Actigraphy is a widespread method of recording human motor activity based on collected acceleration data. Analysing such recordings is an active area of research across a range of multidisciplinary fields [1]. One of the most common applications of actigraphy is the description and analysis of the measured subject’s sleep quality [2] and circadian rhythm [3]. Actigraphy is also utilized in psychiatric examinations [4, 5], i.e., to distinguish between similar mental diseases or to recognize behavioural disorders. Besides therapeutic applications, actigraphy is also employed to study human activity patterns [6, 7]. For example, to find regularities in the distribution of the resting and active periods or to examine time and frequency domain fluctuation features of human activity, in which the powerlaw scaling is a recurrent motif.
Actigraphy utilizes a biomedical measurement device, the socalled actigraph. The actigraph is a small, noninvasive tool containing a triaxial acceleration sensor, and is usually attached to the nondominant wrist of the observed subject. The classical actigraphic device generates an activity value for each consecutive, nonoverlapping, equallength time slot (i.e., epoch, typically 1–60 s long) based on the supported activity calculation procedure and on the acceleration signal it measures (usually sampled with 1100 Hz). The device stores the resulting activity signal in its memory. However, actigraphs from different manufacturers compute “activity” differently as they preprocess (e.g., digital filtering) the measured acceleration in varied ways and then calculate nonidentical types of activity signal from it using different sets of operations (i.e., activity metric, which is typically a nonlinear function). As a consequence, human activity measures lack a standardized unit, which – often combined with incomplete methodological descriptions – makes it difficult to reproduce and compare different studies [1, 8–11]. Nowadays, due to technological progress, actigraphs exist that can store the acceleration signal directly, but even in this case, there are several ways of preprocessing this raw motion data and then calculating activity values.
Due to nonstandardized activity determination methods and for greater flexibility, we have recorded raw acceleration signals of 42 healthy individuals in freeliving conditions along three axes on their nondominant wrist at 10 Hz in the ± 8 g measurement range for 10 days long, each. These recordings are publicly available [12] and serve as the basis of our analysis presented in this work, too. From the raw acceleration signals, we were able to generate further nonidentical types of acceleration signals based on different preprocessing methods established in the actigraphic literature. Subsequently, we could calculate numerous types of activity signals by applying different activity metrics to the already preprocessed acceleration signals. In our previous study [1], we analysed these activity signals in order to investigate how similar activity values different activity calculation procedures generate on the basis of time and frequency domain correlations. From the correlation coefficients calculated between the different types of temporal activity signals and between their power spectral densities, an identical correlation pattern was obtained in the time and frequency domain. The correlation pattern suggests that there may be major differences between the activity signals calculated in different ways, however, most activity signals showed strong similarities when calculated from identically preprocessed acceleration signals.
Our previous work has established the opportunity to examine the general patterns of human activity by analysing the spectrum of different types of actigraphic acceleration and activity signals in such a comprehensive way that was lacking in the literature so far. In our current work, our goal is to answer whether human actigraphic acceleration and activity signals (i.e., the human activity patterns in general) share the same spectral characteristics, what these spectral characteristics look like, and what they tell us about the patterns of everyday activity. On the one hand, assessing the effect of the steps of the activity calculation procedure on the observed spectral characteristic helps to understand the frequency domain relationships between acceleration and activity signals in more depth beyond the correlational similarities we had previously explored. On the other hand, it provides greater insight into what kind of fluctuations are present in human motor activity on a more profound level.
Analysis of human activity patterns
In recent years, significant advances have been made in the study of temporal and spatial patterns of daily human dynamics [13]. In the case of human mobility, scientists have already found powerlaw scaling through statistical analysis [14–16] (e.g., the spatial probability distribution of travel patterns [17]). In one of our previous works [18] we have presented that the minutely calculated displacement in human location data contains 1/ftype noise above the frequency of the daily rhythmicity which is a special form of powerlaw scaling in the frequency domain. Beyond human mobility, powerlaw scaling also exists in the patterns of human activity (e.g., the distribution of passive periods of human activity follows power law [6]). Fluctuations in activity signals and their complexity have also been investigated in many cases mainly for medical and diagnostics purposes [19–21]. Such studies are typically conducted using two analytical methods: frequencydomain description through the Power Spectral Density (PSD or S(f)) and timedomain investigation based on the fluctuation function (F(n)) which is resulting from the Detrended Fluctuation Analysis (DFA).
Considering the frequencydomainbased analytical approach, fluctuations whose power spectral density S(f) is inversely proportional to the frequency are called 1/f noises [19, 22] (a.k.a. pink noises, or flicker noises). In other words, 1/f noises’ PSD follows S(f) ∝ 1/f𝛽 powerlaw scaling, where 𝛽 = 1. In most areas, 1/f type noise is associated with exponent values under more severe constraints (0.8 < 𝛽 < 1.2 [23]), while in others it is identified under less strict constraints (0.5 < 𝛽 < 1.5 [24]). Time series that exhibit such spectral properties have longterm correlations. Moreover, the integral of 1/f noise’s spectral density (i.e., the power) over equally spaced intervals on a logarithmic scale (e.g., decades) is constant, independently of the given scale [25], while the spectrum decays following a straight line with a slope of −𝛽 on loglog scales. In addition to this frequencydomain scalefree nature, what is intriguing about 1/f noise is that there are numerous complex systems that at first glance may appear to be very different from each other, yet they produce this type of fluctuations. Such noise has been observed in several humanmade and natural phenomena, such as semiconductors [26], urban traffic [27], heart rate [28], EEG signals [29], and human activity, as explained later. To date, there is no agreed explanation or general mathematical model that implies the frequency of occurrence and universality of such noise.
Using DFA, one can compute a socalled fluctuation function F(n) of the analysed time series [30]. The algorithm splits the cumulative sum of the analysed time series into nonoverlapping, equalwidth boxes. The trend of each box is estimated by piecewise fitting (i.e., linear or polynomial), and then the rootmeansquare deviation is calculated between the cumulative sum and the trend. The process is repeated over different window sizes n resulting in a fluctuation function F(n). Similar to S(f), F(n) is mainly visualized on loglog scales. If examining 1/f type noise with DFA, the resulting fluctuation function should follow F(n) ∝ n𝛼 powerlaw scaling, where 𝛼 = 1. Both PSD and the fluctuation function describe the correlations in the analysed time series. The 𝛼 exponent of F(n) ∝ n𝛼 and the 𝛽 exponent of S(f) ∝ 1/f𝛽 are mathematically related to each other as 𝛽 = 2 𝛼 − 1 [31] and both can be estimated by linear fitting over an adequate range of the logtransformed fluctuation function and power spectral density, respectively. As can be seen, PSD examines the scalefree nature of time series in the frequency domain, while DFA does it in the time domain.
Even though actigraphic recordings are usually several days long, the fluctuation functions of activity signals in the relevant studies are generally evaluated over timescales ranging only from minutes to multiple hours [32], while activity signals recorded during sleep [19, 20, 33] and wakefulness [34, 35] are typically analysed separately. Although these DFAbased studies identified powerlaw scaling, they were typically limited to a given activity type and to assess how different diseases (i.e., Alzheimer's disease [34], KleinLevin disease [32], depression [36], bipolar disorder [37], autism spectrum disorder [20]) or even aging [38] break down the patterns compared to control groups without giving a detailed description of the general fluctuation patterns of human activity. Regarding frequencydomain analysis, two studies [39, 40] have already noted in the case of two given types of activity signals (up to 1week long time series, examined over their entire length) that they contain 1/f fluctuations above the frequency of the daily rhythmicity without giving further details on the general spectral characteristic. In conclusion, the studies found that longterm correlations and selfaffinity exist in given types of human activity signals, which is an indicator of complex underlying mechanisms and regulations. However, there are many different ways of determining activity values. Therefore, it is not selfevident that the observed features are indeed inherent to human activity, or that they are artefacts of the utilized activity calculation procedure.
The question arises whether, if the activity signals have such properties, these might already be present in the acceleration signals or just the usage of nonlinear operations of the activity metrics brings this phenomenon. In the literature, we found one study [31] that investigated fluctuations in actigraphic acceleration signals instead of activity signals. Although acceleration signals can also be different types depending on the preprocessing method, they analysed the fluctuations in only two given types of acceleration signals to identify sleepwake transitions using Welch’s method, which differs from the usual approach used for the analysis of activity signals. Because of their specific purposes, their analysed acceleration signals were significantly shorter than the activity signals commonly studied in the literature, so their spectra were limited to a narrower frequency band. Yet, they found 1/f noise in this frequency range of actigraphic acceleration signals recorded during wakefulness, they also confirmed their findings by DFA. This also raises the question of the extent to which their recognition can be generalised to actigraphic acceleration signals preprocessed in other ways.
In conclusion, previous studies have already partially examined the scalefree nature of human activity by analysing

activity signals’ fluctuation functions for box widths of less than 24 hours typically separating into sleep and wakefulness for medical purposes,

two given types of multidaylong activity signals’ PSD over the entire frequency range,

and acceleration signals’ fluctuations both with DFA and PSD (using Welch’s method) over a narrower timescale and frequency range, respectively.
As we have shown above, the question arises whether the observed fullspan (i.e., the entire frequency range) spectral characteristics of certain types of activity signals depend on the activity calculation methods, and may the spectral characteristics of the activity signals differ from the ones observed in the acceleration signals, which were so far only described over a narrower frequency band. In this article, we aim to fill these gaps by giving the general spectral characteristic of multidaylong actigraphic recordings measured on freeliving, healthy subjects, and assessing the possible differences caused by different acceleration signal processing techniques and activity metrics using PSD and DFA examination methods without separating sleep and wakefulness. Our analysis also provides insight into the relationship between the acceleration signals and the activity signals calculated from them.