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Should You Add Nonlinear HRV Metrics to Your Practice?

Updated: Apr 30

strange attractor

From Schrödinger's (1944) perspective, life is often aperiodic (e.g., oscillations occur without a fixed period) and oscillates between randomness and periodicity. Nonlinearity means a relationship between variables cannot be plotted as a straight line. Poincaré and detrended fluctuation analysis plots illustrate nonlinear HRV.

Nonlinear measurements allow us to quantify the unpredictability of a time series, which results from the complexity of the mechanisms that regulate heart rate variability. When the same processes generate them (e.g., respiratory sinus arrhythmia and the baroreceptor reflex), nonlinear indices correlate with specific frequency and time domain measurements. While stressors and disorders like diabetes can depress some nonlinear metrics, elevated values do not always signal health. For example, in post-myocardial infarction (post-MI) patients, increased nonlinear HRV is an independent risk factor for mortality (Stein et al., 2005).

The Case for Adding Nonlinear HRV Metrics

HRV time-domain (RMSSD) and frequency-domain (LF power) measurements are linear metrics. Linear analysis assumes that the system being studied responds in a way that is directly proportional to the input. In contrast, nonlinear analysis is used when the relationship between input and output is not proportional, often resulting in more complex behaviors that cannot be explained by linear models alone. Nonlinear heart rate variability (HRV) metrics have emerged as valuable tools for assessing various physiological and pathological conditions, and their integration into clinical practice could enhance patient care. Nonlinear HRV metrics capture the complexity and unpredictability of heart rate dynamics, which are not fully described by traditional linear metrics. These nonlinear properties may more accurately reflect the underlying health of the autonomic nervous and cardiovascular systems. There is no universally best linear HRV metric for diagnosing, prognosizing, and assessing a client's progress. Likewise, there is no universally best nonlinear HRV index. We should evaluate the clinical or performance utility of HRV metrics, application by application. We should avoid a false choice between linear and nonlinear indices. In the future, we may be able to combine them for greater predictive power.

Clinical Applications

Studies have shown that nonlinear HRV metrics can be highly informative in specific clinical settings. For instance, in cardiosurgical patients, nonlinear HRV metrics were among the most informative in predicting depressive symptoms, with a model accuracy of 86.75% for discriminating between depressed and non-depressed patients (Gentili et al., 2017a, 2017b). This suggests that nonlinear HRV metrics could play a significant role in the early diagnosis of depression in patients with coronary heart disease (CHD), who are at greater risk of adverse cardiac events when affected by postoperative depression.

In the context of workload assessment for emergency physicians, nonlinear HRV metrics, particularly Permutation entropy (PE), showed superior validity compared to linear metrics and mean heart rate, indicating that these measures could more accurately reflect the physiological demands placed on physicians during emergency care (Schneider et al., 2017).

Furthermore, nonlinear HRV metrics are reliable even when derived from short-term recordings, which is practical for clinical settings where long-term monitoring may not be feasible (Maestri et al., 2007; Perkiömäki et al., 2001). This reliability is crucial for ensuring the measurements can be trusted when making clinical decisions.

The prognostic value of nonlinear HRV metrics has also been demonstrated in populations with chronic congestive heart failure (CHF), where measures such as detrended fluctuation analysis (DFA) were predictive of survival, complementing traditional HRV measures (Ho et al., 1997). This highlights the potential of nonlinear HRV metrics to contribute to risk stratification and management strategies in CHF patients.

Moreover, nonlinear HRV measures are sensitive to autonomic changes during conditions such as central hypovolemia, although they may not correlate as strongly with sympathetic responses as linear HRV indices (Rickards et al., 2008). This suggests that combining linear and nonlinear HRV metrics could provide a more comprehensive assessment of autonomic function.

Performance Applications

In performance, the study of immediate and long-term effects of different exercise modalities on HRV has revealed that high-intensity interval training had a more pronounced impact on neurocardiac activity than moderate-intensity endurance training, as indicated by both linear and nonlinear HRV measures (Perkins et al., 2017). The utility of nonlinear dynamics to characterize fluctuations and complexity in HRV during acute recovery from exercise has also been piloted, indicating that fitness levels may influence cardiac autonomic nervous system modulation as observed through these nonlinear metrics (Berry et al., 2017).

Additionally, HRV-guided training, where daily exercise prescriptions are based on individual changes in HRV, has effectively improved cardiorespiratory fitness, demonstrating the practical application of HRV in individualizing endurance training (Kiviniemi et al., 2007).

Real-time estimation of aerobic threshold and exercise intensity distribution using fractal correlation properties of HRV has been explored in field applications, highlighting the potential of using these metrics for on-the-spot training adjustments without needing prior laboratory testing (Gronwald et al., 2021).

Lastly, HRV, including nonlinear dynamics, has been recommended for efficient exercise prescription and training control in the general population and high-performance athletes (Gallo-Villegas et al., 2020).

The Case for Caution in Adding Nonlinear HRV Metrics

Karemaker (2020) observed that HRV entropy measures have yielded few practical applications and are not consistently superior to time- and frequency-domain measurements:

Comparisons between more classical statistical- or frequency analysis derived- and entropy-derived measures do, however, not always favor the newer ones (Zhang et al., 2013). This may, partly, be due to the fact that in many applications, entropy-measurements require rather long recordings before reaching a stable value. In critical situations this is a serious drawback. However, when the requirement of more time and more data points it is not an issue, entropy analysis may be a viable option to dig deeper in the complexities of the heart rate signal at hand.

HRV Nonlinear Metrics

We will review Poincaré plots, S, SD1, SD2, SD1/SD2, Approximate Entropy (ApEn), Sample Entropy (SampEn), Multiscale Entropy (MSE), Detrended Fluctuation Analysis (DFA), α1 and α2, Correlation Dimension, and Permutation Entropy (PE).

Poincaré plot

A Poincaré plot (return map) is graphed by plotting every R-R interval against the prior interval, creating a scatter plot. Poincaré plot analysis allows researchers to visually search for patterns buried within a time series (a sequence of successive values). Poincaré plot analysis is insensitive to changes in trends in the R-R intervals (Behbahani et al., 2012).


S, SD1, SD2, and SD1/SD2

We can analyze a Poincaré plot by fitting an ellipse (curve that resembles a squashed circle) to the plotted points. After fitting the ellipse, we can derive three nonlinear measurements: S, SD1, and SD2.

Poincare plot

S measures the ellipse area (representing total HRV) and correlates with baroreceptor reflex sensitivity (BRS), LF and HF power, and RMSSD.

SD1, the standard deviation (hence SD) of the distance of each point from the y = x-axis, specifies the ellipse's width. SD1 measures short-term HRV in milliseconds and correlates with BRS and HF power. The RMSSD is identical to the nonlinear metric SD1, reflecting short-term HRV (Ciccone et al., 2017).

SD2, the standard deviation of each point from the y = x + average R-R interval, specifies the ellipse's length. SD2 measures short- and long-term HRV in milliseconds and correlates with LF power and BRS (Brennan et al., 2001; Brennan et al., 2002; Tulppo et al., 1996; Tulppo et al., 1998).

Kubios incorporates SD1 in its parasympathetic index and SD2 in its sympathetic index (Tarvainen & Niskanen, 2023). The data were recorded from a healthy young man lying supine for 5 minutes.


The ratio of SD1/SD2, which measures the unpredictability of the R-R time series, is used to measure autonomic balance when the monitoring period is sufficiently long and there is sympathetic activation. SD1/SD2 is correlated with the LF/HF ratio (Behbahani et al., 2012; Guzik et al., 2012).

Approximate Entropy

Approximate entropy (ApEn) measures the regularity and complexity of a time series. ApEn was designed for a brief time series in which some noise may be present and makes no assumptions regarding underlying system dynamics (Kuusela, 2013). Applied to HRV data, large ApEn values indicate low predictability of fluctuations in successive R-R intervals (Beckers, Ramaekers, & Aubert, 2001). Small ApEn values mean the signal is regular and predictable (Tarvainen & Niskanen, 2012).

Sample Entropy

Sample entropy (SampEn)was designed to provide a less biased and more reliable signal regularity and complexity measure than ApEn (Lippman et al., 1994). SampEn values are more stable than ApEn for the same participant over successive days (Richman & Moorman, 2000). SampEn values are interpreted and used like ApEn and may be calculated from a much shorter time series of fewer than 200 values (Kuusela, 2013).

Multiscale Entropy

Multiscale Entropy (MSE) extends entropy analysis using resampling to increase the time series' data points to reveal slower fluctuations (Costa et al., 2002, 2005).

Detrended Fluctuation Analysis (DFA)

Detrended fluctuation analysis (DFA) extracts the correlations between successive R-R intervals over different time scales. This analysis results in slope α1, which represents short-term fluctuations, and slope α2, which describes long-term fluctuations. The short-term correlations extracted using DFA reflect the baroreceptor reflex, while long-term correlations reflect the regulatory mechanisms limiting the beat cycle's fluctuation. DFA is designed to analyze a time series that spans several hours of data (Kuusela, 2013).


Correlation Dimension

The correlation dimension estimates the minimum number of variables required to construct a system dynamics model. The more variables needed to predict the time series, the greater its complexity.

The correlation dimension allows us to construct a system dynamics model using the fewest predictor variables.

An attractor, depicted below, is a set of values towards which a variable in a dynamic system converges over time. Graphic © zentilia/


The correlation dimension measures a system's attractor dimension, which can be an integer or fractal (Kuusela, 2013). Below is a correlation dimension plot.

correlation dimension

Permutation Entropy

Permutation entropy (PE) is a measure of complexity derived from the concept of time series analysis. It is particularly effective for evaluating heart rate variability (HRV). PE quantifies a time series' unpredictability by examining the order relations between its values rather than their numerical differences.

PE involves transforming the time series into a sequence of ordinal patterns. Each time series segment is ranked by its value, and these ranks form patterns. The relative frequencies of these ordinal patterns are then used to calculate the entropy, with more irregular and complex data leading to higher entropy values.

PE has proven useful in diagnosing cardiovascular diseases, as different pathological conditions reflect distinct complexities in heart rate dynamics. For example, it has been applied to detect changes in HRV associated with cardiovascular autonomic neuropathy in diabetes mellitus patients, showing significant differences in PE values between healthy individuals and those with the condition (Naranjo et al., 2017).

PE has also been utilized in sleep studies to differentiate between sleep stages based on HRV, demonstrating its capability to reflect autonomic nervous system changes during different sleep phases (Ravelo-García et al., 2015).

One of PE's limitations is its sensitivity to noise, which can lead to inaccurate entropy estimates. This is particularly challenging when analyzing short or noisy data series, where traditional methods may offer more robust results (Porta et al., 2015).

Standard PE assumes that all values in the series are unique, which is rarely the case in real-world data. Modified approaches, such as the one proposed by Bian et al. (2012), handle equal values more accurately by mapping them to the same symbol, thus retaining more of the series' intrinsic complexity.

In conclusion, permutation entropy is a robust tool for analyzing the complexity of heart rate variability and has diverse applications in biomedical research and diagnostics. However, its effectiveness can be affected by the quality of the data and the specific methodology employed.


Should you add nonlinear HRV metrics to your practice? You will need at least 5 minutes of baseline data. If your data acquisition software does not report nonlinear metrics, you must export the interbeat intervals into a program like Kubios, which performs detailed HRV artifacting and analysis.


Your answer may depend on your client populations and whether specific nonlinear metrics can aid assessment or track progress. Although the strongest case can be made in cardiovascular disorders and emergency medicine, nonlinear HRV metrics may, in the future, contribute to a more comprehensive autonomic assessment. Kubios' sympathetic and parasympathetic indices provide an example of this approach. For most providers, nonlinear metrics do not yet provide game-changing information missing from conventional time- and frequency-domain measurements. Investigators continue to discover which metrics are most suited to answering particular clinical and performance questions We encourage you to follow journals like Applied Psychophysiology and Biofeedback for future developments. Finally, we invite you to contribute to the HRV literature through case studies of the populations you train incorporating nonlinear metrics.



If nonlinear measurements of HRV are reduced in diseases like diabetes, is increased nonlinearity always desirable?

No. The answer depends on which processes generate the nonlinearity. While increased nonlinear HRV measurements could signal improved health in a diabetic client with better glucose control, it could signal a greater risk of death in a post-MI patient.


aperiodic: oscillations have no fixed period.

approximate entropy (ApEn): a nonlinear index of HRV that measures the regularity and complexity of a time series.

attractor: a set of values towards which a variable in a dynamic system converges over time.

attractor dimension: the number of variables required to model a dynamic system.

central hypovolemia: a condition with a decreased volume of blood circulating in the body's central blood vessels, including the arteries, veins, and capillaries that serve vital organs such as the heart and brain. This reduction in central blood volume is typically a result of fluid loss from the intravascular space (the space within blood vessels).

correlation dimension: nonlinear index of HRV that estimates the minimum number of variables required to construct a model of the studied system.

detrended fluctuation analysis (DFA): nonlinear index of HRV that extracts the correlations between successive R-R intervals over different time scales and yields estimates of short-term (α1) and long-term (α2) fluctuations.

ellipse: a curve that resembles a squashed circle, fitted to a Poincaré plot to calculate nonlinear HRV measurements SD1, SD2, SD1/SD2.

frequency-domain metrics: measurements that quantify the absolute or relative power distribution into four frequency bands.

interbeat interval: the time interval between the peaks of successive R-spikes (initial upward deflections in the QRS complex). An IBI is also called the NN (normal-to-normal) interval after removing artifacts.

linear analysis: calculations that assume that the system being studied responds in a way that is directly proportional to the input.

multiscale entropy (MSE): an extension of ApEn that uses resampling to increase the number of data points in a time series to detect slower fluctuations.

nonlinear measurements: indices that quantify the unpredictability of a time series, which results from the complexity of the mechanisms that regulate the measured variable.

nonlinear analysis: calculations that assume that the relationship between input and output is not proportional.

nonlinearity: a relationship between variables that cannot be plotted as a straight line.

permutation entropy (PE): a measure of complexity derived from the concept of time series analysis, particularly effective for evaluating heart rate variability (HRV).

Poincaré plot analysis (PPA): a visual display that plots every R-R interval against the prior interval, creating a scatterplot to identify patterns buried within a time series.

R-R interval: the time between consecutive heartbeats, also called the interbeat interval (IBI), measured in milliseconds.

S: a nonlinear HRV index that measures the ellipse area, which represents total HRV.

sample entropy (SampEn): a nonlinear HRV index designed to provide a less biased signal regularity and complexity measure than ApEn.

SD1: the standard deviation of the distance of each point from the y = x-axis that measures short-term HRV.

SD2: the standard deviation of each point from the y = x + average RR interval that measures short- and long-term HRV.

SD1/SD2: a ratio that measures the unpredictability of the R-R time series and autonomic balance under appropriate monitoring conditions.

time-domain metrics: measurements, like the RMSSD, that quantify the variability in interbeat interval measurements (IBI).

time series: a sequence of successive values. For example, a 24-hour record of R-R intervals.


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