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Related Concept Videos

Microsoft Excel: Pearson's Correlation01:18

Microsoft Excel: Pearson's Correlation

Microsoft Excel is a powerful tool for statistical analysis, including calculating Pearson's correlation coefficient, which measures the strength and direction of a linear relationship between two continuous variables. Pearson's correlation coefficient, often denoted as "r," ranges from -1 to 1. A value close to 1 indicates a strong positive correlation, meaning as one variable increases, the other does too. A value close to -1 indicates a strong negative correlation, implying that as one...
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Poisson's ratio is a material property that indicates their stress response. It explains the connection between the elongation or compression a material undergoes in the direction of an applied force and the contraction or expansion it experiences perpendicular to that force. When a slender bar is loaded axially, it stretches in the direction of the force and contracts laterally. Poisson's ratio is the negative ratio of this lateral contraction to the axial elongation. The negative sign ensures...

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Correlation Structure of Fractional Pearson Diffusions.

Nikolai N Leonenko1, Mark M Meerschaert, Alla Sikorskii

  • 1Cardiff School of Mathematics, Cardiff University, Senghennydd Road, Cardiff CF24 4YH, UK.

Computers & Mathematics with Applications (Oxford, England : 1987)
|October 4, 2013
PubMed
Summary
This summary is machine-generated.

This study introduces fractional Pearson diffusions, a novel class of stochastic processes. We derive a formula showing these diffusions exhibit long-range dependence due to fractional derivatives.

Keywords:
Mittag-Leffler functionPearson diffusioncorrelation functionfractional derivative

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Area of Science:

  • Stochastic processes
  • Fractional calculus
  • Mathematical physics

Background:

  • Pearson diffusion is a stochastic solution to diffusion equations with polynomial coefficients.
  • Fractional Pearson diffusion extends this concept by incorporating Caputo fractional derivatives.
  • Understanding these processes is crucial for modeling complex systems with memory effects.

Purpose of the Study:

  • To develop an explicit formula for the covariance function of a fractional Pearson diffusion in its steady state.
  • To analyze the dependence properties of fractional Pearson diffusions.

Main Methods:

  • Utilizing Caputo fractional derivatives to define fractional Pearson diffusions.
  • Deriving an explicit formula for the covariance function using Mittag-Leffler functions.
  • Analyzing the asymptotic behavior of the correlation function.

Main Results:

  • An explicit formula for the covariance function of fractional Pearson diffusions in steady state was developed.
  • The formula is expressed in terms of Mittag-Leffler functions.
  • Fractional Pearson diffusions demonstrate long-range dependence.

Conclusions:

  • Fractional Pearson diffusions exhibit long-range dependence.
  • The correlation function decays as a power law, with the exponent determined by the fractional derivative order.
  • This finding has implications for modeling systems with long-term memory.