Estimation of Number of PARAFAC Components Introduction Encountered in a variety of applications mobile communications, spectroscopy, multi-dimensional medical imaging, finances, food industry, and the estimation of the parameters of the dominant multipath components from MIMO channel sounder measurements Since the measured data is multi-dimensional, traditional approaches require stacking the dimensions into one highly structured matrix In [Haardt, Roemer, Del Galdo, 2008] we have shown how an HOSVD based low-rank approximation of the measurement tensor leads to an improved signal subspace estimate can be exploited in any multi-dimensional subspace-based estimation scheme to achieve this goal, it is required to estimate the model order of the multi-dimensional data

Estimation of Number of PARAFAC Components State-of-the-art model order estimation techniques for PARAFAC data [Bro, Kiers, 2003] include methods such as LOSS function, RELFIT and CORE CONSISTENCY DIAGNOSTICS (CORCONDIA) CORCONDIA method is iterative and subjective, very high computational complexity depends on subjective interpretation of the Core Consistency evaluation of CORCONDIA in terms of Probability of Detection (PoD) is difficult To avoid this subjectivity, we propose two versions of CORCONDIA T-CORCONDIA Fix performs a one-dimensional search for the threshold coefficients, but as consequence the PoD varies for different numbers of paths T-CORCONDIA Var performs a multi-dimensional search for threshold coefficients, but we restrict the PoD to be similar for different numbers of paths

Model Order Estimation The R-Dimensional Exponential Fitting Test (R-D EFT) [da Costa, Haardt, Roemer, Del Galdo, 2007] is a multi-dimensional extension of the Modified Exponential Fitting Test (M-EFT) and is based on the HOSVD of the measurement tensor, also enables us to improve the model order estimation step only one set of eigenvalues is available in the matrix case applying the HOSVD, we obtain R sets of n-mode singular values of the measurement tensor that are combined to form global eigenvalues improve the model order selection accuracy of EFT significantly as compared to the matrix case Inspired by the good performance of R-D EFT, the R-D Akaike Information Criterion (R-D AIC) and R-D Minimum Description Length (R-D MDL) were developed We compare the performance between the multi-dimensional techniques based on HOSVD and the traditional solution for estimating the model order of a PARAFAC tensor based on CORCONDIA

Operations on Tensors and Matrices Unfoldings n-mode product i.e., all the n-mode vectors multiplied from the left-hand-side by 3 1 2

PARAFAC data model Noiseless data representation For R = 3 in case of noiseless data and : Problem

Core Consistency Diagnostics (estimating the factors) Therefore, we define the following function: Alternating Least Squares for R = 3 Core Consistency Definition The closer is to , the greater is the probability of being less or equal than the model order. where and

Core Consistency Diagnostics Example of Core Consistency Analysis is defined as the threshold distance between and Example: Hypothesis:

Core Consistency Diagnostics T-CORCONDIA Fix Example: Example of 4-way PARAFAC:

Core Consistency Diagnostics T-CORCONDIA Var Example:

Exponential Fitting Test and R-D Eigenvalues The eigenvalues of the sample covariance matrix d = 2, M1 = 8, SNR = 0 dB, M2 = 10 Finite SNR, Finite sample size M - d noise eigenvalues that can be approximated by an exponential profile d signal plus noise eigenvalues In the R-D case, we have a measurement tensor This allows to define the r-mode sample covariance matrices The eigenvalues of are denoted by for They are related to the higher-order singular values of the HOSVD of through , In the HOSVD approach, we are limited to the cases, where .

Modified Exponential Fitting Test (M-EFT) M-EFT algorithm (1) Set the number of candidate noise eigenvalues to P = 1 (2) Estimation step: Estimate noise eigenvalue Mr - P (3) Comparison step: Compare estimate with observation. If set P = P + 1, go to (2). (4) The final estimate is (modification w.r.t. original EFT) In general:

Modified Exponential Fitting Test (M-EFT) Determining the threshold coefficients Every threshold-based detection scheme: we follow the CFAR approach (constant false alarm ratio), where is set manually (i.e., 10-6), and then For every P: vary and determine numerically the probability to detect a signal in noise-only data. Then choose such that the desired is met.

R-D Exponential Fitting Test R-D exponential profile The r-mode eigenvalues exhibit an exponential profile for every r Assume . Then we can define global eigenvalues The global eigenvalues also follow an exponential profile, since The product across modes enhances the signal-to-noise ratio and improves the fit to an exponential profile

R-D Exponential Fitting Test Adaptive definition of the global eigenvalues In general, the assumption is not fulfilled Without loss of generality, assume: Start by estimating d with the M-EFT method considering only If we can take advantage of the second unfolding. We therefore run a 2-D EFT on If the new estimate we can continue considering the first three unfoldings, i.e., we use a 3-D EFT on We continue until or

Simulations Comparing the performance

Conclusions and Main References In this contribution, we generalize the data model proposed in [da Costa, Haardt, Roemer, Del Galdo, 2007] to the PARAFAC data model and we apply successfully the extended model order estimation schemes called R-D EFT, R-D MDL, and R-D AIC. We also propose two versions of T-CORCONDIA, a non-subjective form of CORCONDIA [Bro, Kiers, 2003]. T-CORCONDIA Fix performs a one-dimensional search for the calculation of the threshold coefficients, and its drawback is a different Probability of Detection for each number of sources. T-CORCONDIA Var uses a multi-dimensional search, and it finds a similar profile for all the Probability of Detection curves for different numbers of sources. Note that all the HOSVD-based techniques outperform T-CORCONDIA for the PARAFAC data model. Note also that the R-D methods that are based on the HOSVD have a much lower computational complexity. [da Costa, Haardt, Roemer, Del Galdo, 2007]: Ehanced model order estimation using higher-order arrays. In Proc. 41st Asilomar Conference on Signals, Systems, and Computers, pages 412-416, Pacific Grove, CA, USA, November 2007. [Haardt, Roemer, Del Galdo, 2008]: Higher-order SVD based subspace estimation to improve the parameter estimation accuracy in multi-dimensional harmonic retrieval problems. IEEE Trans. Signal Processing, vol. 56, pp. 3198-3213, July 2008. [Bro, Kiers, 2003]: A new efficient method for determining the number of components in PARAFAC models. Journal of Chemometrics, vol. 17, pp. 274-286, 2003.