Generalised smoothing in functional data analysis

The incorporation of model-based penalties in a penalised regression frame- work (generalised smoothing) has been the subject of many publications, most notably: Cao and Ramsay (2007); Heckman and Ramsay (2000); Ramsay and Silverman (2005); Ramsay et al. (2007). Generalised smooth- ing facilitates the estimation of the parameters of an ordinary di erential equation (ODE) from noisy data without the speci cation of an explicit expression of the functional entity described by the ODE. This is a notable consequence of the smoothing procedure however it is not its primary aim. Generalised smoothing aims to obtain an estimated functional entity that adheres to the data and incorporates domain speci c information de ned by the ODE. The existing methodology for the estimation of the param- eters in generalised smoothing is hindered by the absence of an explicit expression in terms of the parameters of the ODE for the penalty within penalised tting criterion. The aim of this research is to obtain this ex- plicit expression for penalties de ned by B{spline basis functions in order to facilitate the development of the estimation procedure. The recursive algorithm developed by de Boor (2001) is the predominant methodology for the evaluation of B-spline basis functions over a given in- terval. While this algorithm is a fast and numerically stable method for nding a point on a B-spline curve given the domain, it does not explicitly provide knowledge of the internal structure of the B-spline functions. This work introduces an alternative representation of B{spline basis functions in terms of the underlying polynomials that comprise the B{spline. This alterative representation of B{spline basis functions produces generalised penalties which can be written explicitly in terms of the parameters of the ODE. A joint estimation strategy in which the penalised least squares cri- terion is minimised with respect to the parameters of the B-spline and the parameters of the ODE is developed. Finally this joint estimation strat- egy is shown to produce estimates of both parameters that have a higher accuracy and are more computationally e cient than estimates developed by existing methods

A comparison of the experimental and simulated data after BHQ880 treatment.

<p>A comparison of the experimental and simulated data after BHQ880 treatment.</p

Schematic of the multi-scale modeling of OBs, OCs and MMs.

<p><b>Intracellular scale</b>: describes the communication among myeloma cells, osteoclasts and osteoblasts and their ‘phenotypic’ switches. <b>Intercellular scale</b>: describes the dynamics of molecules in signaling pathways for each cell after receiving cytokine stimulation from other cells and the specific migration rules for cells. <b>Tissue scale</b>: describes the diffusion of drugs and cytokines.</p

3D snapshots of the tumor system with single-agent BHQ880 treatment at (a) time step 40, (b) time step 50 and (c) time step 120.

<p>3D snapshots of the tumor system with single-agent BHQ880 treatment at (a) time step 40, (b) time step 50 and (c) time step 120.</p

The effect of BHQ880 on the number of OBs, OCs and MMs.

<p>The effect of BHQ880 on the number of OBs, OCs and MMs.</p

The effect of GCs on the number of OBs, OCs and MMs.

<p>The effect of GCs on the number of OBs, OCs and MMs.</p

The effect of BHQ880 + GCs, BHQ880 + Lidamycin,GCs + Lidamycin, and BHQ880 + GCs + Lidamycin on the number of OBs, OCs and MMs.

<p>The effect of BHQ880 + GCs, BHQ880 + Lidamycin,GCs + Lidamycin, and BHQ880 + GCs + Lidamycin on the number of OBs, OCs and MMs.</p

The effects of BHQ880, GCs and Lidamycinon restoring the balance between OCs and OBs and killing MMs.

<p>The effects of BHQ880, GCs and Lidamycinon restoring the balance between OCs and OBs and killing MMs.</p

3D snapshots of the tumor system with single-agent GC treatment at (a) time step 60, (b) time step 65 and (c) time step 120.

<p>3D snapshots of the tumor system with single-agent GC treatment at (a) time step 60, (b) time step 65 and (c) time step 120.</p

The dynamics of OBs, OCs and MMs without drug treatment.

<p>The dynamics of OBs, OCs and MMs without drug treatment.</p