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The Dirichlet Labeling Process

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The Dirichlet Labeling Process for Functional Data Analysis XuanLong Nguyen & Alan E. Gelfand Duke University Machine Learning Group Presented by Lu Ren – PowerPoint PPT presentation

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Title: The Dirichlet Labeling Process


1
The Dirichlet Labeling Process for Functional
Data Analysis
XuanLong Nguyen Alan E. Gelfand Duke
University Machine Learning Group Presented by Lu
Ren
2
Outline
  • Introduction
  • Formalizing the model
  • Properties of the Labeling Process
  • Identifiability
  • Model fitting and inference
  • Applications
  • Conclusions

3
Introduction
  • Functional Data
  • Suppose we have a collection of functions,
    each viewed as a stochastic process realization
    with observations at a common set of locations.
  • e.g., a random curve or surface.
  • 2. Dirichlet Labeling Process
  • For a particular process realization, we
    assume that the observation at a given location
    can be allocated to separate groups via a random
    allocation process.
  • 3. The Primary Objective
  • Examine clustering of the set of curves.

4
Introduction
4. The connections with other models
  • Dirichlet Process (DP) mixture model
  • global clustering
  • Dependent Dirichlet Process (DDP) mixture model
  • local clustering
  • Generalized Spatial DP mixture model
  • thresholding latent Gaussian process

5
Model Formalization
Noisy curve realizations
over Obtained at local sites The corresponding
latent curves Each curve is described by
the label function
Dirichlet labeling process generates a random
distribution and also a marginal multinomial
distribution with
for
6
Model Formalization
Assume a collection of canonical species
is realized at each location by indexing
with the labels, i.e., if
.
Or, it is equivalent to
7
Model Formalization
is a random probability measure on

where is a base measure on
and constructed such that
  1. has a uniform marginal distribution at every
    location
  2. inherits the spatial dependence structure via
    on .

Denote by the finite-dimensional
distributions of . Let
and consider
where denotes the cumulative distribution
function at for .
8
Model Formalization
The vector has uniform marginals and induces a
joint distribution function denoted by
on . Let
be an increasing sequence of threshold
in such that for
. If define
, then
Discretize into hyper-cubes then
So an drawn from yields a label

9
Model Formalization
Similarly, we define an auxiliary variables
on for such that
where
According to the definition of DP,
10
Properties
1. Properties of .
2. Properties of .
Assume is a mean-zero, isotropic Gaussian
process
with covariance function
11
Properties
Under the assumptions on , the quantile
threshold functions are constant with
respect to and the sequence satisfies
.
12
Identifiability
  1. Larger will lead to more smooth learned
    canonical curves but weakly distinguishable,
    while smaller will make the curves
    posteriors cover different regions in the
    function space.
  2. As is close to 0, label switching is
    discouragedglobal clustering if the curve
    realizations tend to switch often, the canonical
    curves become more weakly identified.
  3. Similar locations tend to be (correctly) assigned
    the same labels, but it is possible that the
    whole segment is incorrectly labeled relatively
    to some other segments.
  • strong constraints (ordering of label values)
    can be imposed upon.
  • The model identifiability cannot be ensured with
    constraints but
  • the mixing for posterior inference would be
    expected to improve.

13
Model fitting and inference
The joint distribution associated with model
parameters
For canonical curves, the prior for vector
is normal with mean and covariance
matrix
The full conditional for still has a Gaussian
form, but it has a high dimension for large data
set .
The inference of the label vectors is
dependent on the Polya urn sampling scheme and in
terms of and
14
Applications
1. Synthetic Data
  • Specify locations
  • while leave other 20 locations for validation
    purposes.
  • for are iid drawn
    from at locations
    , where
  • are constructed by
    .
  • The data collection is
    obtained by mixing with
  • an independent error process drawn from
    .

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17
Applications
2. Progesterone modeling
  • The data records the natural logarithm of the
    progesterone metabolite, during a monthly cycle
    for 51 female subjects.
  • Each cycle ranges from -8 to 15 (8 days
    pre-ovulation to 15 days post-ovulation).
  • There are total of 88 cycles the first 66
    cycles belong to non-contraceptive group, the
    remaining 22 cycles belong to the contraceptive
    group.
  • We also consider a modified data with the curves
    of the contraceptive group are down-shifted by 2.
  • We focus our analysis to the case k2.

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21
Applications
  • 3. Image modeling
  • 80 color images with each size equal to
    .
  • Each image is represented by a surface
    realization , where is the color
    intensity of the location .

  • represents the RGB color intensity.
  • We introduce canonical species curves.

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Conclusions
  • The Dirichlet labeling process provides a highly
    flexible prior for modeling collections of
    functions.
  • The inter-relationships between these parameters
    are complex with regard to process behavior.
  • MCMC inference is proved to have a fast mixing
    and yields good results.
  • The model is applied on two real applications.
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