ilmenau university of technology communications research laboratory 1 estimation of number of...

Ilmenau University of TechnologyCommunications Research Laboratory

1Estimation of Number of PARAFAC ComponentsEstimation of Number of PARAFAC Components

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

IntroductionIntroduction


2

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

Estimation of Number of PARAFAC ComponentsEstimation of Number of PARAFAC Components


3Model Order EstimationModel 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


4Operations on Tensors and MatricesOperations on Tensors and Matrices

• n-mode product

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

11 2233

• Unfoldings


5

For R = 3 in case of noiseless data and :

PARAFAC data modelPARAFAC data model

Noiseless data representationNoiseless data representation

ProblemProblem


6Core Consistency DiagnosticsCore Consistency Diagnostics

The closer is to , the greater is the probability of being less or equal than the model order.

Alternating Least Squares for Alternating Least Squares for R R = 3= 3

Core Consistency DefinitionCore Consistency Definition

where and

(estimating the factors)

Therefore, we define the following function:


7

Example of Core Consistency AnalysisExample of Core Consistency Analysis

Hypothesis:Hypothesis:

Core Consistency DiagnosticsCore Consistency Diagnostics

is defined as the threshold

distance between and

Example:



T-CORCONDIA FixT-CORCONDIA Fix

Example of 4-way PARAFAC:

Example:



T-CORCONDIA VarT-CORCONDIA Var


10

d = 2, M1 = 8, SNR = 0 dB, M2 = 10

Exponential Fitting Test and R-D EigenvaluesExponential Fitting Test and R-D Eigenvalues

The eigenvalues of the sample covariance matrixThe eigenvalues of the sample covariance matrix Finite SNR, Finite sample size

M - d noise eigenvalues thatcan be approximated byan 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 .

1 2 3 4 5 6 7 80

2

4

6

8

10

Eigenvalue index i i

,


11

M-EFT algorithmM-EFT algorithm

Modified Exponential Fitting Test (M-EFT)Modified Exponential Fitting Test (M-EFT)

In general:

(modification w.r.t. original EFT)

(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


12

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.

Modified Exponential Fitting Test (M-EFT)Modified Exponential Fitting Test (M-EFT)

Determining the threshold coefficientsDetermining 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


13RR-D Exponential Fitting Test-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

RR-D exponential profile-D exponential profile


14RR-D Exponential Fitting Test-D Exponential Fitting Test

Adaptive definition of the global eigenvaluesAdaptive 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


15

Comparing the performanceComparing the performance

SimulationsSimulations


16 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.

Conclusions and Main ReferencesConclusions and Main References

ilmenau university of technology communications research laboratory 1 estimation of number of...

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