MinimumVarianceAnalysis

The main purpose of minimum or maximum variance analysis (MVA) is to find an estimator for the direction normal $\hat{𝐧}$ to one-dimensional structure, by minimisation of

$$ σ^2=\frac{1}{M} ∑_{m=1}^M | (𝐁^{(m)}-⟨𝐁⟩) · \hat{𝐧}|^2. $$

Quickstart

using Pkg; Pkg.add("MinimumVarianceAnalysis")
using MinimumVarianceAnalysis
using MinimumVarianceAnalysis: normal, normal_component_error

Bl = cosd.(0:30:180)
Bm = sind.(0:30:180)
Bn = 0.1 .* ones(7)
B = hcat(Bl, Bm, Bn)  # rows are samples

F = mva_eigen(B; dim = 1)    # MVAResult: values (λ₁ ≥ λ₂ ≥ λ₃), eigenvector columns `vectors`, mean, n, field
normal(F)                    # minimum-variance direction for magnetic-field MVA
issuccess(F)                 # normal well determined if |λ₂/λ₃| > r0 (default 5)
normal_component_error(F)    # statistical error of ⟨B⟩·normal
mva(B; dim = 1)              # rotate samples into variance frame
mva(B, F; dim = 1)           # rotate any data (e.g. velocity) into F's frame

For electric-field maximum variance analysis, pass field = :E:

E = hcat(0.1 .* ones(7), 0.05 .* ones(7), cosd.(0:30:180))
F = mva_eigen(E; dim = 1, field = :E)
normal(F)    # maximum-variance direction; issuccess uses |λ₁/λ₂|

Containers that carry their times (DimArray, SpaceDataModel's DataVariable) default to their time dimension. Unitful, DynamicQuantities and FlexUnits inputs are supported:

using Unitful

F = mva_eigen(B .* u"nT"; dim = 1)
F.values    # eigenvalues carry squared field units
F.vectors   # eigenvectors are dimensionless directions

Reference

• Sonnerup, B. U. Ö., & Scheible, M. (1998). Minimum and maximum variance analysis. ISSI Scientific Reports Series, 1, 185–220.

Features and Roadmap

• Maximum Variance Analysis on Magnetic Field (MVAB)
  • Constraint $〈B₃〉 = 0$
• Maximum Variance Analysis on Electric Field (MVAE)
  • Transformation to/from moving frame of reference
• Minimum Variance Analysis on Mass Flux (MVAρv)
• Application to 2-D Structures

Notes

• Anisotropic fluctuations have been shown to lead to larger errors in normal determinations.