Generalized Scale Invariance in the Atmosphere and Fractal Models of Rain

The value of the global Hubble parameter corresponding to astrophysical observations was determined theoretically without using ʌСDM models. A nonlinear fractal model of the connection between the distance to the observed galaxy and its coordinate is proposed. Distance is defined as a fractal measure, the measurement scale of which, in contrast to the known fractal models, corresponds to the deviation of the desired measure itself from its fixed value (radius of zero gravity), relative to which the scale invariance is assumed. We used the dimension of our proposed specific anisotropic fractal, which simulates the increase in the distance to the observation point. It is shown that this dimension is also the maximum dimension of the strange attractor of the phase portrait of the equation of gravitational waves and sets of galaxies from different catalogs.

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D-test: A New Test for Analyzing Scale Invariance Using Symbolic Dynamics and Symbolic Entropy

Methodology ◽

10.1027/1614-2241/a000026 ◽

2011 ◽

Vol 7 (3) ◽

pp. 88-95 ◽

Cited By ~ 2

Author(s):

Jose A. Martínez ◽

Manuel Ruiz Marín

Keyword(s):

Symbolic Dynamics ◽

Scale Invariance ◽

Rating Scales ◽

Rating Scale ◽

Marketing Research ◽

Response Patterns ◽

Measurement Scales ◽

Improve Measurement ◽

Entropy Measures ◽

The Difference

The aim of this study is to improve measurement in marketing research by constructing a new, simple, nonparametric, consistent, and powerful test to study scale invariance. The test is called D-test. D-test is constructed using symbolic dynamics and symbolic entropy as a measure of the difference between the response patterns which comes from two measurement scales. We also give a standard asymptotic distribution of our statistic. Given that the test is based on entropy measures, it avoids smoothed nonparametric estimation. We applied D-test to a real marketing research to study if scale invariance holds when measuring service quality in a sports service. We considered a free-scale as a reference scale and then we compared it with three widely used rating scales: Likert-type scale from 1 to 5 and from 1 to 7, and semantic-differential scale from −3 to +3. Scale invariance holds for the two latter scales. This test overcomes the shortcomings of other procedures for analyzing scale invariance; and it provides researchers a tool to decide the appropriate rating scale to study specific marketing problems, and how the results of prior studies can be questioned.

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