MinimumVarianceAnalysis

Estimate the direction normal $𝐧̂$ to a one-dimensional structure using minimum or maximum variance analysis (MVA).

Main Functions: mva, mva_eigen, normal

Estimate the direction normal $𝐧̂$ to a one-dimensional structure using minimum or maximum variance analysis (MVA).

Main Functions: mva, mva_eigen, normal

Calculate the composite statistical error estimate for ⟨B·x₃⟩: |Δ⟨B·x₃⟩| = √(λ₃/(M-1) + (Δφ₃₂⟨B⟩·x₂)² + (Δφ₃₁⟨B⟩·x₁)²)

Parameters:

• λ₁, λ₂, λ₃: eigenvalues in descending order
• M: number of samples
• B: mean magnetic field vector
• x₁, x₂, x₃: eigenvectors

E_x1_error(λ₁, λ₂, λ₃, M, E, x₁, x₂, x₃)

Calculate the composite statistical error estimate for ⟨E·x₁⟩ (the mean electric field along the maximum variance / normal direction):

$|Δ⟨\mathbf{E}·\mathbf{x}_1⟩| = \sqrt{\frac{λ_1}{M-1} + (Δφ_{12}⟨\mathbf{E}⟩·\mathbf{x}_2)^2 + (Δφ_{13}⟨\mathbf{E}⟩·\mathbf{x}_3)^2}$

Parameters:

• λ₁, λ₂, λ₃: eigenvalues in descending order
• M: number of samples
• E: mean electric field vector
• x₁, x₂, x₃: eigenvectors

check_mva_eigen(F; r0=5, verbose=false, field = :B)

Check the quality of the MVA result.

If λ₁ ≥ λ₂ ≥ λ₃ are 3 eigenvalues of the constructed matrix M. For MVAB, a good indicator of nice results should have $|λ₂ / λ₃| > r₀$ (default $r₀ = 5$).

For MVAE, a reliable normal direction requires the maximum eigenvalue $λ₁$ to be well-separated from the intermediate eigenvalue $λ₂$. The ratio $|λ₁ / λ₂| > r₀$ is used as a quality indicator.

convection_efield(v, B; dim=nothing)

Compute the convection electric field $\mathbf{E} = -\mathbf{v} × \mathbf{B}$ from plasma velocity v and magnetic field B.

This can be used as a proxy for the measured electric field when direct measurements are unavailable.

mva(V, F=V; dim=nothing, kwargs...)

Transform a timeseries V into the LMN coordinate system based on the minimum/maximum variance analysis of reference field F along the dim dimension (time).

See also: mva_eigen, transform

mva_eigen(x::AbstractMatrix; dim = nothing, sort=(;), check=false) -> F::Eigen

Perform minimum variance analysis of the magnetic field B or maximum variance analysis of the electric field E when field=:E.

x varies along the dim dimension.

Return Eigen factorization object F which contains the eigenvalues in F.values and the eigenvectors in the columns of the matrix F.vectors. The kth eigenvector can be obtained from the slice F.vectors[:, k].

Set check=true to check the reliability of the result.

Notes

For a one-dimensional current layer, the tangential electric field components are approximately constant across the boundary, while the normal component exhibits the largest variation. Therefore, the eigenvector corresponding to the maximum eigenvalue $λ_1$ (first column of F.vectors) gives an estimate of the boundary normal direction.

normal(F::Eigen; field=:B)

Return the boundary normal eigenvector from an MVA result.

• field=:B (MVAB): minimum variance direction (last eigenvector)
• field=:E (MVAE): maximum variance direction (first eigenvector)

transform(A, mat::AbstractMatrix; dim=nothing)

Transform A into a new coordinate system using transformation matrix mat along the dim dimension (time).

Δφij(λᵢ, λⱼ, λ₃, M)

Calculate the phase error between components i and j according to: |Δφᵢⱼ| = |Δφⱼᵢ| = √(λ₃/(M-1) * (λᵢ + λⱼ - λ₃)/(λᵢ - λⱼ)²)

Parameters:

• λᵢ: eigenvalue i
• λⱼ: eigenvalue j
• λ₃: smallest eigenvalue (λ₃)
• M: number of samples

Δφij(λᵢ, λⱼ, λ₃, M)

Calculate the phase error between components i and j according to: |Δφᵢⱼ| = |Δφⱼᵢ| = √(λ₃/(M-1) * (λᵢ + λⱼ - λ₃)/(λᵢ - λⱼ)²)

Parameters:

• λᵢ: eigenvalue i
• λⱼ: eigenvalue j
• λ₃: smallest eigenvalue (λ₃)
• M: number of samples

Calculate the composite statistical error estimate for ⟨B·x₃⟩: |Δ⟨B·x₃⟩| = √(λ₃/(M-1) + (Δφ₃₂⟨B⟩·x₂)² + (Δφ₃₁⟨B⟩·x₁)²)

Parameters:

• λ₁, λ₂, λ₃: eigenvalues in descending order
• M: number of samples
• B: mean magnetic field vector
• x₁, x₂, x₃: eigenvectors

E_x1_error(λ₁, λ₂, λ₃, M, E, x₁, x₂, x₃)

Calculate the composite statistical error estimate for ⟨E·x₁⟩ (the mean electric field along the maximum variance / normal direction):

$|Δ⟨\mathbf{E}·\mathbf{x}_1⟩| = \sqrt{\frac{λ_1}{M-1} + (Δφ_{12}⟨\mathbf{E}⟩·\mathbf{x}_2)^2 + (Δφ_{13}⟨\mathbf{E}⟩·\mathbf{x}_3)^2}$

Parameters:

• λ₁, λ₂, λ₃: eigenvalues in descending order
• M: number of samples
• E: mean electric field vector
• x₁, x₂, x₃: eigenvectors