Comparing spectra and coherences for groups of unequal size

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Abstract

Spectra and coherences are standard measures of association within and between time series. These measures have several advantages over their time-domain counterparts, not the least of which is the ability to derive and estimate confidence intervals. However, comparing spectra and coherences between two groups of observation is a problem that has not received much attention. This problem is important in neuroscience since it is often of great interest to determine whether the estimates differ between distinct experimental/behavioral conditions. Here we propose one approach to this problem. Based on the known distributional properties of spectral and coherence estimates, we derive a test for equality of two spectral or coherence estimates. The test is applicable to unequal sample sizes. We also derive jackknifed estimates of the variance of the proposed test statistic. We suggest that comparing the estimates obtained from the jackknife procedure with the theoretical estimates provides a robust means of determining whether the data in question shows non-Gaussian or non-stationary behavior. Finally, we present applications of the method to simulated and real data.

Introduction

Measuring the autocorrelation of a single stochastic process, or the strength of association between two stochastic processes is a problem that frequently occurs in neuroscience. It is particularly important in understanding recordings from multiple electrodes. The frequency domain spectrum and the coherency are the fundamental measures used in a large majority of signal processing applications, and have been shown to be useful in the analysis of neural data. For example, local field potential spectra from the lateral intraparietal area of macaques have been shown to exhibit directional tuning in a memory guided saccade task (Pesaran et al., 2002). Spectra and coherences of neural activity from area V4 of macaques have been shown to be affected by the attentional state of the monkey (Fries et al., 2001, Womelsdorf et al., 2006). Spectral measures have also been found useful in the study of Parkinson's disease (Llinas et al., 1999) and in the study of birdsong (Tchernichovski et al., 2004). Finally, spectra have also been used as the basis for a novel algorithm to predict sac-cadic eye movements in monkeys from recorded neural activity (Bokil et al., 2006a). A short review of some of these developments can be found in Bokil et al. (2006b).

Frequency domain measures have the particular advantage that they treat point processes (e.g., spike trains) and continuous processes (e.g., local field potential measurements) in the same way. Spectra can therefore be computed for spike trains (Jarvis and Mitra, 2001, Rosenberg et al., 1989), as well as for local field potentials (LFP) (Pesaran et al., 2002), and coherency estimates can be computed directly for pairs of spike trains (Jarvis and Mitra, 2001, Rosenberg et al., 1989), spike-LFP pairs (Pesaran et al., 2002) as well as LFP–LFP combinations. Also, the magnitudes of the complex coherency, namely the coherence, is a well normalised quantity that can be pooled across recordings. Finally, working in the frequency domain has the advantage that confidence intervals on the estimated quantities are relatively easy to construct (Brillinger, 1975).

One technical problem that has not been satisfactorily treated so far is the comparison of spectra and coherences from two groups with unequal number of trials. Such a case may arise, for example, in testing whether the underlying population showed significant change with a change in a stimulus parameter or a behavioral state variable (such as attentional state). Since the number of trials in the two conditions may not always be the same, it is necessary to compare estimated quantities from unequally sized groups. However, spectral and coherence estimates are biased and the bias depends on size of the group. Therefore, the comparison of these quantities between the two groups is a somewhat nontrivial problem. As an example of this unequal bias, Fig. 1 shows the estimated coherence for two groups with 5 and 50 trials, respectively.

The coherences were computed by averaging over 1000 realizations of 5 and 50 pairs of Gaussian time series, respectively, each with population coherence 0.5. In contrast to the coherence estimated from 50 trials, the estimated coherence for the group with 5 trials shows substantial deviation from the population coherence.

In this paper, we propose a statistical test for the equality of two spectral or coherence estimates. In contrast to previous studies that addressed this question (Amjad et al., 1997, Brillinger, 1975, Chapter 8 in), our work explicitly addresses the issue of unequal bias in the two estimates. For Gaussian data, our test statistics are shown to be distributed as a unit normal when the two population spectra or coherences are equal. In addition, we derive jackknifed estimates of variance of this statistic based on the multi-group jackknife procedure of Arvesen (1969). The jackknife provides a robust estimate of the variance, free of distributional assumptions. Therefore, inconsistency between the jackknifed estimates and the unit normal distribution can be used as a diagnostic of non-Gaussian behavior. The utility of the method is illustrated by applications to simulated and neurobiological time series data.

Section snippets

Method

We begin our discussion with the multi-taper spectral estimation method which is our method of choice for estimating spectra and coherences. Following this, we discuss the proposed test statistics in Section 2.2 and the jackknifed estimates of the variance of the proposed statistics in Section 2.3. Finally, Section 2.4 details the procedure to test the null hypothesis (H0) of equal spectra or coherences.

Analysis of simulated data

If X is a white, Gaussian time series with unit variance, Y=aX+1a2η where a is between 0 and 1, and η is Gaussian random noise with unit variance, independent of X, then the population spectrum of X and Y are both given by Spop = 1, and population coherence between X and Y is given by Cpop = a. Thus, a measures the population coherence between X and Y.

We first verify that Δx and Δy are indeed distributed as N(0, 1) when the null hypothesis is known to be true for all frequencies. We generated 1000

Analysis of experimental data

We now apply the method discussed here to two neurophysiological datasets. The first set consists of simultaneous Magnetoenecephalography (MEG, acquired over the left motor cortex) and bipolar surface Electromyographic (EMG) recordings of a human subject who periodically extended his right wrist for intermittent periods of 8 s. The subject's behavior can therefore be categorized into two conditions: (i) relaxation condition, when the subject's wrist was relaxed, and (ii) isometric contraction,

Conclusion

To conclude, we have developed a method for comparing spectra and coherences from two groups with unequal number of trials. The method provides a statistical test for equality of estimated quantities in different experimental conditions, based on the assumption that the observed neural data is Gaussian. In addition, we also provide jackknifed estimates of the variance of the proposed statistic. Since the jackknife is robust, distribution-free method for estimating the variance, deviation of the

Acknowledgement

This work is supported by the NIH (R01-MH71744 and R01-MH62528).

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