Time series can contain drift, which may be generally defined as a low-frequency process that is not considered to belong to the measured signal of interest. Drift may occur secondary to a variety of hardware-related factors including changes in electronic and thermal environments (Wilmshurst, 1990). Reducing this type of drift can be done at hardware, firmware or software levels and remains an ongoing area of research (e.g. Velásquez et al., 2019). In human experiments, fatigue and other behavioral factors can also produce drift (Song et al., 2009) and this type of drift must be dealt with at the software stage. This paper focuses on the influence of drift on statistical comparisons of time series and offers software for the proposed drift corrections.

In the human biomechanics literature several movement tasks — especially treadmill gait — involve measurements over several minutes or even seconds, and in these measurements drift is generally reduced through high-pass filtering (Murphy and Robertson, 1993, Valentin and Zsoldos, 2016) or detrending (Fig. 1). Both high-pass filtering and detrending regard low-frequency content as noise and are unconcerned with the electromechanical or physiological nature of that noise. Detrending is particularly important to detrended fluctuation analysis (DFA), which has become an important analysis framework for investigating the structure of motor variability (Ducharme and van Emmerik, 2018, Ravi et al., 2020); detrending is useful for motor variability studies, for example postural sway studies, to dissociate low- from higher-frequency content, which are associated with very different (e.g. high- vs. low-level) neural control processes. Unlike the simple long-term detrending depicted in Fig. 1, DFA instead uses local detrending to identify and remove multiple trends that occur in adjacent movement segments. While detrending is often employed in routine analyses, we are unaware of explicit previous reports of trend prevalence, nor of the prevalence of inferential errors which may occur if detrending is not employed.

A key limitation to existing detrending approaches is that they do not consider cycle-level drifts (Fig. 2). In cyclical tasks like gait different parts of the cycle may drift differently and even oppositely (Fig. 2a-b). As a plausible biomechanical example, consider ground reaction force (GRF) in treadmill walking which increases posteriorly in early stance over 100 cycles and also increases anteriorly in late stance; these opposite changes must occur to maintain constant walking speed. For example, maximal Floquet multipliers (representing the stability of the system at certain instants of the gait cycle) have been found to vary within the gait cycle (Kang and Dingwell, 2009), thus emphasizing the need to consider trends as non-constant within the cycle. Moreover, simple simulation shows that trial- and cycle-level detrending can generally yield qualitatively different results (Fig. 2c-d). We are unaware of any previous study that has considered cycle-level drift.

The purposes of this study were: (1) to describe a generic cycle-level detrending procedure, (2) to estimate the prevalence with which significant cycle-level trends exist in an open treadmill GRF dataset, and (3) to estimate the frequency with which these cycle-level trends can qualitatively affect hypothesis testing results. Software implementing all subsequently described detrending procedures is available at https://github.com/0todd0000/detrend1d along with notebooks and data files that replicate this paper’s main results.