Longterm trends change the overall but not the local SD/variance of a time series. As a consequence trends affect mostly MSE_{μ} not MSE_{σ} when the r value is calculated as a percentage of the time series’ SD. MSE_{μ} and MSE_{σ} values are much less effected by trends in implementations that use a fixed r value.
Figure 4 illustrates the effects that linear trends superimposed on normalized 1/f noise fluctuations have on MSE_{μ} and MSE_{σ} values. We considered trends obtained by the concatenation of two linear segments. In one case (red line) the values of the function increase linearly between 4 and 8 for the first 30,000 and then decrease linearly from 8 to to 4 over the following 20,000 data points. In the other case (green line), the rates of increase/decrease were doubled. The superimposition of each of these trends on a 1/f noise time series resulted in signals A and B shown on the first and second panels, respectively.

The SD of the original 1/f noise time series is one. The SDs of time series A and B are 1.3 (30% larger than the SD of the original series) and 2.3 (130% larger than the SD of the original series), respectively.
In MSE_{μ} analyses using the 20% of SD criterion, the absolute values of r are 0.2, 0.26 and 0.46 for the original, time series A and B, respectively. The increase in the absolute value of r due to the trends results in a higher number of matches and consequently in lower entropy values. Specifically, for scale 1, in the case of 1/f noise, the number of matches with m = 2 and m = 3 were 28,348,070 and 5,827,479, respectively. For time series A (B), the number of matches with m = 2 and m = 3 were 38,606,695 (60,511,987) and 10,973,294 (27,887,816), respectively. The effects of trends on MSE_{μ} can be obviated by using a fixed r value. However, when using r = 20% of SD, a barely noticeable trend can significantly change the values of entropy.
In MSE_{σ} analysis, the r value is a percentage of the SD of the SD coarsegrained time series obtained with a window of 5 data points. Since the slow trend has only a small effect on the SD coarsegrained time series, the changes in the entropy values are negligible.
Figure 5 shows the results of MSE_{μ} and MSE_{σ} analyses of the RR interval time series from the groups of healthy young and older subjects and patients with CHF, using r = 20% SD. The first 50,000 data points of each recording (≈ 14hours) were selected for analysis independent of the starting time (not available in most of these cases). As such, the time series may include both awake and sleep periods.

Healthy individuals are more likely to have more pronounced circadian variations than patients with CHF. Thus, the RR interval time series from the former group are more likely to exhibit trends of higher amplitude than the latter. These trends can justify the apparent contradictory MSE_{μ} results that indicate higher dynamical complexity for the CHF than the healthy older group. Such a conjecture is supported by the MSE_{μ} results obtained using a fixed r value as well as the those of MSE_{σ} analyses using both fixed and variable r values. These findings highlight the need to analyze the data using different approaches in order to gain a deeper understanding of the properties of the dynamics and minimize the impact of known and unsuspected confounders.
The presence of outliers can also significantly affect MSE_{σ}, MSE_{σ2} and MSE_{MAD} analyses. For this reason it is important to filter the time series prior to performing these analyses. The results presented here of the RR interval time series analyses were obtained using the filter described below. Alternatively, the use of a metric such as median absolute deviation for coarsegraining (not implemented here) could be considered.
Madalena Costa (mcosta3@bidmc.harvard.edu)