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Related Concept Videos

Coefficient of Correlation01:12

Coefficient of Correlation

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 strength of the linear...
Calibration Curves: Correlation Coefficient01:10

Calibration Curves: Correlation Coefficient

In a linear calibration curve, there is a value called the calibration coefficient, denoted by 'r,' which measures the strength and the direction of association between two variables. The correlation coefficient value ranges from −1 to +1. A value of +1 indicates a perfect positive linear correlation, −1 denotes a perfect negative correlation, and 0 implies no correlation between the two variables. A positive correlation value establishes that as one variable increases, the other increases, and...
Correlation and Regression00:53

Correlation and Regression

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 negative...
Partial Fractions01:28

Partial Fractions

A partial fraction is a component of a rational expression represented as the sum of simpler fractions. When a rational function is expressed as a ratio of two polynomials, it can often be decomposed into a sum of fractions whose denominators are simpler polynomials, typically linear or irreducible quadratic factors. This process is called partial fraction decomposition, and it is used to simplify complex expressions for integration, solving equations, or analysis.Partial fraction decomposition...
Correlation01:09

Correlation

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:
Calculating and Interpreting the Linear Correlation Coefficient01:11

Calculating and Interpreting the Linear Correlation Coefficient

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. Hence, it is also known as the Pearson product-moment correlation coefficient. It can be calculated using the following equation:

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Dual-Color Fluorescence Cross-Correlation Spectroscopy to Study Protein-Protein Interaction and Protein Dynamics in Live Cells
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Fractional correlation.

D Mendlovic, H M Ozaktas, A W Lohmann

    Applied Optics
    |October 22, 2010
    PubMed
    Summary
    This summary is machine-generated.

    Researchers introduced optical interpretations of the fractional-Fourier-transform operator, defining a new fractional correlation operator. This operator offers novel applications in shift-variant image detection due to its non-shift-invariant properties.

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    Area of Science:

    • Optics and Photonics
    • Image Processing
    • Signal Processing

    Background:

    • Recent advancements have introduced optical interpretations for the fractional-Fourier-transform operator.
    • Conventional correlation is a fundamental tool in image processing and signal analysis.

    Purpose of the Study:

    • To define and explore a fractional correlation operator based on the fractional-Fourier-transform.
    • To investigate the properties and potential applications of this new fractional correlation operator, particularly in image detection.

    Main Methods:

    • Defining the fractional correlation operator in two ways, ensuring consistency with conventional correlation.
    • Analyzing the shift-invariance property of the fractional correlation operation.
    • Suggesting and demonstrating a bulk-optics implementation using computer simulations.

    Main Results:

    • Two definitions of the fractional correlation operator were established, both consistent with conventional correlation.
    • Fractional correlation was found to be not always shift-invariant.
    • Computer simulations demonstrated the feasibility of a bulk-optics implementation.

    Conclusions:

    • The fractional correlation operator, derived from optical fractional-Fourier-transform interpretations, provides a new framework for correlation analysis.
    • The non-shift-invariant nature of fractional correlation opens avenues for novel applications, such as shift-variant image detection.
    • A practical bulk-optics implementation of fractional correlation is feasible, as shown through simulations.