Global Sensitivity of Time Courses and Functional Outputs

Why the Output Representation Matters

A QSP model can produce a trajectory rather than one scalar prediction: for example, a simulated biomarker concentration \(Y(t;X)\) over time, with \(X\) denoting uncertain model inputs. A global sensitivity analysis asks how variation in those inputs contributes to variation in an output. For a trajectory, however, “the output” might mean its value at one time, an integral over a time window, or a score representing a pattern across the whole curve. These are different outputs, and their sensitivity measures need not give the same parameter ranking.

The practical choice is not merely a matter of presentation. A time-integrated summary can cancel an early increase against a later decrease. Principal-component compression can preserve most overall trajectory variance while discarding a smaller, biologically relevant transient pattern. A defensible analysis therefore defines its output representation before interpreting sensitivity rankings, and checks whether the representation retains the dynamics relevant to the question.

Three Ways to Represent a Time Course

Suppose the model produces \(Y(t;X)\) on a time interval \([0,T]\), and the uncertain inputs have specified distributions. For the Sobol decompositions below, assume the inputs \(X_1,\ldots,X_d\) are independent, the model output has finite variance, and the model is evaluated consistently for the input values being compared. The Sobol indices describe variation under these chosen input distributions; they are not distribution-free properties of the model.

For any scalar output \(Z(X)\) with positive variance, the first-order Sobol index for \(X_i\) is the fraction of output variance explained by conditioning on \(X_i\) alone. The total-order index is the fraction attributable to \(X_i\) and all its interactions with other inputs. These definitions use independent inputs.

Here \(X_{-i}\) denotes all inputs except \(X_i\). The first-order index does not include variance due only to interactions involving \(X_i\), whereas the total-order index does. An index is undefined if the output has zero variance. For a pointwise analysis, apply these definitions separately to \(Z=Y(t_k;X)\) at each chosen time. For an integral or a score, apply them to that scalar output instead. The resulting indices are dimensionless, but they describe different questions.

Pointwise and Integrated Sensitivity

Pointwise indices show how input importance changes over time. They can reveal that one parameter matters mainly during an early response while another matters during a later phase. They also depend on the selected time grid: sparse sampling may miss a short-lived peak or reversal. Plotting indices across time is often more informative than reporting only a ranking averaged over the grid.

A functional output such as an area under the curve (AUC) is useful when the scientific question concerns accumulated exposure or total response. Its definition must specify the interval and weighting. For example, \(Z(X)=\int_0^T Y(t;X)\,dt\) has units of the output multiplied by time. A weighted integral may emphasize a time window, but the weights should be chosen for a scientific reason rather than to manufacture a desired ranking.

A key feature of an integral is that changes at different times can reinforce or cancel. To see why its sensitivity cannot generally be obtained by simply integrating pointwise indices, define \(m(t)=\operatorname{E}[Y(t;X)]\) and \(f_i(t,X_i)=\operatorname{E}[Y(t;X)\mid X_i]-m(t)\). Then

and the first-order variance component for \(X_i\) in that integral is

The latter expression includes covariance between the input’s conditional effects at different times. It is not, in general, the integral of the pointwise first-order variance components. The same issue applies to total-order effects. Thus, the sensitivity of an integrated output must be calculated for that integrated output, not inferred by averaging pointwise indices.

Principal-Component Scores as Sensitivity Targets

A trajectory can be sampled at \(K\) times and represented by a vector \(\mathbf{Y}(X)=(Y(t_1;X),\ldots,Y(t_K;X))^\mathsf{T}\). Principal-component analysis (PCA) identifies orthogonal directions of variation in such vectors. In a global sensitivity analysis, the PCA basis is typically estimated from centered trajectories generated over the chosen input distribution. For a unit-length component direction \(\mathbf{v}_r\), its score is \(Z_r(X)=\mathbf{v}_r^\mathsf{T}(\mathbf{Y}(X)-\boldsymbol{\mu})\), where \(\boldsymbol{\mu}=\operatorname{E}[\mathbf{Y}(X)]\). Sobol indices can then be computed for each score.

A score can summarize a coordinated pattern across time, such as a rise followed by a fall. But PCA ranks directions by their contribution to total trajectory variance, not by their biological relevance or by how strongly a particular parameter affects them. A low-variance component can represent a short-lived transient that is essential to the question. Sensitivity of retained scores answers questions about those scores; it does not automatically establish that omitted parts of the trajectory are insensitive.

Practical Checks and Failure Modes

The representation and numerical calculation should be checked together. For a pointwise analysis, inspect the time grid around fast changes, peaks, and reversals. For an integral, verify that the integration interval, units, and numerical quadrature reflect the intended quantity. For PCA, report the centering, time grid, scaling, and retained variance, and inspect whether the discarded components contain patterns of interest. Scaling matters: PCA on raw measurements can be dominated by time points or state variables with larger numerical scales.

Estimate the indices using a consistent global sensitivity method and assess Monte Carlo error, especially for small indices or near-ties in rankings. Numerical ODE error can also distort small transient effects; compare solver tolerances or refine the output grid when the result is sensitive to numerical resolution. Use the same input distributions and trajectory processing in comparisons, since changing either can change the sensitivity conclusions.

The main alternatives answer complementary questions. Selected-time analyses retain direct temporal interpretation but require a deliberate choice of times and may produce many rankings. Integrated summaries give a compact answer for a specified aggregate but can conceal timing and cancellation. PCA scores summarize dominant patterns without requiring a hand-chosen scalar summary, but depend on the sampled trajectory ensemble, scaling, and retention rule. None is universally preferable: a robust assessment may report pointwise indices alongside a scientifically motivated functional output and use PCA as a complementary description.

• Do not average pointwise Sobol indices and call the result an AUC index. Compute the index from the integrated output because temporal covariance and cancellation affect its variance.
• Do not equate variance retained by PCA with sensitivity retained. A discarded low-variance component may be strongly controlled by a parameter and may encode a transient of interest.
• Do not interpret a time-course ranking without its input distributions. The indices describe parameter effects under the stated distributions and can change when those distributions change.
• Do not overlook time discretization. A sparse grid or inaccurate quadrature can erase a transient before the sensitivity calculation begins.