Dense Sampling: What Becomes Visible and What Does Not

What Dense Sampling Adds

A concentration-time record is not the latent pharmacokinetic state itself. Write the observation at time \(t_i\) as \(Y_i=X(t_i)+\epsilon_i\), where \(X(t)\) is the changing concentration state and \(\epsilon_i\) is measurement error. Dense sampling gives many nearby observations of the evolving state, but it does not make the error in any one assay vanish. The central question is therefore not whether a dense record looks smoother, but which features of the record can be attributed to a time-varying state and which remain uncertain because of observation error.

The useful gain is temporal information: nearby measurements can collectively indicate the local level and movement of a continuous state. At the same time, a single observed jump between adjacent measurements can be dominated by two independent assay errors. A high sampling frequency is not, by itself, a label attached to each observed fluctuation.

Why Adjacent Changes Can Be Misleading

Suppose the state has locally diffusive variation: over a short interval of length \(\Delta\), its increment has variance approximately \(q\Delta\), where \(q\) is the local process-variation rate. Assume the state increment has mean zero for this calculation; a deterministic mean change can be added separately. The observed increment is

The two terms are independent under the assumptions, and the error difference has variance \(2\tau^2\). Thus, if the state increment variance is \(q\Delta\),

As sampling becomes denser, \(q\Delta\) shrinks while \(2\tau^2\) does not. Consequently, the raw changes between immediately adjacent observations can become increasingly dominated by assay error, even while the full sequence contains more information about the underlying trajectory. This distinction is essential: adjacent differences focus on a very short time scale, whereas estimation of the local state can use a neighborhood of observations.

Local Estimation of a Continuous State

A simple way to express how dense sampling can reveal a state value is to average observations in a shrinking time window around a fixed time \(t\). The window must shrink so that it does not blend together state values from a substantial time span. At the same time, it must contain an increasing number of observations so that independent assay errors average out. The following result makes those requirements precise.

Proof. Decompose the estimation error into a state component and an assay-error component:

For the first term, every included time satisfies \(|t_{i,n}-t|\le h_n\). Hence its absolute value is at most \(\sup_{|s-t|\le h_n}|X(s)-X(t)|\), which tends to zero almost surely by continuity at \(t\) and \(h_n\to0\). For a regular grid, the number \(m_n\) of points in the window tends to infinity because \(h_n/\delta_n\to\infty\). Independence and the common variance assumption give

Chebyshev’s inequality therefore implies that the second term converges in probability to zero. The first term also converges in probability to zero because it converges almost surely to zero. Their sum converges in probability to zero, proving the claim. \(\square\)

This is a consistency result at a fixed time, not a promise that every fluctuation in a finite record can be assigned correctly. The shrinking window balances two errors: a wide window can smooth over real temporal change, while a narrow window may contain too few observations to reduce assay noise. The theorem guarantees convergence under its assumptions; it does not supply an optimal window width for a particular data set.

Covariance Can Show What a Single Reading Cannot

At one sampling time, the observation is a sum of state and error. Its variance alone generally cannot determine how much comes from each source. Information across time can help because the state may be correlated across nearby times while independent assay errors are not. For example, suppose the state is stationary with covariance \(\operatorname{Cov}(X(t),X(t+s))=v\exp(-\lambda|s|)\), where \(v>0\) and \(\lambda>0\). With independent assay error of variance \(\tau^2\), the observation variance is \(v+\tau^2\), whereas the covariance between distinct observations separated by lag \(s>0\) is \(v\exp(-\lambda s)\). The assay contribution appears in the variance but not in this nonzero-lag covariance.

What Dense Sampling Does Not Identify

A single observed deviation cannot generally be classified as process variation or assay error from its size alone. At one time, for example, a normally distributed observation with variance 1 could be represented by independent state and error components with variances 0.8 and 0.2, or by components with variances 0.3 and 0.7. Both decompositions have the same marginal variance. Temporal dependence, repeated information, and scientifically defensible model assumptions are needed to distinguish the sources.

Even when a model separates their variance patterns, it estimates distributions or parameters; it does not turn each measured point into a known state value plus a known error value. A local estimate can make the state near a time more precise, but any particular assay error remains unobserved. Nor does a dense design resolve arbitrarily rapid state changes: the usable temporal resolution depends on the state dynamics, assay precision, and the time scale over which observations can be meaningfully smoothed.

The covariance interpretation also depends on the assumptions. If assay errors are serially correlated, their covariance can persist across nearby times and imitate state dependence. If error variance changes with concentration, a constant \(\tau^2\) is inadequate. If the state is not continuous, the local averaging result need not estimate its value at a jump. A dense sampling schedule cannot repair a misspecified observation model.

Practical Comparisons and Diagnostics

Sparse sampling can leave too few observations at nearby lags to characterize temporal dependence. Dense sampling supplies that local information, but interpreting every adjacent difference as a state change is a poor alternative: under independent assay error, the difference contains two error terms and retains variance \(2\tau^2\) even as the time interval shrinks. Local smoothing targets the state level instead, while covariance-based modeling uses dependence across lags to separate components under a specified model. These approaches answer related but different questions; none makes assumptions unnecessary.

Several checks help reveal when an interpretation is failing:

• Compare the variability of adjacent differences across sampling intervals. If very short-interval differences remain highly variable, that is consistent with an error contribution that does not shrink with the interval, though it is not by itself proof of independent assay error.
• Inspect estimated covariance across several nonzero lags, not just one. A proposed state covariance pattern should explain the observed decay or persistence across those lags.
• Repeat state estimation with more than one plausible window width. Large changes in the estimated trajectory indicate a resolution-versus-noise trade-off that should be reported rather than hidden.
• Check whether the assumed assay errors are plausibly independent and have stable variance. Correlated or concentration-dependent errors can distort the apparent temporal pattern.

The main limitation is that dense sampling improves temporal information only within the chosen observation and state models. It does not identify a unique decomposition from the observed curve without assumptions, and arbitrarily fine sampling can make raw adjacent changes less—not more—informative about process variation.