Overview of Mathematical Relations Between Poincaré Plot Measures and Time and Frequency Domain Measures of Heart Rate Variability

The Poincaré plot was introduced as a tool to analyze heart rate variations caused by arrhythmias. Later, it was applied to time series with normal beats. The plot shows the relationship between the inter-beat interval (IBI) of one beat to the next. Several parameters were developed to characterize this relationship. The short and long axis of the fitting ellipse, <i>SD</i>1 and <i>SD</i>2, respectively, their ratio, and their product are used. The difference between the IBI of a beat and m beats later are also studied, <i>SD</i>1(<i>m</i>) and <i>SD</i>2(<i>m</i>). We studied the mathematical relations between heart rate variability measures and the Poincaré measures in the time (standard deviation of IBI, <i>SDNN</i>, root mean square of successive differences, <i>RMSSD</i>) and frequency domain (power in low and high frequency band, and their ratio). We concluded that <i>SD</i>1 and <i>SD</i>2 do not provide new information compared to <i>SDNN</i> and <i>RMSSD</i>. Only the correlation coefficient <i>r</i>(<i>m</i>) provides new information for <i>m</i> > 1. Novel findings are that <i>ln</i>(<i>SD</i>2(<i>m</i>)/<i>SD</i>1(<i>m</i>)) = <i>tanh</i>^-1(<i>r</i>(<i>m</i>)), which is an approximately normal distributed transformation of <i>r</i>(<i>m</i>), and that <i>SD</i>1(<i>m</i>) and <i>SD</i>2(<i>m</i>) can be calculated by multiplying the power spectrum by a weighing function that depends on <i>m</i>, revealing the relationship with spectral measures, but also the relationship between <i>SD</i>1(<i>m</i>) and <i>SD</i>2(<i>m</i>). Both lagged parameters are extremely difficult to interpret compared to low and high frequency power, which are more closely related to the functioning of the autonomic nervous system.