Electrocardiogram classification using delay differential equations

Abstract

Time series analysis with nonlinear delay differential equations (DDEs) reveals nonlinear as well as spectral properties of the underlying dynamical system. Here, global DDE models were used to analyze 5 min data segments of electrocardiographic (ECG) recordings in order to capture distinguishing features for different heart conditions such as normal heart beat, congestive heart failure, and atrial fibrillation. The number of terms and delays in the model as well as the order of nonlinearity of the model have to be selected that are the most discriminative. The DDE model form that best separates the three classes of data was chosen by exhaustive search up to third order polynomials. Such an approach can provide deep insight into the nature of the data since linear terms of a DDE correspond to the main time-scales in the signal and the nonlinear terms in the DDE are related to nonlinear couplings between the harmonic signal parts. The DDEs were able to detect atrial fibrillation with an accuracy of 72%, congestive heart failure with an accuracy of 88%, and normal heart beat with an accuracy of 97% from 5 min of ECG, a much shorter time interval than required to achieve comparable performance with other methods.

Figure 1

Error ρ and coefficient a for the linear DDE $\dot{x}=a\hspace{0.17em}{x}_{\tau }$ with $\tau =10\hspace{0.17em}\delta t$ vs. frequency f₃ for the signal $x(t)=\cos ({\omega }_{1}t)+\cos ({\omega }_{2}t)+\cos ({\omega }_{3}t)$ with $f_1=31\hspace{0.17em}\text{Hz}$ and $f_2=69\hspace{0.17em}\text{Hz}$ ($\omega =2\pi f$).

Figure 2

Error ρ and coefficient a for the linear DDE $\dot{x}=a\hspace{0.17em}{x}_{\tau }$ with $\tau =10\hspace{0.17em}\delta t$ vs. frequency $f_3$. White noise η was added to the signal: $D=\cos ({\omega }_{1}t)+\cos ({\omega }_{2}t)+\eta$, where the signal-to-noise ration SNR = −10 dB and $f_1=31\hspace{0.17em}\text{Hz}$ and $f_2=69\hspace{0.17em}\text{Hz}$ ($\omega =2\pi f$). The coefficient a and error ρ were then plotted for $x(t)=\mathcal{D}+\cos ({\omega }_{3}t)$ with f₃ varying between 0 and 150 Hz.

Figure 3

Error and coefficient a for the nonlinear DDE $\dot{x}=a\hspace{0.17em}{x}_{{\tau }_1}\hspace{0.17em}{x}_{{\tau }_2}$ with ${\tau }_1=5\hspace{0.17em}\delta t$ and ${\tau }_2=11\hspace{0.17em}\delta t$ vs. frequency f₃ for the signal $x(t)=\cos ({\omega }_{1}t)+\cos ({\omega }_{2}t)+\cos ({\omega }_{3}t)$ with $f_1=31\hspace{0.17em}\text{Hz}$ and $f_2=69\hspace{0.17em}\text{Hz}$ ($\omega =2\pi f$).

Figure 4

Distances d from the separating hyperplanes for all 5 min data windows for all 15 subjects for the three conditions (left plots) for the best DDE models reported in Tab. TABLE IV.. The mean value for each subject is shown as a black line. The histograms of these plots are shown on the right column. Blue refers to NSR, red to AF, and green to CHF. The black horizontal lines indicate the separating lines between the conditions selected by SVD. The vertical lines separate the subjects.

Figure 5

Receiver operating characteristic (ROC curves) and A' (area under the ROC curve) for the classifiers NSR vs. AF, NSR vs. CHF, AF vs. CHF, NSR vs. (AF, CHF), AF vs. (NSR,CHF), and CHF vs. (NSR,AF). The abbreviations N, A, and C in the legend correspond to NSR, AF, and CHF.