Forecasting Concentrations from Filtered PK States

From a filtered state to a concentration forecast

A Kalman filter summarizes information through the current observation time \(t\) by a filtered state estimate and its uncertainty. Forecasting asks a different question: given that summary and a specified future dose or input, what concentrations are plausible at a later time? The answer requires propagating both the state mean and its covariance, then mapping the forecast state to the concentration of interest.

Write \(\mathcal Y_t\) for all observations available through time \(t\). Suppose the filtered state has conditional distribution \(X_t\mid\mathcal Y_t\sim N(m_{t|t},P_{t|t})\). A forecast is conditional on these observations and on the future inputs treated as known. In a linear Gaussian model, its mean and covariance are exact. With nonlinear dynamics or a nonlinear concentration mapping, a local linearization generally produces an approximation, not an exact distribution.

Linear Gaussian propagation

Consider a state transition over one forecast step,

Here \(A_k\) maps the current state to the next state, \(b_k\) contains known inputs such as a specified dose contribution, and \(Q_k\) is the process-noise covariance. Assume that, conditional on information at time \(t\), future process noises are mutually independent, independent of \(X_t\), and Gaussian with the stated covariances. These assumptions make the propagated state distribution Gaussian.

Proof. Conditional on \(\mathcal Y_t\), the state is Gaussian and independent of \(W_k\). An affine transformation of a Gaussian vector is Gaussian, and the sum of independent Gaussian vectors is Gaussian. Taking conditional expectations gives

For the covariance, the deterministic vector \(b_k\) contributes nothing, and independence makes the cross-covariance terms zero. Thus

This gives the stated conditional Gaussian distribution and completes the proof. \(\square\)

For several steps, apply the recursion at each step using the transition and inputs appropriate to that interval. For a constant transition \(A\), zero input, and constant process covariance \(Q\), repeated substitution gives

The covariance expression follows by induction: it holds at \(j=1\) by the one-step result. If it holds at \(j\), then \(P_{t+j+1|t}=AP_{t+j|t}A^{\mathsf T}+Q\). Substituting the expression at \(j\) gives the initial-covariance term \(A^{j+1}P_{t|t}(A^{j+1})^{\mathsf T}\) and the sum with powers \(r=1,\ldots,j\), plus the new \(Q\) term for \(r=0\). This is the formula at \(j+1\). The mean formula follows by applying \(m_{k+1|t}=Am_{k|t}\) repeatedly. \(\square\)

Suppose the latent concentration is a linear function of the state, \(C_k=H_kX_k+d_k\), where \(H_k\) is a row vector for a scalar concentration and \(d_k\) is known. Its conditional mean and variance are

If the future measurement is \(Y_k=C_k+V_k\), with independent \(V_k\sim N(0,R_k)\), its predictive variance is \(s^2_{C,k|t}+R_k\). Include \(R_k\) only when the target is a future assay result; it does not belong in the variance of the latent concentration itself.

has exactly 95% conditional coverage under that Gaussian model. Here \(\Phi\) is the standard normal distribution function. The commonly rounded multiplier 1.96 gives approximately 95% coverage, not exactly 95%.

Forecasting over multiple steps

A multi-step forecast must include process noise from every intervening transition. For time-varying transitions, the one-step recursion is usually safest to implement: use the appropriate \(A_k,b_k,Q_k\) at each step, then compute concentration from the resulting state distribution at the requested horizon. The accumulated noise contribution is transformed by all later transitions, so it need not have the same size at the final time as when it first entered the state.

Local linearization for nonlinear PK models

For a nonlinear transition \(X_{k+1}=f_k(X_k)+W_k\), a common forecast approximation expands \(f_k\) to first order around the current forecast mean \(m_{k|t}\). Define \(A_k=Df_k(m_{k|t})\), the Jacobian matrix of first derivatives. The local approximation is \(f_k(X_k)\approx f_k(m_{k|t})+A_k(X_k-m_{k|t})\). It gives

A nonlinear concentration function \(h(X)\) can be handled similarly: set \(H=Dh(m)\), where \(m\) and \(P\) are the forecast state mean and covariance at the target time. Then \(E[h(X)]\approx h(m)\) and \(\operatorname{Var}(h(X))\approx HPH^{\mathsf T}\). These are first-order approximations; they are not generally exact moments or guarantees of Gaussian prediction-interval coverage. For nonlinear dynamics, re-evaluate the Jacobian at the current forecast mean at each step.

Diagnostics, alternatives, and limitations

Process noise adds uncertainty at each step, but total forecast variance need not increase monotonically with horizon. The transition can contract uncertainty. For example, if \(A=0.5\), \(P_{t|t}=100\), and \(Q=0.1\), then the next variance is \(0.5^2(100)+0.1=25.1\), which is less than 100. Always inspect the propagated covariance rather than assuming that longer horizons necessarily produce wider intervals.

Check that covariance matrices remain symmetric and positive semidefinite, and verify that dose inputs enter at the correct event times. For a scalar concentration, inspect the forecast mean, standard deviation, and interval against plausible concentration scales. A Gaussian interval may extend below zero even when concentration must be nonnegative; that signals a limitation of the Gaussian approximation, not a physical negative concentration. In a nonlinear forecast, large curvature over the range of state uncertainty, strongly asymmetric simulated predictions, or sensitivity to the linearization point are warnings that a local approximation may be poor.

The main alternatives answer related but different needs. A linear Gaussian forecast gives exact conditional moments and intervals under its assumptions. A local linear forecast is computationally efficient for nonlinear models but may miss curvature and skewness. Simulation from the filtered state followed by propagation through the specified nonlinear model can represent non-Gaussian concentration predictions more directly, at additional computational cost. Whichever method is used, state clearly whether the interval concerns latent concentration or a future observation, and which future inputs are conditioned on.