1

Model Order Selection Rules For Covariance
Structure Classification

arXiv:1704.05927v1 [] 19 Apr 2017

V. Carotenuto, Member, IEEE, A. De Maio, Fellow, IEEE, D. Orlando, Senior Member, IEEE, P. Stoica, Fellow,
IEEE

Abstract—The adaptive classification of the interference covariance matrix structure for radar signal processing applications
is addressed in this paper. This represents a key issue because
many detection architectures are synthesized assuming a specific
covariance structure which may not necessarily coincide with the
actual one due to the joint action of the system and environment
uncertainties. The considered classification problem is cast in
terms of a multiple hypotheses test with some nested alternatives
and the theory of Model Order Selection (MOS) is exploited
to devise suitable decision rules. Several MOS techniques, such
as the Akaike, Takeuchi, and Bayesian information criteria
are adopted and the corresponding merits and drawbacks are
discussed. At the analysis stage, illustrating examples for the
probability of correct model selection are presented showing the
effectiveness of the proposed rules.

I. N OTATION
In the sequel, vectors and matrices are denoted by boldface
lower-case and upper-case letters, respectively. The symbols
det(·), Tr (·), ⊗, (·)∗ , (·)T , (·)† denote the determinant,
trace, Kronecker product, complex conjugate, transpose, and
conjugate transpose, respectively. As to numerical sets, R is
the set of real numbers, RN ×M is the Euclidean space of
(N × M )-dimensional real matrices (or vectors if M = 1), C
is the set of complex numbers, and CN ×M is the Euclidean
space of (N × M )-dimensional complex matrices (or vectors
if M = 1). The symbols ℜ {z} and ℑ {z} indicate the real
and imaginary parts of the complex number z, respectively. I N
stands for the N ×N identity matrix, while 0 is the null vector
or matrix of proper dimensions. We denote by J ∈ RN ×N
a permutation matrix such that J (l, k) = 1 if and only if
l + k = N + 1. Given a matrix A = [a1 , . . . , aM ] ∈ CN ×M ,
vec (A) = [aT1 , aT2 , . . . , aTM ]T ∈ CN M×1 , while given a
vector a ∈ CN ×1 , diag (a) ∈ CN ×N indicates the diagonal
matrix whose ith diagonal element is the ith entry of a.
The Euclidean norm of a vector is denoted by k · k. We
write M ≻ 0 if M is positive definite. Let f (x) ∈ R be
a scalar-valued function of vector argument, then ∂f (x)/∂x
denotes the gradient of f (·) with respect to x arranged in a
column vector, while ∂f (x)/∂xT is its transpose. Moreover, if
b belongs to the domain of f (·), then the gradient of f (·) with
x
V. Carotenuto and A. De Maio are with the Dipartimento di Ingegneria Elettrica e delle Tecnologie dell’Informazione, Università degli Studi
di Napoli “Federico II”, via Claudio 21, I-80125 Napoli, Italy. E-mail:
ademaio@unina.it, vincenzo.carotenuto@unina.it.
D. Orlando is with Università degli Studi “Niccolò Cusano”, via Don Carlo Gnocchi 3, 00166 Roma, Italy. E-mail:
danilo.orlando@unicusano.it.
P. Stoica is with the Department of Information Technology, Uppsala University, P O Box 337, SE-751 05, Uppsala, Sweden. E-mail:
peter.stoica@it.uu.se.

b is denoted by ∂f (b
respect to x and evaluated at x
x)/∂x. For
N ×N
a finite set A, |A| stands for its cardinality. U (N ) ⊂ C√
denotes the set of all N × N unitary matrices and j = −1.
For two sets, A and B, A× B denotes their Cartesian product.
The (k, l)-entry (or l-entry) of a generic matrix A (or vector
a) is denoted by A(k, l) (or a(l)). Given two statistical
hypotheses Hi and Hj , then Hi ⊂ Hj means that Hi is
nested into Hj . The acronym i.i.d. means independent and
identically distributed while the symbol E[·] denotes statistical
expectation. Finally, we write x ∼ CN N (m, M ) if x is a
complex circular N -dimensional normal vector with mean m
and covariance matrix M ≻ 0, x ∼ N N (m, M ) if x is a
N -dimensional normal vector with mean m and covariance
matrix M ≻ 0, and ϕ ∼ U(0, 2π) if ϕ is a random variable
uniformly distributed in (0, 2π).
II. I NTRODUCTION , M OTIVATION ,
F ORMULATION

AND

P ROBLEM

Consider a radar system equipped with N ≥ 2 (spatial
and/or temporal) channels. The echoes from the cell under
test (CUT) are downconverted to baseband, pre-processed,
properly sampled, and organized to form a N -dimensional
vector, z say, referred to as primary data or CUT sample. A
set of secondary data, z 1 , . . . , z K , with K > N , statistically
independent of z, is also acquired in order to make the
system adaptive with respect to the unknown Interference
Covariance Matrix (ICM), M ≻ 0. As is customary, these
data are assumed to share the same ICM as z and are obtained
exploiting echoes from range cells in the proximity of the CUT
within the reference window [1]–[11].
To accomplish the detection task which is typical of the
search process, the radar signal processor solves a testing
problem applying a decision rule computed from the collected
data (decision statistic). From a mathematical viewpoint, target
detection can be formulated in terms of a binary hypothesis test
and tools provided by the Decision Theory can be exploited
to solve it. Several design criteria have been adopted in this
respect: the Generalized Likelihood Ratio Test (GLRT) [1],
[12]–[15], the Wald test [16]–[18], [18]–[20], the Rao test1
[7], [11], [18]–[20], [22], and the Invariance Principle [23]–
[28].
Usually a given design technique is applied under specific
assumptions on the ICM structure which are tantamount to
incorporating some degree of a priori knowledge at the design
stage. Specifically, certain structures of the covariance M
1 Note that GLRT, Wald test, and Rao test, under mild conditions, are
asymptotically equivalent [21].

2

can be induced by the interference type, the geometry of the
system array, and/or uniformity of the transmitted pulse train.
In the most general case, M ∈ CN ×N is Hermitian, but it is
well-known that:
ground clutter, observed by a stationary monostatic radar,
often exhibits a symmetric power spectral density centered around the zero-Doppler frequency implying that
the resulting ICM is real, i.e., M ∈ RN ×N [29];
• from a theoretical point of view, symmetrically spaced
linear arrays or pulse trains induce a persymmetric structure on M [30]; the following two cases are possible
– M ∈ CN ×N is Hermitian and persymmetric (or
centrohermitian) if and only if M = J M ∗ J ;
– M ∈ RN ×N is symmetric and persymmetric (or
centrosymmetric) if and only if M = J M J .
For each of the mentioned scenarios, there exist examples of
adaptive detectors in the literature [4], [5], [31]. The knowledge about the environment as well as the structure of the
ICM can guide the system operator towards the most appropriate decision scheme. In this regard, the primary sources of
available information are directly related to the system and/or
to the operating scenario. However, there exist a plethora
of causes that introduce uncertainty and make the nominal
assumptions no longer valid. For instance, array calibration
errors would produce residual imbalances among channels that
can heavily degrade the ICM persymmetric structure. Another
example concerns the level of symmetry of ground clutter
power spectral density which can be altered by the possible
presence of a dominating Doppler or some discretes with a
given velocity. This motivates the need for a classifier capable
of inferring the ICM structure over the range bins of the
system reference window. Its output could then be fed to a
selector choosing the most suitable detection scheme as shown
in Figure 1.
A possible approach to handle the mentioned classification
problem is based on its formulation in terms of a multiple
hypothesis test and on the use of model order selection (MOS)
rules, since each possible choice for M represents a model
with a given number of parameters [32]–[39]. Following this
idea, it is worth making explicit the relationship between
parameters and model. To this end, note that the number of
parameters introduced by the specific structure of M can be
stacked into a vector θi ∈ Rmi ×1 , where mi depends on
the specific scenario. Since the entries of θi parameterize M ,
this dependence is denoted using the notation M (θ i ). Finally,
the considered models (or hypotheses) are representative of
combinations among the possible assumptions on the clutter
spectrum (symmetry around zero-Doppler or the lack of the
mentioned symmetry) and the system configuration (persymmetry).
In summary, the problem at hand is tantamount to choosing
among the following hypotheses:

N ×N
is Hermitian unstructured,

H1 : M (θ 1 ) ∈ C

H : M (θ ) ∈ RN ×N is symmetric unstructured,
2
2
(1)
H3 : M (θ 3 ) ∈ CN ×N is centrohermitian,



H4 : M (θ 4 ) ∈ RN ×N is centrosymmetric.
•

The number of unknown parameters under each hypothesis is
given by:

m1 = N 2
under H1 ,





m2 = N (N + 1)/2
under H2 ,




m3 = N (N + 1)/2
under H3 ,

 
(2)
N
N



+1
if N is even



2
2


2

under H4 .
m4 = 


N
+
1





if N is odd
2

For the sake of clarity, the proofs of (2) for the cases centrohermitian and centrosymmetric are provided in Appendix
A.
Hereafter, for brevity, we omit the dependence on θi letting
M i = M (θi ) and X i = M −1 (θ i ).
Before concluding this section, a few remarks are in order.
First, notice that different models could have the same number
of parameters but, as shown in the next sections, this is not
a limitation since classification rules exploit specific estimates
corresponding to the different structures reflecting the assumed
hypothesis. Second, it is possible to identify nested hypotheses
among those listed in (1), for instance H2 ⊂ H1 , H3 ⊂ H1 ,
H4 ⊂ H2 , etc.
In the next section, several MOS classification algorithms
for problem (1) are briefly described highlighting the respective design assumptions, which might not be always met
in the considered radar application. The latter observation
means that the behavior of these classification rules versus the
parameters of interest deserves a careful investigation. Section
IV provides closed-form expressions for the classification
statistics discussed in Section III. Concretely, these statistics
are computed according to two approaches. The first exploits
the overall data matrix which also comprises the CUT, whereas
the second neglects the CUT and uses secondary data only.
The performances of the considered selectors are analyzed
in Section V, where the figure of merit is the probability of
correct classification as a function of the number of data used
for estimation. Finally, concluding remarks and future research
tracks are given in Section VI. Mathematical derivations are
confined to the appendices.
III. M ODEL O RDER S ELECTION C RITERIA

The aim of this section is twofold. The first part provides
useful preliminary definitions, while the second part presents
a brief review of the adopted selection criteria for problem
(1). Subsequent developments assume that z k ∼ CN N (0, M ),
k = 1, . . . , K, and z ∼ CN N (αv, M ), where α = αre +
jαim , αre , αim ∈ R, is an amplitude factor accounting for
target response and propagation effects and v ∈ CN ×1 is the
nominal steering vector. Finally, the vectors z 1 , . . . , z K , z are
assumed to be statistically independent.
Now, denote by Z = [z 1 , . . . , z K ] ∈ CN ×K the entire
secondary data matrix and let pi be the parameter vector under
the Hi hypothesis, i = 1, . . . , 4. Observe that
• if the CUT is incorporated into the classification rules,
then pi = [θTi αT ]T ∈ Rni ×1 , where α = [αre αim ]T ∈
R2×1 , ni = mi + 2; in this case, we let Zc = {z, Z};

3

if the the classification rules are devised from Z only,
then pi = θ i ∈ Rni , where ni = mi ; here we let Zc =
{Z}.
Because the derivation of the MOS criteria requires the computation of the maximum likelihood estimates (MLE) of the
unknown parameters as well as suitable estimates of the Fisher
Information Matrix (FIM), let us provide the expressions of the
probability density functions (pdfs) of z, z k , k = 1, . . . , K,
Z, and the joint pdf of z and Z = [z 1 , . . . , z K ] ∈ CN ×K
under the considered hypotheses, namely, ∀i = 1, . . . , 4:

exp −(z − αv)† X i (z − αv)
, (3)
f (z; pi , Hi ) =
π N det(M i )
o
n
exp −z †k X i z k
f (z k ; θi , Hi ) =
, k = 1, . . . , K, (4)
π N det(M i )
•

f (Z; pi , Hi ) =

K
Y

f (z k ; θi , Hi )

k=1

=

(

)K
 1
Tr [X i S]
exp − K
π N det(M i )

f (z, Z;pi , Hi ) = f (z; pi , Hi )

K
Y

(5)

f (z k ; θi , Hi )

k=1

=

o K+1
n

1
 exp − K+1
Tr [X i (S α + S)] 


π N det(M i )

(6)



where S α = (z − αv)(z − αv)† and S = ZZ † .
Finally, denote by s(pi , Hi ; z) = log f (z; pi , Hi ),
s(θ i , Hi ; z k ) = log f (z k ; θi , Hi ), k = 1, . . . , K, and let

s(pi , Hi ; z, Z) = log f (z, Z; pi , Hi ) ,



if the CUT is included,
s(pi , Hi ; Zc ) =


s(pi , Hi ; Z) = log f (Z; pi , Hi ) ,

if the CUT is excluded,
(7)
denote the log-likelihood functions2.
The remainder of this section is focused on MOS criteria.
Several of such criteria have been developed for the selection
of an estimated best approximating model from a set of
candidates [40]; most of them rely on minimization of the
Kullback-Leibler (KL) discrepancy. A well-known rule is the
Akaike Information Criterion (AIC), which, with reference to
problem (1), can be formulated as
Hbi = arg min {−2s(b
pi , Hi ; Zc ) + 2ni , } ,
H

(AIC) (8)

bi
where Hbi is the estimated model, H = {H1 , . . . , H4 }, and p
is the MLE of pi . The main drawback of this rule is its nonzero probability of overfitting [33] due to the penalty term 2ni
being too small for high-order models, especially for nested
hypotheses. To overcome this limitation, an empirical modification of AIC has been proposed in [41]. This rule, referred
2 Observe

that α is a nuisance parameter with respect to problem (1).

to as Generalized Information Criterion (GIC), corrects the
penalty term of AIC via a factor (1 + ρ) with ρ > 1, namely
Hbi = arg min {−2s(b
pi , Hi ; Zc ) + (1 + ρ)ni }
H

(GIC).
(9)

Note that if we set ρ = 1 GIC reduces to AIC.
The Takeuchi Information Criterion (TIC), whose main goal
is to extend AIC to mismodeling scenarios, has the following
form [40]:
n
o
b i (b
b −1
Hbi = arg min −2s(b
pi , Hi ; Zc ) + 2Tr [J
pi )I
pi )] ,
i (b
H
(10)
where I i (b
pi ) ∈ Rni ×ni is the negative Hessian of the logbi , namely the observed FIM,
likelihood function evaluated at p
whose expression is
∂ 2 s(b
p i , Hi ; Z c )
b i (b
I
pi ) = −
,
∂pi ∂pTi

(TIC)

(11)

b i (b
and J
pi ) is the sample FIM, viz.

pi , Hi ; z)
∂s(b
pi , Hi ; z) ∂s(b
b i (b
J
pi ) =
∂pi
∂pTi
#
"
K
X
bi , Hi ; z k ) ∂s(θ
b i , Hi ; z k )
∂s(θ
.
+
∂pi
∂pTi
k=1

when z and Z are both considered or
"
#
K
X
bi , Hi ; z k ) ∂s(θ
b i , Hi ; z k )
∂s(θ
.
∂pi
∂pTi
k=1

(12)

(13)

when only Z is considered. Note that, given the true model
b i (b
parameter vector p̄ and the true hypothesis H̄, I
pi ) and
b i (b
J
pi ) are estimators of

 2
∂ s(p̄, H̄; Zc )
(14)
I(p̄) = −E
∂ p̄∂ p̄T
and


∂s(p̄, H̄; Zc ) ∂s(p̄, H̄; Zc )
,
J (p̄) = E
∂ p̄
∂ p̄T


(15)

respectively. It is important to observe that, in general, I(p̄)
will not equal J (p̄) when the model is misspecified. However,
if the model is correctly specified, then by the Information
Matrix Equivalence Theorem [42], the information matrix can
be expressed in either Hessian form, I(p̄), or in the outer
product form, J (p̄).
Both the AIC (along with its generalization) and TIC are
derived under the assumption of large samples. To relax this
requirement, the corrected AIC (AICc) has been devised:


(K + 1)N
Hbi = arg min −2s(b
pi , Hi ; Zc ) + 2ni
H
(K + 1)N − ni − 1
(16)
It is important to note that in the considered framework the
AICc is essentially a heuristic rule since it has been originally
proposed for linear regression models [43] and later extended
to the case of nonlinear regression and autoregressive time
series [44], which neither covers the scenarios considered
herein.
Finally, other selection rules, such as the Bayesian Information Criterion (BIC), can be obtained according to a Bayesian

(AIC

4

framework. The BIC has been derived as an asymptotic evaluated irrespective of the current steering direction. The
approximation to a transformation of the Bayesian posterior above strategies are described in the next two subsections,
probability of a candidate model [45]. In large-sample settings, whereas the last subsection provides the expression of BIC
BIC selects the model which is a posteriori most probable. It is for large values of K.
also worth mentioning that, under some regularity conditions,
BIC minimizes the KL discrepancy [33], [40]. An alternative
A. MOS Decision Rules Using the Entire Data Matrix
formulation of BIC can be obtained relaxing the large-sample
It follows from Section III that the ingredients needed to
requirement and assuming a noninformative prior for both
construct
a MOS decision rule are the MLEs of the unknown
the parameter vector θi and the model Hi . Under the above
parameters,
the log-likelihood functions, and the matrices
hypotheses, BIC can be expressed as [33], [46], [47]
b i (pi ) and J
b i (pi ). Evidently the mathematical expressions
n
o
I
b
Hbi = arg min −2s(b
pi , Hi ; Zc ) + log[det(I i (b
pi ))] ,
(BIC)
for all the above quantities depend on which model (Hi ) is
H
(17) assumed.
The log-likelihood functions can be easily obtained from
which, for large samples and the herein considered context,
(3), (4), and (6), namely
reduces to (see Subsection IV-C)
s(pBIC)
(Asymptotic
i , Hi ; z) = −N log π −log det(M i )−Tr {X i S α } , (19)
(18)
s(θi , Hi ; z k ) = −N log π − log det(M i ) − Tr {X i S k } ,
We note, once again, that even though different models can
(20)
share the same number of parameters, the considered selection
s(pi , Hi ; z, Z) = − (K + 1) [N log π + log det(M i )]
criteria are still capable of discriminating between the different
(21)
− Tr {X i S} − Tr {X i S α } ,
hypotheses since they use the specific MLEs together with
the corresponding log-likelihood function under the current where S = z z † .
k
k k
hypothesis.
The next step towards the derivation of the MOS statisAlso note that the definition of large or small samples, which
tics consists in evaluating the gradients of s(p, Hi ; z) and
is important for some of the previous criteria, depends on
s(p, Hi ; z k ), k = 1, . . . , K, which are required to compute
the ratio between the number of parameters, ni , and number b
J i (pi ). More precisely, observe that
of data, (K + 1)N or KN . Moreover, for the considered


∂s(pi , Hi ; z)
application, ni depends on N . Thus, the behavior of these
∂s(pi , Hi ; z) 

criteria might change according to the specific application and,
i
=  ∂s(p∂θ
(22)

∂pi
i , Hi ; z)
for this reason, has to be investigated.
∂α
For the problem under consideration, the ratio between the
number of parameters and the number of samples approaches and


∂s(θi , Hi ; z k )
zero as the number of homogeneous secondary data, K,
∂s(θi , Hi ; z k ) 
.
(23)
=
∂θi
increases. However, this situation might not be realizable
∂pi
0
in practical scenarios with the consequence that the large
samples assumption would be no longer valid. Finally, the In Appendix B, it is shown that
presence of outliers, clutter-edges, and/or regions with highly
∂s(pi , Hi ; z)
varying reflectivity can make the assumption that the true
=
∂θi
model belongs to the family of candidates fail. Thus, given

these uncertainty factors, it is worthwhile investigating the
−{{vec [X i ]}† C i }T + C †i [X ∗i ⊗ X i ] vec [S α ],


(24)

considered MOS rules to determine which one performs better
if M i is Hermitian,
than the others. This is the scope of the next sections.

−{{vec [X i ]}T C i }T + C Ti (X i ⊗ X i )vec [S α ],


if M i is symmetric,
IV. C OMPUTATION OF MOS D ECISION RULES
2
where C i ∈ CN ×mi is a transformation matrix that depends
This section contains the derivation of the explicit exon the specific structure of M i and on how θi is defined (see
pressions of the aforementioned classification rules. Specifialso Appendix B),
cally, we follow two approaches: Approach A jointly exploits
∂s(θi , Hi ; z k )
secondary and primary data; whereas Approach B relies on
=
secondary data only. The the former processes an additional
∂θi

†
∗
†
T
data vector (primary data) with respect to the latter, but the

 −{{vec [X i ]} C i } + C i [X i ⊗ X i ] vec [S k ],
(25)

number of unknown parameters increases due to the presence
if M i is Hermitian,
T
of the target complex amplitude. Moreover, the estimate of
T
T

 −{{vec [X i ]} C i } + C i (X i ⊗ X i )vec [S k ],

the target response represents an additional computational
if M i is symmetric,
load for the rules based on the full data, which requires the
computation of the decision statistics for each look direction. and



In contrast to this, Approach B does not depend on the system
∂s(pi , Hi ; z)
−αre v † X i v + ℜ z † X i v
=
2
. (26)
steering vector and, hence, the classification schemes can be
−αim v † X i v − ℑ z † X i v
∂α
Hbi = arg min {−2s(b
pi , Hi ; Zc ) + mi log(K)} .
H

5

Now, we move to the evaluation of the Hessian of
s(pi , Hi ; z, Z), which can be partitioned as follows

and

2

α
bre =

b i (pi ) = − ∂ s(pi , Hi ; z, Z)
I
∂pi pTi
 2

∂ s(pi , Hi ; z, Z) ∂ 2 s(pi , Hi ; z, Z)


∂θi αT
∂θi θ Ti

= −
2
2
 ∂ s(pi , Hi ; z, Z) ∂ s(pi , Hi ; z, Z) 
∂αθ Ti

H θθ,i H Tαθ,i
=−
,
H αθ,i H αα,i

α
bim =

∂ααT



(27)

where
H θθ,i =
 †

C i {X ∗i ⊗ [(K + 1)X i − X i (S + S α )X i ]



−[X i (S + S α )X i ]∗ ⊗ X i }C i , if M i is Hermitian,

C Ti {X i ⊗ [(K + 1)X i − X i (S + S α )X i ]



−X i (S + S α )∗ X i ⊗ X i }C i , if M i is symmetric,

H αα,i = −2v† X i vI 2 , and if M i is Hermitian
 n
2αre C †i [X ∗i ⊗ X i ]vec [vv † ]

oT

 −2ℜ{C †i [X ∗i ⊗ X i ]vec [vz † ]}

H αθ,i =  n

2αim C †i [X ∗i ⊗ X i ]vec [vv † ]

oT

+2ℑ{C †i [X ∗i ⊗ X i ]vec [vz † ]}

while if M i is symmetric
 n
2αre C Ti [X i ⊗ X i ]vec [vv † ]

oT

 −2ℜ{C Ti [X i ⊗ X i ]vec [vz † ]}

H αθ,i =  n

2αim C Ti [X i ⊗ X i ]vec [vv † ]

oT

+2ℑ{C Ti [X i ⊗ X i ]vec [vz † ]}

(28)

(29)

α
b=

−1

c v
v† M
1

,

−1

−1

−1

−1

c ℑ{z} − ℑ{v}T M
c ℜ{z}
ℜ{v}T M
2
2

c 2 ℑ{v}
c 2 ℜ{v} + ℑ{v}T M
ℜ{v}T M

,

(33)

.

(34)

c ze
v† M
3
−1

c3 v
v†M

−1

,

and α
bim = −j

c zo
v† M
3
−1

c3 v
v† M

,

(36)

where z e = (z + J z ∗ )/2 and z o = (z − J z ∗ )/2.
Finally, the estimates under H4 can be obtained exploiting
the results in [31], namely
n
o
c 4 = 1 ℜ ZZ † + J (ZZ † )∗ J ,
(37)
M
2K
α
bre =

c 4 Z e]
Tr [V † M

−1

−1

c V]
Tr [V † M
4

,

α
bim = −j

c 4 Z o]
Tr [V † M
−1

c V]
Tr [V † M
4

, (38)

where V = [ℜ{v} ℑ{v}], Z e = [ℜ{z e } ℑ{z e }], and Z o =
[ℜ{z o } ℑ{z o }].
B. MOS Decision Rules Using Secondary Data Only
(30)

−1

c1 z
v† M

−1

−1

The proofs of the above statements are provided in Appendix
C.
The final step consists in replacing the unknown parameters,
namely α and θi , with suitable estimates. Forasmuch as the
ML estimates of the unknown parameters are not always
available in closed form (to our best knowledge), we replace
them with consistent estimates as follows. For the ICM, we
use the ML estimates obtained from secondary data only.
As to α, its estimate is obtained according to the ML rule
assuming known ICM and, then, replacing the ICM with the
corresponding consistent estimate. Thus, when the ICM is
unstructured, namely under H1 , the estimates of the M and
α are [2]
c 1 = 1 ZZ † ,
M
K

−1

c ℑ{v}
c ℜ{v} + ℑ{v}T M
ℜ{v}T M
2
2

−1







.




−1

The persymmetric structure of the ICM, which occurs under
H3 , yields the following estimates [4]
h
i
c 3 = 1 ZZ † + J (ZZ † )∗ J ,
(35)
M
2K
α
bre =











−1

c ℜ{z} + ℑ{v}T M
c ℑ{z}
ℜ{v}T M
2
2

(31)

respectively. When H2 is assumed, the ICM is unstructured
and real. Thus, following the lead of [5], we use the following
estimates
n
o
c 2 = 1 ℜ ZZ †
(32)
M
K

Here we derive the expressions for the terms needed to
compute the MOS rules based on secondary data only. To this
end, we rely on the previous results. More precisely, first recall
that pi = θi , i = 1, . . . , 4, and the log-likelihood function is
given by (see (20))
s(θi , Hi ; Z) =
− K [N log π + log det(M i )] − Tr {X i S} .

(39)

Moreover, the observed FIM and the sample FIM become
2 b
bi ) = − ∂ s(θ i , Hi ; Z) ,
b i (θ
I
∂θi ∂θTi

and
bi ) =
b i (θ
J

#
"
K
X
b i , Hi ; z k )
bi , Hi ; z k ) ∂s(θ
∂s(θ
.
∂θi
∂θTi

(40)

(41)

k=1

bi is the ML estimate of θi under Hi .
respectively, where θ
Note that, as opposed to Approach A, in this case closed form
expressions for the ML estimates are available and they are
precisely given by the expressions presented in the previous
subsections (see (31)-(37)). Finally, to evaluate the gradient
of s(pi , Hi ; z k ), we can use (25) and for the Hessian of
s(θi , Hi ; Z), we use (28) after replacing S + S α with S.

6

C. BIC for Large K
In this subsection, we specialize (17) in the limit of
K → +∞. To this end, we first consider Approach A and
approximate the penalty term of BIC as
b i (pi )] =
log det[I

log det[−H θθ,i ] + log det[−H αα,i + H αθ,i H −1
θθ,i H θα,i ]


H θθ,i
= mi log(K + 1) + log det −
(K + 1)


(K + 1) −1
H θθ,i H θα,i
+ log det −H αα,i + H αθ,i
K +1
K→+∞

≈

mi log(K) + O(1),

(42)

where O(1) represents a term that tends to a constant as K →
+∞. The limiting approximation in (42) was obtained using
the following asymptotic equalities
K→+∞
1 K→+∞
1
(S + zz † ) ≈ M ,
S ≈ M,
(43)
K +1
K
in the expression of H θθ,i /(K + 1), see (28), and observing
that H αα,i , (29), and (30) do not depend on K. As a
consequence, the following equalities hold
H αθ,i
H θα,i
lim
= lim
=0
(44)
K→+∞ K + 1
K→+∞ K + 1
−H θθ,i
= C,
(45)
K +1
where C ≻ 0 does not depend on K. Therefore, neglecting the
O(1) term, (17) becomes (18). Observe that the above criterion
is also valid in the case where the CUT is not used (i.e.,
Approach B). As a matter of fact, the expression of asymptotic
BIC for the latter case can be obtained considering H θθ,i only
and repeating the above arguments.
lim

K→+∞

V. N UMERICAL E XAMPLES AND D ISCUSSION
This section is devoted to the analysis of the classification
schemes presented in the previous sections. The metric used
to assess their performance is the Probability of Correct
Classification (Pcc ) estimated under each hypothesis by means
of standard Monte Carlo counting techniques over 1000 independent trials.
The interference is modeled as circular complex normal
random vectors with the following covariance matrix
M i = Ai Ri A†i + σn2 I,

i = 1, . . . , 4

(46)

σn2 I

where
represents the thermal noise component with σn2
being its power, Ri accounts for the clutter contributions
and incorporates the clutter power, and Ai is a matrix factor
modeling possible array channel errors as, for instance, amplification and/or delay errors, calibration residuals, and mutual
coupling [29]. The specific instances of Ai and Ri depend on
which hypothesis is in force as shown below.
Different interference sources (with exponentially shaped
covariance) are encompassed by Ri , whose (h, k)th entry has
the following expression
R(h, k) =

L
X
i=1

|h−k| j2π(h−k)fl

CNRl ρl

e

,

(47)

where, for the lth interference source, CNRl > 0 is the Clutterto-Noise Ratio, ρl is the one-lag correlation coefficient, and fl
is the normalized Doppler frequency. Finally, L is the number
of interference sources. For each hypothesis, we choose Ri
and Ai as follows
• under H1 : A1 = I + σd W 1 , fl 6= 0, ∀l = 1, . . . , L,
where σd > 0 and W 1 (h, k) ∼ CN 1 (0, 1) i.i.d.;
• under H2 : A2 = I + σd W 2 , fl = 0, ∀l = 1, . . . , L,
where σd > 0 and W 2 (h, k) ∼ N 1 (0, 1) i.i.d.;
• under H3 : A3 = I, fl 6= 0, ∀l = 1, . . . , L;
• under H4 : A4 = I, fl = 0, ∀l = 1, . . . , L.
√
As to the target signature, we choose α = SNRejϕ with
ϕ ∼ U(0, 2π) and SNR= 10 dB is the Signal-to-Noise Ratio,
whereas, the steering vector v is chosen such that
i
(N −1) T
1 h −j2πfv (N −1)
2
· · · e−j2πfv 1 ej2πfv · · · ej2πfv 2
v= √
e
N
(48)
assuming N odd and fv = 0.01. Finally, two study cases are
considered: Case 1 assumes L = 1, i.e., only one clutter source
is considered; Case 2 considers L = 2, i.e., two clutter types
with different powers are assumed. The latter case can arise in
scenarios where the radar swath contains an edge separating
two types of clutter sources (e.g., ground and sea clutter). The
considered parameter settings are described in Table I.
Figures 2 and 3 refer to Case 1 and contain the Pcc curves
for Approach A and B, respectively. Inspection of the first
figure highlights that AICc and GIC with ρ = 4 exhibit
poor performance under H1 for K < 3N and under H2 for
K < 2N . This behavior is presumably due to the fact that in
the current context AICc, as already stated, is heuristic, while
the performance of GIC depends on the value of ρ. Moreover,
under H1 and H4 , BIC requires K > 2N secondary data
to achieve reasonable classification performances. Recall that
BIC uses an estimate of the FIM. The remaining classification
schemes guarantee a Pcc above 0.7 over the considered range
of values for K. The described trend remains the same in
Figure 3 except for a performance degradation for some
architectures (such as AIC and TIC) when K is low. The
behavior of the considered rules can also be studied analyzing
the classification percentages for each hypothesis. To this end,
in Figure 4, we plot the percentages of classification by means
of histograms for Approach A and assuming K = 25. The
inspection of the figure shows that under H1 (or H2 ), some
MOS rules decide for H3 (or H4 ) and vice versa. In other
words, the misclassification occurs between H1 and H3 or
between H2 and H4 . Finally, note that including the CUT
in the MOS classification rules (Approach A) leads to better
performances than those obtained by means of Approach B.
In Figures 5 and 6, the Pcc curves for Case 2 are reported.
The behavior of the classification rules is similar to that
observed in the previous figures with the difference that BIC
suffers performance degradation for low values of K under
H1 only.
From the inspection of all the above figures, it turns out
that there does not exist a specific choice which provides the
highest Pcc under all the considered settings and parameters
range. However, the analysis underlines that the classification

7

performances of some rules, in particular the AICc and GIC
with ρ = 4, are poor for low values of K and this drawback could be a reason to discard these architectures when
K ≤ 2N and for the considered parameters setting. In contrast
to this, TIC and BIC classification schemes are capable of
guaranteeing Pcc > 0.8 when K ≥ 2N in all the considered
conditions. However, these rules become somewhat unstable
when K < 2N ; this behavior may be due to the fact that
the observed and sample FIM are less reliable when K takes
on relatively small values. Finally, the Asymptotic BIC and
GIC with ρ = 2 provide the highest performance even for
low values of K. The similarity in performance of these rules
is due to the penalty terms whose values are close to each
other for the considered parameters (i.e., log(K) ∈ [3, 3.8]
for K ∈ [20, 45]). However, the hyperparameter ρ of GIC
is a degree of freedom that has to be suitably set (in fact,
the GIC with ρ = 4 has the worst performance), and there
does not exist a general tuning criterion which allows us to
choose the best value for ρ. On the other hand, the asymptotic
BIC, which does not require any hyperparameter setting, stems
as a reasonable operational choice at least for the considered
scenarios.
VI. C ONCLUSIONS
This paper has considered the interference covariance structure classification which is of primary concern in some radar
signal processing applications. Starting from a set of multivariate radar observations, the classification has been formulated
as a multiple hypotheses test with some nested instances
characterized by a different number of parameters. Several
MOS rules, based on different theoretical criteria, have been
devised to perform the covariance structure selection. Besides,
the possibilities of using primary and secondary data or only
secondary vectors to implement the classification rules have
been considered. At the analysis stage their performance has
been assessed in correspondence of two different operational
scenarios highlighting the merits and the drawbacks connected
with each approach. The classification curves, the complexity
as well as the stability, has singled out the Asymptotic BIC
based on secondary data only as the recommended selector
for the considered scenarios.
Finally, two possible future research tracks deserve attention. First of all, we will study the effect of the proposed MOS
techniques for ICM structure selection on the performance of
target detection. Some preliminary results in this direction are
encouraging: they show that using the proposed techniques
leads to performances close to those of the oracle target
detector that knows the actual structure of the ICM. Then,
the analysis on real radar data is essential to finally establish
the effectiveness of the proposed approach.
A PPENDIX A
PARAMETERS WHEN M IS
C ENTROHERMITIAN OR C ENTROSYMMETRIC
Assume that M ∈ RN ×N is centrosymmetric with N even
and let m = N/2; then M can be partitioned as follows [48]


JAJ B T
,
(49)
M=
B
A
N UMBER

OF

where A ∈ Rm×m is symmetric, B ∈ Rm×m is persymmetric, and J is an m-dimensional permutation matrix. It is clear
that
• the number of parameters defining A is m(m + 1)/2;
• the number of parameters defining B is m(m + 1)/2.
Thus, M can be represented by means of


N N
+1
(50)
m(m + 1) =
2
2
parameters.
In the case where N is still even and M ∈ CN ×N is
centrohermitian, M has the following representation


J A∗ J B †
,
(51)
M=
B
A
where A ∈ Cm×m is Hermitian and B ∈ Cm×m persymmetric. It follows that
2
• the number of parameters defining A is m ;
• the number of parameters defining B is m(m + 1).
The total number of parameters is
m(m + 1) + m2 =

N
(N + 1).
2

(52)

In order to complete the proof, assume that N is odd and let
m = (N −1)/2. Following the lead of [49], a centrosymmetric
M ∈ RN ×N can be partitioned as


JAJ c
BT
M =  cT
(53)
c cT J  ,
B
Jc A

where A ∈ Rm×m is symmetric, B ∈ Rm×m is persymmetric, c ∈ R, and c ∈ Rm×1 . It turns out that the total number
of parameters is
2

N +1
.
(54)
m(m + 1) + m + 1 =
2

Finally, assume that M ∈ CN ×N is centrohermitian; then it
can be partitioned as [49]


J A∗ J c
B†
M =  c†
(55)
c c† J  ,
B
Jc A

where A ∈ Cm×m is Hermitian, B ∈ Cm×m is persymmetric,
c ∈ R, and c ∈ Cm×1 . As a consequence, the number of
parameters characterizing M is
2m2 + 3m + 1 = N (N + 1)/2.

G RADIENT

OF THE

(56)

A PPENDIX B
L OG -L IKELIHOOD F UNCTIONS

As a preliminary remark, observe that the ICM is always either Hermitian or symmetric. Let us first focus on s(pi , Hi ; z)
and evaluate the first derivative of this function with respect to
the lth component of pi . It follows that two cases are possible:
pi (l) is a component of θi or pi (l) is a component of α.

8

As for the first case, it is possible to show that
where the following equality has been used
∂s(pi , Hi ; z)
∂
∂
=−
{log det(M i )} −
{Tr [X i S α ]}
"
#
∂θi (l)
∂θi (l)
∂θi (l)
T
∂M
(θ
)
i



∂
T
vec
=
= − vec (X i )T
{vec [M i ]}
∂θi (l)
∂θi (l)





∂
∂M ∗ (θ i )
∂M i

=
{[vec (M i )]∗ } =

vec
,
(57)
+ Tr X i S α X i


∂θ
(l)
∂θ
i
i (l)
∂θi (l)



∗
where the last equality comes from equations A.390 and A.391
C i el,i , if M i is Hermitian,

of [50]. The above equation can be further simplified observing
vec ∂M i = ∂ [C i θ i ] =



∂θi (l)
∂θi (l)
that


C i el,i , if M i is symmetric.
vec [M i ] = C i θ i ,
(58)

(65)

2

where C i ∈ CN ×mi is a transformation matrix that depends
on the specific structure of M i and on how θi is defined. For
instance, if M i is Hermitian unstructured with N = 3 and


M (1, 1)
ℜ{M(2, 1)}


ℑ{M(2, 1)}


ℜ{M(3, 1)}



θi = 
(59)
ℑ{M(3, 1)} ,
 M (2, 2) 


ℜ{M(3, 2)}


ℑ{M(3, 2)}
M (3, 3)

then



1
0

0

0

Ci = 
0
0

0

0
0

It follows that

0
1
0
1
0
0
0
0
0

0
j
0
−j
0
0
0
0
0

0
0
1
0
0
0
1
0
0

0
0
j
0
0
0
−j
0
0

0
0
0
0
1
0
0
0
0

0 0
0 0
0 0
0 0
0 0
1 j
0 0
1 −j
0 0


0
0

0

0

0
.
0

0

0
1

As a final step, we evaluate the gradient of s(pi , Hi ; z) with
respect to α. To this end, observe that
∂
∂s(pi , Hi ; z)
=
{−Tr [X i S α ]} =
∂α
∂α
∂  †
αz X i v + α∗ v † X i z − αα∗ v † X i v .
∂α

(66)

Using the above equation, the gradient with respect to α can
be expressed as in (26).
(60)

H ESSIAN

∂
[C i θi ] = C i el,i ,
(61)
∂θi (l)
where el,i is the lth elementary vector of size mi . Moreover,
let A, B, C, and D be generic matrices whose sizes are such
that the product ABCD makes sense and yields a square
matrix; then the following equality holds [51]
Tr (ABCD) = [vec (DT )]T (C T ⊗ A)vec (B).

Hence, exploiting (64), it is not difficult to obtain (24).
Following the same line of reasoning and replacing S α with
S k , it is possible to prove (25).

(62)

Thus, the second term of (57) can be recast as


∂M i
Tr X i S α X i
∂θi (l)
#)T
(
"
∂M T (θ i )
[(X i )T ⊗ X i ]vec [S α ] (63)
= vec
∂θi (l)
Gathering the above results and accounting for M i being
symmetric or Hermitian, (57) becomes

†
∗ T
T

− {vec [X i ]} C i el,i + el,i C i ×

[X ∗ ⊗ X ]vec [S ], if M is Hermitian,
i
α
i
i
(57) =
(64)
− {vec [(X i )]}T C i el,i + eTl,i C Ti ×



[X i ⊗ X i ]vec [S α ], if M i is symmetric,

OF THE

A PPENDIX C
L OG -L IKELIHOOD F UNCTION

In this appendix we derive the Hessian of the log-likelihood
function s(pi , Hi ; z, Z). To this end, consider H θθ,i , whose
(l, m)-entry can be written as
∂ 2 s(pi , Hi ; z, Z)
=
∂θi (l)∂θi (m)
∂2
− (K + 1)
[log det(M i )]
∂θi (l)∂θi (m)
∂2
−
[Tr (X i (S α + S))]
∂θi (l)∂θi (m)


∂M i
∂M i
Xi
= (K + 1)Tr X i
∂θi (l)
∂θi (m)


2
∂ Mi
− (K + 1)Tr X i
∂θi (l)∂θi (m)
 

∂
∂M i
+
Tr X i (S α + S)X i
,
∂θi (m)
∂θi (l)

H θθ,i (l, m) =

(67)

where the last equality comes from the application of (A.391)
and (A.393) in [50]. Now, let us focus on the last term of (67)

9

and exploit (A.391) of [50] to obtain

 

∂M i
∂
Tr X i (S α + S)X i
∂θi (m)
∂θi (l)



∂
∂M i
= Tr
X i (S α + S)X i
∂θ (m)
∂θi (l)

 i
∂M i
∂M i
Xi
Xi
= −Tr (S α + S)X i
∂θi (l)
∂θi (m)
n
+ Tr X i (S α + S)×

o
∂X i ∂M i
∂2M i
+ Xi
∂θi (m) ∂θi (l)
∂θi (l)∂θi (m)


∂M i
∂M i
Xi
Xi
= −Tr (S α + S)X i
∂θi (l)
∂θi (m)


∂M i
∂M i
Xi
Xi
− Tr (S α + S)X i
∂θi (m)
∂θi (l)


2
∂ Mi
.
+ Tr X i (S α + S)X i
∂θi (l)∂θi (m)

where
(68)

The terms involving the second-order derivative of M i can be
discarded because

vec




∂ 2 vec [M i ]
∂2M i
=
∂θi (l)∂θi (m)
∂θi (l)∂θi (m)
∂ 2 vec [C i θi ]
=
= 0.
∂θi (l)∂θi (m)

(69)

Thus, the (l, m)-entry of H θθ,i can be recast as



∂M i
∂M i
Xi
H θθ,i (l, m) = (K + 1)Tr X i
∂θi (l)
∂θi (m)


∂M i
∂M i
Xi
− Tr X i (S α + S)X i
∂θi (l)
∂θi (m)


∂M i
∂M i
Xi
− Tr X i (S α + S)X i
∂θi (m)
∂θi (l)


∂M i
∂M i
= (K + 1)Tr X i
F iX i
∂θi (m)
∂θi (l)


∂M i
∂M i
.
(70)
Xi
− Tr X i (S α + S)X i
∂θi (m)
∂θi (l)

where


(S α + S)
.
F i = IN − Xi
(K + 1)

The above expression can be further simplified exploiting (62).
More precisely, the first term becomes


∂M i
∂M i
(K + 1)Tr X i
F iX i
∂θi (m)
∂θi (l)

†



 ∗

∂M i

(K + 1) vec
X i F̄ i ⊗ X i


∂θi (l)






∂M i

×vec
,

∂θi (m)
=


T h
i

∂M i



(K
+
1)
vec
F̃
⊗
X
X
i
i
i


∂θi (l)





∂M i


×vec
,
∂θi (m)


†
(K + 1) [C i el,i ] X ∗i F̄ i ⊗ X i C i em,i ,



if M is Hermitian,
i
=
(71)
T

(K
+
1) [C i el,i ] [X i F̃ i ⊗ X i ]C i em,i ,



if M i is symmetric.


((S α + S)X i )∗
,
F̄ i = I N −
K +1


(S α + S)∗ X i
.
F̃ i = I N −
K +1

Using the same line of reasoning, it is possible to recast the
last term as follows


∂M i
∂M i
Xi
Tr X i (S α + S)X i
∂θi (m)
∂θi (l)

†

∂M i



[X ∗i ⊗ X i (S α + S)X i ]
vec


∂θ
(l)

i





×vec ∂ M i
∂ θ i (m)
= 

T

∂M i



vec
[X i ⊗ X i (S α + S)X i ]


∂θi (l) 





×vec ∂ M i
∂ θ i (m)

†
[C i el,i ] [(X i )∗ ⊗ X i (S α + S)X i ]C i em,i ,



if M is Hermitian,
i
=
(72)
T

[C
e
i l,i ] [X i ⊗ X i (S α + S)X i ]C i em,i ,



if M i is symmetric.

Summarizing, if M i is Hermitian H θθ,i can be written as
H θθ,i = (K + 1)C †i [X ∗i F̄ i ⊗ X i ]C i

− C †i [(X i )∗ ⊗ X i (S α + S)X i ]C i ,

(73)

whereas if M i is symmetric we have that
H θθ,i = (K + 1)C Ti [X i F̃ i ⊗ X i ]C i

− C Ti [X i ⊗ X i (S α + S)X i ]C i .

(74)

Next, consider H αα,i and observe that the gradient of (26)
with respect to αT is


∂s(pi , Hi ; z, Z)
∂
= H αα,i
∂αT
∂α
 †

(75)
v X iv
0
†
= −2
=
−2v
X
vI
.
i
2
0
v†X iv

10

As a final step towards the evaluation of H i , we derive
the expression for H αθ,i . More precisely, exploiting previous
results we get


∂
∂s(pi , Hi ; z, Z)
∂α
∂θTi (l)


(76)
2αre Tr [A1 ] − 2ℜ {Tr [A2 ]}
=
,
2αim Tr [A1 ] + 2ℑ {Tr [A2 ]}
where A1 = X i vv † X i ∂ X i and A2 = X i vz † X i ∂ X i .
∂ θ i (l)
∂ θ i (l)
Now, assume that the ICM is Hermitian; then using (61), (62),
and (65) the above equation can be recast as


∂s(pi , Hi ; z, Z)
∂
∂α
∂θTi (l)
o 

n
(77)
2αre (C i el,i )† Φ̄i − 2ℜ (C i el,i )† Φ̃i
o ,
n
=
2αim (C i el,i )† Φ̄i + 2ℑ (C i el,i )† Φ̃i

where Φ̄i = (X ∗i ⊗ X i )vec [vv † ], and Φ̃i
X i )vec [vz † ]. As a consequence,


∂s(pi , Hi ; z, Z)
∂
∂α
∂θTi
 n
n
ooT
†
†
2α
C
Φ̄
−
2ℜ
C
Φ̃
re
i
i
i
i

= n
ooT
n
†
+2αim C i Φ̄i + 2ℑ C †i Φ̃i

= (X ∗i ⊗




.

(78)

On the other hand, if the ICM is symmetric, then (76) becomes


∂
∂s(pi , Hi ; z, Z)
∂α
∂θTi (l)
n

o 
(79)
2αre (C i el,i )T Ψ̄i − 2ℜ (C i el,i )T Ψ̃i
o


n
,
=
2αim (C i el,i )T Ψ̄i + 2ℑ (C i el,i )T Ψ̃i

where Ψ̄i = (X i ⊗ X i )vec [vv † ], and Ψ̃i
X i )vec [vz † ]. As a consequence,


∂
∂s(pi , Hi ; z, Z)
∂α
∂θTi
 n
ooT
n
T
T
 2αre C i Ψ̄i − 2ℜ C i Ψ̃i
= n
ooT
n
2αim C Ti Ψ̄i + 2ℑ C Ti Ψ̃i

= (X i ⊗




.

(80)

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11

[36]
[37]

[38]

[39]

[40]

[41]

[42]

[43]

[44]
[45]
[46]

[47]

[48]
[49]

[50]
[51]

Parameter
N
σd
ρ1
f1
CN R1 [dB]
ρ2
f2
CN R2 [dB]

Case 1 (L = 1)
13
0.15
0.85
0.285
30
-

Case 2 (L = 2)
13
0.15
0.85
0.285
20
0.93
0.05
30

TABLE I: Parameters setting.

Fig. 1: Block diagram of a two-stage detection architecture
exploiting the covariance structure classifier.

100

100

90

90

Percentage of Correct Detections

[35]

80
70
60
50
AIC
AICc
GIC ( = 2)
GIC ( = 4)
Asymptotic BIC
BIC
TIC

40
30
20
10
0
15

20

25

30

35

40

45

80
70
60
50
AIC
AICc
GIC ( = 2)
GIC ( = 4)
Asymptotic BIC
BIC
TIC

40
30
20
10
0
15

50

20

25

30

K

(a) Hypothesis 1.
100

90

90

80
70
60
50
AIC
AICc
GIC ( = 2)
GIC ( = 4)
Asymptotic BIC
BIC
TIC

40
30
20
10

20

25

30

40

45

50

(b) Hypothesis 2.

100

0
15

35

K

Percentage of Correct Detections

[34]

Percentage of Correct Detections

[33]

Percentage of Correct Detections

[32]

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35

40

K

(c) Hypothesis 3.

45

80
70
60
50
AIC
AICc
GIC ( = 2)
GIC ( = 4)
Asymptotic BIC
BIC
TIC

40
30
20
10

50

0
15

20

25

30

35

40

45

50

K

(d) Hypothesis 4.

Fig. 2: Pcc versus K for Study Case 1 and Approach A
(primary and secondary data).

100

100

90

90

90

80
70
60
50
AIC
AICc
GIC ( = 2)
GIC ( = 4)
Asymptotic BIC
BIC
TIC

40
30
20
10
0
15

20

25

30

35

40

45

80
70
60
50
AIC
AICc
GIC ( = 2)
GIC ( = 4)
Asymptotic BIC
BIC
TIC

40
30
20
10
0
15

50

20

25

30

K

40

45

80
70
60
50
AIC
AICc
GIC ( = 2)
GIC ( = 4)
Asymptotic BIC
BIC
TIC

40
30
20
10
0
15

50

20

25

30

K

(a) Hypothesis 1.

35

40

45

80
70
60
50

30
20
10
0
15

50

(a) Hypothesis 1.
100

90

90

AIC
AICc
GIC ( = 2)
GIC ( = 4)
Asymptotic BIC
BIC
TIC

40
30
20
10
0
15

20

25

30

35

40

K

(c) Hypothesis 3.

45

70
60
50
AIC
AICc
GIC ( = 2)
GIC ( = 4)
Asymptotic BIC
BIC
TIC

40
30
20
10

50

0
15

20

25

30

35

40

45

80
70
60
50
AIC
AICc
GIC ( = 2)
GIC ( = 4)
Asymptotic BIC
BIC
TIC

40
30
20
10
0
15

50

20

25

30

K

30

35

40

45

50

40

45

80
70
60
50
AIC
AICc
GIC ( = 2)
GIC ( = 4)
Asymptotic BIC
BIC
TIC

40
30
20
10
0
15

50

20

25

30

K

(d) Hypothesis 4.

35

40

45

50

K

(c) Hypothesis 3.

(d) Hypothesis 4.

Fig. 5: Pcc versus K for Study Case 2 and Approach A
(primary and secondary data).

Percentage of Correct Detections

Fig. 3: Pcc versus K for Study Case 1 and Approach B
(secondary data only).

35

100

100

90

90

Percentage of Correct Detections

50

80

Percentage of Correct Detections

100

90

Percentage of Correct Detections

100

60

25

(b) Hypothesis 2.

90

70

20

K

100

80

AIC
AICc
GIC ( = 2)
GIC ( = 4)
Asymptotic BIC
BIC
TIC

40

K

(b) Hypothesis 2.

Percentage of Correct Detections

Percentage of Correct Detections

35

Percentage of Correct Detections

100

90

Percentage of Correct Detections

100

Percentage of Correct Detections

Percentage of Correct Detections

12

80
70
60
50
AIC
AICc
GIC ( = 2)
GIC ( = 4)
Asymptotic BIC
BIC
TIC

40
30
20
10
0
15

20

25

30

35

40

45

80
70
60
50
AIC
AICc
GIC ( = 2)
GIC ( = 4)
Asymptotic BIC
BIC
TIC

40
30
20
10
0
15

50

20

25

30

K

100

100

90

90

80
70
60
50
AIC
AICc
GIC ( = 2)
GIC ( = 4)
Asymptotic BIC
BIC
TIC

40
30
20
10

20

25

30

40

45

50

(b) Hypothesis 2.

Percentage of Correct Detections

Percentage of Correct Detections

(a) Hypothesis 1.

0
15

35

K

35

40

K

(c) Hypothesis 3.

45

80
70
60
50
AIC
AICc
GIC ( = 2)
GIC ( = 4)
Asymptotic BIC
BIC
TIC

40
30
20
10

50

0
15

20

25

30

35

40

45

50

K

(d) Hypothesis 4.

Fig. 6: Pcc versus K for Study Case 2 and Approach B
(secondary data only).
(a) Hypothesis 1.

(b) Hypothesis 2.

(c) Hypothesis 3.

(d) Hypothesis 4.

Fig. 4: Percentage of classification for each hypothesis assuming Approach A and K = 25.

