In
probability theory
Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set o ...
and related fields, the life-time of correlation measures the timespan over which there is appreciable
autocorrelation
Autocorrelation, sometimes known as serial correlation in the discrete time case, is the correlation of a signal with a delayed copy of itself as a function of delay. Informally, it is the similarity between observations of a random variable ...
or
cross-correlation
In signal processing, cross-correlation is a measure of similarity of two series as a function of the displacement of one relative to the other. This is also known as a ''sliding dot product'' or ''sliding inner-product''. It is commonly used fo ...
in
stochastic process
In probability theory and related fields, a stochastic () or random process is a mathematical object usually defined as a family of random variables. Stochastic processes are widely used as mathematical models of systems and phenomena that appea ...
es.
Definition
The
correlation coefficient
A correlation coefficient is a numerical measure of some type of correlation, meaning a statistical relationship between two variables. The variables may be two columns of a given data set of observations, often called a sample, or two components ...
''ρ'', expressed as an
autocorrelation function
Autocorrelation, sometimes known as serial correlation in the discrete time case, is the correlation of a signal with a delayed copy of itself as a function of delay. Informally, it is the similarity between observations of a random variable ...
or
cross-correlation function
In signal processing, cross-correlation is a measure of similarity of two series as a function of the displacement of one relative to the other. This is also known as a ''sliding dot product'' or ''sliding inner-product''. It is commonly used f ...
, depends on the lag-time between the times being considered. Typically such functions, ''ρ''(''t''), decay to zero with increasing lag-time, but they can assume values across all levels of correlations: strong and weak, and positive and negative as in the table.
The life-time of a correlation is defined as the length of time when the correlation coefficient is at the strong level.
[Buda, Andrzej; Jarynowski, Andrzej (2010) ''Life-time of correlations and its applications vol.1'', p.9, łogów: Wydawnictwo Niezależne] The durability of correlation is determined by signal (the strong level of correlation is separated from weak and negative levels). The mean life-time of correlation could measure how the durability of correlation depends on the window width size (the window is the length of time series used to calculate correlation).
References
Stochastic processes
Machine learning
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