Sparse representation and overcomplete dictionary learning for anomaly detection in electrocardiograms
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Abstract
In the hereby work, we present the use of sparse representation and overcomplete dictionary learning method for examining the case of anomaly detection in an electrocardiographic record. The above mentioned signal was introduced in a form of correct electrocardiographic morphological structures and outliers which describe different sorts of disorders. In the course of study, two sorts of dictionaries were used. The first consists of atoms created with the use of differently parameterized analytic Gabor functions. The second sort of dictionaries uses the modified Method of Optimal Directions to find a dictionary reflecting proper structures of an electrocardiographic signal. In addition, in this approach, the condition of decorrelation of dictionary atoms was introduced for the sake of gaining more precise and optimal representation. The dictionaries obtained in these two ways became a basis for the analyzed sparse representation of electrocardiographic record. During the anomaly detection process, which was based on decomposition of the analyzed signal into correct values and outliers, a modified alternating minimization algorithm was used. A commonly accessible base of data of electrocardiograms, that is MIT-BIH Arrhythmia Database, was utilized to examine the conduct of the recommended method. The effectiveness of the solution, which validated itself in searching of anomalies in the analyzed electrocardiographic record, was confirmed by experiment results.
Keywords
Electrocardiogram Sparse representation Dictionary learning Electrocardiographic signal analysis Anomaly detection MIT-BIH Arrhythmia Database1 Introduction
In numerous fields of medical diagnosis, measurement and analysis of biomedical signals are essential elements supplying information concerning manner and well-being of organ functions in living organisms. One of the oldest and best-known techniques, both in the aspect of range of supplied diagnostic data and interpretation abilities, is electrocardiography.
Heart illnesses are cardiological systems’ deviations that occur more and more frequently, and they affect not only older people. The significance of the problem can be confirmed by constantly changing social structure, where larger and larger percent of the population created by group of retired people is more exposed to cardiological problems. The threats defined in such a way doubtlessly become a factor motivating to involve intensive research work in the area [1].
Analysis of a human electrocardiographic (ECG) signal consists in reflecting the electrical activity of the heart muscle cells. Their ability to create electrical impulses influences working of heart as a blood-transporting pump in the human circulatory system, which enables other biological systems within a human body to work efficiently. ECG signals that are predominantly recorded as changes in electrical potentials in time intervals are obtained from different typical points of a human organism (these places are directly tied with 3, 6 or 12—channel obtaining of ECG signals). Signals acquired in such a way contain crucial diagnostic set of data. Unfortunately, the standard record of ECG signals indicates that they are vulnerable to different types of disturbances, which makes it hard or not possible to elicit important and credible set of information in medical diagnostic processes [2, 3].
Morphological fluctuations of shape, or disturbances of the arrangement of the electrocardiographic signal, may occur out of different morbid or physiological reasons. In an ECG record, these changes most frequently occur as disturbances of shape or placement of waves P and T, and segments ST or PQ. Anomalies may also appear in localization and structure of the QRS complexes, which are a crucial factor for calculating the R–R interval. Disturbances of the correct heartbeat rhythm, directly bonded with its irregular contractions, may be characteristic for a disease unit called arrhythmia. One of the most frequently occurring factors of heart rhythm distortion, widely spread among old people, are premature ventricular complexes (PCVs). They are created as an outcome of ectopic hyperactivity of myocardium, and they are the cause of distortions in its synchronization processes [5, 6].
Until now there have been published numerous experimental works directly describing different stages of processing, analysis and detection of given structures in an ECG signal [7, 8]. It was performed by means of: (i) classical methods of signal processing, (ii) fuzzy sets techniques, (iii) artificial neural networks, (iv) support vector classifiers, (v) genetic algorithms and (vi) hybrid techniques [9, 10, 11, 12]. Often these techniques create basis for solutions using machine learning methods for effective diagnosis of heart diseases [13]. Fuzzy sets or fuzzy-based machine learning is an approach commonly used for effective classification of disease alterations reflected in the analyzed ECG signals. For that reason, these techniques utilize smooth variables with membership functions [14]. With reference to these solutions, Behadada and Chikh [15] suggested to use fuzzy decision tree as methodology to diagnose the cardiac abnormalities (PVC events). Another example, presented by Lei et al. [14], developed an adaptive fuzzy ECG classifier for enhancing the performance of conventional classifiers. Further research over fuzzy-based techniques developed through the incorporation of neural networks, such as fuzzy inference networks [14]. Ceylan et al. [16], on the other hand, used Fuzzy Clustering Neural Network Algorithm for classification in ECG analysis achieving accuracy of 99% with acquisition of high clustering performance. However, the training time was large in this case, which limited the benefits. Also, Vishwa et al. [17] proposed an automated neural network (ANN) on which classification system of cardiac arrhythmia (which used multi-channel ECG recordings) was based. To validate the method a Normal Sinus Rhythm (NSR) database was applied. The above-mentioned approach proved the accuracy of 96.77% on MIT-BIH database. Another research performed by Anuradha and Reddy [18] made neural networks for optimizing the classification of ECG signals better. The authors suggested an ANN-based classifier for cardiac arrhythmia. The above-mentioned classifier employed a combination of wavelets back-propagation algorithms (BPA). The outcome of 90.56% of accuracy was achieved, and the classifier effectively detected the activities that can be considered abnormal. Another attitude toward ECG disturbances detection problems is solutions using support vector classifiers, based on statistical learning theory. In a study by Kampouraki et al. [19], the time series of heartbeat are classified with the implementation of support vector machines (SVM). The researchers have revealed that the SVM classifier offers a better functionality than those based on neural networks as it classifies the ECG signals with minimal signal-to-noise ratio. On the other hand, Nasiri et al. [20] proposed an up-to-date attitude for feature extraction and ECG classification for cardiac arrhythmia disease. The SVM classifier was engaged for optimization of the approach by detecting the best subset of feature, which provides the optimal classification of the signal. The researchers have utilized SVM with genetic algorithm (GA), where GA is deployed to enhance the generalization performance of the SVM classifier. Additionally, Priyadharshini and Kumar [21] conducted a research to enhance the classification of Arrhythmia from ECG signals. For this purpose, an improvised genetic algorithm (IGA) was developed and Naïve Bayes classification algorithm was employed for the research purpose. Tending to achieve better classification results for ECG signals, solutions based on hybrid techniques begun developing intensively. Dalal and Birok [22] in their research concentrated on a hybrid classifier by using the principal component analysis (PCA) and neuro-fuzzy classifier. The accuracy of the hybrid approach was estimated to the level of 96%. Another work by Bensujin and Hubert [23] examined and determined (using classification) the ST-segment elevation myocardial infarction (STEMI) in the ECG signal of a person. Remaining solutions, based on hybrid detection methods, were predominantly used for detection of heart rate, ischemic and extraction of STEMI [24].
The contemporary promising attitude toward analyzing and detecting anomalies in an ECG signal is the machine learning methods, based on the sparse representations of the signals realized with the use of dictionaries with redundancy, describing essential structures of the tested signal. However, the above-mentioned methods are used predominantly at the initial stages of the ECG signal analysis, that is either quality enhancement or detection of the QRS complex [25]. The problem of anomaly detection from the perspective of electrocardiography record should by no means be referred to in the same way as the common attitude to anomaly detection solutions. This matter has already been addressed in detail in the available scientific literature [26, 27]. Numerous anomaly detection methods for symbolic sequences were proposed, among others, by Chadola survey [28]. However, anomaly behavior and the sole process of its detection in the sparse representation of an ECG signal is not so frequently described [29, 30].
In our article, we suggest a new approach, creating our contribution to anomaly detection problem, which acquire the reader with the use of resistant-to-outlier dictionary learning method and sparse representation of a signal for specific time series describing the analyzed ECG signals. Dictionary learning process was carried out with the use of modified Method of Optimal Directions (mMOD), where the decorrelation of atoms of dictionary created for the sake of gaining more precise and optimal representations was a significant condition. Anomaly detection is carried out as a solution to the sparse representation of an electrographic record with the modified alternating minimization algorithm (AMA). Our modification consisted in the use of classical version of AMA(normally used for solving structured convex optimization problems, where the sum of two functions is to be minimized) for estimation of sparse representation of the analyzed signal, which is described by means of the proposed model. As a final outcome, we receive the electrocardiographic signal decomposition onto the correct values and outliers (anomalies).
The structure of this work is arranged in a specific manner. Firstly, the introduction is provided. In Sect. 2, we present motivation and related work. Next, in Sect. 3, an ECG signal’s sparse representation is discussed in detail. Then, in Sect. 4, we show the Gabor dictionary which consists of atoms created by using differently parameterized analytic Gabor functions and resistant-to-outlier dictionary learning method based on modified Method of Optimal Directions estimation. Next, Sect. 5 presents the details of the proposed solution consisting of formulation of the sparse representation problem with outliers and the modified AMA solution for optimization issue of sparse representation. Section 6 provides implementation details and experimental results, which are followed by conclusions.
2 Motivation and related work
Linear extensions of the analyzed ECG signal in relation to a defined set of basic functions in many cases presents significant limitations, i.e., they do not exactly represent important features of decomposed ECG signal by means of a small number of linear extension coefficients [31]. What is more, if the structure elements of analyzed ECG signal differ noticeably from the scaling factor of the basic function, the coefficients of linear extension will not constitute optimal representation of such a signal [32]. Therefore, the length-changing signals require applying of basic functions with numerous different scales. A significant limitation of such an approach is bonding of frequency and scale parameters [33].
In case of complex signal structures, such as ECG records, we are not able to define optimal parameters of above-mentioned elements for chosen basic functions. In such a situation, the best solution seems to be introduction of more varied and numerous function sets, called dictionaries with redundancy [34]. These functions are pointed out to fit the character of the analyzed signal best. Such a dictionary can be chosen by: (i) constructing the dictionary on the basis of a mathematical model of data (using already-known forms of models such as wavelets, wavelet packets, contourlets, curvelets or Gabor [34, 35]), or (ii) learning the dictionary using a training dataset (which is the most frequent part of a processed signal) [36]. The most typical method of a learning dictionary is the Method of Optimal Directions (MOD) that was proposed by Engan et al. [37] and was one of the first implemented methods, known as a sparsification process. Equally popular—though different in its character—is K-means singular value decomposition, with its name contracted to K-SVD algorithm, which was published by Ahron et al. [38]. The essential discrepancy between the two methods consists in the manner of maintaining the dictionary as current, i.e., some update all atoms at once (MOD), while others update atoms one after another (K-SVD). Within the field of sparse representation of biomedical signals [39], in particular electrocardiographic records, the aforementioned methods of dictionary creation were most often utilized for compression, analysis or detection of defined ECG signal’s features [37, 40, 41]. In particular, the improved Method of Optimal Directions method versions (e.g., k-coefficient Lipschitzian mappings for sparsity algorithm [40]) often showed an advantage over the K-SVD algorithm in the aspect of precision and effectiveness in creation of representation of the decomposed electrocardiographic signal.
Realization of sparse representation as decomposition of the signal in relation to the dictionary requires quick searching and optimal matching of the elements (i.e., atoms) which best present particular characteristics of the examined signal [31]. For the so-defined problem, the pace of decreasing of the residual’s norm depends on correlation between the following signal’s residues and chosen elements of the dictionary. Assuming that the analyzed signal is a sum of components of high energy, which are dictionary elements, the correlation coefficients of the signal and its residues are significant. Thus, their norm decreases fast because high-energy components are important parts of a signal well correlated with selected atoms of the dictionary [34, 35]. This is why there is a need to define a measure of quality of such a match, and an algorithm ensuring finding best, from this measure’s perspective, representation of the signal appeared. In general, finding such an optimal representation is NP-hard [42]. The issue may be presented as sparse solutions of systems of equations [43] or adaptive greedy approximations [44]. Then, iterative methods are appropriate computational solutions, i.e., different variants of orthogonal matching pursuit (OMP) or basis pursuit (BP) [45]. However, in case of complex representations of a signal (in particular, when we want to present the analyzed signal as a composition of its elements, for instance correct values and anomalies) we must use stronger tools, such as alternating direction method of multipliers (ADMM) [46] or computationally easier alternating minimization algorithm [47]. Then, we will be able to solve given optimization tasks as unconstrained convex problem.
Anomaly detection (that is, particular types of disturbances) in an ECG signal can be realized by different means whose details can be found in numerous papers [26, 27, 28]. This issue is directly connected to a difficulty of an unequivocal definition of the notion of an outlier in the analyzed dataset. The most often adopted definition is the Hawking’s [48], where an outlier (an abnormal observation) is an observation which deviates so much from others that an assumption arises that it was created as a result of a mechanism of a different character. Some authors point that an outlier is also an observation which not only differentiates from the remaining ones, but also it interrupts relations in the analyzed dataset [27].
Currently, a promising approach toward the analysis and anomaly detection in an electrocardiogram is methods based on the idea of sparse representation of signals with the use of dictionaries with redundancy. They allow, in the process of decomposition, to separate essential construction features of the analyzed signal with respect to the character of the exploited dictionary. These methods, however, are most often used only in the initial stages of the ECG signal analysis, i.e., (i) improvement of quality, (ii) compression or (iii) detection of QRS complex [25, 49]. Nevertheless, they are seldom used in the process of widely understood anomaly detection in an ECG signal [29, 30]. This fact has become an inspiration for the solution presented in this paper, which consists of formulation of the problem’s sparse representation with outliers and the modified AMA method for solving the optimization problem of sparse representation.
3 Sparse representation model of electrocardiographic signal
In numerous cases, representations of an ECG signal performed in the form of linear extensions against a defined set of base functions correctly placed in time and/or frequency are often described as not precise and optimal enough. Therefore, a better solution is to employ more diverse, numerous and adjusted to the signal’s character sets of functions, usually referred to as dictionaries with redundancy [42]. In result, more universal and flexible representations are achieved.
Equation (1) presents sparse representation of \(s_{i}\) signal, received by means of the minimal number of decomposition coefficients \(c_{q}\) and the \(d_{q}\) atoms of \(D\) dictionary corresponding with them (when we assume the specific level of \(\xi\) precision). Optimal representation of an ECG signal which is described as such subset of dictionary \(D\) elements, of which linear combination reflects the biggest rate of the signal \(s_{i}\) energy among all the subsets of the same count. Finding such a representation is computationally NP-hard [45]; hence, we are satisfied with iterative adaptive solution (which is suboptimal), the so-called orthogonal matching pursuit algorithm [50].The result of work of such an algorithm is the projection of structural elements of signal on chosen dictionary elements, called atoms.
The OMP algorithm produces three important elements of information: (i) the set of projection coefficients \(C = \left\{ {c_{0} ,c_{1} , \ldots ,c_{p} , \ldots ,c_{Q - 1} } \right\}\), (ii) the set of residues \(RS = \left\{ {r^{0} s,r^{1} s, \ldots ,r^{p} s, \ldots ,r^{Q - 1} s} \right\}\) and (iii) the list of dictionary elements chosen to approximate \(s_{i}\), represented as \(D = \left\{ {d_{0} ,d_{1} , \ldots ,d_{p} , \ldots ,d_{Q - 1} } \right\}\).
Then we calculate the new coefficient \(c_{p}\) described by (3) and new residual \(r^{p} s\) described by (5). Next, we update the set of coefficients \({\text{C}}_{p} = {\text{C}}_{p - 1} \cup c_{p}\) and set of residual \(R^{p} s = R^{p - 1} s \cup r^{p} s\).
The pace of residuum norm’s fall depends on the correlation between consecutive residues of the signal and selected atoms of the dictionary. If the ECG signal is a sum of high-energy components constituting dictionary atoms, then the signal correlation coefficient and its residues are significant. Consequently, their norm falls rapidly because the high-energy components are the signal’s structural elements, well correlated with the chosen atoms of the dictionary.
3.1 The proposed model of an ECG signal decomposition
3.2 Sparse representation of the proposed model with outliers
The solution caused by problem (8) involves sparse \(C\). On the other hand, it enables nonzero columns in \(O\) in case some outliers cannot be represented solely by \(D\).
The most obvious discrepancy between the models (8) and (9) consists in the fact that each signal in SVM is assumed as a single measurement connected to an exceptional nonzero sample of its sparse representation (that is a unique set of atoms). Nevertheless, what we do in MMV is joint-sparsity for the collection \(S\).
4 Dictionaries for sparse representations of ECG signals
However, decomposition of the signal \(s_{i}\) toward the dictionary requires then continuous seeking and matching its appropriate atoms, which reflect best the necessary attributes of the analyzed ECG signal. This matching ought to be performed to maximize correspondence between the chosen dictionary’s atoms and the remaining part of the analyzed signal. Thus, it is strongly recommended to define a measure of quality and an algorithm which would ensure finding the best signal’s representation for that measure [42]. However, to perform such an aimed goal it is advisable, first of all, to define dictionary elements (atoms), which show the necessary attributes of the decomposed signal in the best possible manner. These dictionaries can be created with the use of properly parametrized analytic functions (e.g., Gabor functions [35]) or formed as approximated reflections of defined analyzed signal structures (e.g., Method of Optimal Directions [37]).
4.1 Gabor’s functions dictionary
As far as the proposed method is analyzed, it was suggested to use a waveform (atoms) from a Gabor dictionary in the form of translation, dilation and modulation of a Gaussian window function \(g\left( x \right)\). We may then define a set of dictionary atoms’ indices, i.e.,\(\gamma \in \left\{ {s,p,k,\phi } \right\}\), where index \(\gamma \in Z\) and \(Z = R^{ + } \times R^{3}\), parameters responsible for dilation, \(p\) is translation, \(k\) is modulating frequency, whereas \(\phi\) is phases and \(\phi \in \left[ {0, 2\pi } \right[\).
In our work, we used the idea of dictionary initially suggested in the quoted work [35]. The parameters of the above-mentioned atoms were pointed out from dyadic sequences of integers. We assume that the signal analyzed is limited and it consists of \(N = 2^{L}\) samples, whereas \(L\) is an integer number. Then scale \(s\), which corresponds to an atom’s width in time, is derived from the dyadic sequence \(s = 2^{j}\), where \(0 \le j \le L\) and \(j\) is octave. Parameters \(0 \le p < 2^{ - j} N\) and \(0 \le k < 2^{j}\), which correspond to an atom’s position in time and frequency, are sampled for each octave \(j\) with interval \(2^{j}\). The parameter \(\phi\), on the other hand, defines phase shifts and assumes the \(M\) number of particular values, i.e., \(\phi = 2\pi m/M\), where \(m = 0,1, \ldots M - 1.\) For such parameters changeability, we obtain a limited dictionary, but in return we get calculation simplicity. Hence, we construct Gabor’s dictionary as a function family \(\left\{ {g_{\gamma } \left( x \right)} \right\}\) for appointed parameters \(\gamma = \left\{ {s,p,k,\phi } \right\}\). As a result, we obtain the dictionary consisting of \(\left( {1 + \log_{2} N} \right)NM\) waveforms (i.e., atoms).
It is also worth noticing that if we define resolution as distance in time (\(\Delta t\)) or in frequency (\(\Delta f\)) between the neighboring atoms for the defined subset of Gabor’s dictionary (when we assume that fs is sampling frequency of the analyzed signal) we obtain \(\Delta t = 2^{j} /f_{s}\) and \(\Delta f = f_{s} /2^{j}\). This means that resolution depends on octave, which defines atom’s “width” in time and in frequency. Thus, the range of time variability of dictionary atoms’ defines their ability of evaluation of “width” in time of signal structure element (which is represented by this atom). The change in resolution is gained by changing sampling frequency of \(p\) and \(k\) parameters.
In Fig. 3a, atoms’ selected examples from Gabor’s dictionary are presented.
4.2 Dictionary learning methods of resistant-to-outlier
The issue of dictionary learning in the scope of sparse representation has recently encouraged numerous researchers to analyze the presented field [34]. What makes dictionary learning algorithms different is their approach toward the dictionary adjustment process, i.e., some update all atoms at once (e.g., Method of Optimal Directions [37]), while others update atoms one after another (e.g., K-means singular value decomposition [38]).
The MOD is utilized to resolve the issue of optimization, described in Eq. (12). This process is performed with minimizing the objective function iteratively over one variable, while the remaining two are fixed. For the first, \(D\) and \(O\) are initialized; next minimization over \(C\) is performed—the iterative optimization starts. The regular course of initializing \(D\) imposes the use of a predefined dictionary, e.g., Gabor’s [35]. However, \(O\) is initialized by using the zero matrix.
- Stage I—Sparse Coding At this stage, the decomposition coefficients \(c_{i}\) are gathered in the overcomplete dictionary \(D\) and signal \(s_{i}\). Each phase consists in finding the smallest possible quantity of coefficients which will fulfill Eq. (13). The given \(D\) is known. The OMP algorithm [50] is utilized to estimate \(I\) sparse coefficients \(c_{i}\) for each example of signal \(s_{i}\), by estimation of$$c_{i} \leftarrow \mathop {\arg \hbox{min} }\limits_{{c_{i} }} \left\| {s_{i} - Dc_{i} - o_{i} } \right\|_{2}^{2} \quad {\text{s}} . {\text{t}}\quad \left\| {c_{i} } \right\|_{0} \le T,\quad i = 1,2, \ldots ,I.$$(13)
- Stage II—Outlier Update At this stage, we update the \(o\) outlier vector (every step consists in searching the minimum number of outliers) to fulfill Eq. (14).$$o_{i} \leftarrow \mathop {\arg \hbox{min} }\limits_{{o_{i} }} \left[ {\left\| {s_{i} - Dc_{i} - o_{i} } \right\|_{2}^{2} + \lambda \left\| {o_{i} } \right\|_{2} } \right],\quad i = 1,2, \ldots ,I.$$(14)To perform an update of an outlier vector, it is necessary to solvewhere \(r = s - Dc\) is the residual vector. When the derivative of the objective function is set to be equal to zero, at the optimal point \(\hat{o}\), the result is$$\mathop {\hbox{min} }\limits_{o} \left[ {\left\| {r - o} \right\|_{2}^{2} + \lambda \left\| {o_{2} } \right\|} \right],$$(15)$$\hat{o} = \left\{ {\begin{array}{*{20}c} {\left( {1 - \frac{\lambda }{{2\left\| r \right\|_{2} }}} \right)r,} & {{\text{if}}\;\left\| r \right\|_{2} > \frac{\lambda }{2}} \\ {0,} & {\text{otherwise}} \\ \end{array} } \right\}.$$(16)
- Stage III—Dictionary Update At this stage, we bring the dictionary \(D\) atoms up to date. We calculate the alternative values of atoms and decomposition coefficients to decrease the probability of an error within the range of the signal \(S\) itself and its sparse representation \(D*C\) with outliers.$${\text{D}} \leftarrow { \arg }\mathop {\hbox{min} }\limits_{D} \mathop \sum \limits_{i = 1}^{Q} \left\| {s_{i} - Dc_{i} - o_{i} } \right\|_{2}^{2}$$(17)
All dictionary atoms are brought up to date in this manner. Iteration which follows the three above stages enables forming a dictionary (without outliers) that estimates the given signal \(S\) in a sparse and precise manner, which then results in the mMOD algorithm being a dictionary \(D\) (resistant-to-outlier used in the modified AMA method) including frequently correlated atoms which reflect the examined signal \(S\) in connection with its sparse representation \(D*C\).
We suggest to resolve the described issue by introducing a decorrelation phase in them MOD loop after the update of the dictionary. In Fig. 3b, an instance of the INC-mMOD atoms from dictionary \(D\) is illustrated.
5 The proposed solution: sparse representation with outlier
Representation of the signal \(S\), displayed by Eq. (7) inspires the solution of problems connected with sparse representations of SVM and MMV types (Sect. 3.2), which are illustrated by the examples (8)–(9), and also obtaining a resistant-to-outlier dictionary \(D\) (Sect. 4). In order to carry out such a task, the alternating minimization algorithm was modified so as to achieve a set of outliers \(O\) describing anomalies in the signal \(S\).
5.1 The problem formulation
We suggest the modified alternating minimization algorithm [47], described by Eq. (24), as a solution to the problem since this algorithm is simpler than alternating direction method of multipliers.
5.2 Modified AMA for solving the optimization problem
As it is easy to notice, the change in Eq. (24) into a constrained issue, the first and second terms of Eq. (24) have been decoupled. Such operation lets evade necessity of an iterated-shrinkage solution for \(C\) [56]. By adding the auxiliary variable \(C\), we achieved a close-form solution for \(C\), as well as a one-shot shrinkage solution for \({\text{V}}\).
As outcomes of the AMA performance, we obtain: (i) a set of coefficients \(C\), which together with the dictionary \(D\) (achieved with the use of various versions of the MOD algorithm or presented set of analytical Gabor functions) represent the correct structure of the signal \(S\), (ii) outliers (anomalies included in the signal \(S\)) reflected by the matrix \(O\). The former and the latter represent the suggested model of data, which was described by Eq. (2).
6 Experiments
In order to carry out the suggested experiments, the above-described methodology of detecting different sort of disturbances (i.e., anomalies) in the analyzed electrographic record was implemented. The basis of the explored solution was the following ideas: the proposed model of ECG signal decomposition onto the correct values and outliers (Sect. 3.1), dictionaries for sparse representations (Sects. 4.1 and 4.2) and the modified alternating minimization algorithm (Sect. 5.2) to solve the unconstrained convex problem described by Eq. (24). To define the efficiency of the presented solution, we used commonly accessible and usually explored MIT-BIH Arrhythmia Database constituting the set of test ECG signals. In a particularly detailed manner, the results of anomaly detection for signals marked with #108, #109 and #203 (which contain various morphological structure disturbances) and four arrhythmia types (i.e., premature ventricular contractions, atrial premature contractions, paced beats and ventricular flutter) were presented.
6.1 Experimental data included in the MIT-BIH Arrhythmia Database
The input of experimental material for analysis is expected to consist of electrocardiographic records from a public and reference cardiological database which is a set of tested ECG signals. The database contains correct courses and measurements replicating basic and complex models of typical myocardial dysfunctions. The collected records consist of a series of electrical activity measurements of the heart with the use of an electrocardiograph for diverse signal acquisition methods, both in terms of amount and the length of the study. In the diagnostic aspect, each of the records gathered here is supplemented with a prepared by specialists, careful cardiological analysis, which allows to verify the results of the proposed analyzing methods.
Summary of datasets divided into the analyzed categories
ECG signal type | Annotation | Total | Training | Training ratio | Testing |
---|---|---|---|---|---|
Normal beat (NSR) | N | 75,017 | 11,253 | 23 | 63,764 |
Premature ventricular contractions (PVC) | V | 7129 | 2495 | 35 | 4634 |
Atrial premature contractions (APC) | A | 2546 | 891 | 35 | 1655 |
Paced beat (PACE) | P | 7024 | 2458 | 35 | 4566 |
Ventricular flutter (VF) | ! | 472 | 236 | 50 | 236 |
Total | 92,188 | 17,333 | 18.80 | 74,855 |
We also used the whole available MIT-BIH database (together with localized and described disorders) for verification of the presented solution for four different types of arrhythmia, i.e., (i) premature ventricular contractions—that creates one of the most common heart rhythm irregularities, characterized as early beat or skipped beat which interrupts correct heart rhythm, (ii) atrial premature contractions—they are equally common abnormalities of heart rhythm, which appear as premature or overlooked beats happening more often than normal heartbeats, (iii) paced beats—beats which occur at particular speed and further at greater velocity and (iv) ventricular flutters—very frequent ventricular contractions (250–350 beats/min), which do not allow the identification of the QRS complex.
6.2 Experiments results
For the sake of performing presented experiments, the simulations were carried out on a i7 dual-processor Dell work station composed of eight cores processor and 32 GB of RAM in connection with interactive environment for conducting scientific calculations MATLAB, as well as SparseLab, Sparse Optimization Toolboxes and MOD-Toolbox for solution implementation.
In the described work, we assume that the analyzed ECG signal is limited to five-minute recordings. For the purpose of further analysis, the signal is subdivided into segments (256 elements each). The ECG segments we receive in the process of decomposition of the ECG signal fall into all possible 256-samples windows. The presumed window’s size equaled: \(N = 256\) samples, and it was achieved as a result of numerous experiments conducted (where each particular window should include periodic structure of the analyzed ECG signal). However, if we chose a smaller window, the substantial part of analyzed ECG signal structure would be reduced (i.e., would not reflect the correct structure of a unit of the signal). Consequently, a bigger-size window could include more than one structure unit (i.e., repeated wave structure).
Another considered attitude toward solution of the problem of ECG signal sparse representation was the use of dictionary, whose elements were not differently parametrized analytic functions, but selected structures of the analyzed signal. The supremacy of such a solution consisted in better and sparser representation of an ECG signal, i.e., we were able to represent the analyzed signal with the same accuracy (as in the case of Gabor’s dictionary) but by means of a smaller number of better matched atoms. Such a solution lets us create a dictionary containing important (crucial) structural elements of the analyzed ECG signal. To minimize the influence of possible distortions of an ECG signal on the content of the created dictionary, we optionally employed a resistant-to-outlier dictionary learning method, based on a modified Method of Optimal Directions estimation.
The MOD algorithm (with modifications) was utilized to resolve the optimization problem, i.e., searching the proper dictionary for sparse representation of analyzed ECG signal. This process was done by minimizing the objective function iteratively over one variable (the remaining two are fixed). For the first, dictionary \(D\) and outliers \(O\) are initialized, then minimization over coefficients set \(C\) was carried out—the iterative optimization begins. The usual course of initializing dictionary \(D\) imposes the use of a predefined dictionary, e.g., (i) Gabor’s or (ii) the dictionary is constructed of atoms chosen at random from the training signals. The second option should not be used for our solution-finding process because certain outliers might be treated as atoms. Consequently, they could possibly affect the course of whole process (in the sequel iterations). Outliers \(O\) are initialized by using the zero matrix. For the sake of explanation, at first iteration step, we used all the training signals as not including “outliers,” i.e., distortions of an ECG signal. For starting the MOD algorithm, we used the above-described Gabor’s dictionary.
In case of dictionaries created on the basis of ECG training signals, using Method of Optimal Directions, we considered the three following options: (i) application of classical solution MOD, (ii) use of modified MOD (marked as mMOD) by adding outlier update step for gaining resistant-to-outlier dictionary or (iii) improved solution of mMOD with incoherent dictionary learning step (named INC-mMOD). In the process of dictionary creation on the basis of ECG training signal samples with the use of two modified and a classical versions of Method of Optimal Directions, the following parameter values were used: \(I = 64\), \(T = 8\), \(\lambda = 0.1\) and \(\mu_{0} = 0.2\). The \(I\) and \(T\) values were assumed as commonly occurred in solutions based on classical MOD algorithms. On the other hand, the \(\lambda\) parameter value was estimated on the basis of experiment, i.e., a point was searched in which an error of sparse representation of the correct ECG signal (for the normal sinus rhythm) does not significantly change its value any longer; it was then accepted as \(\lambda\) parameter value. Then, for the training data, each exceeding of \(\lambda\) threshold value was classified as an outlier. Otherwise, it was treated as a proper data vector. Another noticed dependence was that: the smaller \(\lambda\) value is, the larger number of points was accepted as outliers. In case of \(\mu_{0}\) parameter, responsible for coherence of dictionary, the \(\mu_{0}\) value was assumed as a compromise between redundancy of a dictionary and diligent fulfilling decorrelation condition of its atoms. The count of created dictionaries was 1000 atoms. In Fig. 3b, an instance of the INC-mMOD atoms from dictionary D (received on the basis of the training ECG signal including the normal beat) is illustrated. It seems obvious that the atoms of this dictionary reflect (differently translated) characteristic structures of standard ECG signal.
Results of the reconstruction process for ECG signals
Dictionary | PRMSD | PRMSD | PRMSDN | PRMSDN |
---|---|---|---|---|
Mean (%) | SD | Mean (%) | SD | |
Gabor | 18.21 | 4.91 | 18.25 | 4.92 |
MOD | 17.79 | 5.07 | 17.84 | 5.06 |
mMOD | 16.24 | 5.18 | 16.29 | 5.20 |
INC-mMOD | 15.89 | 5.34 | 15.93 | 5.32 |
For the sake of anomaly detection, we used the proposed data model (7), and for solving the problem of sparse representation with separated outliers (17), we tested selected ECG signals (i.e., #108, #203, #109) and four types of arrhythmia. To solve the problem (19), we used the modified AMA method with the following dictionaries: Gabor’s, MOD, mMOD and INC-mMOD, created on the basis of the training normal beat ECG signal. The set of parameters: \(\nu^{0} = 1.0\), \(\delta = 1.5\), \(\alpha = 1.0\), \(\beta = 2.0\), \(\varepsilon = 0.005\), \(d = 1 \,{\text{or}} \,d = 2\) were used for performing modified AMA method. The initial value of penalty parameter \(\nu^{0}\) was described neutrally (i.e., so as no initial punishment at the start of AMA modified algorithm was introduced). On the other hand, the value of parameter \(\delta\) was defined so as to guarantee of convergence of (29) dependence, on condition that the sub-problem is solved, ensuring high precision at every iteration. The value of \(\alpha\) parameter is standard for the presented AMA method, while the value \(\beta\) (that is twice bigger as \(\alpha\)), was chosen so that the process of anomaly (outliers) detection was preferred. The \(d\) parameter, on the other hand, which is a switch, chooses SVM or MMV realizations. The threshold parameter \(\varepsilon\) being stop criterion of the modified AMA was appointed in order to stop algorithm, when there already exists a lack of significant fluctuations of searched values.
Occurrences of anomalies in the analyzed signals (#108, #203 and #109) were detected as columns in the matrix \(O\), with the use of \(\ell_{2}\) norm, and the threshold value \(\vartheta = 0.2\) as irregular heartbeats locations. The value of \(\vartheta\) parameter was experimentally defined as a compromise between the maximization of value of detection rate (DR) and minimization of values of false positive rate (FP) for analyzed ECG signals.
Results of anomaly detection in ECG signals for SVM
Signal | Gabor | MOD | mMOD | INC-mMOD | ||||
---|---|---|---|---|---|---|---|---|
DR (%) | FP (%) | DR (%) | FP (%) | DR (%) | FP (%) | DR (%) | FP (%) | |
#108 | 92.53 | 2.89 | 94.21 | 3.25 | 95.53 | 2.99 | 95.58 | 2.87 |
#203 | 90.32 | 4.47 | 92.78 | 3.72 | 94.46 | 3.75 | 94.45 | 3.81 |
#109 | 91.64 | 3.50 | 93.55 | 3.14 | 96.71 | 2.90 | 96.76 | 2.85 |
Results of anomaly detection in ECG signals for MMV
Signal | Gabor | MOD | mMOD | INC-mMOD | ||||
---|---|---|---|---|---|---|---|---|
DR (%) | FP (%) | DR (%) | FP (%) | DR (%) | FP (%) | DR (%) | FP (%) | |
#108 | 91.87 | 3.96 | 93.78 | 3.19 | 94.77 | 3.01 | 95.29 | 2.89 |
#203 | 90.02 | 4.49 | 91.84 | 3.75 | 92.59 | 3.93 | 93.56 | 3.75 |
#109 | 90.71 | 4.27 | 94.05 | 3.38 | 95.12 | 3.25 | 96.01 | 2.93 |
While analyzing results presented in Tables 3 and 4, it is worth noticing that in case of SVM approach we generally obtain better DR and FP outcomes. This situation is possibly due to the fact that it is better to apply sparsity in every line of the matrix of representation coefficients of the analyzed signal (SVM), than only in the formerly defined number of them (MMV). Therefore, in the following part of the experiment we will be using only SVM approach described by formula (24) for parameter \(d = 1\).
In case of searching for four chosen types of arrhythmia (i.e., premature ventricular contractions, atrial premature contractions, paced beats and ventricular flutter) in testing ECG signals, the way of conducting was of two stage. In the first step, the anomalies, searched with the use of described the modified AMA method for four dictionary types, i.e., Gabora, MOD, mMOD and INC-MMOD, were detected. All dictionaries were trained on the basis of training set including the normal sinus rhythm (NSR). Occurrences of anomalies in the tested signals were detected as columns in the \(O\) matrix, with the use of \(\ell_{2}\) norm, and the threshold value \(\vartheta = 0.2\) as irregular heartbeats locations. Then, in the second step, we classified detected anomalies with the use of the orthogonal matching pursuit algorithm and properly trained dictionaries. The training process consisted in creating of dictionaries on the basis of a training set (see Table 1) including researched types of arrhythmia. In such a way, the four groups of dictionaries were formed (i.e., Gabor’s, MOD, mMOD and INC-MMOD), each of which included four dictionaries (reflecting disturbances) ascribed to every of them (i.e., PVC, APC, PACE and VF). When anomaly was detected (in the first step), its classification consisted in creating sparse representations using OMP algorithm and trained dictionaries. Next, we searched which of them (sparse representations of detected anomaly for a particular dictionary) minimizes the dependence (1), i.e., represents the structure of detected disturbance in the best way. The dictionary used then appointed the type of detected arrhythmia.
Results of different types of arrhythmia detection in ECG signals
Arrhythmia | Gabor | MOD | mMOD | INC-mMOD | DeepLSTMs [60] | |||||
---|---|---|---|---|---|---|---|---|---|---|
Precision | FPR | Precision | FPR | Precision | FPR | Precision | FPR | Precision | FPR | |
PVCs | 96.74 | 1.29 | 98.21 | 0.82 | 98.97 | 0.89 | 99.13 | 0.49 | 99.09 | 0.47 |
APCs | 88.23 | 0.97 | 90.73 | 0.89 | 92.05 | 1.02 | 92.17 | 0.83 | 92.31 | 0.79 |
PACEs | 92.16 | 1.63 | 94.02 | 1.15 | 94.46 | 0.75 | 99.24 | 0.59 | 99.30 | 0.07 |
VFs | 90.55 | 2.05 | 93.89 | 1.73 | 94.78 | 1.90 | 95.50 | 1.02 | 95.59 | 1.18 |
While analyzing the results included in Table 5, one can notice an advantage of the solution with the use of a dictionary created by means of a modified version of Method of Optimal Directions and improved by incoherent dictionary learning step. It is connected to creation of resistant-to-outlier dictionary and introduction of the decorrelation condition. The resulting dictionary was probably characterized by best matching to researched structures of ECG signals. On the other hand, weak signals for Gabor’s dictionary were caused by not sufficient flexibility of dictionary’s atoms (that is Gabor’s functions) with regard to the decomposed morphological structures of the ECG signals. Analyzing the achieved results included in Table 5 (for four presented arrhythmia types detected with the use of INC-mMOD) with results published in the literature [60], it can be stated that they are comparable, that is they correspond to a similar level of values.
As it is easy to notice, there exists significant influence of researched parameters for achieved precision of detection of type PVC disturbances, i.e., along with increase in the researched parameters value the precision of gained detection decreases (approximately hyperbolically or parabolic). The influence of remaining parameters of presented method is significantly smaller, and it is connected to a larger degree with stability and numerical convergence.
7 Conclusions
Medicine nowadays provides us with numerous myocardium diagnostic methods, beginning with physical examinations, through biochemical until diagnostic imaging. What is of utter importance is that in the diagnose process a quick and noninvasive diagnostic method needs to be applied frequently. Such a situation causes an electrocardiograph to be used, in a hospital or a surgery, in most cases. Commonly understood medical care would definitely benefit from the fact of such diagnostics being widely accessible, including patients realizing survey self-sufficiently or receiving exact results of the ECG signal from the cardiological monitoring center.
As far as automatic analysis of ECG signals is analyzed, the need of using complex methods in terms of processing, analysis and recognition of signals, is of crucial importance, where the process of detection and identification of selected ECG parameters is not to be overestimated.
In this article, we described how the sparse representation of electrocardiograms and outlier dictionary learning method for the analyzed electrocardiographic records work. Next, the modified MOD algorithm was used to obtain proper structure of the dictionary. A dictionary created in such manner formed basis for the use of sparse representation of the analyzed electrocardiographic record. The presented anomaly detection method was created as a solution to the task of sparse representation of the electrographic record, where correct values and outliers were used for the sake of description. To carry out the above mentioned task, the modified ADA algorithm was used. Effectiveness of the method presented was examined with the use of MIT-BIH Arrhythmia Database. The effectiveness of the described solution was confirmed by the obtained experiment outcomes.
However, detailed results of the performed tests showed diverse efficacy and correctness of the described solution. The outcomes achieved for the chosen signals #108, #203 and #109 did not present significant differences in DR and FP values. It was certainly caused by correct match of the presented solution for detection of particular but simple types of disorders (anomalies). However, while analyzing the obtained results for the whole MIT-BIH base in context of the four arrhythmia types (i.e., premature ventricular contractions, atrial premature contractions, paced beats and ventricular flutter), there could be noticed the following trend: the more complex the arrhythmia, efficacy of the proposed solution became slightly worse. Most probably, this phenomenon was connected to the fact that the presented solution was aiming at as sparse representation of the processed signals as possible. As a result, it could lead to hiding, in some cases, more complex types of disturbances. In spite of this situation, the achieved outcomes were comparable to the ones presented in the field literature.
Notes
Compliance with ethical standards
Conflict of interest
The author declares that he has no conflicts of interest.
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