Fig 1.
(A) The tangent at the optimum is perpendicular to its residual vector . Boundaries of the model manifold, shown in orange, are marked by black segments. In the parameterization by the model parameter θ the boundaries are reached in the limit θ → ±∞. (B) The values of χ2 on the interval [−∞, ∞] are shown as graph, illustrating the boundedness of the function.
Fig 2.
(A) The shape of the landscape is visualized by solid and dashed contour lines for the original and Riemann approximated χ2, respectively. The colored lines represent paths that are optimal with respect to the parameter k1 for any given value of parameter A0. (B) The non-quadratic χ2 turns into a quadratic function in Riemannian Normal Coordinates. The paths computed for (A) are shown in the new coordinates as colored lines. (C) The χ2 values along the exact and the approximated parameter paths agree well indicating that confidence intervals derived from either objective function coincide. Thresholds for different confidence levels are depicted in gray.
Fig 3.
The profile likelihood for the parameter k3 and initial amount S indicates finite confidence intervals according to both, the original (yellow) and Riemann-approximated (red) χ2.
The estimated parameter values are depicted as black dots. Practically non-identifiable parameters k1 and k2 do not even exceed the 68% confidence thresholds towards large values, correctly recognized by the approximation.