From Fractal Geometry to Statistical Fractal

Padua, R.; Borres, Mark S.

doi:10.32871/rmrj1301.01.09

Cited by 4 publications

(3 citation statements)

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“…Another is to interpret the time-varying probabilities of word occurrences as a fractal time series. As the recursive procedure for constructing the hierarchical probability dis-Journal of Data Analysis and Information Processing tribution shows, the obtained probability distribution can be regarded as a statistical fractal [12] [13], so the time-varying probabilities thus obtained can be considered as a fractal time series [14] [15]. Any relations between fractal dimension of time-varying probabilities and characteristics of dynamic correlations may provide new insights into written texts.…”

Section: Discussionmentioning

confidence: 99%

Origin of Dynamic Correlations of Words in Written Texts

Ogura¹,

Amano²,

Kondo³

2019

JDAIP

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In a previous study, we introduced dynamical aspects of written texts by regarding serial sentence number from the first to last sentence of a given text as discretized time. Using this definition of a textual timeline, we defined an autocorrelation function (ACF) for word occurrences and demonstrated its utility both for representing dynamic word correlations and for measuring word importance within the text. In this study, we seek a stochastic process governing occurrences of a given word having strong dynamic correlations. This is valuable because words exhibiting strong dynamic correlations play a central role in developing or organizing textual contexts. While seeking this stochastic process, we find that additive binary Markov chain theory is useful for describing strong dynamic word correlations, in the sense that it can reproduce characteristics of autocovariance functions (an unnormalized version of ACFs) observed in actual written texts. Using this theory, we propose a model for time-varying probability that describes the probability of word occurrence in each sentence in a text. The proposed model considers hierarchical document structures such as chapters, sections, subsections, paragraphs, and sentences. Because such a hierarchical structure is common to most documents, our model for occurrence probability of words has a wide range of universality for interpreting dynamic word correlations in actual written texts. The main contributions of this study are, therefore, finding usability of the additive binary Markov chain theory to analyze dynamic correlations in written texts and offering a new model of word occurrence probability in which common hierarchical structure of documents is taken into account.

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Section: Discussionmentioning

confidence: 99%

Origin of Dynamic Correlations of Words in Written Texts

Ogura¹,

Amano²,

Kondo³

2019

JDAIP

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“…(All fractals show power-law distributions, but power-law distributions are not necessarily generated by fractal processes.) Yet, it is common in the literature to refer to non-spatial power-law phenomena as a fractal, such as time series [36,37], complex networks [38], and demographic distributions [39,40].…”

Section: What Fractal Signature Are Found In Urban Fabric?mentioning

confidence: 99%

An Overview of Fractal Geometry Applied to Urban Planning

Jahanmiri

Parker

2022

Land

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Since computing advances in the last 30 years have allowed automated calculation of fractal dimensions, fractals have been established as ubiquitous signatures of urban form and socioeconomic function. Yet, applications of fractal concepts in urban planning have lagged the evolution of technical analysis methods. Through a narrative literature review around a series of “big questions” and automated bibliometric analysis, we offer a primer on fractal applications in urban planning, targeted to urban scholars and participatory planners. We find that developing evidence demonstrates linkages between urban history, planning context, and urban form and between “ideal” fractal dimension values and urban aesthetics. However, we identify gaps in the literature around findings that directly link planning regulations to fractal patterns, from both positive and normative lenses. We also find an increasing trend of most literature on fractals in planning being published outside of planning. We hypothesize that this trend results from communication gaps between technical analysts and applied planners, and hope that our overview will help to bridge that gap.

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“…income distribution). This idea of migrating from fractal geometry to fractal statistics was uncovered by Padua and Borres (2013). A natural framework for studying these data is to assume a Pareto distribution: which we shall henceforth refer to as a fractal distribution with dimension λ, where λ takes values between integral values, and θ is the minimum of random variables {x 1 ,x 2 ,…,x n }.…”

Section: Introductionmentioning

confidence: 99%

Uniform Minimum Variance Unbiased Estimator of Fractal Dimension

Maureal

Castillano

Padua³

2021

RMRJ

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The paper introduced the concept of a fractal distribution using a power-law distribution. It proceeds to determining the relationship between fractal and exponential distribution using a logarithmic transformation of a fractal random variable which turns out to be exponentially distributed. It also considered finding the point estimator of fractional dimension and its statistical characteristics. It was shown that the maximum likelihood estimator of the fractional dimension λ is biased. Another estimator was found and shown to be a uniformly minimum variance unbiased estimator (UMVUE) by Lehmann-Scheffe’s theorem.

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From Fractal Geometry to Statistical Fractal

Cited by 4 publications

References 0 publications

Origin of Dynamic Correlations of Words in Written Texts

Origin of Dynamic Correlations of Words in Written Texts

An Overview of Fractal Geometry Applied to Urban Planning

Uniform Minimum Variance Unbiased Estimator of Fractal Dimension

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