Efficient syncope prediction from resting state clinical data using wavelet bispectrum and multilayer perceptron neural network

Abstract

Neurally mediated syncope (NMS) is the most common type of syncope, and head up tilt test (HUTT) is, so far, the most appropriate tool to identify NMS. In this work, an effort to predict the NMS before performing the HUTT is attempted. To achieve this, the heart rate variability (HRV) at rest and during the first minutes of tilting position during HUTT was analyzed using both time and frequency domains. Various features from HRV regularity and complexity, along with wavelet higher-order spectrum (WHOS) analysis in low-frequency (LF) and high-frequency (HF) bands were examined. The experimental results from 26 patients with history of NMS have shown that at rest, a time domain entropy measure and WHOS-based features in LF band exhibit significant differences between positive and negative HUTT as well as among 10 healthy subjects and NMS patients. The best performance of multilayer perceptron neural network (MPNN) was achieved by using an input vector consisted of WHOS-based HRV features in the LF zone and systolic blood pressure from the resting period, yielding an accuracy of 89.7%, assessed by 5-fold cross-validation. The promising results presented here pave the way for an early prediction of the HUTT outcome from resting state, contributing to the identification of patients at higher risk NMS.

Appendices

Appendix A

1.1 A.1 Wavelet higher-order spectral analysis

• 1) The continuous wavelet transform (CWT) is defined as [2]

  $$ \begin{array}{@{}rcl@{}} W_{x}(a,b)=\frac{1}{\sqrt{a}}{\int}_{-\infty}^{+\infty}x(t)\psi^{*}\left( \frac{t-b}{a}\right)dt, \end{array} $$

  (A1)

  where x(t) is the signal in time domain, * is the complex conjugate, and ψ(t) is the mother wavelet scaled by a factor α, α > 0 and dilated by a factor b.

• 2) The bispectrum is defined as [48]

  $$ \begin{array}{@{}rcl@{}} B(\omega_{1} , \omega_{2}) = E \lbrace X_{i}(\omega_{1}) X_{i}(\omega_{2}) X_{i}^{*}(\omega_{1} + \omega_{2})\rbrace, \end{array} $$

  (A2)

  where X(ω_i),i = 1,2, is the complex Fourier coefficient of the process {x(k)} at frequencies ω_i and X^∗(ω_i) is its complex conjugate [47].

• 3) The wavelet bispectrum is defined as [62]

  $$ WB_{x}(a_{1},a_{2})={\int}_{T}W_{x}^{*}(a,\tau) W_{x}(a_{1},\tau) W_{x}(a_{2},\tau)d\tau, $$

  (A3)

  where the integration is done over a finite time interval T : τ₀ ≤ τ ≤ τ₁ and α, α₁,α₂ satisfy the subsequent rule:

  $$ \begin{array}{@{}rcl@{}} \frac{1}{a}=\frac{1}{a_{1}}+ \frac{1}{a_{2}}. \end{array} $$

  (A4)

• 4) The instantaneous wavelet bispectrum IWBS can be defined as a time dependent complex quantity and is defined as [62]

  IWB_x(a₁,a₂,t)

  $$ \begin{array}{@{}rcl@{}} =|{IWB_{x}(a_{1},a_{2},t)}|e^{j\angle IWB_{x}(a_{1},a_{2},t)}=A_{x} e^{j \varphi_{x}}. \end{array} $$

  (A5)

• 5) The instantaneous wavelet biamplitude WBSAmp from which it is possible to infer the relative strength of the interaction given by [62]

  $$ \begin{array}{@{}rcl@{}} A_{x}(a_{1},a_{2},t)=|{IWB_{x}(a_{1},a_{2},t)}|. \end{array} $$

  (A6)

• 6) The maximum instantaneous wavelet biamplitude of the local peaks (LP) in the time interval t is defined as [60]

  $$ A_{x}^{{LP_{i}}}(\omega_{c_{1}} ,\omega_{c_{2}},t)=A_{x}^{{LP_{i}}}(\omega_{1} , \omega_{2},t)|_{A_{x}^{{LP_{i}}}(\omega_{1} , \omega_{2})=max}, $$

  (A7)

  and $c^{i}_{LP}=(\omega _{c_{1}} ,\omega _{c_{2}},t)^{i}, i=1,2,....,n$ is the position where peak i has its maximum value and n is the number of LP of each time interval.

Appendix B

1.1 B.1 Multilayer perceptron neural network (MPNN)

Multilayer perceptron neural networks (MPNNs) are feed-forward artificial neural networks with applications in pattern recognition and classification. The architecture of MPNN consists of three sequential layers: input, one or more hidden, and output layer. The input layer receives outward inputs and the hidden layer through neurons executes processing and transmission of the input information to the output, which the last produces the classification results. The neural network model is described from a nonlinear function with a set of input variables x_i and a set of output variables y_k, controlled by a vector w of adjustable parameters. The overall network function is [8]:

$$ y_{k}(x,w)=\sigma\left( \sum\limits_{j=1}^{M}w_{kj}^{(2)}h\left( \sum\limits_{j=1}^{D}w_{ji}^{(1)}x_{i}+w_{j0}^{(1)}\right)+w_{k0}^{(2)}\right), $$

(B1)

where $w_{ji}^{(1)}, w_{j0}^{(1)}$ are weights and bias similarly in the 1st layer, h(⋅) a differentiable nonlinear activation function, $w_{kj}^{(2)}, w_{k0}^{(2)}$ weights and bias in the 2nd layer, and $\sigma (a)=\frac {1}{1+e^{-a}}$ a sigmoid activation function for binary classification.

The information is sent alternately forwards and backwards through the network and this procedure is known as error back-propagation, which describes the training of MLP using gradient descent applied to a sum-of-squares error function [8]:

$$ \begin{array}{@{}rcl@{}} E(w)=\frac{1}{2}\sum\limits_{n=1}^{N}({y(x_{n},w)-t_{n}})^{2}, \end{array} $$

(B2)

where a training set comprising a set of input vectors {x_n},n = 1,...,N, together with a set of target vectors {t_n}. An iterative procedure for minimization of an error function is used through training algorithms to adjust the weights.