Conceptual Modellingand Hypothesis Formation Research Methods CPE 401 / 6002 / 6003

Professor Will Zimmerman

Motivation

Hypotheses are just “scientific gossip”.

When we spread or hear gossip concerning other people, we are expressing a belief about the state of affairs. Beliefs are strong biases that influence the direction events take, even if ultimately disproved.

Forming hypotheses is central to the scientific method, but not controlled by any intellectual framework that is imposed externallyon the scientist. The scientist has a model of how they perceivereality, and hypotheses are formed from the model or theory.

Agenda

I. ModellingA. Basic PrinciplesB. Conceptual modelC. Mathematical model

1. Distributed rate / conservation law2. Phenomenological / semi-empirical rate law3. Inverse methods

D. Physical models and miniaturesE. Simulation – discrete events, agents, and rulesF. Statistical hypothesis testing

AIMS•Introduce philosophy of modelling•Discuss modelling tools – intuition and dimensional analysisOBJECTIVES•Gain facility with Mathematica

Basic principles Understanding the problem

Description of “what goes on” and “how it happens”. Processes and mechanisms

Defining the problem What do we want to know – aims and objectives

Draw a process diagram Create a model and identify hypotheses Analyse the model dimensionlessly Make assumptions / idealisations about important features to simplify

the model Develop an experimental plan to test the hypotheses / model

predictions

Conceptual model

Make a schematic diagram Define a frame of reference or relative viewpoint /

basis for description Consider analogies to other systems / pattern

matching Define the system (boundaries, open or closed) Identify mechanisms of change or important forces Make hypotheses

Mathematical modelling I: Distributed model / field theory

Identify conserved quantities (e.g. mass, energy, momentum) Write conservation laws / rate laws (transport equations) Close the undetermined quantities by relating to principle field

variables (constitutive properties, equations of state) Treat boundary and source / sink conditions Simplify the model by neglecting unimportant features

(idealization, approximation) Solve the model for all field quantities. Use the field quantities to compute the desired measures.

Input: Physical Parameters

Output: measures / factors

Mathematical modelling II: Phenomenological (lumped parameter) models

Determine whether desired measures (“global knowledge”) are much simpler than field variables (“detailed knowledge”) and if the distributed model is too complicated / inefficient to solve.

Derive model equations from the conservation laws for the desired measures, lumping some features of the detailed model into undetermined features of the model.

Derive expressions for the lumped parameters (e.g. MTC, HTC) or find experimental (empirical correlations) for the desired measures directly.

Solve the model equations for the desired measures directly.

Input: Physical and Empirical Parameters

Output: measures / factors

New paradigm for model building: “All models are inverse problems”

Forward Problem

Computemodel

Inputparameters

Computepredictions

Inverse Problem

Identify parameters

Run model“backwards”

Makemeasurements

Data assimilation permits the inference of the parameters

Example: two phase reactor

Physical models and miniatures

Prepare a geometrically similar model system, perhaps using different constituents / materials in order to determine experimentally the variation of desired measures or global or emergent behaviour to changes.

Analyze dimensionally and match dimensionless groups of the miniature system with those of the macroscale system so that generalized forces are similarly represented in both systems. The desired measures of the miniature system must then match the macroscale system.

Examples: model basins, wind tunnels, wave tanks, pilot plants

Simulation Make assumptions (simple rules) about microscopic behaviour that mimics the real

physical mechanisms, e.g. Probability distribution Deterministic chaos Fractal growth Networks with nodal interaction rules Cellular automata Relaxation toward local equilibria

Apply boundary forcing, random initial conditions, and respond to “events” or imposed “stresses” by the rules.

Compute global properties of the evolution steps or event stages to find emergent, organized, global behaviour. Major output: understanding of complexity.

Examples: molecular dynamics, systems biology, percolation in networks, pattern formation, coherent structures and self-organized structures.

Compare with experiment or observation. Predict sensitivity to changes in external conditions. Compute stability of states.

Amenable to inverse methods

Statistical hypothesis testing paradigm (Fisher)

Formulate a null hypothesis (opposite of the theory which is to be supported).

Chose a measure of deviation from this ideal state with assumed distribution.

Collect experimental data of the distribution of this measure. Compare actual distribution with the assumed distribution (P

value) Argue that the P value is sufficiently large that the null

hypothesis is false, therefore supporting the theory. Essential to replicate experimental conditions and take

multiple samples.

Summary

Central to the scientific method is model building and hypothesis formation.

Issues of conceptualizaton of models and types of models are important in directing hypothesis testing.

Mathematical and computational models are especially straightforward in principle to test, but suffer from the “lack of parameters” problem or the complexity problem.

Inverse methods can assimilate experimental data to form semi-empirical models with robust predictive properties and identify parameters.