# A Maximum Entropy Approach for Predicting Epileptic Tonic-Clonic Seizure

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Method

**x**from a dynamical system D : ℝ

^{S}→ ℝ

^{S}, the corresponding time series consists of a sequence of measurements {v(t

_{n}), n = 1, ··· ,N} on a system considered to be in a state described by

**x**(t

_{n}) ∈ ℝ

^{S}at discrete times t

_{n}, where N is the length of the time series.

_{i}(t) = v(t − (i − 1)Δ), for i = 1, ··· ,d. Δ is the time lag and d is the embedding dimension of the reconstruction. T represents the anticipation time and it is of fundamental importance for a prediction model.

_{k}≤ d with 1 ≤ k ≤ n

_{p}and n

_{p}being an adequately chosen polynomial degree so as to series-expand the mapping F

^{*}. The number of parameters in Equation (3) corresponding to the terms of degree k depends on the embedding dimension and can be calculated using combination with repetitions,

**a**, adopts the form

_{c}, whose n-th row is [1,v

_{i}

_{1}(t

_{n}),v

_{i}

_{1}(t

_{n})v

_{i}

_{2}(t

_{n}), ··· ,v

_{i}

_{1}(t

_{n})v

_{i}

_{2}(t

_{n}), ··· ,v

_{inp}(t

_{n})] (cf. Equation (3)) and (

**v**

_{T})

_{n}= v(t

_{n}+ T), for n = 1, ··· ,M. In this work, we use the maximum entropy principle to characterize important probability distributions. Shannon’s entropy, defined for a discrete random variable, can be extended to situations for which the random variable under consideration is continuous.

**a**is realized with probability P(

**a**). Thus, a normalized probability distribution over the possible sets

**a**is introduced,

**a**= da

_{1}da

_{2}·· da

_{Nc}and N

_{c}is the number of parameters of the model.

**a**) subject to the requirement that the associated entropy H be maximized, since this is the best way of avoiding any bias. The expectation value of

**a**is defined by

**a**with probability density function p(

**a**) on I, where I = (−∞,∞), the entropy is given by

**a**〉 of

**a**. The method for solving this constrained optimization problem is to use Lagrange multipliers for each of the operating constraints and maximize the following functional with respect to P(

**a**),

_{0}and λ are Lagrange multipliers associated, respectively, with the normalization condition and with the constraints, i.e., Equation (9) and Equation (8).

**a**) we get

_{0}(

**a**) is chosen to be proportional to exp( $\text{exp}(-{\scriptstyle \frac{1}{2}}{\mathbf{a}}^{t}{[{\sigma}^{2}]}^{-1}\mathbf{a})$), where

**σ**

^{2}is the covariance matrix, a Gaussian form for the probability distribution P(

**a**) is obtained, with

^{t}(WW

^{t})

^{−1}is known as the Moore–Penrose pseudo-inverse of the matrix W (see [7] and references therein). Consequently, this result shows that the maximum entropy principle coincides with a least square criterion. Once the pertinent parameters are determined, they are used to predict M

_{P}

**new**series’ values,

_{P}×N

_{c}.

## 3. Application to EEG Time Series

#### 3.1. Clinical Data and Experimental Setup

#### 3.2. Maximum Entropy Prediction

_{p}= 3. The frequency sample is Δ = 9.765 × 10

^{−3}Hz. We considered 3000 series values, which corresponds to approximately 30 s, to obtain the parameter vector

**a**. Using a d = 2 embedding dimension, the number of estimated parameters is N

_{c}= 14. We consider an anticipation time T ≈ 1 s, corresponding to 60 samples.

_{s}= 81.5 s) one second before it happens.

## 4. Conclusions

## Acknowledgments

## Conflicts of Interest

## References

- Rosso, O.A.; Martín, M.T.; Plastino, A. Brain electrical activity analysis using wavelet-based informational tools (II): Tsallis non-extensivity and complexity measures. Physica A
**2003**, 320, 497–511. [Google Scholar] - Takens, F. Dynamical Systems and turbulence. In Dynamical Systems and Turbulence, Warwick 1980; Proceedings of a Symposium Held at the University of Warwick 1979/80, Coventry, UK, 1979, Rand, D., Young, L.-S., Eds.; Lecture Notes in Mathematics, Volume 898; Springer: Berlin/Heidelberg, Germany, 1981. [Google Scholar]
- Diambra, L.; Plastino, A. Modelling time series using information theory. Phys. Lett. A
**1996**, 216, 278–282. [Google Scholar] - Shannon, C.E. A Mathematical Theory of Communication. Bell Syst. Tech. J
**1948**, 27. [Google Scholar] - Jaynes, E.T. Information Theory and Statistical Nechanics II. Phys. Rev
**1957**, 108, 171–190. [Google Scholar] - Mallat, S. A Wavelet Tour of Signal Processing, 2nd ed.; Academic Press: San Diego, CA, USA, 1999. [Google Scholar]
- Martín, M.T.; Plastino, A.; Vampa, V.; Judge, G. A parametric information-theory model for predictions in time series. Physica A
**2014**, 405, 63–69. [Google Scholar] - Rosso, O.A.; Martín, M.T.; Plastino, A. Evidence of self-organization in brain electrical activity using wavelet-based informational tools. Physica A
**2005**, 347, 444–464. [Google Scholar]

**Figure 1.**Scalp EEG signal for a generalized epileptic tonic-clonic seizure. Preseizure stage (0–81 s), seizure start (81 s), tonic-clonic phase (82–155 s), seizure end (155 s), and postseizure (155–180 s).

© 2014 by the authors; licensee MDPI, Basel, Switzerland This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/3.0/).

## Share and Cite

**MDPI and ACS Style**

Martín, M.T.; Plastino, A.; Vampa, V.
A Maximum Entropy Approach for Predicting Epileptic Tonic-Clonic Seizure. *Entropy* **2014**, *16*, 4603-4611.
https://doi.org/10.3390/e16084603

**AMA Style**

Martín MT, Plastino A, Vampa V.
A Maximum Entropy Approach for Predicting Epileptic Tonic-Clonic Seizure. *Entropy*. 2014; 16(8):4603-4611.
https://doi.org/10.3390/e16084603

**Chicago/Turabian Style**

Martín, Maria Teresa, Angelo Plastino, and Victoria Vampa.
2014. "A Maximum Entropy Approach for Predicting Epileptic Tonic-Clonic Seizure" *Entropy* 16, no. 8: 4603-4611.
https://doi.org/10.3390/e16084603