A core course on Modeling kees van Overveld Week-by-week summary

A Core Course on Modeling Chapter 1- No Model Without a Purpose    Summary     2 A model  clearly defined purpose; purposes are: explanation, prediction (two cases!), compression, abstraction, unification, communication, documentation, analysis, verification, exploration, decision, optimization,specification, realization, steering and control. Modeling dimensions: static - dynamic: does time play a role? continuous - discrete: 'counting' or 'measuring'? numeric - symbolic: manipulating numbers or expressions? geometric - non-geometric: do features from 2D or 3D space play a role? deterministic - stochastic: does probability play a role? calculating - reasoning: rely on numbers or on propositions? black box - glass box: start from data or from causal mechanisms? Modeling is a process involving 5 stages: define: establish the purpose conceptualize: in terms of concepts, properties and relations formalize: in terms of mathematical expressions execute: (often involves running a computer program) conclude: adequate presentation and interpretion after the party… 2

A Core Course on Modeling Chapter 2- The Art of Omitting      Summary      3 Conceptual model consists of concepts, representing entities; Concept: bundle of properties, each consisting of a name and a set of values (type): Concepts + relations = conceptual model (entity-relationship graph) establish concepts; establish properties; establish types of properties; establish relations. Quantities are properties, disregarding the concept they are a property of; Mathematical operations on quantities: what ordering is offered by the quantity’s type? Nominal (no order), partial ordering or total ordering, interval scale, ratio scale; Measuring =counting the number of units in the measured item. Sets of units with fixed ratio: dimension is an equivalence class on units; Using dimensions, the form of a mathematical relationships can often be checked or even derived (dimension anlysis or dimension synthesis.) 3

A Core Course on Modeling Chapter 3- Time for Change      Summary      4 State = snapshot of a conceptual model at some time point; State space = collection of all states; Change = transitions between states; state chart = graph; nodes (states) and arrows (transitions); Behavior = path through state space; State space explosion: number of states is huge for non trivial cases Projection: given a purpose, distinguish exposed and hidden properties or value sets; Multiple flavors of time: partially ordered time, e.g. specification and verification; totally ordered time, e.g. prediction, steering and control; For totally ordered time, a recursive function Qi+1 = F(Qi , Qi-1 , Qi-2, …. , Pi , Pi-1 , Pi-2 , …) can be used to unroll a behavior arbitrary intervals: just evaluate recursive functions equal intervals: sometimes closed form evaluation is possible (e.g.,: periodic financial transactions); equal, small intervals: approximation, sampling & sampling error (examples: moving point mass, … ); aliasing infinitesimal intervals: continuous time, DE’s (examples: mass-spring system); 4

A Core Course on Modeling Chapter 4-The Function of Functions 5 Conceptual model  formal model : not in a formally provable correct way; Appropriate naming Structure Chain of dependencies: the formal model as a directed acyclic graph; What mechanism? What quantities drive this mechanism? What is the qualitative behavior of the mechanism? What is the mathematical expression to describe this mechanism? To-do-list : all intermediate quantities are found and elaborated in turn; Formation of mathematical expressions: dimensional analysis  mathematical expressions, e.g in the case of proportionality the Relation Wizard can help finding appropriate fragments of mathematics; the Function Selector can help finding an appropriate expression for a desired behavior; wisdom of the crowds can help improve the accuracy of guessed values; 5

A Core Course on Modeling Chapter 5-Roles of Quantities in a Functional Model 6 functional model helps distinguish input (choice) and output (from purpose); Building a functional model as a graph shows roles of quantities. These are: Cat.-I : free to choose; Models for (design) decision support: the notion of design space; Choice of cat.-I quantities: no dependency-by-anticipation; Cat.-II : represents the intended output; The advantages and disadvantages of lumping and penalty functions; The distinction between requirements, desires, and wishes; The notion of dominance to express multi-criteria comparison; Pareto front; Cat.-III : represents constraints from context; Cat.-IV : intermediate quantities; For optimization: use evolutionary approach; Approximate the Pareto front using the SPEA algorithm; Local search can be used for post-processing. 6

A Core Course on Modeling Chapter 6-Models and Confidence 7 Validation and verification: is this the right model / is the model right? Modeling involves uncertainty because of different causes: Differences between accuracy, precision, error; Uncertainty  distributions of values rather than a single value (normal, uniform); Confidence for black box models: Common features of aggregation: average, standard deviation and correlation; Validation of a black box model: Residual error: how much of the behavior of the data is captured in the model? Distinctiveness: how well can the model distinguish between different modeled systems? Common sense: how plausible are conclusions, drawn from a black box model? Confidence for glass box models: Structural validity: do we believe the behavior of the mechanism inside the glass box? Quantitative validity: what is the numerical uncertainty of the model outcome? Sensitivity analysis and the propagation of uncertainty in input data; Sensitivity analysis to decide if a model should be improved. 7

A Core Course on Modeling Chapter 7-A working model – and then? 8 Leading question: to what extent has the initial problem been solved? Taxonomy of criteria for modeling: Input or output side? Modeled system or stakeholders? Qualitative or quantitative? Resulting criteria: Genericity: how many different modeled systems can we handle? Scalability: how large can the size of the problem be? Specialization: how much should the intended audience know? Audience (size): how large can the intended audience be? Convincingness: how plausible are the assumptions? Distinctiveness: e.g., how accurate, how certain, how decisive can the model outcome be? Surprise: to what extent can the model outcome give new insight? Impact: how big can the consequences of the model outcome be? Criteria for modeling quality are related to purposes. 8