A selection of time-domain and frequency-domain measures of HRV were considered. The selected analysis parameters and their computation are presented shortly in the following.
Time-Domain Measures of HRV
The time-domain parameters considered in this study were the mean RR interval (\(\overline{\hbox{RR}}\)), standard deviation of RR intervals (SDNN), and root mean square of successive differences (RMSSD). Dynamics of these parameters were computed in a 60 s moving window (moved with 1-s steps) for the whole RR interval series. The mean RR interval at time t was obtained by averaging the RR intervals within the moving window, i.e.,
$$ \overline{\hbox{RR}}(t) = \frac{1}{L}\sum_{j=1\atop t_j\in {\rm{w}}_{t}}^L \hbox{RR}_j $$
(1)
where L is the number of RR intervals within a time window w
t
= [t − 30, t + 30] s and RR
j
is the jth RR interval observed at time t
j
. The SDNN at time t was similarly defined as
$$ \hbox{SDNN}(t) = \sqrt{\frac{1}{L-1}\sum_{j=1\atop t_j\in{\rm w}_{t}}^L (\hbox{RR}_j-\overline{\hbox{RR}}(t))^2}.$$
(2)
SDNN reflects the overall (both short-term and long-term) variability within the RR interval series, whereas RMSSD which was defined accordingly as
$$ \hbox{RMSSD}(t) = \sqrt{\frac{1}{L-1}\sum_{j=1\atop t_j\in {\rm w}_{t}}^{L-1} (\hbox{RR}_{j+1} - \hbox{RR}_j)^2} $$
(3)
is a measure of short-term variability. It should be noted that the number of RR intervals (L) within the time window w
t
may change from one time instant to another because of changes in the mean heart rate.
Frequency-Domain Measures of HRV
The frequency-domain parameters considered in this study were LF and HF component powers as well as the total spectral power. These power values as a function of time were obtained from the time-varying spectra which were estimated using the Kalman smoother spectrum estimation method. In the spectrum estimation, the data is assumed to be equidistantly sampled, and thus the RR interval series were converted into equidistantly sampled signals by using interpolation methods. The Kalman smoother spectrum estimation method is shortly described in the following (for details see Tarvainen et al.37).
In the method, the equidistantly sampled RR series was first modeled with a time-varying AR model of order p defined as
$$ x_t = -\sum_{j=1}^p a_t^{(j)}x_{t-j} + e_t $$
(4)
where x
t
is the modeled signal (i.e., the RR series), a
(j)
t
is the value of jth AR parameter at time t and e
t
is observation error. By denoting
$$ H_t = (x_{t-1},\ldots,x_{t-p}) $$
(5)
$$ \theta_t = (-a_t^{(1)},\ldots,-a_t^{(p)})^T $$
(6)
the time-varying AR model can be written in the form
$$ x_t = H_t\theta_t + e_t $$
(7)
which is a linear observation model. Kalman smoother algorithm is based on the so-called state-space formalism which means that in addition to observation model, the evolution of the state (i.e., AR parameters) is modeled. Here a random walk model
$$ \theta_{t+1} = \theta_t + w_t $$
(8)
where w
t
is the state noise term, was used. Equations (7) and (8) form the state-space signal model for x
t
and the evolution of the AR parameters can be estimated by using the Kalman smoother algorithm.
Kalman Smoother Algorithm
The Kalman smoother algorithm consists of a Kalman filter algorithm and a fixed-interval smoother. The Kalman filtering problem is to find the linear mean square estimator \(\hat{\theta}_t\) for state θ
t
given the observations \(x_1,x_2,\ldots,x_t. \) Kalman filter equations can be summarized as2
$$ C_{\tilde{\theta}_{t\vert t-1}} = C_{\tilde{\theta}_{t-1}} + C_{w_{t-1}} $$
(9)
$$ K_t = C_{\tilde{\theta}_{t\vert t-1}} H_t^T(H_t C_{\tilde{\theta}_{t\vert t-1}} H_t^T +C_{e_t})^{-1} $$
(10)
$$ \hat{\theta}_t = \hat{\theta}_{t-1} + K_t(x_t-H_t\hat{\theta}_{t-1}) $$
(11)
$$ C_{\tilde{\theta}_t} = (I-K_tH_t)C_{\tilde{\theta}_{t\vert t-1}} $$
(12)
where \(\tilde{\theta}_t=\theta_t - \hat{\theta}_t\) is the state estimation error, \(\tilde{\theta}_{t\vert t-1}=\theta_t - \hat{\theta}_{t-1}\) is the state prediction error, K
t
is the Kalman gain vector, and \({C_{e_{t}}}\;\text{and}\;{C_{w_{t}}}\) are the observation and state noise covariances, respectively.
The fixed-interval smoothing problem is to find estimates \(\hat{\theta}_{t}^{\rm S}\) (S denotes smoothed estimates) for each state θ
t
given all the observations \(x_1,x_2,\ldots,x_N. \) Fixed-interval smoothing equations can be summarized as2
$$ \hat{\theta}_{t}^{\rm S} = \hat{\theta}_t + A_t (\hat{\theta}_{t+1}^{\rm S} - \hat{\theta}_t) $$
(13)
$$C_{\tilde{\theta}_t^{\rm S}} = C_{\tilde{\theta}_t}+A_t(C_{\tilde{\theta}^{\rm S}_{t+1}} - C_{\tilde{\theta}_{t+1\vert t}})A_{t}^{T} $$
(14)
where \(A_t = C_{\tilde{\theta}_t}C_{\tilde{\theta}_{t+1\vert t}}^{-1}. \) The smoother algorithm is initialized with the Kalman filter estimates, i.e., \(\hat{\theta}_{N}^{\rm S}=\hat{\theta}_N\) and \(C_{\tilde{\theta}_{N}^{\rm S}} = C_{\tilde{\theta}_N}. \) That is, the smoothed estimates \(\hat{\theta}_t^{{\rm S}}\) are obtained by first running the Kalman filter algorithm forward in time and then the smoother algorithm backwards in time.
Adaptation of the Algorithm
The terms effecting the adaptation of Kalman smoother algorithm are the state noise covariance (\({C_{w_{t}}}\)), observation noise covariance (\({C_{e_{t}}}={\sigma_{e_{t}}^{2}}\)) and the variance of the modeled signal (\({\sigma_{x_{t}}^{2}}\)). In order to control the adaptation of the algorithm, the following solution was adopted. First, the observation noise variance was estimated recursively at every step of the Kalman filter equations as
$$ \hat{\sigma}_{e_t}^2 = 0.95 \hat{\sigma}_{e_{t-1}}^2 + 0.05 \epsilon_t^2 $$
(15)
where \(\epsilon_t\) is the one step prediction error \(\epsilon_t = x_t-H_t\hat{\theta}_{t-1}. \) Secondly, the state noise covariance matrix was selected to be diagonal \({C_{w_{t}}} = {\sigma_{w_{t}}^{2}}I,\;\text{and}\;{\sigma_{w_{t}}^{2}}\) was adjusted at every step of the Kalman filter equations as
$$ \hat{\sigma}_{w_t}^2 = \hbox{UC} \hat{\sigma}_{e_t}^2/\hat{\sigma}_{x_t}^2 $$
(16)
where \(\hat{\sigma}_{x_t}^2\) is the estimated variance of the observed RR series at time t and UC is an update coefficient through which the adaptation of the algorithm is adjusted. Therefore, the adaptation of the algorithm is adjusted with a single parameter (UC) and a fixed value of UC can be used for all data.
Time-Varying Spectrum Estimation
Once the time evolution of the AR parameters is estimated with the Kalman smoother, the time-varying spectrum estimates can be obtained as
$$ P_t(f) = \frac{\hat{\sigma}_{e_t}^2/f_s}{\vert 1+\sum_{j=1}^p \hat{a}_t^{(j)} e^{-i2\pi jf/f_s}\vert^2} $$
(17)
where f
s
is the sampling frequency, \(\hat{a}_t^{(j)}\) is the jth AR parameter estimate at time t, and \(\hat{\sigma}_{e_t}^2\) is the variance of the estimated observation error process at time t. Note that \(\hat{\sigma}_{e_t}^2\) is not the variance estimated iteratively in the Kalman filter forward run, but it is the posterior error variance evaluated based on the smoothed estimates.
The total spectral power as a function of time can be computed from (17) as
$$ P_t^{\rm Tot.} = \sum_{f=0}^{f_s/2} P_t(f)\Updelta f $$
(18)
where \(\Updelta f\) is the frequency grid interval.
LF and HF band powers could be computed similarly by summing over the frequency points within the predefined frequency bands. However, here the LF and HF powers are computed from decomposed spectral components.
Spectral Decomposition and Component Powers
The AR spectrum given in (17) can be decomposed into components by presenting it in the factored form
$$ P_t(f) = \frac{\hat{\sigma}_{e_t}^2/f_s}{\prod_{j=1}^{p} (z-\alpha_t^{(j)})(1/z - \alpha_t^{(j)*})} $$
(19)
where \(z=e^{i2\pi f/f_{s}}, \alpha_{t}^{(j)}\) are the time-varying roots (poles) of the AR polynomial, and * denotes complex conjugate. Now, consider a pole α
(j)
t
positioned at frequency f
j
. The spectrum of this single component P
(j)
t
(f) in the vicinity of f
j
can be estimated by assuming the effect of other poles on the spectrum to be constant (for details see Tarvainen et al.37). Once the spectral components for each time instant t are estimated, LF and HF powers can be computed by integrating the components positioned within the defined frequency bands.
In order to capture the RSA component always within the HF band, the HF (and also LF) band limits are adjusted according to the respiratory frequency as follows. The lower limit of the HF band was fixed to be not higher than 0.15 Hz, but in the case of slowed down respiratory dynamics, the lower limit was adjusted to be 0.025 Hz below the estimated respiratory frequency (but not less than 0.125 Hz). The lower limit of the HF band was however forced to be at least 0.025 Hz higher than the peak frequency of the LF component observed at previous time step. If the difference between respiratory rate and LF peak frequency would be less than 0.025 Hz, LF and HF powers would not be computed because the separation of the components could not be trusted. The upper limit of the HF band was dynamically adjusted to be 0.125 Hz above the estimated respiratory frequency (but at least 0.4 Hz). This dynamic adjustment of the HF band ensures that the RSA component is always captured within the HF band. For other recent methods for fixing the HF band according to respiratory frequency see Bailón et al.3 and Goren et al.17 For the LF band, the lower limit was fixed to 0.04 Hz and the upper limit was between always 0.125–0.15 Hz according to the lower limit of the HF band.
The LF and HF powers are then obtained as
$$ P_{t}^{\rm LF} = \sum_{\left\{j \vert \angle\alpha_t^{(j)}\in \hbox{LF band}\right\}} \left[\sum_{f=0}^{f_s/2} P_t^{(j)}(f)\Updelta f\right] $$
(20)
$$ P_{t}^{\rm HF} = \sum_{\left\{j \vert \angle\alpha_t^{(j)}\in \hbox{HF band}\right\}}\left[ \sum_{f=0}^{f_s/2} P_t^{(j)}(f)\Updelta f\right] $$
(21)
where P
(j)
t
(f) are the spectral components whose peak frequencies (i.e., the angles of the corresponding poles \(\angle\alpha_t^{(j)}\)) are within the specific band.
In addition to the absolute band powers, the LF and HF powers are commonly presented in normalized units (n.u.) as
$$ P_{t}^{\rm LF} (\hbox{n.u.}) = \frac{P_{t}^{\rm LF}}{P_{t}^{\rm Tot.} - P_{t}^{\rm VLF}} 100\% $$
(22)
$$ P_{t}^{\rm HF} (\hbox{n.u.}) = \frac{P_{t}^{\rm HF}}{P_{t}^{\rm Tot.} - P_{t}^{\rm VLF}} 100\% $$
(23)
where P
VLF
t
is the power of the very low frequency (VLF) band (0–0.04 Hz). These normalized power values reveal the relative strengths of LF and HF components. In addition, the ratio between LF and HF powers is often considered as an index of sympatho-vagal balance.