We will now discuss various aspects of our time-dependent complexity method based on the results and figures presented in the previous section. First of all, we argue why a static complexity metric is not sufficient to characterise the observed changes in the fractal dimension of an activity pattern over time. We also explore the following hypothesis: weeks in which a patient showed decreased functioning are coupled with a lower complexity of the week’s activity pattern. Further, we investigate whether the complexity evolution carries any trends which could not be readily extracted from the activity pattern itself and discuss which relations might arise between the two signals and why. Next, we study the extent of scale-dependent variations in fractal dimension and provide an intuitive interpretation of the scaling behaviour. We then compare our method to other similar studies which attempt to characterise the complexity of CFS patients’ activity patterns. Finally, we discuss the limitations of our work and provide guidelines for future work.
Within-patient variations in complexity over time
Figure
4 illustrates how the fractal dimension of an activity pattern can evolve over time. For some patients, the fractal dimension tends towards both the minimal and maximal theoretical value (respectively equal to 1.0 and 1.5) at different points throughout the recording period. For others, the fractal dimension does not change as drastically, but still seems to evolve over time. We can observe a daily oscillation in the signal, which is more regular for some patients than for others. We refer the interested reader to the supplementary material (see [
30]) for a detailed explanation of this phenomenon. In summary, it arises due to an alternation between sleep and wake time. Because we move our sliding window at the small resolution of 5 minutes, these day-night rhythms are present in the outcome of the method. Such momentary oscillations in the fractal dimension are not necessarily informative of longer-term patterns in the behaviour of patients, and when disregarding them we can still observe considerable changes in complexity within the range of days and weeks. From now on, when we talk about temporal changes in complexity, we are referring to these macro-trends, such as the rise and fall of the fractal dimension over multiple days.
While application of the allometric aggregation method to the full activity pattern at once will implicitly average out some more unreliable fluctuations and artefacts, information about the longer-term changes in complexity will be lost as well. Indeed, even though a weekly static fractal dimension seems to (partially) summarise the observed complexity evolution throughout the week, it does not do these variations justice. The time-dependent complexity method allows us to capture how the fractal dimension evolves over a period of several days and weeks. At the same time, studying the general course of this complexity signal can still provide an idea of the overall complexity of the full activity pattern in the long term. From these observations, we learn that a temporal averaging of the fractal dimensions obtained in smaller windows (i.e. the mean of the outcome of the AAA method for all 3-day windows in one week of recording, which would be the average of the t-AAA curve in one week), is not the same as performing the AAA method to get the fractal dimension over one larger window (i.e. the AAA method applied to one week of recording, depicted in the dashed line).
Apart from illustrating the variations in complexity over time, Fig.
4 also reveals the potential of a time-dependent complexity characterisation to capture the personalised nature of CFS. One patient’s complexity signal might behave more erratically than the other (e.g., patient 6 vs. patient 4), reflecting different properties of their activity patterns and possible differences in underlying disease states. Of course, it is not surprising that the properties of the activity patterns differ from patient to patient, as anyone’s daily schedules and events that might interrupt these are entirely different. Apart from this, however, patients can also experience various inhibitions as a result of CFS, which we can expect to leave different marks on the activity pattern and in turn on its fractal properties. Both of these aspects give reason to study the complexity evolution
within patients, rather than comparing
across them.
Charts like the ones presented in Fig.
4 could form a useful point for discussion between patients and their clinicians. We imagine the chart could be annotated with further information on patient functioning, important life events, and possibly other temporal signals such as the activity counts and their extracted properties. Even in cases where it proves difficult to identify a mapping from trends in complexity (or other temporal quantifiers anyone might want to investigate) to changes in patient functioning, the ensuing discussion could still enhance the mutual understanding between patient and clinician and provide them with a fresh outlook on the patient’s personal triggers that perpetuate their symptoms.
Relation between complexity and patient functioning
As outlined in the
Background section, we have reason to believe that the evolution of the complexity of the activity pattern of a CFS patient might form a way to quantify changes in their underlying disease state. This forms the motivation to investigate whether changes in complexity (here, measured in terms of fractal dimension) can be related to changes in high-level patient functioning. Prior studies [
20,
21] investigated the hypothesis that CFS patients show signs of reduced complexity compared to healthy controls. We propose to move from their idea of comparing a
single quantification of complexity
between patients to comparing the complexity over
different windows in time but
within a given patient. To this end, we formulate the following hypothesis:
\(\mathcal {H}\): Periods of decreased functioning in CFS patients are associated with a higher fractal dimension (corresponding to a reduced complexity) of the corresponding activity pattern.
To study this hypothesis, we would ideally compare the time-dependent complexity signal as depicted in Fig.
4 to a fine-grained (i.e. in the order of days) evolution of patient functioning. Currently, we only have a rough ranking of each patient’s three weeks in terms of functioning, meaning we cannot perform such a fine-grained comparison. Instead, we turn to the rank correlations between weekly static complexity and weekly functioning reported in Table
2. While many of these correlation coefficients are positive, which is in line with the hypothesis, the rankings of the weeks in terms of functioning and complexity do not consistently align. For this reason, hypothesis
\(\mathcal {H}\) has to be rejected based on the data we currently have. Since we do not have a more fine-grained indication of patient functioning on the daily level, we also refrain from speculating about whether sub-weekly trends in complexity (e.g. the rise and fall of the fractal dimension within a single week, which often occurs in Fig.
4) could possibly be related to changes in the patient’s functioning during that week.
From Table
2 it is also clear that the weekly activity counts mostly show a negative correlation with weekly functioning, though again both rankings are not consistently aligned. This shows that the activity pattern in its simplest form is not highly indicative of patient functioning either.
Relation between physical activity and complexity
We can ask ourselves whether the time-dependent complexity signal encodes any trends which could not readily be extracted from the activity pattern itself. This question can be partially answered by studying the linear relation between the two signals. As illustrated by the last column in Table
1, the linear correlation coefficient obtained for samples extracted from both signals varies across patients. For some patients, the activity counts within a window are strongly correlated with the fractal dimension obtained for that part of the activity sequence, while for others this correlation is much less strong. For patients of the former type (for example, patient 4), the time-dependent complexity method might show less potential for revealing novel properties of the activity pattern, again emphasising the personalised care framework in which this method should be viewed.
Whether weak or strong, the correlations which are not close to zero are all negative. This means that segments with a higher total of activity counts are usually paired with lower fractal dimensions, suggesting the presence of stronger fractal correlations (generated by a more regular process) in these sequences than for their less active counterparts. Indeed, if there is a higher total activity in the studied 3-day segment, the variation of the aggregated activity counts signal (i.e., when taking subsequent steps in the allometric aggregation procedure) is expected to increase more rapidly relative to the total activity, compared to segments with a lower total activity. This results in a steeper slope on the log-log plot and a lower fractal dimension. Intuitively, we can interpret this as the fractal organisation of movement being less similar to that of a random process when it is part of a more physically intense activity. We hypothesise that such activities are, on average, executed with more purpose and structure than a sedentary activity which is interrupted by more randomly dispersed movements.
At the same time, there are several instances where higher activity counts do not result in higher complexity (otherwise the correlation coefficients in Table
1 would be much closer to -1). While investigating the activity signals in detail, we realised that relatively short-term high peaks in activity counts can interrupt the more regular organisation of the activity sequence. This then seems to lead to a momentary increase of the fractal dimension, contradicting the general observed trend of activity sequences with higher counts being paired with lower fractal dimensions. Since patients indicated the type of activities undertaken throughout the day in the daily surveys, we were usually able to identify these momentary increases in activity counts as intensive sports, such as fitness or running. Indeed, as these relatively constrained movements bear less relation to the surrounding free-range movements, the assessment of such a sequence’s complexity across timescales will be more similar to that of an uncorrelated random process (which, in theory, has a fractal dimension of 1.5).
In order to leverage the potential of the time-dependent complexity method to reveal novel properties of patients’ activity patterns, insight into which aspects of the activity pattern directly influence the obtained fractal dimension is needed. It might be appropriate to remove or reduce large momentary peaks in activity counts
4 before calculating the fractal dimension, as their influence might distort the obtained complexity disproportionately. In this way, the obtained fractal dimensions would disregard extreme interruptions in activity counts, and the signal might be more reflective of the fractal correlations present in the free-range activity and movements of the patients. The part of the remaining trend which can be attributed to the described negative linear correlation between activity counts and the fractal dimension could then be removed, leaving only the part of the observed trend which reveals fractal properties of the activity signal which were not obvious from the activity pattern itself. We would expect this latter part to be more reflective of underlying changes in the regulation of the patient’s complex adaptive system, and exposing it could allow the identification of stronger relations with patient functioning and their general disease state.
Scale-dependent variations in complexity
The goal of this section is to dig deeper into the properties of the AAA method that are related to variations across scales. We both aim to provide intuitive explanations for some phenomena observed in the scaling behaviour of the activity sequences we studied, as well as motivate some of our choices regarding the parameters of the (t-)AAA method that were used to obtain the results reported in previous sections.
Apart from its variation across time, we have also shown that the fractal dimension can vary according to the scale parameter (represented by
n, the level of aggregation in the allometric aggregation algorithm). Of course, as the complexity metric is designed to capture the fractal correlations across scales, the slope obtained around a particular value of the scaling parameter
n will partially depend on the strength of the correlations on all scales, from
\(n_{min}\) to
\(n_{max}\). It is the overall coherence of the variations within various scales, ranging from a level of minutes to a level of hours, that decides whether the time series shows any sense of self-similarity and to what extent. However, at the same time, the lower range of scales may still exhibit a different strength of fractal correlations than the higher range of scales, which is illustrated in Fig.
2: the relation between mean and variance is better captured by a polynomial than by a single straight line. In this case, we see that lower scales exhibit steeper slopes, corresponding to lower fractal dimensions. The discussion that follows links the inner workings of the allometric aggregation method to an intuitive explanation for this observation.
Imagine dealing with a time series generated by a random uncorrelated process, characterised by a certain mean and variance. When aggregating the samples into blocks of size n, we essentially take the sum of n independent and identically distributed variables with identical mean and variance. Basic statistics allow obtaining the mean and variance of the resulting aggregated time series as the original mean and variance both multiplied by n, respectively. On a log-log scale, this results in a slope of 1 (and a fractal dimension of 1.5), since both mean and variance increase with the same amount. When we are dealing with a process that exhibits some regularity, consecutively generated samples will not be independent anymore. As we aggregate the series into blocks of n samples, the mean will still increase with factor n, but now the variance will increase with an additional amount proportional to the covariance between the consecutive samples (reflecting the strength of the dependence between the samples generated by the process). On a log-log scale, this results in a slope that is steeper than 1. This translates into a fractal dimension lower than 1.5, reflecting that there is some regularity in the signal, as is indeed the case for the activity patterns we present here. Since the slope in the lower scaling ranges is steeper than in the higher ranges, we can state that the signal shows more regularity for these lower scales. Our intuition would indeed confirm that there should be more regularity in movements when these are compared from one minute to the next, as particular activities often span a time-frame larger than a couple of minutes.
At higher scales, the slope becomes lower, resulting in a higher fractal dimension. Suppose, for example, that we are considering 3-hour scales. This effectively means that we are comparing consecutive 3-hour segments and calculating their variation in terms of total aggregated activity counts enclosed in these segments. We can expect that the properties of this time series correspond more to those of a random signal than was the case for lower scales, as we may be comparing across activities less related to one another. However, the fact that the slope of the mean-variance relation still remains higher than one shows that some regularity is still present in the signal at this scale. There is usually still some purpose behind a person moving from activity to activity, explaining why activity patterns show a degree of self-similarity even on higher timescales.
The question that follows is which range of scales is more important when analysing the complexity of our activity patterns. We can argue that scales of a different order of magnitude each capture a different notion of complexity. The way the strength of the mean-variance relation changes across scales as we move to higher ranges also contains information about the complexity of a time series. When we discard information on the self-similarity of the signal when viewing it on lower scales, we discard information about how the subject’s movement is hindered when operating within a specific activity. On the other hand, when discarding the fractal dimension obtained for higher scales, we discard indications of the activity pattern’s self-similarity across higher-level tasks and daily schedules. In the current work, we have made the decision to focus on the fractal dimension obtained for higher scales (\(\sim\) 3 hours) when presenting our time-dependent complexity characterisation, in order to simplify the presentation and analysis of these results. However, future work should focus on developing a temporal complexity representation which also contains information about the signal’s scale-dependent behaviour, without disregarding the fractal properties in the lower range of scales (\(\sim\) minutes).
Apart from taking into account these considerations on the changes in fractal dimension across scales, there is also a trade-off to consider when choosing the width of the window used to obtain the time-dependent complexity signal. The slope of the mean-variance relation can only be calculated reliably if enough segments are created in the higher-level aggregation steps. This induces a limitation on the width of the window which selects the part of the activity series to feed into the algorithm. For example, when we aim to study the fractal correlations up to a scale of 3 hours, the window should not become much smaller than 24 hours. While a larger window allows for more reliable estimates, it also results in a slower manifestation of the changes in the activity series’ properties. Intuitively, it is not always clear which choice of window and scale is most appropriate in light of this trade-off. From experience, we recommend the window size to be at least 8 times as large as the maximum scale (\(n_{max}\)) one desires to study. The step size, which we fixed to 5 minutes, can be chosen freely according to the desired granularity of the complexity signal, although it will impact the execution time.
The 3-day sliding window we chose to apply is sufficiently large in relation to the 3-hour scale we focus on, which would require a 24-hour window at least. Going for a larger window increases the reliability of the estimated means and variances in the AAA algorithm, reducing the amount of sharp momentary peaks in the complexity signal which would make it harder to extract and interpret a global trend in complexity. A larger window also decreases the magnitude of the daily oscillations which arise due to an alternation between sleep and wake time, a phenomenon we have pointed out briefly when discussing the within-patient variations in complexity. Furthermore, we believe this window size better reflects the level at which we desire to observe changes in complexity for these patients. We do not want momentary changes in the behaviour of the patients, which may not bear a relation to changes in the manifestation of their illness, to have an immediate impact on the complexity characterisation. Rather, we want to observe slower but more permanent changes in complexity, which might still be triggered by a singular event in the patient’s timeline, but which bear a more long-lasting impact. We believe that a window of 3 days is long enough to average out the effect of irrelevant momentary changes, while the resolution of the information contained in the resulting complexity signal is still high enough compared to its full length of 3 weeks.
Comparison with other studies
We identified only two other studies which investigated the complexity of CFS patients’ activity patterns in particular. Ohashi et al. [
20] used two techniques, detrended fluctuation analysis (DFA) and wavelet transform modulus maxima (WTMM), to estimate the fractal scaling exponent of 14-day physical activity time series. They reported that CFS patients’ activity patterns showed indications of reduced complexity compared to healthy controls. Burton et al. [
21] reported similar observations, using the allometric aggregation method (cf. Algorithm 1) to estimate the fractal dimension of 12-day activity patterns. What both studies have in common, is the fact that they characterised the fractal properties of the 2-week long time series using a single static complexity value per patient, as opposed to the time-dependent characterisation presented here. Both studies were only able to significantly discriminate very inactive CFS patients from healthy controls based on the fractal properties of their activity time series.
As motivated previously, we decided to use the same metric as Burton et al. [
21] to allow for a closer comparison. Furthermore, we followed Burton et al. [
21] in converting the raw acceleration recordings to activity counts. Our observations suggest that their results should be interpreted with caution. Table
1 illustrates how the fractal dimensions obtained for the first, second and third week of the recording are relatively far apart for some patients. Some variations are around the same order of magnitude as the differences reported by Burton et al. [
21] between the average fractal dimension obtained for their control group (1.14), their active CFS patient group (1.16) and their pervasively inactive CFS patient group
5 (1.21) (though the scale for which these dimensions were obtained is unclear). This indicates a lack of robustness of the static allometric aggregation method, especially for its application in a diagnostic context: the fractal dimension of a week-long activity pattern might classify a subject to fall within the CFS patient group one week, but not the next.
The observed temporal variations in complexity might partially explain why little statistical difference was found between the fractal dimension of active CFS patients and controls. As opposed to the method applied by Burton et al. [
21], a time-dependent complexity characterisation of the activity pattern over multiple weeks does not discard information on the variations in fractal dimension. While identifying new methods for diagnosis of CFS is not the focus of the current work, we can argue that retaining this information might lead to a more robust complexity-based diagnostic method.
We can also question whether comparison between groups of patients and controls, based on fractal dimensions obtained for a particular scale, is warranted. Burton et al. [
21] did not provide a motivation for the scale upon which they evaluated the fractal dimension, and it is unclear whether multiple scales were tried and how their results would differ in these cases. In any case, Fig.
3 illustrates how comparisons between patients can differ when the fractal correlation strength in different scaling ranges is explored. The same might be the case if we compared these patients with healthy controls: a patient might have a lower fractal dimension on a scale of 30 minutes but a higher one on a scale of 3 hours. It is certainly necessary to stick to one particular setting for the scale parameter to ensure comparability of fractal dimensions (as was done by Burton et al. [
21]), but apart from this it would also be advised to explore various scale settings and report whether this impacts the significance of the results.
Both Ohashi et al. [
20] and Burton et al. [
21] only reported significant differences for so-called pervasively inactive CFS patients when comparing them to a group of healthy controls. Patients who are relatively active despite their CFS diagnosis could not easily be identified. The activity patterns for the inactive patient group are likely very static (showing permanently low activity counts), and in this case it makes sense that the fractal dimension obtained for such activity patterns would be relatively static as well, allowing them to be discriminated from the group of healthy subjects. Additionally, in the case of these pervasively inactive patients, we see that Burton et al. [
21] were initially able to identify a significant difference in mean activity counts between this group of CFS patients and their matched healthy counterparts. It begs the question whether a complexity-based separation of these groups is even needed, if the activity pattern in itself already suffices for this purpose. Though we cannot directly compare the patients included in our trial to theirs, based on the interviews and objective activity counts, we would not classify any of the patients included in our pilot trial as pervasively inactive. It is exactly for this rather active group of patients that we expect an evolution of the complexity to be more informative than a single static complexity metric.
Limitations and future work
We have proposed an extension of the allometric aggregation method to capture the evolution of complexity over time. While we applied it to activity sequences in the context of CFS patients, the t-AAA method is sufficiently general to be applied to any time signal obtained in- or outside a clinical context. Future work could explore its potential to extract temporal variations in complexity from other time series. For example, as fractal properties of HRV signals have been shown to contain information which can be linked to heart failure [
35], we can expect quantification of variations in the complexity of the HRV signal to be useful in contexts outside of CFS as well.
Apart from showing that the fractal dimension evolves over time, we also showed that it varies across scales: the fractal properties are different when evaluated around scales at the level of minutes vs. hours. We made a choice to focus on the evolution of the complexity on a 3-hour scale, but the fractal behaviour of the activity pattern in the lower scale range is different, and its variation over time likely contains complementary information. In the future, we should try to find a representation of the complexity and its variation over time which takes this scale-dependent behaviour into account as well.
Furthermore, additional insight is needed into the relationship between certain properties of the activity pattern and the influence these have on the fractal dimension which is derived from it. In our comparison of complexity with activity, we discussed the possible distortion of complexity as a result of large momentary peaks in activity counts (which might occur when a patient engages in heavy exercise). We also described the observed overall negative linear correlation between activity counts and the fractal dimension. Given these observations, we propose that future research should jointly model complexity and activity to account for the relation between both.
Seeing as we were unable to confirm our hypothesis that periods of decreased functioning are related to a decrease in complexity of the CFS patient’s activity pattern, it seems unlikely that the proposed complexity metric (applied to physiological outputs such as activity) can adequately quantify a patient’s state of well-being. We foresee that future research should investigate the potential of a multi-modal approach, where various metrics that can be continuously tracked (e.g. activity counts and derived features, possibly including complexity) are combined into one monitoring system. Instead of focusing on a predefined metric, such as the method of allometric aggregation, we should look into machine learning-based methods to obtain a personalised temporal representation of each patient, which can serve as a quantification of the underlying properties of the patient’s CAS.
Finally, it would be useful to repeat the conducted analysis for a larger and more diverse group of patients, e.g. including patients who have only recently received a diagnosis and have not yet developed many coping mechanisms. The purpose of this extension would not be to compare results across patients, but rather to have a broader view on the extent of within-patient variations in complexity and functioning. We pose that such a follow-up study should collect more fine-grained (i.e. on the daily level) indications of general functioning. This would allow a more direct comparison with any fine-grained temporal quantifiers derived from the activity pattern, facilitating the development of multi-modal temporal representations that are predictive of the personal functioning.