1 Multidimensional Model Order Selection

2 Motivation  Stock Markets: One example of [1] [1]: M. Loteran, “Generating market risk scenarios using principal components analysis: methodological and practical considerations”, in the Federal Reserve Board, March,  Information: Long Term Government Bond interest rates. Canada, USA, 6 European countries and Japan.  Result: by visual inspection of the Eigenvalues (EVD). Three main components: Europe, Asia and North America.

3 Motivation  Ultraviolet-visible (UV-vis) Spectrometry [2] [2]: K. S. Von Age, R. Bro, and P. Geladi, “Multi-way analysis with applications in the chemical sciences,” Wiley, Aug Non-identified substance Radiation Wavelength Oxidation state pH samples  Result: successful application of tensor calculus. In [2], the model order is estimated via the core consistency analysis (CORCONDIA) by visual inspection.

4 Motivation  Sound source localization [3]: A. Quinlan and F. Asano, “Detection of overlapping speech in meeting recordings using the modified exponential fitting test,” in Proc. 15 th European Signal Processing Conference (EUSIPCO 2007), Poznan, Poland. Microphone array Sound source 1 Sound source 2  Applications: interfaces between humans and robots and data processing.  MOS: Corrected Frequency Exponential Fitting Test [3]

5 Motivation  Wind tunnel evaluation [4]: C. El Kassis, “High-resolution parameter estimation schemes for non-uniform antenna arrays,” PhD Thesis, SUPELEC, Universite Paris-Sud XI, (Wind tunnel photo provided by Renault)  MOS: No technique is applied. [4] Wind Array Source: Carine El Kassis [4].

6 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) Motivation  Channel model

7 Motivation  An unlimited list of applications  Radar;  Sonar;  Communications;  Medical imaging;  Chemistry;  Food industry;  Pharmacy;  Psychometrics;  Reflection seismology;  EEG;  …

8 Outline  Motivation  Introduction  Tensor calculus  One dimensional Model Order Selection  Exponential Fitting Test  Multidimensional Model Order Selection (R-D MOS)  Novel contributions R-D Exponential Fitting Test (R-D EFT) Closed-form PARAFAC based model order selection (CFP-MOS)  Comparisons  Conclusions

9 Outline  Motivation  Introduction  Tensor calculus  One dimensional Model Order Selection  Exponential Fitting Test  Multidimensional Model Order Selection (R-D MOS)  Novel contributions R-D Exponential Fitting Test (R-D EFT) Closed-form PARAFAC based model order selection (CFP-MOS)  Comparisons  Conclusions

10 Introduction  The model order selection (MOS)  is required for the principal component analysis (PCA).  is the amount of principal components of the data.  has several schemes based on the Eigenvalue Decomposition (EVD).  can be estimated via other properties of the data, e.g., removing components until reaching the noise level or shift invariance property of the data.  The multidimensional model order selection (R-D MOS)  requires a multidimensional structure of the data, which is taken into account (this additional information is ignored by one dimensional MOS).  gives an improved performance compared to the MOS.  based on tensor calculus, e.g., instead of EVD and SVD, the Higher Order Singular Value Decomposition (HOSVD) [5] is computed. [5]: L. de Lathauwer, B. de Moor, and J. Vanderwalle, “A multilinear singular value decomposition”, SIAM J. Matrix Anal. Appl., vol. 21(4), 2000.

11 Introduction  A large number of model order selection (MOS) schemes have been proposed in the literature. However,  most of the proposed MOS schemes are compared only to Akaike’s Information Criterion (AIC) [6] and Minimum Description Length (MDL) [6];  the Probability of correct Detection (PoD) of these schemes is a function of the array size (number of snapshots and number of sensors).  In [7], we have proposed expressions for the 1-D AIC and 1-D MDL. Moreover, for matrix based data in the presence of white Gaussian noise, the Modified Exponential Fitting Test (M-EFT)  outperforms 12 state-of-the-art matrix based model order selection techniques for different array sizes.  For colored noise, the M-EFT is not suitable, as well as several other MOS schemes, and the RADOI [8] reaches the best PoD according to our comparisons. [6]: M. Wax and T. Kailath “Detection of signals by information theoretic criteria”, in IEEE Trans. on Acoustics, Speech, and Signal Processing, vol. ASSP-33, pp , [7]: J. P. C. L. da Costa, A. Thakre, F. Roemer, and M. Haardt, “Comparison of model order selection techniques for high-resolution parameter estimation algorithms,” in Proc. 54th International Scientific Colloquium (IWK), (Ilmenau, Germany), Sept [8]: E. Radoi and A. Quinquis, “A new method for estimating the number of harmonic components in noise with application in high resolution radar,” EURASIP Journal on Applied Signal Processing, 2004.

12 Introduction  One of the most well-known multidimensional model order selection schemes in the literature is the Core Consistency Analysis (CORCONDIA) [9]  a subjective MOS scheme, i.e., depends on the visual interpretation.  In [10], we have proposed the Threshold-CORCONDIA (T-CORCONDIA)  which is non-subjective, and its PoD is close, but still inferior to the 1-D AIC and 1-D MDL.  By taking into account the multidimensional structure of the data, we extend the M-EFT to the R-D EFT [10] for applications with white Gaussian noise.  For applications with colored noise, we proposed the Closed-Form PARAFAC based Model Order Selection (CFP-MOS) scheme,  which outperforms the state-of-the-art colored noise scheme RADOI [11]. [9]: R. Bro and H.A.L. Kiers. A new efficient method for determining the number of components in PARAFAC models. Journal of Chemometrics, 17:274–286,2003. [10]: J. P. C. L. da Costa, M. Haardt, and F. Roemer, “Robust methods based on the HOSVD for estimating the model order in PARAFAC models,” in Proc. 5-th IEEE Sensor Array and Multichannel Signal Processing Workshop (SAM 2008), (Darmstadt, Germany), pp , July [11]: J. P. C. L. da Costa, F. Roemer, and M. Haardt, “Multidimensional model order via closed-form PARAFAC for arbitrary noise correlations,” submitted to ITG Workshop on Smart Antennas 2010.

13 Outline  Motivation  Introduction  Tensor calculus  One dimensional Model Order Selection  Exponential Fitting Test  Multidimensional Model Order Selection (R-D MOS)  Novel contributions R-D Exponential Fitting Test (R-D EFT) Closed-form PARAFAC based model order selection (CFP-MOS)  Comparisons  Conclusions

14 Tensor algebra  3-D tensor = 3-way array  n-mode products between and  Unfoldings M1M1 M2M2 M3M3 “1-mode vectors” “2-mode vectors” “3-mode vectors” i.e., all the n-mode vectors multiplied from the left-hand-side by 1 2

15 The Higher-Order SVD (HOSVD)  Singular Value Decomposition  Higher-Order SVD (Tucker3) “Full HOSVD” Low-rank approximation (truncated HOSVD) “Economy size HOSVD” “Full SVD”  Matrix data model signal part noise part rank d  Tensor data model signal part noise part rank d “Economy size SVD” Low-rank approximation

16 Outline  Motivation  Introduction  Tensor calculus  One dimensional Model Order Selection  Exponential Fitting Test  Multidimensional Model Order Selection (R-D MOS)  Novel contributions R-D Exponential Fitting Test (R-D EFT) Closed-form PARAFAC based model order selection (CFP-MOS)  Comparisons  Conclusions

17 Exponential Fitting Test (EFT)  Observation is a superposition of noise and signal  The noise eigenvalues still exhibit the exponential profile [12,13]  We can predict the profile of the noise eigenvalues to find the “breaking point”  Let P denote the number of candidate noise eigenvalues. choose the largest P such that the P noise eigenvalues can be fitted with a decaying exponential d = 3, M = 8, SNR = 20 dB, N = 10 [12]: J. Grouffaud, P. Larzabal, and H. Clergeot, “Some properties of ordered eigenvalues of a wishart matrix: application in detection test and model order selection,” in Proceedings of the IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP’96). [13]: A. Quinlan, J.-P. Barbot, P. Larzabal, and M. Haardt, “Model order selection for short data: An exponential fitting test (EFT),” EURASIP Journal on Applied Signal Processing, 2007

18 Exponential Fitting Test (EFT)  Start with P = 1  Predict M-1 based on M  Compare this prediction with actual eigenvalue  relative distance:  In our case it agrees, we continue d = 3, M = 8, SNR = 20 dB, N = 10

19 Exponential Fitting Test (EFT)  Now, P = 2  Predict M-2 based on M-1 and M  relative distance d = 3, M = 8, SNR = 20 dB, N = 10

20 Exponential Fitting Test (EFT)  Now, P = 3  Predict M-3 based on M-2, M-1, and M  relative distance d = 3, M = 8, SNR = 20 dB, N = 10

21 Exponential Fitting Test (EFT)  Now, P = 4  Predict M-4 based on M-3, M-2, M-1, and M  relative distance d = 3, M = 8, SNR = 20 dB, N = 10

22 Exponential Fitting Test (EFT)  Now, P = 5  Predict M-5 based on M-4, M-3, M-2, M-1, and M  relative distance  The relative distance becomes very big, we have found the break point. d = 3, M = 8, SNR = 20 dB, N = 10

23 Outline  Motivation  Introduction  Tensor calculus  One dimensional Model Order Selection  Exponential Fitting Test  Multidimensional Model Order Selection (R-D MOS)  Novel contributions R-D Exponential Fitting Test (R-D EFT) Closed-form PARAFAC based model order selection (CFP-MOS)  Comparisons  Conclusions

24 R-D Exponential Fitting Test  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  r-mode eigenvalues

25 R-D Exponential Fitting Test  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 profile

26 R-D Exponential Fitting Test  Comparison between the global eigenvalues profile and the profile of the last unfolding  R-D exponential profile

27 R-D Exponential Fitting Test  Is an extended version of the M-EFT operating on the  Exploits the fact that the global eigenvalues still exhibit an exponential profile  The enhanced SNR and the improved fit lead to significant improvements in the performance  Is able to adapt to arrays of arbitrary size and dimension through the adaptive definition of global eigenvalues  R-D EFT

28 Outline  Motivation  Introduction  Tensor calculus  One dimensional Model Order Selection  Exponential Fitting Test  Multidimensional Model Order Selection (R-D MOS)  Novel contributions R-D Exponential Fitting Test (R-D EFT) Closed-form PARAFAC based model order selection (CFP-MOS)  Comparisons  Conclusions

29  Another way to look at the SVD  decomposition into a sum of rank one matrices  also referred to as principal components (PCA)  Tensor case: SVD and PARAFAC ++ = ++ =

30 HOSVD and PARAFAC  HOSVD  PARAFAC  Identity tensor  Core tensor Core tensor usually is full. R-D STE [14] Identity tensor is always diagonal. CFP-PE [15] [14]: M. Haardt, F. Roemer, and G. Del Galdo, ``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 , July [15]: J. P. C. L. da Costa, F. Roemer, and M. Haardt, “Robust R-D parameter estimation via closed-form PARAFAC,” submitted to ITG Workshop on Smart Antennas 2010.

31 Closed-form solution to PARAFAC  The task of PARAFAC analysis: Given (noisy) measurements and the model order d, find such that Here is the higher-order Frobenius norm (sum of squared magnitude of all elements). [16]:F. Roemer and M. Haardt, “A closed-form solution for multilinear PARAFAC decompositions,” in Proc. 5- th IEEE Sensor Array and Multichannel Signal Processing Workshop (SAM 2008), (Darmstadt, Germany), pp , July  Our approach: based on simultaneous matrix diagonalizations (“closed-form”).  By applying the closed-form PARAFAC (CFP) [16]  R*(R-1) simultaneous matrix diagonalizations (SMD) are possible;  R*(R-1) estimates for each factor are possible;  selection of the best solution by different heuristics (residuals of the SMD) is done

32  For P = 2, i.e., P < d Closed-form PARAFAC based Model Order Selection + = + =  Assuming that d = 3, and solutions with the two smallest residuals of the SMD.  Using the same principle as in [17], the error is minimized when P = d.  Due to the permutation ambiguities, the components of different tensors are ordered using the amplitude based approach proposed in [18]. [17]:J.-M. Papy, L. De Lathauwer, and S. Van Huffel, “A shift invariance-based order-selection technique for exponential data modelling,” in IEEE Signal Processing Letters, vol. 14, No. 7, pp , July [18]:M. Weis, F. Roemer, M. Haardt, D. Jannek, and P. Husar, “Multi-dimensional Space-Time-Frequency component analysis of event-related EEG data using closed-form PARAFAC,” in Proc. IEEE Int. Conf. Acoust., Speech, and Signal Processing (ICASSP), (Taipei, Taiwan), pp , Apr  For P = 4, i.e., P > d ++ = + ++ = +

33 Outline  Motivation  Introduction  Tensor calculus  One dimensional Model Order Selection  Exponential Fitting Test  Multidimensional Model Order Selection (R-D MOS)  Novel contributions R-D Exponential Fitting Test (R-D EFT) Closed-form PARAFAC based model order selection (CFP-MOS)  Comparisons  Conclusions

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52 Outline  Motivation  Introduction  Tensor calculus  One dimensional Model Order Selection  Exponential Fitting Test  Multidimensional Model Order Selection (R-D MOS)  Novel contributions R-D Exponential Fitting Test (R-D EFT) Closed-form PARAFAC based model order selection (CFP-MOS)  Comparisons  Conclusions

53 Conclusions  State-of-the-art one dimensional and multidimensional model order selection techniques were presented;  For one dimensional scenarios:  in the presence of white Gaussian noise Modified Exponential Fitting Test (M-EFT)  in the presence of severe colored Gaussian noise RADOI  For multidimensional scenarios:  in the presence of white Gaussian noise R-dimensional Exponential Fitting Test (R-D EFT)  in the presence of colored noise Closed-form PARAFAC based Model Order Selection (CFP-MOS) scheme  The mentioned schemes are applicable to problems with a PARAFAC data model, which are found in several scientific fields.

54 Thank you for your attention! Vielen Dank für Ihre Aufmerksamkeit!