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Difference Equations and Recurrece Relations

Visual modeling. Visual modeling is the graphic representation of objects and systems of interest using graphical languages.

Visual modeling

Visual modeling languages may be General-Purpose Modeling (GPM) languages (e.g., UML, Southbeach Notation, IDEF) or Domain-Specific Modeling (DSM) languages (e.g., SysML). They include industry open standards (e.g., UML, SysML), as well as proprietary standards, such as the visual languages associated with VisSim, MATLAB and Simulink, OPNET, and NI Multisim. Rgraphviz. Bioconductor version: Release (2.14) Interfaces R with the AT and T graphviz library for plotting R graph objects from the graph package.

Rgraphviz

Author: Jeff Gentry, Li Long, Robert Gentleman, Seth Falcon, Florian Hahne, Deepayan Sarkar, Kasper Daniel Hansen Maintainer: Kasper Daniel Hansen <khansen at jhsph.edu> To install this package, start R and enter: Graphviz. Teacher package: Mathematical Modelling. September 2007 This is the second installment of a new feature in Plus: the teacher package.

Teacher package: Mathematical Modelling

Every issue contains a package bringing together all Plus articles about a particular subject from the UK National Curriculum. Whether you're a student studying the subject, or a teacher teaching it, all relevant Plus articles are available to you at a glance. Models in Science. First published Mon Feb 27, 2006; substantive revision Mon Jun 25, 2012 Models are of central importance in many scientific contexts.

Models in Science

The centrality of models such as the billiard ball model of a gas, the Bohr model of the atom, the MIT bag model of the nucleon, the Gaussian-chain model of a polymer, the Lorenz model of the atmosphere, the Lotka-Volterra model of predator-prey interaction, the double helix model of DNA, agent-based and evolutionary models in the social sciences, and general equilibrium models of markets in their respective domains are cases in point.

Scientists spend a great deal of time building, testing, comparing and revising models, and much journal space is dedicated to introducing, applying and interpreting these valuable tools. In short, models are one of the principal instruments of modern science. Philosophers are acknowledging the importance of models with increasing attention and are probing the assorted roles that models play in scientific practice. Mathematical model. A mathematical model is a description of a system using mathematical concepts and language.

Mathematical model

The process of developing a mathematical model is termed mathematical modelling. Mathematical models are used not only in the natural sciences (such as physics, biology, earth science, meteorology) and engineering disciplines (e.g. computer science, artificial intelligence), but also in the social sciences (such as economics, psychology, sociology and political science); physicists, engineers, statisticians, operations research analysts and economists use mathematical models most extensively.

A model may help to explain a system and to study the effects of different components, and to make predictions about behaviour. Mathematical models can take many forms, including but not limited to dynamical systems, statistical models, differential equations, or game theoretic models. These and other types of models can overlap, with a given model involving a variety of abstract structures. Control Systems/Signal Flow Diagrams - Wikibooks, collection of open-content textbooks.

Signal Flow Diagrams[edit] Signal Flow Diagrams are another method for visually representing a system.

Control Systems/Signal Flow Diagrams - Wikibooks, collection of open-content textbooks

Signal Flow Diagrams are especially useful, because they allow for particular methods of analysis, such as Mason's Gain Formula. Control theory. The concept of the feedback loop to control the dynamic behavior of the system: this is negative feedback, because the sensed value is subtracted from the desired value to create the error signal, which is amplified by the controller.

Control theory

The usual objective of a control theory is to calculate solutions for the proper corrective action from the controller that result in system stability, that is, the system will hold the set point and not oscillate around it. The inputs and outputs of a continuous control system are generally related by differential equations. If these are linear with constant coefficients, then a transfer function relating the input and output can be obtained by taking their Laplace transform. Graphical Models. By Kevin Murphy, 1998.

Graphical Models

"Graphical models are a marriage between probability theory and graph theory. They provide a natural tool for dealing with two problems that occur throughout applied mathematics and engineering -- uncertainty and complexity -- and in particular they are playing an increasingly important role in the design and analysis of machine learning algorithms. Fundamental to the idea of a graphical model is the notion of modularity -- a complex system is built by combining simpler parts. Probability theory provides the glue whereby the parts are combined, ensuring that the system as a whole is consistent, and providing ways to interface models to data. The graph theoretic side of graphical models provides both an intuitively appealing interface by which humans can model highly-interacting sets of variables as well as a data structure that lends itself naturally to the design of efficient general-purpose algorithms.

gRaphical models in R. Graph algebra (social sciences) Signal-flow graph. A signal-flow graph (SFG) is a special type of block diagram[1]—and directed graph—consisting of nodes and branches.

Signal-flow graph

Its nodes are the variables of a set of linear algebraic relations. An SFG can only represent multiplications and additions. Courtney Brown, Ph.D. Selected Bibliography for Modeling Social Phenomena Abraham, Ralph H. and Christopher D.

Courtney Brown, Ph.D.

Shaw. 1992. Dynamics: The Geometry of Behavior, Second Edition. Redwood City, California: Addison-Wesley. Graph Algebra and Social Theory : SAGE Research Methods Online. Little Green Book Nearly everything that occurs in the universe can be considered a part of some system, and that certainly includes human behavior and, potentially, human attitudes as well. But this does not mean that systems theory, and thus graph algebra, is appropriate for use in all situations. There are many competing approaches to the study of social and political phenomena, and systems theory using graph algebra is only one such approach. Graph Algebra: Mathematical Modeling with a Systems Approach. From the publisher's description of the book: Graph Algebra: Mathematical Modeling with a Systems Approach introduces a new modeling tool to students and researchers in the social sciences.

Derived from engineering literature that uses similar techniques to map electronic circuits and physical systems, graph algebra utilizes a systems approach to modeling that offers social scientists a variety of tools that are both sophisticated and easily applied. Key Features: Designed for readers in the social sciences with minimal mathematical training: In this volume, the author assists readers in developing their own difference and differential equation model specifications. Incorporates Social Theory: This book describes an easily applied language of mathematical modeling that uses boxes and arrows to develop very sophisticated algebraic statements of social and political phenomena.

15595_Chapter_1.pdf (objeto application/pdf) Systems Analysis for Social Scientists (Comparative studies in behavioral science) (9780471175094): Fernando Cortes, Adam Przeworski, John Sprague. Dynamic Modeling 1: Linear Difference Equations. (This is the first in a series on the use of Graph Algebraic models for Social Science.) Linear Difference models are a hugely important first step in learning Graph Algebraic modeling. That said, linear difference equations are a completely independent thing from Graph Algebra. Dynamic Modeling 2: Our First Substantive Model. (This is the second of a series of ongoing posts on using Graph Algebra in the Social Sciences.)

First-order linear difference equations are powerful, yet simple modeling tools. They can provide access to useful substantive insights to real-world phenomena. They can have powerful predictive ability when used appropriately. Additionally, they may be classified in any number of ways in accordance with the parameters by which they are defined. And though they are not immune to any of a host of issues, a thoughtful approach to their application can always yield meaningful information, if not for discussion then for further refinement of the model.

Dynamic Modeling 3: When the first-order difference model doesn’t cut it. Data must be selected carefully. The predictive usefulness of the model is grossly diminished if outliers taint the available data. Figure 1, for instance, shows the Defense spending (as a fraction of the national budget) between 1948 and 1968. Note how the trend curve (as defined by our linear difference model from the last post: see appendix for a fuller description) is a very poor predictor. Whatever is going on here isn’t a first-order process. Courtney Brown, Ph.D. Help! My model fits too well!