ECG Heart Rate Variability (HRV)

Reference: Chapter 8 of Bioelectrical Signal Processing in Cardiac and Neurological Applications (Biomedical Engineering) by Leif Sörnmo and Pablo Laguna

1. Overview

Heart rate variability (HRV) is a hot topic in medical research, given its noninvasive nature and the rich insights it offers into the autonomic nervous system and related diseases. The main idea is to examine the time intervals between successive heartbeats (RR intervals) and use those for various types of analyses. Let's break down the key points.

1.1 Key Concepts

RR Intervals: These are the time intervals between successive heartbeats. This series forms the raw material for HRV analysis.
Instantaneous Heart Rate: This is another way to represent heart rhythm, and it's derived from RR intervals.
Spectral Analysis: This technique identifies oscillations in heart rhythm, often due to respiratory activity or blood pressure fluctuations. Peaks in the estimated power spectrum can quantify these.
Low-frequency Components: These are typically below 0.5 Hz and are important for HRV. They aren't affected by baseline wander removal techniques.

1.2 Important Sections of HRV Analysis

Data Acquisition: The quality and type of data you collect will significantly impact the HRV analysis. This is often the first step.
RR Interval Conditioning: Raw RR intervals often require some conditioning or preprocessing to be useful for analysis.
Time Domain Measures: These are basic metrics like mean and standard deviation of RR intervals, providing a simple but useful analysis.
Advanced HRV Measures: These are derived from signals representing heart rhythm and quantify the correlations between different RR intervals.
Spectral Analysis Techniques: Here, you'll find more advanced ways to analyze heart rhythm, taking into account that it's derived from unevenly sampled signals.
Ectopic Beat Correction: Sometimes heartbeats are irregular (ectopic beats). These need to be identified and corrected for in the RR interval series.
Physiological Interactions: This section explores how heart rate interacts with other physiological signals and how these interactions can be modeled mathematically.

1.3 Points to Remember

Baseline Wander: Removing this from ECG signals doesn't affect HRV because it is more about the morphology of the ECG, while HRV focuses on the timing of heartbeats.
Physiological Background: For a deeper understanding of why HRV is important, a review of the physiological context is advised (found on page 431).

2. Acquisition and RR Interval Conditioning

2.1 Fiducial Point and RR Intervals

HRV analysis usually starts with the series of occurrence times produced by a QRS detector. While it might be more accurate to use the P wave as the fiducial point of the heartbeat, it's often impractical due to the P wave's low amplitude or even its absence in some cases. Thus, RR intervals—time between successive QRS complexes—are commonly used for HRV analysis. This is based on the assumption that PR intervals are relatively fixed, making RR intervals a good reflection of sinoatrial node activity.

2.2 Data Acquisition Requirements

When acquiring ECG data, the sampling rate is crucial. For resting ECG analysis, a sampling rate between 250-500 Hz is generally sufficient. However, some clinical HRV studies use Holter recordings with lower sampling rates (100-125 Hz), which can distort the high-frequency components in spectral analysis.

2.3 Artifact Handling and Conditioning

General Artifacts

Holter recordings are particularly prone to artifacts like noise bursts, which can cause both false positives and false negatives in QRS detection. To make HRV analysis more reliable, it's important to exclude these "non-normal" RR intervals. This set of intervals is known as normal-to-normal intervals (NN intervals).

Automated Exclusion

Manual editing of a long Holter recording is time-consuming, so automated procedures have been developed. One common approach is to consider an RR interval as abnormal if it deviates more than 20% from the mean of preceding intervals. There are other, more complex methods, but they don't necessarily yield better results.

Ectopic Beats

Ectopic beats, or abnormal heartbeats, are another type of artifact that can affect HRV. They require special handling, including corrections to both the preceding and subsequent RR intervals.

2.4 Sensitivity to Artifacts

Different HRV metrics have varying sensitivities to artifacts. Simple time domain measures are less sensitive compared to those derived from power spectral analysis. Sometimes, the only feasible approach for spectral analysis might be to use artifact-free, minute-length ECG segments.

3. Time Domain Measures

Types of Measures and Their Importance

Time domain measures in HRV analysis are often based on RR intervals and can be divided into long-term and short-term measures, reflecting primarily sympathetic and parasympathetic activity respectively. This becomes critical when analyzing Holter recordings, which may require segment-wise analysis.

HRV can have large variability throughout the day, so it's important to consider the time of day when analyzing long-term measures.

RR_intervals_day_night

Basic Measures

SDNN

The standard deviation of all NN intervals (SDNN) gives a basic measure of HRV. However, it's a rough estimate for long-term recordings, as heart rate changes significantly during different parts of the day.

SDANN

The standard deviation of the average NN intervals in 5-minute segments (SDANN) offers another long-term measure and mainly captures circadian rhythms.

Short-term Measures

rMSSD

Root mean-square of successive differences between adjacent NN intervals (rMSSD) focuses on short-term variability. It is sensitive to high-frequency fluctuations in the heart rate but also more vulnerable to artifacts.

pNN50

The percentage of adjacent NN intervals that differ by more than 50 ms (pNN50) is another measure of short-term variability. It's less sensitive to artifacts than rMSSD.

Special Cases

Time domain measures that focus on long-term variability have proven useful in specific patient groups, such as alcoholics and diabetics with neuropathy, for detecting autonomic dysfunction. They're also good for predicting mortality rates in post-myocardial infarction patients.

Geometric Measures

TINN

Another angle comes from geometric measures like the Triangular Interpolation Index (TINN). Here, a triangle is fitted to the RR interval histogram. The base width of the triangle serves as a robust measure of HRV.

Summary of Time Domain Measures

Measure	Definition
SDNN	Standard deviation of all NN intervals
SDANN	Standard deviation of the average NN intervals in all 5-minute segments of the entire ECG recording.
rMSSD	Root mean-square of successive differences of adjacent NN intervals.
pNN50	Percentage of pairs of adjacent NN intervals differing by more than 50 ms.
TINN	Triangular interpolation index. The base of a triangle fitted to the RR interval histogram (see text and Figure 8.2).

Robustness and Limitations

Histogram-based methods like TINN are robust against artifacts and ectopic beats but are best suited for 24-hour Holter recordings. They can sometimes misrepresent variability in cases where the RR interval histogram is bimodal or multimodal.

Beyond Normal Rhythms

Histogram methods aren't limited to normal rhythms; they've been applied to study arrhythmias like atrial fibrillation.

RR_interval_histogram

(a) There are two peaks in the histogram. The left corresponds to T waves and the right to QRS complexes.
(b) Since T waves are not of interest in HRV analysis, it is removed from the histogram based on TINN.

Summary

Time domain measures offer a wide range of options for HRV analysis, each with its own advantages and limitations. While basic measures like SDNN and SDANN are straightforward but may lack nuance, more specialized methods like rMSSD, pNN50, and TINN offer deeper insights but come with their own sets of challenges, particularly regarding data quality and artifact handling.

4. Heart Rhythm Representations

4.1 Introduction

Objective and Scope

The aim of heart rhythm representation is to craft a signal that accurately mirrors heart rhythm variability (HRV) and is amenable to various analytical techniques. This representation can either be interval-based or rate-based, the latter being defined by the inverse of RR intervals.

Foundations

The series of QRS occurrence times provides the basis for constructing the heart rhythm signal. These times are unevenly spaced, thus creating a need for regularizing the sampling rate to align with techniques that require an evenly spaced signal. It's vital not to mix this with the ECG digitization rate, which is often much higher (500-1000 Hz) compared to the rate required for HRV analysis (a few Hz).

Regularization of Sampling Rate

The ECG's high sampling rate (500-1000 Hz) captures the nuances of the QRS complex, while for HRV analysis, a much lower rate suffices. This lower rate is not just adequate for HRV study but also beneficial in reducing data volume.

Ambiguity in Performance Metrics

Determining how well a heart rhythm representation captures HRV is not straightforward. However, mathematical models like the Integral Pulse Frequency Modulation (IPFM) model have proven useful for this.

Conceptual Importance

Differentiating between series based on RR intervals and those based on QRS occurrence times is essential for conceptual clarity. Though they are interconvertible, they serve different analytical needs and carry different information.

4.2 The Integral Pulse Frequency Modulation Model (IPFM)

Conceptual Overview

The IPFM model is a key framework for understanding heart rate variability (HRV). It helps simulate an event series like the timings of heartbeats and ties this to a continuous-time input signal, which usually has physiological significance.

Working Principle

In the IPFM model, an input signal is integrated until it reaches a threshold $R$ . At this point, an event is generated, and the integrator is reset to zero. The input signal is the sum of a DC level $m0$ and a time-dependent modulating function $m(t)$ . This modulating function is crucial: it represents physiological changes affecting heart rate.

Physiological Interpretation

From a biological standpoint, the integrator's output can be seen as the membrane potential of a sinoatrial pacemaker cell. When the threshold is reached, it initiates a new heartbeat. Here, $m0$ defines the mean heart rate, and $m(t)$ is modulated by the autonomic activity on the sinoatrial node.

Mathematical Description

The IPFM model can be mathematically described by a set of equations that help understand the variability of heart rate intervals. In simpler terms, $m(t)$ dictates how much the length between two events changes, and $R$ becomes a measure for the mean repetition frequency ( $FI$ ) in hertz.

Generalization

The model can be extended to a continuous-time function, allowing it to represent heart rhythms more broadly. This leads to the development of something known as the heart timing signal.

Performance Evaluation

When it comes to analyzing HRV, $m(t)$ is generally assumed to be a sum of $P$ sinusoids. This is a practical model to simulate HRV caused by factors like respiration and blood pressure variations.

Limitations and Applications

While the IPFM model is widely used for simulation studies and gaining insights into HRV mechanisms, it's not an exact representation of sinoatrial activity. More advanced models may provide a closer approximation. Additionally, its utility isn't confined to HRV; it's also useful in other biomedical contexts like neurophysiology.

4.3 Interval Series Representions

The IPFM model aims to simulate event series like heartbeats. The input is a signal that gets integrated over time. When the integral reaches a threshold $R$ , an "event" (like a heartbeat) is generated. The goal is to infer the modulating function $m(t)$ which describes how the event series deviates from a regular pattern.

Physiologically, this can model how a pacemaker cell in the heart charges up until it reaches a threshold, triggering a heartbeat.
Mathematically, it gives you a framework for understanding the variations between successive events.

Interval Series Representations

You're looking at various ways to represent heart rhythm:

Interval Tachogram ( $diT(k)$ ): It represents the time between heartbeats.
Inverse Interval Tachogram ( $diiT(k)$ ): This is more of an instantaneous heart rate measure.

Both $diT(k)$ and $diiT(k)$ are problematic for spectral analysis as they aren't evenly sampled in time.

Interval Function ( $diF(t)$ ) and its Inverse: These are continuous-time versions of the tachogram and its inverse, respectively.

Interpolation and Resampling

The text also talks about resampling these signals to get evenly spaced data points, which is essential for any frequency domain analysis. Zero-order (or sample-and-hold) interpolation is the simplest but introduces high-frequency components.

Key Points

Estimation of $m(t)$ : The end goal is often to understand this modulation function, as it could carry significant information about physiological states.
Limitations and Extensions: The IPFM model is useful but not the end-all-be-all; more advanced models exist for capturing the nuances in the data.

Applications

Heart Rate Variability (HRV) studies, which have implications for understanding stress, health conditions, etc.
Cross-correlation with other physiological signals for more comprehensive health analysis.

4.4 Event Series Representation

Overview:

Traditional ways of representing heart rhythm, like tachograms, have limitations such as not being evenly sampled in time. Event series representation is a newer approach that seeks to improve this.

Key Concepts:

Redundancy in Traditional Methods: Tachograms are limited because they are closely tied to the actual occurrence times of heartbeats and the intervals between them. This is redundant and can make further analysis tricky.
New Perspective with Event Series: Unlike tachograms, the event series representation takes its cues from the heartbeats themselves, making it easier to analyze the low-frequency components that truly characterize heart rate variability.
Lowpass Filtering for Improved Representation: To isolate the useful low-frequency components, the event series data is filtered using a linear, time-invariant lowpass filter. The cut-off frequency is generally well below the average heart rate to preserve important variations.
Discrete-Time Version and Digital Techniques: In practice, the event series data is digitized for computational ease. Special algorithms can be used to improve efficiency and reduce aliasing distortion, which is a common problem in digital signal processing.

Challenges and Solutions:

Non-causal Computation: The lowpass filtered event series is non-causal. This isn’t a big issue for offline ECG signal analysis, which is most often the case.
Avoiding Aliasing: When converting from continuous to discrete-time, care is taken to avoid aliasing by choosing an appropriate sampling rate. Several algorithmic solutions exist to mitigate this problem.
Improved Filtering Techniques: More advanced methods, like windowing techniques, can be employed to improve the filter design and avoid issues like poor stopband attenuation.

4.5 Heart Timing Representation

Overview:

While previous methods focus on heart rhythm, the Heart Timing Representation targets the underlying modulating function that controls heartbeat intervals. This approach is rooted in the IPFM (Integral Pulse Frequency Modulation) model and uses an unevenly sampled signal, denoted as $d^u_{HT}(t)$ , to achieve this aim.

Key Concepts:

Connection to IPFM Model: The Heart Timing Signal is explicitly based on the IPFM model and aims to estimate the modulating function $m(t)$ . This modulating function directly influences the intervals between heartbeats.
Calculating $d^u_{HT}(t)$ : The signal $d^u_{HT}(t)$ is defined as the deviation between the actual and expected heartbeat times, normalized by the average heartbeat interval. The expected times are calculated from the IPFM model.
Power Spectrum Estimate: By employing Fourier Transform techniques on $d^u_{HT}(t)$ , the power spectrum of $m(t)$ can be estimated. This is helpful for understanding the modulating function in the frequency domain.
Bandlimit Considerations: Both $m(t)$ and $d^u_{HT}(t)$ are assumed to be band-limited, meaning they don't contain frequencies higher than a specified cutoff. This makes it easier to fully retrieve $d^u_{HT}(t)$ from the time instants.

eart_rhythm_representations

Strengths and Weaknesses:

Superior Performance for Modulating Function: When the goal is to understand $m(t)$ , $d^u_{HT}(t)$ offers advantages over other heart rhythm representations. However, this superior performance is within the framework of IPFM, which may not fully capture human heart rate variability (HRV).
Clinical Validity: While $d^u_{HT}(t)$ works well in a model-based scenario, its effectiveness in real-world clinical settings is yet to be demonstrated. The model might have inherent limitations in capturing the complexities of HRV in humans.

5. Spectral Analysis of Heart Rate Variability Simplified

5.1 Overview

What's Happening in the Heart?

Healthy hearts show variability in the heart rate that tends to oscillate. Different bodily systems like temperature regulation, blood pressure control, and respiration contribute to this variability. These systems have their characteristic frequency ranges:

Temperature: Below 0.05 Hz
Blood Pressure: Around 0.1 Hz
Respiration: 0.2 to 0.4 Hz

ECG_power_spectrum

(a) resting position
(b) 90-degree head-up tilt

Why It Matters?

Understanding these oscillations can help identify heart-related pathologies. However, it's not always easy to pinpoint these oscillations, especially those related to temperature and blood pressure.

Autonomic Balance

Instead, researchers often measure the power of low- and high-frequency components. The balance between these frequencies can reveal the state of the autonomic nervous system:

Low-Frequency Power: Increased Sympathetic Activity
High-Frequency Power: Increased Parasympathetic Activity

The ratio between these powers is a common index for evaluating autonomic balance in clinical studies.

Stationarity and Spectral Analysis

Stationarity, meaning the signal's statistical properties don't change over time, is a key consideration. Practically, analysts look for abnormal heartbeats, as they can skew the analysis.

Sampling Constraints

It's important to note that you can't reliably analyze frequencies higher than half the mean heart rate. Going beyond this can result in misleading information.

Methodology and Tools

There are various approaches to the spectral analysis:

Classical methods: Such as the periodogram
Model-based methods: Like Autoregressive (AR) modeling

Both approaches are well-studied and straightforward to apply.

interrelationships_heart_rhythm_representations

The above figure illustrates how different heart rhythm analysis methods are connected, focusing on how they handle signal samples. The classical periodogram uses evenly spaced samples, while the other methods work with unevenly spaced samples. Both lowpass filtering and interpolation are combined into one step because interpolation acts like a type of lowpass filtering.

Interpolation and Resampling

For unevenly sampled signals, you can use methods designed for such data, like Lomb's periodogram. Alternatively, you can interpolate and resample the data to make it evenly spaced before applying classical methods.

5.2 Direct Estimation from Unevenly Spaced Samples

Introduction

When working with heart rate variability (HRV), you'll often encounter unevenly sampled data. It's important to know how to accurately perform spectral analysis on such data sets.

The Fourier Transform with Uneven Sampling

We usually employ the Fourier Transform to get a frequency representation of a signal. But what happens when the data is unevenly spaced in time? Technically, you'd represent the unevenly sampled signal as a combination of the sampling function and the actual continuous-time heart rate signal.

The sampling function itself is a set of impulse functions at the times where the event (a heart beat, in this case) occurs. This leads to a Fourier Transform that's influenced by both the sampling and the actual signal. In simpler terms, the frequency spectrum you get isn't just a clean representation of the heart rate but is somewhat convoluted by how the heart rate was sampled.

Working with a Finite Set of Events

When dealing with real-world data, you have a limited set of heart beats, which you can think of as a "window" into the continuous-time signal. This windowing itself affects the spectral representation.

Non-Periodic Fourier Transform

It's important to note that the Fourier Transform derived from unevenly spaced samples is not periodic. This is unlike what you'd expect from evenly spaced, discrete-time signals. Nonetheless, you're still constrained by the same Nyquist limit, where frequencies of interest should be less than half the mean heart rate.

Resampling: A Special Case

In some instances, you might want to convert your uneven samples into a set of evenly spaced samples through interpolation. Once you do that, the convolution we talked about earlier simplifies. The resulting spectrum would then look more like what you'd expect from an evenly sampled signal.

Using well-known sampling theory, you'd find that if you've bandlimited your signal correctly, your resampling won't introduce aliasing or distortions.

Final Thoughts

Analyzing unevenly spaced HRV data is more complicated than dealing with evenly spaced samples. The convolution effects and non-periodic nature of the Fourier Transform should be considered for accurate interpretation. However, resampling can simplify this, as long as you heed the standard rules from sampling theory.

5.3 The Spectrum of Counts

Introduction

In the study of heart rate variability (HRV), the "spectrum of counts" holds a special place. This refers to the Fourier Transform of the event series, which in this case is made up of heart beats. Understanding this spectrum can offer significant insights into the characteristics of HRV.

What is the Spectrum of Counts?

The spectrum of counts is essentially the Fourier Transform of the event series (the times when heartbeats occur). This spectrum is crucial because it is one part of what makes up the overall spectrum of the heart rate signal when it is unevenly sampled. In simpler terms, it’s one of the elements that modify the spectrum of the original, continuous-time HRV signal.

Deviations from Mean Heart Rate

An analytical expression of the spectrum of counts becomes clearer when the event times are expressed as deviations from the mean heart rate. A mathematical transformation allows these deviations to be represented in terms of a modulating function. This modulating function describes the variability in the inter-beat intervals and is part of the Integrated Pulse Frequency Modulation (IPFM) model of HRV.

Reducing Complexity

Complex mathematical manipulations allow us to express the spectrum of counts in a more straightforward form. The result reveals a relationship between the spectrum of counts and a frequency-modulated function. This simplification assumes that the variability in heart rate is small compared to the mean heart rate, which is usually the case in HRV studies.

Unwanted Terms in the Spectrum

One of the challenges is that the spectrum of counts includes some unwanted terms. These can distort the spectrum and lead to misleading interpretations. For example, there is a DC component that can leak into adjacent low-frequency areas of the spectrum. Unlike in other scenarios, you can’t simply subtract this DC component when you’re dealing with an event series.

Special Case: Sine Modulation

When the modulating function is a sine wave, the spectrum of counts can be determined analytically. This spectrum will include some additional frequency peaks that are multiples of the mean heart rate. The amplitude of these extra peaks will vary depending on how much the heart rate is modulated.

Convolution Effects Revisited

As mentioned before, the overall spectrum of the unevenly sampled HRV signal is a convolution of the spectrum of counts and the original spectrum. Understanding the spectrum of counts helps clarify why the observed spectrum from unevenly sampled data might deviate from what you'd expect from a clean, continuous-time signal.

5.4 Lomb's Periodogram

Lomb's periodogram is a valuable tool for analyzing signals that are unevenly sampled. It provides an alternative to traditional periodograms that often require the data to be interpolated and resampled. Both types of periodograms aim to measure how much of a certain frequency is present in a signal, but Lomb's approach does so without making assumptions about how the signal was generated.

Conceptual Foundation

The core idea behind Lomb's method is to find a best-fitting sinusoidal model for the given signal. This is done by minimizing the squared error between the observed signal and the sinusoidal model. The resulting value, referred to as the energy, quantifies how much of the specified frequency is present in the data.

To find the energy, Lomb's approach solves a least-squares problem. This involves mathematical optimization, where the goal is to find the model parameters that minimize the error between the model and the observed data. In mathematical terms, this can be formulated as a matrix equation, but the key takeaway is that this optimization leads to an energy value for each frequency of interest.

Improvements and Advantages

Lomb's original paper proposed a clever tweak: introducing a delay parameter in the sinusoidal model. This adjustment simplifies the calculations and makes Lomb's periodogram translation-invariant. In simpler terms, this means the periodogram's outcome remains consistent regardless of the time positioning of the observed samples.

Moreover, Lomb's method is not restricted by the Nyquist frequency, which is the upper limit for traditional methods. However, one must be cautious when analyzing frequencies above this limit, as it could lead to misleading results due to aliasing.

Computational Aspects

Because Lomb's periodogram involves quite a bit of computation, fast algorithms similar to the Fast Fourier Transform have been developed. It is also versatile enough to revert to a classical periodogram when the signal is evenly sampled, making it a flexible tool for various kinds of data.

Practical Considerations

In terms of performance, Lomb's periodogram is comparable to classical periodograms. The choice between the two often comes down to specific needs, such as computational resources and how the data was sampled. For example, if your signal has gaps or irregular intervals, Lomb's method might be more suitable.

Examples and Applications

Various studies have shown that Lomb's periodogram performs well in spectral analysis across different domains. For example, in the context of heart rate variability, Lomb's method provides insights that are largely consistent with those from traditional methods. This makes it a reliable option for spectral analysis in various fields.

6. Clustering of Heartbeat Morphologies for HRV Analysis

6.1 Overview

When analyzing Heart Rate Variability (HRV), it's essential first to label sinus beats—these are the heartbeats that are key to understanding the sinus rhythm. This labeling is done through a clustering method similar to the one used for classifying Motor Unit Action Potentials (MUAPs) in intramuscular EMG signal analysis. However, unlike in MUAP clustering, where every cluster holds equal weight, in HRV only the sinus beat cluster is of primary concern. Even though we don't have prior knowledge on which cluster holds the sinus beats, it can often be identified as the one with the most beats.

Features and Measures for Clustering

Clustering relies on features that describe the shape and possibly the rhythm of the heartbeats. The time-domain representation is often used in ECG signal processing, and it's particularly effective when paired with a cross-correlation coefficient for measuring how similar patterns are.

Before clustering, the ECG signal is usually bandpass filtered to minimize the impact of baseline wander and EMG noise. Using cross-correlation as a similarity measure ensures that the clustering is insensitive to variations in the amplitude of the QRS complex in the heartbeats. This is useful in HRV analysis, where we're mainly interested in the timing of the sinus beats and not their amplitude.

However, if the goal is to reduce noise by averaging the sinus beats in a cluster, this amplitude insensitivity can be a downside. If heartbeats in the cluster have widely varying QRS amplitudes, the average can be misleading.

Alternative Measures and Improved Accuracy

Another approach for feature extraction in heartbeat clustering is to use basis function representation. In this method, metrics like the Mahalanobis distance are commonly used for measuring pattern similarity.

Improving the timing accuracy of each heartbeat ('tk') is closely tied to the clustering process. Once you have a well-defined cluster, you can align each heartbeat more precisely in time. This is particularly relevant when using cross-correlation for similarity measurement, as the heartbeats are shifted in time to maximize the cross-correlation value.

Ultimately, after the clustering is done, you can fine-tune the timing of the beats in the sinus cluster using advanced methods like Woody's method.

6.2 Correlation-based Correction

When conducting a spectral analysis of heart rate signals through a nonparametric approach, it's often crucial to account for irregular or ectopic beats. In this context, we use an adjusted version of the interval tachogram, denoted as $diT(k)$ , to improve the reliability of our spectral analysis.

Underlying Rhythms and Missing Data

The interval tachogram primarily captures the sinus rhythm, denoted as $dsR(k)$ . However, during intervals where an ectopic beat occurs, this value is considered missing or undefined. To proceed with the analysis, we assume that the occurrence of ectopic beats and the sinus rhythm are statistically independent. This assumption allows us to isolate the correlation function of the sinus rhythm from that of ectopic beats.

Modified Correlation Estimates

Given this assumption of independence, we can derive a new estimator for the correlation function of the desired sinus rhythm. To avoid division by zero or other computational errors, certain conditions must be met, particularly when calculating the correlation estimate of the binary variables that denote the presence or absence of ectopic beats.

Special Considerations for Lag

In cases where the correlation estimate for ectopic beats approaches zero—commonly in instances of large time lags—the computation can become problematic. Here, a truncated version of the correlation function might be more appropriate for periodogram computation. This is generally not an issue for short time lags and is particularly relevant when analyzing ECGs that only occasionally include ectopic beats.

6.3 Interpolation-based Correction

An alternative strategy for dealing with ectopic beats in heart rate data is to use interpolation. This method focuses on the interval function, $d_{IF}(tk)$ , or its inverse, $d_{IIF}(tk)$ , to fill the data gap caused by an ectopic beat.

Interpolation Parameters

When an ectopic beat occurs in the interval between $tk_-$ and $tk_+$ , interpolation must rely on the samples adjacent to these times, bypassing the interval that contains the ectopic beat itself. Essentially, you'd use the data from $d_{IF}(tk_-)$ and onwards from $d_{IF}(tk_+2)$ to estimate what the missing data would likely have been if the ectopic beat had not occurred.

Linear and Polynomial Interpolation

Linear interpolation is the simplest way to fill this gap, requiring only two adjacent samples. However, for more robust results, higher-order polynomial interpolation can be used. This more advanced method would take additional samples from both sides of the ectopic beat into account, providing a potentially more accurate estimation.

Integration into Analysis Pipeline

Once the new interpolated data points are generated, they are integrated back into the original interval function. This corrected signal is then ready for further analysis, offering a more complete and less distorted view of the heart rate data.

6.4 The Heart Timing Signal and Ectopic Beats

When studying heart timing signals, it's crucial to account for the disturbances introduced by ectopic beats, unpredictable events that can skew your analysis. This section delves into a method that modifies the original definition of the heart timing signal to accommodate the occurrence of such beats.

Conceptual Foundation: Parameter $s$

In the modified approach, a parameter $s$ is introduced. This parameter captures the "jump" in the signal that happens when an ectopic beat occurs and resets the natural rhythm at the sinoatrial (SA) node. A value of $s$ close to zero implies that the event is likely an artifact, whereas a value close to one suggests a true ectopic beat followed by a compensatory pause.

Two-Pronged Estimation: Forward and Backward

The key challenge is to estimate this parameter $s$ accurately, as well as to adapt our previous estimation methods for heart timing signals. Two estimators are employed: a "forward estimator," based on data preceding the ectopic beat, and a "backward estimator," based on data following it. These estimators provide two different but related trajectories of the heart timing signal, offset by the parameter $s$ .

Overlapping to Estimate $s$

To estimate $s$ , we first extend these two trajectories until they overlap. This involves assuming that the natural (sinus) rhythm both precedes and follows the period disrupted by the ectopic beat. Once overlapping segments are obtained, the difference between the two trajectories gives us the desired parameter $s$ .

Error Minimization

The next step is to find the value of $s$ that minimizes the error between the two estimated trajectories. In technical terms, this is done through a least squares (LS) criterion. Practically speaking, this criterion measures how well the forward and backward estimators agree when they overlap.

Refining RR Interval Estimation

Upon determining $s$ , the average RR interval length can be recalibrated to account for the perturbation introduced by the ectopic beat. This adjusted signal can then be interpolated and resampled for more refined heart rate variability (HRV) analysis.

Practical Implications

It's worth noting that this technique avoids some pitfalls of interpolation-based methods, particularly the artificial increase in low-frequency components. The method becomes increasingly valuable as the frequency of ectopic events rises, offering a robust way to maintain the integrity of heart timing signal analysis.

7. Interaction with Other Physiological Signals

The variation in heart rate is not an isolated phenomenon; it is influenced by other physiological signals like respiration and blood pressure. Understanding these relationships is key to gaining deeper insights into the cardiovascular system.

Closed-Loop Identification Problem

To study these interactions, we can use multivariate signal models that consider multiple physiological variables at once. This approach is termed a "closed-loop identification problem" because the system is identified based on real-time measurements of these physiological signals, rather than in isolation.

Role of Baroreceptor Reflex

The baroreceptor reflex plays a vital role in this interaction by regulating blood pressure. These specialized nerve cells sense blood pressure changes and adjust the heart rate accordingly through a feedback loop. So, it's not surprising that the variations in heart rate and blood pressure are closely linked.

Analyzing Interaction Dynamics

By simultaneously measuring heart rate and blood pressure, we can delve into the dynamics of these interactions. This measurement is crucial for understanding mechanisms like the baroreceptor reflex. Interpolation and resampling techniques may be necessary to align these unevenly sampled signals for further analysis.

Cross-Power Spectrum

Before we delve into modeling, let's introduce the concept of the cross-power spectrum. This is a tool to gauge the correlation between two signals, like heart rate and blood pressure, across different frequencies. It is often normalized to produce what's called the magnitude squared coherence, which indicates how strongly the signals are correlated at specific frequencies.

Autoregressive Modeling

Autoregressive (AR) models can be particularly useful for studying these interactions. Specifically, a bivariate AR model has been found effective in capturing how heart rate and blood pressure oscillate around their mean values. These models account for the feedback mechanisms between the two variables.

Expanding to Multiple Signals

This approach can also be extended to include more physiological signals, such as respiration. However, not all interactions are meaningful. For instance, while respiration affects both heart rate and blood pressure, the reverse is not significantly true.