Regularisation

Simply stated regularisation is a method of translating an ill-posed problem into a more tractable well-posed one by approximation or constraints. Consider a discrete

linear system, with zero initial conditions,

y=F u, F ∈Rm×n (7.2)

where yis an output vector, length m,F a convolution matrix and uthe vector of the unknown input, of length n. If a noisy output of the system (ˆy) is available it would be expected that the predicted output y be sufficiently close to the measured output, accounting for noise ()

||y−yˆ||2 < . (7.3)

A naive solution to this, if F is square and invertible, is to ignore the presence of noise in the measured signal (ˆy) and to invert the convolution matrix to give an estimate for the input

ˆ

u=F−1yˆ; (7.4)

however, due to the noise present the solutions of this process are ill-conditioned, leading to negative values and oscillations. Alternatively, to recover the vector u a least squares minimisation approach could be employed

min

u ||yˆ−F u||2, F ∈R m×n

m > n (7.5)

whereuare parameters relating to the input function, not system parameters as would be seen in a standard parameter estimation procedure. This may still result in an ill- posed and ill-conditioned problem [Sparacino et al., 2001] as the optimisation process is free to alter the parameters, possibly into unfeasible regions, to obtain the closest fit to the measured output. To alleviate this problem a priori knowledge of the input function is used to form additional constraints on the solution in order to stabilise the deconvolved input. The constraints are formed by an additional function, that will provide a mapping from the input vector u to a scalar value which encapsulates thea priori information regarding the input signal, referred to asg(u) in the following text.

Section: 7.2.1 150 Using regularisation is beneficial when recovering biomedical signals due to the existence of two obvious constraints: Firstly, due to the distributive nature of drug kinetics the in vivo immune system response in terms of the synthesis of the immune complex would be ‘smooth’ over the measured response time. Secondly, the input signal is by definition positive, and a negative synthesis is already accounted for in the model by clearance.

In order to deconvolve the input vectoru from noisy data both the constraintg(u) and distance metric between the observered and simulated output f(u) need to be minimised, which is a constrained optimisation problem.

min

u f(u) subject to g(u) =c. (7.6)

It should be noted at this point that the object of this work is not to evaluate con- strained optimisation, nor evaluate the performance of algorithms to solve the problems specified. A successful outcome is a working prototype that demonstrates the use of deconvolution to highlight possible synthesis and chemotherapy effects. Nonetheless, the question remains as to the form the distance metric (f(u)) and constraint function (g(u)) should take. These will be explored in the following section.

7.2.1

Maximum Entropy signal recovery

The use of Maximum Entropy to recover signal data originates in the field of image analysis [Cornwell and Evans, 1985; Skilling and Bryan, 1984], where it is used to filter images to remove high-levels of noise. However, it has also been used in the recovery of biomedical signals [Charter and Gull, 1987]. An excellent review and comparison of methods can be found in Madden et al. [1995]. The ‘entropy’ in the title refers to the amount of uncertainty present in a signal. The concept behind Maximum Entropy is to produce an input signal that maximises the uncertainty, thus creating a signal that has the minimal assumptions whilst maintaining an acceptable fit to an observed output. If the input signal (u(t)) is discretised into piecewise function, the entropy of the signal can be used [Skilling and Bryan, 1984]

xi = ui Σui , S(x) = − N X i=1 xiln( xi ri ). (7.7)

where xi presents the normalised value of the input at the ith sample, and ri is a base-

line value that the production should take in the presence of no other information. The values ofxi are assumed to be positive, since a negative production is unfeasible. In the

above equations rn is calculated using a nearest neighbour average ((xi−1 +xi+1)/2).

This discourages adjacent elements of the input function from large variation, thus smoothing the recovered input signal. At the sample points i = 0 and i = N the average is taken of adjacent samples, e.g. (x0+x1)/2.

During the Maximum Entropy signal reconstruction the entropy of the input signal is maximised under the constraint that the output of the system should ‘match’ the real data measurements. A common form used to model this constraint is the χ2 metric. This is given by:

χ2 = N X i=1 (D1[i]−c1[i])2 σ2 i , E(χ2) = N (7.8)

where E denotes the expected value, N the number of samples, andD1[i] andc1[i] are

the measurements taken and the predicted value at the ith _{sample,t =τ}

i. This can be

seen as a weighted least squares estimator with weights of 1

σi.