Measuring two-dimensional complexity: A conceptual structure

Betak, John

doi:10.1016/0031-3203(72)90002-7

Cited by 6 publications

(8 citation statements)

References 7 publications

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“…With respect to decomposition, geographers extended the use of information theory in linguistics to issues of spatial complexity (Dacey 1970). Betak used information theory constructs applied to syntactic complexity and context-free grammar to describe two dimensional languages for polygons (Betak 1970(Betak , 1972 and to assess environmental structure (Betak 1974). In the former he used information theory to decompose syntactic complexity into four elements: redundancy, nesting, structure and transformation.…”

Section: Level Of Servicementioning

confidence: 99%

Traffic Grammar and Algorithmic Complexity in Urban Freeway Flow Patterns

2007

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This paper uses techniques from formal language theory to describe the linear spatial patterns in urban freeway traffic flows in order to understand and analyze "hidden order" in such high volume systems. A method for measuring randomness based on algorithmic entropy is introduced and developed. These concepts are operationalized using Pincus' approximate entropy formulation in an appropriate illustration. These measures, which may be viewed as counterintuitive, are believed to offer robust and rigorous guidance to enhance the overall understanding of efficiency in urban freeway traffic systems. Utilization of such measures should be facilitated by information generated by real time intelligent transportation systems (ITS) technologies and may prove helpful in real time traffic flow management.

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Section: Level Of Servicementioning

confidence: 99%

Traffic Grammar and Algorithmic Complexity in Urban Freeway Flow Patterns

2007

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“…There are several ways of specifying syntactic complexity, in both natural languages [Z] and in two-dimensional languages [5], and a unique characterization of complexity is not possible. Olson [22] has "The base of the logarithm is rather arbitrary; however, for communication problems it is convenient to use the binary base" €24, p. 781.…”

Section: Complexity and Redundancy In Mapsmentioning

confidence: 99%

Complexity and Redundancy in Binary Maps

Gatrell

1977

Geographical Analysis

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The complexity of binary maps that is provided by the areal arrangement of colors is considered, and measured using information theory. In addition, information theory provides other measures that have an interpretation in a map context. One of these, redundancy, is examined and found to bear a striking empirical relationship to a spatial autocorrelation statistic. It is argued that spatial autocorrelation is, conceptual1 y as well as empirically, the two-dimensional equivalent of redundancy. It too measures the extent to which the occurrence of an event (color) in an areal unit constrains, or makes more probable, the occurrence of an event in a neighboring areal unit.The observation that maps are devices for the communication of information has gained wide currency in the cartographic literature [ 19, 231. As natural languages fulfill the same purpose, it is reasonable to argue that maps may be conceived, in a general sense, as output from languages. Given such a conceptualization it seems worthwhile to explore the possibilities of using the ideas and methods of linguistics in map analysis. In this paper, information theory is used to examine two syntactic or grammatical aspects of maps, complexity and redundancy.Harvey asserts that map syntax embraces "the internal structure of the map statements and its basic form as an abstract calculus" [15, pp. 373-741. In natural language it is concerned, broadly speaking, with the way in which elements of the language are put together, that is, with the rules of sentence construction. Extending this into two dimensions, it can be argued that map syntax involves notions of spatial association, periodicity [23, pp. 5-61, and the search for the rules that generate a map pattern. INFORMATION CONTENT AND REDUNDANCY IN ONE AND Two DIMENSIONS Information theory [27] is becoming well known in the geographic literature [3, 171. It can be used to measure the uncertainty or "informa-*This paper is based on my Master's thesis, "Complexity and Redundancy in Choropleth Maps," The Pennsylvania State University, 1974. I would like to thank Professor Peter Gould and Dr. Conrad Strack for their helpful advice and encouragement.Anthony C. Gatrell is lecturer in geography, Uiiicersity of Salford.

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“…As Betak has noted, the condition of generality is satisfied [2]. That is, the grammars are constructed in accordance with a specific theory of linguistic structure in which component terms, such as "phrase" and "sentence," are defined independently of the language.…”

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confidence: 99%

“…Dacey's set of languages for simple polygons is found to satisfy the requirements of context-free grammars. Betak found that three measures of syntactical complexity have application to these languages[ 2 ] .…”

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Two‐Dimensional Syntactic Complexity Measures: A Preliminary Evaluation*

Betak

1973

Geographical Analysis

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The purposes of this paper are twofold: (1) preliminary evaluation of the construct of two-dimensional syntactic complexity presented by Betak [ 2 ] , and (2) preliminary evaluation of Dacey's [ 4 ] languages for simple polygons with respect to grammatical adequacy. The evaluation is done by considering evidence and observations from the psychological literature on physical measures of twodimensional complexity [3, 6, 71.The paper is divided into three parts: (1) discussion of observed relationships between the syntactic measures and the physical measures, (2) evaluation of the validity and adequacy of the syntactic measures, and (3) conclusions and implications for further research. SYNTACTIC MEASURES VIS-A-VIS PHYSICAL MEASURESSeveral measures of two-dimensional syntactic complexity are presented by Betak [ 2 ] . These measures include: (1) information and redundancy, (2) degree of nesting, (3) structural complexity, and (4) transformational complexity. The following discussion considers these measures in terms of available evidence, drawn from psychology, regarding physical measures of two-dimensional complexity.

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Measuring two-dimensional complexity: A conceptual structure

Cited by 6 publications

References 7 publications

Traffic Grammar and Algorithmic Complexity in Urban Freeway Flow Patterns

Traffic Grammar and Algorithmic Complexity in Urban Freeway Flow Patterns

Complexity and Redundancy in Binary Maps

Two‐Dimensional Syntactic Complexity Measures: A Preliminary Evaluation*

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