**6. Statistical analysis of ECG detection methods**

ECG signals are predominantly non-Gaussian (Rizk and Zgallai, 1999 and Rizk et al., 1995), and exhibit quadratic and higher-order non-linearities supported by third- and fourth-order statistics, respectively. ECG signals do contain measurable quantities of quadratic and cubic non-linearities. Such quantities if not synthesised and removed before any further processing for the purpose of signal identification and classification could lead to poor performance with regard to fetal QRS-complex detection rates. The non-linearity in the ECG signal can be detected using the bicoherence squared. The bicoherence squared has peaks at the frequency pairs of (6 Hz,15 Hz) and (14 Hz,14 Hz) for the fetal scalp cardiac cycle, (15 Hz,15 Hz) for the maternal chest cardiac cycle, and (7.5 Hz,7.5 Hz) for the maternal transabdominal cardiac cycle. These bicoherence peaks support non-linearity (Zgallai, 2007). There is a general consensus that individual cardiac cycles are locally stationary. However, when applying a highly dimensional signal such as the transabdominal ECG that have several individual non-linear and deterministic signals overlapping both in the time and frequency domains, all coexisting in a cocktail of noise and motion artefact, it is prudent to re-examine the validity of the stationarity assumption in relation to such signals. It is only natural to expect that the proximity of two non-linear signals such as the maternal and fetal QRS-complexes would result in non-linear (quadratic and higher-order) coupling and this in turn would invoke non-stationarity. The bispectral OT region is insepcted (Nikias and Petropulu, 1993) for the maternal bispectral contour maps at a level of -30 dB. When the two R-waves of the maternal and fetal QRS-complexes are separated by 200 msec, the resultant bispectrum does not support the OT region (Zgallai, 2007). However, the situation is totally different when the two R-waves are as close as 35 msec. The OT region of the bispectrum is fully occupied and non-stationary (Zgallai, 2007). Hence, conventional signal processing techniques to separate the maternal and fetal QRS-complexes cannot be used. This problem has been adequately solved by linearising (at least removing quadratic coupling) the transabdominal signal before attempting to separate individual QRS-complexes.

Correlartion-based second-order statistics do not show any distinguishable features that could be used to differentiate between maternal QRS-complex, fetal heartbeat with maternal contribution, and QRS-free ECG contributions. The FFT method reveals a fetal scalp electrode ECG principal spectral peak at 30 Hz (Zgallai, 2007). The FFT method for the transabdominal cardiac cycle reveals the maternal principal spectral peak of 15 Hz (Zgallai, 2007). However, the FFT does not clearly show fetal spectral peak from the segmented transabdominal signal. There could be a shallow peak as low as 28 Hz or a shifted peak as high as 42 Hz (Zgallai, 2007).

Statistical analysis of ECG data, including Pearson's correlation analysis and higher order moments have been carried out (Rizk and Zgallai, 1999). The value of Pearson's productmoment correlation coefficient for both the third-order cumulant and the bispectral contour method is within the range of -0.1 to +0.1.

The Receiver Operating Characeteristics (ROC) analysis has been used to statistically analyse the results of the two propsoed detection methods, third-order cunulant and bispectral contour, compared to the second-order statistics method. The Area Under Curve (AUC) has been used as a measure for diagnostic accuracy and discriminating power. The second-order statistics-based, third-order cumulant slice, and the bispectral controue methods have AUC values of 0.731, 0.794, and 0.843, respectively. This suggests that the third-order cumulant is a better detection method than second-order statistics-based, and that the bispectral contour method outperforms the third-order cumulant method.

Youden's index, defined as sensitivty + specificity – 1, has also been used for the detection methods. The second-order statistics-based methods have indices in the range of 0.42 to 0.55. The third-order cumulant method has an index of 0.72. The bispectral contour method has an index of 0.80. This suggests that the third-order cumulant is a better detection method than second-order statistics-based, and that the bispectral contour method outperforms the third-order cumulant method.

Also, the Partial Area Under Curve (PAUC) measure has been used for a False-Positive Rate (FPR) of 10% and sensitivty larger than 75%. The second-order statistics-based method gives a PAUC of 0.043. The third-order cumulant method has a corresponding value of 0.125 whilst that of the bispectral contour method is 0.137. This suggests that the third-order cumulant is a better detection method than second-order statistics-based, and that the bispectral contour method outperforms the third-order cumulant method.
