Parametrising correlation matrices
Introduction
In applications of matrices, there are many settings in which the rows and columns have distinct meaning. For example, in a survey of people, giving a numerical score between and for their rating of different movies, there is an matrix – the data matrix – such that the rows correspond to the people and the columns to the movies. The th column is then the vector of scores given for movie , and its th entry is the score given by person . Let denote the average of the scores in column , and let denote the vector with all entries equal to 1. The recentred, zero mean score vectors are then specified as , and the recentred data matrix is
The sample covariance matrix is specified in terms of as Note that is a symmetric matrix, and its entry in rows and column gives the sample covariance between the scores of movies and . In the case that the rows and/or columns of are drawn from a vector Gaussian distribution with given covariance, (1.2) is referred to a Wishart matrix. For such random matrices, a vast number of theoretical results have been assembled, and applied settings identified, since the pioneering paper [23]; see, e.g., [16].
Natural from the viewpoint of data analysis is to further refine (1.2) by forming the sample correlation matrix Here, as well as the original score vectors being centred by subtracting their mean, each has been scaled to correspond to a unit vector. One sees immediately that the entries of are all equal to unity on the diagonal, while on the off diagonal, in accordance with the Cauchy–Schwarz inequality, they all have modulus less than or equal to 1. Moreover, the decomposition where shows that , like , is positive definite but now with bounded entries .
This latter feature, although making some aspects of theoretical analysis more difficult (e.g., studies of the eigenvalues) allows for a distinct set of questions to be posed. For example, in the case of correlation matrices the volume of the natural embedding in —referred to as an elliptope [13] and to be denoted —is well defined. Knowledge of this volume allows an answer to the question: if the strictly upper triangular entries of (1.3) are chosen uniformly at random in the range , what is the probability that is a valid correlation matrix (i.e., is positive definite) [6]?
A direct approach to this question requires a parametrisation of the space of correlation matrices. Two such parametrisations are available in the literature, both applying to the lower triangular matrix in the Cholesky factorisation One of these use hyperspherical co-ordinates in to parametrise row () [6], [18], [19], [20], [21], and the other makes use of a sequence of partial correlations [10], [11], [14]. The latter method yielded the first direct computation of the volume [11] where It is only in the last few years that this same formula (in equivalent forms) was derived using the hyperspherical parametrisation [6], [19].
Indirect computations of are also possible. Such a method, giving a formula equivalent to (1.6) actually predates the work [11]—this is due to Wong et al. [24]. As implied by a comment in [19, 2nd paragraph of Introduction] the same result, again deduced indirectly, follows from the still earlier work of Muirhead [16, p. 148].
The circumstances just described suggest a number of follow up problems. The most immediate is to relate the hyperspherical and partial correlation parametrisations. To give a satisfactory account on this point, a self contained theory relating to the latter must be developed. Moreover, the hyperspherical parametrisation gives a different viewpoint on known results [16] for the marginal distribution of the elements of (1.3), when chosen uniformly at random, and similarly for the moments of .
The literature cited above is restricted to the case of real entries. Complex valued covariance matrices, and thus complex valued correlation matrices, are well motivated from the viewpoint of their application in wireless communication; see for example [22]. Thus, in addition to addressing the above problems when the correlation matrices have real entries, we consider too the case of complex (and quaternion) entries.
Section snippets
Cholesky factorisation and parametrisations
Let be an positive definite matrix with all diagonal entries equal to unity, as is consistent with (1.3). Introduce the Cholesky factorisation (1.5) with Well established theory (see, e.g., [9]) gives that this is unique for positive definite subject to the requirement that
The fact that the diagonal entries in (1.3) are all equal to unity implies that the sum of the squares of the non-zero entries along each row of is
The Jacobian, hyperspherical parametrisation of determinant and some consequences
Let denote the Jacobian (absolute value of the determinant of the Jacobian matrix) for the change of variables as implied by (1.3), (1.5), (2.4). It is shown in [19] and [6] that upon ordering the entries i.e., reading sequentially along rows of the strictly lower triangular portion of , and similarly ordering the angles, the Jacobian matrix is lower triangular. Its determinant and thus the Jacobian can be read off as equal to
Random correlation matrices with complex or quaternion entries
A correlation matrix can be constructed out of a data matrix with complex entries by replacing in (1.4) by . As mentioned in the Introduction, this is of interest in studies in wireless communications engineering . Of less practical interest, but still of theoretical relevance within random matrix theory (see, e.g., [7]) is to form correlation matrices out of data matrices with entries having 2 × 2 block structure Such 2 × 2 matrices form a representation of quaternions,
CRediT authorship contribution statement
Peter J. Forrester: Conceptualization.
Acknowledgments
This work is part of a research program supported by the Australian Research Council (ARC) through the ARC Centre of Excellence for Mathematical and Statistical frontiers (ACEMS). PJF also acknowledges partial support from ARC grant DP170102028, and JZ acknowledges the support of a Melbourne postgraduate award, and an ACEMS, Australia top up scholarship.
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