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Correlations02:20

Correlations

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Correlation means that there is a relationship between two or more variables (such as ice cream consumption and crime), but this relationship does not necessarily imply cause and effect. When two variables are correlated, it simply means that as one variable changes, so does the other. We can measure correlation by calculating a statistic known as a correlation coefficient. A correlation coefficient is a number from -1 to +1 that indicates the strength and direction of the relationship between...
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Correlation01:09

Correlation

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In statistics, two variables are said to be correlated if the values of one variable are associated with the other variable. Depending on the relationship between two variables, correlation can be of three types– positive correlation, negative correlation, and zero correlation.
Two variables, for example, a and b, are said to be positively correlated if both variables move in the same direction. In other words, a positive correlation exists between two variables, a and b, if:
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Correlation and Regression00:53

Correlation and Regression

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In statistics, correlation describes the degree of association between two variables. In the subfield of linear regression, correlation is mathematically expressed by the correlation coefficient, which describes the strength and direction of the relationship between two variables. The coefficient is symbolically represented by 'r' and ranges from -1 to +1. A positive value indicates a positive correlation where the two variables move in the same direction. A negative value suggests a...
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Coefficient of Correlation01:12

Coefficient of Correlation

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The correlation coefficient, r, developed by Karl Pearson in the early 1900s, is numerical and provides a measure of strength and direction of the linear association between the independent variable x and the dependent variable y.
If you suspect a linear relationship between x and y, then r can measure how strong the linear relationship is.
What the VALUE of r tells us:
The value of r is always between –1 and +1: –1 ≤ r ≤ 1.
The size of the correlation r indicates the...
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Correlation and Causation01:27

Correlation and Causation

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Statistical tests can calculate whether there is a relationship, or correlation, between independent and dependent variables. An indirect relationship of the variables signifies a correlation, while a direct relationship shows causation. If it is determined that no connection exists between the variables, then the correlation is a coincidence.
Correlation versus Causation
If the dependent variable increases or decreases when the independent variable increases, there is a positive or negative...
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Correlation of Experimental Data01:23

Correlation of Experimental Data

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Dimensional analysis simplifies complex physical problems and guides experimental investigations, but it does not provide complete solutions. It identifies the dimensionless groups that influence a phenomenon, but experimental data is needed to establish the specific relationships and validate theoretical predictions.
For example, a spherical particle moving through a viscous fluid experiences drag. Dimensional analysis shows that the drag force depends on the particle's diameter, velocity,...
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A Psychophysics Paradigm for the Collection and Analysis of Similarity Judgments
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A non-hierarchical correlation discrete variable representation.

Roman Ellerbrock1, Uwe Manthe2

  • 1Department of Chemistry and The PULSE Institute, Stanford University, Stanford, California 94305, USA.

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|April 9, 2022
PubMed
Summary
This summary is machine-generated.

A new non-hierarchical correlation discrete variable representation (CDVR) method improves multi-configurational time-dependent Hartree (MCTDH) calculations. This approach reduces computational cost while maintaining accuracy for complex quantum dynamics simulations.

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Area of Science:

  • Quantum Chemistry
  • Computational Physics
  • Theoretical Chemistry

Background:

  • The multi-configurational time-dependent Hartree (MCTDH) method is a powerful tool for simulating quantum dynamics.
  • Efficient evaluation of potential matrix elements is crucial for MCTDH performance.
  • The correlation discrete variable representation (CDVR) facilitates MCTDH calculations but often employs a hierarchical grid structure.

Purpose of the Study:

  • To introduce a non-hierarchical correlation discrete variable representation (CDVR) approach for MCTDH calculations.
  • To reduce the computational resources required by MCTDH while maintaining accuracy.
  • To improve the scalability of quantum dynamics simulations.

Main Methods:

  • Development of a non-hierarchical CDVR approach utilizing identically structured quadratures across all layers.
  • Implementation within the MCTDH framework for evaluating potential matrix elements.
  • Testing the method on the photodissociation of NOCl and vibrational states of CH3.

Main Results:

  • The non-hierarchical CDVR significantly reduces the number of required grid points compared to hierarchical CDVR.
  • Demonstrated superior scaling properties for the new approach.
  • Achieved identical results to hierarchical methods for representations with the same topology.

Conclusions:

  • The non-hierarchical CDVR offers a more efficient and scalable alternative for MCTDH calculations.
  • The method accurately models quantum dynamics, as evidenced by tests on NOCl photodissociation and CH3 vibrations.
  • This advancement is expected to benefit complex quantum system simulations.