Atrial fibrillation detection by DFA and SDCST methods

Abstract

Many cardiac disorders were diagnosed by analyzing an electrocardiogram signal, in particular, atrial fibrillation. We join the SDCST method with the Detrended Fluctuation Analysis (DFA) and the backpropagation net to identify atrial fibrillation in one hundred ECG signals obtained from Physionet Challenge 2017 database. The accuracy of the proposed classifier parameter is 97% for the training set and 95% for the test set.

Keywords

Atrial fibrillation ECG DFA SDCST

1. Introduction

The electrocardiogram signal (ECG) can diagnose several cardiac disorders (Andreotti et al., 2017), one of these diseases is a particular case of cardiac arrhythmia called atrial fibrillation. Estimates indicate that two percent of the general population may present episodes of atrial fibrillation and, there is a strong correlation between this type of anomaly with mortality risks (Behar et al., 2017).

In this context, there is paroxysmal atrial fibrillation that not always occurs during a conventional ECG test. Then the Holter Monitor appears like an efficient tool (Fan et al., 2018). In possession of these data, the cardiologists must search for anomalies. However, atrial fibrillation can occur in a short time, which makes this work exhausting and subject to errors. For this reason, the proposition of methods that identify atrial fibrillation on short-term ECG signals, thirty or sixty seconds, is of extreme importance.

Aligned with this objective, the Physionet has proposed a challenge (Clifford et al., 2017) that had as target classifies whether a short ECG lead recording shows atrial fibrillation. Among the papers that emerged from this challenge, we can cite Datta et al. (2017) that incorporate to their methodology the multi-layer cascade binary classification approach and, Teijeiro et al. (2017) that uses a set of high-level and clinically meaningful features provided by the abductive interpretation of the records.

The signal denoising by clustering and soft thresholding (SDCST) (Vargas and Veiga, 2018) clusters the detail wavelet coefficients in two sets and, respectively, shrink them according to estimated thresholds. Here we join the SDCST method with the Detrended Fluctuation Analysis (DFA) and the backpropagation net to identify atrial fibrillation in two hundred ECG signals obtained from Physionet Challenge 2017 database.

The novelty of our method is the union of these three theories. For example, Teijeiro et al. (2017) and Datta et al. (2017) use as input parameters the general features of the ECG signal. Here, we use the Hurst Exponent value (calculated by the DFA method) of the time series determined by the SDCST method.

The DFA is a method that detects the self-affinity in a time series. We can see applications of this method in ECG signals (Linhares, 2016, 2018), in econophysics (Kutner et al., 2019) and, in seismic processing (Linhares, 2017).

Our work also differs from the method introduced by Linhares (2016), because Linhares uses directly the Hurst Exponent value obtained from the RR-interval of the ECG series submitted to her SDFA method. We apply the DFA directly in two time series and fit a backpropagation net to classify healthy and unhealthy signals. Beyond that, Linhares has identified a general type of arrhythmia, processing a long time ECG signals. Here we detect the specific atrial fibrillation arrhythmia, processing short-time ECG signals.

In Section 2, we review some basic properties about the SDCST method, DFA and backpropagation net. In Section 3, we present the proposed SDCST Atrial Fibrillation Classifier (SDCST-ACF) and in Section 4 we show the application. In Section 5 we summarize the major conclusions of this study.

2. Basic concepts

In this section, we introduce the basic concepts about the SDCST method, DFA, and backpropagation net. We begin with the SDCST method in Section 2.1.

2.1 SDCST method

The Signal Denoising by Clustering and Soft-thresholding (SDCST) method uses the discrete wavelet transform, Hidden Markov Models, and the Viterbi algorithm to cluster the detail wavelet coefficients in groups of signal and noise.

After that, we submit each of these sets to thresholding processes and sum these two pre-processed sets.

Over the result of this sum, we apply the Inverse Discrete Wavelet Transform and obtain the clean signal estimate. For details about the SDCST method see Vargas and Veiga (2018).

2.2 Detrended fluctuation analysis

The Detrended Fluctuation Analysis (DFA) method, proposed by Peng et al. (1994), is a method useful for analysing time series that appear to be long-memory processes. Given a time series $\{X_{t}\}_{t=1}^{n}$ , the Detrended Fluctuation Analysis (DFA) consists on five steps. In the first one, for each $t\in\{1,2,\ldots,n\}$ , we calculate

$\displaystyle Y_{t}=\sum_{j=1}^{t}X_{j}.$ (1)

In the second step we divide the time series $\{Y_{t}\}_{t=1}^{n}$ into $\left\lfloor\frac{n}{l}\right\rfloor$ nonoverlapping blocks, each containing $l$ observations, where $\lfloor\cdot\rfloor$ indicates the integer part function. In the third step, for each block, one fits a least-squares line to the data. In the fourth step, in each block we calculate

$\displaystyle Z_{t}^{l}=Y_{t}-Y_{t}^{l},$ (2)

where $Y_{t}^{l}$ denotes the adjusted fit on each block. Let $g(n)$ be the maximum block size. In the fifth step, for each $l\in\{4,5,\ldots,g(n)\}$ , we calculate the root mean square fluctuation

$\displaystyle F(l)=\sqrt{\frac{1}{\widetilde{n}}{\sum_{t=1}^{\widetilde{n}}{(Z% _{t}^{l})}^{2}}},\text{ where }\widetilde{n}=l\left\lfloor\frac{n}{l}\right\rfloor.$ (3)

If we apply the natural logarithm to each $F(l)$ and each $l$ , that is $\ln(F(l))$ versus $\ln(l)$ , we will verify a linear relationship between these two quantities. The linear coefficient obtained from the linear regression of these points is an estimate for Hurst’s exponent.

2.3 Backpropagation net

The backpropagation net is a method of artificial neural network training that calculates a gradient. This gradient allows us to estimate the network weights. Here we use a neural network with two hidden layers, the first contains five neurons and the second one contains two neurons. The input layer contains two neurons and, the output layer contains one neuron. The Fig. 3 represents this neural network.

The neural net is fitted to the data by the backpropagation of the error, see Fausett et al. (1994) for details. Which these estimated weights we can apply the backpropagation algorithm by the following steps:

Each input unit $X_{1}$ , $X_{2}$ broadcasts input signal to hidden units.

Each hidden unit $Z_{1},Z_{2},\ldots,Z_{5}$ computes input signal

$\displaystyle z_{in_{h}}=u_{0h}+x_{1}u_{1h}+x_{2}u_{2h},h\in\{1,2,\ldots,5\},$

applies activation function to compute output signal

$\displaystyle z_{h}=f(z_{in_{h}}),$

and sends its output signal to the units in the second hidden layer.

Each hidden unit $ZZ_{1},ZZ_{2}$ computes input signal

$\displaystyle zz_{in_{j}}=v_{0j}+\sum_{h=1}^{5}z_{h}v_{hj},∼{}∼{}j\in\{1,2\}$

applies activation function to compute output signal

$\displaystyle zz_{j}=f(zz_{in_{j}})$

and sends its output signal to output units.

Each output unit $Y_{1}$ sums weighted input signal

$\displaystyle y_{in_{1}}=w_{01}+w_{11}zz_{1}+w_{21}zz_{2}$

and applies activation function to compute its output signal

$\displaystyle y_{1}=g(y_{in_{1}}).$

3. Atrial fibrillation classifier

In this section we present the SDCST Atrial fibrillation classifier (SDCST-ACF). Given a sequence of ECG signals $\bm{x}_{1},\ldots,\bm{x}_{n}$ we apply, in the first step, the SDCST method obtaining the values of the smoothed signals $\hat{\bm{x}}_{1},\ldots,\hat{\bm{x}}_{n}$ as well as the noise estimates

$\displaystyle\hat{\bm{e}}_{1}=\hat{\bm{x}}_{1}-\bm{x}_{1},\ldots,\hat{\bm{e}}_% {n}=\hat{\bm{x}}_{n}-\bm{x}_{n}.$

The second step consists in to calculate the Hurst exponents $H_{\hat{\bm{x}}_{1}},\ldots,H_{\hat{\bm{x}}_{n}}$ and $H_{\hat{\bm{e}}_{1}},\ldots,H_{\hat{\bm{e}}_{n}}$ .

Let:

•
$\mu_{{H}_{\hat{\bm{x}}}}$ the mean of the values $H_{\hat{\bm{x}}_{1}},\ldots,H_{\hat{\bm{x}}_{n}}$ ;
•
$\sigma_{H_{\hat{\bm{x}}}}$ the standard deviation of the values $H_{\hat{\bm{x}}_{1}},\ldots,H_{\hat{\bm{x}}_{n}}$ ;
•
$\mu_{H_{\hat{\bm{e}}}}$ the mean of the values $H_{\hat{\bm{e}}_{1}},\ldots,H_{\hat{\bm{e}}_{n}}$ ;
•
$\sigma_{H_{\hat{\bm{e}}}}$ the standard deviation of the values $H_{\hat{\bm{e}}_{1}},\ldots,H_{\hat{\bm{e}}_{n}}$ .

In the third step, we obtain the normalized Hurst exponents $\tilde{H}_{\hat{\bm{x}}_{1}},\ldots,\tilde{H}_{\hat{\bm{x}}_{n}}$ $\tilde{H}_{\hat{\bm{e}}_{1}},\ldots,\tilde{H}_{\hat{\bm{e}}_{n}}$ by the Eqs (4) and (5).

$\displaystyle\tilde{H}_{\hat{\bm{x}}_{i}}=\frac{H_{\hat{\bm{x}}_{i}}-\mu_{H_{% \hat{\bm{x}}}}}{\sigma_{H_{\hat{\bm{x}}}}},∼{}∼{}i\in\{1,2,\ldots,n\}.$ (4) $\displaystyle\tilde{H}_{\hat{\bm{e}}_{i}}=\frac{H_{\hat{\bm{e}}_{i}}-\mu_{H_{% \hat{\bm{e}}}}}{\sigma_{H_{\hat{\bm{e}}}}},∼{}∼{}i\in\{1,2,\ldots,n\}.$ (5)

Table 1
Hurst exponent values given in the second step of the Section 3, healthy ECG signals

$A$ $i$ $H_{\hat{x}_{i}}$ $H_{\hat{e}_{i}}$

00001 1 0.712 0.487

00002 2 0.736 0.250

00006 3 0.718 0.092

00007 4 0.670 0.073

00011 5 0.769 0.519

00012 6 0.708 0.281

00014 7 0.770 0.484

00016 8 0.685 0.218

00018 9 0.766 0.483

00019 10 0.684 0.077

00021 11 0.672 0.453

00025 12 0.768 0.536

00028 13 0.722 0.475

00031 14 0.791 0.555

00033 15 0.787 0.572

00034 16 0.782 0.586

00035 17 0.698 0.352

00037 18 0.718 0.539

00040 19 0.738 0.265

00042 20 0.753 0.544

00045 21 0.730 0.458

00046 22 0.756 0.565

00048 23 0.767 0.545

00049 24 0.775 0.329

00050 25 0.745 0.555

00051 26 0.683 0.489

00052 27 0.790 0.545

00053 28 0.756 0.363

00057 29 0.883 0.529

00059 30 0.735 0.524

00060 31 0.737 0.233

00062 32 0.732 0.249

00063 33 0.689 0.053

00064 34 0.782 0.542

00068 35 0.777 0.559

00072 36 0.784 0.567

00073 37 0.752 0.553

00076 38 0.724 0.042

00080 39 0.623 0.208

00081 40 0.674 0.421

00084 41 0.711 0.243

00085 42 0.787 0.568

00089 43 0.734 0.531

00091 44 0.680 0.056

00093 45 0.751 0.529

00094 46 0.777 0.548

00097 47 0.728 0.040

00098 48 0.671 0.292

00099 49 0.732 0.540

00104 50 0.753 0.304

Table 2
Hurst exponent values, see the second step of Section 3, unhealthy ECG signals

$A$ $i$ $H_{\hat{x}_{i}}$ $H_{\hat{e}_{i}}$

00004 51 0.739 0.441

00009 52 0.747 0.554

00015 53 0.864 0.559

00027 54 0.753 0.370

00071 55 0.669 0.044

00087 56 0.768 0.457

00101 57 0.743 0.037

00102 58 0.733 0.282

00107 59 0.623 0.501

00128 60 0.749 0.065

00132 61 0.650 0.225

00141 62 0.627 0.047

00155 63 0.720 0.399

00156 64 0.685 0.506

00208 65 0.744 0.504

00216 66 0.712 0.046

00225 67 0.761 0.153

00231 68 0.549 0.044

00247 69 0.732 0.042

00253 70 0.711 0.215

00267 71 0.646 0.036

00271 72 0.657 0.037

00301 73 0.679 0.040

00321 74 0.751 0.422

00395 75 0.807 0.539

00397 76 0.787 0.092

00405 77 0.926 0.589

00422 78 0.674 0.048

00432 79 0.534 0.032

00438 80 0.594 0.062

00439 81 0.695 0.459

00441 82 0.843 0.552

00473 83 0.781 0.452

00486 84 0.682 0.056

00509 85 0.676 0.062

00519 86 0.749 0.530

00520 87 0.752 0.021

00524 88 0.787 0.555

00542 89 0.741 0.434

00551 90 0.698 0.043

00567 91 0.744 0.412

00587 92 0.675 0.275

00591 93 0.853 0.532

00592 94 0.586 0.069

00611 95 0.803 0.554

00613 96 0.736 0.169

00614 97 0.629 0.080

00621 98 0.716 0.049

00624 99 0.718 0.057

00625 100 0.670 0.044

Denote $\tilde{\bm{H}}_{\hat{\bm{x}}}=(\tilde{H}_{\hat{\bm{x}}_{1}},\ldots,\tilde{H}_{% \hat{\bm{x}}_{n}})$ and $\tilde{\bm{H}}_{\hat{\bm{e}}}=(\tilde{H}_{\hat{\bm{e}}_{1}},\ldots,\tilde{H}_{% \hat{\bm{e}}_{n}})$ . In the fourth step we fit a backpropagation net (see Section 2.3) considering the activation functions given by the Eqs (6) and (7). In the input layer we consider $\mathbf{X}_{1}=\tilde{\bm{H}}_{\hat{\bm{x}}}$ and $\mathbf{X}_{2}=\tilde{\bm{H}}_{\hat{\bm{e}}}$ . And in the output layer we consider $\mathbf{Y}_{1}=0$ for healthy ECG signals and, $\mathbf{Y}_{1}=1$ for unhealthy ECG signals.

$\displaystyle f(x)=\tanh(0.5x).$ (6) $\displaystyle g(x)=x.$ (7)

In the fifth step we fit a backpropagation net considering the $n$ input data $(\tilde{H}_{\hat{\bm{x}}_{1}},\tilde{H}_{\hat{\bm{e}}_{1}}),\ldots,(\tilde{H}_% {\hat{\bm{x}}_{n}},\tilde{H}_{\hat{\bm{e}}_{n}})$ and the $n$ output data $\mathbf{Y}_{1},\ldots,\mathbf{Y}_{n}$ . This training step gives to us the values of the weights $u$ , $v$ and $w$ detailed in the Section 2.3 and graphicaly represented by the Fig. 3.

In both the training and predict stage, the output values need to be denormalized. We do this by multiplying the normalized output by 0.5 and add to this result the value 0.5. Then, we obtain the output $\bm{Y}_{i},i\in\{1,2,\ldots,n\}$ .

Then, in the sixth step, we use the values previously estimated for the weights to apply the algorithm described in Section 2.3. The value of the atrial fibrillation classifier parameter $C_{i},i\in\{1,2,\ldots,n\}$ is calculated in the last step of this algorithm (see Eq. (8)). If the $\mathbf{Y}_{i}$ is less than 0.5, then we classify the ECG signal as normal. If the $\mathbf{Y}_{i}$ is greater or equal to 0.5, then we classify the ECG as atrial fibrillation.

$\displaystyle C_{i}=\begin{cases}\text{Normal}&\text{if }\mathbf{Y}_{i}<0.5,\\ \text{Atrial fibrillation}&\text{if }\mathbf{Y}_{i}\geqslant 0.5.\end{cases}$ (8)
4. Applications

$A$	$i$	$H_{\hat{x}_{i}}$	$H_{\hat{e}_{i}}$
00001	1	0.712	0.487
00002	2	0.736	0.250
00006	3	0.718	0.092
00007	4	0.670	0.073
00011	5	0.769	0.519
00012	6	0.708	0.281
00014	7	0.770	0.484
00016	8	0.685	0.218
00018	9	0.766	0.483
00019	10	0.684	0.077
00021	11	0.672	0.453
00025	12	0.768	0.536
00028	13	0.722	0.475
00031	14	0.791	0.555
00033	15	0.787	0.572
00034	16	0.782	0.586
00035	17	0.698	0.352
00037	18	0.718	0.539
00040	19	0.738	0.265
00042	20	0.753	0.544
00045	21	0.730	0.458
00046	22	0.756	0.565
00048	23	0.767	0.545
00049	24	0.775	0.329
00050	25	0.745	0.555
00051	26	0.683	0.489
00052	27	0.790	0.545
00053	28	0.756	0.363
00057	29	0.883	0.529
00059	30	0.735	0.524
00060	31	0.737	0.233
00062	32	0.732	0.249
00063	33	0.689	0.053
00064	34	0.782	0.542
00068	35	0.777	0.559
00072	36	0.784	0.567
00073	37	0.752	0.553
00076	38	0.724	0.042
00080	39	0.623	0.208
00081	40	0.674	0.421
00084	41	0.711	0.243
00085	42	0.787	0.568
00089	43	0.734	0.531
00091	44	0.680	0.056
00093	45	0.751	0.529
00094	46	0.777	0.548
00097	47	0.728	0.040
00098	48	0.671	0.292
00099	49	0.732	0.540
00104	50	0.753	0.304

$A$	$i$	$H_{\hat{x}_{i}}$	$H_{\hat{e}_{i}}$
00004	51	0.739	0.441
00009	52	0.747	0.554
00015	53	0.864	0.559
00027	54	0.753	0.370
00071	55	0.669	0.044
00087	56	0.768	0.457
00101	57	0.743	0.037
00102	58	0.733	0.282
00107	59	0.623	0.501
00128	60	0.749	0.065
00132	61	0.650	0.225
00141	62	0.627	0.047
00155	63	0.720	0.399
00156	64	0.685	0.506
00208	65	0.744	0.504
00216	66	0.712	0.046
00225	67	0.761	0.153
00231	68	0.549	0.044
00247	69	0.732	0.042
00253	70	0.711	0.215
00267	71	0.646	0.036
00271	72	0.657	0.037
00301	73	0.679	0.040
00321	74	0.751	0.422
00395	75	0.807	0.539
00397	76	0.787	0.092
00405	77	0.926	0.589
00422	78	0.674	0.048
00432	79	0.534	0.032
00438	80	0.594	0.062
00439	81	0.695	0.459
00441	82	0.843	0.552
00473	83	0.781	0.452
00486	84	0.682	0.056
00509	85	0.676	0.062
00519	86	0.749	0.530
00520	87	0.752	0.021
00524	88	0.787	0.555
00542	89	0.741	0.434
00551	90	0.698	0.043
00567	91	0.744	0.412
00587	92	0.675	0.275
00591	93	0.853	0.532
00592	94	0.586	0.069
00611	95	0.803	0.554
00613	96	0.736	0.169
00614	97	0.629	0.080
00621	98	0.716	0.049
00624	99	0.718	0.057
00625	100	0.670	0.044

We consider one hundred ECG signals for training, fifty healthy and fifty unhealthy, from Physionet Challenge 2017 database (Clifford et al., 2017). The ECG names start with the letter “A” followed by five digits.

Then we apply the steps given by the Section 3. The Tables 2 and 2 show the values of the Hurst exponent given in the second step of the Section 3.

Table 3
Output values $\bm{Y}_{i},i\in\{1,2,\ldots,100\}$

$\bm{Y}_{1}$	$\bm{Y}_{2}$	$\bm{Y}_{3}$	$\bm{Y}_{4}$	$\bm{Y}_{5}$	$\bm{Y}_{6}$	$\bm{Y}_{7}$	$\bm{Y}_{8}$	$\bm{Y}_{9}$	$\bm{Y}_{10}$	$\bm{Y}_{11}$	$\bm{Y}_{12}$	$\bm{Y}_{13}$	$\bm{Y}_{14}$	$\bm{Y}_{15}$	$\bm{Y}_{16}$	$\bm{Y}_{17}$	$\bm{Y}_{18}$	$\bm{Y}_{19}$	$\bm{Y}_{20}$
0.46	0.13	0.15	0.43	0.13	0.17	0.18	0.13	0.13	0.13	0.15	0.13	0.18	0.13	0.13	0.13	0.13	0.38	0.13	0.13
$\bm{Y}_{21}$	$\bm{Y}_{22}$	$\bm{Y}_{23}$	$\bm{Y}_{24}$	$\bm{Y}_{25}$	$\bm{Y}_{26}$	$\bm{Y}_{27}$	$\bm{Y}_{28}$	$\bm{Y}_{29}$	$\bm{Y}_{30}$	$\bm{Y}_{31}$	$\bm{Y}_{32}$	$\bm{Y}_{33}$	$\bm{Y}_{34}$	$\bm{Y}_{35}$	$\bm{Y}_{36}$	$\bm{Y}_{37}$	$\bm{Y}_{38}$	$\bm{Y}_{39}$	$\bm{Y}_{40}$
0.13	0.13	0.13	0.13	0.37	0.47	0.13	0.13	0.15	0.13	0.14	0.13	0.33	0.13	0.13	0.13	0.13	0.15	0.18	0.14
$\bm{Y}_{41}$	$\bm{Y}_{42}$	$\bm{Y}_{43}$	$\bm{Y}_{44}$	$\bm{Y}_{45}$	$\bm{Y}_{46}$	$\bm{Y}_{47}$	$\bm{Y}_{48}$	$\bm{Y}_{49}$	$\bm{Y}_{50}$	$\bm{Y}_{51}$	$\bm{Y}_{52}$	$\bm{Y}_{53}$	$\bm{Y}_{54}$	$\bm{Y}_{55}$	$\bm{Y}_{56}$	$\bm{Y}_{57}$	$\bm{Y}_{58}$	$\bm{Y}_{59}$	$\bm{Y}_{60}$
0.14	0.13	0.13	0.37	0.26	0.13	0.14	0.15	0.13	0.13	0.98	0.63	0.98	0.91	0.74	0.95	0.91	0.98	0.98	0.91
$\bm{Y}_{61}$	$\bm{Y}_{62}$	$\bm{Y}_{63}$	$\bm{Y}_{64}$	$\bm{Y}_{65}$	$\bm{Y}_{66}$	$\bm{Y}_{67}$	$\bm{Y}_{68}$	$\bm{Y}_{69}$	$\bm{Y}_{70}$	$\bm{Y}_{71}$	$\bm{Y}_{72}$	$\bm{Y}_{73}$	$\bm{Y}_{74}$	$\bm{Y}_{75}$	$\bm{Y}_{76}$	$\bm{Y}_{77}$	$\bm{Y}_{78}$	$\bm{Y}_{79}$	$\bm{Y}_{80}$
0.98	0.92	0.90	0.53	0.98	0.98	0.91	0.91	0.90	0.91	0.82	0.79	0.81	0.98	0.98	0.91	0.92	0.66	0.91	0.84
$\bm{Y}_{81}$	$\bm{Y}_{82}$	$\bm{Y}_{83}$	$\bm{Y}_{84}$	$\bm{Y}_{85}$	$\bm{Y}_{86}$	$\bm{Y}_{87}$	$\bm{Y}_{88}$	$\bm{Y}_{89}$	$\bm{Y}_{90}$	$\bm{Y}_{91}$	$\bm{Y}_{92}$	$\bm{Y}_{93}$	$\bm{Y}_{94}$	$\bm{Y}_{95}$	$\bm{Y}_{96}$	$\bm{Y}_{97}$	$\bm{Y}_{98}$	$\bm{Y}_{99}$	$\bm{Y}_{100}$
0.61	0.98	0.93	0.32	0.34	0.86	0.91	0.13	0.98	0.96	0.98	0.98	0.98	0.85	0.98	0.89	0.98	0.98	0.93	0.74

Figure 1.

Graphical Representation of Tables 2–3.

Table 4

Hurst exponent values given in the second step of the Section 3, test set, healthy ECG signals

$A$	$i$	$H_{\hat{x}_{i}}$	$H_{\hat{e}_{i}}$
00435	1	0.687	0.250
00436	2	0.726	0.053
00437	3	0.743	0.571
00440	4	0.692	0.294
00447	5	0.672	0.071
00448	6	0.692	0.290
00449	7	0.689	0.162
00452	8	0.768	0.480
00454	9	0.686	0.226
00455	10	0.730	0.275
00457	11	0.763	0.536
00458	12	0.763	0.524
00462	13	0.610	0.226
00463	14	0.757	0.520
00467	15	0.718	0.075
00469	16	0.677	0.264
00472	17	0.707	0.560
00475	18	0.658	0.208
00481	19	0.666	0.053
00483	20	0.731	0.521
00487	21	0.752	0.547
00488	22	0.746	0.544
00491	23	0.732	0.536
00492	24	0.770	0.509
00496	25	0.755	0.573
00501	26	0.733	0.497
00505	27	0.736	0.525
00508	28	0.783	0.566
00510	29	0.761	0.543
00516	30	0.779	0.536
00517	31	0.728	0.500
00518	32	0.726	0.528
00528	33	0.770	0.562
00532	34	0.677	0.068
00538	35	0.698	0.287
00540	36	0.737	0.557
00543	37	0.769	0.528
00552	38	0.728	0.506
00553	39	0.720	0.541
00555	40	0.731	0.529
00560	41	0.780	0.323
00564	42	0.725	0.542
00569	43	0.680	0.475
00571	44	0.722	0.204
00574	45	0.640	0.305
00575	46	0.730	0.494
00576	47	0.744	0.556
00580	48	0.663	0.074
00581	49	0.711	0.538
00582	50	0.663	0.047

Table 5

Hurst exponent values, see the second step of Section 3, test set, unhealthy ECG signals

$A$	$i$	$H_{\hat{x}_{i}}$	$H_{\hat{e}_{i}}$
00264	51	0.634	0.048
00493	52	0.770	0.430
00662	53	0.770	0.505
00725	54	0.642	0.032
01013	55	0.534	0.034
01019	56	0.695	0.040
01024	57	0.715	0.033
01041	58	0.723	0.038
01052	59	0.750	0.036
01087	60	0.736	0.507
01092	61	0.694	0.454
01104	62	0.688	0.037
01153	63	0.715	0.053
01163	64	0.785	0.526
01198	65	0.761	0.506
01230	66	0.794	0.524
01240	67	0.661	0.049
01268	68	0.703	0.542
01290	69	0.773	0.420
01302	70	0.563	0.079
01303	71	0.626	0.055
01313	72	0.688	0.037
01324	73	0.711	0.042
01342	74	0.605	0.052
01355	75	0.716	0.042
01371	76	0.702	0.523
01383	77	0.753	0.503
01433	78	0.822	0.542
01443	79	0.727	0.411
01449	80	0.559	0.027
01450	81	0.736	0.451
01501	82	0.780	0.511
01562	83	0.748	0.428
01584	84	0.660	0.014
01589	85	0.710	0.050
01642	86	0.625	0.516
01685	87	0.705	0.042
01691	88	0.662	0.031
01693	89	0.664	0.041
01753	90	0.757	0.448
01774	91	0.688	0.456
01776	92	0.673	0.046
01784	93	0.744	0.423
01828	94	0.546	0.071
01869	95	0.743	0.510
01883	96	0.793	0.373
01892	97	0.731	0.317
01930	98	0.713	0.050
01941	99	0.739	0.462
01986	100	0.756	0.034

Then, we apply the fifth step of the Section 3 to obtain the output values $\bm{Y}_{i},i\in\{1,2,\ldots,100\}$ presented by the Table 3. As we can see, by the Eq. (8), the accuracy of the proposed classifier parameter is of 97%. Figure 1 shows a graphical representation of the values presented by Tables 2–3.

To illustrate the generalizability of the method, we present to the classifier a test set consisting of 100 ECG signals whose the neural network did not have access to. In this case, the results are presented in the Tables 5–6. Figure 2 shows a graphical representation of these results. The classifier accuracy for the test set is 95%.

Table 6

Output values, test set, $\bm{Y}_{i},i\in\{1,2,\ldots,100\}$

$\bm{Y}_{1}$	$\bm{Y}_{2}$	$\bm{Y}_{3}$	$\bm{Y}_{4}$	$\bm{Y}_{5}$	$\bm{Y}_{6}$	$\bm{Y}_{7}$	$\bm{Y}_{8}$	$\bm{Y}_{9}$	$\bm{Y}_{10}$	$\bm{Y}_{11}$	$\bm{Y}_{12}$	$\bm{Y}_{13}$	$\bm{Y}_{14}$	$\bm{Y}_{15}$	$\bm{Y}_{16}$	$\bm{Y}_{17}$	$\bm{Y}_{18}$	$\bm{Y}_{19}$	$\bm{Y}_{20}$
0.27	0.40	0.35	0.21	0.32	0.20	$-$ 0.09	0.46	0.22	0.41	0.26	0.36	0.55	0.44	0.48	0.41	0.51	0.47	0.49	0.34
$\bm{Y}_{21}$	$\bm{Y}_{22}$	$\bm{Y}_{23}$	$\bm{Y}_{24}$	$\bm{Y}_{25}$	$\bm{Y}_{26}$	$\bm{Y}_{27}$	$\bm{Y}_{28}$	$\bm{Y}_{29}$	$\bm{Y}_{30}$	$\bm{Y}_{31}$	$\bm{Y}_{32}$	$\bm{Y}_{33}$	$\bm{Y}_{34}$	$\bm{Y}_{35}$	$\bm{Y}_{36}$	$\bm{Y}_{37}$	$\bm{Y}_{38}$	$\bm{Y}_{39}$	$\bm{Y}_{40}$
0.33	0.42	0.33	0.40	0.27	0.41	0.39	0.22	0.25	0.32	0.36	0.28	0.16	0.39	0.16	0.39	0.30	0.35	0.33	0.31
$\bm{Y}_{41}$	$\bm{Y}_{42}$	$\bm{Y}_{43}$	$\bm{Y}_{44}$	$\bm{Y}_{45}$	$\bm{Y}_{46}$	$\bm{Y}_{47}$	$\bm{Y}_{48}$	$\bm{Y}_{49}$	$\bm{Y}_{50}$	$\bm{Y}_{51}$	$\bm{Y}_{52}$	$\bm{Y}_{53}$	$\bm{Y}_{54}$	$\bm{Y}_{55}$	$\bm{Y}_{56}$	$\bm{Y}_{57}$	$\bm{Y}_{58}$	$\bm{Y}_{59}$	$\bm{Y}_{60}$
0.48	0.28	0.29	0.17	0.34	0.40	0.35	0.47	0.48	0.57	0.94	0.79	0.42	0.99	0.95	0.79	0.94	0.68	0.97	0.42
$\bm{Y}_{61}$	$\bm{Y}_{62}$	$\bm{Y}_{63}$	$\bm{Y}_{64}$	$\bm{Y}_{65}$	$\bm{Y}_{66}$	$\bm{Y}_{67}$	$\bm{Y}_{68}$	$\bm{Y}_{69}$	$\bm{Y}_{70}$	$\bm{Y}_{71}$	$\bm{Y}_{72}$	$\bm{Y}_{73}$	$\bm{Y}_{74}$	$\bm{Y}_{75}$	$\bm{Y}_{76}$	$\bm{Y}_{77}$	$\bm{Y}_{78}$	$\bm{Y}_{79}$	$\bm{Y}_{80}$
0.60	0.89	0.69	0.50	0.52	0.70	0.68	0.52	0.80	0.93	0.97	0.88	0.87	1.06	0.80	0.56	0.55	0.88	0.64	1.04
$\bm{Y}_{81}$	$\bm{Y}_{82}$	$\bm{Y}_{83}$	$\bm{Y}_{84}$	$\bm{Y}_{85}$	$\bm{Y}_{86}$	$\bm{Y}_{87}$	$\bm{Y}_{88}$	$\bm{Y}_{89}$	$\bm{Y}_{90}$	$\bm{Y}_{91}$	$\bm{Y}_{92}$	$\bm{Y}_{93}$	$\bm{Y}_{94}$	$\bm{Y}_{95}$	$\bm{Y}_{96}$	$\bm{Y}_{97}$	$\bm{Y}_{98}$	$\bm{Y}_{99}$	$\bm{Y}_{100}$
0.58	0.51	0.85	1.08	0.77	0.50	0.94	0.87	0.65	0.72	0.53	0.68	0.82	0.86	0.54	0.76	0.53	0.74	0.59	1.22

Figure 2.

Graphical representation of Tables 5–6.

Figure 3.

Graphical representation of the backpropagation net.

5. Conclusions

In this paper, we presented a new classifier parameter for the detection of atrial fibrillation of short-time ECG signals (between 30 and 60 seconds). This classifier parameter, based on the Detrended Fluctuation Analysis (DFA), SDCST method, and backpropagation net presented an accuracy of 97% and 95% for the training set and test set, respectively.

This article does not contain any studies with human participants or animals performed by any of the authors.

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Atrial fibrillation detection by DFA and SDCST methods

Abstract

Keywords

1. Introduction

2. Basic concepts

2.1 SDCST method

2.2 Detrended fluctuation analysis

3. Atrial fibrillation classifier

Table 3 Output values 𝒀 i , i ∈ { 1 , 2 , … , 100 }

References

Table 3
Output values $\bm{Y}_{i},i\in\{1,2,\ldots,100\}$