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

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2 A Core Course on Modeling after the party… Summary Week 1- No Model Without a Purpose 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 - sampled - 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

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3 A Core Course on Modeling Summary Conceptual model consists of concepts, some represent 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: ordering 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 derived. Conceptual model consists of concepts, some represent 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: ordering 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 derived. Week 2- The Art of Omitting

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4 A Core Course on Modeling Summary 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; A recursive function Q i+1 = F(Q i, Q i-1, Q i-2, …., P i, P i-1, Pi-2, …) to unroll a behavior; equal intervals: closed form evaluation (e.g.,: periodic financial transactions; sampling); equal, small intervals: approximation, sampling error (examples: moving point mass, … ); infinitesimal intervals: continuous time, DEs (examples: mass-spring system); Week 3- Time for Change

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5 A Core Course on Modeling 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; Week 4-The Function of Functions

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6 A Core Course on Modeling 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. Week 5-Roles of Quantities in a Functional Model

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7 A Core Course on Modeling Week 6-Models and Confidence Modeling involves uncertainty because of different causes: Differences between accuracy, precision, error; Uncertainty distributions of values rather than a single value (normal, uniform); The notions of distance and similarity; 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.

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8 A Core Course on Modeling Week 7-A working model – and then? Leading question: to what extent has the initial problem been solved? Approach: cat.-II quantities to assess the quality of the modeling process Taxonomy: Input or output side? Modeled system or stakeholders? Qualitative or quantitative? Resulting cat.-II quantities: 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? 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? Cat.-II quantities for modeling quality are related to purposes.

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