# Chapter 1

## Medium-term monitoring of chosen features

The combination of breathing rate and amplitude allows for the detection of a greater number of respiratory mechanics adaptation in short-term changes. After implementation of these features in the monitoring device, we analyse if they are also relevant for the detection of changes related to slower processes, where changes take place along the days.

In this part of the work, focus is given in the variations of the respiratory profile that could be related to acute exacerbation events. To do so, a new study protocol was developed and implemented for the medium-term monitoring of COPD patients.

In this chapter, the protocol and the acquired data are described in detail.

### 1.1 Data acquisition

COPD patients were monitored during their stay at the Respiratory and neuro-respiratory follow-up and rehabilitation care unit (SSR respiratoire) under the same study protocol described in Part II.

In the week of admission, patients were equipped with one or two TeleOx® devices according to the number of oxygen sources used by the patient. Devices were collected before hospital discharge. All devices had the latest firmware version implemented, which means that, every 5 minutes, breathing rate, inspiratory amplitude, oxygen flow, patient presence and signal quality were recorded.

Typically, the SSR respiratoire receives patients that have recently experienced an exacerbation requiring hospitalisation. After a first hospitalisation, patients are transferred to the SSR respiratoire for rehabilitation. Thus, at arrival, time from last exacerbation is variable.

After admission, patients are examined by the healthcare team and personal objectives of rehabilitation are decided. Globally, the goal of hospitalization is to help patients to evolve towards a more stable state of health. It can be achieved by increasing the patient's autonomy to execute daily activities, as treatment administration or walking, or even by weight gain or loss, among other examples.

During this period, patients may present acute events. Luckily, they are closely followed by the medical staff, that can intervene whenever needed. Still, exacerbations can have different levels of severity. They are often treated on the unit, but in case of severity a transfer to an intensive care unit may be necessary.

At the end of stay, TeleOx® data was retrieved, together with exacerbation dates and
6MWT results from patients medical records. Thus, each recording, composed by TeleOx®
data and medical data, corresponds to a patient's stay in the SSR respiratoire.

1.2 Data treatment

Every day, a TeleOx® records 288 data points. Each point is a vector, containing the time, status (with information about oxygen source type, patient presence and signal quality), oxygen flow rate, median breathing rate and median amplitude. In other words, TeleOx® recordings contain three time series (oxygen flow, breathing rate and amplitude), where status offers information about why a point is missing or not.

Whenever more than one TeleOx® is used to monitor the same patient at the same time, time series are concatenated with respect to time.

Acquired data are treated so only valid periods are kept for posterior analysis. Valid
periods are characterized by the presence of the patient, spontaneous breathing and a good
signal quality.

In the context of this data acquisition, the use of pulsed oxygen source is rare. Patients were equipped with an oxygen cylinder and/or wall oxygen supply. As previously discussed, recorded signals under pulsed oxygen flow do not correspond to the spontaneous breathing of the patient. In these cases, TeleOx® will not try to quantify the patient's inspiratory amplitude. For those reasons, points identified as such by TeleOx® firmware are not considered valid measures for posterior analysis.

In some of the recordings, patients plugged a TeleOx® in the oxygen circuit of their NIV machines. TeleOx® embedded algorithm identifies the continuous oxygen flow and quantifies the breathing rate (that may or may not be defined by the NIV configuration). The amplitude, however, is biased by the ventilation support machine and presents values much higher than spontaneous breathing. Since TeleOx® firmware was not projected to treat those situations, NIV points are eliminated with a posterior data treatment.

NIV associated points are considered to be all points with amplitude above 300 Pa, where no gradual increase of amplitude was observed.

To treat both situations, amplitude measures are analysed in a daily basis. For a given day, if 99% of all measured amplitudes are above 300 Pa, all points are considered NIV associated. Else, an agglomerative clustering is performed over the amplitude measures to identify two groups of points. Clusters should be separated by at least 30 Pa to be considered different clusters. If 99% of all points in a cluster are above 300 Pa, they are considered NIV related.

Moreover, as it was observed in Part II, extreme amplitude measures can be originated from artifacts from the oxygen source. Thus, a final filtering is executed using DBSCAN algorithm to eliminate outliers according to breathing rate and amplitude values [58]. Daily points are clustered with DBSCAN, by considering that a point is a core point when there is at least 1 other point around it within a distance threshold of 100. Each day, points outside of the biggest cluster are considered outliers.

**1.3 Results**

Recordings were retrieved between June 2019 and April 2021. In total 63 patients (39 male) were included, from which 6 had two or three periods of follow-up because of re-hospitalisation. This resulted in 70 periods of recording which are described in Table 1.1. 29 recordings include at least one exacerbation, with 36 total exacerbations.
<table><caption>Table 1.1 – Clinical characteristics of COPD patients during periods of follow-up</caption><thead><tr><th></th><th>Total<br>(n = 70)</th><th>With exacerbation<br>(n = 29)</th><th>Without exacerbation<br>(n = 41)</th></tr></thead><tbody><tr><td>Men</td><td>42 (60.0%)</td><td>22 (75.9%)</td><td>20 (48.8%)</td></tr><tr><td>Age (years)</td><td>69.0 ± 7.8</td><td>69.3 ± 7.9</td><td>68.8 ± 7.8</td></tr><tr><td>FEV1 after bronchodilator (%) <sup>1</sup></td><td>34.3 ± 16.4<br>(24 missing<sup>2</sup>)</td><td>34.0 ± 16.5<br>(11 missing<sup>2</sup>)</td><td>34.5 ± 16.3<br>(13 missing<sup>2</sup>)</td></tr><tr><td>Recording duration<br>(days)</td><td>31.5 ± 15.7</td><td>29.1 ± 13.8</td><td>33.2 ± 16.7</td></tr></tbody></table>

Figure 1.1 is an example of one week of recording, during which the patient used NIV with oxygen supply every night. Colors indicate the results from the agglomerative clustering method. Pink points are considered NIV related and are further removed from analysis.

<figure><img src="image_48.png" /><figcaption>Figure 1.1 – Time series from TeleOx® for a week from recording 13. Patient was on NIV during sleep. Colors indicate daily clusters. Pink points are points identified as NIV related.</figcaption></figure>

In this example, the two distinct types of periods are visually perceptible. Diurnal periods are identified by variable breathing rate and amplitude below 140 Pa. Nocturnal periods are distinguishable by significantly higher amplitudes combined with more constant breathing rate. Indeed, the patient's ventilation support was configured to trigger respiratory cycles for a breathing rate of at least 12 breaths per minute.

Series from a week of recording from another patient are presented in Figure 1.2. Although some observed amplitudes exceed 300 Pa, we don't have any evidence of it belonging to NIV related points. On the contrary, it appears to be a real indicator of the patient's respiratory profile as its evolution is gradual through time. With the agglomerative clustering algorithm, no points are identified as NIV related within this period of recording.

<footnote n="1">¹ FEV1 values were extracted from the spirometry examination performed closest to the recording period.</footnote>

<footnote n="2">² For some patients, there was no spirometry result available in the medical record.</footnote>
<figure><img src="image_49.png" /><figcaption>Figure 1.2 – Time series from TeleOx® for a week from recording 22. Patient was on oxygen therapy during day and night. Although amplitude measures are high, there is no evidence that the TeleOx® was plugged in a NIV machine.</figcaption></figure>

After filtering NIV related points, a final filtering step is executed to eliminate outliers that may be related to artefacts in the oxygen source. Figure 1.3 gives an example from another patient. After combining data from recordings with two TeleOx® devices and testing for the presence of NIV related periods, two points are identified as outliers. Outliers are then removed from latter analysis.

<figure><img src="image_50.png" /><figcaption>Figure 1.3 – Time series from TeleOx® for a week from recording 42. Patient was on oxygen therapy during day and night. Detected outliers are indicated in orange.</figcaption></figure>

The treated series can also be visualised in terms of their daily distribution. In Figure 1.4, the previously presented recording examples are presented as daily distributions of breathing rate and amplitude.
<figure><img src="image_51.png" /><figcaption>Figure 1.4 – Daily distributions corresponding to valid data from previously presented weeks of recordings from TeleOx®</figcaption></figure>

1.4 Discussion

It is interesting to notice the periodicity of the series presented above (Figures 1.1, 1.2 and 1.3). Patients typically present lower and less variate breathing rate and/or amplitudes during nocturnal periods.

TeleOx<sup>®</sup> data allows for observations concerning the differences of the non pharmaco-
logical treatments used by each patient. Oxygen adherence and flow rate illustrate different
habits and needs.

In addition to the treatment administration, TeleOx® data provides information about the ways of adapting to daily changes. In this regard, differences between patients are already present in the plots of the time series data (Figures 1.1, 1.2 and 1.3). Still, the individual profiles are put in evidence by observing the daily distributions of breathing rate and amplitude in Figure 1.4.

In these recordings, for example, we observe that in Figure 1.4a, changes in the breathing rate are common, while the amplitude diverges very little. Inversely, in Figure 1.4b it is mainly the amplitude that changes, while keeping the breathing rate in a narrower range. In Figure 1.4c, breathing rate and amplitude changes seem to be equally present, although maybe in lesser intensity than the first two individuals.

1.5 Conclusion and perspectives in data treatment

With TeleOx<sup>®</sup>, we were able to record respiratory data from 63 patients diagnosed with COPD during their stay in the rehabilitation unit SSR respiratoire.

After data treatment, we observe that these recordings are characterized by daily sea-
sonality. This seasonality is probably related to a combination of factors, as treatments
choice, daily routine and respiratory adaptation capacities. Besides, at least at first glance,
patterns seem personal.
Thus, in the next chapter, we seek at describing typical respiratory profiles present in
the data.
# Chapter 2

## Automatic detection of respiratory profiles

Previous observation pointed to the presence of daily seasonality in the recordings. Every day, patients need to adapt to different conditions. At the *SSR respiratoire* a daily planning is proposed for the patients. Still, participation to activities vary from patient to patient and day to day.

In addition, when comparing rest and exercise in recordings from healthy subjects and COPD patients in Part II, we observed that the interaction between breathing rate and amplitude is different for each individual.

Thus, in this chapter we study which are the respiratory profiles in the dataset. First, we identify and describe observed patterns. Then, we discuss if there are respiratory profiles that could be associated with more or less stable states.

### 2.1 Methods

Two sets of the data are chosen to i. analyse all profiles in the data and ii. examine only profiles related to exacerbation or baseline periods. Unsupervised learning is used in both sets to reveal the types of respiratory profiles.

#### 2.1.1 Dataset

A first clustering is done using all non-overlapping weeks available in the data. Weeks are considered as seven consecutive days starting at any day of the week. A week is only considered valid if there are at least 5 days with 4 hours of data or more. Weeks are chosen trying to maximize the number of weeks available. Thus, whenever a period of consecutive days is missing, the following week starts in the first day containing sufficient data.

A second clustering is performed containing a single week from each recording. Recordings with no exacerbation are used to extract weeks of stability (baseline). Baseline week is the last week of those recordings, only if there are at least 10 days between first day of baseline and day of admission. For the other recordings, days of exacerbation together with the 6 preceding days are extracted as prodromal weeks. For both baseline and prodrome, if less than 5 days with at least 4 hours of valid recording data are available, week is dismissed from analysis.
## 2.1.2 Barycenters

In both clustering sets, for each selected week, daily distributions of breathing rate and amplitude were used to compute the corresponding Wasserstein barycenter.

Based on the Wasserstein distance between discrete distributions, the Wasserstein barycenter is a weighted average distribution $\nu$ that is at midway from a given set of distributions $\{\mu_j\}$. The Wasserstein barycenter of the distributions $\{\mu_j\}$ is estimated with Equation 2.1.1, where the weights $\rho_j \ge 0$ and $\sum_j \rho_j = 1$.

$$ \nu = \min_{\nu} \sum_j \rho_j W(\nu, \mu_j) \qquad (2.1.1) $$

To ensure that breathing rate and amplitude have the same impact, the cost function used is the Mahalanobis distance.

Each estimated barycenter has 288 points, independently of the number of valid measures available in the original daily distributions. This corresponds to a complete day of oxygen therapy adherence, where a valid measure is recorded every 5 minutes.

## 2.1.3 Clustering

Barycenters of daily distributions within a week are then clustered using agglomerative clustering based on the Wasserstein distance. The linkage metrics single, average, complete and ward are tested with 2 to 10 clusters.

Results with different linkage metrics and number of clusters are compared using the silhouette coefficient. This coefficient allows for quantifying the quality of the clustering. For each element, the silhouette coefficient is computed according to Equation 2.1.2, where $a(i)$ is the mean distance between element $i$ and all other elements in the same cluster $C_k$ (Equation 2.1.3) and $b(i)$ is the mean distance between element $i$ and all other elements in the nearest cluster (Equation 2.1.4).

$$ S(i) = \frac{b(i) - a(i)}{\max(a(i), b(i))} \qquad (2.1.2) $$

$$ a(i) = \begin{cases} \frac{1}{|C_k|-1} \sum_{j \in C_k, j \neq i} d(x_i, x_j) & \text{if } |C_k| > 1 \\ 0 & \text{if } |C_k| = 1 \end{cases} \qquad (2.1.3) $$

$$ b(i) = \min_{C_k \neq C_{k'}} \frac{1}{|C_{k'}|} \sum_{j \in C_{k'}} d(x_i, x_j) \qquad (2.1.4) $$

The silhouette coefficient attached to each element is $-1 \le s(i) \le 1$. Silhouette coefficients close to 1 indicate that the element is well matched with its cluster. At the opposite, a negative silhouette coefficient indicates that the element is closer to the neighbouring cluster.

The silhouette score associated with the complete classification is simply the mean of the silhouette coefficients for each element in the dataset.

Each cluster is represented by the Wasserstein barycenter of all weekly barycenters assigned to the cluster. The barycenters patterns are discussed qualitatively.

Then, clusters completeness is analysed with respect to the original recordings or with the presence of an exacerbation within the week. For both classifications, the completeness criteria is computed [80], which gives a sort of agreement between known classes and the clustering. More precisely, in a perfectly complete clustering, the completeness score is equal to 1 and all elements from a given class are assigned to the same cluster.
## 2.2 Results

### 2.2.1 Week mean distribution

Figure 2.1 gives as examples the barycenters of the three weeks presented in the previous chapter.

<figure><img src="image_52.png" /><figcaption>Figure 2.1 – Wasserstein barycenters from weeks presented in Figure 1.4</figcaption></figure>

The barycenters are consistent with the daily distributions observed in the previous chapter: the leftmost barycenter indicates important variation in the breathing rate combined with quite constant amplitude; the barycenter in the center has a more variable breathing amplitude with little changes in the breathing rate; the rightmost barycenter indicates variations in both breathing rate and amplitude.

### 2.2.2 Considering all weeks

A first analysis concerns all weeks in the dataset. A total of 232 weeks are extracted. On average, each recording had 4 ± 2.03 weeks included.

An exacerbation onset is present in 20 of those recordings, though its position within the week is variable. Thus, a week that includes an exacerbation also includes a period of prodrome and/or recovery.

The estimated silhouette scores for each linkage method and number of cluster is given in Figure 2.2.

<figure><img src="image_53.png" /><figcaption>Figure 2.2 – Silhouette coefficients for the agglomerative clustering of all weeks, using different linkage methods.</figcaption></figure>

Best results were obtained when using ward, average or complete linkages. Little improvement is observed when the number of clusters is above 5. Thus, the classification with 5 clusters using the complete linkage is retained for further analysis.
The silhouette score for this classification is 0.29. The barycenters of the five obtained clusters are presented in Figure 2.3.

<figure><img src="image_54.png" /><figcaption>Figure 2.3 – Wasserstein barycenters representing the clusters for all weeks.</figcaption></figure>

Clusters 1, 2 and 3 are characterised by the presence of higher amplitudes. Weeks in cluster 1 and 2 had high and low breathing rates respectively. Cluster 3 is represented by weeks with even higher amplitude, combined with moderate breathing rates.

Clusters 4 and 5 present more “flat” patterns. Breathing in those weeks is shallower, with a wide variety of breathing rates (cluster 4) or rather low breathing rates (cluster 5).

The sizes (number of weeks with exacerbation) of the clusters are: 49(3), 20(1), 17(3), 53(2) and 93(11). There are weeks containing exacerbations in every cluster.

The completeness related to the occurrence of exacerbations is 0.008. Completeness is also computed with respect to the corresponding recordings. The completeness score is 0.898. This value indicates that almost all recordings have all weeks classified in the same cluster. Only 12 recordings have weeks classified in two different clusters. From these, 7 have at least one exacerbation during follow-up and 5 have no exacerbation.

### 2.2.3 Baseline and prodrome weeks

In a second analysis, focus is given to the weeks of “stability” (baseline) and the weeks preceding exacerbation (prodrome), hence, the weeks are chosen according to the patient’s state.

There are 15 weeks for prodrome and 29 weeks of baseline periods. Each week is transformed in its corresponding Wasserstein barycenter before being clustered with unsupervised agglomerative algorithms. Figure 2.4 presents the silhouette coefficients for the different linkage methods used.

<figure><img src="image_55.png" /><figcaption>Figure 2.4 – Silhouette coefficients for the agglomerative clustering of weeks from baseline and prodrome periods, using different linkage methods.</figcaption></figure>

Best silhouette coefficients are obtained when using the average linkage method. However, this method has the tendency of creating clusters which only contain one element.
This artifact can increase the performance according to the silhouette coefficient, while not giving interesting information about the similarity between the elements.

The next best classification seems to be obtained using the complete linkage method with 6 clusters. The silhouette score is 0.31. Its dendrogram is presented in Figure 2.5 with the corresponding clusters barycenters in Figure 2.6.

<figure><img src="image_56.png" /><figcaption>Figure 2.5 – Dendrogram of the unsupervised clustering of baseline and prodrome weeks. Leafs labels are the recording identifiers and are presented in black if baseline or red if prodromal period.</figcaption></figure>

<figure><img src="image_57.png" /><figcaption>Figure 2.6 – Barycenters representing the clusters of the baseline and prodrome weeks. Colors correspond to the branches’ colors in the dendrogram of Figure 2.5.</figcaption></figure>

Clusters 3 and 4 include 2 or 1 week, respectively. In these three weeks, recording distributions have the particularity of happening at low breathing rate (mostly below 20 cycles/minute). Cluster 4 (or recording 30) presents few points in the low left corner of the graphic. Instead, the patient's respiratory profile can be described as slow and deep.
The other clusters 1, 2, 5 and 6 have more representative weeks. Cluster 1 is composed by 5 weeks, where patients have moderate breathing rate and amplitudes that vary along the days. In cluster 2 (8 weeks), there are patients who breathe at higher rates and also use the amplitude to adapt their breathing. Cluster 3, also with 8 representatives, includes weeks where patients present shallow breathing.

Finally, the 20 weeks belonging to cluster 6 have the particularity of breathing at low rate, rarely passing the 30 cycles/minute. In this group, very profound breathing is not common, but the adaptation of the amplitude between low and moderate levels is frequent.

Moreover, 10 out of the 15 prodromal weeks in this analysis are classified in cluster 6. The associated hyper-geometric test shows a *P* = .043, suggesting that this pattern may be frequent before exacerbations.

The other 5 exacerbations are divided between clusters 1, 2 and 4. The completeness score concerning prodrome class is 0.10.

## 2.3 Discussion

Different respiratory profiles are distinguished using an unsupervised learning method based in the Wasserstein distance and barycenter. Respiratory patterns differ in both breathing rate and amplitude. This agrees with results from Part II, showing that COPD patients also have ways of adapting any of these features.

The continuous follow-up enabled a more detailed description of the respiratory profiles. Analysis showed that profiles also differ in the ranges of the features. Thus, the description of days based on a single measure seems insufficient to capture some types of changes.

The classification of all weeks in the data points to the individuality of the respiratory profiles. Indeed, weeks belonging to the same recordings tend to be very similar. They are mostly classified in the same cluster, which is confirmed by the high completeness score.

In both analysis, the cluster that contains more exacerbations is also the one that contained more weeks overall. They are represented in brown in Figures 2.3 and 2.6. The barycenters of these clusters are very similar, with breathing rates below 30 cycles/minute and amplitudes below 100Pa.

The presence of most exacerbations in these groups may seem contradictory with previous hypothesis in the literature that would point to an increase in the breathing rate preceding an exacerbation. One possible explanation for the prevalence of weeks with exacerbation in these clusters would be a limitation in exercising. While patients are encouraged to exercise daily in the rehabilitation program, they may be excused from the activities when they feel less willing.

Another consideration is that this cluster presents the second shallower breathing profile. Shallow breathing may be associated with lung hyperinflation and its consequent limitation in the $V_T$. However, without access to data to confirm the presence of hyperinflation in these groups, it remains an hypothesis.

Moreover, exacerbations are present in all or most clusters, being expressed by very low completeness scores. These low scores suggest that there is no unique profile that is related to exacerbation or stability.

## 2.4 Conclusion

The unsupervised classification of respiratory profiles shows a variety of patterns among the recordings. The patterns are mainly consistent for the same individual throughout the follow-up period. In fact, most recordings consisted of a single pattern type.
The respiratory profiles could not be easily associated with exacerbation or stability, which suggests that population-based algorithms may be not adapted for exacerbation detection. Our hypothesis is that it is the changes from patient's own baseline that may be indicative of exacerbations, thus they need to be identified individually.
# Chapter 3

## Automatic detection of respiratory changes

The use of Wasserstein barycenters to identify types of respiratory profiles within the recordings indicates that most patients present a consistent weekly profile during their stay in the rehabilitation unit and that there is no common respiratory profile for exacerbations.

In this chapter, we focus in respiratory changes at individual level, by using personal Wasserstein barycenters to describe reference distributions of breathing rate and amplitude.

First, to validate the capacity of detecting individual changes, we work on the detection of changes that happen within a day by comparing sleep and wakefulness periods. Then, we use individual barycenters to analyse the evolution of the patients in the rehabilitation program.

### 3.1 Detection of sleep and wakefulness

It has been previously described that the respiration of healthy individuals differ between sleep and wakefulness. Sleep stages have their particularities, but overall sleep is characterized by slower, shallower and less variable respiration, compared to wakefulness [38]. These changes were indeed observed in the series presented in Chapter 1.

The objective of this section is to characterize the individual respiratory profiles during day and night periods and use them to confirm the capacity to automatically identify sleep and wakefulness.

#### 3.1.1 Method

The individual diurnal and nocturnal respiratory profiles are extracted from the patient's reference week. We choose the reference week as the last week of recording, which should correspond to the closest from stability for patients in the rehabilitation unit. Only patients with at least 5 nights with nocturnal data in the reference week are kept in this analysis.

Reference points are extracted from this week according to the time of measuring. Expressly, diurnal and nocturnal points are extracted from periods between 9am and 7pm and between 11pm and 5am, respectively. Since the real bedtimes are unknown, uncertain periods are avoided in the reference distributions.

For each recording, the reference diurnal distribution is the Wasserstein barycenter of the daily diurnal distributions with respect to the breathing rate and the amplitude. Likewise, the reference nocturnal distribution is the Wasserstein barycenter of the daily nocturnal distributions. Although these barycenters represent shorter periods of time, the
number of points used is also 288. Those distributions represent what a day or night period is expected to look like for this individual.

The next step consists of comparing the series to these reference distributions. A sliding window is used to extract periods from the recording. The breathing rate-amplitude distribution within a period is compared to the day and the night reference distributions using the Wasserstein distance. Different window sizes are tested.

Period's distribution that resembles more to the day reference distribution is more likely to have been recorded while the patient was awake. In opposition, a period's distribution that match the night reference distribution was probably recorded during sleep. To verify the performance, sleep time for all subjects is stipulated from 10pm to 7am. This period is only a rough estimation and do not correspond exactly to the truth for all individuals.

## 3.1.2 Results

Most patients in the dataset were not under oxygen therapy during the night. Some don't need ventilatory support during sleep, while others used NIV. 15 recordings contain sufficient nocturnal points and are used in the following analysis.

The window size is chosen to improve the accuracy at capturing the distribution around a point while not losing in specificity by smoothing too much. Figure 3.1 gives the area under the ROC curve (AUC) for different window sizes for classifying day and night periods for all patients combined.

<figure><img src="image_58.png" /><figcaption>Figure 3.1 – Performances for day and night detection according to the size of the window used</figcaption></figure>

Between 30 minutes and 2 hours, the performance of the method increases with the window size. With windows between 3 and 6 hours, the performance of the method is rather stable and higher than shorter and longer windows. In fact, short windows capture less points to characterise the distribution of the period, while longer windows include more points in the distribution that do not belong to the same state.

A window size of 4 hours is chosen, because it is an intermediate length between those achieving the highest performances. Thus, each point in the series, together with the points measured up to 2 hours before or later, is compared to the reference distributions for day and night.

The method is executed for each subject separately. Predictions are compared to theoretical nocturnal periods (from 10pm to 7am). The performances for each recording are presented in Figure 3.2.

Over all patients, the AUC is 0.833. Using zero crossing as the threshold, nights are detected with 87.6% sensitivity, 55.9% specificity and 77.0% accuracy.
<figure><img src="image_59.png" /><figcaption>Figure 3.2 – Performance for each recording with sliding window of 4 hours. Red and black lines represent recordings with and without exacerbations, respectively.</figcaption></figure>

## Example of sleep and wakefulness detection

The last week from recording 42 is presented in Figure 3.3. Reference points for day and night periods are put in evidence in orange and blue, respectively. Gray points are not used in the estimation of the reference distribution.

<figure><img src="image_60.png" /><figcaption>Figure 3.3 – Last week from recording 42. Reference points for day and night are plotted in orange and blue respectively. Gray points belong to uncertain periods and are not used as reference.</figcaption></figure>

Figure 3.4 presents the diurnal and nocturnal barycenters for the reference points from the recording 42. As expected, diurnal reference distribution reveals higher variability, with higher values of breathing rate and amplitude, than nocturnal distribution.

Distributions present an overlapping area at low breathing rate and amplitude. This
<figure><img src="image_61.png" /><figcaption>Figure 3.4 – Barycenters for day (orange) and night (blue) from last week of recording 42</figcaption></figure>

demonstrates that is is not possible to identify a single point as sleep or wakefulness by itself. Instead, it is necessary to analyse each point according to its neighboring distribution (in time).

In Figure 3.5, each point is presented as the distance to the diurnal distribution minus the distance to the nocturnal distribution. In this way, positive scores indicate points that are closer to the nocturnal distribution, while negative scores indicate points that are closer to the diurnal distribution.

<figure><img src="image_62.png" /><figcaption>Figure 3.5 – Complete night detection.</figcaption></figure>

Considering the difference between the reference distributions

Rationally, the detection of nocturnal periods would be more difficult whenever the diurnal and nocturnal reference distributions are more similar. In Figure 3.6, the AUC for each recording is plotted in function of the distance between the reference distributions for day and night. As expected, the two recordings with lower difference between the reference distributions are also the ones with the lowest performance.

It is also interesting to note the high correlation between the AUC and the distributions distance for the recordings without exacerbation in Figure 3.6. Three out of the five recordings with exacerbations present a different behaviour. They have lower performance for what would have been expected given the distance between their reference distributions.

Recording 22 (Figure 3.7a) belongs to a patient with GOLD score 4D. Some nights have missing data, specially in the 10 days preceding the exacerbation. After exacerbation, short periods of sleep are identified during the day, while some nights have little to no points detected as such.

Recording 38 (Figure 3.7b) comes from a patient with GOLD 4D that would often sleep under NIV. Still, some data concerning his night periods are available. Even those points rarely pass the threshold for night detection.
<figure><img src="image_63.png" /><figcaption>Figure 3.6 – Performance in function of the distance between reference distributions for day and night. Red and dark points represent recordings with and without exacerbations, respectively.</figcaption></figure>

Recording 30 (Figure 3.7c) is from a patient with less severe COPD (GOLD 2B). However, this patient was under sedative medication (depakote, valium, buprenorphine, risperidone, paroxetine, mianserine), which could explain periods of sleep during daytime. Besides, a polygraphy revealed nocturnal hypoventilation, that can be related to a poor quality of sleep during the night.

### 3.1.3 Discussion

Sleep disorders are common among COPD patients, reflecting in poor sleep quality during the night and sleepiness during the day. Recordings where day and night periods are harder to identify may indicate patients with less well defined sleep and wakefulness periods.

In fact, poor sleep quality has been associated with higher risk of exacerbation in COPD patients [89]. Hence, the sleep quality is probably an interesting parameter to monitor in COPD patients. Unfortunately, this method has not been further developed hereafter because of the reduced amount of nocturnal data recovered. Still, these primary results show the feasibility of the monitoring with TeleOx® devices since diurnal and nocturnal data can be recorded without intervention with a single device, simplifying further comparison between day and night periods.

In the future, a new data collection on a larger sample of patients under oxygen therapy during sleep would allow to evaluate the prediction of exacerbations based on sleep-wake differences. More precise knowledge of the patient's sleeping habits could also bring more precision to the detection of daytime naps, for instance.

Meanwhile, the results show that it is possible to differ day and night periods according to the distributions of breathing rate and amplitude measures from the TeleOx® monitoring device.

## 3.2 Detection of post-exacerbation and stable weeks

In this second approach, we use the barycenters to model the evolution of patients during their stay in the rehabilitation unit. In fact, the objective of the rehabilitation is to help patients to recover from previous exacerbation and improve their health status before medical discharge. Thus, we test if this evolution can be observed in their respiratory pattern.
<figure><img src="image_64.png" /><figcaption>Figure 3.7 – Complete night detection for three recordings with exacerbation. First day of exacerbation is represented in red. Reference nocturnal periods are indicated by blue areas.</figcaption></figure>

### 3.2.1 Methods

Recordings from patients who did not exacerbate during the follow-up period are included in this analysis. These recordings are chosen because they are more likely to present a monotonic evolution. The occurrence of an exacerbation, on the other hand, represent a decline in health status, possibly preceded and/or followed by periods of improvement.

For each recording, two reference respiratory profiles are estimated using a Wasserstein barycenter. The first reference distribution concerns the state at arrival at the rehabilitation unit. This state is presumably the less stable health status, being closer to the previous exacerbation. The barycenter is estimated from daily distributions in breathing rate and amplitude extracted from the first week of recording, as previously described.

The second reference distribution describes the patient's state just before medical discharge. Without the advent of an exacerbation, the last week in the rehabilitation center is expected to be the best state of health of the patient under monitoring. The barycenter is computed from the daily distributions from the last week of recording.

Once the two reference distributions estimated, daily distributions are compared to both barycenters. A transition score for each day is defined as the distance from the initial reference distribution minus the distance from the final reference distribution. The
transition score is used to estimate to which week a day is closer.

Days from the first and last weeks are used to confirm the correctness of the method.

## Variations of the method

The proposed method is compared with state of the art used descriptors. To do so, we test if changes could also be detected based on breathing rate alone and/or on a single measure per day.

Three other methods are tested: i. a method based on reference barycenters of the daily distributions of breathing rate measures only; ii. a method based in a single daily measure of breathing rate, from which the weekly mean is used as reference; iii. a method based on a single daily measure of breathing rate and amplitude from which the weekly mean is used as reference.

For the methods based on a single daily measure, we reduce the daily measures of breathing rate and amplitude to a daily mean. Means from the first week are used to compute the initial reference mean. Similarly, daily means from the last week are used to compute the final reference mean. Each day is classified as initial or final according to the minimal euclidean distance to the references.

### 3.2.2 Results

After exclusion of recordings containing exacerbation events or that lasted less than 2 weeks, 34 recordings are included in the analysis. Length of stay varies from 15 to 62 days, which means that the distance between the first and last week of follow-up differ between records.

The accuracies for all recordings combined are given in Table 3.1.

<table><caption>Table 3.1 – Accuracy for detection of initial and final daily profiles for all recordings combined with different methods and features</caption><thead><tr><th></th><th>Breathing rate</th><th>Breathing rate and amplitude</th></tr></thead><tbody><tr><td>Based on mean</td><td>70.7%</td><td>72.67%</td></tr><tr><td>Based on barycenter</td><td>77.0%</td><td>82.7%</td></tr></tbody></table>

The addition of amplitude measures allows for the detection of changes that would not have been observable with the breathing rate alone. Moreover, the use of distribution establishes more detail of the respiratory profile, revealing changes that would be hidden with a single daily measure.

The individual accuracies for the detection of initial and final days using the four methods are compared in Figure 3.8.

The individual accuracies are compared using a one-sided T-test for paired samples. The use of barycenters based on breathing rate distributions only is more efficient than the use of means of breathing rate alone (Figure 3.8a, *P* = .001), but it is similar to the use of means of breathing rate and amplitude combined (Figure 3.8b, *P* = 0.09).

The use of barycenters based on breathing rate and amplitude outperforms barycenters of breathing rate alone (Figure 3.8c, *P* = .01). Barycenters of breathing rate and amplitude also achieve higher accuracies than the use of means of breathing rate alone (Figure 3.8d, *P* < .001) or combined with the amplitude (Figure 3.8e, *P* < .001).

According to the most complete method, based on the distributions of breathing rate and amplitude, most days of all recordings are correctly classified as belonging to the first
<figure><img src="image_65.png" /><figcaption>Figure 3.8 – Comparison of individual accuracies for methods using daily distribution or mean of breathing rate alone or combined with the amplitude.</figcaption></figure>

or the last week. Figure 3.9 shows the individual accuracies with respect to the distances between the reference barycenters. Recordings where at least one day in first and last weeks had less than 12 recorded points are presented in red. The number of daily valid points allows for an estimation of the daily oxygen compliance. Since TeleOx® takes new measures every 5 minutes, 12 daily points indicate one hour of oxygen use.

<figure><img src="image_66.png" /><figcaption>Figure 3.9 – Comparison of the individual accuracies for detecting initial and final status and the distance between initial and final reference distributions. Points presented in black represent weeks where all days had at least 12 recorded points. Points presented in red indicate recordings where at least one day presented less than 12 recorded points.</figcaption></figure>

Among recordings with good compliance, individual accuracies and the distances between the barycenters are correlated (R=0.708, *P* < .001).
## Comparison with clinical evolution

In Figure 3.10, the weeks classification performances with the barycenter of breathing rate and amplitude for all patients are compared to the change in the their 6MWT relative results (the percentages relative to expected distances). Points are colored according to the GOLD levels at each recording. The accuracies from recordings at GOLD 1, 3 and 4 are not different between the groups ($P = .81$ with Kruskal-Wallis H-test).

<figure><img src="image_67.png" /><figcaption>Figure 3.10 – Comparison of the individual accuracies for detecting initial and final status and the absolute difference between initial and final 6MWT relative results. Points color indicate the GOLD level of the patient during follow-up.</figcaption></figure>

The Spearman correlation between the capacity of separating days from initial and final weeks and the evolution with respect with the 6MWT is not statistically significant ($P = .65$).

## Examples of good and poor classification

Figure 3.11 is an example of the series and resulting barycenters from first and last weeks.

Daily distributions are compared to both barycenters. The differences of the Wasserstein distances for recording 52 are presented in Figure 3.12a. The higher the score, the closer a day is from the last week, compared to the first week. For the recording 52, 100% of the days from first and last week are correctly identified.

A gradual evolution of respiratory profiles from the initial pattern to the final pattern is observed in the recording 52. This shows that there is a change in the respiratory profile of the patient and that this change is well captured by the barycenters.

On the opposite, in recording 27, only 64.3% of the days from first and last weeks are correctly identified. Days in this recording (Figure 3.12b) present comparable distances from first week barycenter and last week barycenter. In fact, the reference barycenters for this recording are quite similar (Figure 3.13).

The contrast between these two recordings can also be observed by directly comparing Wasserstein distances of days from first and last weeks. Distance matrices are built by computing the distance between each pair of days from the same recording. The resulting matrices for initial and final weeks from recordings 52 and 27 are presented in Figure 3.14. This verifies that initial and final weeks in recording 52 contain great similarity within the week, but they all differ from days in the other week. Initial and final days in recording 27 are all similar at the same level to each other.

Moreover, the patients that generated the recordings 27 and 52 present different clinical profiles.

Recording 27 was obtained from a patient at GOLD 3. At the beginning of follow-up, the patient was able to walk 240 meters during the 6MWT, which corresponds to 53% of
<figure><img src="image_68.png" /><figcaption>(a) Series from first week</figcaption></figure>

<figure><img src="image_69.png" /><figcaption>(b) Barycenter for first week</figcaption></figure>

<figure><img src="image_70.png" /><figcaption>(c) Series of the last week</figcaption></figure>

<figure><img src="image_71.png" /><figcaption>(d) Barycenter of the last week</figcaption></figure>

Figure 3.11 – First and last weeks for recording 52, corresponding to initial and final days of stay in the rehabilitation unit.

<figure><img src="image_72.png" /><figcaption>(a) Recording 52</figcaption></figure>

<figure><img src="image_73.png" /><figcaption>(b) Recording 27</figcaption></figure>

Figure 3.12 – Transition score based on barycenters of first and last weeks of recording. Barycenters are estimated for each recording individually.

the distance expected for someone of the same gender, age, height and weight. At the end of stay, the patient walked 360 meters, which represents 80% of the expected distance.

Recording 52, on the other hand, comes from a patient at GOLD 4, revealing higher severity. The results on 6MWT went from 60m (16%) to 205m (56%). These results point to a greater progress in the patient's ability to exercise.
<figure><img src="image_74.png" /><figcaption>Figure 3.13 – Initial and final reference barycenters of recording 27. Barycenters are estimated from first and last weeks of recording, respectively.</figcaption></figure>

<figure><img src="image_75.png" /><figcaption>Figure 3.14 – Wasserstein distances between each pair of days from first and last weeks of recording.</figcaption></figure>

### 3.2.3 Discussion

The detection of post-exacerbation (initial) and stable (final) weeks is tested with 4 methods that differ with respect to the feature used (breathing rate alone or combined with amplitude) and to the number of daily measures (single daily measure or multiple daily measures).

In the single measure context, we opt to keep the daily mean, which is deduced from the complete distribution. In some of previous studies based on breathing rate measured with NIV, it was not clear how the breathing rate was extracted from the period of follow up [16, 15, 44]. In other studies, when intervention was needed for measuring physiological indicators, measures were commonly taken punctually in the day or week [88, 43, 68, 49, 60]. This can make the comparison of daily measures more difficult since the conditions around the measurement may vary day-to-day.
Concerning the results obtained with the present dataset, there is a significant improvement when combining the features breathing rate and amplitude and comparing a distribution of points. In fact, when data is limited to a single data point, much of the information contained in the daily respiratory distribution is missed. Thus, the comparison of more complete representations of the patient's respiratory profile increases the chances of changes in the respiratory profile being detected.

With the method based in the daily distributions of breathing rate and amplitude, more than 80% of the days were correctly classified. Moreover, a correlation between the accuracy of the method and the difference between the reference distributions is observed. Similarly to the detection of sleep periods, the fact that the reference distributions are similar manifest a lack of change between the compared states.

In the case of recordings with poor oxygen compliance, the accuracies are lower than expected, given the distances between the reference barycenters. In fact, one limitation of the Wasserstein distance and barycenter is that all distributions being compared must have the same sum of weights. For this reason, we stipulate that each point's weight is $1/N$, where $N$ is the number of points in that day. Thus, the lowest the compliance in a day, the heaviest is the weight attributed to its points. Additionally, the distributions from low compliance days may be less representative of the complete day. The combination of these two factors yields in biased distances, revealing that this method may be less adapted to low compliances or even to highly variable compliances.

No correlation was found between the capacity to detect changes and the patient's GOLD level.

The performance of the change detection was also compared to the patient's evolution in the 6MWT. In fact, during their stay in the *SSR respiratoire*, patients healthy status are expected to evolve positively. One way of quantifying this evolution is by performing a 6MWT in the first week and another 6MWT at the last week before departure. However, there is no improvement in the detection of initial and final weeks in cases of better progress in the 6MWT.

One explanation is that rehabilitation progress manifests itself in different ways for each patient. Some patients will in fact improve their capacity to walk, while others may reduce the oxygen usage, gain weight, lose weight, sleep better, etc. The 6MWT is just one way of assessing the patient's advancement.

## 3.3 Conclusion

These analysis indicate that the continuous monitoring of breathing rate and inspiratory amplitude provide sufficient information for the identification of periods of change within a day or along days.

Changes in the respiratory profile have been observed for most of the patients who do not exacerbate during their hospitalisation in the rehabilitation center. The method proposed is based on the modeling of two reference respiratory profiles which is only possible in a posteriori analysis. For real life exacerbation detection, the respiratory profile associated to the acute event is not known until it takes place.

Moreover, methods based on distribution barycenters have two major pitfalls. First, the performance is not satisfactory for days with low oxygen therapy compliance, because of the limited number of measuring points. Secondly, the temporal characteristic of the series is not taken into account by this method. Thus, useful information associated to the order of the measures might be missing from the model. Thus, a new modeling approach that takes into account data temporality is proposed in the following chapter.
# Chapter 4

## Taking temporality into account
using a Hidden Markov Model

We previously described the modeling of the reference distribution using the Waserstein barycenter. In this method the temporal character of the series is not taken into account.

In this chapter, we describe a different modeling approach, based on Hidden Markov Models (HMM). In these models, we consider that the observations actually depend on non-observable states and that the progression of hidden states is a markovian process.

For interpretation purposes, we assume that the diurnal data can be explained by two hidden states corresponding to rest and effort. If this is true, the parameters of HMM have real meanings.

When the current state of a subject is rest, it is more likely that he/she remains at rest, than that he/she starts exercising. Similarly, the probabilities of remaining in effort while one is already in effort is greater than the probability of changing state. This would result in transition matrices with strong diagonal coefficients and weak extra-diagonal coefficients.

Moreover, we can imagine that the observations of breathing rate and breathing amplitude will depend on the physiological state of the subject. When at rest, one tends to breath slower and/or shallower than when exercising. In the HMM, the emission distributions give the relation between the hidden states and the observations.

### 4.1 Estimation of model parameters

For a given recording, the training set is composed by observations series for each day of the week $O = (o_M, o_{Tu}, o_W, o_{Th}, o_F, o_{Sa}, o_{Su})$, although up to two days can be missing. These series consists of points designated by measures of breathing rate, amplitude and oxygen flow.

Nocturnal points are eliminated from the series. The detection of nocturnal points is done automatically with previously described algorithm, whenever there is sufficient nocturnal data for this detection. Else, fixed times (22pm and 7am) are used.

The observed series are the ordered pair (breathing rate, amplitude) or ordered triple (breathing rate, amplitude, oxygen flow). Since the variables are continuous, the emission probabilities come from multivariate normal distributions, which means that, for a state $j$, the emission probability $b_j$ is a multivariate normal distribution with mean $\mu_j$ and covariance matrix $\Sigma_j$. ($b_j(o_t) = P(o_t|s_t = j) = N(o; \mu_j, \Sigma_j)$).

The parameters $\Theta$ of the individual model are computed with the class *Gaussian HMM* from the *hmmlearn* library on python.
4.1.1 Modeling example

The last week of recording 52 is used to model its baseline respiratory profile with HMM. For
this example, only breathing rate and amplitude are used, to simplify data visualisation.
The series used to train the HMM correspond to those presented in Figure 3.11c. The
resulting model is described by:

* the initial probabilities $\pi = [1.0, 0]$, that indicates that daily series always start at state 0.

* the transition matrix $A = \begin{bmatrix} 0.82 & 0.18 \\ 0.28 & 0.72 \end{bmatrix}$, which reads: the probability of staying at state 0 is of 0.82, against 0.18 chances of changing from state 0 to state 1. Similarly, the probability of staying at state 1 is greater than the probability of changing from state 1 to state 0.

* the emission distribution $b_0$ associated to state 0 is given by the normal distribution with $\mu_0 = [27.83 \ 39.55]$ and $\Sigma_0 = \begin{bmatrix} 22.51 & 22.74 \\ 22.74 & 89.57 \end{bmatrix}$. The emission distribution $b_1$ associated to state 1 is given by the normal distribution with $\mu_1 = [27.45 \ 89.57]$ and $\Sigma_1 = \begin{bmatrix} 8.78 & -10.25 \\ -10.25 & 1516.13 \end{bmatrix}$. The distributions can be better visualised in Figure 4.3, where the 95% confidence intervals of both reference distributions are represented by the dashed ellipses.

<figure><img src="image_76.png" /><figcaption>Figure 4.1 – Schematic representation of HMM trained for recording 52 baseline week. The ordered pair (breathing rate, amplitude) at instant *t* is indicated as ($br_t$, $amp_t$)</figcaption></figure>

Using the estimated parameters and the Viterbi algorithm, we can estimate the hidden
states of each of the points in the daily series. The series from Figure 3.11c are presented
in Figure 4.2, this time colored according to the predicted state.

All daily points from the last week are plotted in breathing rate-amplitude plane in
Figure 4.3, colored by attributed hidden states. The position of the points can be visualised
with respect to the confidence intervals of the reference distributions.

As we can observe in Figure 4.2, points from the same hidden state tend to be grouped
in time. That is because the transition matrix attributes greater probabilities for staying in
the same state. Moreover, we can observe in Figure 4.3 that some orange points may seem
closer to the blue distribution (and vice-versa). Indeed, the HMM combines the emission
probability with the transition probability, finding the series of states that are more likely
given the observations and the model parameters.
<figure><img src="image_77.png" /><figcaption>Figure 4.2 – Series from last weeks of recording 52. Each point is colored according to the state that has been attributed to it by the Viterbi algorithm. State 0 is indicated in blue and state 1 in orange.</figcaption></figure>

<figure><img src="image_78.png" /><figcaption>Figure 4.3 – Daily distributions from last weeks of recording 52. Each point is colored according to the state that has been attributed to it by the Viterbi algorithm. State 0 is indicated in blue and state 1 in orange. The 95% confidence intervals of the respective reference distributions are represented by the dashed ellipses.</figcaption></figure>

## 4.2 Comparing with real physiological states

The predicted states using the described model are compared to an estimation of the true physiological state of the individuals. To compare the series of predicted states with the truth about exercising status, we use data from patients who were equipped with two TeleOx® devices: one for a stationary source, used exclusively in the patient's room, and one for a portable source, used outside the room. This is the best proxy of the patients
activity, although the patient can perform physical activities inside their room or rest outside.

Individual models with two hidden states are trained over daily series extracted from the last week of each recording. Hidden states are estimated using the individual models and the Viterbi algorithm. Hidden states 0 and 1 are associated with stationary or portable sources so their agreement is maximized.

31 recordings are retained for this analysis, from patients who were equipped with two TeleOx®. The accordance between predicted states and the oxygen source used are variable among the patients. The ROC curves for each recording is presented in Figure 4.4.

<figure><img src="image_79.png" /><figcaption>Figure 4.4 – Individual ROC curves for the agreement between predicted hidden states and oxygen source used during the last week of recording</figcaption></figure>

All patients combined, the models based on breathing rate and amplitude present 61.0% accuracy with the oxygen source. The accuracy of models based on observations of breathing rate, amplitude and oxygen flow is 71.2%.

The addition of the oxygen flow enhances the agreement between the predicted hidden states and the oxygen sources used. Indeed, patients often adjust the oxygen source to higher flow rates while exercising.

For some of the recordings, the agreement is poor in both tested cases. This could be because the use of oxygen source for labelling the points is not a good proxy of the physiological change for these patients or because the states estimated by the model do not correspond to the physiological states.

The time series contain missing points that are ignored with this method. Since what connects a point to the following one in the HMM is its hidden state, we assume that the state did not change during the missing period. In practice, each point depends on the hidden state of the last recorded point. Nevertheless, this assumption is perhaps not optimal since the lack of recordings may be an information in its own. For example, missing periods can indicate periods where the oxygen therapy is not needed, because the patient at rest or on NIV.

Still, even if the trained model do not correspond directly to rest and effort states, this method allows for describing the respiration profile according to its sequences and
observations. From this method, we want to retrieve information about changes in the respiratory profile along the days.

## 4.3 Initial and final profiles

In the previous chapter, daily series are compared to reference barycenter distributions using the Wasserstein distance, in order to quantify the evolution of the respiration through follow up days.

Here, a similar method is tested. Two reference models are fitted for each recording without exacerbation. The first model, based on the first week of follow-up represent the initial respiratory profile of the patient. A second model, fitted over the final week of recording indicate the respiratory profile before medical release.

Hereafter, a few metrics are tested to the detection of initial and final respiratory profiles.

### 4.3.1 Log-likelihood under the HMM

The log-likelihood of daily observation series under each reference model is estimated using the forward algorithm. Each day is classified as belonging to the first or last week, according to which one it is more likely. Like before, first and last weeks are used to quantify the method accuracy.

With this method, 86.0% of the days are correctly identified as first or last week when the HMM is trained over observations of breathing rate and amplitude, all recordings combined. When oxygen flow is also considered, the accuracy is of 87.5%.

Individual accuracies when observations are the ordered pair (breathing rate, amplitude) or the ordered triple (breathing rate, amplitude, oxygen flow) are statistically identical (two-sided paired t-test, *P* = .590, Figure 4.5a).

Moreover, individual accuracies are not different from those obtained with the method based on barycenters of breathing rate and amplitude. Accuracies for barycenter-based method and HMM with observations of breathing rate and amplitude are presented in Figure 4.5b (*P* = .063). Accuracies for barycenter-based method and HMM with observations of breathing rate, amplitude and oxygen flow are presented in Figure 4.5c (*P* = .105).

<figure><img src="image_80.png" /><figcaption>Figure 4.5 – Comparison of individual accuracies for supervised methods for classification of initial and final days. Compared methods use the Wasserstein distance from barycenters, the log-likelihood with HMM on the ordered pair and the log-likelihood with HMM on the ordered triple.</figcaption></figure>
In this supervised context, initial and final profiles can be well identified based on the
log-likelihood under the HMM, as much as with the Wasserstein distance from barycenters.
Still, the HMM also considers the oxygen flow used by the patient and the temporal
characteristics of the series. Moreover, this modeling makes it possible to describe the
respiratory profile of an individual in relation to an estimation of the physiological states
of rest and effort. These characteristics could help highlight changes in the respiratory
profile that would not be possible by considering the distribution of points in the breathing
rate-amplitude plane alone.

4.3.2 Cross-entropy and Kullback-Leibler divergence

The similarity between the observed distributions and the emission distribution of the
underlying HMM is also quantified using the cross-entropy (CE), the Kullback-Leibler
divergence (KL) and the symmetrization of the cross entropy (CE<sub>symm</sub>), as proposed by
[111]. These metrics are considered as *static models* because the temporal variations of the
model are not explicitly considered.

The emission distributions of the underlying HMM are continuous multivariate distri-
butions Q that describe the expected probabilities according to the model. However, the
empirical distributions P are discrete, given by the daily observations. Therefore discrete
approximations p and q of P and Q are defined, on which the discrete similarity measures
can be applied.

To do so, the breathing rate-amplitude plan is partially covered by a grid of size
*N<sub>grid</sub>*×*N<sub>grid</sub>* such that all observation data are covered (example in Figure 4.6). For each
cell (i, j) of the grid, *p*<sub>i,j</sub> is the fraction of observations in the cell and *q*<sub>i,j</sub> is the area of
the cell multiplied by the HMM emission distribution in its center. It is the easiest way
to approximate the probability that a random value emitted by the HMM falls in the cell.
Although *q* is not really a distribution, we use it as such.

<figure><img src="image_81.png" /><figcaption>Figure 4.6 – Representation of the cross-entropy estimation with *N*<sub>grid</sub> = 10. The scatter plot represent the empirical distribution of daily observations *p*, colored by predicted state. Dashed-ellipses represent the corresponding emission distributions *q*, which are used to estimate *q*<sub>i,j</sub> at each cell in the grid.</figcaption></figure>

Based on the grid, the dissimilarity measures are given by the following equations.
$$CE = - \sum_{i,j}^{N_{grid}} p_{i,j} \log(q_{i,j})$$

$$KL = \sum_{i,j}^{N_{grid}} p_{i,j} \log(\frac{p_{i,j}}{q_{i,j}})$$

$$CE_{symm} = \sum_{i,j}^{N_{grid}} (p_{i,j} - q_{i,j}) \log(\frac{p_{i,j}}{q_{i,j}})$$

The dissimilarity metrics are only defined in the cells where $p_{i,j} \neq 0$ and $q_{i,j} \neq 0$. Thus, the absence of observations in a region where points are highly probable do not increase the dissimilarity score. For this reason, the size of the cell should not be too small. On the other hand, cells that cover a large area will lose in precision. In this sense, the chosen size of the grid was 10x10.

For a same grid, $CE$ and $KL$ only differ by $\log(p_{i,j})$, which is a constant for a given day. Thus, dissimilarity is quantified with $CE$ and $CE_{symm}$ only.

The dissimilarity measures are computed for each daily series with respect to the initial and final HMM based on the ordered pair (breathing rate, amplitude). Ordered triples are not used with this metrics because adding a new dimension increases the computation time while increasing the proportion empty cells.

Days from first and last weeks are used to verify the identification of both respiratory profiles. Lower dissimilarity scores indicate that the day is closer to that profile.

The classification abilities are tested with respect to $CE$ and $CE_{symm}$ over the days from first and last weeks. All recordings combined, 80.7% and 78.7% of the days from first and last weeks are correctly identified using $CE$ and $CE_{symm}$, respectively.

Individual accuracies with $CE$ and $CE_{symm}$ are similar ($P = .307$, Figure 4.7a). However, they are different to those obtained with the log-likelihood under the HMM based on breathing rate and amplitude. The performances are superior when using the log-likelihood in both cases ($P = .011$ and $P < .001$, Figures 4.7b and 4.7c, respectively).

<figure><img src="image_82.png" /><figcaption>Figure 4.7 – Comparison of individual accuracies for supervised methods for classification of initial and final days. Compared methods use the log-likelihood, $CE$ and $CE_{symm}$ with HMM on the ordered pair.</figcaption></figure>

Cross-entropy metrics are less accurate at separating initial and final days when compared to the log-likelihood. In fact, $CE$ and $CE_{symm}$ take into account the predicted
hidden states and emission probabilities, but the observations order is not taken into ac-
count. Moreover, these metrics are dependent on the number of points observed, since less
points can lead to more cells with $p_{i,j} = 0$.

4.4 Conclusion

Using the HMM allows to consider the temporal aspect of the data, while staying flexible
with respect to the activity times. In fact, while patients follow proposed exercising sched-
ules in the *SSR respiratoire*, this is not expected in other contexts, specially when patients
are at home.

The estimated hidden states seem to be related to the physiological states of the pa-
tients, although more in-depth studies are required to confirm this hypothesis. The estim-
ation of periods of rest and effort can be useful for the monitoring of the patient’s ability
to adapt between these two physiological states. For example, a patient who is frequently
active, goes from rest to effort and from effort to rest multiple times in a day. In case of an
acute exacerbation, this patient can experience increased fatigue, reducing daily activities
and possibly staying at rest more often during the days. Another hypothesis is that the
exacerbation would lead to increased respiratory effort, and that rest states would become
more rare from a respiratory point of view.

Indeed, exacerbations have numerous ways of showing and, for this reason, we hypothesized that the detection of respiratory changes is in the center of interests. In a supervised context, where two reference profiles are known a priori, the three metrics defined (Wasserstein distance from reference barycenters, log-likelihood under the HMM and symmetric cross-entropy between observation and emission distribution) are capable of identifying different respiratory profiles.

The methods presented are a proof of concept for showing that changes in the res-
piratory profiles can be detected. However, the methods present a major pitfall that is
training and testing the models over the same data. Still, classifications are not at ran-
dom, indicating that these metrics contain pertinent information about the resemblance of
the empirical distribution of daily series and the theoretical distribution of the reference
models.

A larger dataset with more accurate labeling would be of great value in refining the
methods, according to minor or relevant changes. In addition, since these changes can
occur in different directions, methods for change detection from a baseline breathing profile
should be considered.

Thus, the next chapter can now propose to test a novelty detection approach for the
detection of exacerbation events based on the previous results and the validation of the
algorithms.
# Chapter 5

## Exacerbation and novelty detection

The methods described in the preceding chapters are based on the definition of two reference patterns. Each of those patterns is represented by a model which is helpful in guiding the classification of days in one of those two classes.

In the context of the detection of exacerbations, modeling two opposite classes is not pertinent nor possible, since exacerbation events are rare and variable. Thus, hereafter, a novelty detection approach is tested for the detection of pre- and post-exacerbation profiles. This approach consists of training a single model for each recording so it describes the patient’s “normal” respiratory profile. By comparing daily observations to this unique reference model, our aim is to define when these changes are important enough to characterize an “abnormal”, or “atypical” profile.

In this chapter, previously described methods and metrics are tested about exacerbation detection, in their one-class form.

### 5.1 One-class metrics

For the one-class methods, only the reference data is used to train the reference model and to define the patient’s “normality”. Hereafter, all the previously proposed models are used to describe the normal days: the Wasserstein barycenter, HMM based on the ordered pair (breathing rate and amplitude) and HMM based on the ordered triple (breathing rate, amplitude and oxygen flow).

Daily anomaly scores like the Wasserstein distance, the Kullback-Leibler divergence, the cross-entropy and the log-likelihood are estimated according to the reference model.

Then, normality boundaries are estimated for each anomaly score. Since patients’ baselines are variable, the thresholds are also defined individually. Two methods are used for estimating the thresholds. The first method is based on samplings from the reference model and the second method is based on the anomaly score estimated for the training data.

#### 5.1.1 Threshold estimation from model sampling

With this approach, daily anomaly scores are compared to the reference distribution of expected anomaly scores given the model. This method is adapted for anomaly scores from parametric models, from which samples can be randomly sampled. In our case, those are the HMM and its emission distributions.

Series samples can be generated from the reference HMM with the same length as the daily series to be compared. The log-likelihood of the sampled series are estimated,
generating a reference distribution of normality. The log-likelihood of the observed daily series can then be compared to the distribution, resulting in an approximation of the chances of observing it given the model.

For example, the last week from recording 52 was used to fit the reference HMM. One of its daily series contained 157 observations and its log-likelihood under the reference model was estimated to -1200.6. Different number of samples of length 157 were generated from the reference model in order to build the expected log-likelihood distribution (Figure 5.1). We can therefore estimate an empirical p-value $p = P(X < x_{obs})$, where X is the log-likelihood distribution and $x_{obs}$ is the observed log-likelihood for the daily series. In other words, $p$ is the fraction of log-likelihood from X that are lesser than $x_{obs}$. Whenever all log-likelihood from X are greater than $x_{obs}$, we use $p = 1/n$, to conform to the level of precision.

<figure><img src="image_83.png" /><figcaption>Figure 5.1 – log-likelihood distributions for series of length 157 with different number of samples. Log-likelihoods associated with $p = 0.05$ are indicated with a green bar.</figcaption></figure>

For distributions estimated based in 100, 1000 and 10000 samples, the estimated $p$ for this day from recording 52 was 0.3, 0.272 and 0.287 respectively. Thus, it is most likely a “normal” day.

A similar approach can be used with CE, KL and symmetric CE scores. In this case, samples of same length and same number of points at each state are generated from the emission distributions. Examples of symmetric CE-score distributions for series of length 157, where 117 points are at state 0, are presented in Figure 5.2. The score associated with $p = 0.05$ is indicated in the right tail, because daily series that diverge from the reference distribution present greater score.

The values of $p$ associated with the symmetric CE were 0.64, 0.587 and 0.5768. Once again, in every case, it is most likely that this day from recording 52 comes from the reference model.

The main drawback of this method for estimating the boundaries of normality is the computation time, that increases significantly with the precision desired. In the results presented hereafter, the number of samples generated from the reference HMM was 1000.

### 5.1.2 Threshold estimation from training anomaly scores

In this approach, the boundaries of normality are defined according to the anomaly scores computed for the training series. For each score and each individual, we use the mean $\mu$ and the standard deviation $\sigma$ to estimate the lower and upper thresholds of normality.

$$ \lambda = \mu \pm 1.96\sigma $$
<figure><img src="image_84.png" /><figcaption>Figure 5.2 – Symmetric cross-entropy distributions for series of length 107, with different number of samples. Log-likelihoods associated with $p = 0.05$ are indicated with a green bar.</figcaption></figure>

The anomaly scores estimated for each test daily series are then compared to these thresholds. If a daily anomaly score falls outside the boundaries, the series is considered anomalous.

With this method, other anomaly scores can also be used. Thus, the distance from reference barycenter, the adherence to oxygen therapy and the proportion of points in each state according to HMM are also used as anomaly scores.

Moreover, this method is similar to those proposed in the literature [16, 15, 44] for the monitoring of the breathing rate, where limits of normality are defined according to the 25^th^ and 75^th^ percentiles of the parameter at baseline. With a similar approach, we compute the daily mean of breathing rate and estimate the thresholds in the same way as for the other metrics.

## 5.2 Novelty detection of initial profiles

First, the novelty detection method is tested with recordings without exacerbations, as in previous chapters. Here, instead of using the initial and final weeks to describe two reference profiles, only the final week is used as reference.

Since the final week of recording is the last week before medical discharge, it is considered the most stable medical state for patients who do not exacerbate during stay. For this reason, days in the final week are used to train the reference model.

In opposition, the first week of recording corresponds to the beginning of the rehabilitation program and is considered as a recovery phase. Thus, the proposed methods are tested concerning the detection of days in the initial week as anomalies, with respect to the final week.

Tables 5.1 and 5.2 present the performances for the novelty detection methods using normality boundaries defined by model sampling and from training anomaly scores. The sensitivity and specificity are the abilities to detect initial and final days, respectively.

Because the models are trained and tested with the final weeks, the specificity is quite good for almost all methods tested. In the case of Table 5.2, normality boundaries are defined using $\mu \pm 1.96\sigma$. Assuming that the anomaly scores of the training days are normally distributed, about 95% of the days should be contained within the boundaries, which correspond to observed results.

When estimating the normality boundaries by sampling from reference model (Table
<table><caption>Table 5.1 – Detection of initial profiles based on reference models from final days for patients who do not exacerbate. Normality boundaries are defined by sampling from reference model.</caption><thead><tr><th>Anomaly score</th><th>Accuracy</th><th>Sensitivity</th><th>Specificity</th></tr></thead><tbody><tr><td>log-likelihood with HMM with ordered pairs</td><td>64.0%</td><td>41.8%</td><td>87.1%</td></tr><tr><td>Kullback-Leibler divergence</td><td>57.2%</td><td>88.4%</td><td>25.0%</td></tr><tr><td>Cross-entropy</td><td>59.4%</td><td>57.3%</td><td>61.6%</td></tr><tr><td>Symmetric cross-entropy</td><td>59.6%</td><td>85.3%</td><td>33.0%</td></tr><tr><td>log-likelihood with HMM with ordered triples</td><td>74.3%</td><td>58.6%</td><td>90.6%</td></tr></tbody></table>

<table><caption>Table 5.2 – Detection of initial profiles based on reference models from final days for patients who do not exacerbate. Normality boundaries are defined from training series as $\mu \pm 1.96\sigma$</caption><thead><tr><th>Anomaly score</th><th>Accuracy</th><th>Sensitivity</th><th>Specificity</th></tr></thead><tbody><tr><td>log-likelihood with HMM with ordered pairs</td><td>63.2%</td><td>32.3%</td><td>95.1%</td></tr><tr><td>Kullback-Leibler divergence</td><td>62.9%</td><td>34.5%</td><td>92.4%</td></tr><tr><td>Cross-entropy</td><td>62.1%</td><td>28.9%</td><td>96.4%</td></tr><tr><td>Symmetric cross-entropy</td><td>64.5%</td><td>36.2%</td><td>93.8%</td></tr><tr><td>log-likelihood with HMM with ordered triples</td><td>74.1%</td><td>52.6%</td><td>96.4%</td></tr><tr><td>Distance from reference barycenter</td><td>73.7%</td><td>51.3%</td><td>96.9%</td></tr><tr><td>Oxygen therapy compliance</td><td>64.0%</td><td>34.5%</td><td>94.6%</td></tr><tr><td>Proportion of points in state 0 according to HMM with ordered pairs</td><td>64.5%</td><td>33.2%</td><td>96.9%</td></tr><tr><td>Proportion of points in state 0 according to HMM with ordered triples</td><td>70.4%</td><td>45.3%</td><td>96.4%</td></tr><tr><td>Breathing rate daily mean</td><td>66.7%</td><td>36.2%</td><td>98.2%</td></tr></tbody></table>

5.1), the limits are defined for unilateral $P = .05$. The performances obtained for the log-likelihood scores are similar to those obtained in Table 5.2. However, there is an important decrease in specificity for methods based on cross-entropy (KL, CE and $CE_{symm}$) with normality boundaries set from reference model sampling. In fact, many of the reference days are identified as anomalies which may indicate that observations don't behave the same as samples from the emission distributions. Still, the normality boundaries defined with training anomaly scores for this metrics contain most of the days used as baseline.

Compared to the supervised methods, the accuracies are lower. That reduction is related to less initial days being identified as such, which was expected. Still, most initial days are correctly identified in methods based on the log-likelihood with HMM with ordered triples or based on the Wasserstein distance from reference barycenter.

## 5.3 Novelty detection of prodromal profiles

Novelty detection methods are also applied in recordings containing exacerbation events. Similar to recordings without exacerbation, the reference week is defined as the last week of follow-up, because it is usually the best state of the patient before medical discharge.

There are 25 recordings containing an exacerbation, from which, four of the recordings present an exacerbation in the last days of follow-up. These recordings are excluded from the analysis, since all weeks are considered inappropriate for the estimation of baseline.

Yet another 5 recordings had data completely missing in the week preceding exacerbation.
tion and, for this reason, are also not included in the analysis.

From the remaining 16 recordings containing an exacerbation, the days from final week are used for training the novelty detection methods. We consider the prodromal periods, which include the exacerbation day and the six days preceding it, as the “atypical” days that should be detected with the proposed methods. In some cases, exacerbations take place in the beginning of follow-up, resulting in incomplete prodromal weeks. Each recording contains at least 3 days of prodrome.

The performances obtained for the novelty detection in the recordings containing exacerbations are presented in Tables 5.3 and 5.4. The results are similar to those obtained with recordings without exacerbation, although most accuracies are slightly higher.

<table><caption>Table 5.3 – Detection of prodrome profiles based on reference models from final days for patients who exacerbate during follow-up. Normality boundaries are defined by sampling from reference model.</caption><thead><tr><th>Anomaly score</th><th>Accuracy</th><th>Sensitivity</th><th>Specificity</th></tr></thead><tbody><tr><td>log-likelihood with HMM with ordered pairs</td><td>65.9%</td><td>39.2%</td><td>89.8%</td></tr><tr><td>Kullback-Leibler divergence</td><td>59.5%</td><td>90.7%</td><td>31.5%</td></tr><tr><td>Cross-entropy</td><td>71.2%</td><td>63.9%</td><td>77.8%</td></tr><tr><td>Symmetric cross-entropy</td><td>62.4%</td><td>85.6%</td><td>41.7%</td></tr><tr><td>log-likelihood with HMM with ordered triples</td><td>77.1%</td><td>68.0%</td><td>85.2%</td></tr></tbody></table>

<table><caption>Table 5.4 – Detection of prodrome profiles based on reference models from final days for patients who exacerbate during follow-up. Normality boundaries are defined from training series as $\mu \pm 1.96\sigma$</caption><thead><tr><th>Anomaly score</th><th>Accuracy</th><th>Sensitivity</th><th>Specificity</th></tr></thead><tbody><tr><td>log-likelihood with HMM with ordered pairs</td><td>69.3%</td><td>36.1%</td><td>99.1%</td></tr><tr><td>Kullback-Leibler divergence</td><td>73.2%</td><td>48.5%</td><td>95.4%</td></tr><tr><td>Cross-entropy</td><td>72.7%</td><td>45.4%</td><td>97.2%</td></tr><tr><td>Symmetric cross-entropy</td><td>74.6%</td><td>51.5%</td><td>95.4%</td></tr><tr><td>log-likelihood with HMM with ordered triples</td><td>72.7%</td><td>43.3%</td><td>99.1%</td></tr><tr><td>Distance from reference barycenter</td><td>79.5%</td><td>62.9%</td><td>94.4%</td></tr><tr><td>Oxygen therapy compliance</td><td>70.2%</td><td>38.1%</td><td>99.1%</td></tr><tr><td>Proportion of points in state 0 according to HMM with ordered pairs</td><td>66.8%</td><td>33.0%</td><td>97.2%</td></tr><tr><td>Proportion of points in state 0 according to HMM with ordered triples</td><td>75.6%</td><td>54.6%</td><td>94.4%</td></tr><tr><td>Breathing rate daily mean</td><td>71.2%</td><td>40.2%</td><td>99.1%</td></tr></tbody></table>

Figure 5.3a indicates the number of “atypical” days detected for each recording based on the daily means of breathing rate. Although this method detects 40% of the prodromal days, it can not detect any abnormal day within 6 out of 16 recordings.

In comparison, the highest accuracy is obtained when using the Wasserstein distance from reference barycenter, with more than 60% of the prodromal days correctly identified. Figure 5.3b shows the classification for this method. 11 out of 16 recordings had most of the prodromal days correctly identified, while only recording 30 had all prodromal days detected as “normal”.

Yet another method that obtained good performance is the one based on the sampling of log-likelihood from reference HMM with ordered triples. Figure 5.3c presents the clas-
<figure><img src="image_85.png" /><figcaption>(a) According to daily mean of breathing rate measures</figcaption></figure>

<figure><img src="image_86.png" /><figcaption>(b) According to the Wasserstein distance from the reference barycenter</figcaption></figure>

<figure><img src="image_87.png" /><figcaption>(c) According to sampling of the log-likelihood from the reference HMM with ordered triples</figcaption></figure>

Figure 5.3 – Number of prodromal days detected as “atypical” by recording. Red and blue bars represent number of days detected as “atypical” and “normal”, respectively. Gray bars indicate prodromal days that are not available in the recordings.

sification of prodromal days based on this method. With this method, most days are also identified as “atypical”. Interestingly, this method does not identify any “atypical” day in recording 38, while all days from recording 30 are classified as such.

This results confirm that different methods capture different changes in the respiratory profile. Because exacerbations can manifest themselves in different ways, it could be interesting to consider a combination of anomaly scores.

For example, the methods distance from barycenter and log-likelihood from the HMM with ordered triples can be combined by considering that if a day is “atypical” according to one of the methods, then the day is “atypical”. In this dataset, this would increase the accuracy to 84.4% (with 88.7% sensitivity and 80.6% specificity). Yet another approach is to combine multiple metrics and to accept the most voted classification.

The combination of metrics can be useful to improve the detection of different types of profile changes. However, attention must be given to the risk of overfitting. Thus, with this limited dataset, no specific conclusion can be driven.

## 5.4 Detection of less stable days

Finally, all available days in the recordings are used to analyse how methods classify days that are not prodrome neither baseline. The novelty detection methods are designed to
identify if days are “normal” or “atypical”, thus a new reference labelling of the days is used.

A day is labelled “atypical” if they are in a prodromal or recovery phase. These phases are considered to include the weeks preceding and following an exacerbation day. The first weeks of all recordings are also considered to be “atypical”, since, as previously stated, patients often arrive in the rehabilitation unit in post-exacerbation. All other days are considered as “normal”, although not necessarily used as training data, as explained above.

The dataset used includes recordings with and without exacerbations and which a final week that can be used as baseline. This results in a dataset with 50 recordings.

With this new labeling, the accuracies achieved for all patients combined are presented in Tables 5.5 and 5.6 with thresholds chosen from sampling from reference model or from training anomaly scores.

<table><caption>Table 5.5 – Performances with normality boundaries defined by sampling from reference model</caption><thead><tr><th>Anomaly score</th><th>Accuracy</th><th>Sensitivity</th><th>Specificity</th></tr></thead><tbody><tr><td>log-likelihood with HMM with ordered pairs</td><td>63.5%</td><td>37.4%</td><td>75.9%</td></tr><tr><td>Kullback-Leibler divergence</td><td>40.3%</td><td>89.1%</td><td>17.2%</td></tr><tr><td>Cross-entropy</td><td>53.3%</td><td>57.0%</td><td>51.5%</td></tr><tr><td>Symmetric cross-entropy</td><td>42.2%</td><td>85.6%</td><td>21.6%</td></tr><tr><td>log-likelihood with HMM with ordered triples</td><td>63.6%</td><td>62.4%</td><td>64.2%</td></tr></tbody></table>

<table><caption>Table 5.6 – Performances with normality boundaries defined from training series as $\mu \pm 1.96\sigma$</caption><thead><tr><th>Anomaly score</th><th>Accuracy</th><th>Sensitivity</th><th>Specificity</th></tr></thead><tbody><tr><td>log-likelihood with HMM with ordered pairs</td><td>64.8%</td><td>35.9%</td><td>78.5%</td></tr><tr><td>Kullback-Leibler divergence</td><td>64.5%</td><td>42.0%</td><td>75.1%</td></tr><tr><td>Cross-entropy</td><td>64.3%</td><td>38.5%</td><td>76.5%</td></tr><tr><td>Symmetric cross-entropy</td><td>63.4%</td><td>42.6%</td><td>73.4%</td></tr><tr><td>log-likelihood with HMM with ordered triples</td><td>64.5%</td><td>49.8%</td><td>71.5%</td></tr><tr><td>Distance from reference barycenter</td><td>62.5%</td><td>55.4%</td><td>65.9%</td></tr><tr><td>Oxygen therapy compliance</td><td>65.1%</td><td>38.1%</td><td>77.9%</td></tr><tr><td>Proportion of points in state 0 according to HMM with ordered pairs</td><td>64.9%</td><td>34.8%</td><td>79.2%</td></tr><tr><td>Proportion of points in state 0 according to HMM with ordered triples</td><td>67.1%</td><td>47.4%</td><td>76.4%</td></tr><tr><td>Breathing rate daily mean</td><td>63.3%</td><td>37.2%</td><td>75.7%</td></tr></tbody></table>

There is a decrease in all accuracies, mainly related to a decrease in the specificities. Indeed, until now, the days used to test the classification of the normal group are the same days used for training. Moreover, the choice of the reference period is arbitrary and its relevance is questionable. Results presented in Tables 5.5 and 5.6 consider a real test set, with days never used for training the model. Concerning the detection of “atypical” days, the detection is slightly reduced.

Hereafter, detailed results based in the Wasserstein distance from reference barycenter are presented. This method is chosen because it achieves the best results with the prodromal profiles. The scores of days labeled as “normal” are presented in blue, while those from “atypical days” are presented in red. The normality boundaries, which are individual, are presented as dashed black lines.
### 5.4.1 Results for recordings with exacerbation

Recordings with exacerbation have more precise labelling of atypical days. Although real duration of prodromal and recovery phases is unknown, the exacerbation date is an indicator of a known event. In the figures that present daily novelty scores, the days of exacerbation onset are indicated with a red bar.

The first group of recordings with exacerbations is presented in Figure 5.4. This recordings have in common that all or most days preceding an exacerbation are detected as “atypical”, while final days mostly stay within the normality boundaries.

<figure><img src="image_88.png" /></figure>

Figure 5.4 – Novelty scores based on the Wasserstein distance from reference barycenters for recordings where prodrome is well identified. Horizontal dashed-black lines indicate the individual normality boundaries. The scores of days labeled as “normal” are presented in blue, while those from “atypical days” (prodrome, recovery and exacerbation days) are presented in red.

According to the scores, recordings 17 and 68 present recovery periods longer than a week. Others (21, 22, 28 and 69) present a more abrupt change in the profile, going back under the normality boundary within a few days. This should not be surprising, since the literature describes that patients can take multiple weeks to recover from an exacerbation [75, 86].
Two exacerbations were recorded during recording 37. The first one has a prodrome that is considered atypical, while the second does not. The patient was at GOLD 4D and his medical record indicates modest efficacy of the rehabilitation program. Since the patient is a frequent exacerbator, it is hard to say if the final week of recording could really be considered as a reference for stability.

The second group of recordings with exacerbations contains the recordings for which the detection of prodrome and recovery phases are less evident (Figure 5.5). In each of these recordings, at least a few days of prodrome or recovery were identified as “atypical”. However, there is no clear evolution pattern.

<figure><img src="image_89.png" /><figcaption>Figure 5.5 – Novelty scores based on the Wasserstein distance from reference barycenters for recordings where prodrome is poorly identified. Horizontal dashed-black lines indicate the individual normality boundaries. The scores of days labeled as “normal” are presented in blue, while those from “atypical days” (prodrome, recovery and exacerbation days) are presented in red.</figcaption></figure>

For comparison, those same recordings are considered using the anomaly score of the log-likelihood from the reference HMM with ordered triples (Figure 5.6). For this type of normality boundaries, days with scores below 0.05 are considered “atypical”. We observe that, with this method, prodrome and recovery phases are better identified for recordings 29, 30, 46 and 61.
<figure><img src="image_90.png" /><figcaption>Figure 5.6 – Novelty scores based on the log-likelihood from HMM with ordered triples for recordings where prodrome is not well identified with the Wasserstein distance from reference barycenters. Horizontal dashed-black line indicate the 0.05 normality boundary (scores below the line are considered “atypical”). The scores of days labeled as “normal” are presented in blue, while those from “atypical days” (prodrome, recovery and exacerbation days) are presented in red.</figcaption></figure>

Both methods allow for the detection of different types of changes in the respiratory profile. Although only these two methods have the results presented here, other methods can complete them, by highlighting yet other kinds of changes. More in-depth studies can be carried out to identify which methods are most relevant in the population or whether the most relevant method should be chosen individually.

5.4.2 Results for recordings without exacerbation

Back to the results achieved with the Wasserstein distance from the reference barycenter,
the following figures contemplate the recordings which do not include an exacerbation. Still,
the recordings are considered to contain periods of recovery at the beginning of follow-up.

Figure 5.7 presents the results for the recordings where little to no change is detected
with the chosen method. First and last days are sometimes detected as “atypical”, which
is expected since they can be incomplete according to the time the monitoring device was connected to the oxygen source.

This lack of significant changes related to the recovery period could be explained by i. the changes cannot be captured by the novelty detection method or ii. recording started when patients had already achieved medical stability.

<figure><img src="image_91.png" /><figcaption>Figure 5.7 – Novelty scores based on the Wasserstein distance from reference barycenters for recordings where days stay within the normality boundaries. Horizontal dashed-black lines indicate the individual normality boundaries. The scores of days labeled as “normal” are presented in blue, while those from “atypical days” (prodrome, recovery and exacerbation days) are presented in red.</figcaption></figure>

Other recordings, as those presented in Figure 5.8, present a distinct profile in the beginning of follow-up. Those first days may be related to a short or finishing recovery period. For some, the change is quite abrupt, raising the question about the changes being related to treatment adaptations, proposed after an initial period of evaluation in the rehabilitation unit.

In the next group of recordings (Figure 5.9), a distinct initial period is also detected. However, in those examples, the evolution towards the final profile is longer and more gradual. With current data, it is not possible to confirm if this group and the preceding one have different evolution profiles because of changes in the treatment or because of
<figure><img src="image_92.png" /><figcaption>Figure 5.8 – Novelty scores based on the Wasserstein distance from reference barycenters for recordings where first days are identified as “atypical”. Horizontal dashed-black lines indicate the individual normality boundaries. The scores of days labeled as “normal” are presented in blue, while those from “atypical days” (prodrome, recovery and exacerbation days) are presented in red.</figcaption></figure>

rehabilitation success.

The final group (Figure 5.10) contains recordings where days are sporadically identified as “atypical”. Some of the recordings (24, 42, 65 and 66) present a subtle slope, where days get more similar to the final profile. Days are occasionally "atypical", but it probably only indicate punctual changes. Their relevance for patient’s health should be considered in further studies.

Hypothesis for explaining these highly variable recordings include: i. last week is not a good choice of reference days for this patient; ii. variability belongs to the patient’s normal state and one week of reference days was not capable of capturing this characteristics; iii. days are variable because the patient is still at recovery.

In this study, periods used as reference were chosen as the more likely to correspond to the patient’s baseline. However, the patients were still at hospital when these periods were recorded, indicating that these days are possibly not adequate. A longer monitoring of patients at home could lead to a better identification of periods of stability. Moreover,
<figure><img src="image_93.png" /><figcaption>Figure 5.9 – Novelty scores based on the Wasserstein distance from reference barycenters for recordings where longer periods of initial days are identified as “atypical”. Horizontal dashed-black lines indicate the individual normality boundaries. The scores of days labeled as “normal” are presented in blue, while those from “atypical days” (prodrome, recovery and exacerbation days) are presented in red.</figcaption></figure>

it should be considered if the reference is static or if it evolves through time.

In this study, only static reference periods are used because of limited recording duration. Reference periods of one week are considered so variations between week days and weekends are taken into account. However, it may not be sufficient, as an important day-to-day variability may be normal for some of the patients. Thus, longer-term monitoring of the patients should also consider other durations of baseline period.

A dataset with longer monitoring duration also opens to the possibility of adapting the reference models through time. Indeed, patients are ever changing and some of this changes can be considered normal, or even healthy. The methods proposed here can be easily adapted to consider new reference data within a sliding window.
<figure><img src="image_94.png" /><figcaption>Figure 5.10 – Novelty scores based on the Wasserstein distance from reference barycenters for recordings where days identified as “atypical” are shuffled. Horizontal dashed-black lines indicate the individual normality boundaries. The scores of days labeled as “normal” are presented in blue, while those from “atypical days” (prodrome, recovery and exacerbation days) are presented in red.</figcaption></figure>

5.5 Conclusion

In this chapter, the previously proposed methods were tested in their novelty detection
form. The results point to a moderate accuracy, with anomaly scores that are only capable
of detecting part of the “atypical” days.

Still, the ability to detect abnormalities is similar to, and sometimes better than, that obtained by the method based on a single measurement of respiratory rate per day. Moreover, we show that different abnormality scores are able to identify different types of changes, which may be particular for a patient or for an exacerbation. Thus, further research should be conducted to better understand their advantages and complementarity.

Furthermore, it should be noted that the choice of reference periods is questionable, which may influence significantly the capacity of the methods of identifying relevant changes.
# Chapter 6

## Conclusion and perspectives

This thesis includes different clinical studies with the aim of improving the detection of changes in breathing patterns in COPD patients on LTOT. Although the current dataset only allows for limited conclusion with respect to the detection of exacerbation events, the present study raises new approaches for this problem.

Firstly, the study showed that it is possible to monitor COPD patients with a non invasive and automatic device. Indeed, TeleOx® presents numerous advantages for this task, it is easy to use and adapted for long-term monitoring, it does not add any new con- straint for patients and health providers and it concerns most COPD patients. Moreover, the device was improved by the addition of the measures of inspiratory amplitude and by the automation of data recovery.

Secondly, we show that the poor previously published results in following only the breathing rate can be improved by monitoring the breathing rate combined with other parameters, which leads to a more complete understanding of the patient's respiratory adaptations.

Thirdly, the continuous monitoring of breathing rate combined with the inspiratory amplitude and the oxygen flow provides more detailed information of the patient's respir- atory profile, compared to single daily measures. The study of daily distribution of the features showed that the respiratory profiles vary more between patients than between the days of a same patient. This points to the importance of using artificial intelligence methods that take into account the patient's individuality. Moreover, by measuring sev- eral points per day, the dynamics between days and within days wan be better explained. Indeed, the identification of periods of rest, effort and sleep also lead to a more complete description of the patient's respiratory profile.

Finally, the monitoring of these features and the diversity of anomaly scores proposed lead to the detection of different types of changes in the patient's respiratory profile. Thus, the results obtained allow us to consider the perspective of applying the techniques de- veloped in this thesis to the early detection of exacerbations at home. Still, their relevance and the preference for certain methods for the early detection of exacerbations at home needs to be confirmed with new monitoring data.

Indeed, the main limitation in the clinical validation of the present study concerns the choice of the reference data for training the individual models. In the novelty detection approach, the choice of the reference data is very important since it is the only reference used in both model training and thresholds estimation. In this study, the last week of follow- up was used as the best proxy of the patient's baseline. However, it most probably does not correspond to the patient's reality since they are still hospitalized in the rehabilitation unit.
For further improvements of the described methods, a new data acquisition from patients home monitored is considered. Ideally, monitoring should start when patients are at their real and stable baseline state. This will allow for the description of a pertinent patient's normality model.

Moreover, in the current dataset, few days could be assumed to really belong to the normal state of the patients. While in this dataset, all days were recorded after a recent exacerbation, the new data acquisition should favor longer recordings for inclusion of longer periods of stability. The stability periods are very important for the adjustment of the models with regard to false alarms, which can depreciate the functioning of a medical decision support tool.

Besides, the present study was based on data acquisition in the hospital. On one hand, this allowed for a more precise identification of exacerbation onset dates, while patients at home can postpone looking for a doctor. On the other hand, this means that healthcare providers could intervene quickly, inhibiting exacerbation development. So, changes in the respiratory profiles could be more substantial in the home monitoring context.

Concerning the normal frequency of changes within the days, it is hard to say if it is expected to be higher at home or at hospital. While at the rehabilitation unit, patients followed the rehabilitation program, so most week days were organized similarly. However, patients are closely followed by the medical staff and adaptations in the treatments may occur more often then at home. Hence, the duration of the baseline period should be reconsidered in future studies.

Complementary studies should be lead concerning the relevance of the difference between sleep and wakefulness patterns and to confirm the relationship between hidden states and physiological states of rest and effort. Additionally, other anomaly detection methods can be considered, like the goodness-of-fit tests based in the Wasserstein distance, as proposed by [39]. Future work on the development of the proposed methods also include considering a sliding period of reference and combinations of the proposed anomaly scores.
References

[1] Davies Adeloye et al. ‘Global and regional estimates of COPD prevalence: Systematic review and meta-analysis’. In: *Journal of global health* 5.2 (2015).

[2] Charu C Aggarwal. *Outlier Analysis*. 2nd ed. Springer, 2017.

[3] James P Allinson et al. ‘Combined impact of smoking and early-life exposures on adult lung function trajectories’. In: *American journal of respiratory and critical care medicine* 196.8 (2017), pp. 1021–1030.

[4] Nicolino Ambrosino et al. ‘Tele-monitoring of ventilator-dependent patients: a European Respiratory Society Statement’. In: *European respiratory journal* 48.3 (2016), pp. 648–663.

[5] Nicolino Ambrosino et al. ‘Telemedicine in chronic obstructive pulmonary disease’. In: *Breathe* 12.4 (2016), pp. 350–356.

[6] Julio Ancochea et al. ‘Efficacy and costs of telehealth for the management of COPD: the PROMETE II trial’. In: *European Respiratory Journal* 51.5 (2018).

[7] Alessandra Angelucci and Andrea Aliverti. ‘Telemonitoring systems for respiratory patients: technological aspects’. In: *Pulmonology* 26.4 (2020), pp. 221–232.

[8] ANTADIR. *Observatoire des patients de l'ANTADIR. Données au 31 décembre 2015*. URL: https://www.antadir.com/uploads/editor/file/site-web-observatoire-31.12.2015.pdf.

[9] NR Anthonisen et al. ‘Antibiotic therapy in exacerbations of chronic obstructive pulmonary disease’. In: *Annals of internal medicine* 106.2 (1987), pp. 196–204.

[10] A Anzueto, I Leimer and S Kesten. ‘Impact of frequency of COPD exacerbations on pulmonary function, health status and clinical outcomes’. In: *International journal of chronic obstructive pulmonary disease* 4 (2009), p. 245.

[11] Tarig Ballal et al. ‘A pilot study of the nocturnal respiration rates in COPD patients in the home environment using a non-contact biomotion sensor’. In: *Physiological measurement* 35.12 (2014), p. 2513.

[12] Miguel T Barbosa et al. ‘Telemedicine in COPD: An Overview by Topics’. In: *COPD: Journal of Chronic Obstructive Pulmonary Disease* 17.5 (2020), pp. 601–617.

[13] Douglas Bates et al. ‘Fitting linear mixed-effects models using lme4’. In: *Journal of Statistical Software* 67.1 (2015), pp. 1–48.

[14] J Martin Bland and Douglas G Altman. ‘Measuring agreement in method comparison studies’. In: *Statistical methods in medical research* 8.2 (1999), pp. 135–160.

[15] Sophie Blouet et al. ‘Prediction of severe acute exacerbation using changes in breathing pattern of COPD patients on home noninvasive ventilation’. In: *International journal of chronic obstructive pulmonary disease* 13 (2018), p. 2577.
[16] Jean-Christian Borel et al. ‘Parameters recorded by software of non-invasive ventilators predict COPD exacerbation: a proof-of-concept study’. In: *Thorax* 70.3 (2015), pp. 284–285.

[17] George EP Box et al. *Time series analysis: forecasting and control*. John Wiley & Sons, 2015.

[18] Christopher Burton, Hilary Pinnock and Brian McKinstry. ‘Changes in telemonitored physiological variables and symptoms prior to exacerbations of chronic obstructive pulmonary disease’. In: *Journal of telemedicine and telecare* 21.1 (2015), pp. 29–36.

[19] PMA Calverley and NG Koulouris. ‘Flow limitation and dynamic hyperinflation: key concepts in modern respiratory physiology’. In: *European Respiratory Journal* 25.1 (2005), pp. 186–199.

[20] Anthony E Camilli et al. ‘Longitudinal changes in forced expiratory volume in one second in adults: effects of smoking and smoking cessation’. In: *American Review of Respiratory Disease* 135.4 (1987), pp. 794–799.

[21] M Chatwin et al. ‘Randomised crossover trial of telemonitoring in chronic respiratory patients (TeleCRAFT trial)’. In: *Thorax* 71.4 (2016), pp. 305–311.

[22] T Conroy et al. ‘Recommandations pour la pratique clinique: prise en charge par chimiothérapie palliative de première ligne des patients atteints d’un cancer colorectal métastatique’. In: *Bulletin du cancer* 93.2 (2006), pp. 197–200.

[23] Christopher B Cooper et al. ‘Statistical process control improves the feasibility of remote physiological monitoring in patients with chronic obstructive pulmonary disease’. In: *International journal of chronic obstructive pulmonary disease* 14 (2019), p. 2485.

[24] Josephine M Cranston et al. ‘Domiciliary oxygen for chronic obstructive pulmonary disease’. In: *Cochrane database of systematic reviews* 4 (2005).

[25] Gavin C Donaldson et al. ‘Factors associated with change in exacerbation frequency in COPD’. In: *Respiratory research* 14.1 (2013), pp. 1–9.

[26] Mark D Eisner et al. ‘An official American Thoracic Society public policy statement: Novel risk factors and the global burden of chronic obstructive pulmonary disease’. In: *American journal of respiratory and critical care medicine* 182.5 (2010), pp. 693–718.

[27] Paul L Enright and Duane L Sherrill. ‘Reference equations for the six-minute walk in healthy adults’. In: *American journal of respiratory and critical care medicine* 158.5 (1998), pp. 1384–1387.

[28] Majid Ezzati and Alan D Lopez. ‘Estimates of global mortality attributable to smoking in 2000’. In: *The lancet* 362.9387 (2003), pp. 847–852.

[29] Atena Roshan Fekr, Katarzyna Radecka and Zeljko Zilic. ‘Design and evaluation of an intelligent remote tidal volume variability monitoring system in e-health applications’. In: *IEEE journal of biomedical and health informatics* 19.5 (2015), pp. 1532–1548.

[30] Gary T Ferguson. ‘Why does the lung hyperinflate?’ In: *Proceedings of the American Thoracic Society* 3.2 (2006), pp. 176–179.
[31] Miguel Angel Fernandez-Granero, Daniel Sanchez-Morillo and Antonio Leon-Jimenez. ‘An artificial intelligence approach to early predict symptom-based exacerbations of COPD’. In: *Biotechnology & Biotechnological Equipment* 32.3 (2018), pp. 778–784.

[32] Miguel Angel Fernandez-Granero et al. ‘A machine learning approach to prediction of exacerbations of chronic obstructive pulmonary disease’. In: *International Work-Conference on the Interplay Between Natural and Artificial Computation*. Springer. 2015, pp. 305–311.

[33] C Fuhrman, M-C Delmas et al. ‘Épidémiologie descriptive de la bronchopneumopathie chronique obstructive (BPCO) en France’. In: *Revue des Maladies Respiratoires* 27.2 (2010), pp. 160–168.

[34] GJ Gibson. ‘Lung volumes and elasticity.’ In: *Clinics in chest medicine* 22.4 (2001), pp. 623–35.

[35] GOLD. *Global strategy for the diagnosis, management, and prevention of Chronic Obstructive Pulmonary Disease*. Tech. rep. 2021. URL: https://goldcopd.org/wp-content/uploads/2020/11/GOLD-REPORT-2021-v1.1-25Nov20_WMV.pdf.

[36] Diane R Gold et al. ‘Effects of cigarette smoking on lung function in adolescent boys and girls’. In: *New England Journal of Medicine* 335.13 (1996), pp. 931–937.

[37] Daniel C Grinnan and Jonathon Dean Truwit. ‘Clinical review: respiratory mechanics in spontaneous and assisted ventilation’. In: *Critical Care* 9.5 (2005), pp. 1–13.

[38] Guillermo Gutierrez et al. ‘Respiratory rate variability in sleeping adults without obstructive sleep apnea’. In: *Physiological reports* 4.17 (2016), e12949.

[39] Marc Hallin, Gilles Mordant and Johan Segers. *Multivariate Goodness-of-Fit Tests Based on Wasserstein Distance*. Working Papers ECARES 2020-06. ULB – Université Libre de Bruxelles, Mar. 2020. URL: https://ideas.repec.org/p/eca/wpaper/2013-303372.html.

[40] Matthew J Hegewald and Robert O Crapo. ‘Socioeconomic status and lung function’. In: *Chest* 132.5 (2007), pp. 1608–1614.

[41] Maarten van der Heijden et al. ‘An autonomous mobile system for the management of COPD’. In: *Journal of biomedical informatics* 46.3 (2013), pp. 458–469.

[42] Ike Iheanacho et al. ‘Economic burden of Chronic Obstructive Pulmonary Disease (COPD): a systematic literature review’. In: *International journal of chronic obstructive pulmonary disease* 15 (2020), p. 439.

[43] Morten H Jensen et al. ‘Moving prediction of exacerbation in chronic obstructive pulmonary disease for patients in telecare’. In: *Journal of telemedicine and telecare* 18.2 (2012), pp. 99–103.

[44] Weipeng Jiang et al. ‘Day-to-Day Variability of Parameters Recorded by Home Non-invasive Positive Pressure Ventilation for Detection of Severe Acute Exacerbations in COPD’. In: *International Journal of Chronic Obstructive Pulmonary Disease* 16 (2021), p. 727.

[45] Marjan Kerkhof et al. ‘Association between COPD exacerbations and lung function decline during maintenance therapy’. In: *Thorax* 75.9 (2020), pp. 744–753.

[46] HA Kerstjens et al. ‘Decline of FEV1 by age and smoking status: facts, figures, and fallacies’. In: *Thorax* 52.9 (1997), p. 820.
[47] Romain Kessler et al. 'COMET: a multicomponent home-based disease-management programme versus routine care in severe COPD'. In: *European Respiratory Journal* 51.1 (2018).

[48] Robab Kohansal et al. 'The natural history of chronic airflow obstruction revisited: an analysis of the Framingham offspring cohort'. In: *American journal of respiratory and critical care medicine* 180.1 (2009), pp. 3–10.

[49] Thomas Kronborg et al. 'Population exacerbation incidence contains predictive information of acute exacerbations in patients with chronic obstructive pulmonary disease in telecare'. In: *International journal of medical informatics* 111 (2018), pp. 72–76.

[50] Bernd Lamprecht et al. 'COPD in never smokers: results from the population-based burden of obstructive lung disease study'. In: *Chest* 139.4 (2011), pp. 752–763.

[51] C Laurendeau et al. 'Prise en charge et coûts de la bronchopneumopathie chronique obstructive en France en 2011'. In: *Revue des Maladies Respiratoires* 32.7 (2015), pp. 682–691.

[52] Qiancheng Liang et al. 'Research on non-contact monitoring system for human physiological signal and body movement'. In: *Biosensors* 9.2 (2019), p. 58.

[53] Rafael Lozano et al. 'Global and regional mortality from 235 causes of death for 20 age groups in 1990 and 2010: a systematic analysis for the Global Burden of Disease Study 2010'. In: *The lancet* 380.9859 (2012), pp. 2095–2128.

[54] Carlo Massaroni et al. 'Contact-based methods for measuring respiratory rate'. In: *Sensors* 19.4 (2019), p. 908.

[55] Colin D Mathers and Dejan Loncar. 'Projections of global mortality and burden of disease from 2002 to 2030'. In: *PLoS medicine* 3.11 (2006), e442.

[56] Goeffrey J McLachlan. 'Mahalanobis distance'. In: *Resonance* 4.6 (1999), pp. 20–26.

[57] Jere Mead. 'Control of respiratory frequency'. In: *Journal of applied physiology* 15.3 (1960), pp. 325–336.

[58] Kishan G Mehrotra, Chilukuri K Mohan and HuaMing Huang. *Anomaly detection principles and algorithms*. Springer, 2017.

[59] Joanna Miłkowska-Dymanowska et al. 'A pilot study of daily telemonitoring to predict acute exacerbation in chronic obstructive pulmonary disease'. In: *International journal of medical informatics* 116 (2018), pp. 46–51.

[60] Mas S Mohktar et al. 'Predicting the risk of exacerbation in patients with chronic obstructive pulmonary disease using home telehealth measurement data'. In: *Artificial intelligence in medicine* 63.1 (2015), pp. 51–59.

[61] Joseph M Montserrat et al. 'Evaluation of nasal prongs for estimating nasal flow.' In: *American journal of respiratory and critical care medicine* 155.1 (1997), pp. 211–215.

[62] Patrick B Murphy et al. 'Effect of home noninvasive ventilation with oxygen therapy vs oxygen therapy alone on hospital readmission or death after an acute COPD exacerbation: a randomized clinical trial'. In: *Jama* 317.21 (2017), pp. 2177–2186.

[63] Christopher JL Murray et al. 'Global, regional, and national disability-adjusted life years (DALYs) for 306 diseases and injuries and healthy life expectancy (HALE) for 188 countries, 1990–2013: quantifying the epidemiological transition'. In: *The Lancet* 386.10009 (2015), pp. 2145–2191.
[64] Christopher JL Murray et al. 'The state of US health, 1990-2010: burden of diseases, injuries, and risk factors'. In: *Jama* 310.6 (2013), pp. 591-606.

[65] Laura C Myers et al. 'Reducing Readmissions for Chronic Obstructive Pulmonary Disease in Response to the Hospital Readmissions Reduction Program'. In: *Annals of the American Thoracic Society* ja (2021).

[66] Andrea Nicolò et al. 'The Importance of Respiratory Rate Monitoring: From Health-care to Sport and Exercise'. In: *Sensors* 20.21 (2020), p. 6396.

[67] Denis E O'Donnell, Katherine A Webb and J Alberto Neder. 'Lung hyperinflation in COPD: applying physiology to clinical practice'. In: *COPD Research and Practice* 1.1 (2015), pp. 1-12.

[68] Peter Orchard et al. 'Improving prediction of risk of hospital admission in chronic obstructive pulmonary disease: application of machine learning to telemonitoring data'. In: *Journal of medical Internet research* 20.9 (2018), e263.

[69] World Health Organization. *Projections of mortality and causes of death, 2016 and 2060*. URL: http://www.who.int/healthinfo/global_burden_disease/projections/en/ (visited on 01/02/2021).

[70] H Passing and W Bablok. 'A new biometrical procedure for testing the equality of measurements from two different analytical methods. Application of linear regression procedures for method comparison studies in clinical chemistry, Part I'. In: (1983).

[71] Gabriel Peyré, Marco Cuturi et al. 'Computational optimal transport: With applications to data science'. In: *Foundations and Trends in Machine Learning* 11.5-6 (2019), pp. 355-607.

[72] Eric Polley et al. *SuperLearner: Super Learner Prediction*. R package version 2.0-24. 2018. URL: https://CRAN.R-project.org/package=SuperLearner.

[73] Eva Prescott and Jørgen Vestbo. 'Socioeconomic status and chronic obstructive pulmonary disease'. In: *Thorax* 54.8 (1999), pp. 737-741.

[74] Ph H Quanjer et al. *Lung volumes and forced ventilatory flows*. Vol. 6. Suppl 16. Eur Respiratory Soc, 1993, pp. 5-40.

[75] Klaus F Rabe et al. 'Global strategy for the diagnosis, management, and prevention of chronic obstructive pulmonary disease: GOLD executive summary'. In: *American journal of respiratory and critical care medicine* 176.6 (2007), pp. 532-555.

[76] Julien Rabin et al. 'Wasserstein barycenter and its application to texture mixing'. In: *International Conference on Scale Space and Variational Methods in Computer Vision*. Springer. 2011, pp. 435-446.

[77] Chotirat Ann Ratanamahatana and Eamonn Keogh. 'Making time-series classification more accurate using learned constraints'. In: *Proceedings of the 2004 SIAM international conference on data mining*. SIAM. 2004, pp. 11-22.

[78] N Roche et al. 'Données françaises dans l'enquête internationale Continuing to confront COPD (C2C)'. In: *Revue des Maladies Respiratoires* 34.3 (2017), pp. 180-187.

[79] Juan Luis Rodriguez Hermosa et al. 'Compliance and utility of a smartphone app for the detection of exacerbations in patients with chronic obstructive pulmonary disease: cohort study'. In: *JMIR mHealth and uHealth* 8.3 (2020), e15699.
[80] Andrew Rosenberg and Julia Hirschberg. 'V-measure: A conditional entropy-based external cluster evaluation measure'. In: *Proceedings of the 2007 joint conference on empirical methods in natural language processing and computational natural language learning (EMNLP-CoNLL)*. 2007, pp. 410–420.

[81] Noah Rubio et al. ‘Home monitoring of breathing rate in people with chronic obstructive pulmonary disease: observational study of feasibility, acceptability, and change after exacerbation’. In: *International journal of chronic obstructive pulmonary disease* 12 (2017), p. 1221.

[82] Philippe Salamitou and Xuan Loc Le. *Oxygen therapy monitoring device and method*. U.S. Patent App. 15/760,102. Sept. 2018.

[83] Philippe Salamitou and Xuan Loc Le. *Oxygen therapy monitoring device and method*. European Patent App. 16775584.2. Apr. 2019.

[84] Sundeep S Salvi and Peter J Barnes. ‘Chronic obstructive pulmonary disease in non-smokers’. In: *The lancet* 374.9691 (2009), pp. 733–743.

[85] M Schmidt et al. ‘Variabilité ventilatoire et assistance ventilatoire en réanimation’. In: *Réanimation* 23.1 (2014), pp. 17–24.

[86] Terence AR Seemungal et al. ‘Time course and recovery of exacerbations in patients with chronic obstructive pulmonary disease’. In: *American journal of respiratory and critical care medicine* 161.5 (2000), pp. 1608–1613.

[87] G Segrelles Calvo et al. ‘A home telehealth program for patients with severe COPD: the PROMETE study’. In: *Respiratory medicine* 108.3 (2014), pp. 453–462.

[88] Syed Ahmar Shah et al. ‘Exacerbations in chronic obstructive pulmonary disease: identification and prediction using a digital health system’. In: *Journal of medical Internet research* 19.3 (2017), e69.

[89] Matthew Shorofsky et al. ‘Impaired sleep quality in COPD is associated with exacerbations: the CanCOLD cohort study’. In: *Chest* 156.5 (2019), pp. 852–863.

[90] Thomas Similowski. *BPCO: un enjeu pour la médecine générale*. Elsevier Masson, 2011.

[91] J Soler et al. ‘Validation of respiratory rate measurements from remote monitoring device in COPD patients’. In: *Respiratory medicine and research* 76 (2019), pp. 1–3.

[92] Ministère des Solidarités et de la Santé. *La télésurveillance : ETAPES*. URL: https://solidarites-sante.gouv.fr/soins-et-maladies/prises-en-charge-specialisees/telesante-pour-l-acces-de-tous-a-des-soins-a-distance/article/la-telesurveillance-etapes* (visited on 21/06/2021).

[93] Beni Solow and Andrew Sandham. ‘Nasal airflow characteristics in a normal sample’. In: *The European Journal of Orthodontics* 13.1 (1991), pp. 1–6.

[94] Paul Sorlie et al. ‘Influence of cigarette smoking on lung function at baseline and at follow-up in 14 years: the Framingham Study’. In: *Journal of chronic diseases* 40.9 (1987), pp. 849–856.

[95] T Soumagne and B Degano. ‘La distension pulmonaire dans la bronchopneumopathie chronique obstructive: physiopathologie, conséquences sur la mécanique ventilatoire et stratégies thérapeutiques’. In: *La Lettre du pneumologue* 17.5 (2014), pp. 170–177.
[96] Loredana Stendardi, Barbara Binazzi and Giorgio Scano. 'Exercise dyspnea in patients with COPD'. In: *International journal of chronic obstructive pulmonary disease* 2.4 (2007), p. 429.

[97] Samy Suissa, Sophie Dell'Aniello and Pierre Ernst. 'Long-term natural history of chronic obstructive pulmonary disease: severe exacerbations and mortality'. In: *Thorax* 67.11 (2012), pp. 957-963.

[98] IB Tager et al. 'The natural history of forced expiratory volumes. Effect of cigarette smoking and respiratory symptoms'. In: *Am Rev Respir Dis* 138 (1988), pp. 837-849.

[99] K Song Tan et al. 'Real-time vision based respiration monitoring system'. In: *2010 7th International Symposium on Communication Systems, Networks & Digital Signal Processing (CSNDSP 2010)*. IEEE. 2010, pp. 770-774.

[100] Mingyan Teng. 'Anomaly detection on time series'. In: *2010 IEEE International Conference on Progress in Informatics and Computing*. Vol. 1. IEEE. 2010, pp. 603-608.

[101] Marion Teulier et al. 'Acute bronchodilation increases ventilatory complexity during resting breathing in stable COPD: toward mathematical biomarkers of ventilatory function?' In: *Respiratory physiology & neurobiology* 185.2 (2013), pp. 477-480.

[102] Michael J Tipton et al. 'The human ventilatory response to stress: rate or depth?' In: *The Journal of physiology* 595.17 (2017), pp. 5729-5752.

[103] Mai Tsutsui, Firoozeh Gerayeli and Don D Sin. 'Pulmonary Rehabilitation in a Post-COVID-19 World: Telerehabilitation as a New Standard in Patients with COPD'. In: *International Journal of Chronic Obstructive Pulmonary Disease* 16 (2021), p. 379.

[104] Oliver D Tupper et al. 'Effect of tele-health care on quality of life in patients with severe COPD: a randomized clinical trial'. In: *International journal of chronic obstructive pulmonary disease* 13 (2018), p. 2657.

[105] Mark J Van der Laan, Eric C Polley and Alan E Hubbard. 'Super learner'. In: *Statistical applications in genetics and molecular biology* 6.1 (2007).

[106] Paul P Walker et al. 'Telemonitoring in chronic obstructive pulmonary disease (CHROMED). A randomized clinical trial'. In: *American journal of respiratory and critical care medicine* 198.5 (2018), pp. 620-628.

[107] JR Wheatley, TC Amis and LA Engel. 'Nasal and oral airway pressure-flow relationships'. In: *Journal of Applied Physiology* 71.6 (1991), pp. 2317-2324.

[108] Tom MA Wilkinson et al. 'Early therapy improves outcomes of exacerbations of chronic obstructive pulmonary disease'. In: *American journal of respiratory and critical care medicine* 169.12 (2004), pp. 1298-1303.

[109] M Wysocki et al. 'La variabilité du comportement ventilatoire: perspectives pour le malade de réanimation'. In: *Réanimation* 15.1 (2006), pp. 7-13.

[110] Aina M Yañez et al. 'Monitoring breathing rate at home allows early identification of COPD exacerbations'. In: *Chest* 142.6 (2012), pp. 1524-1529.

[111] Dit-Yan Yeung and Yuxin Ding. 'Host-based intrusion detection using dynamic and static behavioral models'. In: *Pattern recognition* 36.1 (2003), pp. 229-243.

[112] Sabine C Zimmermann et al. 'Day-to-day variability of forced oscillatory mechanics for early detection of acute exacerbations in COPD'. In: *European Respiratory Journal* 56.3 (2020).