Universidade de Brasília Laboratório de Processamento de Sinais em Arranjos 1 Subspace based Multi-Dimensional Model Order Selection in Colored Noise Scenarios João Paulo C. Lustosa da Costa, Florian Roemer, Dominik Schulz, and Rafael Timóteo de Sousa Jr. University of Brasília (UnB) Department of Electrical Engineering (ENE) Laboratory of Array Signal Processing PO Box 4386 Zip Code , Brasília - DF Homepage:

Universidade de Brasília Laboratório de Processamento de Sinais em Arranjos Outline Motivation Data Model and Goal ESTER R-D ESTER Simulations Conclusions 2

Universidade de Brasília Laboratório de Processamento de Sinais em Arranjos Outline Motivation Data Model and Goal ESTER R-D ESTER Simulations Conclusions 3

Universidade de Brasília Laboratório de Processamento de Sinais em Arranjos 4 Motivation The model order selection (MOS) problem is encountered in a variety of signal processing applications including radar, sonar, communications, channel modeling, malicious activity detection in computer networks [1], medical imaging, and the estimation of the parameters of the dominant multipath components from MIMO channel measurements. Not only for signal processing applications, but also in several science fields, e.g., chemistry, food industry, stock markets, pharmacy and psychometrics, the MOS problem is investigated. It has been shown in [2] that for data contaminated by white noise, the R-D Exponential Fitting Test (R-D EFT) outperforms all the other model order selection schemes proposed in the literature. Since for data contaminated by colored noise, the R-D EFT is not applicable, the Closed-Form PARAFAC based Model Order Selection (CFP-MOS) scheme has been proposed [2]. [1]: B. M. David et al., "Blind Automatic Malicious Activity Detection in Honeypot Data," The International Conference on Forensic Computer Science (ICoFCS) [2]: J. P. C. L. da Costa, F. Roemer, M. Haardt, and R. T. de Souza Jr., ``Multi-Dimensional Model Order Selection,'' EURASIP Journal on Advances in Signal Processing 2011:26, July 2011, Springer publisher.

Universidade de Brasília Laboratório de Processamento de Sinais em Arranjos 5 Motivation The computational complexity of CFP-MOS is prohibitive for several applications. Therefore, another multidimensional model order selection scheme for colored noise scenarios is required. In this paper, we propose a multi-dimensional extension of Estimation Error (ESTER) [3]. Since in [4] ESTER is outperformed by Subspace based Automatic Model Order Selection (SAMOS) scheme, we also compare it to our proposed R-D ESTER. [3]: R. Badeau, B. David, and G. Richard, Selecting the modeling order for the ESPRIT high resolution method an alternative approach, In Proc. IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP 2004), Montreal, Canada May [4]: J.-M. Papy, L. de Lathauwer, and S. Van Huffel, A shift invariance-based order-selection technique for exponential data modeling, in IEEE Signal Processing Letters, Vol. 14, pp , Jul

Universidade de Brasília Laboratório de Processamento de Sinais em Arranjos Outline Motivation Data Model and Goal ESTER R-D ESTER Simulations Conclusions 6

Universidade de Brasília Laboratório de Processamento de Sinais em Arranjos 7 Data Model and Goal Noiseless case Our objective is to estimate d from the noisy observations. Matrix data model ++ =

Universidade de Brasília Laboratório de Processamento de Sinais em Arranjos 8 Colored noise model Data Model and Goal

Universidade de Brasília Laboratório de Processamento de Sinais em Arranjos Receive array: 1-D or 2-D Frequency Time Transmit array: 1-D or 2-D Direction of Arrival (DOA) Delay Doppler shift Direction of Departure (DOD) Example of MultiDimensional Data Example of Multi-Dimensional Data Channel model 5

Universidade de Brasília Laboratório de Processamento de Sinais em Arranjos Data Model and Goal Noiseless data representation Problem where is the colored noise tensor. = + + Our objective is to estimate d from the noisy observations. 17

Universidade de Brasília Laboratório de Processamento de Sinais em Arranjos Outline Motivation Data Model and Goal ESTER R-D ESTER Simulations Conclusions 11

Universidade de Brasília Laboratório de Processamento de Sinais em Arranjos ESTER (1) Estimation of Signal Parameters via Rotation Invariance Techniques ESPRIT 12 d Shift Invariance Equation

Universidade de Brasília Laboratório de Processamento de Sinais em Arranjos ESTER (2) Estimation of Signal Parameters via Rotation Invariance Techniques ESPRIT 13 The steering matrix A and the first d eigenvectors U of the covariance matrix generate the same subspace Note that U s is related to the low-rank approximation. Spatial frequencies

Universidade de Brasília Laboratório de Processamento de Sinais em Arranjos ESTER (3) [3] ESTER Algorithm [3] 14

Universidade de Brasília Laboratório de Processamento de Sinais em Arranjos Outline Motivation Data Model and Goal ESTER R-D ESTER Simulations Conclusions 15

Universidade de Brasília Laboratório de Processamento de Sinais em Arranjos 16 Operations on Tensors (Multidimensional Arrays) Unfoldings Concatenation

Universidade de Brasília Laboratório de Processamento de Sinais em Arranjos 17 Operations on Tensors and Matrices n-mode product i.e., all the n-mode vectors multiplied from the left-hand-side by 1 2 3

Universidade de Brasília Laboratório de Processamento de Sinais em Arranjos 18 Review: The SVD of Matrices Full SVD Economy size SVD Low-rank approximation

Universidade de Brasília Laboratório de Processamento de Sinais em Arranjos 19 Extension to the HOSVD of Tensors Full HOSVD Low-rank approximation (truncated HOSVD) Economy size HOSVD

Universidade de Brasília Laboratório de Processamento de Sinais em Arranjos R-D ESTER (1) R-D ESTER Algorithm 20

Universidade de Brasília Laboratório de Processamento de Sinais em Arranjos Outline Motivation Data Model and Goal ESTER R-D ESTER Simulations Conclusions 21

Universidade de Brasília Laboratório de Processamento de Sinais em Arranjos Simulations 22

Universidade de Brasília Laboratório de Processamento de Sinais em Arranjos Simulations 23

Universidade de Brasília Laboratório de Processamento de Sinais em Arranjos Outline Motivation Data Model and Goal ESTER R-D ESTER Simulations Conclusions 24

Universidade de Brasília Laboratório de Processamento de Sinais em Arranjos Conclusions In this work, we have proposed a multi-dimensional extension of the ESTER (R-D ESTER) scheme. By taking into account the multi-dimensional structure of the data, the R-D ESTER outperforms significantly showing its matrix based version. Similarly to the R-D EFT, the R-D ESTER is also based on the HOSVD. 25