# Stochastic Modeling and Optimal Time-Frequency Estimation of Task-Related HRV
## 1 Introduction  
Extensive research has been dedicated to understand and describe the complex interplay of factors that affect heart rate variability (HRV), the physiological phenomenon of the variation in the time interval between consecutive heartbeats. The dynamic task-specific vagal control of the heart represented by task-related HRV changes is especially valuable in exploring the flexible adaption of the organism to physical and mental challenges [1; 2]. Age has been identified as the most critical variable affecting HRV, with the spectral power of HRV decreasing with increasing age [3; 4; 5]. Nevertheless, several other variables play a role, such as gender, body-mass index (BMI), anxiety and stress levels [6; 7]. State-of-the-art spectral methods have been widely applied in the analysis of HRV, especially the periodogram and the Welch method [8]. These methods are suitable for HRV data measured in resting conditions, and for extracting the traditional high-frequency and low-frequency HRVpower measures. However, these measures often do not provide accurate results in task-related HRV measurements [2; 9; 10]. Task-related HRV data are usually non-stationary signals and therefore require time-frequency (TF) spectral estimators, e.g., the spectrogram or other [PERSON]'s class TF smoothing kernel estimators, which allow for capturing changes in the spectral power and frequency content over time [11]. For HRV estimation, generally smoothed kernel estimators are applied to capture the TF variations [2]. Other approaches include wavelet based TF analysis [10], as well as time-varying multitaper spectral analysis in order to decrease the variance of the estimate [12]. In recent years, several studies investigated and challenged the traditional high-frequency and low-frequency HRV power measures, not suitable for time-varying signals [13]. A crucial factor influencing HRV is the respiratory sinus arrhythmia (RSA), i.e., the fluctuation in the heart rate corresponding to breathing, through which the heart rate increases during inspiration and decreases during expiration [14]. The respiratory frequency information is of utmost importance in the analysis of HRV and has been included in several novel approaches. Metronome guided breathing is used in [15] to investigate the effects of respiration on HRV indices, finding that the breathing frequency needs to be considered as a contributor when analyzing HRV measures. Moreover, the results in [16] support that joint analysis of respiration and HRV obtains a more reliable characterization of autonomic nervous response to stress. The respiration rate changes with exercise levels and can increase above the traditional high-frequency band. Therefore new respiration based frequency analysis bands have been suggested for running and cycling stress testing [2]. Respiration based frequency bands are also considered in the analysis of HRV in relation to emotion recognition [9], where the respiration frequency is extracted and used to define a subject-based time-varying frequency band. These studies justify the increased interest in the estimation of the respiratory frequency from the HRV signal when the respiratory information is not present [17]. In [18], the HRV is decomposed into a component that is correlated with the respiratory frequency and one residual component: the latter is found to have more discrimination power than traditional HRV analysis to monitor mental stress. A similar approach is proposed in [10], where the heart rate is decomposed into a respiration-locked component and a respiration-unrelated component using empirical mode decomposition. In this paper, we propose a stochastic model and its mean square error (MSE) optimal TF spectral estimator for task-related HRV signals, measured during a novel chirp-breathing task, where the 92 test participants (TPs) were instructed to increase their breathing frequency accordingly to a metronome. These measurements allow the examination of the cardiac regulation mediated by the dynamics of the peripheral nervous system.