\nu$, the signal has to be
represented also in between the sample points. A monomial approximation function
based on a Moving Least Squares approach is introduced for
that purpose.

The value of the electrical potential at arbitrary time $t_0$ is approximated.
Denote the vector of $n$ known values near $t_0$ by $\boldsymbol{f}$ (called
<i>support</i>), and the times at which they were measured by
$\boldsymbol{t}$. The approximation $\hat{f}$ of $\boldsymbol{f}$ is introduced as a linear
combination of $m$ in general arbitrary basis functions $(b_j)_{j=1}^m$,
however in this work only monomials are considered.
\[\hat{f} = \sum_{j=1}^m\alpha_jb_j \]

The most widely used approach to solve above problem and find the appropriate
$\hat{f}$ is to minimize the weighted 2-norm of the error, also known as the
Weighted Least Squares (WLS) method.
\[ \|\boldsymbol{f} - \hat{f}(\boldsymbol{t})\|_w^2 = \sum_{i=1}^n (f_i -\hat{f}(t_i))^2 w(t_i), \]
In the above equation $w$ is a nonnegative weight function.

The only unknown quantities are the $m$ coefficients $\boldsymbol{\alpha}$ of the linear
combination, which can be expressed as a solution of an overdetermined linear
system $W\!B\boldsymbol{\alpha} = W\!\boldsymbol{f}$, where $W$ is the $n\times n$ diagonal
weight matrix, $W_{ii} = \sqrt{w(t_i)}$ and $B$ is the $n\times m$ collocation
matrix, $B_{ij} = b_j(t_i)$. There are different approaches towards finding the
solution. The fastest and also the least stable and accurate is to solve the
Normal System $B^\mathsf{T} W^\mathsf{T} WB\boldsymbol{\alpha} = B^\mathsf{T} W^\mathsf{T} W\boldsymbol{f}$, a more
expensive but also more stable is via QR decomposition, and finally the most
expensive and also the most stable is via SVD
decomposition. The resulting vector $\boldsymbol{\alpha}$ is then
used to calculate $\hat{f}(t)$ for any given $t$. The derivatives are
approximated simply by differentiating the approximating function, $\hat{f}' =
\sum_{j=1}^m\alpha_jb_j'$.

The WLS approximation weights the influence of support points using the weight
function $w$. Usually, such a weight is chosen that points closest to $t_0$ are
more important in the norm in comparison to the nodes far away. Naturally such
approximation is valid only as long as the evaluation point is close to the
$t_0$.
A more general approach is a [[Moving Least Squares (MLS)|Moving Least Square (MLS)]] approximation, where
coefficients $\alpha$ are not spatially independent any more, but are recomputed
for each evaluation point. Naturally, such an approach is way more expensive,
but also more precise. A comparison of both methods, i.e. WLS and MLS, is shown
in <xr id="fig:mlswlsHeartvar"/>.

<figure id="fig:mlswlsHeartvar">
[[File:mlswlsHeartvar.png|600px|thumb|upright=2|alt= Beat graph|<caption>MLS and WLS approximation of a heartbeat-like function
$ f(x) = \frac{\sin x}{x}\frac{\left| x+8\right| - \left| x-5\right| +26
}{13 ((\frac{x-1}{7})^4+1)}+\frac{1}{10} $,
with measurements taken at points $ \{-14, \ldots, 24\} $.</caption>]]
</figure>

← Older revision		Revision as of 14:39, 24 February 2018
Line 1:		Line 1:
	We used MLS and WLS aproximation to extract heart rate variability from a wearable ECG sensor.		We used MLS and WLS aproximation to extract heart rate variability from a wearable ECG sensor.
−	~~[[:File:heartratevar.pdf\|Full paper available for download here.]]~~
−	~~[[:File:heartratevar_pres.pdf \| Presentation available for download here.]]~~

	The code can be found in <syntaxhighlight lang="bash" inline> /EKG/detect.cpp </syntaxhighlight>.		The code can be found in <syntaxhighlight lang="bash" inline> /EKG/detect.cpp </syntaxhighlight>.

@@ Line 307: / Line 307: @@
 and the method is unstable, which explains the loss of precision for orders larger than $12$.
 {| class="wikitable"
-|+ Results and errors of the RRI and HRV detection.
+|+ Table 1: Results and errors of the RRI and HRV detection.
 |-
 ! quantity [s]
@@ Line 326: / Line 326: @@
 |$e_M $||$0$||$0.007778$||$0.000829$
 |}

← Older revision		Revision as of 17:50, 7 November 2016
Line 285:		Line 285:
	The middle two coarse detection columns continue off the chart, but are not shown completely for clarity.</caption>]]		The middle two coarse detection columns continue off the chart, but are not shown completely for clarity.</caption>]]
	</figure>		</figure>
		+
		+	Results of RRI and HRV detection by global and local search are presented in
		+	Table 1. The generated times were taken as precise and the
		+	algorithm was run to produce global and local approximations. Then the average
		+	RRI time and HRV were calculated for each data set separately. The average RRI
		+	time is estimated very well with both methods, but the precision of the global
		+	method is not satisfactory when measuring heart rate variability. The precision
		+	is significantly improved using the local search. A chart showing the average
		+	error of the detected times is shown in <xr id="fig:allerrs"/>. It can be seen
		+	that MLS performs better on average, but WLS is very close and this loss of
		+	precision is a reasonable tradeoff.
		+
		+	<figure id="fig:allerrs">
		+	[[File:allerrs.png\|600px\|thumb\|center\|upright=2\|alt= Comparison of WLS and MLS errors\|<caption> Comparison of WLS and MLS errors using different orders and
		+	support sizes.</caption>]]
		+	</figure>
		+
		+	The red values in <xr id="fig:allerrs"/> indicate invalid region, having more basis functions than support points.
		+	Both MLS and WLS have the same region of validity, being precise in the predicted regime.
		+	For very high order approximation the condition number of the matrix becomes critical
		+	and the method is unstable, which explains the loss of precision for orders larger than $12$.
		+	{\| class="wikitable"
		+	\|+ Results and errors of the RRI and HRV detection.
		+	\|-
		+	! quantity [s]
		+	! generated
		+	! coarse
		+	! fine
		+	\|-
		+	\|$\bar{r} $\|\|$0.861136$\|\|$0.861139$\|\|$0.861136$
		+	\|-
		+	\|$e_{\bar{r}}$\|\|$0$\|\|$3.34 \cdot 10^{-6}$\|\|$2.83 \cdot 10^{-8}$
		+	\|-
		+	\|$h$\|\|$0.004102$\|\|$0.005324$\|\|$0.004137$
		+	\|-
		+	\|$e_h $\|\|$0$\|\|$0.001222$\|\|$3.52 \cdot 10^{-5}$
		+	\|-
		+	\|$e_a $\|\|$0$\|\|$0.002969$\|\|$0.000263$
		+	\|-
		+	\|$e_M $\|\|$0$\|\|$0.007778$\|\|$0.000829$
		+	\|}

@@ Line 140: / Line 140: @@
 approximation is valid only as long as the evaluation point is close to the
 $t_0$.
-A more general approach is a [[Moving Least Squares (MLS)|Moving Least Square (MLS)]] approximation, where
+A more general approach is a [[Weighted Least Squares (WLS)|Moving Least Square (MLS)]] approximation, where
 coefficients $\alpha$ are not spatially independent any more, but are recomputed
 for each evaluation point. Naturally, such an approach is way more expensive,

@@ Line 11: / Line 11: @@
 =Detection of heart rate variability<br>from a wearable differential ECG device=
-[mailto:jure.slak@student.fmf.uni-lj.si Jure Slak], [mailto:gregor.kosec@ijs.si Gregor Kosec],
+[mailto:jure.slak@ijs.si Jure Slak], [mailto:gregor.kosec@ijs.si Gregor Kosec],
 Jožef Stefan Insitute, Department of Communication Systems, Ljubljana.

@@ Line 1: / Line 1: @@
 We used MLS and WLS aproximation to extract heart rate variability from a wearable ECG sensor.
-[http://www-e6.ijs.si/ParallelAndDistributedSystems/MeshlessMachine/technical_docs/html/heartratevar.pdf Full paper available for download here.]
+[[:File:heartratevar.pdf|Full paper available for download here.]]
-[http://www-e6.ijs.si/ParallelAndDistributedSystems/MeshlessMachine/technical_docs/html/heartratevar_pres.pdf Presentation available for download here.]
+[[:File:heartratevar_pres.pdf | Presentation available for download here.]]
 The code can be found in <syntaxhighlight lang="bash" inline> /EKG/detect.cpp </syntaxhighlight>.
@@ Line 153: / Line 153: @@
 }{13 ((\frac{x-1}{7})^4+1)}+\frac{1}{10} $,
 with measurements taken at points  $ \{-14, \ldots,  24\} $.</caption>]]
 </figure>