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Difference Equations and Recurrece Relations
Bioconductor version: Release (2.11) Interfaces R with the AT and T graphviz library for plotting R graph objects from the graph package. Users on all platforms must install graphviz; see the README file, available in the source distribution of this file, for details. Author: Jeff Gentry, Li Long, Robert Gentleman, Seth Falcon, Florian Hahne, Deepayan Sarkar, Kasper Daniel Hansen Maintainer: Kasper Daniel Hansen <khansen at jhsph.edu>
September 2007 This is the second installment of a new feature in Plus : the teacher package. 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.
First published Mon Feb 27, 2006; substantive revision Mon Jun 25, 2012 Models are of central importance in many scientific contexts.
A mathematical model is a description of a system using mathematical concepts and language. 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 .
[ edit ] Signal Flow Diagrams Signal Flow Diagrams are another method for visually representing a system. Signal Flow Diagrams are especially useful, because they allow for particular methods of analysis, such as Mason's Gain Formula .
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.
By Kevin Murphy, 1998. "Graphical models are a marriage between probability theory and graph theory.
A signal-flow graph (SFG) is a special type of block diagram [ 1 ] —and directed graph —consisting of nodes and branches. Its nodes are the variables of a set of linear algebraic relations. An SFG can only represent multiplications and additions.
Selected Bibliography for Modeling Social Phenomena Abraham, Ralph H. and Christopher D. Shaw. 1992. Dynamics: The Geometry of Behavior, Second Edition . Redwood City, California: Addison-Wesley.
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.
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:
(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.
(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.
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.
This is sort-of related to my sidelined study of graph algebra. I was thinking about data I could apply a first-order linear difference model to, and the stock market came to mind. After all, despite some black swan sized shocks, what better predicts a day’s closing than the previous day’s closing? So, I hunted down the data and graphed exactly that: