# Fuzzy Linguistic Protoforms to Summarize Heart Rate Streams of Patients with Ischemic Heart Disease

Authors: María Dolores Peláez-Aguilera; Macarena Espinilla; María Rosa Fernández Olmo; Javier Medina
Journal: Complexity
 (2019)

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## Abstract

Cardiac rehabilitation is a key program which significantly decreases mortality rates in high-risk patients with ischemic heart disease. Due to the huge lack of accessibility to such programs at Health Centers, outdoor-based programs for cardiac rehabilitation have been proposed as an excellent tool to improve accessibility for patients at Health Centers. These outdoor-based programs make use of wrist-worn devices for real-time monitoring of rehabilitation sessions based on clinical guidelines. In this way, a greater number of patients can fortunately gain access to the rehabilitation program. However, this advantage also means that the cardiac rehabilitation team has to monitor a greater number of sessions due to the increase of the number of benefited patients, so the team members spend a lot of time analyzing each patient’s sessions. In this paper, we present a methodology to evaluate heart rate streams of patients with ischemic heart disease using a linguistic approach. This innovative methodology manages relevant linguistic descriptions (protoforms) for the cardiac rehabilitation team to identify sessions with interest indicators by means of linguistic summaries. Therefore, the analysis process is automated in a comprehensible way, offering short linguistic descriptions to the cardiac rehabilitation team, who promptly provide feedback to their patients. In order to show the great efficiency and effectiveness of the proposed methodology, we have used and applied this methodology to real data provided by patients of an outdoor cardiac rehabilitation program run by the Health Council of the Andalusian Health Service (Spain).


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## Body

## 1. Introduction

The Internet of Things (IoT) paradigm [1] is based on the idea of multiple devices located around the world working to acquire information and store it in order to subsequently process and analyze this data with the aim of providing intelligent services [2].In this context, there are open challenges to extract rich information from the huge amount of data from sensor sources in large-scale deployment within the revolutionary paradigm of IoT. To alleviate this limitation, multiple areas in the research field have been involved, such as data filtering, data aggregation, semantic analysis, and information utilization [3].On the other hand, linguistic descriptions of data generate natural language texts [4] that convey the most relevant information contained, and sometimes hidden, in the data. Protoforms and fuzzy logic, which were proposed by Zadeh [5, 6] as a useful knowledge model for reasoning [7], summarization [8], and aggregation [9] of data under uncertainty, are modeled by fuzzy sets whose degree of truth to fuzzy sets is defined by membership functions.Both IoT paradigm and fuzzy linguistic models have been successfully proposed for managing uncertainty and vagueness in an interpretable way, which is a key issue to obtain high performance and results [10]. So, the use of protoforms and fuzzy logic has provided brilliant results in IoT systems in multiple areas with sensor data streams, such as weather forecasting [11], predicting of demand for urgent care in smart cities [12], fever medication control [13], visual scenes [14], or monitoring of patients with preeclampsia in wearable devices [15]. So, the fuzzy logic has been demonstrated as a useful tool to deal with the uncertainty in the complex Internet of Things systems. In the green multimodal routing problem to improve the reliability of the routes, the fuzzy logic was proposed to model the uncertainty in a piecewise linear function to represent the road traffic congestion [16]. The approaches for activity recognition based on fuzzy logic have provided excellent results in the optimization of the configuration of a heterogeneous architecture of sensors [17]. In the context of robot manipulators’ systems with multiple sensors and actuators, a fuzzy control scheme with adaptation algorithms has been proposed to manage the uncertainty of the information [18]. To predict the available maximal data transfer rate of single-pair high-speed digital subscriber line connections from measured frequency dependent electrical parameters of wire pairs, an approach based on fuzzy logic has been proposed [19].This paper falls in the research field of linguistic descriptions of data with protoforms and fuzzy logic applied to e-health solutions based on complex IoT systems, specifically, in cardiovascular diseases, which represent the main health problem in developed countries according to the World Health Organization (WHO) [20] and where fuzzy logic has been shown to work as an effective modeling tool in cardiac rehabilitation [21, 22].In the health field, secondary prevention programs and Cardiac Rehabilitation Units (CRU) have been developed in several countries [23, 24], having been proven as the most effective tool to improve prognosis. Cardiac rehabilitation (CR) is defined as the sum of activities required to influence the underlying cause of heart disease favorably, as well as to ensure the best physical, social, and mental condition of patients, enabling them to occupy a normal place in society by their own means [25].In previous work [22], an outdoor cardiac rehabilitation program (CRP) for patients was embedded in a wrist-worn device with a heart rate sensor for personalized care. Outdoor CRPs have increased the accessibility of cardiac rehabilitation programs due to the fact that they overcome several limitations, such as lack of time, commodities, geographical area, and access to health services [24, 26]. In [22], an outdoor program was designed and supervised remotely by the cardiac rehabilitation team by means of a wearable mobile-cloud platform for collecting and synchronizing data between patients and the cardiac rehabilitation team. To do so, a linguistic approach based on fuzzy logic was proposed in order to model the cardiac rehabilitation protocol and the expert knowledge from the cardiac rehabilitation team.A great impact under the outdoor CRP is that the number of benefited patients has drastically increased. This positive fact, however, means that the health team must monitor a greater number of sessions. In this way, the team members spend a lot of time analyzing the patients’ sessions in a home-based CRP and they are overwhelmed with the huge amounts of information generated by each patient’s wrist-worn device.In order to solve this limitation, in this paper we present a methodology that generates textual information, summaries, from the heart rate streams of patients with ischemic heart disease by means of protoforms and fuzzy logic. So, this paper presents a methodology to summarize patients’ rehabilitation sessions, offering understandable information for the cardiac rehabilitation team. The key points of the proposed methodology are the following:(i)
To allow the cardiac rehabilitation team to supervise a huge number of sessions and patients by means of linguistic summaries, which integrate an intuitive representation(ii)
To model a proposed methodology where the linguistic summaries are focused on rich expressiveness, including linguistic temporal terms and linguistic quantifiers by means of linguistic aggregation operators(iii)
To provide a flexible linguistic methodology where the cardiac rehabilitation team intuitively defines the key interest indicators using protoforms based on expert knowledge in order to recover and dynamically select the rehabilitation sessions that suit and match the expert criteriaThe proposed methodology is applied to real data provided by several patients of a cardiac rehabilitation program run by the Health Council of the Andalusian Health Service (Spain) in order to show its efficiency and effectiveness.The remainder of this paper is structured as follows: Section2 presents the novel methodology to generate linguistic summaries of the rehabilitation sessions from the heart rate streams of patients with ischemic heart disease for the cardiac rehabilitation team. Section 3 presents a case study to show the utility and applicability of the proposed methodology with real data from the rehabilitation sessions of three patients freely participating in cardiac rehabilitation programs provided by the Health Council of the Andalusian Health Service in the Region of Jaén (Spain). Finally, some concluding remarks are pointed out in conjunction with future works.


## 2. Methodology

In this section, we describe a methodology to generate linguistic summaries of the rehabilitation sessions (RSs) from the heart rate streams (HRS) of patients with ischemic heart disease that follow an outdoor CRP. The HRS data are collected from real patients wearing a high-quality wrist-worn device, which has improved the quality of heart rate measurements and their health applications [27]. It is noteworthy that the linguistic modeling developed in this work has been defined by health experts in the CRP to summarize the sessions with interest indicators in cardiac rehabilitation.For this purpose, we will present data processing of HRS from rehabilitation sessions through three stages. In the first stage, the raw data from heart rate streams is initially preprocessed using a previous approach [22]. In this stage, a fuzzy model is proposed to monitor the heart rate under a linguistic approach in real time by means of three representative terms and their membership functions,low,adequate, andhigh, as well as short-term fuzzy temporal windows (FTWs). The linguistic terms are computed in real time within the wrist-worn device in order to advise patients while they are undergoing the rehabilitation session. The main key points are briefly described in Section 2.1.In this work, we present a methodology to compute linguistic summaries identifying key health indicators using expert knowledge from the linguistic terms computed in the first stage. Unlike the first stage, which is computed in real time while the patient is doing the exercise and without the complete information of the session, the summaries are computed for the cardiac rehabilitation team in a centralized way to evaluate a huge number of patients and sessions. For this purpose, we present a flexible model which selects and recovers the sessions according to expert criteria linguistically and intuitively.So, in the second stage, we integrate an interpretable approach for the cardiac rehabilitation team that models knowledge linguistically. To do so, we use ad hoc protoforms. Protoforms are a general form of linguistic data summery [8].The protoforms proposed in this methodology define linguistic summaries from the HRS of the sessions by means of long-term fuzzy temporal windows and fuzzy quantifiers, which provide rich expressiveness in the model.In the third stage, we present the computing of a unique aggregation degree for the protoforms which describes the rehabilitation sessions summarizing the relevance and impact of the protoforms, during the complete rehabilitation sessions. To do so, two semantics of aggregation operators are described:maximum andamplitude.In Figure1, we describe the three stages of the proposed methodology to summarize the rehabilitation sessions of patients by means of protoforms using a linguistic approach.Figure 1
Architecture of the proposed methodology. First(2.1), the heart rate streams generated by wearable devices are computed to determine if the adherence to rehabilitation programs is adequate in real time. Second (2.2), linguistic summaries identify health indicators defined by experts using protoforms. Third (2.3), a final aggregation degree from the protoforms describes the rehabilitation sessions of the patients.


### 2.1. Real-Time Monitoring of Heart Rate Streams

In work [22], real-time monitoring of heart rate streams was proposed. First, three intuitive termslow,adequate, andhigh defined with three membership functions by means of fuzzy sets were proposed in order to describe the variableheart rate (HR).The HR was measured by a pair valuesi¯={si,ti}, where si represents a given value in the HRS and ti is its time stamp. Hence, the HRS of a session is composed of a set of measured values SHRS={s0¯,…,si¯,…,sn¯}, which are collected by the heart rate sensor of the wearable device.Second, the fuzzification of HR,hri¯, is determined by (i) Optimal Heart Rate Training Zones (OHRTZ) where the patient must develop the sessions, which is represented as a discrete HR range within [r+∗,r-∗], and (ii) the Ventilatory Thresholds [VT1,VT2], which represent the aerobic-anaerobic thresholds for the performance of an efficient and safe physical activity. So, the termslow,adequate, andhigh of the variable HR si are described by a fuzzy set characterized by a membership function whose shape corresponds to the trapezoidal functions (TS, TR, and TL are described in Abbreviations) of(1)μadequateSHRS=TSSHRSVT1,r-∗,r+∗,VT2,VT1<r-∗<r+∗<VT2μhighSHRS=TRSHRSr+∗,VT2,VT2>r+∗μlowSHRS=TLSHRSVT1,r-∗,VT1<r-∗.In Figure2, we show the representation of the fuzzy sets (membership functions) of HR for the three linguistic terms:low,adequate, andhigh. As noted in work [22], the thresholds of TS r+∗(ti),r-∗(ti),VT1(ti),VT2(ti) could progressively increase from basal state-defining time-dependent terms, but for the sake of simplicity here we write r+∗,r-∗,VT1,VT2.Figure 2
Representation with trapezoidal membership functions of the linguistic termslow,adequate, andhigh of the HR variable by means of Optimal Heart Rate Training Zones and Ventilatory Thresholds.Third, a fuzzy temporal window [28, 29] to model the HRS was proposed in order to weight fuzzy linguistic terms based on fuzzy temporal linguistic terms and provide flexibility in the presence of eventual signal loss or variance in the sample rate. The FTWs are described straightforwardly according to the distance of the current time t0 to a given time stamp ti as Δti=ti-t0. In this work, the use of FTWs is introduced also to describe temporal evaluation of long and middle terms in SHRS.Fourth, the degrees of the fuzzy linguistic termsV={Vlow, Vadequate, Vhigh} are weighted by the degree of their time stamps evaluated by the FTW Tk:(2)Vr∩Tksi¯=Vrsi∩TkΔti∈0,1Vr∪TkSHRS=⋃si¯∈SHRSVr∩Tksi¯∈0,1.A Fuzzy Weighted Average (FWA) [30] was proposed as an operation to model the t-norm and conorm: (3)Vr∪TkSHRS=1∑tiSHRSTkΔti∑tiSHRSTkΔtiVrsi×TkΔti,∈0,1.Under evaluation, the experts defined and selected the most adequate size for the FTWsTlow, Tadequate, and Thigh in order to weight the terms Vlow, Vadequate, and Vhigh, respectively. An embedded application in the wrist-worn device computes the degree of the termslow,adequate, andhigh in real time for each si¯ using(4)lowsi¯=Vlow∪Tlowsi¯adequatesi¯=Vadequate∪Tadequatesi¯highsi¯=Vhigh∪Thighsi¯.Finally, the degrees of the fuzzy linguistic termslow(si¯), adequate(si¯), and high(si¯) are computed within the wrist-band device: (i) showing a visually interpretable colored circle which represents whether the current HR si¯ islow,adequate, orhigh during its FTWs and (ii) alerting the patient through sensor vibration in the wrist-band when si¯ is computed as high adequate(si¯)<high(si¯) in its FTWs.


### 2.2. Fuzzy Linguistic Summaries of Cardiac Rehabilitation Sessions

In the previous section, we described the real-time evaluation of HR in a wrist-band application using a clinical-based protocol for monitoring and advisingSHRS. Here, we detail a methodology to generate linguistic summaries and identify key interest indicators using expert knowledge from the fuzzy linguistic terms computed in the wrist-worn devices, which describe the real-time adherence and performance of the patient in his/her HRS.For this purpose, we start from the degree of the termsadequate, low, and high described in (4) for each si¯ within the data stream SHRS. An example is shown in Figure 3, where a timeline with a real HRS is plotted using gradual colorsblue, green, and red based on the degree of the fuzzy linguistic terms low(si¯), adequate(si¯), and high(si¯), respectively.Figure 3
A timeline with a real HRS. The time and value configure the points in the chartsi¯={si,ti}, which are plotted using gradual colorsblue, green, and red based on the degree of the terms low(si¯), adequate(si¯), and high(si¯), respectively.


#### 2.2.1. Protoforms for Describing Heart Rate Streams

The aim of the proposed methodology is to generate linguistic summaries from the rehabilitation sessions of patients. For this purpose, in this second stage we process the fuzzy linguistic termslow(si¯), adequate(si¯), and high(si¯) from the data stream SHRS, which are described in the previous section.First, in order to integrate an interpretable and rich-expressive approach to model the expert knowledge linguistically, we introduce an ad hocprotoformPo in the form of(5)Posi¯:QkLiTj,where(i)
Li defines a fuzzy linguistic term to evaluate the data stream. Here, Li is straightforwardly related to fuzzy terms low(si¯), adequate(si¯), and high(si¯)(ii)
Tj defines a fuzzy temporal term where the term Li is aggregated. The use of FTWs, which were introduced in the previous section, is extended to generate linguistic summaries of middle-long temporal terms from SHRS. The aggregation of Li over Tj for a given si¯ is computed by (4) as Li∪Tj(si¯)(iii)
Qk defines a fuzzy quantifier (FQ) to evaluate the impact and fulfillment of the linguistic term Li within the temporal window Tj [28]. A FQ applies a transformation μQK:[0,1]→[0,1] to the aggregated temporal degree of μQK(Vr∪Tk(si¯)The aim of modeling knowledge through protoforms is allowing the rehabilitation team to define key interest indicators using expert intuitive representations of temporal and quantification terms linguistically. An example of protoform ismost of the time (Qk) HR is adequate (Li) for around 40-60 minutes (Tj). We note that the protoforms are suitable to linguistically describe the impact and fulfillment of a fuzzy linguistic term in a fuzzy temporal window in more detail, but subsequently they can be renamed to a shorter linguistic description, such asadequate HR in session.In Figure4, the protoformmost of the time HR is adequate for around 30-50 minutes is shown, whose degree is represented with a real SHRS. We also plot the degree of the protoform Po(si¯) in relation to the term adequate(si¯) to describe the importance of the middle-long FTW and the FQ which enable the linguistic interpretability of the sentenceadequate heart rate in session in the sensor stream. For the sake of simplicity, the shapes of the fuzzy membership functions of the example are detailed in Section 3.Figure 4
A timeline with the degrees of the termadequate(si¯) (dotted line) and protoformmost of the time HR is adequate for around 30-50 minutesPo(si¯) (thick line) on a real HRS. Based on its semantics, the degree of the protoform increases when the patient has maintained an adequate HR for 30 minutes.Second, protoformsPo(si¯) can be combined using fuzzy logical operators to increase the linguistic capabilities of the model. So, we briefly introduce the following basic operations, which could be straightforwardly increased with advanced fuzzy operations in other contexts:(i)
Fuzzy negation operator, which is represented as the complement¬ by the fuzzy function ¬Po(si¯)=1-Po(si¯)(ii)
Fuzzy union operator, which is represented by the t-normPo∧Pq(si¯)=Po(si¯)∧Pq(si¯). The semantic function proposed for the fuzzy union operator is min: Po∧Pq(si¯)=min⁡{Po(si¯),Pq(si¯)}(iii)
Fuzzy intersection operator, which is represented by the conormPo∨Pq(si¯)=Po(si¯)∨Pq(si¯). The semantic function proposed for the fuzzy intersection operator is max: Po∧Pq(si¯)=max⁡{Po(si¯),Pq(si¯)}


#### 2.2.2. Aggregation Operation for Protoforms

As we detailed in Section2.2.1, protoforms are defined to represent the health indicators from the rehabilitation sessions linguistically by means of expert knowledge. Although the evaluation of protoforms is properly computed Po(si¯) throughout the data stream SHRS, an aggregation degree is proposed here in order to summarize the relevance of a protoform.First, in (6) we describe the aggregation operation ∪(Po) of the protoform Po, which computes a single degree from the degree of protoform Po(si¯) over SHRS, as(6)∪Po=⋃si¯SHRSPosi¯∈0,1.In order for the aggregated degree of the aggregation operator∪(Po) to keep its semantic integrity with the degree of the protoforms, we define two properties which the aggregation operation should assess:(i)
∪0: zero-aggregation from zero stream. If the degree of the protoform is zero in the stream, the aggregation is zero: Po(si¯)=0,∀si¯∈SHRS→∪(Po)=0. It is a necessary condition of boundary property [31](ii)
∪+: positive-aggregation from nonzero stream. If there is a nonzero degree of the protoform in the stream, the aggregation is nonzero: ∃Po(si¯)>0,si¯∈SHRS→∪(Po)≠0→∪(Po)>0We note the two properties have been determined based on the semantics of the protoforms to describe the HRS, which guarantee that the number of instances which match to a given protoform∪(Po)>0 is equal although the semantics of the aggregation varies. Specifically, the aggregation degree is zero if and only if the protoform is not representative in the session ∪(Po)=0↔Po(si¯)=0,∀si¯∈SHRS; in another case ∪(Po)>0. We note that, in different fields and other contexts, further properties in aggregation operators can be selected [32].Second, we propose two semantics to aggregate the degree of the protoformPo(si¯):maximum andamplitude.(i)
On the first hand, themaximum aggregation operator max⁡(Po) computes the maximal value of degree of the protoform, which is shown in (7)max⁡Po=max⁡Pos0¯,…,Posi¯∈0,1.The semantics ofmax⁡(Po) describes the maximal truth degree of the protoform providing an intuitive representation as aggregation, which has been widely used in fuzzy logic as aggregation of rules within Mamdani-type inference models [33]. It fulfills the properties of zero-aggregation ∪0 and positive-aggregation ∪+.(8)∪0:Posi¯=0,∀si¯∈SHRS≡∪Po=max⁡0,…,0=0∪+:∃Posi¯+>0,si¯∈SHRS≡∪Po=max⁡0,Posi¯+,…,0>0(ii)
On the other hand, theamplitude aggregation operator (Po) describes the persistence and presence of the protoform degree Po(si¯) throughout the rehabilitation session. For this purpose, the fuzzy quantification of the weight of the protoform W(Po) degree within the HRS is proposed in (9)WPo=∑si¯SHRSsi¯SHRSPo=QWPo∈0,1.First, the weight of the protoform degreeW(Po), which represents a suitable measure as fuzzy aggregation [34], is computed as the relation between the sum of the degrees ∑si¯SHRSsi¯ regarding the norm (or size) of the complete HRS SHRS. Second, a fuzzy quantification is provided by a FQ which transforms W(Po) into (Po)=Q(W(Po)) through the fuzzy membership function μQ:[0,1]→[0,1]. For the sake of simplicity, we refer to μQ as Q.To guarantee that(Po) fulfills the properties of zero-aggregation and positive-aggregation, the membership function Q of the FQ should assess the following properties: Q is a monotone function x≤y→Q(x)≤Q(y), Q(0)=0, and limx→0+Q(x)>0:(10)∪0:Q0=0,Posi¯=0,∀si¯∈SHRS≡WPo=0→QWPo=0∪+:limx→0+Qx>0,x≤y→Qx≤Qy,∃Posi¯>0,si¯∈SHRS≡∑si¯SHRSsi¯>0→WPo>0→QWPo>0


## 2.1. Real-Time Monitoring of Heart Rate Streams

In work [22], real-time monitoring of heart rate streams was proposed. First, three intuitive termslow,adequate, andhigh defined with three membership functions by means of fuzzy sets were proposed in order to describe the variableheart rate (HR).The HR was measured by a pair valuesi¯={si,ti}, where si represents a given value in the HRS and ti is its time stamp. Hence, the HRS of a session is composed of a set of measured values SHRS={s0¯,…,si¯,…,sn¯}, which are collected by the heart rate sensor of the wearable device.Second, the fuzzification of HR,hri¯, is determined by (i) Optimal Heart Rate Training Zones (OHRTZ) where the patient must develop the sessions, which is represented as a discrete HR range within [r+∗,r-∗], and (ii) the Ventilatory Thresholds [VT1,VT2], which represent the aerobic-anaerobic thresholds for the performance of an efficient and safe physical activity. So, the termslow,adequate, andhigh of the variable HR si are described by a fuzzy set characterized by a membership function whose shape corresponds to the trapezoidal functions (TS, TR, and TL are described in Abbreviations) of(1)μadequateSHRS=TSSHRSVT1,r-∗,r+∗,VT2,VT1<r-∗<r+∗<VT2μhighSHRS=TRSHRSr+∗,VT2,VT2>r+∗μlowSHRS=TLSHRSVT1,r-∗,VT1<r-∗.In Figure2, we show the representation of the fuzzy sets (membership functions) of HR for the three linguistic terms:low,adequate, andhigh. As noted in work [22], the thresholds of TS r+∗(ti),r-∗(ti),VT1(ti),VT2(ti) could progressively increase from basal state-defining time-dependent terms, but for the sake of simplicity here we write r+∗,r-∗,VT1,VT2.Figure 2
Representation with trapezoidal membership functions of the linguistic termslow,adequate, andhigh of the HR variable by means of Optimal Heart Rate Training Zones and Ventilatory Thresholds.Third, a fuzzy temporal window [28, 29] to model the HRS was proposed in order to weight fuzzy linguistic terms based on fuzzy temporal linguistic terms and provide flexibility in the presence of eventual signal loss or variance in the sample rate. The FTWs are described straightforwardly according to the distance of the current time t0 to a given time stamp ti as Δti=ti-t0. In this work, the use of FTWs is introduced also to describe temporal evaluation of long and middle terms in SHRS.Fourth, the degrees of the fuzzy linguistic termsV={Vlow, Vadequate, Vhigh} are weighted by the degree of their time stamps evaluated by the FTW Tk:(2)Vr∩Tksi¯=Vrsi∩TkΔti∈0,1Vr∪TkSHRS=⋃si¯∈SHRSVr∩Tksi¯∈0,1.A Fuzzy Weighted Average (FWA) [30] was proposed as an operation to model the t-norm and conorm: (3)Vr∪TkSHRS=1∑tiSHRSTkΔti∑tiSHRSTkΔtiVrsi×TkΔti,∈0,1.Under evaluation, the experts defined and selected the most adequate size for the FTWsTlow, Tadequate, and Thigh in order to weight the terms Vlow, Vadequate, and Vhigh, respectively. An embedded application in the wrist-worn device computes the degree of the termslow,adequate, andhigh in real time for each si¯ using(4)lowsi¯=Vlow∪Tlowsi¯adequatesi¯=Vadequate∪Tadequatesi¯highsi¯=Vhigh∪Thighsi¯.Finally, the degrees of the fuzzy linguistic termslow(si¯), adequate(si¯), and high(si¯) are computed within the wrist-band device: (i) showing a visually interpretable colored circle which represents whether the current HR si¯ islow,adequate, orhigh during its FTWs and (ii) alerting the patient through sensor vibration in the wrist-band when si¯ is computed as high adequate(si¯)<high(si¯) in its FTWs.


## 2.2. Fuzzy Linguistic Summaries of Cardiac Rehabilitation Sessions

In the previous section, we described the real-time evaluation of HR in a wrist-band application using a clinical-based protocol for monitoring and advisingSHRS. Here, we detail a methodology to generate linguistic summaries and identify key interest indicators using expert knowledge from the fuzzy linguistic terms computed in the wrist-worn devices, which describe the real-time adherence and performance of the patient in his/her HRS.For this purpose, we start from the degree of the termsadequate, low, and high described in (4) for each si¯ within the data stream SHRS. An example is shown in Figure 3, where a timeline with a real HRS is plotted using gradual colorsblue, green, and red based on the degree of the fuzzy linguistic terms low(si¯), adequate(si¯), and high(si¯), respectively.Figure 3
A timeline with a real HRS. The time and value configure the points in the chartsi¯={si,ti}, which are plotted using gradual colorsblue, green, and red based on the degree of the terms low(si¯), adequate(si¯), and high(si¯), respectively.


### 2.2.1. Protoforms for Describing Heart Rate Streams

The aim of the proposed methodology is to generate linguistic summaries from the rehabilitation sessions of patients. For this purpose, in this second stage we process the fuzzy linguistic termslow(si¯), adequate(si¯), and high(si¯) from the data stream SHRS, which are described in the previous section.First, in order to integrate an interpretable and rich-expressive approach to model the expert knowledge linguistically, we introduce an ad hocprotoformPo in the form of(5)Posi¯:QkLiTj,where(i)
Li defines a fuzzy linguistic term to evaluate the data stream. Here, Li is straightforwardly related to fuzzy terms low(si¯), adequate(si¯), and high(si¯)(ii)
Tj defines a fuzzy temporal term where the term Li is aggregated. The use of FTWs, which were introduced in the previous section, is extended to generate linguistic summaries of middle-long temporal terms from SHRS. The aggregation of Li over Tj for a given si¯ is computed by (4) as Li∪Tj(si¯)(iii)
Qk defines a fuzzy quantifier (FQ) to evaluate the impact and fulfillment of the linguistic term Li within the temporal window Tj [28]. A FQ applies a transformation μQK:[0,1]→[0,1] to the aggregated temporal degree of μQK(Vr∪Tk(si¯)The aim of modeling knowledge through protoforms is allowing the rehabilitation team to define key interest indicators using expert intuitive representations of temporal and quantification terms linguistically. An example of protoform ismost of the time (Qk) HR is adequate (Li) for around 40-60 minutes (Tj). We note that the protoforms are suitable to linguistically describe the impact and fulfillment of a fuzzy linguistic term in a fuzzy temporal window in more detail, but subsequently they can be renamed to a shorter linguistic description, such asadequate HR in session.In Figure4, the protoformmost of the time HR is adequate for around 30-50 minutes is shown, whose degree is represented with a real SHRS. We also plot the degree of the protoform Po(si¯) in relation to the term adequate(si¯) to describe the importance of the middle-long FTW and the FQ which enable the linguistic interpretability of the sentenceadequate heart rate in session in the sensor stream. For the sake of simplicity, the shapes of the fuzzy membership functions of the example are detailed in Section 3.Figure 4
A timeline with the degrees of the termadequate(si¯) (dotted line) and protoformmost of the time HR is adequate for around 30-50 minutesPo(si¯) (thick line) on a real HRS. Based on its semantics, the degree of the protoform increases when the patient has maintained an adequate HR for 30 minutes.Second, protoformsPo(si¯) can be combined using fuzzy logical operators to increase the linguistic capabilities of the model. So, we briefly introduce the following basic operations, which could be straightforwardly increased with advanced fuzzy operations in other contexts:(i)
Fuzzy negation operator, which is represented as the complement¬ by the fuzzy function ¬Po(si¯)=1-Po(si¯)(ii)
Fuzzy union operator, which is represented by the t-normPo∧Pq(si¯)=Po(si¯)∧Pq(si¯). The semantic function proposed for the fuzzy union operator is min: Po∧Pq(si¯)=min⁡{Po(si¯),Pq(si¯)}(iii)
Fuzzy intersection operator, which is represented by the conormPo∨Pq(si¯)=Po(si¯)∨Pq(si¯). The semantic function proposed for the fuzzy intersection operator is max: Po∧Pq(si¯)=max⁡{Po(si¯),Pq(si¯)}


### 2.2.2. Aggregation Operation for Protoforms

As we detailed in Section2.2.1, protoforms are defined to represent the health indicators from the rehabilitation sessions linguistically by means of expert knowledge. Although the evaluation of protoforms is properly computed Po(si¯) throughout the data stream SHRS, an aggregation degree is proposed here in order to summarize the relevance of a protoform.First, in (6) we describe the aggregation operation ∪(Po) of the protoform Po, which computes a single degree from the degree of protoform Po(si¯) over SHRS, as(6)∪Po=⋃si¯SHRSPosi¯∈0,1.In order for the aggregated degree of the aggregation operator∪(Po) to keep its semantic integrity with the degree of the protoforms, we define two properties which the aggregation operation should assess:(i)
∪0: zero-aggregation from zero stream. If the degree of the protoform is zero in the stream, the aggregation is zero: Po(si¯)=0,∀si¯∈SHRS→∪(Po)=0. It is a necessary condition of boundary property [31](ii)
∪+: positive-aggregation from nonzero stream. If there is a nonzero degree of the protoform in the stream, the aggregation is nonzero: ∃Po(si¯)>0,si¯∈SHRS→∪(Po)≠0→∪(Po)>0We note the two properties have been determined based on the semantics of the protoforms to describe the HRS, which guarantee that the number of instances which match to a given protoform∪(Po)>0 is equal although the semantics of the aggregation varies. Specifically, the aggregation degree is zero if and only if the protoform is not representative in the session ∪(Po)=0↔Po(si¯)=0,∀si¯∈SHRS; in another case ∪(Po)>0. We note that, in different fields and other contexts, further properties in aggregation operators can be selected [32].Second, we propose two semantics to aggregate the degree of the protoformPo(si¯):maximum andamplitude.(i)
On the first hand, themaximum aggregation operator max⁡(Po) computes the maximal value of degree of the protoform, which is shown in (7)max⁡Po=max⁡Pos0¯,…,Posi¯∈0,1.The semantics ofmax⁡(Po) describes the maximal truth degree of the protoform providing an intuitive representation as aggregation, which has been widely used in fuzzy logic as aggregation of rules within Mamdani-type inference models [33]. It fulfills the properties of zero-aggregation ∪0 and positive-aggregation ∪+.(8)∪0:Posi¯=0,∀si¯∈SHRS≡∪Po=max⁡0,…,0=0∪+:∃Posi¯+>0,si¯∈SHRS≡∪Po=max⁡0,Posi¯+,…,0>0(ii)
On the other hand, theamplitude aggregation operator (Po) describes the persistence and presence of the protoform degree Po(si¯) throughout the rehabilitation session. For this purpose, the fuzzy quantification of the weight of the protoform W(Po) degree within the HRS is proposed in (9)WPo=∑si¯SHRSsi¯SHRSPo=QWPo∈0,1.First, the weight of the protoform degreeW(Po), which represents a suitable measure as fuzzy aggregation [34], is computed as the relation between the sum of the degrees ∑si¯SHRSsi¯ regarding the norm (or size) of the complete HRS SHRS. Second, a fuzzy quantification is provided by a FQ which transforms W(Po) into (Po)=Q(W(Po)) through the fuzzy membership function μQ:[0,1]→[0,1]. For the sake of simplicity, we refer to μQ as Q.To guarantee that(Po) fulfills the properties of zero-aggregation and positive-aggregation, the membership function Q of the FQ should assess the following properties: Q is a monotone function x≤y→Q(x)≤Q(y), Q(0)=0, and limx→0+Q(x)>0:(10)∪0:Q0=0,Posi¯=0,∀si¯∈SHRS≡WPo=0→QWPo=0∪+:limx→0+Qx>0,x≤y→Qx≤Qy,∃Posi¯>0,si¯∈SHRS≡∑si¯SHRSsi¯>0→WPo>0→QWPo>0


## 2.2.1. Protoforms for Describing Heart Rate Streams

The aim of the proposed methodology is to generate linguistic summaries from the rehabilitation sessions of patients. For this purpose, in this second stage we process the fuzzy linguistic termslow(si¯), adequate(si¯), and high(si¯) from the data stream SHRS, which are described in the previous section.First, in order to integrate an interpretable and rich-expressive approach to model the expert knowledge linguistically, we introduce an ad hocprotoformPo in the form of(5)Posi¯:QkLiTj,where(i)
Li defines a fuzzy linguistic term to evaluate the data stream. Here, Li is straightforwardly related to fuzzy terms low(si¯), adequate(si¯), and high(si¯)(ii)
Tj defines a fuzzy temporal term where the term Li is aggregated. The use of FTWs, which were introduced in the previous section, is extended to generate linguistic summaries of middle-long temporal terms from SHRS. The aggregation of Li over Tj for a given si¯ is computed by (4) as Li∪Tj(si¯)(iii)
Qk defines a fuzzy quantifier (FQ) to evaluate the impact and fulfillment of the linguistic term Li within the temporal window Tj [28]. A FQ applies a transformation μQK:[0,1]→[0,1] to the aggregated temporal degree of μQK(Vr∪Tk(si¯)The aim of modeling knowledge through protoforms is allowing the rehabilitation team to define key interest indicators using expert intuitive representations of temporal and quantification terms linguistically. An example of protoform ismost of the time (Qk) HR is adequate (Li) for around 40-60 minutes (Tj). We note that the protoforms are suitable to linguistically describe the impact and fulfillment of a fuzzy linguistic term in a fuzzy temporal window in more detail, but subsequently they can be renamed to a shorter linguistic description, such asadequate HR in session.In Figure4, the protoformmost of the time HR is adequate for around 30-50 minutes is shown, whose degree is represented with a real SHRS. We also plot the degree of the protoform Po(si¯) in relation to the term adequate(si¯) to describe the importance of the middle-long FTW and the FQ which enable the linguistic interpretability of the sentenceadequate heart rate in session in the sensor stream. For the sake of simplicity, the shapes of the fuzzy membership functions of the example are detailed in Section 3.Figure 4
A timeline with the degrees of the termadequate(si¯) (dotted line) and protoformmost of the time HR is adequate for around 30-50 minutesPo(si¯) (thick line) on a real HRS. Based on its semantics, the degree of the protoform increases when the patient has maintained an adequate HR for 30 minutes.Second, protoformsPo(si¯) can be combined using fuzzy logical operators to increase the linguistic capabilities of the model. So, we briefly introduce the following basic operations, which could be straightforwardly increased with advanced fuzzy operations in other contexts:(i)
Fuzzy negation operator, which is represented as the complement¬ by the fuzzy function ¬Po(si¯)=1-Po(si¯)(ii)
Fuzzy union operator, which is represented by the t-normPo∧Pq(si¯)=Po(si¯)∧Pq(si¯). The semantic function proposed for the fuzzy union operator is min: Po∧Pq(si¯)=min⁡{Po(si¯),Pq(si¯)}(iii)
Fuzzy intersection operator, which is represented by the conormPo∨Pq(si¯)=Po(si¯)∨Pq(si¯). The semantic function proposed for the fuzzy intersection operator is max: Po∧Pq(si¯)=max⁡{Po(si¯),Pq(si¯)}


## 2.2.2. Aggregation Operation for Protoforms

As we detailed in Section2.2.1, protoforms are defined to represent the health indicators from the rehabilitation sessions linguistically by means of expert knowledge. Although the evaluation of protoforms is properly computed Po(si¯) throughout the data stream SHRS, an aggregation degree is proposed here in order to summarize the relevance of a protoform.First, in (6) we describe the aggregation operation ∪(Po) of the protoform Po, which computes a single degree from the degree of protoform Po(si¯) over SHRS, as(6)∪Po=⋃si¯SHRSPosi¯∈0,1.In order for the aggregated degree of the aggregation operator∪(Po) to keep its semantic integrity with the degree of the protoforms, we define two properties which the aggregation operation should assess:(i)
∪0: zero-aggregation from zero stream. If the degree of the protoform is zero in the stream, the aggregation is zero: Po(si¯)=0,∀si¯∈SHRS→∪(Po)=0. It is a necessary condition of boundary property [31](ii)
∪+: positive-aggregation from nonzero stream. If there is a nonzero degree of the protoform in the stream, the aggregation is nonzero: ∃Po(si¯)>0,si¯∈SHRS→∪(Po)≠0→∪(Po)>0We note the two properties have been determined based on the semantics of the protoforms to describe the HRS, which guarantee that the number of instances which match to a given protoform∪(Po)>0 is equal although the semantics of the aggregation varies. Specifically, the aggregation degree is zero if and only if the protoform is not representative in the session ∪(Po)=0↔Po(si¯)=0,∀si¯∈SHRS; in another case ∪(Po)>0. We note that, in different fields and other contexts, further properties in aggregation operators can be selected [32].Second, we propose two semantics to aggregate the degree of the protoformPo(si¯):maximum andamplitude.(i)
On the first hand, themaximum aggregation operator max⁡(Po) computes the maximal value of degree of the protoform, which is shown in (7)max⁡Po=max⁡Pos0¯,…,Posi¯∈0,1.The semantics ofmax⁡(Po) describes the maximal truth degree of the protoform providing an intuitive representation as aggregation, which has been widely used in fuzzy logic as aggregation of rules within Mamdani-type inference models [33]. It fulfills the properties of zero-aggregation ∪0 and positive-aggregation ∪+.(8)∪0:Posi¯=0,∀si¯∈SHRS≡∪Po=max⁡0,…,0=0∪+:∃Posi¯+>0,si¯∈SHRS≡∪Po=max⁡0,Posi¯+,…,0>0(ii)
On the other hand, theamplitude aggregation operator (Po) describes the persistence and presence of the protoform degree Po(si¯) throughout the rehabilitation session. For this purpose, the fuzzy quantification of the weight of the protoform W(Po) degree within the HRS is proposed in (9)WPo=∑si¯SHRSsi¯SHRSPo=QWPo∈0,1.First, the weight of the protoform degreeW(Po), which represents a suitable measure as fuzzy aggregation [34], is computed as the relation between the sum of the degrees ∑si¯SHRSsi¯ regarding the norm (or size) of the complete HRS SHRS. Second, a fuzzy quantification is provided by a FQ which transforms W(Po) into (Po)=Q(W(Po)) through the fuzzy membership function μQ:[0,1]→[0,1]. For the sake of simplicity, we refer to μQ as Q.To guarantee that(Po) fulfills the properties of zero-aggregation and positive-aggregation, the membership function Q of the FQ should assess the following properties: Q is a monotone function x≤y→Q(x)≤Q(y), Q(0)=0, and limx→0+Q(x)>0:(10)∪0:Q0=0,Posi¯=0,∀si¯∈SHRS≡WPo=0→QWPo=0∪+:limx→0+Qx>0,x≤y→Qx≤Qy,∃Posi¯>0,si¯∈SHRS≡∑si¯SHRSsi¯>0→WPo>0→QWPo>0


## 3. Case Study

In this section, we present a case study which illustrates the proposed methodology. The data of the rehabilitation sessions correspond to three patients who freely participated in the projectMonitoring of Patients with Ischemic Heart Disease within Outdoor Cardiac Rehabilitation Programs of the Council of Health for the Andalusian Health Service in the Region of Jaén (Spain). They had a wrist-worn heart rate device (Polar M600) (https://www.polar.com/es/productos/sport/M600-GPS-smartwatch (accessed on 10/14/2018)), which collected the heart rate data during the rehabilitation sessions using a wearable application.141 rehabilitation sessions were collected (48, 55, and 38, respectively, for the three patients) from April to August 2018. The duration of the sessions was defined and adapted to patient evolution by the cardiac rehabilitation team, varying from 30 minutes to 80 minutes. A total of 639.709 heart rate samples were collected.


### 3.1. Real-Time Monitoring of Heart Rate Streams

The application embedded in the wrist-worn heart rate device collected the heart rate of patients and advised patients during their outdoor rehabilitation sessions. For this purpose, the application computes fuzzy linguistic termsadequate, high, and low using a short FTW. Their membership functions were obtained from the previous work [22] and are described in Table 1.Table 1
Membership functions for the terms adequate, low, and highμV and their respective FTWs μT.
Term μ V μ T adequate T S ( s i ) [ V T 1 , r - ∗ , r + ∗ , V T 2 ] T L ( Δ t i ) [ 3 s , 5 s ]  low T L ( s i ) [ V T 1 , r - ∗ ] T L ( Δ t i ) [ 3 s , 5 s ]  high T R ( s i ) [ r + ∗ , V T 2 ] T L ( Δ t i ) [ 0 s , 1 s ]The values of OHRTZ[r+∗,r-∗] and Ventilatory Thresholds [VT1,VT2] were adapted for each patient based on an initial controlled stress test at the Health Center.


### 3.2. Protoforms for Describing Heart Rate Streams

The protoforms enable the rehabilitation team to define key health indicators linguistically using expert knowledge. In Table2, some examples of the protoforms defined by the cardiac rehabilitation team are described.Table 2
Textual description in natural language: short linguistic description and related protoforms.
Icon Id short linguistic descriptions Protoforms: Q k   L i   T j P 1 Adequate HR in session At least half of the time the HR is adequate for around 25-50 minutes  P 2 High HR in session is worrying Most of the time the HR is high for around 1-3 minutes  P 3 Low HR intensity in session Most of the time the HR is low for around 15-25 minutes  P 4 Unstable HR progression in session While a part of the time the HR is high in the last 2 minutes ∩ While a part of the time the HR was low 1-3 minutes agoNext, the membership functions of the FTWs and FQs were straightforwardly defined by both the computer science team and cardiac rehabilitation team of the project. They have been defined by different shapes of trapezoidal membership functions, whose values are shown in Table3.Table 3
Trapezoidal membership functions for FTWs and FQ of protoforms.
Textual description in natural language Type μ T / μQ For around 25-50 minutes T j T L ( Δ t i ) [ 25 m , 50 m ]  For around 15-25 minutes T j T L ( Δ t i ) [ 15 m , 25 m ]  In the last 2 minutes T j T L ( Δ t i ) [ 1 m , 2 m ]  Around 1-3 minutes ago T j T S ( Δ t i ) [ 0 m , 1 m , 2 m , 3 m ]  At least half of the time Q k T R ( ( V r ∪ T k ( s i - ) ) [ 0.25,0.75 ]  While a part of the time Q k T R ( ( V r ∪ T k ( s i - ) ) [ 0.25,0.5 ]  Most of the time Q k T R ( ( V r ∪ T k ( s i - ) ) [ 0.5,1 ]In Figure5, we show the computing of the protoform Po in real rehabilitation sessions, including the degree of the term Li. We note that the protoforms are also useful to determine the region of interests over SHRS indicating the ranges where the truth degree is activated Po(si¯)>0.Figure 5
Example of the protoforms in real rehabilitation sessions. (a) A session with predominance of adequate adherence and some high rates. (b) The degree of the termadequate is represented by the dotted line; the degree of the protoformat least half of the time the HR is adequate for around 25-50 minutes is represented by the thick line. The aggregated degreesmaximum andamplitude are shown in the bottom. (c) The degree of the termhigh is represented by the dotted line; the degree of the protoformmost of the time the HR is high for around 1-3 minutes is represented by the thick line. The aggregated degreesmaximum andamplitude are shown in the bottom. (d) A session with predominance of low adherence and some high rates. (e) The degree of the termlow is represented by the dotted line; the degree of the protoformmost of the time the HR is low for around 15-25 minutes is represented by the thick line. The aggregated degreesmaximum andamplitude are shown in the bottom. (f) The degrees of the termshigh andlow are represented by the dotted line in yellow and blue, respectively; the degree of the protoformpart of the time the HR is high in the last 2 minutes∩part of the time the HR was low around 1-3 minutes ago is represented by the thick line. The aggregated degreesmaximum andamplitude are shown in the bottom.


### 3.3. Aggregation Operation for Protoforms

In this section, we describe the results of the aggregation of the protoform in the HRS of patients, which determine a single and descriptive degree from the protoform degreePo(si¯) over the heart rate stream SHRS.In this work, two semantics to aggregate the degree of the protoform are proposed:maximummax⁡(Po) andamplitude(Po), which represent the maximal truth degree of the protoform and the presence of the protoform throughout the rehabilitation session, respectively.Themaximum aggregation operator is not parametric. Conversely, theamplitude aggregation operator was modeled using expert knowledge of the computer science team as well as the cardiac rehabilitation team, who determined the membership function of the fuzzy quantifiers, μQ, for each protoform (Po), which are described in Table 4.Table 4
Membership functions for fuzzy quantifiers ofamplitude aggregation operator.
Textual description in natural language μ Q adequate heart rate in session T R ( W ( P o ) ) [ 0,0.5 ]  the high rate is worrying T R ( W ( P o ) ) [ 0,0.05 ]  the session presents low intensity T R ( W ( P o ) ) [ 0,0.25 ]  the session has unstable rates T R ( W ( P o ) ) [ 0,0.01 ]We note that (i) the membership functions are in the shape ofTR(x)[0,α] to fulfill the properties of zero-aggregation and positive-aggregation described in Section 2.2.2 and (ii) the quantification membership functions of protoforms where a high rate is involved relate short weights to relevant amplitudes, due to the fact that high rates are very significant and worrying in SHRS. An example of real sessions and the aggregation operatorsmaximum andamplitude for the proposed protoforms is presented in Figure 5.In Table5, we summarize the number and percentage of RS which have been recovered for each protoform ∪(Po)>0 and user. We note the descriptive summary which represents the aggregation in determining and differentiating the performance of the patients within the rehabilitation program.Table 5
Metrics in the aggregation of protoforms from rehabilitation sessions by patient (number of sessionsN and percentage % from the RS |RS| total in parentheses).
Total P 1 P2 P 3 P 4  Patient R S N ( % ) N ( % ) N ( % ) N ( % )  1 49 2(4%) 24(48%) 49(100%) 9(18%)  2 56 36(64%) 44(78%) 48(85%) 26(46%)  3 39 36(92%) 35(90%) 4(10%) 7(18%)We note, based on the properties described for the aggregation operators, that bothmaximummax⁡(Po) andamplitudePo recover same HRs.Finally, as each aggregation operator determines a degree∪(Po)∈[0,1] which can be filtered by a threshold α using a straightforward α-cut in order to recover more descriptive and relevant sessions which match the protoform Po, this intuitive value α can be also modified by experts when analyzing the summaries of the patients to filter and select the RSs. In Table 6, we present the number of RSs recovered by α-cut for each protoform in function of the values of α={0,0.5,0.9}.Table 6
Metrics in the aggregation of protoforms from rehabilitation sessions (number of sessions and percentage from the total of RS in parentheses).
α-cut max amp max amp max amp max amp  α=0 74(51%) 74(51%) 103(72%) 103(72%) 101(70%) 101(70%) 42(29%) 42(29%)  α=0.5 46(32%) 51(35%) 51(35%) 81(56%) 72(48%) 87(51%) 15(10%) 29(20%)  α=0.9 32(22%) 32(23%) 40(27%) 65(45%) 63(43%) 60(42%) 13(9%) 20(14%)


## 3.1. Real-Time Monitoring of Heart Rate Streams

The application embedded in the wrist-worn heart rate device collected the heart rate of patients and advised patients during their outdoor rehabilitation sessions. For this purpose, the application computes fuzzy linguistic termsadequate, high, and low using a short FTW. Their membership functions were obtained from the previous work [22] and are described in Table 1.Table 1
Membership functions for the terms adequate, low, and highμV and their respective FTWs μT.
Term μ V μ T adequate T S ( s i ) [ V T 1 , r - ∗ , r + ∗ , V T 2 ] T L ( Δ t i ) [ 3 s , 5 s ]  low T L ( s i ) [ V T 1 , r - ∗ ] T L ( Δ t i ) [ 3 s , 5 s ]  high T R ( s i ) [ r + ∗ , V T 2 ] T L ( Δ t i ) [ 0 s , 1 s ]The values of OHRTZ[r+∗,r-∗] and Ventilatory Thresholds [VT1,VT2] were adapted for each patient based on an initial controlled stress test at the Health Center.


## 3.2. Protoforms for Describing Heart Rate Streams

The protoforms enable the rehabilitation team to define key health indicators linguistically using expert knowledge. In Table2, some examples of the protoforms defined by the cardiac rehabilitation team are described.Table 2
Textual description in natural language: short linguistic description and related protoforms.
Icon Id short linguistic descriptions Protoforms: Q k   L i   T j P 1 Adequate HR in session At least half of the time the HR is adequate for around 25-50 minutes  P 2 High HR in session is worrying Most of the time the HR is high for around 1-3 minutes  P 3 Low HR intensity in session Most of the time the HR is low for around 15-25 minutes  P 4 Unstable HR progression in session While a part of the time the HR is high in the last 2 minutes ∩ While a part of the time the HR was low 1-3 minutes agoNext, the membership functions of the FTWs and FQs were straightforwardly defined by both the computer science team and cardiac rehabilitation team of the project. They have been defined by different shapes of trapezoidal membership functions, whose values are shown in Table3.Table 3
Trapezoidal membership functions for FTWs and FQ of protoforms.
Textual description in natural language Type μ T / μQ For around 25-50 minutes T j T L ( Δ t i ) [ 25 m , 50 m ]  For around 15-25 minutes T j T L ( Δ t i ) [ 15 m , 25 m ]  In the last 2 minutes T j T L ( Δ t i ) [ 1 m , 2 m ]  Around 1-3 minutes ago T j T S ( Δ t i ) [ 0 m , 1 m , 2 m , 3 m ]  At least half of the time Q k T R ( ( V r ∪ T k ( s i - ) ) [ 0.25,0.75 ]  While a part of the time Q k T R ( ( V r ∪ T k ( s i - ) ) [ 0.25,0.5 ]  Most of the time Q k T R ( ( V r ∪ T k ( s i - ) ) [ 0.5,1 ]In Figure5, we show the computing of the protoform Po in real rehabilitation sessions, including the degree of the term Li. We note that the protoforms are also useful to determine the region of interests over SHRS indicating the ranges where the truth degree is activated Po(si¯)>0.Figure 5
Example of the protoforms in real rehabilitation sessions. (a) A session with predominance of adequate adherence and some high rates. (b) The degree of the termadequate is represented by the dotted line; the degree of the protoformat least half of the time the HR is adequate for around 25-50 minutes is represented by the thick line. The aggregated degreesmaximum andamplitude are shown in the bottom. (c) The degree of the termhigh is represented by the dotted line; the degree of the protoformmost of the time the HR is high for around 1-3 minutes is represented by the thick line. The aggregated degreesmaximum andamplitude are shown in the bottom. (d) A session with predominance of low adherence and some high rates. (e) The degree of the termlow is represented by the dotted line; the degree of the protoformmost of the time the HR is low for around 15-25 minutes is represented by the thick line. The aggregated degreesmaximum andamplitude are shown in the bottom. (f) The degrees of the termshigh andlow are represented by the dotted line in yellow and blue, respectively; the degree of the protoformpart of the time the HR is high in the last 2 minutes∩part of the time the HR was low around 1-3 minutes ago is represented by the thick line. The aggregated degreesmaximum andamplitude are shown in the bottom.


## 3.3. Aggregation Operation for Protoforms

In this section, we describe the results of the aggregation of the protoform in the HRS of patients, which determine a single and descriptive degree from the protoform degreePo(si¯) over the heart rate stream SHRS.In this work, two semantics to aggregate the degree of the protoform are proposed:maximummax⁡(Po) andamplitude(Po), which represent the maximal truth degree of the protoform and the presence of the protoform throughout the rehabilitation session, respectively.Themaximum aggregation operator is not parametric. Conversely, theamplitude aggregation operator was modeled using expert knowledge of the computer science team as well as the cardiac rehabilitation team, who determined the membership function of the fuzzy quantifiers, μQ, for each protoform (Po), which are described in Table 4.Table 4
Membership functions for fuzzy quantifiers ofamplitude aggregation operator.
Textual description in natural language μ Q adequate heart rate in session T R ( W ( P o ) ) [ 0,0.5 ]  the high rate is worrying T R ( W ( P o ) ) [ 0,0.05 ]  the session presents low intensity T R ( W ( P o ) ) [ 0,0.25 ]  the session has unstable rates T R ( W ( P o ) ) [ 0,0.01 ]We note that (i) the membership functions are in the shape ofTR(x)[0,α] to fulfill the properties of zero-aggregation and positive-aggregation described in Section 2.2.2 and (ii) the quantification membership functions of protoforms where a high rate is involved relate short weights to relevant amplitudes, due to the fact that high rates are very significant and worrying in SHRS. An example of real sessions and the aggregation operatorsmaximum andamplitude for the proposed protoforms is presented in Figure 5.In Table5, we summarize the number and percentage of RS which have been recovered for each protoform ∪(Po)>0 and user. We note the descriptive summary which represents the aggregation in determining and differentiating the performance of the patients within the rehabilitation program.Table 5
Metrics in the aggregation of protoforms from rehabilitation sessions by patient (number of sessionsN and percentage % from the RS |RS| total in parentheses).
Total P 1 P2 P 3 P 4  Patient R S N ( % ) N ( % ) N ( % ) N ( % )  1 49 2(4%) 24(48%) 49(100%) 9(18%)  2 56 36(64%) 44(78%) 48(85%) 26(46%)  3 39 36(92%) 35(90%) 4(10%) 7(18%)We note, based on the properties described for the aggregation operators, that bothmaximummax⁡(Po) andamplitudePo recover same HRs.Finally, as each aggregation operator determines a degree∪(Po)∈[0,1] which can be filtered by a threshold α using a straightforward α-cut in order to recover more descriptive and relevant sessions which match the protoform Po, this intuitive value α can be also modified by experts when analyzing the summaries of the patients to filter and select the RSs. In Table 6, we present the number of RSs recovered by α-cut for each protoform in function of the values of α={0,0.5,0.9}.Table 6
Metrics in the aggregation of protoforms from rehabilitation sessions (number of sessions and percentage from the total of RS in parentheses).
α-cut max amp max amp max amp max amp  α=0 74(51%) 74(51%) 103(72%) 103(72%) 101(70%) 101(70%) 42(29%) 42(29%)  α=0.5 46(32%) 51(35%) 51(35%) 81(56%) 72(48%) 87(51%) 15(10%) 29(20%)  α=0.9 32(22%) 32(23%) 40(27%) 65(45%) 63(43%) 60(42%) 13(9%) 20(14%)


## 4. Conclusions and Future Works

Complex IoT systems for e-health solutions allow us to reach a greater number of patients. However, this positive fact generates a vast amount of information that must be analyzed by the health team. This paper has been focused on the real-time monitoring of an outdoor cardiac rehabilitation program for patients with ischemic heart disease, which was designed and supervised remotely by the cardiac rehabilitation team. In this program, wearable wrist-worn devices with heart rate sensors integrating a high-quality protocol based on clinical guidelines were used to monitor the heart rate of patients in a personalized way. A wearable mobile-cloud platform was defined for collecting and synchronizing data between patients and the cardiac rehabilitation team to provide feedback.The main motivation behind this work has been to provide the cardiology rehabilitation team with linguistic summaries of the rehabilitation sessions based on the heart rate streams of patients. In order to address this challenge, a methodology has been proposed in this paper, which is based on the use of the linguistic descriptions of data with protoforms and fuzzy logic of the heart rate streams of patients in order to provide linguistic summaries with rich expressiveness of interest indicators. So, the proposed methodology models short descriptions such asthe high rate is worrying in the session orthe session presents low intensity.The proposed methodology enables (i) a fast analysis process to monitor a higher number of benefited patients and (ii) identification of sessions with interest indicators for the cardiac rehabilitation team to provide feedback. To do so, on the one hand, linguistic temporal terms and linguistic quantifiers have been used on linguistic aggregation operators in the heart rate streams of patients. On the other hand, flexible linguistic modeling has been defined in the proposed methodology where the cardiac rehabilitation team intuitively defines the key interest indicators using protoforms by means of expert knowledge in order to recover and dynamically select the rehabilitation sessions which suit and match the expert criteria.In future works, the methodology will be extended to automatically generate linguistic recommendations for the patients in further sessions by means of machine learning techniques and based on the knowledge of the cardiac rehabilitation team.



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*Source: 2694126-2019-01-01.xml*