Clustering with multiple distance metrics - multi-level ...jornsten/FMSJornsten09.pdf · distance...
Transcript of Clustering with multiple distance metrics - multi-level ...jornsten/FMSJornsten09.pdf · distance...
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Jornsten.Clustering with
multipledistance metrics
Outline
ClusterAnalysis
Within-clusterparameteriza-tions
Between-clusterparameteriza-tions:Multi-levelmixturemodeling
Results
Conclusion andFuture work
Clustering with multiple distance metrics
- multi-level mixture models with profile
transformations.
Rebecka Jornsten
Department of Mathematical StatisticsChalmers and Gothenburg University
[email protected], http://www.stat.rutgers.edu/∼rebecka
September 16, 2009
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Jornsten.Clustering with
multipledistance metrics
Outline
ClusterAnalysis
Within-clusterparameteriza-tions
Between-clusterparameteriza-tions:Multi-levelmixturemodeling
Results
Conclusion andFuture work
1 Cluster Analysis
2 Within-cluster parameterizations
3 Between-cluster parameterizations:Multi-level mixture modeling
4 Results
5 Conclusion and Future work
![Page 3: Clustering with multiple distance metrics - multi-level ...jornsten/FMSJornsten09.pdf · distance metrics Outline Cluster Analysis Within-cluster parameteriza-tions Between-cluster](https://reader034.fdocuments.in/reader034/viewer/2022042219/5ec5d1e1acb7740ab05168bc/html5/thumbnails/3.jpg)
Jornsten.Clustering with
multipledistance metrics
Outline
ClusterAnalysis
Within-clusterparameteriza-tions
Between-clusterparameteriza-tions:Multi-levelmixturemodeling
Results
Conclusion andFuture work
Cluster analysis
General problem
Popular approach for dimension reduction
Wide range of applications: engineering, geologicaldata, social networks, high-throughput biology
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Jornsten.Clustering with
multipledistance metrics
Outline
ClusterAnalysis
Within-clusterparameteriza-tions
Between-clusterparameteriza-tions:Multi-levelmixturemodeling
Results
Conclusion andFuture work
Thinking about the problems in termsof Model Selection
Clustering1 How many clusters?
2 Can we find an efficient within-cluster parameterization?(E.g., are cluster profiles flat, or increasing?)
3 Can we find an efficient between-clusterparameterization? (Do cluster profiles share a similarshape, or coincide for a subset of data dimensions?)
![Page 5: Clustering with multiple distance metrics - multi-level ...jornsten/FMSJornsten09.pdf · distance metrics Outline Cluster Analysis Within-cluster parameteriza-tions Between-cluster](https://reader034.fdocuments.in/reader034/viewer/2022042219/5ec5d1e1acb7740ab05168bc/html5/thumbnails/5.jpg)
Jornsten.Clustering with
multipledistance metrics
Outline
ClusterAnalysis
Within-clusterparameteriza-tions
Between-clusterparameteriza-tions:Multi-levelmixturemodeling
Results
Conclusion andFuture work
Clustering gene expression data
Tasks
To group genes by similarity of expression profiles acrossexperimental conditions
Assign functions to genes - ’guilt by association’
Goals
We need an objective interpretation of cluster profiles -gene function inferences
Avoid ”Waste of parameters”
![Page 6: Clustering with multiple distance metrics - multi-level ...jornsten/FMSJornsten09.pdf · distance metrics Outline Cluster Analysis Within-cluster parameteriza-tions Between-cluster](https://reader034.fdocuments.in/reader034/viewer/2022042219/5ec5d1e1acb7740ab05168bc/html5/thumbnails/6.jpg)
Jornsten.Clustering with
multipledistance metrics
Outline
ClusterAnalysis
Within-clusterparameteriza-tions
Between-clusterparameteriza-tions:Multi-levelmixturemodeling
Results
Conclusion andFuture work
Within-cluster parameterizations
Mathematical Formulation
For each gene g we observe a feature vector xg :
xg | Rg = k ∼ MVN(µk ,Σk),
where Rg is the cluster membership indicator.
We seek efficient/interpretable parameterizations of thecluster profiles; µk = Wθk
A sparse representation of µk is obtained if we set someof θk to 0.
The design matrix W is chosen to represent a scientificquestion of interest.
Fitting the model: EM with a modified M-step incorporatingthe θk constraints (see Jornsten and Keles, Biostatistics,2008).
![Page 7: Clustering with multiple distance metrics - multi-level ...jornsten/FMSJornsten09.pdf · distance metrics Outline Cluster Analysis Within-cluster parameteriza-tions Between-cluster](https://reader034.fdocuments.in/reader034/viewer/2022042219/5ec5d1e1acb7740ab05168bc/html5/thumbnails/7.jpg)
Jornsten.Clustering with
multipledistance metrics
Outline
ClusterAnalysis
Within-clusterparameteriza-tions
Between-clusterparameteriza-tions:Multi-levelmixturemodeling
Results
Conclusion andFuture work
Within-cluster parameterizations
Model Selection
Searching for an optimal sparse within-cluster representation:
Classification EM (CEM) approach, see Jornsten andKeles, 2008.
Simultaneous selection via rate-distortion theory, seeJornsten, JCGS, 2009.
![Page 8: Clustering with multiple distance metrics - multi-level ...jornsten/FMSJornsten09.pdf · distance metrics Outline Cluster Analysis Within-cluster parameteriza-tions Between-cluster](https://reader034.fdocuments.in/reader034/viewer/2022042219/5ec5d1e1acb7740ab05168bc/html5/thumbnails/8.jpg)
Jornsten.Clustering with
multipledistance metrics
Outline
ClusterAnalysis
Within-clusterparameteriza-tions
Between-clusterparameteriza-tions:Multi-levelmixturemodeling
Results
Conclusion andFuture work
Multi-level Mixture models:Between-cluster parameterizations
Cell-line data
There seem to be moreneuron-specific clusters thanglia specific
MIXL: Mixture models formulti-factor experiments
——————————————
mRNA expression time courseafter spinal cord injury
Clusters seem to share a similarshape, and differ in terms ofoffset and scale.
MIXT : Mixture models withprofile transformations.
Example 1:
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Example 2:
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Jornsten.Clustering with
multipledistance metrics
Outline
ClusterAnalysis
Within-clusterparameteriza-tions
Between-clusterparameteriza-tions:Multi-levelmixturemodeling
Results
Conclusion andFuture work
MIXT - Model formulation
Multi-level structure
For each gene g we observe a feature vector xg .
Indicators Rg = k,Ug = l : cluster k and sub-clusterl ∈ {1, · · · , Lk}.
xg | Rg = k,Ug = l ∼ MVN(Aklµk + bkl ,AklΣkA′kl)
Within-cluster parameterization: µk = Wθk
Between-cluster parameterization:affine transformation (Akl , bkl)→ still Gaussian mixture.
Clustering with Multiple distance metrics
Here, consider sign-flip and/or scale transformations:µkl = Tkl(µk) = βkl(µk + αkl), Σkl = β2klΣk
Equivalent to clustering with three distance metricssimultaneously: 1− |correlation|, 1− correlation, andeuclidean distance!
![Page 10: Clustering with multiple distance metrics - multi-level ...jornsten/FMSJornsten09.pdf · distance metrics Outline Cluster Analysis Within-cluster parameteriza-tions Between-cluster](https://reader034.fdocuments.in/reader034/viewer/2022042219/5ec5d1e1acb7740ab05168bc/html5/thumbnails/10.jpg)
Jornsten.Clustering with
multipledistance metrics
Outline
ClusterAnalysis
Within-clusterparameteriza-tions
Between-clusterparameteriza-tions:Multi-levelmixturemodeling
Results
Conclusion andFuture work
Model fitting
Fitting the model: EM with a profile M-stepincorporating the constraints.
Separate the tasks into
1 The top-level estimation problem: (µk = Wkθk ,Σk)
2 The transformation parameter estimation problem:(αkl , βkl) (where βk1 = 1, αk1 = 0,∀k).
How can we do this? Well, given (αkl , βkl), maximizingthe conditional likelihood translates to an application ofthe inverse profile transformation, and(
xgβkl− αkl
)|Rg = k ,Ug = l ∼ MVN(µk ,Σk).
Solve the aggregate problem first.
![Page 11: Clustering with multiple distance metrics - multi-level ...jornsten/FMSJornsten09.pdf · distance metrics Outline Cluster Analysis Within-cluster parameteriza-tions Between-cluster](https://reader034.fdocuments.in/reader034/viewer/2022042219/5ec5d1e1acb7740ab05168bc/html5/thumbnails/11.jpg)
Jornsten.Clustering with
multipledistance metrics
Outline
ClusterAnalysis
Within-clusterparameteriza-tions
Between-clusterparameteriza-tions:Multi-levelmixturemodeling
Results
Conclusion andFuture work
The aggregate problem
For each cluster k , and its sub-clusters l = 1, · · · , Lk ,compute
T−1kl (xg ) =xgβkl− αkl
1 Update
Σk =
∑g ,l ηgkl(T−1kl (xg )− µk)(T−1kl (xg )− µk)′∑
g ,l ηgkl
2 Condition on Σk , solve for the constrained θk :
θk =
∑g ,l ηgkl(W ′
kΣ−1k Wk)−1W ′kΣ−1k
(T−1kl (xg )
)∑g ,l ηgkl
Note that xg contributes to the top-level parameters inmultiple ways, moderated by the sub-clustermembership probabilities.
![Page 12: Clustering with multiple distance metrics - multi-level ...jornsten/FMSJornsten09.pdf · distance metrics Outline Cluster Analysis Within-cluster parameteriza-tions Between-cluster](https://reader034.fdocuments.in/reader034/viewer/2022042219/5ec5d1e1acb7740ab05168bc/html5/thumbnails/12.jpg)
Jornsten.Clustering with
multipledistance metrics
Outline
ClusterAnalysis
Within-clusterparameteriza-tions
Between-clusterparameteriza-tions:Multi-levelmixturemodeling
Results
Conclusion andFuture work
Estimating the transform parameters
Condition on (µk ,Σk), and consider
maxαkl ,βkl
∑g
ηgkl−1
2
(xg − βklµk − α′kl1J)′Σ−1k (xg − βklµk − α′kl1J)
β2kl
+
−J
2log(β2
kl) + c , where α′kl = βklαkl , c is a constant.
Solutions:
1 α′kl =∑
g ηgkl1′JΣ−1
k xg∑g ηgkl1
′Σ−1k 1J
− βkl1′JΣkµk
1′JΣ−1k 1J
.
2 β2kl − Aβkl − B = 0, where
A =1
J∑
g ηgkl
[1′JΣ−1
k µk
∑g ηgkl1
′JΣ−1
k xg
1′JΣ−1k 1J
−∑g
ηgklx′gΣ−1
k µk
]
B =1
J∑
g ηgkl
[∑g
ηgklx′gΣ−1
k xg −(∑
g ηgkl1′JΣ−1
k xg )2∑
g ηgkl1′JΣ−1
k 1J
]
![Page 13: Clustering with multiple distance metrics - multi-level ...jornsten/FMSJornsten09.pdf · distance metrics Outline Cluster Analysis Within-cluster parameteriza-tions Between-cluster](https://reader034.fdocuments.in/reader034/viewer/2022042219/5ec5d1e1acb7740ab05168bc/html5/thumbnails/13.jpg)
Jornsten.Clustering with
multipledistance metrics
Outline
ClusterAnalysis
Within-clusterparameteriza-tions
Between-clusterparameteriza-tions:Multi-levelmixturemodeling
Results
Conclusion andFuture work
Making it work!
Setting up the model is the easy bit...
Starting values: The search space(K , {Lk , k = 1, · · · ,K}) is huge ...
We parameterize this as (M,K ), where M =∑
k Lk isthe total number of clusters.
1 Initialize with M kmeans clusters
2 Use PAM (robust kmeans) with 1− correlation or1− |correlation| to group the M centroids into Ktop-level clusters.
3 Add an additional random element by initializingΣk = |z | × Cov(x|Rg = k), where z ∼ N(r , s).
Search forward for M, backwards for K ∈ {M, · · · , 1}to minimize BIC.
More details in the papers (regularization, parametervalue initialization,...).
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Jornsten.Clustering with
multipledistance metrics
Outline
ClusterAnalysis
Within-clusterparameteriza-tions
Between-clusterparameteriza-tions:Multi-levelmixturemodeling
Results
Conclusion andFuture work
Clustering results
BIC as a function of the number of clusters,for the standard method (K), the sign-flip (F),
and sign-flip+scale (B) models.
K
K K KK
K
K
K
K
K
K
2 4 6 8 10 12
8900
9000
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BIC
FF F F F F F
F F
BB
B B B B BB B
B
We select 9 clusters with an efficient between-clusterparameterization, compared to 5 using the standard mixturemodel.
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Jornsten.Clustering with
multipledistance metrics
Outline
ClusterAnalysis
Within-clusterparameteriza-tions
Between-clusterparameteriza-tions:Multi-levelmixturemodeling
Results
Conclusion andFuture work
Clustering results
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The MIXT with M = 5 has K = 2 profile clusters, withsub-clusters (1,2,3), where cluster 3 is a sign-flip cluster, andsub-clusters (4,5), both with positive βkl .
![Page 16: Clustering with multiple distance metrics - multi-level ...jornsten/FMSJornsten09.pdf · distance metrics Outline Cluster Analysis Within-cluster parameteriza-tions Between-cluster](https://reader034.fdocuments.in/reader034/viewer/2022042219/5ec5d1e1acb7740ab05168bc/html5/thumbnails/16.jpg)
Jornsten.Clustering with
multipledistance metrics
Outline
ClusterAnalysis
Within-clusterparameteriza-tions
Between-clusterparameteriza-tions:Multi-levelmixturemodeling
Results
Conclusion andFuture work
Clustering resultsThe 9 cluster profiles
k = 1: (1,3,4) sign-flip of (2,5,6,7), k = 2: (8,9)
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Clusters 7,2,5,6: increased level of response - distinctfunctional gene classes in each cluster.Clusters 1,3,4: wave of early immune responseClusters 8,9: under-expression of neuron-specific genes,marker of neuron death after injury.
![Page 17: Clustering with multiple distance metrics - multi-level ...jornsten/FMSJornsten09.pdf · distance metrics Outline Cluster Analysis Within-cluster parameteriza-tions Between-cluster](https://reader034.fdocuments.in/reader034/viewer/2022042219/5ec5d1e1acb7740ab05168bc/html5/thumbnails/17.jpg)
Jornsten.Clustering with
multipledistance metrics
Outline
ClusterAnalysis
Within-clusterparameteriza-tions
Between-clusterparameteriza-tions:Multi-levelmixturemodeling
Results
Conclusion andFuture work
Clustering results
MIXT allows for clustering via multiple distancemetrics simultaneously
Sub-clusters: Mahalanobis distanceTop-level: 1− cor or 1− |cor |.
Compare cluster profiles:
PAM(1− cor) (dash)PAM(1− |cor |) (dash-dot)MIXT top-level profilesµk , k = 1, 2 (solid).
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With MIXT we can choose to investigate clusters ateither level of the modeling hierarchy.
![Page 18: Clustering with multiple distance metrics - multi-level ...jornsten/FMSJornsten09.pdf · distance metrics Outline Cluster Analysis Within-cluster parameteriza-tions Between-cluster](https://reader034.fdocuments.in/reader034/viewer/2022042219/5ec5d1e1acb7740ab05168bc/html5/thumbnails/18.jpg)
Jornsten.Clustering with
multipledistance metrics
Outline
ClusterAnalysis
Within-clusterparameteriza-tions
Between-clusterparameteriza-tions:Multi-levelmixturemodeling
Results
Conclusion andFuture work
Multi-factor MIXT
xg and yg the time-course expression of mild (x) andmoderate (y) injury levels. We assume that
(xg , yg )|Rg = k ,Ug = l ∼
∼ N(
(βkl(µXk + αkl1J), δkl(µ
Yk + γkl1J)),Σkl
),
where
Σkl =
[β2klΣ
Xk βklδklΣ
XYk
βklδklΣYXk δ2klΣ
Yk
].
µYk = µX
k + ∆k = W Xk θX
k + ∆k , where ∆k is a vector ofcluster contrast parameters between the two levels of injury.This parameterization provides a sparse representation of µk
if the cluster profiles µXk ' µY
k .
![Page 19: Clustering with multiple distance metrics - multi-level ...jornsten/FMSJornsten09.pdf · distance metrics Outline Cluster Analysis Within-cluster parameteriza-tions Between-cluster](https://reader034.fdocuments.in/reader034/viewer/2022042219/5ec5d1e1acb7740ab05168bc/html5/thumbnails/19.jpg)
Jornsten.Clustering with
multipledistance metrics
Outline
ClusterAnalysis
Within-clusterparameteriza-tions
Between-clusterparameteriza-tions:Multi-levelmixturemodeling
Results
Conclusion andFuture work
Multi-factor MIXT
The joint maximization with respect to scaletransformation parameters (αkl , βkl , γkl , δkl), given theprofile shape parameters (µk ,Σk), is complex.
We approximate this optimization by de-coupling theproblem in terms of the components (x, y), essentiallyassuming that Σk is block-diagonal.
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Jornsten.Clustering with
multipledistance metrics
Outline
ClusterAnalysis
Within-clusterparameteriza-tions
Between-clusterparameteriza-tions:Multi-levelmixturemodeling
Results
Conclusion andFuture work
Clustering results
Cluster mean profiles. The left sub-panel: mild injury data.The right figure sub-panel: the moderate injury data.
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The MIXT model with M = 7 clusters has K = 4 profileclusters. No sign-flip clusters are detected.
![Page 21: Clustering with multiple distance metrics - multi-level ...jornsten/FMSJornsten09.pdf · distance metrics Outline Cluster Analysis Within-cluster parameteriza-tions Between-cluster](https://reader034.fdocuments.in/reader034/viewer/2022042219/5ec5d1e1acb7740ab05168bc/html5/thumbnails/21.jpg)
Jornsten.Clustering with
multipledistance metrics
Outline
ClusterAnalysis
Within-clusterparameteriza-tions
Between-clusterparameteriza-tions:Multi-levelmixturemodeling
Results
Conclusion andFuture work
Clustering results
We detect several scale-transformation clusters ((1,2), (3,4),and (5,6)).
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The sub-cluster separation for the mild injury level data,|βk1 − βk2|, is smaller than the sub-cluster separation for themoderate injury level data, |δk1 − δk2|.
![Page 22: Clustering with multiple distance metrics - multi-level ...jornsten/FMSJornsten09.pdf · distance metrics Outline Cluster Analysis Within-cluster parameteriza-tions Between-cluster](https://reader034.fdocuments.in/reader034/viewer/2022042219/5ec5d1e1acb7740ab05168bc/html5/thumbnails/22.jpg)
Jornsten.Clustering with
multipledistance metrics
Outline
ClusterAnalysis
Within-clusterparameteriza-tions
Between-clusterparameteriza-tions:Multi-levelmixturemodeling
Results
Conclusion andFuture work
Conclusions
Multi-level mixture models
Between-cluster parameterizations provide an objectiveinference tool for comparing clusters...
... and constitutes a substantial savings in terms of thenumber of model parameters.
We detect clusters that a standard model-basedclustering method may miss.
Multi-level mixture models allow for clustering usingmultiple distance metrics simultaneously.
———————————————————————–
Papers are available athttp://www.stat.rutgers.edu/∼rebecka
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Jornsten.Clustering with
multipledistance metrics
Outline
ClusterAnalysis
Within-clusterparameteriza-tions
Between-clusterparameteriza-tions:Multi-levelmixturemodeling
Results
Conclusion andFuture work
Future Work
More complex experiments - need to automate subsetselection as much as possible (e.g. consider differentparameterizations simultaneously).
Time course experiments: no multi-level structure isknown a-priori. Define clusters by a unique paththrough a set of nodes. This allows clusters to coincidefor a subset of data dimensions.
d=1 d=2 d=3
•Number of nodes K(d)={2,3,2}
•Number of unique paths = 12
µ2(d=1)
µ1(d=1)µ2(d=2)
µ1(d=3)
Path Pl (bold arrows) = {k(d),d=1,…,D}={2,2,1} This path connects the shaded nodes.
µ1(d=2)
µ2(d=3)µ3(d=2)
![Page 24: Clustering with multiple distance metrics - multi-level ...jornsten/FMSJornsten09.pdf · distance metrics Outline Cluster Analysis Within-cluster parameteriza-tions Between-cluster](https://reader034.fdocuments.in/reader034/viewer/2022042219/5ec5d1e1acb7740ab05168bc/html5/thumbnails/24.jpg)
Jornsten.Clustering with
multipledistance metrics
Outline
ClusterAnalysis
Within-clusterparameteriza-tions
Between-clusterparameteriza-tions:Multi-levelmixturemodeling
Results
Conclusion andFuture work
Acknowledgements
Prof. Ron Hart and Loyal Goff, W.M. Keck center forCollaborative Neuroscience, Rutgers
Sunduz Keles, U. Wisconsin, Madison
NSF, EPA
![Page 25: Clustering with multiple distance metrics - multi-level ...jornsten/FMSJornsten09.pdf · distance metrics Outline Cluster Analysis Within-cluster parameteriza-tions Between-cluster](https://reader034.fdocuments.in/reader034/viewer/2022042219/5ec5d1e1acb7740ab05168bc/html5/thumbnails/25.jpg)
Jornsten.Clustering with
multipledistance metrics
Outline
ClusterAnalysis
Within-clusterparameteriza-tions
Between-clusterparameteriza-tions:Multi-levelmixturemodeling
Results
Conclusion andFuture work