Multi-dimensional model order selection

Multi-dimensional model order selection (MOS) techniques achieve an improved accuracy, reliability, and robustness, since they consider all dimensions jointly during the estimation of parameters. Additionally, from fundamental identifiability results of multi-dimensional decompositions, it is known that the number of main components can be larger when compared to matrix-based decompositions. In this article, we show how to use tensor calculus to extend matrix-based MOS schemes and we also present our proposed multi-dimensional model order selection scheme based on the closed-form PARAFAC algorithm, which is only applicable to multidimensional data. In general, as shown by means of simulations, the Probability of correct Detection (PoD) of our proposed multi-dimensional MOS schemes is much better than the PoD of matrix-based schemes. Introduction In the literature, matrix array signal processing techniques are extensively used in a variety of applications including radar, mobile communications, sonar, and seismology. To estimate geometrical/physical parameters such as direction of arrival, direction of departure, time of direction of arrival, and Doppler frequency, the first step is to estimate the model order, i.e., the number of signal components. By taking into account only one dimension, the problem is seen from just one perspective, i.e., one projection. Consequently, parameters cannot be estimated properly for certain scenarios. To handle that, multidimensional array signal processing, which considers several dimensions, is studied. These dimensions can correspond to time, frequency, or polarization, but also spatial dimensions such as oneor two-dimensional arrays at the transmitter and the receiver. With multidimensional array signal processing, it is possible to estimate parameters using all the dimensions jointly, even if they are not resolvable for each dimension separately. Moreover, by considering all dimensions jointly, the accuracy, reliability, and robustness can be improved. Another important advantage of using multi-dimensional data, also known as tensors, is the identifiability, since with tensors the typical rank can be much higher than using matrices. Here, we focus particularly on the development of techniques for the estimation of the model order. The estimation of the model order, also known as the number of principal components, has been investigated in several science fields, and usually model order selection schemes are proposed only for specific scenarios in the literature. Therefore, as a first important contribution, we have proposed in [1,2] the one-dimensional model order selection scheme called Modified Exponential Fitting Test (M-EFT), which outperforms all the other schemes for scenarios involving white Gaussian noise. Additionally, we have proposed in [1,2] improved versions of the Akaike’s Information Criterion (AIC) and Minimum Description Length (MDL). As reviewed in this article, the multi-dimensional structure of the data can be taken into account to improve further the estimation of the model order. As an example of such improvement, we show our proposed R-dimensional Exponential Fitting Test (R-D EFT) for multi-dimensional applications, where the noise is additive white Gaussian. The R-D EFT successfully outperforms the M-EFT confirming that even the technique with the best performance can be improved by taking into account the multi-dimensional structure of the data [1,3,4]. In addition, we also extend our modified versions of AIC and MDL to their respective multi-dimensional versions R-D AIC and R-D MDL. For scenarios with colored noise, we present our proposed multi-dimensional model order selection technique called closed-form PARAFAC-based model order selection (CFP-MOS) scheme [3,5]. * Correspondence: [email protected] University of Brasília, Electrical Engineering Department, P.O. Box 4386, 70910-900 Brasília, Brazil Full list of author information is available at the end of the article da Costa et al. EURASIP Journal on Advances in Signal Processing 2011, 2011:26 http://asp.eurasipjournals.com/content/2011/1/26 © 2011 da Costa et al; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The remainder of this article is organized as follows. After reviewing the notation in second section, the data model is presented in third section. Then the R-dimensional exponential fitting test (R-D EFT) and closedform PARAFAC-based model order selection (CFPMOS) scheme are reviewed in fourth section. The simulation results in fifth section confirm the improved performance of R-D EFT and CFP-MOS. Conclusions are drawn finally. Tensor and matrix notation In order to facilitate the distinction between scalars, matrices, and tensors, the following notation is used: Scalars are denoted as italic letters (a, b, ..., A, B, ..., a, b, ...), column vectors as lower-case bold-face letters (a, b, ...), matrices as bold-face capitals (A, B, ...), and tensors are written as bold-face calligraphic letters (A,B, . . .). Lower-order parts are consistently named: the (i, j)-element of the matrix A is denoted as ai,j and the (i, j, k)-element of a third order tensor X as xi,j,k. The n-mode vectors of a tensor are obtained by varying the nth index within its range (1, 2, ..., In) and keeping all the other indices fixed. We use the superscripts T, H, -1, +, and * for transposition, Hermitian transposition, matrix inversion, the Moore-Penrose pseudo inverse of matrices, and complex conjugation, respectively. Moreover the Khatri-Rao product (columnwise Kronecker product) is denoted by A ◊ B. The tensor operations we use are consistent with [6]: The r-mode product of a tensor A ∈ CI1×I2×···×IR and a matrix U ∈ CJr×Iralong the rth mode is denoted as A×r U ∈ CI1×I2···×Jr ···×IR. It is obtained by multiplying all r-mode vectors of A from the left-hand side by the matrix U. A certain r-mode vector of a tensor is obtained by fixing the rth index and by varying all the other indices. The higher-order SVD (HOSVD) of a tensor A ∈ CI1×I2×···×IR is given by A = S×1U1×2U2 · · · ×RUR, (1) where S ∈ CI1×I2×···×IR is the core-tensor which satisfies the all-orthogonality conditions [6] and Ur ∈ CIr×Ir, r = 1, 2, ..., R are the unitary matrices of r-mode singular vectors. Finally, the r-mode unfolding of a tensor A is symbolized by [A](r) ∈ CIr×(I1I2 ...Ir−1Ir+1...IR), i.e., it represents the matrix of r-mode vectors of the tensor A. The order of the columns is chosen in accordance with [6]. Data model To validate the general applicability of our proposed schemes, we adopt the PARAFAC data model below x0(m1, m2, . . . , mR+1) = d ∑ n=1 f (1) n (m1) · f (2) n (m2) . . . f (R+1) n (mR+1), (2) where f (r) n (mr)is the mrth element of the nth factor of the rth mode for mr = 1, ..., Mr and r = 1, 2, ..., R, R +1. The MR+1 can be alternatively represented by N, which stands for the number of snapshots. By defining the vectors f (r) n = [ f (r) n (1)f (r) n (2) . . . f (r) n (Mr) ]T and using the outer product operator ∘, another possible representation of (2) is given by