Clustering of brain tumours through constrained manifold learning using class information

1
Clustering of brain tumours through constrained
manifold learning using class information

• Raúl Cruz and Alfredo Vellido
• rcruz, avellido_at_lsi.upc.edu

2
Contents

• Introduction
• GTM model
• t-GTM model
• Class-tGTM
• Human brain tumour data
• Experimental results
• Conclusions
• References

3
Introduction

• The cluster structure through Model-based
  clustering p(X).
• In classification, we model the relationship
  between the labels C and the data by p(CX).
• We aim to discover the cluster structure of the
  data while taking into account the available
  class information in a semi-supervised approach.

4
The Generative Topographic Mapping

• The GTM is a non-linear latent variable model
• It defines a mapping from a latent space to the
  data space, generating a probability density
  within the latter
• The GTM can be used for multivariate data
  visualization and clustering
• It can be seen as a constrained Mixture of
  Distributions or as a probabilistic alternative
  to the Self-Organizing Map (SOM).

5
GTM an illustration
Image borrowed from Bishop, C.M., Svensén, M.
Williams, C.K.I., (1998) Developments of the
Generative Topographic Mapping. Neurocomputing,
21(1-3)
6
GTM the algorithm (1)

• A regular grid of nodes ui is defined in latent
  space, with a prior probability

• A regular, fixed grid of non-linear basis
  functions provides the non-linear mapping into
  data space, defined by
• This mapping defines a set of reference vectors
  
• in data space

7
GTM the algorithm (2)

• Each of the reference vectors yk forms the centre
  of an isotropic Gaussian distribution in data
  space
• and, therefore, the data p.d.f. for the GTM model
  is given by
• REMEMBER The GTM can be understood as a
  constrained mixture of distributions.

8
GTM the algorithm (3)

• Given that the GTM is a parametric probability
  density model, it can be fitted to the data. The
  corresponding log-likelihood is defined as
• The EM algorithm can be used to calculate the
  parameters of the model

9
The t-GTM model

• What if outliers are present in data?
• The use of Gaussian in standard GTM is likely
  to negatively bias the estimation of the
  adaptative parameters, distorting the clustering
  results.
• The GTM was redefined as a constrained
  mixture of Student t-distributions the t-GTM
  (Vellido, 2006), aiming to increase the
  robustness of the model towards outliers.

10
The class-tGTM

• Class separability might be improved if the
  clustering model accounted for the available
  class information.
• this can be achieved by modelling the joint
  density

11
Human brain tumour data

• The data used in this study consist of 98 single
  voxel PROBE (PROton Brain Exam system) spectra
  acquired in vivo for five viable tumour types
  astrocytomas, glioblastomas, metastases,
  meningiomas, and oligodendrogliomas (typology
  that will be used in this study as class
  information) and from cystic regions (associated
  to tumours)
• A procedure based on Multivariate Bayesian
  Variable Selection was used elsewhere (Lee et al
  2000), to describe the data set in the form of
  six frequency intensities, corresponding to fatty
  acids, lactate, a compound-unassigned peak,
  glutamine, choline, and taurine-inositol.

12
Experimental results (1)

• The differences in class separability between the
  models with and without class information were
  quantified through the following entropy-like
  measure

where
set of tumour types
is the total number of tumour spectra
is the number of tumour spectra assigned to the
kth cluster
is the number of tumour spectra from tumour type
i assigned to cluster k
,
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Experimental results artificial data (2)

• We used a synthetic data set, consisting of 1,200
  data points, sampled from four neatly separated
  Gaussian distributions with centres located at
  and they were subsequently
  corrupted by four extra features consisting of
  Gaussian noise.

Figure 1. Representation, on the t-GTM
2-dimensional latent space, of the synth data set
described in the main text. The representation is
based on the mean posterior distributions for the
1200 data points. Points belonging to each of the
four original Gaussians (classes) are plotted
using a different symbol. The axes of the plot
are the elements of the latent vector and convey
no meaning by themselves. For that reason, axes
are kept unlabeled. (Left) t-GTM without class
information. (Right) class-t-GTM.
14
Experimental results tumour data (3)
Figure 2 Cysts (circles) vs Tumour tissue
Class-tGTM
t-GTM
Figure 3. The brain tumour MRS data set using
the full tumour typology. Symbols Cysts
(circles) astrocytomas (black dots)
glioblastomas (black rhombus) metastases
(five-pointed stars) meningiomas (white
rhombus) oligodendrogliomas (asterisks).
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Experimental results (4)
Table 1. Entropy results for the different models
and data sets analyzed. In all cases, and for the
sake of brevity, results are reported only for
three values of ? (although results for other
values are consistent with the reported ones).
Brain simplified refers to the simplification in
which data are separated into cysts versus tumour
tissue, with the five tumour types reduced to a
single class. Brain refers to the full tumour
typology.
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Conclusions and future work

• Adding class information to a clustering process
  has the potential of improving the class
  separation provided by the resulting clusters.
• We have clustered human brain tumour MRS data
  using a constrained generative model, in a
  variant known to behave robustly in the presence
  of outliers the t-GTM. This model has been
  extended to account for class information in a
  semi-supervised manner, and experiments carried
  out on both synthetic and the MRS data have shown
  that class-t-GTM improves class discrimination.
• This semi-supervised clustering model might also
  be used to visualize new and unlabelled MR
  spectra and infer their typology.
• An alternative and perhaps more principled way to
  estimate the joint density would entail
  considering a mixture model with both Gaussian or
  t-Student components for the continuous data and
  multinomial components for the class information
  (binary, categorical)

17
References

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References

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