Batch Mean Batch Correlation (BMBC) Theory

This page explains the theoretical foundations of the Batch Mean Batch Correlation (BMBC) algorithm for estimating the variance of the mean estimator in correlated time series.

Introduction

The BMBC algorithm is designed to compute the uncertainty of empirical mean estimators when dealing with correlated data, particularly in time series analysis. Traditional methods assume independent and identically distributed (i.i.d.) samples, but real-world data often exhibits temporal correlation that must be accounted for in uncertainty quantification.

Theoretical Background

For a time series \(Y_1, Y_2, \ldots, Y_N\) with temporal correlation, the empirical mean is:

\[\bar{Y} = \frac{1}{N} \sum_{i=1}^N Y_i\]

The variance of this mean estimator \(\sigma^2_{\mu}\) is not simply \(\sigma^2/N\) as in the i.i.d. case, but must account for the correlation structure.

Batch Means Method

The batch means method divides the time series into \(K\) batches of size \(M\):

\[K = \left\lfloor \frac{N}{M} \right\rfloor\]

For each batch \(b\) (where \(b = 1, 2, \ldots, K\)), we compute the batch mean:

\[\bar{Y}_b = \frac{1}{M} \sum_{i=(b-1)M + 1}^{bM} Y_i\]

The variance between batch means provides an estimate of the variance of the overall mean.

Batch Correlation Analysis

The BMBC algorithm extends the batch means method by analyzing the correlation between consecutive batches. We compute two key quantities:

  1. S₀ - Independent variance component:

\[S_0 = \frac{1}{K-1} \sum_{b=1}^K (\bar{Y}_b - \bar{\bar{Y}})^2\]

  1. S₁ - Correlated variance component:

\[S_1 = \frac{1}{K-1} \sum_{b=1}^{K-1} (\bar{Y}_b - \bar{\bar{Y}})(\bar{Y}_{b+1} - \bar{\bar{Y}})\]

where \(\bar{\bar{Y}}\) is the mean of the batch means.

Optimal Batch Size Selection

The algorithm adaptively selects the batch size \(M\) to achieve a target ratio \(S_1/S_0\) that indicates appropriate decorrelation between batches. The target range is typically between 0.2 and 0.6.

Variance Estimation

The final variance estimate of the mean is:

\[\hat{\sigma}^2_{\mu} = \frac{1}{(K-1)(K-2)} (S_0 + 2S_1)\]

This estimator accounts for both the independent and correlated components of variance.

Block Bootstrap

For uncertainty quantification, the algorithm provides a block bootstrap method that resamples entire batches rather than individual observations, preserving the correlation structure.

References

The BMBC algorithm is based on the following reference [BMBC2023].

[BMBC2023]

Russo, S., Luchini, P., 2017. A fast algorithm for the estimation of statistical error in DNS (or experimental) time averages. Journal of Computational Physics 347, 328–340. https://doi.org/10.1016/j.jcp.2017.07.005

Implementation Details

The OpenTURNS implementation provides:

  • Adaptive batch size selection

  • Automatic correlation detection

  • Block bootstrap resampling

  • Integration with OpenTURNS statistical framework

See Also