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Verotta (1996) provides a detailed review of deconvolution, with a focus on non- parametric methods. The most basic method, quadrature, assumes that the input function is piecewise constant between measurements. The relation between input and measurements can be expressed as a system of linear equations. Since the number of unknowns is equal to the number of equations, the system has a unique solution, provided that it is invertible. While very straightforward, this method does have obvious problems. An input function that is constant between measurements may not be a very realistic model. Additionally, by forcing exact agreement with the data, the method becomes sensitive to noise. This is especially true for systems with lowpass characteristics, which attenuate high frequencies. Any high-frequency noise will become amplified when such a system is inverted. A counterintuitive consequence of this is that a densely sampled dataset is more sensitive to noise than a more sparsely sampled dataset, since the dense dataset can represent higher frequencies. One way to overcome this is to model the input function as being piecewise constant on a set of intervals that are fewer than the number of measurements. This results in an overdetermined linear system, for which the least-squares solution can be obtained in closed form using the Moore-Penrose pseudoinverse (Penrose 1955). This helps to decrease noise sensitivity, at the expense of constraining the functional form of the input function even more. A more realistic-looking function can be obtained by using differentiable basis functions such as B-splines (Boor 1986). Regardless of the choice of basis functions, a closed-form solution is available.

An alternative to the quadrature method, suggested by Verotta (1996), is to usepenalised least-squares, where the methods above are modified by adding a regularisation (penalisation) term, as described in Section 2.2.1. This makes it possible to have a more fine-grained input function parameterisation, and hence this method can be described as nonparametric. A finite difference approximation of the first or second derivative of the input function can be constructed, and its sum of squares can

be used as a regularisation term. Penalisation can be done either on the basis function coefficients or on the function itself. For piecewise constant functions, this makes little difference, but for B-splines, this amounts to making different assumptions. Since the finite difference approximation is a linear operator, the estimation problem is still a quadratic optimisation problem, which has a closed-form solution. It is also possible to add constraints to these problems, that disallow solutions that attain negative values. In addition to making the solution more physiologically plausible, it can also have a regularising effect, as it prevents the input function from oscillating between large positive and negative values. When constraints are added to the problem, it no longer has a closed-form solution. However, it is a convex optimisation problem, for which efficient methods with strong guarantees of finding the optimum exist (Boyd and Vandenberghe 2004).

Verotta (1996) also mentions that penalised least-squares has a Bayesian interpretation, as explained in Section 2.2.2. In this view, the regularisation term is interpreted as a quantity that is proportional to the negative log-prior. Adding a regularisation term, penalising a discrete-time approximation of thejth derivative, is equivalent to modelling the input function as a cascade ofj integrators, driven by white noise referred to as the process noise. The continuous-time setting is conceptually similar, although such an input function would have to be modelled using tools from stochastic calculus (Klebaner 2012; Øksendal 2003) in order to be mathematically rigorous. Seeking least-squares solutions also means that there is an implicit assumption that the measurement noise has a Gaussian distribution. Non-negativity constraints can be interpreted as assigning a prior probability density of zero for any solution that takes negative values.

Similar ideas have been discussed by Sparacino and Cobelli (1996). The application under consideration is the task of estimating insulin secretion rate after a intravenous glucose tolerance test. The input function is modelled as a piecewise constant function, with a considerably larger number of intervals than the number of measurements. To make the estimation problem well-defined, a regularisation term penalising the first derivative is added, equivalent to treating the input function as a Gaussian random walk. By making a Bayesian interpretation, the optimal regularisation parameter can be shown to be the ratio of the measurement noise variance and the process noise variance. In this setting, it is possible to derive optimal settings for the regularisation parameter based on ideas presented by MacKay (1992), using the expected weighted residual sum of squares, and the expected weighted sum of squares of the function estimate. The process and measurement noise variance is then set so that the observed values are equal to the estimated ones. Sparacino and

Cobelli (1996) also mention the possibility of using splines, but consider this to be a parametric method.

De Nicolao et al. (1997) discuss similar nonparametric methods, and also discuss their similarities to discrete-time Kalman filtering. It is noted that while these methods have similar statistical justifications, a drawback with Kalman filtering is that it is not straightforward to add constraints. Furthermore, nonuniform sampling causes the discretised system to be time-varying.