THE PROHOROV METRIC FRAMEWORK AND AGGREGATE DATA INVERSE PROBLEMS FOR RANDOM PDEs.

We consider nonparametric estimation of probability measures for parameters in problems where only aggregate (population level) data are available. We summarize an existing computational method for the estimation problem which has been developed over the past several decades [24, 5, 12, 28, 16]. Theoretical results are presented which establish the existence and consistency of very general (ordinary, generalized and other) least squares estimates and estimators for the measure estimation problem with specific application to random PDEs. AMS Subject Classification: 62G07, 34A55, 46S50, 93E24

[1]  Harvey Thomas Banks,et al.  Use of difference-based methods to explore statistical and mathematical model discrepancy in inverse problems , 2016 .

[2]  R A Albanese,et al.  Physiologically based pharmacokinetic models for the transport of trichloroethylene in adipose tissue , 2002, Bulletin of mathematical biology.

[3]  I. Gary Rosen,et al.  Blind deconvolution for distributed parameter systems with unbounded input and output and determining blood alcohol concentration from transdermal biosensor data , 2014, Appl. Math. Comput..

[4]  G. Wahba.  Splines in Nonparametric Regression , 2006 .

[5]  Suzanne S. Sindi,et al.  Estimating the rate of prion aggregate amplification in yeast with a generation and structured population model , 2018 .

[6]  G. Wahba.  Bayesian "Confidence Intervals" for the Cross-validated Smoothing Spline , 1983 .

[7]  C. Wang,et al.  Deconvolving an estimate of breath measured blood alcohol concentration from biosensor collected transdermal ethanol data , 2008, Appl. Math. Comput..

[8]  Susan E Luczak,et al.  Development of a real-time repeated-measures assessment protocol to capture change over the course of a drinking episode. , 2015, Alcohol and alcoholism.

[9]  H T Banks,et al.  Probabilistic methods for addressing uncertainty and variability in biological models: application to a toxicokinetic model. , 2004, Mathematical biosciences.

[10]  R. Swift,et al.  Direct measurement of alcohol and its metabolites. , 2003, Addiction.

[11]  P. Bushnell,et al.  A physiologically based pharmacokinetic model for trichloroethylene in the male long-evans rat. , 2002, Toxicological sciences : an official journal of the Society of Toxicology.

[12]  Sophia Blau,et al.  Analysis Of The Finite Element Method , 2016 .

[13]  H. T. Banks,et al.  Parameter estimation techniques for interaction and redistribution models: a predator-prey example , 1987, Oecologia.

[14]  Philip Gerlee,et al.  Travelling wave analysis of a mathematical model of glioblastoma growth. , 2013, Mathematical biosciences.

[15]  H. Banks.  A Functional Analysis Framework for Modeling, Estimation and Control in Science and Engineering , 2012 .

[16]  Harvey Thomas Banks,et al.  A comparison of approximation methods for the estimation of probability distributions on parameters , 2007 .

[17]  P. M. Prenter.  Splines and variational methods , 1975 .

[18]  Nathan Louis Gibson,et al.  Terahertz-Based Electromagnetic Interrogation Techniques for Damage Detection , 2004 .

[19]  Bruce G. Lindsay,et al.  A review of semiparametric mixture models , 1995 .

[20]  D. Ba Santa,et al.  'Go or Grow': the key to the emergence of invasion in tumour progression? , 2010 .

[21]  H. Banks,et al.  Estimation of nonlinearities in parabolic models for growth, predation, and dispersal of populations , 1989 .

[22]  Yuanyuan Liang,et al.  Use of continuous transdermal alcohol monitoring during a contingency management procedure to reduce excessive alcohol use. , 2014, Drug and alcohol dependence.

[23]  L. Sheiner,et al.  Modelling of individual pharmacokinetics for computer-aided drug dosage. , 1972, Computers and biomedical research, an international journal.

[24]  R. Swift,et al.  Transdermal measurement of alcohol consumption. , 1993, Addiction.

[25]  Marie Davidian,et al.  Nonlinear models for repeated measurement data: An overview and update , 2003 .

[26]  Harvey Thomas Banks,et al.  EXISTENCE AND CONSISTENCY OF A NONPARAMETRIC ESTIMATOR OF PROBABILITY MEASURES IN THE PROHOROV METRIC FRAMEWORK , 2015 .

[27]  H. T. Banks,et al.  Well-posedness in Maxwell systems with distributions of polarization relaxation parameters , 2005, Appl. Math. Lett..

[28]  Marie Davidian,et al.  Smooth nonparametric maximum likelihood estimation for population pharmacokinetics, with application to quinidine , 1992, Journal of Pharmacokinetics and Biopharmaceutics.

[29]  I. G. Rosen,et al.  Estimating the distribution of random parameters in a diffusion equation forward model for a transdermal alcohol biosensor , 2018, Autom..

[30]  Harvey Thomas Banks,et al.  Experimental design and estimation of growth rate distributions in size-structured shrimp populations , 2009 .

[31]  I. Gary Rosen,et al.  Approximation and convergence in the estimation of random parameters in linear holomorphic semigroups generated by regularly dissipative operators , 2017, 2017 American Control Conference (ACC).

[32]  Shuhua Hu,et al.  Modeling shrimp biomass and viral infection for production of biological countermeasures. , 2006, Mathematical biosciences and engineering : MBE.

[33]  Santosh Kesari,et al.  Malignant gliomas in adults. , 2008, The New England journal of medicine.

[34]  J. Murray,et al.  Virtual and real brain tumors: using mathematical modeling to quantify glioma growth and invasion , 2003, Journal of the Neurological Sciences.

[35]  Olivier Clatz,et al.  Biocomputing: numerical simulation of glioblastoma growth using diffusion tensor imaging , 2008, Physics in medicine and biology.

[36]  Yuanyuan Liang,et al.  Transdermal alcohol concentration data collected during a contingency management program to reduce at-risk drinking. , 2015, Drug and alcohol dependence.

[37]  H. Banks,et al.  Estimating intratumoral heterogeneity from spatiotemporal data , 2018, Journal of mathematical biology.

[38]  K. Painter,et al.  Mathematical modelling of glioma growth: the use of Diffusion Tensor Imaging (DTI) data to predict the anisotropic pathways of cancer invasion. , 2013, Journal of theoretical biology.

[39]  H. Banks,et al.  Estimation of variable coefficients in parabolic distributed systems , 1985 .

[40]  Nancy P Barnett,et al.  A Multimodal Investigation of Contextual Effects on Alcohol’s Emotional Rewards , 2018, Journal of abnormal psychology.

[41]  J. Murray,et al.  A quantitative model for differential motility of gliomas in grey and white matter , 2000, Cell proliferation.

[42]  Zheng Dai,et al.  Using drinking data and pharmacokinetic modeling to calibrate transport model and blind deconvolution based data analysis software for transdermal alcohol biosensors. , 2016, Mathematical biosciences and engineering : MBE.

[43]  Harvey Thomas Banks,et al.  Modeling and Imaging Techniques with Potential for Application in Bioterrorism , 2003 .

[44]  A. Mallet.  A maximum likelihood estimation method for random coefficient regression models , 1986 .

[45]  Yang Kuang,et al.  Mathematically modeling the biological properties of gliomas: A review. , 2015, Mathematical biosciences and engineering : MBE.

[46]  Tracy L. Stepien,et al.  TRAVELING WAVES OF A GOOR-GROW MODEL OF GLIOMA 1 GROWTH , 2018 .

[47]  Marie Davidian,et al.  Nonlinear Models for Repeated Measurement Data , 1995 .

[48]  Hervé Delingette,et al.  Realistic simulation of the 3-D growth of brain tumors in MR images coupling diffusion with biomechanical deformation , 2005, IEEE Transactions on Medical Imaging.

[49]  H. T. Banks,et al.  A Probabilistic Multiscale Approach to Hysteresis in Shear Wave Propagation in Biotissue , 2005, Multiscale Model. Simul..

[50]  Karl Kunisch,et al.  Cubic spline approximation techniques for parameter estimation in distributed systems , 1983 .

[51]  H. Banks,et al.  Quantifying uncertainty in the estimation of probability distributions. , 2008, Mathematical biosciences and engineering : MBE.

[52]  Tracy L. Stepien,et al.  A data-motivated density-dependent diffusion model of in vitro glioblastoma growth. , 2015, Mathematical biosciences and engineering : MBE.

[53]  Hervé Delingette,et al.  Towards an Identification of Tumor Growth Parameters from Time Series of Images , 2007, MICCAI.

[54]  B. Lindsay.  The Geometry of Mixture Likelihoods: A General Theory , 1983 .

[55]  Laura Kay Potter,et al.  Physiologically Based Pharmacokinetic Models for the Systemic Transport of Trichloroethylene , 2001 .

[56]  A. Schumitzky.  Nonparametric EM Algorithms for estimating prior distributions , 1991 .

[57]  A. Gallant,et al.  Nonlinear Statistical Models , 1988 .

[58]  Dahyeon Kang,et al.  Deconvolving the input to random abstract parabolic systems: a population model-based approach to estimating blood/breath alcohol concentration from transdermal alcohol biosensor data , 2018, Inverse problems.

[59]  R. Guillevin,et al.  Simulation of anisotropic growth of low‐grade gliomas using diffusion tensor imaging , 2005, Magnetic resonance in medicine.

[60]  Harvey Thomas Banks,et al.  Estimation of growth rate distributions in size structured population models , 1991 .

[61]  David Frakes,et al.  Mathematical Analysis of Glioma Growth in a Murine Model , 2017, Scientific Reports.

[62]  B. Lindsay.  Mixture models : theory, geometry, and applications , 1995 .

[63]  Sabine Fenstermacher,et al.  Estimation Techniques For Distributed Parameter Systems , 2016 .

[64]  Michael Berens,et al.  A mathematical model of glioblastoma tumor spheroid invasion in a three-dimensional in vitro experiment. , 2007, Biophysical journal.

[65]  H. Banks,et al.  Mathematical and Experimental Modeling of Physical and Biological Processes , 2001 .

[66]  R. Swift,et al.  Transdermal alcohol measurement for estimation of blood alcohol concentration. , 2000, Alcoholism, clinical and experimental research.

[67]  I. Rosen,et al.  Estimation of the Distribution of Random Parameters in Discrete Time Abstract Parabolic Systems with Unbounded Input and Output: Approximation and Convergence. , 2018, Communications in applied analysis.

[68]  Harvey Thomas Banks,et al.  Modeling and Inverse Problems in the Presence of Uncertainty , 2014 .

[69]  Marie Davidian,et al.  The Nonlinear Mixed Effects Model with a Smooth Random Effects Density , 1993 .

[70]  D M Bortz,et al.  Incorporation of variability into the modeling of viral delays in HIV infection dynamics. , 2003, Mathematical biosciences.

[71]  A. Laconti,et al.  Studies on a wearable, electronic, transdermal alcohol sensor. , 1992, Alcoholism, clinical and experimental research.

[72]  Harvey Thomas Banks,et al.  Modeling and estimating uncertainty in parameter estimation , 2001 .

[73]  Zackary R. Kenz,et al.  A review of selected techniques in inverse problem nonparametric probability distribution estimation , 2012 .

[74]  D. Dougherty,et al.  Comparing the detection of transdermal and breath alcohol concentrations during periods of alcohol consumption ranging from moderate drinking to binge drinking. , 2012, Experimental and clinical psychopharmacology.

[75]  Harvey Thomas Banks,et al.  Quantitative modeling of growth and dispersal in population models , 1987 .

[76]  H. Banks,et al.  Electromagnetic inverse problems involving distributions of dielectric mechanisms and parameters , 2006 .

[77]  D. A. Labianca,et al.  The chemical basis of the Breathalyzer: A critical analysis , 1990 .