Introduction

As medical monitoring devices continue to grow in complexity and shrink in size, both the number of possible concurrent measurements and the size of the observable population increase. These factors in turn result in a rise in the amount of data available for analysis, which is driving the need for new processing algorithms. The sheer volume of recorded values makes it difficult to process within a relevant timeline using conventional means, and the intricacies of some of the more obscure variables makes them difficult to interpret at all. Because of this, the need for novel signal processing algorithms that can detect and highlight underlying subtleties and features is becoming ever more apparent. Weight management is one area in which new methods could be applied to great effect. Overweight and obesity among Americans are highly prevalent, affecting roughly 2/3 of the country’s population [1]. Obesity is a major risk factor for cardiometabolic diseases such as type 2 diabetes, hypertension, obstructive sleep apnea, and a variety of other disabling conditions, all of which combine to an impressive burden on the healthcare industry not only in terms of dollars spent but also effort and time expended. Volitional weight loss is attempted by the majority of obese individuals, but a variety of neurobehavioral mechanisms are activated following weight loss that result in weight regain in most instances [2]. However, in most controlled weight loss studies, a proportion of individuals can maintain a significant weight loss [3, 4]. To date, no definitive factors have been identified that can predict weight loss or weight regain. With the emerging monitoring capabilities, though, datasets large enough to investigate are becoming possible. The ability to differentiate between patients who will maintain weight loss and those prone to weight regain would not only allow for a greater understanding of the biology of obesity but also provide a path to tailored interventions in the population. Early interventions in those at high risk have been shown to reduce the risk of diseases such as those mentioned above [5].

This problem, along with many others in the clinical setting where time-series measurements are to be analyzed, stands to benefit from advanced signal processing. A common practice in the medical field, various signal processing algorithms have been applied to a wide variety of situations. From heart rate monitoring to myoelectric signal classification, various techniques are used to analyze time series data. Feature extraction has proved to be an effective approach, commonly used in health applications, and involves calculating variables characteristic of the signal. Another approach, direct comparison methods, analyzes and compares raw signals simultaneously. The power, entropy, and average value of a signal are three common features used in signal processing, and the Pearson correlation coefficient is used when comparing signals and samples directly. The currently used methods, while effective in many applications, cannot always differentiate between two sets of similar signals in complex health applications, especially when the differences among signals are subtle. Indeed, the main shortcoming of existing models in predicting weight maintenance is a clear lack of effective computational approaches to mining available data. Instead, most of the existing predictive models of obesity are based on correlation of weight regain/loss with only a small amount of basic patient information, which result in models with limited predictive capabilities. For instance, a major factor that can help lead the personalization of treatments for obesity is estimation of the type and level of physical activities. However, the motion signals, when analyzed with the conventional signal processing methods, have failed to produce accurate and robust prediction results. The complexity of the data collected for any one patient, let alone that found in the collective data of a large study groups, demands more advanced computational techniques that can extract these subtle patterns and distinguish sub-classes of weight loss and regain. In this study we describe a new approach in signal processing that can detect subtle changes in the behavior of complex signals, in particular motion time-series, and as such distinguish between patient cohorts with different clinical outcome. The proposed approach is based on our extended formulation of the persistent homology theory and introduction of a modified, semimetric version of the Hausdorff distance to analyze data in the feature space [6–12]. We apply and demonstrate the efficacy of the proposed computational methods in differentiating between various levels of weight loss maintenance.

Materials and methods

Signal analysis using persistent homology

Persistent homology is a broad mathematical theory, and one of its applications is examining how the characteristics of an object in a space change based on the spatial resolution used to examine the object. As the resolution changes, persistent homology features, representing the special characteristics of the object, quantify these changes. The transitions in these features can be studied to help develop a better understanding of the object. When applied to time series as objects, persistent homology can be used to extract features of a signal representing the changes in the characteristic patterns of variations observed in the signal at different resolutions [14–21]. Specifically, the persistent homology algorithm converts a signal into points scattered across a min-max plane. By treating each minimum in a time-series as the ‘birth’ of a feature and each maximum as a ‘death’ it is possible to examine the significance of a trend by the persistence of its corresponding min-max pairing. Larger differences between two extrema correspond to more pronounced variation, and any resulting points will be farther from the y = x diagonal. Conversely, a point closer to the diagonal represents a smaller magnitude of change and is more likely to be noise. The resulting min-max plot, or ‘persistence diagram’, represents the characteristics of the input sequence and can be used to compare the differences between the patterns and variations of signals. This information can also be visualized in a ‘barcode’ format, as described in [22]. The process, including the derivation of the persistence diagram, is as follows:

Suppose that f is a real-valued function on the discrete set {1,2,…,n}. To make notation convenient, we also define f(0) = ∞ and f(n+1) = −∞. Also, we modify f to define a function defined by where ε > 0 is infinitesimal. We define the function-value ordering ⊑ on {0,1,…,n+1} as follows. If 0 ≤ i, j ≤ n+1 then we define i ⊑ j if . Equivalently, we have i ⊑ j if f(i)<f(j), or f(i) = f(j) and i < j. The relation ⊑ on {0,1,…,n+1} is a total ordering. So for all i and j we have:

1. i ⊑ i;
2. i ⊑ j or j ⊑ i;
3. if i ⊑ j and j ⊑ i, then i = j;
4. if i ⊑ j and j ⊑ k then i ⊑ k.

We also write i ⊏ j if i ⊑ j and i ≠ j. Define the set of local minima and maxima by (1a) (1b) respectively.

Lemma 1. The sets E_min and E_max have the same number of elements.

proof. It is not hard so see that the smallest element of the set of extrema E = E_min ∪ E_max with respect to the ordering ≤ lies in E_min. Also, the largest element lies in E_max. It is also elementary to see that local maxima and local minima alternate. So the number of local maxima and local minima is the same.

By the lemma, there exists r, a₁, a₂,…,a_r, b₁, b₂,…,b_r such that E_min = {a₁, a₂,…,a_r} and E_max = {b₁, b₂,…,b_r} such that a₁ ⊏ a₂ ⊏ ⋯ ⊏ a_r and b₁ ⊏ b₂ ⊏ ⋯ ⊏ b_r. The proof of the lemma shows that there are permutations τ and γ in the symmetric group S_r such that

Lemma 2. We have a_i ⊏ b_i for all i.

proof. It is clear that a_τ(i) ⊏ b_γ(i) for all i. For j ≤ i we have a_{γ⁻¹ τ(j)} ⊏ b_j ⊑ b_i. So at least i of the a’s are smaller than b_i with respect to ⊏. This implies that a_i ⊏ b_i.

If then we define (2)

We define a permutation σ ∈ S_r and sets U₁,U₂,…,U_r inductively as follows: (3) and (4)

By Lemma 2, the set U_k is nonempty.

Definition 1. The persistence diagram associated to the function f is

The mathematical complexity of the persistent homology approach formulated above might mask its great conceptual potentials to analyze complex signals. Below is a graphical illustration and explanation of this method. As shown in Fig 1, first a given signal is plotted and all local extrema (maxima and minima) are identified. A square plot in the min-max plane is generated and placed such that the vertical axes align between the two graphs, and the line y = x is drawn (Fig 1a). Starting with the smallest minimum (in this case, point 1), a mark is placed on the min axis of the second graph corresponding to the minimum’s y value (Fig 1b). Each local extremum is considered, moving from smallest to largest, and for each minimum another mark is placed (Fig 1c). When a maximum is encountered, that point’s y value is paired with the mark from the most recent minimum, as long as there is no other maximum between the two points. If such a maximum exists, the second most recent minimum is considered, and the process continues until a suitable pairing is found (Fig 1d). The next extremum is then considered, and the algorithm proceeds until all points are paired (Fig 1e). In the event that there are more minima than maxima (or vice versa), the point with the largest x value in the larger set is dropped. Once completed, the points in the min-max plane constitute the signal’s persistence diagram (Fig 1f). As it can be seen, starting from a time-series, persistent homology creates a pattern of dots in the max-min plot that represent the variations of the signal.

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Fig 1. Persistent homology.

https://doi.org/10.1371/journal.pone.0177696.g001

These min-max patterns represent how the variations in signals can be used, as described later, to effectively distinguish between different types of signals (e.g. signals representing ‘success’ and those representing ‘failure’). Next, we describe our method of distinguishing between different patterns/signals by using a measure that quantifies the disparity between different min-max plots.

Results

To begin, the power of each raw armband data was calculated (Table 1). This rather simple and intuitive feature is often used in analysis of activity signals as a main characteristic number describing the data. In this study, however, there is no measurable difference between the power of a participant labeled as a failure and that of a participant labeled as a success (p = 0.326). The sample size was 100, of which 79 were labeled ‘failures’ and the remaining 21 were considered ‘successes’.

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Table 1.

https://doi.org/10.1371/journal.pone.0177696.t001

The entropy and average of signals, also popular in assessment of activity signals, were calculated and analyzed as well (Table 1). Entropy as a measure of disorder and information can often distinguish between functional classes of data, in particular when dealing with biomedical signals, and is heavily used in the signal processing literature. However, as in the case of power, there was no significant difference between the failure and success groups (p = 0.608 and p = 0.262, respectively).

As a final check using current established methods, correlation coefficient calculation was used to analyze the data. For this, each signal was compared to every other signal in the pool and a correlation value was obtained. When armband data from two individuals labeled as failures were compared, the resulting signal was placed into a ‘failure vs failure’ group (N = 3081); likewise, when the movement profiles of two successes were compared, the result was placed into a ‘success vs success’ group (N = 210). When comparing two ‘failure’ signals, the average correlation coefficient was 0.0712, while the average for comparing two ‘success’ signals was 0.0992. While the standard deviations were relatively high, as shown in Table 2, there was a statistically relevant difference between the groups (p = 0.006).

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Table 2.

https://doi.org/10.1371/journal.pone.0177696.t002

Comparing the ‘failure vs success’ group (N = 1659) to the ‘failure vs failure’ group yields another statistically significant difference (p = 0.0004), but comparing it to the ‘success vs success’ group does not (p = 0.1968). This indicates that the failures share less intra-group similarities than do the successes, but the second two cases are indistinguishable under this metric.

Next, the armband data of each of the 100 participants was compared to every other file using our proposed windowed persistent homology method. Persistence diagrams were generated for each of the 350 windows extracted from a signal. These plots were compared to corresponding plots from a second signal using the modified Hausdorff semimetric, and a single number was noted for each window. The result of the algorithm, when applied to two armband/activity files, was a new signal with a length of 350 points, corresponding to the distance between the two input plots over time. The top of Fig 2 shows two armband signals plotted together, one drawn in red and the other in blue. Plotting the measured distance between two corresponding windows over the course of the analysis yields the signal shown in the bottom of Fig 2.

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Fig 2. Data analysis.

https://doi.org/10.1371/journal.pone.0177696.g002

The average value of the resulting signal was placed into a group based on the input signals, again separating ‘failure vs failure’, ‘failure vs success’, and ‘success vs success’. The average of these averages was then calculated and this was used as the characteristic value for each group (Table 3). Using an unpaired t test, the analysis showed that there is a statistically significant difference between not only the two main groups (p = 1.505⁻²³) but between any two of the three (p = 1.661 × 10⁻²⁸ and p = 5.715⁻⁹ for ‘failure vs failure’ vs ‘failure vs success’ and ‘failure vs success’ vs ‘success vs success’, respectively). It should be noted that the smaller average distance between successes when compared to that between failures is further strengthened by the correlation analysis: while the coefficients were small, the successes tended to be more highly correlated with one another than did the failures.

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Table 3.

https://doi.org/10.1371/journal.pone.0177696.t003

These results were consistent across variations in window size: values from 15 to 25 were also tried, with p-values no worse than an order of magnitude higher (p = 1.170 × 10⁻⁸ for ‘failure vs success’ vs ‘success vs success’ with a window size of 16); some comparisons were more significant. Additionally, the method was implemented on ‘blind’ files in which the labels were randomly generated. With this random distribution there were no statistically significant differences between the groups (average p = 0.113 over 100 trials for the main pairing), further reinforcing the notion that our method is capturing some difference in the underlying structures of the signals.

Finally, we applied the same approach but this time used the q-th Wasserstein true distance metric to compare the persistence diagrams (q equal to one; code provided by [23] was used). As show in Table 4, the results are again statistically significant when comparing the ‘failure vs failure’ group to the ‘success vs success’ group (p = 1.231 × 10⁻⁵). While comparing the ‘failure vs failure’ cohort to the ‘failure vs success’ set also yields a significant difference (p = 6.181 × 10⁻⁷), it should be noted that both p-values are larger than their counterparts obtained using the semimetric. In addition, this true metric does not detect a measurable variation between the ‘failure vs success’ and ‘success vs success’ groups (p = 0.012). When run with randomly generated labels, the results are once again not significant for any combination of groups (average p = 0.084 over 100 trials for the main pairing).

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Table 4.

https://doi.org/10.1371/journal.pone.0177696.t004

Despite the lack of complete statistical significance between all pairs, the pattern of higher intra-group similarities in the successes than the failures is continued. This, combined with the results of the ‘blind’ files, lends even more credence to the claim that there is an underlying disparity between the behaviors of the two groups.

Discussion

Fig 3 shows an example comparison from each of the three groupings. As shown in the plots, the general movement profile recorded by the armband sensors varies more heavily within the failure group than in the success group. Intuitively, this indicates that those destined to fail behave in a variety of different ways while those who will lose weight and maintain the loss share a more unified pattern of behavior. The general topographical structures present in the signals represent these overall patterns of activity and are captured in the persistence diagrams; in this sense, persistent homology is ideally suited to uncovering the underlying differences. Because of the larger variation present across the movement profiles of the ‘failures’, the signal-to-signal comparisons yield consistently higher values of both the average distance and the corresponding standard deviation across all metrics (true and semi-) used above.

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Fig 3. Signal comparisons.

https://doi.org/10.1371/journal.pone.0177696.g003

In utilizing our persistent homology algorithm, we are extracting a set of connected components from the time-series data [24]. In looking at this homology group, we are examining the underlying patterns of each individual’s short-term behavior. Intuitively, this shows that the types, frequencies, and amplitudes of movement vary between each group, not just in general trends but also in minute-to-minute fluctuations. Future studies of this phenomenon could lead to discoveries pertaining to physical movement and how it contributes to and informs future weight loss success.

While further analyses with more participants would greatly help strengthen and validate these results, the initial implications are twofold. First, the clear disparity between the two groups indicates that there is a measurable difference in the movement profiles of those that will lose weight and keep it off versus those that will not lose any or, after losing weight, regain a substantial amount. By measuring this contrast a patient could potentially be classified as a ‘success’ or a ‘failure’ before even beginning a diet, in turn leading to more effective and individually tailored interventions. This would help to greatly ease the economic costs associated with overweight and obesity, as well as their related diseases. It would also save both the clinician and the patient time spent pursuing a course of action likely to produce unsatisfactory results, instead allowing them the option to first pursue alternatives. Secondly, and more immediately evident, the results indicate that the windowed persistent homology method, coupled with the modified Hausdorff semimetric, is capable of detecting subtle, underlying differences between signals. This method could potentially be used in other clinical settings where a deeper analysis of a complex signal would result in improved care, as well as other signal processing applications. Relatively tolerant of noise and sampling frequency, our algorithm can be easily applied to short and long time-series alike, drawing out features from the signal useful in exposing subtle differences.

While the exact physical characteristics measured by the persistence diagram remain unclear and will be closely examined in future work, it can be said that the patterns representing more scattering in the persistence diagram represent higher levels of physical activity. In addition to further exploring the physical and physiological implications of an individual persistence diagram, there are a number of modifications to the implemented algorithm that we will investigate in future work. For instance, in our analysis, the inputs were blindly compared. In our future work, we intend to implement a set of alignment procedures in the pre-processing steps. Syncing time of day or sleep/wake cycles between two armband signals before the persistent homology algorithm is applied could help reduce any noise associated with comparing across states (e.g. one participants sleep data with another’s morning routine). Another route for our future investigations will be to implement a dynamic windowing method: by altering the length of each window based on the number of included extrema, we can improve the resolution of our algorithm in areas of high activity (e.g. exercise) without sacrificing accuracy in low-movement periods (e.g. sleep). A third future improvement to the analysis involves error-checking the edge cases. If a window boundary splits a monotonically increasing or decreasing section of the signal, a false maximum and minimum are formed on either side. By implementing a check on the next point outside any given window, we can reduce the number of artificial extrema, thus reducing inaccurate pairings and noise in the persistence diagrams.

Conclusions

To predict a priori weight loss/maintenance success in overweight or obese individuals by applying information learned from one week of simple, noninvasive measures would heavily impact the healthcare industry. With such a high prevalence rate in the country, both the economic burden and the time spent treating overweight, obesity, and their related diseases could be drastically reduced. By facilitating more tailored and individualized treatment, which may include modifying an existing program, identifying alternative modalities, or focusing on other patient issues, countless hours could be saved for both the patient population and the clinicians. The results presented in this paper indicate, through the use of a novel computational method, a measurable contrast between the group of participants able to maintain weight loss and the group unable to do so. By using the windowed persistent homology method defined above and the modified Hausdorff semimetric, a physician could determine whether or not a specific intervention would be effective for a given patient. This project demonstrates the effectiveness of the novel signal processing method and the potential impact it can have on clinical decision making and patient care.

Supporting information

Acknowledgments

We would like to acknowledge the Michigan Center for Diabetes Translational Research, the Blue Care Network, and the Blue Cross Blue Shield of Michigan Foundation.