Conceptual Modelling and 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 externally on the scientist. The scientist has a model of how they perceive reality, and hypotheses are formed from the model or theory.

Agenda I. Modelling A. Basic Principles B. Conceptual model C. Mathematical model 1. Distributed rate / conservation law 2. Phenomenological / semi-empirical rate law 3. Inverse methods D. Physical models and miniatures E. Simulation – discrete events, agents, and rules F. Statistical hypothesis testing AIMS Introduce philosophy of modelling Discuss modelling tools – intuition and dimensional analysis OBJECTIVES 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 Compute model Input parameters Compute predictions Inverse Problem Identify parameters Run model “backwards” Make measurements 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.