Cross-correlation - Wikipedia
Moreover, the averaged cross-correlation function often had an .. Its peak value shows the strength of a linear relationship between x and y. Note that people commonly refer to any relationship as correlation, but commonly -used correlation coefficients in statistics measure the degree of a linear. The calculation and interpretation of the sample product moment correlation coefficient and the linear regression equation are discussed and illustrated.
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Conceived and designed the experiments: Contributed to the writing of the manuscript: Received May 2; Accepted Jul Associated Data Data Availability Statement The authors confirm that all data underlying the findings are fully available without restriction.
The companies, which stock prices are used in the paper, are listed in Table 1. Corresponding time series can be freely downloaded from http: Abstract We study historical correlations and lead-lag relationships between individual stock risk volatility of daily stock returns and market risk volatility of daily returns of a market-representative portfolio in the US stock market.
We focus on the behavior of the cross-correlations at the times of financial crises with significant jumps of market volatility. The observed historical dynamics showed that the dependence between the risks was almost linear during the US stock market downturn of and after the US housing bubble inremaining at that level until Moreover, the averaged cross-correlation function often had an asymmetric shape with respect to zero lag in the periods of high correlation.
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We develop the analysis by the application of the linear response formalism to study underlying causal relations. The calculated response functions suggest the presence of characteristic regimes near financial crashes, when the volatility of an individual stock follows the market volatility and vice versa.
Introduction A financial market is a complex system demonstrating diverse phenomena and attracting attention from a whole spectrum of disciplines ranging from social to natural science . Better understanding of the behavior of financial markets has become an integral part of the discussion on further sustainable economic development.
In this context, proper assessment of financial risks  plays a crucial role: Underestimated risks contribute to financial bubbles with eventual crashes while overestimation of risks might cause inefficiency of financial resource allocations and a slowdown in economic growth, giving rise to periods of stagnation.
Cross-correlation analysis is basically a generalization of standard linear correlation analysis, which provides us with a good place to start.
Suppose we obtain repeated spectra of one of the brighter Seyfert galaxies, and we want to determine whether or not the variations in the H emission line and the optical continuum are correlated which was an interesting question 20 years ago, even before emission-line time delays were considered.
The first thing you would do is plot the H flux against the continuum flux, as in the left-hand panel of Fig.
Statistics review 7: Correlation and regression
A measure of the strength of the correlation is given by the correlation coefficent, 31 where there are N pairs of values xi, yi and their respective means are and. For the data shown in the left panel of Fig. Confidence levels for linear correlation can be found in standard statistical tables 6.
While this is quite a good correlation, we see something more remarkable if we plot both variables as functions of time i. We see that the patterns of variation are very similar, except that the emission-line light curve is delayed in time, or "lagged," relative to the continuum light curve. It is obvious that the correlation between the continuum and emission-line fluxes would be even better if we allowed a linear shift in time between the two light curves in order to line up their prominent maxima and minima.Cross Correlation Function
This is what cross-correlation does. The H emission-line and optical continuum fluxes for Mrkas shown in Fig.
It is clear from the figure that the continuum and emission-line fluxes are well-correlated, and that the correlation can be improved by a linear shift in time of one time series relative to the other. The optimum linear correlation occurs by shifting the emission-line light curve backwards by The first operational problem in computing a cross-correlation is also immediately apparent: Unfortunately, regularly sampled data are almost never found in Astronomy; ground-based programs have weather to contend with, and even satellite-based observations are almost never regularly spaced in time.
The essence of the cross-correlation problem in Astronomy is dealing with time series that are not evenly sampled. Moreover, the light curves are often limited in extent and are noisy.
Correlation and cross-correlations
For well-sampled series as in Fig. The simple, effective solution is to interpolate one series between the actual data points, and use the interpolated points in the cross-correlation. We illustrate this schematically in Fig.