scaled correlation
   HOME

TheInfoList



OR:

In
statistics Statistics (from German language, German: ''wikt:Statistik#German, Statistik'', "description of a State (polity), state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of ...
, scaled correlation is a form of a coefficient of
correlation In statistics, correlation or dependence is any statistical relationship, whether causal or not, between two random variables or bivariate data. Although in the broadest sense, "correlation" may indicate any type of association, in statistics ...
applicable to data that have a temporal component such as
time series In mathematics, a time series is a series of data points indexed (or listed or graphed) in time order. Most commonly, a time series is a sequence taken at successive equally spaced points in time. Thus it is a sequence of discrete-time data. Exa ...
. It is the average short-term correlation. If the signals have multiple components (slow and fast), scaled coefficient of correlation can be computed only for the fast components of the signals, ignoring the contributions of the slow components.Nikolić D, Muresan RC, Feng W, Singer W (2012) Scaled correlation analysis: a better way to compute a cross-correlogram. ''European Journal of Neuroscience'', pp. 1–21, doi:10.1111/j.1460-9568.2011.07987.x http://www.danko-nikolic.com/wp-content/uploads/2012/03/Scaled-correlation-analysis.pdf This filtering-like operation has the advantages of not having to make assumptions about the sinusoidal nature of the signals. For example, in the studies of brain signals researchers are often interested in the high-frequency components (beta and gamma range; 25–80 Hz), and may not be interested in lower frequency ranges (alpha, theta, etc.). In that case scaled correlation can be computed only for frequencies higher than 25 Hz by choosing the scale of the analysis, ''s'', to correspond to the period of that frequency (e.g., ''s'' = 40 ms for 25 Hz oscillation).


Definition

Scaled correlation between two signals is defined as the average correlation computed across short segments of those signals. First, it is necessary to determine the number of segments K that can fit into the total length T of the signals for a given scale s: :K = \operatorname\left(\frac\right). Next, if r_k is Pearson's coefficient of correlation for segment k, the scaled correlation across the entire signals \bar_s is computed as :\bar_s = \frac \sum\limits_^K r_k.


Efficiency

In a detailed analysis, Nikolić et al. showed that the degree to which the contributions of the slow components will be attenuated depends on three factors, the choice of the scale, the amplitude ratios between the slow and the fast component, and the differences in their oscillation frequencies. The larger the differences in oscillation frequencies, the more efficiently will the contributions of the slow components be removed from the computed correlation coefficient. Similarly, the smaller the power of slow components relative to the fast components, the better will scaled correlation perform.


Application to cross-correlation

Scaled correlation can be applied to auto- and
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 order to investigate how correlations of high-frequency components change at different temporal delays. To compute cross-scaled-correlation for every time shift properly, it is necessary to segment the signals anew after each time shift. In other words, signals are always shifted ''before'' the segmentation is applied. Scaled correlation has been subsequently used to investigate synchronization hubs in the visual cortex Folias, S.E., S. Yu, A. Snyder, D. Nikolić, and J.E. Rubin (2013) Synchronisation hubs in the visual cortex may arise from strong rhythmic inhibition during gamma oscillations. ''European Journal of Neuroscience'', 38(6): 2864–2883. Scaled correlation can be also used to extract functional networks.Dolean, S., Dînşoreanu, M., Mureşan, R. C., Geiszt, A., Potolea, R., & Ţincaş, I. (2017, September). A Scaled-Correlation Based Approach for Defining and Analyzing Functional Networks. In International Workshop on New Frontiers in Mining Complex Patterns (pp. 80–92). Springer, Cham.


Advantages over filtering methods

Scaled correlation should be in many cases preferred over signal filtering based on spectral methods. The advantage of scaled correlation is that it does not make assumptions about the spectral properties of the signal (e.g., sinusoidal shapes of signals). Nikolić et al. have shown that the use of
Wiener–Khinchin theorem In applied mathematics, the Wiener–Khinchin theorem or Wiener–Khintchine theorem, also known as the Wiener–Khinchin–Einstein theorem or the Khinchin–Kolmogorov theorem, states that the autocorrelation function of a wide-sense-stationary ...
to remove slow components is inferior to results obtained by scaled correlation. These advantages become obvious especially when the signals are non-periodic or when they consist of discrete events such as the time stamps at which neuronal action potentials have been detected.


Related methods

A detailed insight into a correlation structure across different scales can be provided by visualization using multiresolution correlation analysis.Pasanen, L., & Holmström, L. (2016). "Scale space multiresolution correlation analysis for time series data." ''Computational Statistics'', 1–22.


See also

*
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 ...
*
Coherence (signal processing) In signal processing, the coherence is a statistic that can be used to examine the relation between two signals or data sets. It is commonly used to estimate the power transfer between input and output of a linear system. If the signals are ergodi ...
*
Convolution In mathematics (in particular, functional analysis), convolution is a operation (mathematics), mathematical operation on two function (mathematics), functions ( and ) that produces a third function (f*g) that expresses how the shape of one is ...
*
Correlation In statistics, correlation or dependence is any statistical relationship, whether causal or not, between two random variables or bivariate data. Although in the broadest sense, "correlation" may indicate any type of association, in statistics ...
*
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 ...
*
Phase correlation Phase correlation is an approach to estimate the relative translative offset between two similar images (digital image correlation) or other data sets. It is commonly used in image registration and relies on a frequency-domain representation of the ...
*
Spectral density The power spectrum S_(f) of a time series x(t) describes the distribution of power into frequency components composing that signal. According to Fourier analysis, any physical signal can be decomposed into a number of discrete frequencies, o ...
*
Cross-spectrum In time series analysis, the cross-spectrum is used as part of a frequency domain analysis of the cross-correlation or cross-covariance between two time series. Definition Let (X_t,Y_t) represent a pair of stochastic processes that are jointl ...
*
Wiener–Khinchin theorem In applied mathematics, the Wiener–Khinchin theorem or Wiener–Khintchine theorem, also known as the Wiener–Khinchin–Einstein theorem or the Khinchin–Kolmogorov theorem, states that the autocorrelation function of a wide-sense-stationary ...


References

{{reflist


Free sources

* A free source code for computing scaled cross correlation and an interface for MATLAB can be downloaded here: http://www.raulmuresan.ro/sources/corrlib/ * Simple demo code in python: https://github.com/dankonikolic/Scaled-Correlation Covariance and correlation