Multivariate Complex Normal Distribution Research Articles

Oscillating neural circuits: Phase, amplitude, and the complex normal distribution.

Multiple oscillating time series are typically analyzed in the frequency domain, where coherence is usually said to represent the magnitude of the correlation between two signals at a particular frequency. The correlation being referenced is complex-valued and is similar to the real-valued Pearson correlation in some ways but not others. We discuss the dependence among oscillating series in the context of the multivariate complex normal distribution, which plays a role for vectors of complex random variables analogous to the usual multivariate normal distribution for vectors of real-valued random variables. We emphasize special cases that are valuable for the neural data we are interested in and provide new variations on existing results. We then introduce a complex latent variable model for narrowly band-pass-filtered signals at some frequency, and show that the resulting maximum likelihood estimate produces a latent coherence that is equivalent to the magnitude of the complex canonical correlation at the given frequency. We also derive an equivalence between partial coherence and the magnitude of complex partial correlation, at a given frequency. Our theoretical framework leads to interpretable results for an interesting multivariate dataset from the Allen Institute for Brain Science.

Analysis of Fisher Information and the Cramér–Rao Bound for Nonlinear Parameter Estimation After Random Compression

In this paper, we analyze the impact of compression with complex random matrices on Fisher information and the Cramer–Rao Bound (CRB) for estimating unknown parameters in the mean value function of a complex multivariate normal distribution. We consider the class of random compression matrices whose distribution is right-unitarily invariant. The compression matrix whose elements are i.i.d. standard complex normal random variables is one such matrix. We show that for all such compression matrices, the Fisher information matrix has a complex matrix beta distribution. We also derive the distribution of CRB. These distributions can be used to quantify the loss in CRB as a function of the Fisher information of the noncompressed data. In our numerical examples, we consider a direction of arrival estimation problem and discuss the use of these distributions as guidelines for choosing compression ratios based on the resulting loss in CRB.

Simple algorithm for a maximum-likelihood SAD function.

Recently, the multivariate complex normal distribution has been used to develop a maximum-likelihood probability function for single-wavelength anomalous diffraction phasing and refinement of heavy-atom parameters [Pannu & Read (2004), Acta Cryst. D60, 22-27]. The function accounts explicitly for the correlations between the observed and calculated Friedel mates and their errors. However, the method of derivation of the equation described by Pannu & Read (2004) leads to a complicated likelihood expression that suffers from a number of algorithmic limitations. Here, an alternative derivation of the P(SAD) function is described that leads to simplified algorithmic requirements and that allows an intuitive understanding of the expression.

New ways of looking at experimental phasing.

In the original work by Blow and Crick, experimental phasing was formulated as a least-squares problem. For good data on good derivatives this approach works reasonably well, but we now attempt to extract more information from poorer data than in the past. As in many other crystallographic problems, the assumptions underlying the use of least squares for phasing are not satisfied, particularly for poor derivatives. The introduction of maximum likelihood (and more powerful computers) has led to substantial improvements. For computational convenience, these new methods still make many assumptions about the independence of different measurements and sources of error. A more general formulation for the probability distributions underlying likelihood-based methods for both experimental phasing and molecular-replacement phasing is reviewed. In the new formulation, all the structure factors associated with a particular hkl are considered to be related by a complex multivariate normal distribution. When it is assumed that certain errors are independent, the general formulation reduces to current likelihood targets. However, the new formulation makes the necessary assumptions more explicit and points the way to improving phasing using both isomorphous and anomalous differences.

Application of the complex multivariate normal distribution to crystallographic methods with insights into multiple isomorphous replacement phasing

Probabilistic methods involving maximum-likelihood parameter estimation have become a powerful tool in computational crystallography. At the centre of these methods are the relevant probability distributions. Here, equations are developed based on the complex multivariate normal distribution that generalize the distributions currently used in maximum-likelihood model and heavy-atom refinement. In this treatment, the effects of various sources of error in the experiment are considered separately and allowance is made for correlations among sources of error. The multivariate distributions presented are closely related to the distributions previously derived in ab initio phasing and can be applied to many different aspects of a crystallographic structure-determination process including model refinement, density modification, heavy-atom phasing and refinement or combinations of them. The underlying probability distributions for multiple isomorphous replacement are re-examined using these techniques. The re-analysis requires the underlying assumptions to be made explicitly and results in a variance term that, unlike those previously used for maximum-likelihood multiple isomorphous replacement phasing, is expressed explicitly in terms of structure-factor covariances. Test cases presented show that the newly derived multiple isomorphous replacement likelihood functions perform satisfactorily compared with currently used programs.

Modeling the indoor MIMO wireless channel

This paper demonstrates the ability of a physically based statistical multipath propagation model to match capacity statistics and pairwise magnitude and phase distributions of measured 4 /spl times/ 4 and 10 /spl times/ 10 narrow-band multiple-input multiple-output data (MIMO) at 2.4 GHz. The model is compared to simpler statistical models based on the multivariate complex normal distribution with either complex envelope or power correlation. The comparison is facilitated by computing channel element covariance matrices for fixed sets of multipath statistics. Multipolarization data is used to demonstrate a simple method for modeling dual-polarization arrays.

Pushing the boundaries of molecular replacement with maximum likelihood.

The molecular-replacement method works well with good models and simple unit cells, but often fails with more difficult problems. Experience with likelihood in other areas of crystallography suggests that it would improve performance significantly. For molecular replacement, the form of the required likelihood function depends on whether there is ambiguity in the relative phases of the contributions from symmetry-related molecules (e.g. rotation versus translation searches). Likelihood functions used in structure refinement are appropriate only for translation (or six-dimensional) searches, where the correct translation will place all of the atoms in the model approximately correctly. A new likelihood function that allows for unknown relative phases is suitable for rotation searches. It is shown that correlations between sequence identity and coordinate error can be used to calibrate parameters for model quality in the likelihood functions. Multiple models of a molecule can be combined in a statistically valid way by setting up the joint probability distribution of the true and model structure factors as a multivariate complex normal distribution, from which the conditional distribution of the true structure factor given the models can be derived. Tests in a new molecular-replacement program, Beast, show that the likelihood-based targets are more sensitive and more accurate than previous targets. The new multiple-model likelihood function has a dramatic impact on success.

Moments of the complex multivariate normal distribution

Moments of the complex multivariate normal distribution are obtained by differentiating its characteristic function, applying the differential operators for the differentiation of functions of complex vectors. A recurrence relation for the derivatives of the characteristic function is derived, and explicit expressions for the moments are obtained. Moments up to order 6 are expressed in terms of Hermitian matrices. Moments of Hermitian quadratic forms up to order 3 are also obtained.

The multivariate complex normal distribution-a generalization

The multivariate complex normal distribution usually employed in the literature is a special case since certain restrictions have been imposed on the covariances of the real and imaginary parts of its variables. A more general distribution is proposed of which the usual distribution is shown to be a special case. >

The spin- [formula omitted] implementation of the Gibbs/Band-Park model of thermodynamic equilibrium

Band and Park enunciated some fifteen years ago the problem of estimating a Gibbs distribution over the totality of possible states of a quantum system, so as to yield the expected value of a Hamiltonian. The apparent lack of a suitable prior distribution over these states, as well as the difficulty of parameterizing them, has impeded the implementation of their program. Here - considering the continuum of spin- 1 2 states, parameterized by points of the Riemann sphere - their program is implemented using the principles of (Bayesian) multivariate analysis to find the suitable prior. This turns out to be the square root of the determinant of the Fisher information matrix associated with a complex multivariate normal distribution having the density matrix of a state as its covariance matrix and a priori zero mean vector. For comparison purposes, the model is estimated using a (naive) uniform prior over the Riemann sphere. Relations to the Jaynesian program of finding the state having the expected value of the Hamiltonian and the maximum value of the von Neumann entropy have yet to be resolved.

The role of powers of matrices in determining the distribution of quadratic forms involving the complex normal distribution

By investigating the powers of matrices used in quadratic forms and by replacing the standard trace function with the generalized trace (defined as the sum of the principal minors of a certain order of a matrix), generalizations of necessary and sufficient conditions for a quadratic form to be chi-square are established. Khatri [5] has established several matrix results which can be considered generalizations of Cochran's theorem. These matrix results are further extended here, and their role in determining the statistical distribution of quadratic forms is shown when the quadratic forms involve the complex multivariate normal distribution.

Asymptotic distributions of the sphericity test in a complex multivariate normal distribution

In this paper, asymptotic expansions of the null and non-null distributions of the sphericity test criterion in the case of a complex multivariate normal distribution are obtained for the first time in terms of beta distributions. In the null case, it is found that the accuracy of the approximation by taking the first term alone in the asymptotic series is sufficient for practical purposes. In fact for p - 2. the asymptotic expansion reduces to the first term which is also the exact distribution in this case. Applications of the results to the area of inferences on multivariate time series are also given.

Unbiasedness of Invariant Tests for Manova and Other Multivariate Problems

Let $Y:p \times r$ and $Z:p \times n$ be normally distributed random matrices whose $r + n$ columns are mutually independent with common covariance matrix, and $EZ = 0$. It is desired to test $\mu = 0$ vs. $\mu \neq 0$, where $\mu = EY$. Let $d_1, \cdots, d_p$ denote the characteristic roots of $YY'(YY' + ZZ')^{-1}$. It is shown that any test with monotone acceptance region in $d_1, \cdots, d_p$, i.e., a region of the form $\{g(d_1, \cdots, d_p)\leq c\}$ where $g$ is nondecreasing in each argument, is unbiased. Similar results hold for the problems of testing independence of two sets of variates, for the generalized MANOVA (growth curves) model, and for analogous problems involving the complex multivariate normal distribution. A partial monotonicity property of the power functions of such tests is also given.

Nonnull distribution of likelihood ratio criterion for reality of covariance matrix

In this paper the distribution of the likelihood ratio test for testing the reality of the covariance matrix of a complex multivariate normal distribution is investigated. Some simplifications in the noncentral distribution are made and the noncentral distribution is derived for the special case where the rank of the noncentrality matrix is two. In the null case exact expressions for the distribution are given up to p = 6, and percentage points are tabulated. These percentage points were compared with percentage points derived from an asymptotic expansion of the distribution, and the accuracy of the approximation was found to be sufficient for several practical situations.

Exact distribution of sphericity criterion in the complex case and its percentage points

The exact null distribution of the sphericity criterion for testing against the alternative σ>0 and unknown, in the multivariate complex normal distribution is obtained and percentage points are computed for p=3(1)6, ∑ = 0.05 and 0.01 and various values of the sample size.

Classical Statistical Analysis Based on a Certain Multivariate Complex Gaussian Distribution

N. R. Goodman [4] has discussed some aspects of the complex multivariate normal distribution, in particular, the analogue of the Wishart distribution and of multiple and partial correlations. We shall obtain maximum likelihood estimates for certain parameters and likelihood ratio tests of certain hypotheses arising in the study of such complex multivariate normal distributions. Although the general principles involved in the derivation of the distribution of the associated statistics are well known to mathematicians working on group representations (see, for example, Gelfand and Naimark [3], p. 24), it has seemed desirable to derive the needed results on this type in a way parallel to one method of obtaining the distributions of real multivariate analysis (Deemer and Olkin [2], Olkin [6] and Roy [7]). With the help of the results derived, we can handle the complex variates in the same manner as we do for real variates in the case of Gaussian distributions. Moreover, it can be noted that for every distributional result of classical multivariate Gaussian statistical analysis obtainable in closed (explicit) form, the counterpart analysis for complex Gaussian is also obtainable in closed (explicit) form with necessary changes. It may be pointed out that the non-central distributions in this connection were derived independently by A. T. James [5] with the help of zonal polynomials of hermitian matrices, but we feel that sometimes the derivation of distributions with the help of Jacobian transformations may be useful.