EEG Spike Detection Technique Using Output Correlation Method: A Kalman Filtering Approach

Abstract

This correspondence presents a technique for the electroencephalogram (EEG) spike enhancement and detection, which uses the Kalman filtering (KF) approach based on the output correlation method for the nonstationary signal enhancement. We describe the nonstationary EEG signal in terms of the general Markov model, in which the parameters are considered to be time-varying. In the proposed methodology, neither the process and measurement noise statistics nor the initial Kalman blending factor are stringently required. The EEG epileptic spikes (ESs) are pre-emphasized using the output correlation method, and subsequently, the detection is performed using the decision threshold based on the output of same adaptive filter. We have tested the proposed scheme on the synthetic EEG signal corrupted with randomly occurring triangular spikes. The presented simulation results manifest significant improvement in the signal-to-noise ratio (SNR) due to the modified estimation of time-varying parameters of the general Markov model, which in turn leads to the alleviated number of false-positives (FPs). It is apparent that the real-time EEG signal (rat data) can be analyzed using the proposed EEG epileptic spike enhancement and detection adaptive scheme, which outperforms the conventional KF technique under the different SNR conditions. At 10 dB SNR, the output correlation method provides approximately 40 % reduction in FPs for the triangular spikes in synthetic EEG signal and approximately 27.5 % reduction in FPs for ESs in the rat data as compared to the conventional KF scheme.

Appendices

Appendix 1

In the innovation correlation approach [22], it is apparent that

$$\begin{aligned} \vec {\beta }\left( {l;t+P} \right) =E\left[ {\zeta ^{-}\left( {t+P} \right) \zeta ^{-} \left( {t+P-l} \right) ^\mathrm{T}} \right] \end{aligned}$$

(66)

Substitution of (26) in the above equation results in

$$\begin{aligned} \vec {\beta }\left( {l\ne 0;t+P} \right)&= \vec {C}\left( {t+P} \right) E\left[ {\hat{{e}}^{-}\left( {t+P} \right) \hat{{e}}^{-}\left( {t+P-l} \right) ^\mathrm{T}} \right] \vec {C}^\mathrm{T}\left( {t+P} \right) \nonumber \\&+\, \vec {C}\left( {t+P} \right) E\left[ {\hat{{e}}^{-}\left( {t+P} \right) \vec {V}^\mathrm{T}\left( {t+P-l} \right) } \right] \end{aligned}$$

(67)

$$\begin{aligned} \vec {\beta }\left( {l=0;t+P} \right)&= \vec {C}\left( {t+P} \right) \hat{{P}}_e^- \left( {t+P} \right) \vec {C}^\mathrm{T}\left( {t+P} \right) +\vec {R}=\vec {\beta }\left( {t+P} \right) \end{aligned}$$

(68)

However, the solution of (67) requires $\hat{{e}}^{-}\left( {t+P} \right)$ and $\hat{{e}}^{-}\left( {t+P-l} \right)$. Therefore, by using equation (18) and (21), it can be shown that

$$\begin{aligned} \hat{{e}}^{-}\left( {t+P} \right)&= \vec {A}\left[ {I_P -\vec {K}\left( {t+P-1} \right) \vec {C}\left( {t+P-1} \right) } \right] \hat{{e}}^{-}\left( {t+P-1} \right) \nonumber \\&-\,\vec {A}\left[ {\vec {K}\left( {t+P-1} \right) \vec {V}\left( {t+P-1} \right) } \right] +\vec {W}\left( {t+P} \right) \end{aligned}$$

(69)

Further simplification of (69) is only possible under the static conditions, i.e., $\vec {K}\left( {t+P-1} \right) =\vec {K}\left( {t+P} \right) =\vec {K}_0$ (a priori sub-optimal Kalman gain) and $\vec {C}\left( {t+P-1} \right) =\vec {C}\left( {t+P} \right) =\vec {C}$, which leads to

$$\begin{aligned} \hat{{e}}^{-}\left( {t+P} \right)&= \left[ {\vec {A}\left\{ {I_P -\vec {K}_0 \vec {C}} \right\} } \right] ^{l}\hat{{e}}^{-}\left( {t+P-l} \right) \nonumber \\&+\,\sum _{j=1}^l {\left[ {\vec {A}\left\{ {I_P -\vec {K}_0 \vec {C}} \right\} } \right] ^{j-1}\vec {W}\left( {t+P-j+1} \right) }\nonumber \\&-\,\sum _{j=1}^l {\left[ {\vec {A}\left\{ {I_P -\vec {K}_0 \vec {C}} \right\} } \right] ^{j-1}\vec {A}\vec {K}_0 \vec {V}\left( {t+P-j} \right) } \end{aligned}$$

(70)

This method is not supporting the dynamic conditions arising in the nonstationary EEG signal. Moreover, this approach is extremely cumbersome.

Appendix 2

In the mathematical analysis of EEG signals, we need to calculate $\vec {R}_Y \left( {l;t+P} \right)$, such that

$$\begin{aligned} \vec {R}_Y \left( {l;t+P} \right) =E\left[ {\vec {Y}\left( {t+P} \right) \vec {Y}^\mathrm{T}\left( {t+P-l} \right) } \right] \end{aligned}$$

(71)

It can be calculated using the simple statistical approach [27] as

$$\begin{aligned} \hat{{R}}_Y \left( {l;t+P} \right) =\left( {1\big /{\left( {M+1} \right) }} \right) \sum _{m=l}^M {\vec {Y}\left( {t+P-m+l} \right) \vec {Y}^\mathrm{T}} \left( {t+P-m} \right) \end{aligned}$$

(72)

where the value of M is kept high to give

$$\begin{aligned} \hat{{R}}_Y \left( {l;t+P} \right)&= \left( {1\big /{\left( {M+1} \right) }} \right) \nonumber \\&\times \left[ {\begin{array}{l} \vec {Y}\left( {t+P} \right) \vec {Y}^\mathrm{T}\left( {t+P-l} \right) + \\ \sum _{\bar{{m}}=l}^{M-1} {\vec {Y}\left( {t+P-\bar{{m}}+l-1} \right) \vec {Y}^\mathrm{T}} \left( {t+P-\bar{{m}}-1} \right) \\ \end{array}} \right] \nonumber \\ \end{aligned}$$

(73)

However, it can be approximated as

$$\begin{aligned} \hat{{R}}_Y \left( {l;t+P} \right)&\approx \left[ {\frac{\vec {Y}\left( {t+P} \right) \vec {Y}^\mathrm{T}\left( {t+P-l} \right) }{M+1}} \right] \nonumber \\&\quad +\,\left[ {\hat{{R}}_Y \left( {l;t+P-1} \right) -\frac{\hat{{R}}_Y \left( {l;t+P-1} \right) }{M+1}} \right] \end{aligned}$$

(74)

$$\begin{aligned} \hat{{R}}_Y \left( {l;t+P} \right)&\approx \hat{{R}}_Y \left( {l;t+P-1} \right) \nonumber \\&\quad +\,\left[ {\frac{\vec {Y}\left( {t+P} \right) \vec {Y}^\mathrm{T}\left( {t+P-l} \right) -\hat{{R}}_Y \left( {l;t+P-1} \right) }{M+1}} \right] \end{aligned}$$

(75)