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
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Estimation of Number of PARAFAC Components

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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
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Model Order Estimation

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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
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Operations on Tensors and Matrices
•
Unfoldings
•
n-mode product
i.e., all the n-mode vectors multiplied
from the left-hand-side by
1
2
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PARAFAC data model
Noiseless data representation

For R = 3 in case of noiseless data and
:
Problem
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Core Consistency Diagnostics
Alternating Least Squares for R = 3
(estimating the factors)

Therefore, we define the following function:
Core Consistency Definition
The closer is
to , the greater
is the probability of being less or
equal than the model order.
where
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and
Core Consistency Diagnostics
Example of Core Consistency Analysis

is defined as the threshold
distance between
and
 Example:
Hypothesis:
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Core Consistency Diagnostics
T-CORCONDIA Fix
Example:
 Example of 4-way PARAFAC:
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Core Consistency Diagnostics
T-CORCONDIA Var
 Example:
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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
10
 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
6
i

8
4

This allows to define the r-mode sample
covariance matrices
2
0
,
1
2
3
4
5
6
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Eigenvalue index i

The eigenvalues of

They are related to the higher-order singular values of the HOSVD of

In the HOSVD approach, we are limited to the cases, where
are denoted by
for
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through
.
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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
(modification w.r.t. original EFT)
(3) Comparison step: Compare estimate with observation.
 If
set P = P + 1, go to (2).
(4) The final estimate is

In general:
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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.
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R-D Exponential Fitting Test
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R-D exponential profile

The r-mode eigenvalues exhibit an exponential profile for every r

Assume

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
. Then we can define global eigenvalues
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R-D Exponential Fitting Test
Adaptive definition of the global eigenvalues


In general, the assumption
Without loss of generality, assume:
is not fulfilled
 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
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Simulations
Comparing the performance
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Conclusions and Main References
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
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 higherorder 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.
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