Signal Processing

Bessel's Correction

For a known true mean value $\mu$, the variance of a data set $x_i$ can be estimated as

If the mean value is not know and the true mean value is replaced by the estimated mean derived from the same data set as

then the variance estimate above will be biased. For independent samples, the estimate

is unbiased, where the division by $N-1$ instead of $N$ is widely know as Bessel's correction.

Similar corrections can be made to estimates of the correlation function. Unfortunately, this requires considering that the data samples are correlated — why one would otherwise calculate the correlation function?

relevant literature:

• Vogelsang T J and Yang J (2016): Exactly/nearly unbiased estimation of autocovariances of a univariate time series with unknown mean. Journal of Time Series Analysis 37:723-740
• Nobach H (2017): Practical Realization of Bessel's Correction for a Bias-Free Estimation of the Auto-Covariance and the Cross-Covariance Functions.
• Nobach H (2023): Practical Realization of Bessel's Correction for a Bias-Free Estimation of the Auto-Covariance and the Cross-Covariance Functions. arXiv:2303.11047 [stat.ME]
• Damaschke N, Kühn V, Nobach H (2023): Bias-Free Estimation of the Auto- and Cross-Covariance and the Corresponding Power Spectral Densities from Gappy Data. arXiv:2304.13997 [eess.SP]
• Damaschke N, Kühn V, Nobach H (2024): Bias-Free Estimation of the Covariance Function and the Power Spectral Density from Data with Missing Samples Including Extended Data Gaps. Journal on Advances in Signal Processing, 2024, 17