MSE minimization without heuristics I propose you a solution for the - PDF document

Lesson 11 University of Bergamo Engineering and Management for Health FOR CHRONIC DISEASES MEDICAL SUPPORT SYSTEMS LESSON 11 Bioimaging analysis: analysis of an IVIM dataset to determine patient-specific parameters. Ettore Lanzarone April 15, 2020 MSE minimization without heuristics I propose you a solution for the MSE minimization which exploits the optimizer provided by MATLAB instead of coding an heuristic approach. The code is written under NO SEGMENTATION. Alternative cases (including segmentation )can be easily coded modifying this one. Results are unacceptable in some voxels, then to improve with segmentation or averaging approaches.

Lesson 11 MSE minimization without heuristics Bayesian perspective In case we need a distribution to compare confidence intervals of estimated parameters, the estimation approach can be performed with a Bayesian approach. Let’s see the approach for the segmented case in two cases: 1. Voxel-wise approach in which the estimation is independently performed within each voxel: independent likelihood for each voxel; independent Gaussian prior for each voxel 2. A conditional autoregressive approach to link neighbor voxel independent likelihood for each voxel; conditional autoregressive prior to link the voxels

Lesson 11 Bayesian perspective LIKELIHOOD FUNCTION The likelihood function is based, as usual, on the model structure. First step: Second step: Bayesian perspective LIKELIHOOD FUNCTION The likelihood function is based, as usual, on the model structure. First step: Second step:

Lesson 11 Bayesian perspective First step: Bayesian perspective

Lesson 11 Bayesian perspective Second step: Bayesian perspective

Lesson 11 Bayesian perspective PRIOR DENSITY The prior depends on the approach. As mentioned: 1. Voxel-wise approach in which the estimation is independently performed within each voxel: independent likelihood for each voxel; independent Gaussian prior for each voxel 2. A conditional autoregressive approach to link neighbor voxel independent likelihood for each voxel; conditional autoregressive prior to link the voxels Bayesian perspective Gaussian prior: First step Second step

Lesson 11 Bayesian perspective Conditional autoregressive prior: Bayesian perspective

Lesson 11 Practical lesson For the same real dataset I provided you, I give you the Bayesian code (R script and RSTAN model) both the Gaussian and the CAR approach. I put in the folder also the output files to have an idea of the results with a high number of iterations.