Integrability, neural networks, and the empirical modelling of dynamical systems Oscar Garcia forestgrowth.unbc.ca.

Slides:

Advertisements

Similar presentations

Design of Experiments Questions Network Inference Working Group October 8, 2008.

Advertisements

Notes Sample vs distribution “m” vs “µ” and “s” vs “σ” Bias/Variance Bias: Measures how much the learnt model is wrong disregarding noise Variance: Measures.

2806 Neural Computation Self-Organizing Maps Lecture Ari Visa.

A.M. Alonso, C. García-Martos, J. Rodríguez, M. J. Sánchez Seasonal dynamic factor model and bootstrap inference: Application to electricity market forecasting.

Discrete Choice Model of Bidder Behavior in Sponsored Search Quang Duong University of Michigan Sebastien Lahaie

Sensorimotor Transformations Maurice J. Chacron and Kathleen E. Cullen.

Adjoint Orbits, Principal Components, and Neural Nets Some facts about Lie groups and examples 2.Examples of adjoint orbits and a distance measure 3.Descent.

Quantification of Nonlinearity and Nonstionarity Norden E. Huang With collaboration of Zhaohua Wu; Men-Tzung Lo; Wan-Hsin Hsieh; Chung-Kang Peng; Xianyao.

Professor Walter W. Olson Department of Mechanical, Industrial and Manufacturing Engineering University of Toledo Lumped Parameter Systems.

Human Body Drug Simulation Nathan Liles Benjamin Munda.

TNO orbit computation: analysing the observed population Jenni Virtanen Observatory, University of Helsinki Workshop on Transneptunian objects - Dynamical.

The catchment: empirical model NRML10 Lena - Delta Andrea Castelletti Politecnico di Milano.

A Short Introduction to Curve Fitting and Regression by Brad Morantz

Differential Flatness Jen Jen Chung. Outline Motivation Control Systems Flatness 2D Crane Example Issues Jen Jen Chung | CDMRG2.

PROCESS INTEGRATED DESIGN WITHIN A MODEL PREDICTIVE CONTROL FRAMEWORK Mario Francisco, Pastora Vega, Omar Pérez University of Salamanca – Spain University.

Markov processes in a problem of the Caspian sea level forecasting Mikhail V. Bolgov Water Problem Institute of Russian Academy of Sciences.

D Nagesh Kumar, IIScOptimization Methods: M1L1 1 Introduction and Basic Concepts (i) Historical Development and Model Building.

INTEGRATED DESIGN OF WASTEWATER TREATMENT PROCESSES USING MODEL PREDICTIVE CONTROL Mario Francisco, Pastora Vega University of Salamanca – Spain European.

Chapter 5 NEURAL NETWORKS

A globally asymptotically stable plasticity rule for firing rate homeostasis Prashant Joshi & Jochen Triesch

Arizona State University DMML Kernel Methods – Gaussian Processes Presented by Shankar Bhargav.

Circular statistics Maximum likelihood Local likelihood Kenneth D. Harris 4/3/2015.

Saturation, Flat-spotting Shift up Derivative Weight Decay No derivative on output nodes.

Classification and Prediction: Regression Analysis

Linearizing ODEs of a PID Controller Anchored by: Rob Chockley and Scott Dombrowski.

Javad Lavaei Department of Electrical Engineering Columbia University Joint work with Somayeh Sojoudi Convexification of Optimal Power Flow Problem by.

Gaussian process modelling

Algorithms for a large sparse nonlinear eigenvalue problem Yusaku Yamamoto Dept. of Computational Science & Engineering Nagoya University.

Biological Modeling of Neural Networks: Week 9 – Adaptation and firing patterns Wulfram Gerstner EPFL, Lausanne, Switzerland 9.1 Firing patterns and adaptation.

FORESTRY AND FOREST PRODUCTS Project Level Carbon Accounting Toolkit CSIRO Forestry and Forest Products Department of Forestry, Australian National University.

Biological Modeling of Neural Networks Week 8 – Noisy input models: Barrage of spike arrivals Wulfram Gerstner EPFL, Lausanne, Switzerland 8.1 Variation.

Application of Stochastic Frontier Regression (SFR) in the Investigation of the Size-Density Relationship Bruce E. Borders and Dehai Zhao.

Various topics Petter Mostad Overview Epidemiology Study types / data types Econometrics Time series data More about sampling –Estimation.

GADA - A Simple Method for Derivation of Dynamic Equation Chris J. Cieszewski and Ian Moss.

Boyce/DiPrima 9 th ed, Ch 1.4: Historical Remarks Elementary Differential Equations and Boundary Value Problems, 9 th edition, by William E. Boyce and.

Biological Modeling of Neural Networks Week 8 – Noisy output models: Escape rate and soft threshold Wulfram Gerstner EPFL, Lausanne, Switzerland 8.1 Variation.

The Logistic Growth SDE. Motivation  In population biology the logistic growth model is one of the simplest models of population dynamics.  To begin.

Multivariate Dyadic Regression Trees for Sparse Learning Problems Xi Chen Machine Learning Department Carnegie Mellon University (joint work with Han Liu)

Mechanical Engineering Department Automatic Control Dr. Talal Mandourah 1 Lecture 1 Automatic Control Applications: Missile control Behavior control Aircraft.

Fuzzy Systems Michael J. Watts

Håkon Dahl-Olsen, Sridharakumar Narasimhan and Sigurd Skogestad Optimal output selection for batch processes.

ECE-7000: Nonlinear Dynamical Systems Overfitting and model costs Overfitting  The more free parameters a model has, the better it can be adapted.

Dropout as a Bayesian Approximation

Eucalyptus globoidea productivity in New Zealand Dean Meason, Tobias Herrman, Christine Todoroki.

Linearizability of Chemical Reactors By M. Guay Department of Chemical and Materials Engineering University of Alberta Edmonton, Alberta, Canada Work Supported.

Gaussian Processes For Regression, Classification, and Prediction.

Modeling regional variation in the self-thinning boundary line Aaron Weiskittel Sean Garber Hailemariam Temesgen.

The Unscented Particle Filter 2000/09/29 이 시은. Introduction Filtering –estimate the states(parameters or hidden variable) as a set of observations becomes.

CHEE825 Fall 2005J. McLellan1 Nonlinear Empirical Models.

Dynamic Neural Network Control (DNNC): A Non-Conventional Neural Network Model Masoud Nikravesh EECS Department, CS Division BISC Program University of.

AUTOMATIC CONTROL THEORY II Slovak University of Technology Faculty of Material Science and Technology in Trnava.

Nonlinear balanced model residualization via neural networks Juergen Hahn.

Amir Yavariabdi Introduction to the Calculus of Variations and Optical Flow.

Mean Field Methods for Computer and Communication Systems Jean-Yves Le Boudec EPFL Network Science Workshop Hong Kong July

ETHEM ALPAYDIN © The MIT Press, Lecture Slides for.

State Space Representation

Radar Micro-Doppler Analysis and Rotation Parameter Estimation for Rigid Targets with Complicated Micro-Motions Peng Lei, Jun Wang, Jinping Sun Beijing.

Department of Civil and Environmental Engineering

Ecosystem Demography model version 2 (ED2)

Effective Connectivity

Dynamic Causal Modelling (DCM): Theory

Filtering and State Estimation: Basic Concepts

Modeling in the Time Domain

State Space Analysis UNIT-V.

State Space Method.

Effective Connectivity

Masoud Nikravesh EECS Department, CS Division BISC Program

Wellcome Centre for Neuroimaging at UCL

Physics 451/551 Theoretical Mechanics

Deep learning enhanced Markov State Models (MSMs)

Presentation transcript:

Integrability, neural networks, and the empirical modelling of dynamical systems Oscar Garcia forestgrowth.unbc.ca

Outline Dynamical Systems, forestry example The multivariate Richards model Extensions, Neural Networks Integrability, phase flows Conclusions

An engineer thinks that his equations are an approximation to reality. A physicist thinks reality is an approximation to his equations. A mathematician doesnt care. Anonymous Modelling

All models are wrong, but some are useful. G. E. P. Box

Dealing with Time Processes, systems evolving in time Functions of time Rates (Newton) System Theory (1960s) Control Theory, Nonlinear Dynamics

Dynamical systems Instead of State: Local transition function (rates): : inputs (ODE) Output function: Copes with disturbances

Example (whole-stand modelling)

3-D

Site Eichhorn (1904)

Integration No (Global transition function) Group:

3-D

Equation forms? Theoretical. Empirical. Constraints Simplest, linear: E.g., with Why not Average spacing? Mean diameter? Volume or biomass? Relative spacing?... ?

Multivariate Richards where The (scalar) Bertalanffy-Richards: with

Examples Radiata pine in New Zealand (García, 1984) t scaled by a site quality parameter Eucalypts in Spain – closed canopy (García & Ruiz, 2002)

Parameter estimation Stochastic differential equation: adding a Wiener (white noise) process. Then get the prob. distribution (likelihood function), and maximize over the parameters

Variations / extensions Multipliers for site, genetic improvement Additional state variables: relative closure, phosphorus concentration Those variables in multipliers: with a physiological time such that

Where to from here?

Transformations to linear

Transformations to constant V. I. Arnold Ordinary Differential Equations. The MIT Press, Invariants within a trajectory or flow line

Integrable systems Integrable systems?

Integrability Diffeomorphic to a constant field Integrable?

Modelling Assumption: For a wide enough class of systems there exists a smooth one-to-one transformation of the n state variables into n independent invariants Model (approximate) these transformations Automatic ways of doing this?

Artificial Neural Networks Problem: Not one-to-one

The multivariate Richards network

Estimation Regularization, penalize overparameterization Pruning

Integration No (Global transition function) Group:

Modelling the global T.F. (flow) No (Global transition function) Group:

Arnold No (Global transition function) Group: Phase flow one-parameter group of transformations

3-D

In forest modelling... Algebraic difference equations, Self- referencing functions Examples (A = age) : 1-D. Often confuse integration constants with site-dependent parameters But, perhaps it makes sense, after all? Clutter et al (1983) Tomé et al (2006)

Conclusions / Summary Dynamical modelling with ODEs seem natural, although it is rare in forestry Multivariate Richards, an example of transformation to linear ODEs, or to invariants More general empirical transformations to invariants: ANN, etc. Modelling the invariants themselves, rather than ODEs