Parameter values for dynamical systems described by ODEs can be estimated from data using an objective function. (*A*) In the objective function, each unknown parameter of the ODE system corresponds to a dimension. The surface of the objective function resembles an energy landscape, with the altitude at each point denoting the goodness of fit of a specific set of parameters to data. Here, a three-dimensional slice through a complex objective function (corresponding to two parameters) shows numerous steep inclines/declines, local maxima/minima, and large areas where the objective function is independent of the two parameters displayed. (*B*) The deviation between points of synthetic data and model trajectories can be measured and used to evaluate the parameters. The effect of assuming perfect data means that there is a well-defined minimum that is the “true” parameter set, while the assumption of a variance means that the χ^{2} landscape has realistic values for its peaks and valleys. (*C*) The approximated surface of a particular valley in the complex landscape is shown in blue. (*D*) The curvature of the approximated surface area can be calculated as the second term of the Taylor expansion of the objective function, the Hessian. The eigenvectors of the Hessian represent the short and long axes of the paraboloid, and generally do not point in the direction of any single parameter. (*E*) Short eigenvectors indicate the direction of a steep parabola (large eigenvalue; red), and long eigenvectors indicate the direction of a shallow parabola (small eigenvalue; blue). (*F*) Moving in the direction of either eigenvector in parameter space has different consequences for model trajectories. Moving along a steep eigenvector of a Hessian leads to significant changes in the trajectory (red), while moving along the shallow eigenvector leads to only minor changes (blue), corresponding respectively to large and small changes in the values of the objective function.

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