The Fourier transform is often used to calculate the third-order periodogram, which is an estimate of the bispectrum. However, it has energy leakage at adjacent frequencies, and the third-order periodogram is not a consistent estimator of the bispectrum because the variance does not converge toward zero with increasing sample size.

The Kolmogorov-Zurbenko Fourier Transform (KZFT) is an iteration of the regular Fourier transform. It can overcome the energy leakage by k degrees less where k is the number of iterations in the KZFT algorithm. To obtain a consistent spectrum estimation of a stationary time series, smoothing the periodogram across the frequency is often used. To obtain a consistent estimator of the bispectrum, a semi-adaptive smoothing algorithm is developed to smooth the third-order periodogram calculated with the KZFT. The shape of the spectral window is fixed as a square window. The bandwidth of the spectral window is determined by the optimal squared variation which is a pre-specified ad hop percentage of the total squared variation. The algorithm was applied using simulated data. It separated two close frequency components with very different energies in noisy data to estimate the bispectrum.

In the application, we developed an outlier detection algorithm for automated seizure detection. The data used was electroencephalogram (EEG) data recorded from patients with epilepsy. Using the training set, we converted the raw EEG data into frequency-based features based upon the spectrum, and used stepwise logistic regression for feature selection. In the resulting multi-dimensional feature space, we derived a frequency heuristic, defined as the direction from the center of the non-seizure data to the center of the seizure data. This direction heuristic was used with two other seizure heuristics: (1) a seizure is a sudden change from baseline activity; and (2) a seizure lasts at least 10 seconds. The algorithm was applied to 46 hours of intracranial EEG data and correctly identified eight out of nine seizures, with only one false positive.

Keywords: KZ statistics, KZFT, Third-order periodogram, Bispectrum, Smoothing, Optimal local squared variation, EEG, Seizure detection