Recurrence Quantitative Analysis of Wavelet-Based Surrogate Data for Nonlinearity Testing in Heart Rate Variability

Abstract

Exploring the presence of nonlinearity through surrogate data testing provides insights into the nature of physical and biological systems like those obtained from heart rate variability (HRV). Short-term HRV time series are of great clinical interest to study autonomic impairments manifested in chronic diseases such as the end stage renal disease (ESRD) and the response of patients to treatment with hemodialysis (HD). In contrast to Iterative Amplitude Adjusted Fourier Transform (IAAFT), the Pinned Wavelet Iterative Amplitude Adjusted Fourier Transform (PWIAAFT) surrogates preserve nonstationary behavior in time series, a common characteristic of HRV. We aimed to test synthetic data and HRV time series for the existence of nonlinearity. Recurrence Quantitative Analysis (RQA) indices were used as discriminative statistics in IAAFT and PWIAAFT surrogates of linear stationary and nonstationary processes. HRV time series of healthy subjects and 29 ESRD patients before and after HD were tested in this setting during an active standing test. Contrary to PWIAAFT, linear nonstationary time series may be erroneously regarded as nonlinear according to the IAAFT surrogates. Here, a lower proportion of HRV time series was classified as nonlinear with PWIAAFT, compared to IAAFT, confirming that the nonstationarity condition influences the testing of nonlinear behavior in HRV. A contribution of nonlinearity was found in the HRV data of healthy individuals. A lower proportion of nonlinear time series was also found in ESRD patients, but statistical significance was not found. Although this proportion tends to be lower in ESRD patients, as much as 60% of time series proved to be nonlinear in healthy subjects. Given the important contribution of nonlinearity in HRV data, a nonlinear point of view is required to achieve a broader understanding of cardiovascular physiology.

Keywords: active standing; heart rate variability; hemodialysis; nonlinear dynamics; nonstationarity; recurrence analysis; surrogate data.

FIGURE 1

Full-size panels depict synthetic time series, original (left column), one Iterative Amplitude Adjusted Fourier Transform (IAAFT) surrogate time series (middle column) and one Pinned Wavelet Iterative Amplitude Adjusted Fourier Transform (PWIAAFT) (ρ = 0.01) surrogate time series (right column). AR2s Eq. (1) time series corresponds to top row, while the AR2ns one Eq. (2) is in the bottom row. (A) original AR2s, (B) IAAFT surrogate and (C) PWIAAFT (ρ = 0.01) surrogate. (D) AR2ns Eq. (2) and one example of (E) IAAFT surrogate and (F) PWIAAFT (ρ = 0.01) surrogate. Small-size panels show 8 randomly selected segments of 50 data points obtained from the whole time series. The dashed lines represent the means and the dotted lines indicate one standard deviation. AR2s original, IAAFT and PWIAAFT surrogate series were identified as stationary. Whereas AR2ns original and PWIAAFT surrogate series were regarded as nonstationary, the corresponding IAAFT surrogate was identified as stationary. Time series in all panels are shown as arbitrary units.

FIGURE 2

Time series and recurrence plots (m = 5, τ = 3) for synthetic data AR2s Eq. (1). Original time series (panel A), one surrogate obtained by Iterative Amplitude Adjusted Fourier Transform (IAAFT) (panel B) and one surrogate obtained by Pinned Wavelet Iterative Amplitude Adjusted Fourier Transform (PWIAAFT) (panel C). Time series in all panels are shown as arbitrary units.

FIGURE 3

Time series and recurrence plot (m = 5, τ = 3) for synthetic data AR2ns Eq. (2). Original time series (panel A), one surrogate obtained by Iterative Amplitude Adjusted Fourier Transform (IAAFT) (panel B) and one surrogate obtained by Pinned Wavelet Iterative Amplitude Adjusted Fourier Transform (PWIAAFT) (panel C). Time series in all panels are shown as arbitrary units.

FIGURE 4

Histograms for the laminarity (LAM) values of recurrence plot from of 99 surrogates (orange) obtained with Iterative Amplitude Adjusted Fourier Transform (IAAFT) and Pinned Wavelet Iterative Amplitude Adjusted Fourier Transform (PWIAAFT) techniques. The LAM values measured from the original data are depicted in blue. AR2s, IAAFT (A), PWIAAFT (B); AR2ns, IAAFT (C), PWIAAFT (D).

FIGURE 5

Examples of nonstationary heart rate variability (HRV) time series. Full-size panels show the whole HRV time series in supine position (top row) of (A) healthy subject, (B) end stage renal disease (ESRD) patient before hemodialysis (HD), and (C) ESRD patient after HD (same individual). HRV time series collected at active standing (bottom row) from (D) healthy subject, (E) ESRD patient before HD, and (F) ESRD patient after HD. HRV time series units in all panels are shown as seconds (s).

FIGURE 6

(A–I) Examples of time series and recurrence plots for heart rate variability (HRV) data in supine position. Top row corresponds to a healthy subject (m = 4, τ = 1), (A) original data, (B) Iterative Amplitude Adjusted Fourier Transform (IAAFT) surrogate, and (C) Pinned Wavelet Iterative Amplitude Adjusted Fourier Transform (PWIAAFT) surrogate. Middle row, end stage renal disease (ESRD) patient before hemodialysis (HD) (m = 6, τ = 6), (D) original data, (E) IAAFT surrogate, and (F) PWIAAFT surrogate. Bottom row, ESRD patient after HD (m = 6, τ = 7), (G) original data, (H) IAAFT surrogate, and (I) PWIAAFT surrogate. HRV time series units in all panels are shown as seconds (s).

FIGURE 7

(A–I) Examples of time series and recurrence plots for heart rate variability (HRV) data in active standing. Top row corresponds to a healthy subject (m = 5, τ = 10), (A) original data, (B) Iterative Amplitude Adjusted Fourier Transform (IAAFT) surrogate, and (C) Pinned Wavelet Iterative Amplitude Adjusted Fourier Transform (PWIAAFT) surrogate. Middle row, end stage renal disease (ESRD) patient before hemodialysis (HD) (m = 6, τ = 10), (D) original data, (E) IAAFT surrogate, and (F) PWIAAFT surrogate. Bottom row, ESRD patient after HD (m = 8, τ = 6), (G) original data, (H) IAAFT surrogate, and (I) PWIAAFT surrogate. HRV time series units in all panels are shown as seconds (s).

FIGURE 8

Percentage of heart rate variability (HRV) time series of every group that leads to reject the null hypothesis (nonlinearity demonstrated) using Iterative Amplitude Adjusted Fourier Transform (IAAFT) and Pinned Wavelet Iterative Amplitude Adjusted Fourier Transform (PWIAAFT) techniques (bars display the 95% confidence interval). § PWIAAFT vs IAAFT in the same group p < 0.001. There were no significant differences between groups (same position) nor within groups (supine vs active standing, same group).