tensorModelSolution

Solve the tensor noisy-data model with iteratively reweighted least squares.

Developer documentation: this item describes internal implementation details.

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Declaration

 [xi,CmInv,W] = tensorModelSolution(values,designMatrix,distribution,rho_X,Aeq,beq,Aineq,bineq)

Parameters

• values observation values as an N-by-1 vector
• designMatrix splines on the observation grid, N-by-M
• distribution distribution describing the errors
• rho_X optional observation correlation matrix
• Aeq optional equality-constraint matrix
• beq optional equality-constraint values
• Aineq optional inequality-constraint matrix
• bineq optional inequality-constraint values

Returns

• xi fitted tensor spline coefficients
• CmInv inverse coefficient covariance or system matrix
• W final weight matrix or weights

Discussion

This is the core fitting loop behind ConstrainedSpline. At each iteration it solves

\[\min_{\xi}\ (y - \mathbf{B}\xi)^{T} W^{(n)} (y - \mathbf{B}\xi)\]

subject to the supplied equality and inequality constraints, then updates the per-observation variances from the current residuals \(r^{(n)} = y - \mathbf{B}\xi^{(n)}\) through the distribution model.

When rho_X is supplied, the observation covariance is modeled as

\[\Sigma^{(n)}_{ij} = \sigma_i^{(n)} \rho_{ij} \sigma_j^{(n)},\]

and W represents the corresponding weighted solve rather than an explicitly formed inverse matrix.