General spectral characteristics of human activity and its inherent scale-free fluctuations

doi:10.21203/rs.3.rs-2539448/v1

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General spectral characteristics of human activity and its inherent scale-free fluctuations

https://doi.org/10.21203/rs.3.rs-2539448/v1

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Although actigraphy is commonly used in many research areas, the activity calculation methods are not standardized, therefore activity signals can be very different. The scale-free nature of daily human activity has been observed in different aspects; however, the description of its spectral characteristics is incomplete. The presence of 1/f noise in activity or acceleration signals was mostly analysed for short time windows, the complete spectral characteristic has only been examined in the case of certain types of activity signals. To explore the general spectral nature of human activity in greater detail, we have performed Power Spectral Density (PSD) based examination and Detrended Fluctuation Analysis (DFA) on multi-day-long, triaxial actigraphic acceleration signals of 42 healthy, free-living individuals. We generated different types of activity signals from these, using different acceleration preprocessing techniques and activity metrics. We revealed that different types of activity signals’ spectra generally follow a universal characteristic including 1/f noise over frequencies above the circadian rhythmicity. Moreover, we discovered that the PSD of the raw acceleration signal has this same characteristic. Our findings prove that the spectral scale-free nature is generally inherent to the motor activity of healthy, free-living humans, and is not limited to any particular activity calculation method.

Biological sciences/Computational biology and bioinformatics/Power law

Biological sciences/Computational biology and bioinformatics/Scale invariance

Biological sciences/Systems biology/Time series

Biological sciences/Systems biology/Signal processing

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 power-law scaling is a recurrent motif.

Actigraphy utilizes a biomedical measurement device, the so-called actigraph. The actigraph is a small, non-invasive tool containing a triaxial acceleration sensor, and is usually attached to the non-dominant wrist of the observed subject. The classical actigraphic device generates an activity value for each consecutive, non-overlapping, equal-length 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 1-100 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 non-identical 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 non-standardized activity determination methods and for greater flexibility, we have recorded raw acceleration signals of 42 healthy individuals in free-living conditions along three axes on their non-dominant 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 non-identical 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 power-law 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/f-type noise above the frequency of the daily rhythmicity which is a special form of power-law scaling in the frequency domain. Beyond human mobility, power-law 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: frequency-domain description through the Power Spectral Density (PSD or S(f)) and time-domain investigation based on the fluctuation function (F(n)) which is resulting from the Detrended Fluctuation Analysis (DFA).

Considering the frequency-domain-based 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/f^𝛽 power-law 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 long-term 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 log-log scales. In addition to this frequency-domain scale-free 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 human-made 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 so-called fluctuation function F(n) of the analysed time series [30]. The algorithm splits the cumulative sum of the analysed time series into non-overlapping, equal-width boxes. The trend of each box is estimated by piecewise fitting (i.e., linear or polynomial), and then the root-mean-square 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 log-log scales. If examining 1/f type noise with DFA, the resulting fluctuation function should follow F(n) ∝ n^𝛼 power-law scaling, where 𝛼 = 1. Both PSD and the fluctuation function describe the correlations in the analysed time series. The 𝛼 exponent of F(n) ∝ n^𝛼 and the 𝛽 exponent of S(f) ∝ 1/f^𝛽 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 log-transformed fluctuation function and power spectral density, respectively. As can be seen, PSD examines the scale-free 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 DFA-based studies identified power-law scaling, they were typically limited to a given activity type and to assess how different diseases (i.e., Alzheimer's disease [34], Klein-Levin 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 frequency-domain analysis, two studies [39, 40] have already noted in the case of two given types of activity signals (up to 1-week 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 long-term correlations and self-affinity 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 self-evident 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 sleep-wake 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 scale-free 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 multi-day-long 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 full-span (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 multi-day-long actigraphic recordings measured on free-living, 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.

In the field of actigraphy, procedures for calculating human activity are highly diversified, and their details are typically hidden. However, a general processing scheme can be defined in which the raw acceleration signal is converted into an activity signal within a few typical steps (see Fig. 1).

The first step is to preprocess the acceleration signal, which is usually measured along three axes and sampled with 1-100 Hz. During preprocessing, the acceleration signal can be treated in different ways [9, 41–43] (e.g., calculating the magnitude of acceleration, digital filtering, and normalization, as detailed later). The conditioned signal is then segmented into consecutive, non-overlapping time slots (i.e., epochs) of length T_e. The next step is to apply an activity metric [44–48] (several are widely used in the literature, see detailed later), which produces an activity value for each epoch. The resulting activity signal is built by activity values, whose are T_e spaced in time. Most actigraphic devices on the market do the entire procedure, therefore they only store activity values created in a device-specific way instead of a larger amount of acceleration data. In contrast, the actigraphic recordings investigated in this work were obtained using such an actigraph (for more details about the device, see our previous work [1]) that stores the raw acceleration signal recorded along three axes, providing maximum control and flexibility for subsequent calculation of activity values as desired.

One of the main aspects of our previous article [1] was the collection and categorization of activity calculation methods that are widely used in the literature. In the following, we are briefly summarizing them, as both the differently generated acceleration and activity signals’ spectral characteristics are analysed in the present study.

Preprocessing techniques of the acceleration signal

The most straightforward acceleration signal preprocessing approach is to calculate the magnitude of acceleration [43] by taking the square root of the sum of each of the squared axial components. However, this contains the constant gravitational acceleration (i.e., gravity of Earth, g), which can be eliminated in several ways. The simplest way is to normalise the resultant magnitude of acceleration values by subtracting 1 g and then taking the absolute value of the difference. However, this can also be achieved by digital filtering. Actigraphs often use band-pass filters [42, 43] to remove the DC component by filtering the low-frequency range and to eliminate vibrations associated with involuntary motion and other noise by filtering the high-frequency range of the acceleration. Such filtering can be done on the raw axial acceleration values before magnitude calculation [49, 50]. However, it is also possible to filter the magnitude of acceleration already calculated from the raw axial acceleration values [51]. As a result, based on the different acceleration signal preprocessing techniques, different types of acceleration signals can be generated from a single, raw triaxial acceleration data as seen in Fig. 2 with the nomenclature we have introduced previously [1].

For band-pass filtering, a 3rd -order Butterworth digital filter with f_L = 0.25 Hz and f_H = 2.5 Hz was utilized. One of the activity metrics requires a variation of the FMpre acceleration signal, which is produced using a high-pass filter rather than a band-pass filter. High-pass filtering was executed using a 4th -order Butterworth digital filter with f_C = 0.2 Hz.

Activity metrics

An activity signal is built up from a series of activity values, and the activity metric defines how to determine the activity value from the already preprocessed acceleration signal for each epoch. As we mentioned previously, several activity metrics are common in the literature, different actigraphic devices usually implement different metrics (and preprocessings). Table 1. summarizes the activity metrics that we use in the analysis.

Table 1

The most common activity metrics in the literature and their brief description [1].
Activity metric	Definition
PIM (Proportional Integration Method) [44]	It integrates the acceleration signal for a given epoch. In the following, we use the simplest numerical integration. $PIM={T}_{s}\sum _{i=1}^{n}{x}_{i}$ In the formula x₁, x₂, …, x_{(n − 1)}, x_n are the n acceleration values of the given epoch, and T_s is the sampling time of the acceleration signal.
ZCM (Zero Crossing Method) [44]	It counts the number of times the acceleration signal crosses a T_ZCM threshold for each epoch.
TAT (Time Above Threshold) [44]	It measures the length of time that the acceleration signal is above a T_TAT threshold for each epoch.
MAD (Mean Amplitude Deviation) [47]	$MAD=\frac{1}{n}\sum _{i=1}^{n}\left\|{r}_{i}-\stackrel{-}{r}\right\|$ In the formula r₁, r₂, …, r_{n − 1}, r_n are the n magnitude of acceleration values of the given epoch, $\stackrel{-}{r}$ is their arithmetic mean.
ENMO (Euclidean Norm Minus One) [47, 52]	$ENMO=\frac{1}{n}\sum _{i=1}^{n}\text{max}\left({r}_{i}-\text{1,0}\right)$ In the formula r₁, r₂, …, r_{n − 1}, r_n are the n magnitude of acceleration values of the given epoch. The values of r are in g.
HFEN (High-pass Filtered Euclidean Norm) [46]	This metric requires its own, specially preprocessed acceleration signal type (previously defined as HFMpre). $HFEN=\frac{1}{n}{\sum }_{i=1}^{n}{r}_{fi}$ In the formula r_f1, r_f2, …, r_{fn − 1}, r_fn are the n values of the HFMpre acceleration of the given epoch.
AI (Activity Index) [48]	$AI=\sqrt{\text{max}\left(\frac{1}{3}\left(\sum _{m=1}^{3}{\sigma }_{m}^{2}-{\stackrel{-}{\sigma }}^{2}\right),0\right)}$ In the formula m = 1, 2, 3 corresponds to the three axes, ${\sigma }_{m}^{2}$ is the variance of the vector components along the m^th axis of the given epoch, and ${\stackrel{-}{\sigma }}^{2}$ is the variance of the baseline noise of the total measurement data (so-called systematic noise variance).

The T_ZCM and T_TAT threshold values are crucial for the ZCM and TAT activity metrics to work properly, however, they have no universal value in the literature. In our previous work [1], we have established that the standard deviation of the acceleration data can be used as an appropriate threshold level for these level intersection-based metrics. We proceeded accordingly in the current work.

Examined activity signals generated using different preprocessing techniques and activity metrics

The different combinations of preprocessings and activity metrics result in different activity signals. In our previous article [1] we have already investigated in detail which preprocessing techniques are compatible with which activity metrics, therefore, we are limiting the current analysis to the proper combinations of them. In conclusion, 11 different types of acceleration signals and 35 different types of activity signals can be generated from a single raw triaxial acceleration measurement (see Fig. 3). For ease of interpretation, we use the activity metric as an operation and the acceleration signal preprocessing method as its argument to denote activity signals. For example, PIM(UFNM) denotes the activity signal obtained by applying the PIM metric to the unfiltered normalized magnitude of acceleration.

Our results are derived from the analysis of a data package containing 42 acceleration signals, each for one subject. We also used this in our previous work [1], and its description is available in the Materials and methods section.

We executed the following procedure for all the subjects. From a 10-day-long recording, we determined an activity value for each 60-second-long epoch, so the acceleration signal became a 10-day-long activity signal sampled with 1/60 Hz. We executed this minute-based activity calculation process in different ways based on the right combinations of acceleration preprocessing techniques and activity metrics that were presented in Fig. 3. As a result, we were able to generate 35 different types of activity signals (e.g., PIM(UFNM), MAD(FX), etc.) and 11 different types of acceleration signals (e.g., FMpre, UFM, etc.) that all describe the same subject’s 10 days of activity but in altering ways.

Then we determined the power spectral density S(f) and fluctuation function F(n) of each of these acceleration and activity signals using Discrete Fourier Transform (DFT) and DFA, respectively. To compare the different types of activity signals, we averaged the power spectral density and fluctuation function of the activity signals derived from the 42 subjects for each activity signal type. We also executed it in the case of the differently preprocessed – i.e., different types of – acceleration signals. This resulted in a single ensemble-averaged S(f) and F(n) for each different type of acceleration and activity signal based on the 42 subject’s 10-day-long motion. Through the ensemble-averaged functions, the general spectral characteristics and scaling properties of the different types of acceleration and activity signals became analysable. To present the S(f) ∝ 1/f^𝛽 and F(n) ∝ n^𝛼 scaling properties in detail, we utilized multiple approaches. Firstly, we calculated the ensemble-averaged total power of the analysed time series in each log-spaced frequency bin (i.e., the integral of S(f) for each bin). Secondly, we executed linear fitting (between 10⁻⁴ Hz to 10⁻² Hz in the case of the spectral densities, and between 10⁴ s to 10² s in the case of fluctuation functions (the choice of the fitting intervals will be justified later) on the log-transformed and ensemble-averaged S(f) and F(n) functions to estimate the value of their 𝛽 and 𝛼 exponents, respectively. Finally, based on simple numerical derivation, we also calculated the 𝛽 exponent values between every consecutive point of the ensemble-averaged spectral density and fluctuation function which results in 𝛽 exponent curves. A detailed description of generating the PSDs and fluctuation functions can be found in the supplementary material. The calculation of the ensemble-averaged S(f), and F(n), and the 𝛽 exponent curves calculated from them are detailed in the Materials and methods section.

In the following two sections, we present the results based on the analysis of the ensemble-averaged spectral density and fluctuation function first for the activity signals and then for the acceleration signals. In the presentation of our results, we show the corresponding figures for a few activity and acceleration signal types, the rest is available in the supplementary material. To make the resulting characteristics visually comparable, the spectral density and fluctuation functions of the activity signals are visualized on the same scales as for the acceleration signals, even if the PSDs and fluctuation functions of the activity signals are evaluated in a narrower scale due to their significantly lower sampling rate.

Spectral characteristics of the activity signals

Examining the PSDs of activity signals

The ZCM is one of the most common activity metrics, and in addition to PIM, ZCM activity signals’ PSD over broader frequency range has already been investigated by a previous study [6] revealing that they have 1/f characteristic over frequencies above the circadian rhythmicity. The study that discussed this also used epoch length of 60 s, however, they did not report the sampling rate of the acceleration signal, neither what preprocessing they performed on the acceleration signal before applying the metric, nor did they provide the T_ZCM threshold level which is essential for the correct functioning of this activity metric. In light of these uncertainties, the question arises whether the 1/f characteristic is present in all the different types of ZCM activity signals or is only limited to specific acceleration preprocessings. Firstly, we are presenting our results in the case when the ZCM metric is applied on the unfiltered normalized magnitude and on the axially filtered magnitude of acceleration (ZCM(UFNM), and ZCM(FMpre), presented in Fig. 4 and Fig. 5, respectively), the figures for the remaining ZCM activity signals can be found in the supplementary material.

The ZCM(UFNM) and the ZCM(FMpre) activity signals have the same spectral characteristics as seen in Fig. 4 and Fig. 5, respectively. The observed spectral characteristic is composed of the following components. A distinct change in the spectra’s slope can be observed at the corner frequency of approximately 10^− 4 Hz. Above this frequency, the spectrum decays following a straight line on the logarithmic scales and fits perfectly on the 1/f trendline which indicates the existence of 1/f nature. This claim is supported by the fact that the total power in each frequency bin is nearly constant in this region as seen in subplot c) of Fig. 4 and Fig. 5, and the PSD-based 𝛽 exponent curve is also within the tolerance range for 1/f noise. Below the corner frequency, the spectrum exhibits white noise as it breaks away from the trendline and the peaks associated with the 24 and 12-hourly periodicities (around 10⁻⁵ Hz and 10^−4.5 Hz, respectively) are well-exposed. Since the relationship between the fluctuation function's and the power spectral density's scaling exponent is defined as 𝛽 = 2 𝛼 − 1, the fluctuation function follows a trendline with a slope of 1 on the logarithmic scales as expected. Another parallelism is that the fluctuation function also deviates from the trendline above a certain box width and then flattens. Our PSD-based spectral analysis results are consistent with the previously mentioned study. However, we have not found any previous DFA-based analysis of ZCM activity signals. Note that, by examining the ZCM(FMpre) activity signals’ ensemble averaged fluctuation function and the corresponding 𝛽 exponent curve, it can be seen that there is a hump in the fluctuation function at the box width of 24 hours. Compared to ZCM(FMpre), the ZCM(UFNM) activity signals have a less substantial presence of circadian rhythmicity observed in their PSD. In consequence, the hump in the fluctuation function became less perceptible to the eye, yet can be identified by the 𝛽 exponent curve.

Comparing these characteristics with the results for the remaining ZCM activity types in the supplementary material, it can be stated that all the ZCM activity signals calculated from differently preprocessed acceleration signals show similar general spectral characteristics, only at the last half-decade of lower frequencies of the spectrums show slight differences compared to each other. Thus, ZCM activity signals have spectral scale-free, 1/f-like characteristics in general, regardless of the way the acceleration signal was preprocessed. In conclusion, we have successfully reproduced the spectral shape already presented in the literature [6] for ZCM activity signals and even extended it to ZCM activity signals calculated from different types of acceleration signals.

The question arises if activity signals produced by other activity metrics have a similar spectral nature as ZCM activity signals. To provide a complete description of the overall spectral nature of human activity in general, we also examined the ensemble-averaged spectral densities and fluctuation functions of activity signals generated with the additional 6 activity metrics. For example, Fig. 6 depicts the general characteristics in the case of the AI(FXYZ) activity signals, as the AI activity metric has the most distinct working principle compared to other metrics.

As seen, AI(FXYZ) activity signals follow the same spectral characteristics in general as explained above. One can notice that the peak associated with the daily periodicity is more prominent compared to the previously depicted characteristics. Accordingly, the hump in the fluctuation function at around the 24 hours box width is more detectable in this case.

For practical considerations concerning the length of the article, the figures of the remaining activity signal types can be found in the supplementary material.

Spectral comparison of different types of activity signals

To assess how similar spectral characteristics the differently determined activity signals follow, Fig. 7 demonstrates the total power for each frequency bin in the case of the different types of activity signals grouped into subplots by activity metrics. On one hand, the comparison of the power curves within each subplot reveals the different acceleration preprocessing techniques’ impact on the resulting activity signals’ spectral characteristic. On the other hand, the discrepancies in the spectral characteristics caused by the different activity metrics can be assessed by comparing the subplots to each other. Note that although we have analysed 35 types of activity signals in total, Fig. 7 depicts only 25 types: if an activity metric could be applied on the axial acceleration signals separately, it is only represented in the case of the y-axis (anteroposterior axis, i.e., FY or UFY acceleration signals) in the figure. The power curve of the activity signal types determined along the x and z axes can be found in the supplementary material as there is no significant difference compared to those determined along the y axis.

As presented, different types of activity signals can be described with the same spectral characteristic in general which we described above. In greater detail, 29 of the 35 activity signal types have similar and distinct 1/f nature above the corner frequency for the rest of the spectrum, only the last half decade shows slight variations. The remaining 6 activity signal types (PIM(UFM), AI(UFXYZ), MAD(FMpre), and MAD activity signals calculated from the unfiltered axial accelerations) have a slightly different spectral shape as their power curves deviate greatly from the horizontal line. Such dissimilarity in the spectral shape also occurred when applying a given activity metric on differently preprocessed signals. Therefore, the way of preprocessing the acceleration signals has a noticeable effect on the spectral characteristics of the activity signals. However, all of these differences can be explained. By definition, the MAD metric must be applied to the unfiltered magnitude of acceleration (UFM) [47]. If we calculate MAD activity signals according to the metric’s definition, the resulting signals follow the same spectral shape as the other 28 activity signal types. Even though there is no technical barrier to applying the MAD metric to axial acceleration signals or filtered acceleration signals, these combinations produce such activity signals that’s spectral shape differs from the observed general characteristic. The creators of the AI activity metric did not define whether their method requires filtered or unfiltered axial acceleration signals [48]. Nonetheless, if we apply the AI metric to filtered acceleration signals, it already results in activity signals those spectral shapes almost identical to the other 28 activity signal types. The PIM metric can be applied to UFM-type acceleration signals only if the resulting activity signal is corrected afterward [1] (subtracting the integral of g), therefore, the way of correction could be the reason for the observed difference.

To more accurately identify the strength of the scale-free nature of the 29 different types of activity signals which have the same spectral shape, we assessed the 𝛽 exponents of their power-law scaling by linear fitting on log(S(f)) versus log(f) as can be seen in Fig. 8. The fitting was performed between 10⁻⁴ Hz (the approximate corner frequency) and the highest frequency component of the spectrums (which is approximately 10⁻² Hz). Similarly, the linear fitting-based 𝛽 exponent assessment was also performed on the ensemble-averaged fluctuation functions between 10² s and 10⁴ s box widths. We were able to determine the strength of the 1/f nature of the activity signals from two different but comparable perspectives through their PSDs and fluctuation functions.

In general, the values of the 𝛽 exponents determined by both the PSD-based and DFA-based linear fitting indicates indicate clear 1/f fluctuations over the investigated range as they fall between 0.8 and 1.2 in most cases independently of any given activity signal type. Slightly larger exponents were obtained systematically by the PSD-based fitting, the average difference between the cells of the a) and b) matrices is 0.047 ± 0.005 (Mean ± SD). Thus, from two independent perspectives, we showed that 1/f noise is a general feature of human motoric activity signals. Their spectral structure and notable frequency bands are also common: at frequencies below the corner frequency (approximately at 10⁻⁴ Hz) the spectrum flattens and the peaks associated with the 24 and 12-hour periodicities arise, while at higher frequencies 1/f noise dominates. Even though all 35 activity signal types are represented in Fig. 8, the 𝛽 exponents should be treated with caveats in the case of the PIM(UFM), AI(UFXYZ), MAD(FMpre), and MAD activity signals calculated from the unfiltered axial accelerations as the linear fitting was less accurate in their case due to the previously explained discrepancies.

Spectral characteristics of the acceleration signals

Examining the PSDs of acceleration signals

As proved above, the spectral 1/f nature of human activity signals is independent of any given activity metric and is a general attribute. Consequently, the question arises whether the scale-free nature is already present in the acceleration signals and if the acceleration signal preprocessing techniques have an impact on their spectral nature. Note that the activity metrics are mostly nonlinear transformations, so it is not evident how similar are the PSDs of activity and acceleration signals. Moreover, 1/f noises exhibit interesting properties and invariances in special cases for nonlinear operations [53], which deserves further investigation on its own. Although a previous study [31] has investigated the fluctuations of actigraphic acceleration signals, they only examined certain signal types in a narrow frequency band and found 1/f fluctuations in that frequency range of signals recorded during wakefulness, but characteristics over a broader frequency range were not given. To investigate the general full-span spectral nature of acceleration signals, we have performed the same analysis on the different types of multi-day-long acceleration signals as we did on the activity signals. Note that it only makes sense to investigate the spectral characteristics for acceleration signal types where the last step of the preprocessing did not include digital filtering (i.e., UFX, UFY, UFZ, UFM, UFNM, FMpre, and HFMpre), otherwise we would exhibit the characteristics of the digital filter.

Since we began the presentation of our results with the activity signals, we will continue to use the top-down approach. Firstly, we are presenting our results with the type of acceleration signals that requires the most operations to produce. These are the FMpre acceleration signals, which require the band-pass filtering of the raw axial acceleration signals before computing the magnitude of acceleration. As illustrated in Fig. 9, these acceleration signals follow the same spectral characteristic as the activity signals generally. From 10^− 4 Hz onwards in the direction of higher frequencies, the shape of the spectrum follows a straight line of slope − 1 on the logarithmic scales. Over that frequency range, which contains additional higher-frequency components compared to the activity signals because of the higher sampling rate, the spectrum also follows this trendline; although there are small local deviations that do not affect the global trend of the spectrum. At frequencies lower than 10^− 4 Hz the spectrum’s slope is milder and peaks corresponding to 24- and 12-hour periodicities appear. The fluctuation function also matches the general shape of the fluctuation function of the activity signals. The fluctuation function also follows the trendline corresponding to 1/f noise starting from the smallest box width and then diverges from it and flattens permanently. There is also a hump around the box width of 24 hours based on the 𝛽 exponent curve. The results obtained by the two analytical methods (PSD and DFA) indicate that the FMpre acceleration signals follow a similar spectral characteristic as the activity signals, including the existence of 1/f noise.

The HFMpre acceleration signals can be considered as a variation of the FMpre acceleration signals since the only difference is that instead of a band-pass filter, a high-pass filter is used to condition the axial acceleration signals. The figure of the HFMpre acceleration signals can be found in the supplementary material, they follow the same spectral characteristics as FMpre signals.

In Fig. 10, we present our results for the most straightforwardly preprocessed acceleration signal type, which is the magnitude of acceleration calculated from the raw triaxial acceleration data (i.e., the UFM-type acceleration signals). If we narrow the interval of the analysis to the same frequency range we used for the activity signals, the only difference in the spectral characteristics of UFM acceleration signals is that the slope of the linearly decaying section above the corner frequency is a bit milder (approximately − 0.8 instead of -1 on the logarithmic scales). However, at frequencies above about 10^− 2 Hz, the spectral shape of UFM acceleration signals significantly differs compared to FMpre acceleration signals as it curves upwards. The spectral differences between the UFM and FMpre acceleration signals can also be traced in the shape of the fluctuation functions. The UFM acceleration signals’ fluctuation function curves upwards below 10² s, while at larger box widths, its slope is milder compared to the FMpre acceleration signals’ fluctuation function. Overall, the UFM acceleration signals also have spectral 1/f nature and in the frequency range below 10^− 2 Hz, they follow the same characteristics as both the activity signals in general and the FMpre acceleration signals. A study has previously investigated UFM acceleration signals recorded during wakefulness and found that there is 1/f noise in the frequency range of 3.3 ⋅ 10^− 4 Hz to 3.3 ⋅ 10^− 2 Hz (approximately between 10^− 3.5 Hz and 10^− 1.5 Hz) [31] – which overlaps with the range where our measurements’ PSD shows the same nature. A slight difference is that the 𝛽 exponent they found was closer to 1, but this could be explained by the fact that we did not only examine the wakeful segments of the recordings but analysed 10-day-long acceleration signals as a whole.

The UFNM acceleration signal is obtained by subtracting 1 g from the UFM data and taking the absolute value of the difference (the figure related to the UFNM acceleration signals can be found in the supplementary material). It can be said that the spectral shape of the UFNM acceleration signals – including the 𝛽 exponent of the 1/f-like segment – follows the same characteristic compared to the FMpre acceleration signals’ spectral density below 10⁻² Hz, but differs from it at frequencies above 10⁻² Hz.

Finally, out of the x, y, and z-axial accelerations (i.e., the projections of the acceleration vector recorded on the subject's wrist), Fig. 11 shows the ensemble-averaged spectral density and fluctuation function of the UFY acceleration signals (measured on the anteroposterior axis, parallel to the forearm).

The UFY signals’ spectral structure is similar to the ones presented earlier. However, that segment that follows power-law has a slightly larger 𝛽 exponent value, it is around 1.2 based on the 𝛽 exponent curve, and the fitting. Moreover, the shape of the spectrum slightly curves downwards at frequencies above 10⁻¹ Hz. The shape of the fluctuation function reflects the observed spectral characteristics. Note that, compared to the previous spectrums, the UFY acceleration spectrum has the smallest peak associated with a 24-hour periodicity and the peak corresponding to the 12-hour periodicity is not visible. As a consequence, the fluctuation function does not have a perceptible hump around the 24-hour box width even if considering the 𝛽 exponent curve. Compared to the previous DFA results, the UFY acceleration signal’s fluctuation function flattens the earliest in the absence of the hump, at around the window length equal to the reciprocal of the spectrum’s corner frequency.

The figures for the UFX and UFZ acceleration signals can be found in the supplementary material. An order can be established between these axial signal types depending on the strength of circadian rhythmicity observed in their spectrum: while UFY has the smallest, UFX has the largest peak corresponding to 24 hours periodicity. Moreover, UFX acceleration signals (measured on the mediolateral axis, perpendicular to the forearm) exhibit the mildest flattening below the corner frequency among the raw axial acceleration signal types.

Spectral comparison of differently preprocessed acceleration signals

Figure 12 presents the total power in each frequency bin for all the analysed acceleration signal types for the sake of comparability. The slope of the power curves clearly distinguishes the raw axial acceleration signals from the others.

To more accurately identify the strength of the spectral 1/f nature of the 7 different acceleration signal types examined, we have executed the same linear fitting based 𝛽 exponent assessment as we did in the case of the activity signals. Figure 13 summarizes the typical 𝛽 exponent value for each type of acceleration signal. The fitting was performed between 10⁻⁴ and 10⁻² Hz in the case of the PSDs, and between 10² s and 10⁴ s for fluctuation functions, which are the same fitting ranges as we used in the case of activity signals for the ease of comparability.

The 𝛽 exponent values show that the PSD- and the DFA-based examination methods are consistent, as there is no significant difference between fitted exponents, the difference is 0.001 ± 0.013 (Mean ± SD). For the raw axis acceleration signals (UFX, UFY, UFZ), the exponent of the power-law scaling deviates most substantially from the ideal 𝛽 = 1 to such an extent it is hard to state that these signals exhibit 1/f noise, if strictly speaking. However, the raw, axial acceleration signals are only projections of the acceleration vector, whose magnitude (UFM) has clear 1/f characteristic. The exponent of the other acceleration signal types (UFNM, FMpre, HFMpre) generated by more complex preprocessing techniques is even closer to 1. Note that, exponents of 1.5 are also associated with 1/f noise in some fields [24]. In that case, all the examined types of acceleration signals meet the requirements of 1/f noise.

As a result of our analysis, we found that both the differently computed activity signals and the acceleration signal preprocessed in varying ways follow a general spectral characteristic in general and have the same scale-free, 1/f-type nature. The spectral characteristics were investigated using independent time- and frequency-domain analysis methods (DFA, and PSD, respectively) prevalent in the literature, which yielded consistent results. The perceived characteristics subsist of the following components. At a corner frequency of approximately 10^− 4 Hz, the PSDs’ slope significantly changes. For frequencies above the corner frequency, 1/f nature is identified. In the case of a few types of acceleration signals, the 1/f nature does not remain up to the end of the spectrum as the spectrum deviates from the previously followed trendline in the last 1–2 decades (the extent of the deviations depends on the preprocessing of the acceleration signal). The spectrum flattens below the corner frequency and the peaks corresponding approximately to 24- and 12-hour periodicities are exposed. The white-noise nature of the flattening section indicates that the long-range correlations in the signal gradually disappears below the corner frequency (i.e., approximately above period of 3 hours).

In our work, the approximate positions of frequency bands with different spectral properties (i.e., the components of the observed characteristic) were estimated mainly on the basis of PSDs which were also reflected in the results of the DFA. However, further analysis is required to more precisely determine the frequencies that delimit the assessed characteristics (e.g., the value of the corner frequency and its variance from subject to subject). It is also necessary to investigate the technical aspects of this matter with further computer simulations and analyses in the future. Beyond technical-oriented investigations, it may also be beneficial to further examine and model bio- and neurophysiological mechanisms [39, 54–56] (i.e., notable periodicities beyond the circadian rhythm, such as the circatidal clock [57] and the ultradian cyclicity [58]) to explore the underlying control mechanisms that are the causes of the observed characteristics of human activity. For example, what causes the flattening of the spectrum below the corner frequency for both the activity and acceleration signals, and for what reason does the spectrum deviate from the 1/f trendline it previously followed in the last 1–2 decades in the case of given types of acceleration signals. The relevance of such examinations is further strengthened by the fact that the presented scale-free nature and 1/f-type noise are not limited to human acceleration and activity signals. For example, the spectral characteristics of human activity presented in this work highly resemble what we have found for the PSD of minutely calculated displacements in human location data [18], but scale-free fluctuations were also identified in human brain activity [55]. Our results also raise the question of whether it is necessary to analyse activity signals in order to study human motor activity, or whether it may be sufficient to restrict ourselves to examining acceleration signals that have been preprocessed in some way without calculating activity from them if our actigraphic device is capable to record raw acceleration. The clear advantage of activity calculation is that it compresses the actigraphic recordings, so it requires fewer resources to process them while preserving the main spectral features of acceleration signals at lower frequencies. In contrast, acceleration signals carry additional information about the motion patterns due to the higher sampling rate and by avoiding lossy compression.

To demonstrate that the presence of circadian rhythmicity (i.e., the 24-hour periodicity) in itself is not responsible for the flattening of the spectrum at frequencies below the corner frequency we executed a simple computer simulation. We have generated 10 pieces of 80-day-long 1/f noises sampled with 1/60 Hz (similarly sampled than the analysed activity signals) which we multiplied with a PWM signal (high state – 1, low state – 0) whose frequency was 1/24 hours and its duty cycle was 66% to roughly approximate the general daily lifestyle of humans. The ensemble-averaged spectral density and fluctuation function of the generated 1/f noises and the multiplied signals were determined and compared, they are presented in Fig. 14.

Although the presence of periodicity results in the appearance of the peaks associated with 24 hours periodicity and its harmonics – which are also observed for real, measured signals –, they are simply superimposed on the spectra without having a global effect on the spectrum’s characteristics. It can be seen that the strong periodicity simulated by the multiplication of 1/f noise with the PWM signal alone cannot be the reason for the observed spectral flattening of real acceleration and activity signals, as it can be proved analytically, too. The effect of the multiplication with the PWM signal is also noticeable in the case of the fluctuation function, it results in a hump around the box width equal to the period of the PWM signal. We have also observed such a hump in the close region of the box width equal to the period time of the circadian rhythmicity in the case of the analysed real signals, which is consistent with our simulation.

One would think that strong periodicities would not interfere with the detrended fluctuation analysis, however, the distorting effect of dominating periodicities on the fluctuation function of long-correlated time series is a known phenomenon [31, 59, 60]. Due to the dominating periodicity, the fluctuation function deviates upwards from its previously followed trendline around the window length associated with the periodicity and then returns to the previously followed trendline creating a so-called frequency hump. However, in the case of the real acceleration and activity signals we analysed, the fluctuation function does not return to the previously followed trendline after the frequency hump, but it flattens out permanently. This confirms our observation of the existence of a corner frequency based on spectral densities. However, the frequency hump masks the real shape of the fluctuation function in that region where we would expect the distinct change of its scaling exponent to be observed based on the spectrums. In light of these, special attention should be paid to the ensemble-averaged fluctuation function of the UFY acceleration signals (Fig. 5), where the frequency hump is present to the least extent, and therefore, the masking is the least pronounced in the case of these type of acceleration signals’ fluctuation functions. It has been noted by a previous study [31] that PSD-based power-law scaling exponent estimation of actigraphic signals is more reliable because linear fitting to DFA could be biased by these frequency humps. As showed above, we also experienced the distortion effect of frequency humps during our investigation.

To explore the general spectral nature of human activity in greater detail, we have performed PSD and DFA-based analysis on a data package containing triaxial actigraphic acceleration signals of 42 healthy, free-living individuals. From a single subject's 10-day-long recording, we produced different types of acceleration and activity signals and characterized their ensemble-averaged PSDs and fluctuation functions over the entire frequency and box range.

Our main novel finding is that we revealed that both different types of actigraphic acceleration and activity signals’ spectra generally follow a universal characteristic that we described. The assessed spectral characteristics consist of the following components. At about 10^− 4 Hz (i.e., the corner frequency), a distinct change can be observed in the spectrum’ slope on log-log scales. At frequencies higher than the corner frequency, the spectrum distinctly follows S(f) ∝ 1/f^𝛽 power-law scaling, where 𝛽 ≈ 1. At frequencies lower than the corner frequency, the spectrum flattens and two distinct peaks arise, belonging approximately to the 24- and 12-hour periods (i.e., circadian and circatidal rhythmicity). Our results were mainly obtained using the PSD-based analytical method, but the fluctuation function-based analysis yielded consistent results. Therefore, we supported the validity of our results using two different analytical methods.

Based on the general spectral characteristic of the different types of activity and acceleration signals we have analysed, we have proven that 1/f nature (i.e., the presence of 1/f noise for at least 2 decades) and the spectral scale-free property are not limited to a particular acceleration preprocessing technique or activity metric. Consequently, these spectral features are inherently existing in the multi-day motor activity of healthy, free-living humans and do not emerge at any particular stage of the activity determination process. Moreover, our results showed that the measured raw acceleration signal has the same characteristics as the different activity signals calculated from it. Although the presence of 1/f noise in the acceleration signal was known for a shorter period of wakefulness [31], we were the first to identify that the multi-day wrist movement of healthy humans exhibits 1/f noise above the frequency of the daily rhythmicity.

Actigraphic measurement data

The analysed data package contains 42 raw, triaxial acceleration signals of 42 subjects measured during a 10-day observation period, each. The acceleration signals were evenly sampled with 10 Hz in the ± 8 g measurement interval and were measured on different healthy and anonymized human subjects’ non-dominant wrist. The participants were instructed to minimize non-wear time, e.g., taking the device off for bathing to avoid water damage. The description of subject recruitment and the subjects can be found in our previous article [1]. The data package is available [12] at Figshare under CC-BY 4.0 license.

Ethics declaration

The study was carried out as a part of research entitled “Examination neurobiological, cognitive and neurophenomenological aspects of the susceptibilities to mood swings or unusual experiences of healthy volunteer students„, and was approved by the Human Investigation Review Board, University of Szeged, Albert Szent-Györgyi Clinical Centre, Hungary (No 267/2018-SZTE) following its recommendations. All subjects gave written informed consent under the Declaration of Helsinki was informed of their right to withdraw at any time without explanation and they were financially compensated.

Calculation of ensemble-averaged spectral density and fluctuation functions

The definition of power spectral density and detrended fluctuation analysis are detailed in the supplementary material. To determine the ensemble-averaged power spectral density S(f) and fluctuation functions F(n) of a specific acceleration or activity signal type based on 42 subjects’ 10-day-long actigraphic recording we have executed the following procedure.

The logarithmic-binning-based ensemble averaging procedure performs the same steps regardless of whether fluctuation functions or spectral densities are to be processed. It should be noted that the functions to be ensemble-averaged must be normalized in a preliminary step to ensure that their values cover a common scale. To achieve this, we performed sum-based normalization. Since calculating ensemble averages aim to determine the typical shape of the functions on a log-log scale, therefore, only function values for which the argument is greater than 0 are included in the normalization sum (i.e., we excluded the DC-component of the PSDs). The ensemble averaging procedure requires two parameters: the N numbers of spectral densities or fluctuation functions, and B which controls the resolution of the outputted ensemble average. For the sake of simplicity, let’s consider spectral densities, the steps of the algorithm are the following.

For each spectral density S(f), keep only those components for which f > 0 (i.e., omitting the DC component). In the case of fluctuation functions, this step has no effect since fluctuation function F(n) is interpreted only for n ≥ 4. The restriction of argument n is justified in the supplementary material.

Calculate the broadest common frequency range [f_min, f_max] of the spectral densities.

Split the frequency range [f_min, f_max] into bins b₁, …, b_k, such that their width is logarithmically spaced and each frequency decade is split across approximately B bins. The k value can be calculated based on Eq. (1).

$$\begin{array}{c}k=\left[B\left(\text{log}\left({f}_{\text{m}\text{a}\text{x}}\right)-\text{log}\left({f}_{\text{m}\text{i}\text{n}}\right)\right)\right]\#\left(1\right)\end{array}$$

Within each frequency bin, determine the average of the magnitude values of all spectral densities. This produces a single averaged magnitude value for each bin, which is assigned to the geometric mean of the given frequency bin. As a consequence, the resulting ensemble averaged spectral density consist of equally spaced components if visualized on log-log scale while the data reduction level is set by B.

If N = 1 (i.e., the input is a single S(f) or F(n)), the above procedure performs averaging-based noise filtering instead of ensemble averaging. If analysing 1/f fluctuations through spectral densities’ scaling properties, it is also advantageous to calculate the sum in addition to the average in the 4th step. The summation results in the integral of each bin (i.e., the total power in the frequency bins), which must be constant in the case of 1/f noises. Since we used both approaches in our analysis, let LBBA (Logarithmic-Binning Based Averaging) the procedure where averages are determined in the 4th step, and the procedure where sums are determined instead of averages will be referred to as LBBS (Logarithmic-Binning Based Summation) for the ease of differentiation.

Applying the procedures presented so far, the following processing chain is constructed to determine the ensemble-averaged spectral density, total power in frequency bins, and fluctuation function of a given type of acceleration or activity signal (the PIM(UFNM) activity signals were chosen as an example in Fig. 15) based on 42 subjects’ 10-day-long actigraphic recording.

In the processing chain, LBBA and LBBS operations immediately following the PSD calculation aim at noise reduction and to determine the total power in frequency bins beyond data reduction, respectively, both proceeded with B = 100 resolution (i.e., 100 bins in each decade). After normalisation, the ensemble averaged spectral density and power in each frequency bin are determined using LBBA operation with B = 10 resolution.

In the case of fluctuation functions, no intermediate step is required before normalisation, as fluctuation functions were already generated with B = 20 resolution considering the computational resource requirements. The ensemble-averaged fluctuation function is determined afterward the normalisation using LBBA operation with B = 10 resolution to match the resolution of ensemble-averaged power spectral densities.

At the end of the process, for a given type of signal under investigation (PIM(UFNM) activity signal in Fig. 15 as an example) 3 ensemble-averages are obtained: power spectral density, total power in each frequency bin, and fluctuation function; each with a resolution of 10 points per decade on logarithmic scale.

Calculation of 𝛽 exponent curves

To describe the S(f) ∝ 1/f^𝛽 and F(n) ∝ n^𝛼 scaling properties, the 𝛽 and 𝛼 exponents can be evaluated between every consecutive point of the ensemble-averaged spectral density and fluctuation function based on the numerical derivation of the log-transformed data. The 𝛽 exponent between each successive point is calculated using Eq. (2), where y_PSD is N-long ensemble-averaged spectral density’s magnitude values and x_PSD is the corresponding frequencies, while i = 1, …, N − 1.

$$\begin{array}{c}\beta \left[i\right]=-\frac{\text{log}\left({y}_{\text{P}\text{S}\text{D}}\left[i+1\right]\right)-\text{log}\left({y}_{\text{P}\text{S}\text{D}}\left[i\right]\right)}{\text{log}\left({x}_{\text{P}\text{S}\text{D}}\left[i+1\right]\right)-\text{log}\left({x}_{\text{P}\text{S}\text{D}}\left[i\right]\right)}\#\left(2\right)\end{array}$$

If we exploit that 𝛽 = 2 𝛼 − 1, we can calculate the 𝛽 exponents based on the 𝛼 exponents evaluated between every consecutive point of the ensemble-averaged fluctuation functions. Eq. (3) defines this, where y_DFA is the ensemble-averaged N-long fluctuation function’s fluctuation values and x_DFA is the corresponding box widths, while i = 1, …, N − 1.

$$\begin{array}{c}\beta \left[i\right]=2\left(\frac{\text{log}\left({y}_{\text{D}\text{F}\text{A}}\left[i+1\right]\right)-\text{log}\left({y}_{\text{D}\text{F}\text{A}}\left[i\right]\right)}{\text{log}\left({x}_{\text{D}\text{F}\text{A}}\left[i+1\right]\right)-\text{log}\left({x}_{\text{D}\text{F}\text{A}}\left[i\right]\right)}\right)-1\#\left(3\right)\end{array}$$

Using the two formulas results in two exponent curves that describe the scaling properties of a given signal type in relatively high resolution compared to common linear fitting.

Author contributions

B.M. and G.V. designed the study; B.M. carried out the numerical analysis; B.M., Z.G., and G.V. discussed and interpreted the results, B.M. wrote the paper under the supervision of G.V.; B.M., Z.G., and G.V. proofed the paper. All authors approved the final version of the manuscript.

Competing interests

The authors have declared that no competing interests exist.

Data availability statement

The datasets used and/or analysed during the current study are available in the Figshare repository, https://doi.org/10.6084/m9.figshare.16437684.

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No competing interests reported.

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General spectral characteristics of human activity and its inherent scale-free fluctuations

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Abstract

Figures

Introduction

Analysis of human activity patterns

The Different Ways Of Determining Human Activity

Preprocessing techniques of the acceleration signal

Activity metrics

Examined activity signals generated using different preprocessing techniques and activity metrics

Results

Spectral characteristics of the activity signals

Examining the PSDs of activity signals

Spectral comparison of different types of activity signals

Spectral characteristics of the acceleration signals

Examining the PSDs of acceleration signals

Spectral comparison of differently preprocessed acceleration signals

Discussion

Conclusion

Materials And Methods

Actigraphic measurement data

Ethics declaration

Calculation of ensemble-averaged spectral density and fluctuation functions

Calculation of 𝛽 exponent curves

Declarations

Author contributions

Competing interests

Data availability statement

References

Additional Declarations

Supplementary Files

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