1 multidimensional model order selection. 2 motivation stock markets: one example of [1] [1]: m....

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Multidimensional Model Order SelectionMultidimensional Model Order Selection

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MotivationMotivation

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, 1997.

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.

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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. 2004.

Non-identified substanceRadiation

Wa

vele

ng

th

Oxidation statepH

samples

Result: successful application of tensor calculus.

In [2], the model order is estimated via the core consistency

analysis (CORCONDIA) by visual inspection.

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Sound source localization

[3]: A. Quinlan and F. Asano, “Detection of overlapping speech in meeting recordings using the modified exponential fitting test,” in Proc. 15th 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]

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Wind tunnel evaluation

[4]: C. El Kassis, “High-resolution parameter estimation schemes for non-uniform antenna arrays,” PhD Thesis, SUPELEC, Universite Paris-Sud XI, 2009. (Wind tunnel photo provided by Renault)

MOS: No technique is applied. [4]

Wind

Array

Source: Carine El Kassis [4].

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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)

MotivationMotivation Channel model

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

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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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IntroductionIntroduction

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.

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IntroductionIntroduction 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. 387-392, 1974.

[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. 2009.

[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.

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IntroductionIntroduction

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. 510 - 514, July 2008.

[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.

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Tensor algebraTensor algebra

3-D tensor = 3-way array

n-mode products between and

Unfoldings

M1

M2

M3

“1-mode vectors”



i.e., all the n-mode vectors multiplied from the left-hand-side by

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The Higher-Order SVD (HOSVD)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 partsignal part noise partnoise part

rank rank dd

Tensor data model

signal partsignal part noise partnoise part

rank rank dd

“Economy size SVD”

Low-rank approximation

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Exponential Fitting Test (EFT)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 profileof the noise eigenvaluesto 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

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Exponential Fitting Test (EFT)Exponential Fitting Test (EFT)

Start with P = 1

Predict M-1 based on M

Compare this predictionwith actual eigenvalue

relative distance:

In our case it agrees, we continue

d = 3, M = 8, SNR = 20 dB, N = 10

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Now, P = 2

Predict M-2 based on M-1 and M

relative distance

d = 3, M = 8, SNR = 20 dB, N = 10

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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

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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

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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

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RR-D Exponential Fitting Test-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

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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

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Comparison between the global eigenvalues profile and the profile of the last unfolding

R-D exponential profile

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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

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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 PARAFACSVD and PARAFAC

+ +=

+ +=

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HOSVD and PARAFACHOSVD 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. 3198 - 3213, July 2008.

[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.

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Closed-form solution to PARAFACClosed-form solution to PARAFAC

The task of PARAFAC analysis: Given (noisy) measurements

and the model order d, findsuch 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. 487 - 491, July 2008.

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

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For P = 2, i.e., P < d

Closed-form PARAFAC basedClosed-form PARAFAC basedModel Order SelectionModel 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. 473 - 476, July 2007.

[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. 349-352, Apr. 2009.

For P = 4, i.e., P > d

+ += +

+ += +

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ComparisonsComparisons

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ConclusionsConclusions 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.

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

1 multidimensional model order selection. 2 motivation stock markets: one example of [1] [1]: m....

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