Error estimates for minimum variance analysis

Analytical estimates of errors, associated with random statistical noise in magnetic field and other data to which the minimum/maximum variance analysis technique is commonly applied, are derived from first principles. A systematic expansion procedure is used in which the expansion parameter is proportional to the noise amplitude and inversely proportional to the square root of the number of vector data samples, K . The two special cases where the signal‐to‐noise ratio is large and small are considered for arbitrary noise distributions. The ideal case of small errors and isotropic Gaussian noise allows determination of uncertainty cones of elliptic cross section for all three eigenvectors, x 1 , x 2 , x 3 , and uncertainty intervals for all three eigenvalues, λ 1 , λ 2 , λ 3 , of the variance matrix, all in terms of the eigenvalues themselves and the number of data points. Denoting the angular standard deviation (in radians) of vector x i toward or away from vector x j by Δϕ ij these results are Δϕ ij = ±[λ 3 (λ i + λ j − λ 3 )/( K − 1)(λ i − λ j ) 2 ] 1/2 and Δλ i = ±[2λ 3 (2λ i − λ 3 )/( K − 1)] 1/2 where λ 3 is the smallest eigenvalue. The predictions from these formulas compare favorably with results from simple bootstrap numerical error analysis. Applications to maximum variance analysis of electric field data and to the study of anisotropic plasma pressure tensors are discussed as special cases.