Figure 1

Illustration model. The three figures in panel (a) show the dynamics and measurement realization for the small model used for illustration purpose. C(t) is measured and the dynamics of all states, i.e. A(t), B(t), and C(t), is intended to be predicted. Panel (b) shows as an example the prediction profile likelihood (gray dashed curve) and validation profile likelihood (black dashed curve) of A(t = 10). Thresholding yields confidence intervals for prediction (gray vertical lines) and validation (black vertical lines). The threshold and the respective projections correspond to the α = 90% confidence interval. The VCIs are larger than the PCIs, because they account for the measurement error of a validation data point. Panels (c)-(e) show prediction confidence intervals (gray) for the unobserved states A(t), B(t), as well as for the measured state C(t). The prediction profile likelihood functions are plotted as black curves in vertical direction. Non-observability is illustrated in panels (f)-(h). Panel (f) shows a realization of the measurements for a design which does not provide sufficient information about the steady state of C. This leads to a flat prediction profile likelihood for large values for A(t) as shown in panel (g), as well as for B(t) for t > 0 as plotted in panel (h). A flat prediction profile likelihood in turn yields unbounded prediction and validation confidence intervals and non-observability of A(t) and B(t) as indicated by the gray shaded regions.