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Astron. Astrophys. 330, 341-350 (1998)

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3. Correlation of the individual modes

3.1. Correlations two by two

No striking general correlation appears when looking at the set of 18 modes displayed on Fig. 1. Nor does it stand out from the computation of the correlations of these modes, two by two. Although some large correlations are measured ([FORMULA] between the modes [FORMULA], [FORMULA] and [FORMULA], [FORMULA]), even larger anticorrelations are also found ([FORMULA] between the modes [FORMULA], [FORMULA] and [FORMULA], [FORMULA]). No general trend is visible, the mean value being [FORMULA] (Fig. 4).

[FIGURE] Fig. 4. Correlation coefficients of the modes [FORMULA] observed by GOLF, two by two. For each value of n, the long and short ticks correspond to [FORMULA] and [FORMULA], respectively. The symbol [FORMULA] (resp. -) is used for a positive (resp. negative) correlation. The smallest symbols correspond to correlations smaller than the statistical error, intermediate and big symbols correspond to correlations smaller than 2 and 3 statistical errors respectively.

The statistical error of the estimator of the correlation coefficient between exponential distributions, from a sample of p points, scales like [FORMULA]. For our sample of 210 points, no effect smaller than [FORMULA] can therefore be detected. Altogether, [FORMULA] of the couples (17 out of 306 couples) present correlations contained, in absolute value, between 2 and 3 standard deviations, which is not very significant ([FORMULA] would be expected for a normal distribution of the statistical error).

Nevertheless, a more sensitive indicator can be constructed, in order to determine the mean correlation coefficient more accurately.

3.2. Test of the null hypothesis. Comparison with a Gamma distribution

If k distributions are independent (null hypothesis), the variance of their sum should be equal to the sum of their variances. This test was performed by Baudin et al. (1996) with IPHIR data, who normalized the distribution by their level of noise, and found some discrepancy.

A fundamental feature of our method is the use of the exponential nature of each individual energy distribution, in order to compute the standard deviation of our estimate of the correlation, and therefore the confidence level of our conclusions.

We denote by [FORMULA] the sum of k distributions of energy, made of p events, where each of the distributions is normalized by its estimated mean energy. Since each distribution appears to be exponential within the statistical error (Sect. 2), [FORMULA] ought to resemble a Gamma-distribution of order k (denoted by [FORMULA]) if they are independent, or an exponential distribution of mean value k if they are all identical. The null hypothesis can therefore be tested by comparing the observed distribution [FORMULA] with the theoretical [FORMULA] -distribution, using the variance and KS tests.

Denoting by [FORMULA] the correlation coefficient between the modes i and j, and [FORMULA] their mean value, the variance of [FORMULA] is directly related to these correlations:

[EQUATION]

If the modes are independent, the standard deviation [FORMULA] of the variance estimator of [FORMULA] is

[EQUATION]

Consequently, another way of testing the null hypothesis is the comparison of the variance of [FORMULA] with var [FORMULA], in units of the statistical error [FORMULA].

Even if the k distributions defining [FORMULA] are independent and exponential, an additional error of the order of [FORMULA] is introduced in the estimation of the mean value of each exponential distribution. We use a Montecarlo method of [FORMULA] trials made from independent exponential distributions, in order to define the cumulative distributions [FORMULA] for the outcome of the variance test, and [FORMULA] for the outcome [FORMULA] of the KS test. Here again, we have used autocorrelated exponential distributions in the Montecarlo simulations.

Eq. (7) indicates that the variance of the distribution gives a direct measure of the mean correlation among the modes:

[EQUATION]

This formulae, however, is not directly useful without an expression of the statistical error [FORMULA] associated with the estimator of the variance. If the modes are correlated, computing it requires some additionnal assumptions about the properties of the correlation (Sect.  3.4). Nevertheless, [FORMULA] coincides with Eq. (8) to first order. Together with Eq. (9), the smallest correlation detectable with this method scales as follows:

[EQUATION]

which is a factor [FORMULA] smaller than the sensitivity of the correlation coefficient two by two. A better sensitivity is therefore obtained by summing a large number of modes. However, the hypothesis of a constant correlation between the modes might be questionable if the range of frequencies is large, especially since the mean energy and the lifetime of the modes vary significantly with frequency.

3.3. GOLF results

Both tests are of course very sensitive to the presence of a gap in the data. If [FORMULA] of the data were filled with zeros due to an interruption of the instrument, the variance of [FORMULA] would be increased by a factor [FORMULA]. Consequently, we have carefully removed from our samples the points corresponding to these gaps.

The sum of the energies of the 9 modes [FORMULA], [FORMULA], normalized to their mean energy, is shown in Fig. 5. We note in passing that the clear gap in the data appearing around the 156th day confirms the validity of our procedure for the extraction of the energy. As before, four days of signal have been removed from our statistical study to account for the stabilisation of the instrument, resulting in a sample made up of 210 points. Their distribution is successfully compared to a [FORMULA] -distribution in Fig. 6 ([FORMULA] and [FORMULA]). The same test performed on the 9 modes [FORMULA], [FORMULA], obtains [FORMULA] and [FORMULA]. Applied to these 18 modes altogether, the tests confirm again the null hypothesis ([FORMULA] and [FORMULA]).


[FIGURE] Fig. 5. Time evolution of [FORMULA], the sum of the normalized energies of the 9 modes [FORMULA], [FORMULA], observed by GOLF.

[FIGURE] Fig. 6. Histogram (20 bins) of the energy and cumulative distribution of the sum of the 9 modes [FORMULA], [FORMULA] observed by GOLF, compared to a [FORMULA] distribution.

The correlation being low, we may use Eq. (9) with the statistical error given by Eq. (8), and obtain a confirmation of the absence of correlation, with an error bar: [FORMULA] for 9 modes [FORMULA], and [FORMULA] for 18 modes [FORMULA].

3.4. Test of the " [FORMULA] -hypothesis". Correlation due to an additive common signal

A refined estimate of the correlation can be obtained by making some assumptions about its origin. We build in Appendix B a simple model where the excitation is a mixture of two types of sources, which, taken separately, would result in uncorrelated/highly correlated modes energies respectively.

The first type represents the granules, which produce so many excitations per damping time that the correlation among the modes energies is close to zero.

The second type is hypothetical. It could be produced by some isolated events, possibly of magnetic origin, separated by a time comparable to or larger than the damping time of the modes considered.

We assume that the mode response to an excitation is linear, and therefore the response to a mixture of sources is a superposition of the answers to the two types separately. Our model depends on a single parameter [FORMULA], namely the fraction of the energy of each mode due to the second type of sources.

We define in Appendix B the theoretical distribution function [FORMULA] corresponding to such correlated modes energies. If the model is applicable, the distribution [FORMULA] ought to converge, for [FORMULA], towards a well defined distribution denoted by [FORMULA], such that:

[EQUATION]

The variance and KS test can therefore be used, for various values of [FORMULA], in order to test this " [FORMULA] -hypothesis".

With the definition of Appendix B, the correlation coefficient [FORMULA] is related to the coefficient [FORMULA] as follows:

[EQUATION]

The statistical error [FORMULA] associated with the estimator of the variance also depends on [FORMULA] according to Eq. (B6). Eq. (9) can then be used to determine the mean correlation [FORMULA], with a consistent statistical error.

We do not expect higher order moments of the distribution [FORMULA] to be more sensitive to a correlation between the modes, since we prove in Appendix B that they also vary like [FORMULA] at first order.

We also demonstrate that the shape of the cumulative distribution varies like [FORMULA]. The sensitivity limit of the KS test is therefore expected to be comparable to the sensitivity limit of the variance test.

Here again, normalizing the distributions by their estimated mean value introduces a bias, which we take into account using a Montecarlo method. For each value of [FORMULA] considered, we compute from [FORMULA] trials the theoretical distribution [FORMULA], and use [FORMULA] other trials to define the cumulative distributions [FORMULA] and [FORMULA] which are used for our variance and KS tests. For the sake of simplicity, the effect of the autocorrelation of each mode is neglected here. Indeed, we know from Sect.  2.3and 3.2that it introduces very small corrections on the cumulative distributions [FORMULA] and [FORMULA].

We define the error bar of the correlation coefficient as the range of values of [FORMULA] within which the test ([FORMULA] or [FORMULA]) remains inside the upper [FORMULA] region, by analogy with the statistics of normal distributions.

Fig. 7 shows that the variance and KS tests, applied to 18 modes for various values of [FORMULA], stays within the upper [FORMULA] region for [FORMULA], which coincides with the statistical limitation expressed by Eq. (10).

[FIGURE] Fig. 7. Comparison of GOLF data ([FORMULA] points) with a theoretical distribution made of k modes with a uniform correlation [FORMULA], using the variance and KS tests. The vertical dotted line delimits the sensitivity limit ([FORMULA]) defined by Eq. (10) for [FORMULA] modes. It coincides with the value below which the corresponding variance and KS tests remain inside the upper [FORMULA] region.

Nevertheless, while the measurements made by GOLF are compatible with a total lack of correlation of the modes, our tests cannot exclude that up to [FORMULA] of the energy is common to the modes.

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© European Southern Observatory (ESO) 1998

Online publication: January 8, 1998
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