IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE, VOL. 26, NO. 9, SEPTEMBER 2004 1
Simultaneous Feature Selection and Clustering
Using Mixture Models
Martin H.C. Law, Student Member, IEEE, Mario A.T. Figueiredo, Senior Member, IEEE, and
Anil K. Jain, Fellow, IEEE
Abstract—Clustering is a common unsupervised learning technique used to discover group structure in a set of data. While there exist
many algorithms for clustering, the important issue of feature selection, that is, what attributes of the data should be used by the
clustering algorithms, is rarely touched upon. Feature selection for clustering is difficult because, unlike in supervised learning, there
are no class labels for the data and, thus, no obvious criteria to guide the search. Another important problem in clustering is the
determination of the number of clusters, which clearly impacts and is influenced by the feature selection issue. In this paper, we
propose the concept of feature saliency and introduce an expectation-maximization (EM) algorithm to estimate it, in the context of
mixture-based clustering. Due to the introduction of a minimum message length model selection criterion, the saliency of irrelevant
features is driven toward zero, which corresponds to performing feature selection. The criterion and algorithm are then extended to
simultaneously estimate the feature saliencies and the number of clusters.
Index Terms—Feature selection, clustering, unsupervised learning, mixture models, minimum message length, EM algorithm.
1 INTRODUCTION
THE goal of clustering is to discover a “natural” grouping
in a set of patterns, points, or objects, without knowledge
of any class labels. Clustering, or cluster analysis, is
prevalent in any discipline that involves analysis of multivariate
data. It is, of course, impractical to exhaustively list
the numerous uses of clustering techniques. Image segmentation,
an important problem in computer vision, can
be formulated as a clustering problem [21], [28], [55].
Documents can be clustered [23] to generate topical
hierarchies for information access [53] or retrieval [5].
Clustering is also used to perform market segmentation [2],
[11] as well as in biology, e.g., to study genome data [3].
Many clustering algorithms have been proposed in
different application scenarios [25], [29]. They can be
divided roughly into two categories: hierarchical clustering,
which creates a “tree” with branches merging at different
levels, and partitional clustering, which divides the data into
different “flat” clusters. The input of clustering algorithms
can either be a proximity matrix containing the similarities/
dissimilarities between all pairs of points, or a pattern
matrix, where each item is described by a vector of
attributes, also called features. In this paper, we shall focus
on partitional clustering with a pattern matrix as input.
In principle, the more information we have about each
pattern, the better a clustering algorithm is expected to
perform. This seems to suggest that we should use as many
features as possible to represent the patterns. However, this is
not the case in practice. Some features can be just “noise,” thus
. M.H.C. Law and A.K. Jain are with the Department of Computer Science
and Engineering, Michigan State University, 3115 Engineering Building,
East Lansing, Michigan 48824-1226. E-mail: {lawhiu, jain}@cse.msu.edu.
. M.A.T. Figueiredo is with the Instituto de Telecomunicações, Instituto
Superior Técnico, Torre Norte, Piso 10, Av. Rovisco Pais, 1049-001 Lisboa,
Portugal. E-mail: mtf@lx.it.pt.
Manuscript received 15 May 2003; accepted 27 Feb. 2004.
Recommended for acceptance by B.J. Frey.
For information on obtaining reprints of this article, please send e-mail to:
tpami@computer.org, and reference IEEECS Log Number TPAMI-0077-0503.
æ
not contributing to (or even degrading) the clustering
process. The task of selecting the “best” feature subset is
known as feature selection, sometimes as variable selection or
subset selection.
Feature selection is important for several reasons, the
fundamental one being arguably that noisy features can
degrade the performance of most learning algorithms (see
the example in Fig. 1). In supervised learning, it is known
that feature selection can improve the performance of
classifiers learned from limited amounts of data [49]; it
leads to more economical (both in storage and computation)
classifiers and, in many cases, it may lead to interpretable
models. Feature selection is particularly important for data
sets with large numbers of features, e.g., classification
problems in molecular biology may involve thousands of
features [3], [62], and a Web page can be represented by
thousands of different key-terms [58]. Appearance-based
image classification methods may use each pixel as a
feature [6], thus easily involving thousands of features.
Feature selection has been widely studied in the context
of supervised learning (see [7], [24], [33], [34] and references
therein), where the ultimate goal is to select features that
can achieve the highest accuracy on unseen data. Feature
selection has received comparatively very little attention in
unsupervised learning or clustering. One important reason
is that it is not at all clear how to assess the relevance of a
subset of features without resorting to class labels. The
problem is made even more challenging when the number
of clusters is unknown, since the optimal number of clusters
and the optimal feature subset are interrelated, as illustrated
in Fig. 2 (taken from [16]). Note that methods based
on variance (such as principal components analysis) need not
select good features for clustering, as features with large
variance can be independent of the intrinsic grouping of the
data (see example in Fig. 3).
Most feature selection algorithms (such as [9], [33], [47])
involve a combinatorial search through the space of all
feature subsets. Usually, heuristic (nonexhaustive) methods
have to be adopted, because the size of this space is
0162-8828/04/$20.00 ß 2004 IEEE Published by the IEEE Computer Society
2 IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE, VOL. 26, NO. 9, SEPTEMBER 2004
Fig. 1. A uniformly distributed irrelevant feature (x2) makes it difficult for
the Gaussian mixture learning algorithm in [18] to recover the two
underlying clusters. If only feature x1 is used, however, the two clusters
are easily identified. The curves along the horizontal and vertical axes of
the figure indicate the marginal distribution of x1 and x2, respectively.
exponential in the number of features. In this case, one
generally loses any guarantee of optimality of the selected
feature subset.
In this paper, we propose a solution to the feature selection
problem in unsupervised learning by casting it as an
estimation problem, thus avoiding any combinatorial search.
Instead of selecting a subset of features, we estimate a set of
real-valued (actually in ½0; 1Š) quantities (one for each feature)
which we call the feature saliencies. This estimation is carried
out by an EM algorithm derived for the task. Since we are in
the presence of a model-selection-type problem, it is
necessary to avoid the situation where all the saliencies take
the maximum possible value. This is achieved by adopting a
minimum message length (MML, [60], [61]) penalty, as was
done in [18] to select the number of clusters. The MML
criterion encourages the saliencies of the irrelevant features to
go to zero, allowing us to prune the feature set. Finally, we
integrate the process of feature saliency estimation into the
algorithm proposed in [18], thus obtaining a method which is
able to simultaneously perform feature selection and determine
the number of clusters. Although the algorithm is
presented with respect to Gaussian mixture-based clustering,
one can extend it to other types of model-based clustering as
well. The algorithm first appears in [38].
Fig. 2. Number of clusters is interrelated with feature subset used. The
optimal feature subsets for identifying three, two, one clusters in this
data set are fx1;x2g, fx1g, and fx2g, respectively. On the other hand,
the optimal number of clusters for feature subsets fx1;x2g, fx1g, and
fx2g are also three, two, one, respectively.
Fig. 3. Feature x1, although explaining more data variance than feature
x2, is spurious for the identification of the two clusters in this data set.
The remainder of this paper is organized as follows: In
Section 2, we review approaches for feature selection and
previous attempts to solve the feature selection problem in
unsupervised learning. The details of our approach are
presented in Section 3. Experimental results are reported in
Section 4, followed by comments on the proposed algorithm
in Section 5. Finally, we conclude in Section 6 and outline
some future work directions.
2 RELATED WORK
Most of the literature on feature selection pertains to
supervised learning (both classification [24] and regression
[40]). Feature selection algorithms can be broadly divided
into two categories [7], [33]: filters and wrappers. The filter
approaches evaluate the relevance of each feature (subset)
using the data set alone, regardless of the subsequent learning
algorithm. RELIEF [32] and its enhancement [36] are
representatives of this class, where the basic idea is to assign
feature weights based on the consistency of the feature value
in the k nearest neighbors of every data point. Informationtheoretic
methods are also used to evaluate features: the
mutual information between a relevant feature and the class
labels should be high [4]. Nonparametric methods can be
used to compute mutual information involving continuous
features [37]. A feature can be regarded as irrelevant if it is
conditionally independent of the class labels given other
features. The concept of Markov blanket is used to formalize
this notion of irrelevancy in [34].
On the other hand, wrapper approaches [33] invoke the
learning algorithm to evaluate the quality of each feature
(subset). Specifically, a learning algorithm (e.g., a nearest
neighbor classifier, a decision tree, a naive Bayes method) is
run on a feature subset and the feature subset is assessed by
some estimate of the classification accuracy. Wrappers are
usually more computationally demanding, but they can be
superior in accuracy when compared with filters, which
ignore the properties of the learning task at hand [33].
Both approaches, filters and wrappers, usually involve
combinatorial searches through the space of possible
feature subsets; for this task, different types of heuristics,
such as sequential forward or backward searches, floating
search, beam search, bidirectional search, and genetic
search have been suggested [9], [33], [47], [63]. It is also
possible to construct a set of weak (in the boosting sense
[20]) classifiers, with each one using only one feature, and
then apply boosting, which effectively performs feature
LAW ET AL.: SIMULTANEOUS FEATURE SELECTION AND CLUSTERING USING MIXTURE MODELS 3
selection [59]. It has also been proposed to approach feature
selection using rough set theory [35].
All of the approaches mentioned above are concerned
with feature selection in the presence of class labels.
Comparatively, not much work has been done for feature
selection in unsupervised learning. Of course, any method
conceived for supervised learning that does not use the
class labels could be used for unsupervised learning; it is
the case for methods that measure feature similarity to
detect redundant features, using, e.g., mutual information
[53] or a maximum information compression index [42]. In
[16], [17], the normalized log-likelihood and cluster separability
are used to evaluate the quality of clusters obtained
with different feature subsets. Different feature subsets and
numbers of clusters, for multinomial model-based clustering,
are evaluated using marginal likelihood and crossvalidated
likelihood in [58]. The algorithm described in [52]
uses automatic relevance determination priors to select
features when there are two clusters. In [13], the clustering
tendency of each feature is assessed by an entropy index. A
genetic algorithm is used in [31] for feature selection in
k-means clustering. In [56], feature selection for symbolic
data is addressed by assuming that irrelevant features are
uncorrelated with the relevant features. Reference [14]
describes the notion of “category utility” for feature
selection in a conceptual clustering task. The CLIQUE
algorithm [1] is popular in the data mining community and
it finds hyperrectangular shaped clusters using a subset of
attributes for a large database. The wrapper approach can
also be adopted to select features for clustering; this has
been explored in our earlier work [19], [38].
All the methods referred above perform “hard” feature
selection (a feature is either selected or not). There are also
algorithms that assign weights to different features to
indicate their significance. In [43], weights are assigned to
different groups of features for k-means clustering based on
a score related to the Fisher discriminant. Feature weighting
for k-means clustering is also considered in [41], but the goal
there is to find the best description of the clusters after they
are identified. The method described in [46] can be
classified as learning feature weights for conditional
Gaussian networks. An EM algorithm based on Bayesian
shrinking is proposed in [22] for unsupervised learning.
3 EM ALGORITHM FOR FEATURE SALIENCY
In this section, we propose an EM algorithm for performing
mixture-based (or model-based) clustering with feature
selection. In mixture-based clustering, each data point is
modeled as having been generated by one of a set of
probabilistic models [25], [39]. Clustering is then done by
learning the parameters of these models and the associated
probabilities. Each pattern is assigned to the mixture
component that most likely generated it. Although the
derivations below refer to Gaussian mixtures, they can be
generalized to other types of mixtures.
3.1 Mixture Densities
A finite mixture density with K components is defined by
pðyÞ ¼ XK
j¼1
j pðyj jÞ; ð1Þ
where 8j; j 0; P
j j ¼ 1; each j is the set of parameters of
the jth component (all components are assumed to have the
same form, e.g., Gaussian); and f 1; ...; K; 1; ...; Kg
will denote the full parameter set. The goal of mixture
estimation is to infer from a set of N data points
Y¼fy 1; ...; y Ng, assumed to be samples of a distribution
with density given by (1). Each y i is a D-dimensional feature
vector ½yi1; ...;yiDŠ T . In the sequel, we will use the indices i, j,
and l to run through data points (1 to N), mixture components
(1 to K), and features (1 to D), respectively.
As is well-known, neither the maximum likelihood (ML)
estimate,
b
ML ¼ arg maxflog pðYj Þg;
nor the maximum a posteriori (MAP) estimate (given some
prior pð Þ)
b
MAP ¼ arg maxflog pðYj Þþlog pð Þg;
can be found analytically. The usual choice is the EM
algorithm, which finds local maxima of these criteria [39].
This algorithm is based on a set Z¼fz1; ...; zNg of N missing
(latent) labels, where zi ¼½zi1; ...;ziKŠ, with zij ¼ 1 and
zip ¼ 0, for p 6¼ j, meaning that y i is a sample of pð j jÞ. For
brevity of notation, sometimes we write zi ¼ j for such zi. The
complete data log-likelihood, i.e., the log-likelihood if Z was
observed, is
log pðY; Zj Þ¼ XN
X K
i¼1 j¼1
zij log jpðy ij jÞ : ð2Þ
The EM algorithm produces a sequence of estimates
f b ðtÞ; t¼ 0; 1; 2; ...g using two alternating steps:
. E-step: Compute W¼E½ZjY; bðtÞŠ, the expected
value of the missing data given the current parameter
estimate, and plug it into log pðY; Zj Þ, yielding the socalled
Q-function Qð ; bðtÞÞ ¼ log pðY; Wj Þ. Since the
elements of Z are binary, we have
wi;j E zijjY; b h i
ðtÞ ¼ Pr zij ¼ 1j yi; b h i
ðtÞ
¼
bjðtÞ pðyijb jðtÞÞ
PK k¼1bkðtÞ pðyijb kðtÞÞ :
Notice that j is the a priori probability that zij ¼ 1
(i.e., that y i belongs to cluster j), while wij is the
corresponding a posteriori probability, after observing
y i.
. M-step: Update the parameter estimates,
b ðt þ 1Þ ¼arg max fQð ; b ðtÞÞ þ log pð Þg;
ð3Þ
in the case of MAP estimation, or without log pð Þ in
the ML case.
3.2 Feature Saliency
In this section, we define the concept of feature saliency and
derive an EM algorithm to estimate its value. We assume
that the features are conditionally independent given the
(hidden) component label, that is,
pðyj Þ¼ XK
j¼1
j pðyj jÞ ¼ XK
j¼1
Y D
j
l¼1
pðylj jlÞ; ð4Þ
4 IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE, VOL. 26, NO. 9, SEPTEMBER 2004
Fig. 4. A graphical model for the probability model in (5) for the case of four features (D ¼ 4) with different indicator variables. l ¼ 1 corresponds to
the existence of an arc from z to yl, and l ¼ 0 corresponds to its absence. (a) 1 ¼ 1, 2 ¼ 1, 3 ¼ 0, 4 ¼ 1. (b) 1 ¼ 0, 2 ¼ 1, 3 ¼ 1, 4 ¼ 0.
where pð j jlÞ is the pdf of the lth feature in the jth
component. This assumption enables us to utilize the power
of the EM algorithm. In the particular case of Gaussian
mixtures, the conditional independence assumption is
equivalent to adopting diagonal covariance matrices, which
is a common choice for high-dimensional data, such as in
naïve Bayes classifiers, latent class models, as well as in the
emission densities of continuous hidden Markov models.
Among different definitions of feature irrelevancy
(proposed for supervised learning), we adopt the one
suggested in [48], [58], which is suitable for unsupervised
learning: the lth feature is irrelevant if its distribution is
independent of the class labels, i.e., if it follows a common
density, denoted by qðylj lÞ. Let ¼ð 1; ...; DÞ be a set of
binary parameters, such that l ¼ 1 if feature l is relevant
and l ¼ 0, otherwise. The mixture density in (4) can then
be rewritten as
pðyj ; f jg; f jlg; f lgÞ ¼ XK
j¼1
Y D
j
l¼1
½pðylj jlÞŠ l ½qðylj lÞŠ 1 l :
A related model for feature selection in supervised learning
has been considered in [44], [48]. Intuitively, determines
which edges exist between the hidden label z and the
individual features yl in the graphical model illustrated in
Fig. 4, for the case D ¼ 4.
Our notion of feature saliency is summarized in the
following steps: 1) We treat the ls as missing variables and
2) we define the feature saliency as l ¼ Pð l ¼ 1Þ, the
probability that the lth feature is relevant. This definition
makes sense, as it is difficult to know for sure that a certain
feature is irrelevant in unsupervised learning. The resulting
model (likelihood function) is written as (see the proof in
Appendix A)
pðyj Þ¼ XK
j¼1
Y D
j
l¼1
ð5Þ
ð lpðylj jlÞþð1 lÞqðylj lÞÞ; ð6Þ
where ¼ff jg; f jlg; f lg; f lgg is the set of all the
parameters of the model. An intuitive way to see how (6)
is obtained is to notice that ½pðylj jlÞŠ l ½qðylj lÞŠ 1 l can be
written as l pðylj jlÞþð1 lÞ qðylj lÞ, because l is binary.
The form of qð:j:Þ reflects our prior knowledge about the
distribution of the nonsalient features. In principle, it can be
any 1D distribution (e.g., a Gaussian, a student-t, or even a
mixture). We shall limit qð:j:Þ to be a Gaussian, since this
leads to reasonable results in practice.
Equation (6) has a generative interpretation. As in a
standard finite mixture, we first select the component label j
by sampling from a multinomial distribution with parameters
ð 1; ...; KÞ. Then, for each feature l ¼ 1; ...;D,we
flip a biased coin whose probability of getting a head is l; if
we get a head, we use the mixture component pð:j jlÞ to
generate the lth feature; otherwise, the common component
qð:j lÞ is used. A graphical model representation of (6) is
shown in Fig. 5 for the case D ¼ 4.
3.2.1 EM Algorithm
By treating Z (the hidden class labels) and as hidden
variables, one can derive (see details in Appendix B) the
following EM algorithm for parameter estimation:
. E-step: Compute the following quantities:
aij l ¼ P ð l ¼ 1;yiljzi ¼ jÞ ¼ l pðyilj jlÞ; ð7Þ
bij l ¼ P ð l ¼ 0;yiljzi ¼ jÞ ¼ð1 lÞ qðyilj lÞ; ð8Þ
cij l ¼ P ðyiljzi ¼ jÞ ¼aijl þ bijl;
Q
j l
wij ¼ P ðzi ¼ jjyiÞ¼ ð9Þ
cij l
; ð10Þ
P
j j
Q
l cij l
uij l ¼ P ð l ¼ 1;zi ¼ jjyiÞ¼ aij l
wij; ð11Þ
cij l
vij l ¼ P ð l ¼ 0;zi ¼ jjy iÞ¼wij uij l: ð12Þ
. M-step: Reestimate the parameters according to
following expressions:
P
i
bj ¼
wij
P
ij wij
P
i
¼
wij
; ð13Þ
n
P
Mean d
i in jl ¼
uij l yil
P
i uijl
; ð14Þ
P
Var d
i in jl ¼
uij l ðyil ð Mean d in jlÞÞ 2
P
i uij
; ð15Þ
l
P
Mean d
i
in l ¼
ðPj
vij lÞ yil
; ð16Þ
P
Var d
i
in l ¼
ðP
bl ¼
P
i;j uij l
P
ij vij l
P
i;j uij l þ P
i;j vij l
j vij lÞðyil ð Mean d in lÞÞ 2
P
; ð17Þ
ij vij l
P
i;j
¼
uij l
: ð18Þ
n
In these equations, the variable uij l measures how important
the ith pattern is to the jth component, when the lth feature is
used. It is thus natural that the estimates of the mean and the
variance in jl are weighted sums with weight uij l. Similar
LAW ET AL.: SIMULTANEOUS FEATURE SELECTION AND CLUSTERING USING MIXTURE MODELS 5
Fig. 5. A graphical model showing the mixture density in (6). The
variables z, 1; 2; 3; 4 are “hidden” and only y1;y2;y3;y4 are observed.
relationship exists between P
j vij l and l. The term P
ij uij l
can be interpreted as how likely it is that l equals one,
explaining why the estimate of l is proportional to P
ij uij l.
3.3 Model Selection
Standard EM for mixtures exhibits some weaknesses which
also affect the EM algorithm introduced above: it requires
knowledge of K, and a good initialization is essential for
reaching a good local optimum. To overcome these difficulties,
we adopt the approach in [18], which is based on the
MML criterion [61], [60].
The MML criterion for our model (see details in
Appendix C) consists of minimizing, with respect to , the
following cost function (after discarding the order one term)
log pðYj Þþ
K þ D
2
log N þ RX
2
D X
l¼1
K
logðN j lÞ
j¼1
þ SX
2
D
logðNð1 lÞÞ;
l¼1
ð19Þ
where R and S are the number of parameters in jl and l,
respectively. If pð:j:Þ and qð:j:Þ are univariate Gaussians
(arbitrary mean and variance), R ¼ S ¼ 2. From a parameter
estimation viewpoint, (19) is equivalent to a maximum
a posteriori (MAP) estimate,
Fig. 6. The unsupervised feature saliency algorithm.
(
^ ¼ arg max log pðYj Þ RDX
2
K
log
l¼1
j
SX
2
D
RKX
2
logð1
l¼1
lÞ
D
)
log l ;
l¼1
ð20Þ
with the following (Dirichlet-type, but improper) priors on
the js and ls:
pð 1; ...; KÞ / YK
pð 1; ...; DÞ / YD
j¼1
l¼1
RD=2
j ;
RK=2
l ð1 lÞ S=2 :
Since these priors are conjugate with respect to the complete
data likelihood, the EM algorithm undergoes a minor
modification: The M-step (13) and (18) are replaced by
bj ¼ maxðPi
wij
RD
2 ; 0Þ
P
j maxðPi
wij
RD
2 ; 0Þ
maxð
bl ¼
P
i;j uijl
maxð P
i;j uijl
KR
2
KR
2 ; 0Þ
; 0ÞþmaxðP
i;j vijl
ð21Þ
: ð22Þ
S
2 ; 0Þ
In addition to the log-likelihood, the other terms in (19)
have simple interpretations. The term KþD
2 log N is a standard
MDL-type [50] parameter code-length corresponding to K j
values and D l values. For the lth feature in the jth
component, the “effective” number of data points for
estimating jl is N j l. Since there are R parameters in each
jl, the corresponding code-length is R
2 logðN l jÞ. Similarly,
for the lth feature in the common component, the number of
effective data points for estimation is Nð1 lÞ. Thus, there is
a term S
2 logðNð1 lÞÞ in (19) for each feature.
One key property of (21) and (22) is their pruning
behavior, forcing some of the j to go to zero and some of
the l to go to zero or one. This pruning behavior also has
the indirect benefit of protecting us from almost singular
covariance matrices: the weight of such component is
usually very small, and it is likely to be pruned in the next
6 IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE, VOL. 26, NO. 9, SEPTEMBER 2004
Fig. 7. The solid ellipses represent the Gaussian mixture components; the dotted ellipse represents the common density. The number in parenthesis
along the axis label is the feature saliency; when it reaches 1, the common component is no longer applicable to that feature. Thus, in (d), the
common component degenerates to a line; when the feature saliency for feature 1 also becomes 1, as in (f), the common density degenerates to a
point at ð0; 0Þ. (a) The data set, (b) initialization, (c) a snapshot, (d) 2 is “pruned” to 1, (e) a local minimum (K ¼ 5), and (f) the best local minimum.
few iterations. Concerns that the message length in (19) may
become invalid at these boundary values can be circumvented
by the arguments in [18]: When l goes to zero, the
lth feature is no longer salient and l and 1l; ...; Kl are
removed; when l goes to 1, l and l are dropped.
Finally, since the model selection algorithm determines
the number of components, it can be initialized with a large
value of K, thus alleviating the need for a good initialization,
as shown in [18]. Because of this, a component-wise
version of EM can be adopted [10], [18]. The algorithm is
summarized in Fig. 6.
3.4 POSTPROCESSING OF FEATURE SALIENCY
The feature saliencies generated by the algorithm in Fig. 6
attempt to find the best way to model the data, using
different component densities. Alternatively, we can consider
feature saliencies that best discriminate between
different components. This can be more appropriate if the
ultimate goal is to discover well-separated clusters. If the
components are well-separated, each pattern is likely to be
generated by one component only. Therefore, one quantitative
measure of the separability of the clusters is
J ¼ XN
i¼1
log Pðzi ¼ tijy iÞ; ð23Þ
where ti ¼ arg maxj Pðzi ¼ jjy iÞ. Intuitively, J is the sum of
the logarithms of the posterior probabilities of the data,
assuming that each data point was indeed generated by the
component with maximum posterior probability (an im-
plicit assumption in mixture-based clustering). J can then be
maximized by varying
meters fixed.
l while keeping the other para-
Unlike the MML criterion, J cannot be optimized by an
EM algorithm. However, defining
hilj ¼
pðyilj jlÞ qðylj lÞ
lpðyilj jlÞþð1 lÞqðyilj lÞ ;
gil ¼ XK
j¼1
it is easy to show that
@
wijhilj;
log wij ¼ hilj gil;
@ l
@ 2
@ l @ m
log wij ¼ XN
ðgilgim
i¼1
@2 @ 2 log wij ¼
l
Xn
ðg
i¼1
2 il
X K
j¼1
h 2 ilj Þ:
wijhiljhimjÞ; for l 6¼ m;
The gradient and Hessian of J can then be calculated
accordingly, if we ignore the dependence of ti on l. We can
then use any constrained nonlinear optimization software
to find the optimal values of l in ½0; 1Š. We have used the
MATLAB optimization toolbox in our experiments. After
obtaining the set of optimized l, we fix them and estimate
the remaining parameters using the EM algorithm.
LAW ET AL.: SIMULTANEOUS FEATURE SELECTION AND CLUSTERING USING MIXTURE MODELS 7
Fig. 8. Feature saliencies for (a) the 10-D 4 Gaussian data set used in Fig. 7a, and (b) the Trunk data set. The mean values plus and minus one
standard deviation over ten runs are shown. Recall that features 3 to 10 for the 4 Gaussian data set are the noisy features. (a) Features saliencies:
4 Gaussian and (b) features saliencies: trunk.
4 EXPERIMENTAL RESULTS
4.1 Synthetic Data
The first synthetic data set consists of 800 data points from a
mixture of four equiprobable Gaussians Nðmi; IÞ;i¼
f1; 2; 3; 4g, where m1 ¼ 0
3 , m2 ¼ 1
9 , m3 ¼ 6
4 , m4 ¼ 7
10
(Fig. 7a). Eight “noisy” features (sampled from a Nð0; 1Þ
density) are then appended to this data, yielding a set of
800 10-dimensional patterns. We ran the proposed algorithm
10 times, each initialized with K ¼ 30; the common component
is initialized to cover the entire set of data, and the feature
saliency values are initialized at 0.5. The stopping threshold is
10 7 . A typical run of the algorithm is shown in Fig. 7. In all the
10 runs with this mixture, the four components were always
correctly identified. The saliencies of all the 10 features,
together with their standard deviations (error bars), are
shown in Fig. 8a. We can conclude that, in this case, the
algorithm successfully locates the true clusters and correctly
assigns the feature saliencies.
In the second experiment, we consider the Trunk data [24],
[57]: two 20-dimensional Gaussians Nðm1; IÞ and Nðm2; IÞ,
where m1 ¼ð1; 1ffiffi p ; ...;
2
1ffiffiffi p Þ, m2 ¼ m1. Data are obtained by
20
sampling 5,000 points from each of these two Gaussians. Note
that the features are arranged in descending order of
relevance. As above, the stopping threshold is set to 10 7
and the initial values of K to 30. In all the 10 runs performed,
the two components are always detected. The feature
saliencies are shown in Fig. 8b. The lower the rank number,
TABLE 1
Real World Data Sets
the more important is the feature. We can see the general trend
that as the feature number increases, the saliency decreases, in
accordance with the true characteristics of the data.
4.2 Real Data
We tested our algorithm on several data sets with different
characteristics (Table 1). The wine recognition data set
(wine) contains results of chemical analysis of wines grown
in different cultivars. The goal is to predict the type of a
wine based on its chemical composition; it has 178 data
points, 13 features, and three classes. The Wisconsin
diagnostic breast cancer data set (wdbc) was used to obtain
a binary diagnosis (benign or malignant) based on 30
features extracted from cell nuclei presented in an image; it
has 576 data points. The image segmentation data set
(image) contains 2,320 data points with 19 features from
seven classes; each pattern consists of features extracted
from a 3 3 region taken from seven types of outdoor
images: brickface, sky, foliage, cement, window, path, and
grass. The texture data set (texture) consists of 4,000 19dimensional
Gabor filter features from a collage of four
Brodatz textures [27]. A data set (zer) of 47 Zernike
moments extracted from images of handwriting numerals
(as in [26]) are also used; there are 200 images for each digit,
totaling 2,000 patterns. The data sets wine, wdbc, image,
and zernike are from the UCI machine learning repository
(http://www.ics.uci.edu/~mlearn/MLSummary.html).
Normalization to zero mean and unit variance is performed
for all but the texture data set, so as to make the
contribution of different features roughly equal a priori.
Each data set has N data points with D features from c classes. The feature with a constant value in image is discarded.
8 IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE, VOL. 26, NO. 9, SEPTEMBER 2004
Since these data sets were collected for supervised
classification, the class labels are not involved in our
experiment, except for evaluation of the clustering results.
Each data set was first randomly divided into two
halves: one for training, another for testing. The algorithm
in Fig. 6 was run on the training set. The feature saliency
values can be post-processed as described in Section 3.4. We
evaluate the results by interpreting the components as
clusters and compare them with the ground truth labels.
Each data point in the test set is assigned to the component
that most likely generated it, and the pattern is classified to
the class represented by the component. We can then
compute the error rates on the test data. For comparison, we
also run the mixture of Gaussian algorithm in [18] using all
the features, with the number of classes of the data set as a
lower bound on the number of components. This gives us a
fair ground for comparing Gaussian mixtures with and
without feature saliency. In order to ensure that we have
enough data with respect to the number of features for the
algorithm in [18], the covariance matrices of the mixture
components are restricted to be diagonal, but are different
for different components. The entire procedure is repeated
20 times and the results are shown in Table 2. We also show
the feature saliency values of different features in different
runs as gray-level image maps in Fig. 9.
From Table 2, we can see that the proposed algorithm
reduces the error rates when compared with using all the
features for all five data sets. The improvement is more
significant for the image data set, but this may be due to
the increased number of components estimated. The high
error rate for zernike is due to the fact that digit images
are inherently more difficult to cluster: for example, “4” can
be written in a manner very similar to “9” and it is difficult
for any unsupervised learning algorithm to distinguish
among them. The postprocessing can increase the “contrast”
of the feature saliencies, as the image maps in Fig. 9
show, without deteriorating the accuracy. It is easier to
perform “hard” feature selection using these postprocessed
feature saliencies, if this is a must for the application.
5 DISCUSSION
5.1 Complexity
The major computational load in the proposed algorithm is
in the E-step and the M-step. Each E-step iteration
computes OðNDKÞ quantities. As each quantity can be
TABLE 2
The Result of the Proposed Algorithm over 20 Random Runs
“Error” corresponds to the mean of the error rates on the testing set when the clustering results are compared with the ground truth labels.bc denotes
the number of Gaussian components estimated. Note that postprocessing does not change the number of Gaussian components. The numbers in
parenthesis are the standard deviation of the corresponding quantities.
computed in constant time, the time complexity for E-step is
also OðNDKÞ. Similarly, the M-step takes OðNDKÞ time.
The total amount of computation depends on the number of
iterations required for convergence.
At first sight, the amount of computation seems to be
demanding. However, a close examination reveals that each
iteration (E-step and M-step) of the standard EM algorithm
also takes OðNDKÞ time. The value of K in the standard
EM, though, is usually smaller, because the proposed
algorithm starts with a larger number of components. The
number of iterations required for our algorithm is also, in
general, larger because of the increase of the number of
parameters. Therefore, it is true that the proposed algorithm
takes more time than the standard EM algorithm with one
parameter setting. However, the proposed algorithm can
determine both the number of clusters and feature subsets.
If we want to achieve the same goal with the standard
EM algorithm using a wrapper approach, we need to rerun
EM multiple times, with different number of components
and different feature subsets. The computational demand is
much heavier than the proposed algorithm, even with
heuristic search to guide the selection of feature subsets.
Another strength of the proposed algorithm is that by
initialization with a large number of Gaussian components,
the algorithm is less sensitive to the local minimum
problem than the standard EM algorithm. We can further
reduce the complexity by adopting optimization techniques
applicable for standard EM for Gaussian mixture, such as
sampling the data, compressing the data [8], or using
efficient data structures [45], [54].
For the postprocessing step in Section 3.4, each computation
of the quantity J and its gradient and Hessian takes
OðNDKÞ time. The number of iterations is difficult to
predict, as it depends on the optimization routine. However,
we can always put an upper bound on the number of
iterations and trade speed for the optimality of the results.
5.2 Relation to Shrinkage Estimate
One interpretation of (6) is that we “regularize” the distribution
of each feature in different components by the common
distribution. This is analogous to the shrinkage estimator for
covariance matrices of class-conditional densities [15], which
is the weighted sum of an estimate of the class-specific
covariance matrix, and the “global” covariance matrix
estimate. In (6), the pdf of the lth feature is also a weighted
sum of a component-specific pdf and a common density. An
LAW ET AL.: SIMULTANEOUS FEATURE SELECTION AND CLUSTERING USING MIXTURE MODELS 9
Fig. 9. Image maps of feature saliency for different data sets with and without the postprocessing procedure. Feature saliency of 1 (0) is shown as a
pixel of gray level 255 (0). The vertical and horizontal axes correspond to the feature number and the trial number, respectively. (a) wine, proposed
algorithm, (b) wine, after postprocessing, (c) wdbc, proposed algorithm, (d) wdbc, after postprocessing, (e) image, proposed algoriothm, (f) image,
after postprocessing, (g) texture, proposed algorithm, (h) texture, after postprocessing, (i) zernike, proposed algorithm, and (j) zernike, after
postprocessing.
important difference here is that the weight l is estimated
from the data, using the MML principle, instead of set
heuristically, as is commonly done. As shrinkage estimators
have found empirical success to combat data scarcity, this
“regularization” viewpoint is an alternative explanation for
the usefulness of the proposed algorithm.
5.3 Limitation of the Proposed Algorithm
A limitation of the proposed algorithm is the feature
independence assumption (conditioned on the component).
While, empirically, the violation of the independence
assumption usually does not affect the accuracy of a
classifier (as in supervised learning) or the quality of
10 IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE, VOL. 26, NO. 9, SEPTEMBER 2004
clusters (as in unsupervised learning), this has some
negative influence on the feature selection problem.
Specifically, a feature that is redundant because its
distribution is independent of the component label given
another feature cannot be modeled under the feature
independence assumption. As a result, both features are
kept. This explains why, in general, the feature saliencies
are somewhat high. The postprocessing in Section 3.4 can
cope with this problem because it considers the posterior
distribution and, therefore, can discard features that do not
help in identifying the clusters directly.
5.4 Extension to Semisupervised Learning
Sometimes, we may have some knowledge of the class
labels of different Gaussian components. This can happen
when, say, we adopt a procedure to combine different
Gaussian components to form a cluster (e.g., as in [51]), or
in a semisupervised learning scenario, where we can use a
small amount of labelled data to help us identify which
Gaussian component belongs to which class. This additional
information can suggest combination of several
Gaussian components to form a single class/cluster, thereby
allowing the identification of non-Gaussian clusters. The
postprocessing step can take advantage of this information.
Suppose we know there are C classes and the posterior
probability that pattern yi belongs to the cth class, denoted
ric, can be computed as ric ¼ PK j¼1 cjPðzi ¼ jjyiÞ. For
example, if we know that the components 4, 6, and 10 are
from class 2, we can set 2;4 ¼ 2;6 ¼ 2;10 ¼ 1=3 and the
other 2;j to be zero. The postprocessing is modified
accordingly: Redefine ti in (23) to ti ¼ arg maxc ric, i.e., it
becomes the class label for yi in view of the extra
information; replace log P ðzi ¼ tijyiÞ in (23) by log ri;ti . The
gradient and Hessian can still be computed easily after
noting that
@
@ l
@wij
@ l
¼ wij
log ric ¼ 1
@
log wij ¼ wijðhilj
@ l
gilÞ
X
ric
K
j¼1
@
cj wij ¼
@ l
XK
j¼1
cjwij
hilj
ric
gil:
ð24Þ
We can then optimize the modified J in (23) to carry out the
postprocessing.
5.5 A Note on Maximizing the Posterior Probability
The sum of the logarithm of the maximum posterior
probability considered in the postprocessing in Section 3.4
can be regarded as the sample estimate of an unorthodox
type of entropy (see [30]) for the posterior distribution. It
can be regarded as the limit of Renyi’s entropy R ðPÞ when
tends to infinity, where
R ðP Þ¼ 1
1
log XK
j¼1
p i : ð25Þ
When this entropy is used for parameter estimation under
the maximum entropy framework, the corresponding
procedure is closely related to minimax inference. Other
functions on the posterior probabilities can also be used,
such as the Shannon entropy of the posterior distribution.
Preliminary studies show that the use of different types of
entropy does not affect the results significantly.
6 CONCLUSIONS
In this paper, we have presented an EM algorithm to
estimate the importance of different features and the best
number of components for Gaussian-mixture clustering.
The proposed algorithm can avoid running EM many times
with different numbers of components and different feature
subsets, and can achieve better performance than using all
the available features for clustering. The usefulness of the
algorithm was demonstrated on both synthetic and benchmark
real data sets.
There are several avenues for future work. The space
complexity of the proposed algorithm is OðNDKÞ, which
can slow down the algorithm significantly when the data
set (which is of size OðNDÞ) is so large that the intermediate
variables cannot be held in memory. How to extend the
algorithm to cope with this is a challenging problem. We
also may attempt to model the dependency between
different features explicitly. Merging the proposed algorithm,
which is basically a wrapper, with other filter
techniques, can lead to a hybrid algorithm that is applicable
for data sets with enormous numbers of features. We can
replace the mixture of Gaussians by a mixture of multinomial
distribution, thereby making the proposed algorithm
also applicable to categorical data. One may also
extend the current algorithm to handle different salient
features for different components. Finally, principles other
than MML, such as variational Bayes [12], can be adopted to
perform model selection.
APPENDIX A
THE MIXTURE MODEL
Recall (5), which is the conditional density of y, given
¼ð 1; ...; DÞ,
pðyj Þ¼ XK
j¼1
Y D
j
l¼1
ðpðylj jlÞÞ l ðqðylj lÞÞ 1 l :
We treat as a set of missing variables and define
l ¼ P ð l ¼ 1Þ, for l ¼ 1; ...;D, as a set of parameters to
be estimated (the feature saliencies). We assume the ls are
mutually independent and also independent of the hidden
component label z for any pattern y. Thus,
pðy; Þ¼pðyj Þpð Þ
¼
¼ XK
X K
j¼1
j¼1
Y D
j
l¼1
Y D
j
l¼1
ðpðylj jlÞÞ l ðqðylj lÞÞ 1 l
The marginal density for y is
! Y D
l¼1
ð lpðylj jlÞÞ l ðð1 lÞqðylj lÞÞ 1 l :
l
l ð1 lÞ 1 l
ð26Þ
LAW ET AL.: SIMULTANEOUS FEATURE SELECTION AND CLUSTERING USING MIXTURE MODELS 11
pðyÞ ¼ X pðy; Þ
¼ XK
j¼1
¼ XK
j¼1
¼ XK
j¼1
j
XY D
Y D
l¼1
X 1
j
l¼1 l¼0
YD j
l¼1
ð lpðylj jlÞÞ l 1 l
ðð1 lÞ qðylj lÞÞ
ð lpðylj jlÞÞ l ðð1 lÞ qðylj lÞÞ 1 l
pðylj jlÞ l þ qðylj lÞð1 lÞ ;
ð27Þ
which is (6). Note that the features are independent, given
the component label z.
APPENDIX B
DERIVING THE EM ALGORITHM
The complete-data log-likelihood for the model in (6) is
P ðy i;zi ¼ j; Þ¼ j
Y D
l¼1
Define the following quantities:
ð lpðyilj jlÞÞ l ðð1 lÞqðyilj lÞÞ 1 l :
wij ¼ P ðzi ¼ jjyiÞ; uij l ¼ P ðzi ¼ j; l ¼ 1jyiÞ; vij l ¼ Pðzi ¼ j; l ¼ 0jyiÞ: ð28Þ
They are calculated using the current parameter estimate
now . Note that uij l þ vij l ¼ wij and P N
i¼1
P K
j¼1 wij ¼ n. The
expected complete data log-likelihood based on now is
E now½log PðY; z; ÞŠ
¼ X
Pðzi ¼ j; jyiÞ log j þ X
i;j;
log lÞþð1 lÞðlog qðyilj lÞþlogð1 lÞÞ
¼ X
Pðzi ¼ jjyiÞ log j þ XX
i;j
¼ X
j
lðlog pðyilj jlÞþlog lÞ
i;j
l
l
X 1
l¼0
þð1 lÞðlog qðyilj lÞþlogð1 lÞÞ
X
i
wij log j
|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}
part 1
þ XX
vij l log qðyilj lÞ
l
þ X
l
lðlog pðyilj jlÞþ
!
P ðzi ¼ j; ljy iÞ
þ XX
uij l log pðyilj jlÞ
j;l
i;j
|fflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}
part 3
i
|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}
part 2
X
log l uij l þ logð1 lÞ X
vij l
i;j
i;j
|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}
part 4
!
:
The four parts in the equation above can be maximized
separately. Recall that the densities pð:Þ and qð:Þ are
univariate Gaussian and are characterized by their means
and variances. As a result, maximizing the expected
complete data log-likelihood leads to the M-step in (13)-
(18). Finally, observe that
P ð l ¼ 1jzi ¼ j; yiÞ¼ Pð l ¼ 1; yijzi ¼ jÞ
Pðyijzi ¼ jÞ
¼ lpðylj jlÞ Q
l06¼l l0pðyl0j jl0Þþð1 l0Þqðyl 0j l0Þ Q
l0 l0pðyl0j jl0Þþð1 l0Þqðyl 0j l0Þ lpðylj jlÞ
¼
lpðylj jlÞþð1 lÞqðylj lÞ ¼ aij l
:
cij l
Therefore, (11) follows because
uijl ¼ P ð l ¼ 1jzi ¼ j; yiÞP ðzi ¼ jjyiÞ¼ aij l
wij: ð29Þ
APPENDIX C
APPLYING MINIMUM MESSAGE LENGTH
The minimum message length (MML) criterion is given by
(see [18] for details and references)
n
^ ¼ arg min log pð Þ log pðYj Þþ 1
log jIð Þjþ
2
c
1
o
ð30Þ
1 þ log ;
2 12
where is the set of parameter of the model, c is the
dimension of , Ið Þ¼ E½D 2 log pðYj ÞŠ is the (expected)
Fisher information matrix (the negative expected value of
the Hessian of the log-likelihood), and jIð Þj is the
determinant of Ið Þ. The information matrix for the model
(6) is very difficult to obtain analytically. Therefore, as in
[18], we approximate it by the information matrix of the
complete data log-likelihood, Icð Þ. By differentiating the
logarithm of (28), we can show that
h
Icð Þ¼block diag M;
1
; ...;
1ð1 1Þ
cij l
1
Dð1 DÞ ;
1 1Ið 11Þ; ...; 1 DIð 1DÞ;
2 1Ið 21Þ; ...; 2 DIð 2DÞ; ...; K 1Ið K1Þ; ...;
i
K DIð KDÞ; ð1 1ÞIð 1Þ; ...; ð1 DÞIð DÞ ;
ð31Þ
where M is the information matrix of the multinominal
distribution with parameters ð 1; ...; KÞ. The size of Ið Þ is
ðK þ D þ KDR þ DSÞ, where R and S are the number of
parameters in jl and l, respectively. Note that ð lð1
lÞÞ 1 is the information of a Bernoulli distribution with
parameter l. Thus, we can write
jIcð Þj ¼ log Iðf jgÞ þ XD
þ XK
X D
j¼1 l¼1
l¼1
log Ið lÞþR XK
log Ið jlÞþS XD
l¼1
X D
j¼1 l¼1
logð j lÞ
logð1 lÞþ XD
l¼1
Ið lÞ:
ð32Þ
For the prior densities of the parameters, we assume that
different groups of parameters are independent. Specifically,
f jg, l (for different values of l), jl (for different
values of j and l), and l (for different values of l) are
independent. Furthermore, since we have no knowledge
12 IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE, VOL. 26, NO. 9, SEPTEMBER 2004
about the parameters, we adopt noninformative Jeffrey’s
priors (see [18] for details and references) which are
proportional to the square root of the determinant of the
corresponding information matrices. When we substitute
pð Þ and jIð Þj into (30), and drop the order-one term, we
obtain our final criterion, which is (19):
n
^ ¼ arg min log pðYj Þþ 1
ðK þ D þ KDR þ DSÞ
log n þ RX
2
K X
j¼1
D
l¼1
2
logð j lÞþ S
2
X D
l¼1
logð1
o
lÞ :
ACKNOWLEDGMENTS
This research was supported by ONR grant number
N000140410183. M.A.T. Figueiredo’s research was supported
by the (Portugese) Foundation for Science and Technology
under project POSI/33143/SRI/2000.
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Martin H.C. Law received the BEng degree and
MPhil degree in computer science from Hong
Kong University of Science and Technology.
After working in Hong Kong Baptist University
and Hong Kong University of Science and
Technology for more than two years, he moved
to the US and is currently a PhD candidate in the
Department of Computer Science and Engineering
at Michigan State University. His research
interests include data clustering, mixture models,
manifold learning, dimensionality reduction, and kernel methods. He
is a student member of the IEEE.
Mário A.T. Figueiredo (S’87-M’95-SM’2000)
received the EE, MSc, and PhD degrees in
electrical and computer engineering, all from the
Instituto Superior Técnico (I.S.T.), the engineering
school of the Technical University of Lisbon,
Portugal, in 1985, 1990, and 1994, respectively.
Since 1994, he has been with Department of
Electrical and Computer Engineering, I.S.T. He
is also a researcher and area coordinator at the
Institute of Telecommunications, Lisbon. In
1998, he held a visiting position with the Department of Computer
Science and Engineering at Michigan State University, East Lansing,
Michigan. His scientific interests include image processing and analysis,
computer vision, statistical pattern recognition, and statistical learning.
He received the Portuguese IBM Scientific Prize in 1995 for work on
unsupervised image restoration. He is an associate editor of the journals
Pattern Recognition Letters, IEEE Transactions on Image Processing,
and IEEE Transactions on Mobile Computing. He is also guest coeditor
of special issues of the journals IEEE Transactions on Pattern Analysis
and Machine Intelligence and IEEE Transactions on Signal Processing.
He was a cochair of the 2001 and 2003 Workshops on Energy
Minimization Methods in Computer Vision and Pattern Recognition, and
local chair of the 2004 Joint IAPR International Workshops on
Syntactical and Structural Pattern Recognition and Statistical Pattern
Recognition. He has been a member of program committees of several
international conferences, including CVPR, EECV, ICASSP, and ICPR.
He is a senior member of the IEEE.
Anil K. Jain is a University Distinguished
Professor in the Departments of Computer
Science & Engineering and Electrical & Computer
Engineering at Michigan State University. He
was the department chair from 1995-1999. His
research interests include statistical pattern
recognition, exploratory pattern analysis, texture
analysis, document image analysis, and biometric
authentication. Several of his papers
have been reprinted in edited volumes on image
processing and pattern recognition. He received the best paper awards
in 1987 and 1991, and received certificates for outstanding contributions
in 1976, 1979, 1992, 1997, and 1998 from the Pattern Recognition
Society. He also received the 1996 IEEE Transactions on Neural
Networks Outstanding Paper Award. He is a fellow of the IEEE, ACM,
and International Association of Pattern Recognition (IAPR). He has
received a Fulbright Research Award, a Guggenheim fellowship, and
the Alexander von Humboldt Research Award. He delivered the 2002
Pierre Devijver lecture sponsored by the IAPR. He holds six patents in
the area of fingerprint matching. He is the author of the following books:
Handbook of Fingerprint Recognition (Springer 2003) (received the 2003
PSP award from the Association of American Publishers), BIO-
METRICS: Personal Identification in Networked Society (Kluwer
1999), 3D Object Recognition Systems (Elsevier 1993), Markov
Random Fields: Theory and Applications (Academic Press 1993),
Neural Networks and Statistical Pattern Recognition (North-Holland
1991), Analysis and Interpretation of Range Images (Springer-Verlag
1990), Algorithms For Clustering Data (Prentice-Hall 1988), and Real-
Time Object Measurement and Classification (Springer-Verlag 1988).
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