From Spatio-Temporal Data to a
Weighted and Lagged Network
Between Functional Domains:
Applications in Climate and Neuroscience
Ilias Fountalis
PhD Thesis Defense
Examination
2016
Spatio-Temporal Data
Climate
Human Brain
Ecological
Social
Economical
2
Spatio-Temporal Data
Applications
3
Spatio-Temporal Data
Representation
• Embedded in a two-or-three dimensional grid
• Grid cells contain measurements (time-series)
for the variables of interest
• Grid cells to not correspond to functionally
distinct units
4
Spatio-Temporal Data
Functional Components
• Spatio-temporal systems are modular
• Functional components:
–
–
–
–
Spatially contiguous
Functionally homogeneous
Possibly overlapping
Weighted and lagged interactions
5
Thesis Overview
• Geo Cluster (presented in PhD proposal)
– Identifies functional components of a spatio-temporal
system
– Spatially contiguous non-overlapping areas
– Models their interactions as a complete and weighted
network
•
Spatio-temporal network analysis for studying climate patterns (Fountalis et al., Clym. Dyn.
2014)
• ENSO in CMIP5 simulations (presented in PhD proposal)
– Evaluation of cutting edge climate models
– Ranking models in terms of their ability to reproduce the
climate of the past
– Investigating model trajectories under future climate
warming scenarios
•
ENSO in CMIP5 simulations: network connectivity from the recent past to the twenty-third
century (Fountalis, et al., Clym. Dyn. 2015)
6
Thesis Overview
• δ-MAPS (the focus of this talk)
– Identifies domains: Distinct semi-autonomous
components of the system
• Spatially contiguous, possibly overlapping regions
– Infers their potentially lagged and weighted
interactions
• Applied to
– Climate data
– Resting state fMRI
•
δ-MAPS: From spatio-temporal data to a weighted and lagged network between functional
domains (Fountalis et al., submitted to KDD’16)
7
Outline
•
•
•
•
•
Related Work
–
–
–
Methods for Analyzing Spatio-Temporal Data
Synthetic Data
Method Limitations
–
–
–
–
Method Overview
Domain Identification
Network Inference
Application on Synthetic Data
δ-MAPS
Applications in Climate Science
Applications in Neuroscience
Conclusions & Future Work
8
Outline
•
•
•
•
•
Related Work
–
–
–
Methods for Analyzing Spatio-temporal Data
Synthetic Data
Method Limitations
–
–
–
–
Method Overview
Domain Identification
Network Inference
Application on Synthetic Data
δ-MAPS
Applications in Climate Science
Applications in Neuroscience
Conclusions & Future Work
9
Methods for Analyzing
Spatio-Temporal Data
• Objective
– Infer the functional components of a spatio-temporal
system and study their interactions
• Methods
– Multivariate statistical methods
• Principal Component Analysis (PCA)/Empirical Orthogonal
Function (EOF) Analysis
• Independent Component Analysis (ICA)
– Clustering
• Spatial contiguity constraints
• Drawbacks
– Illustration of limitations in a data set for which we
know the ground truth
10
Outline
•
•
•
•
•
Related Work
–
–
–
Methods for Analyzing Spatio-Temporal Data
Synthetic Data
Method Limitations
–
–
–
–
Method Overview
Domain Identification
Network Inference
Application on Synthetic Data
δ-MAPS
Applications in Climate Science
Applications in Neuroscience
Conclusions & Future Work
11
Synthetic Data Generation
Setup
• Synthetic Component
– Modeled as a circle of radius rp
– Core of radius rc
– Time series at core modulated by a factor
• Decay Function:
•
Connecting components i,j:
•
Final steps:
– Given source signals yi(t), yj(t)
– xi(t) = (1-α)yi(t) ± αyj (t+τ)
– α: controls the strength of the connection
– Additive superimposition of component time series
– Addition of white Gaussian noise N(0,1)
12
Synthetic Data Generation
Setup
13
Outline
•
•
•
•
•
Related Work
–
–
–
Methods for Analyzing Spatio-Temporal Data
Synthetic Data
Method Limitations
–
–
–
–
Method Overview
Domain Identification
Network Inference
Application on Synthetic Data
δ-MAPS
Applications in Climate Science
Applications in Neuroscience
Conclusions & Future Work
14
Dimensionality Reduction:
PCA/EOF
• PCA/EOF analysis:
– Identifies orthogonal components of high-energy
content in terms of the variance of the signal
– Variance of the field dominated by few “modes of
variability”, masking weaker regions of interest
– Orthogonality constraint difficult to be
interpreted physically
– See also: A cautionary note on the interpretation of EOFs (Dommenget and Latif,
J. Clym. 2002)
15
Dimensionality Reduction:
ICA
• ICA
– Separates a mixed signal into independent non Gaussian
subcomponents
– No orthogonality constraints
– Cannot determine the variance, sign or correct ordering of
the independent components
– Identified components are noisy
– Difficult to interpret functional structure if we do not know
the ground truth
–
See also: Modulation of temporally coherent brain networks estimated using ICA at rest and 16
during cognitive tasks (Calhoun et. al., Humm. Brain Mapp. 2008)
Dimensionality Reduction:
Clustering
• Clustering
–
–
–
–
–
Many flavors (Spectral, Agglomerative, Region Growing)
Typically require as an input # of clusters
Each grid cell belongs to a cluster
No spatial contiguity guarantees
Normalized cut group clustering of resting state fMRI data (Van De Heuvel et al., PLoS ONE,
2008)
• K-Means example
– Clusters correspond to noise
– Separate components are joined to the same cluster
– Cannot separate between local diffusion and remote interactions
17
Dimensionality Reduction:
Spatial-Clustering
• Geo Cluster:
– Spatial contiguous clustering
– Automatically identifying number
of underlying components
– Clusters are used as the nodes of a functional
network
– Applied extensively to investigate the Earth’s
climate and evaluate climate models
• Spatio-temporal network analysis for studying climate
patterns (Fountalis et al., Clym. Dyn. 2014)
• ENSO in CMIP5 simulations: network connectivity from
the recent past to the twenty-third century (Fountalis, et
al., Clym. Dyn. 2015)
– Does not allow overlap between identified clusters18
Outline
•
•
•
•
•
Related Work
–
–
–
Methods for Analyzing Spatio-Temporal Data
Synthetic Data
Method Limitations
–
–
–
–
Method Overview
Domain Identification
Network Inference
Application on Synthetic Data
δ-MAPS
Applications in Climate Science
Applications in Neuroscience
Conclusions & Future Work
19
δ-MAPS: Method Overview
20
Outline
•
•
•
•
•
Related Work
–
–
–
Methods for Analyzing Spatio-Temporal Data
Synthetic Data
Method Limitations
–
–
–
–
Method Overview
Domain Identification
Network Inference
Application on Synthetic Data
δ-MAPS
Applications in Climate Science
Applications in Neuroscience
Conclusions & Future Work
21
δ-MAPS:
Notations
• Spatio-temporal field X(t)
• Embedded in a grid
– Modeled as a planar graph G(V,E)
• Similarity between grid cells i,j
– Pearson correlation:
22
δ-MAPS:
Domain Constraints
• Domain: Spatially contiguous set of grid cells that
participate in the same function
• Homogeneity of a domain
• Homogeneity constraint
–
> than a threshold δ
23
δ-MAPS:
Domain Constraints
• Domains have an epicenter of action
• K-neighborhood ΓK(i) of grid cell i
– K nearest grid cells to i, including i
• Local homogeneity of grid cell i:
• Domain core, cell at which local homogeneity is
– Local maximum
– Larger than δ
24
δ-MAPS:
Problem Statement
• A domain is spatially contiguous (IG(A) = 1) if it forms a
connected component in G
• Given cell c: core of domain A
• A homogeneity threshold δ
• Domain must satisfy:
(1)
•
•
Exact boundaries of a domain are unknown
Domain identification problem:
– Given field X(t) on spatial grid G, core cell c
and threshold δ
– Identify domain A as the maximum-sized set of cells that
satisfies (1)
25
δ-MAPS:
Problem Statement
• Domain must satisfy:
(1)
• Domain identification problem:
– Given field X(t) on spatial grid G, core cell c and threshold δ
– Identify domain A as the maximum-sized set of cells that
satisfies (1)
•
Problem is NP-Complete
– Reduction of densest connected k-subgraph to domain
identification problem
–
•
The complexity of clustering in planar graphs (Keil and Brecht, J. Combin. Math. Combin. Comput ,1991)
Greedy algorithm for domain identification
– Identify seeds (cores)
– Iteratively expand and merge seeds to identify domains
26
δ-MAPS:
Seed Selection
•
Seed: Grid cell including its local neighborhood
– Must satisfy:
• Local maximum:
• a
• Single domain can have more than
one seeds
– Noise
– Overlapping regions
27
δ-MAPS:
Domain Identification
• Input: Sets of seeds S
• Iterative process
– (1) Merging and (2) Expansion of domains
28
δ-MAPS:
Domain Identification
• Merging:
– Domains can be merged if
• Spatially adjacent
• a
– Merge first the two domains with max
– Terminate when no merging is possible
29
δ-MAPS:
Domain Identification
• Merging:
– Domains can be merged if
• Spatially adjacent
• a
– Merge first the two domains with max
– Terminate when no merging is possible
30
δ-MAPS:
Domain Identification
• Expansion:
–
–
–
–
Domains sorted by homogeneity
Expand by considering all adjacent grid cells
Expand by adding grid cell with max
After each expansion check if merging is possible
31
δ-MAPS:
Domain Identification
• Expansion:
–
–
–
–
Domains sorted by homogeneity
Expand by considering all adjacent grid cells
Expand by adding grid cell with max
After each expansion check if merging is possible
32
δ-MAPS:
Domain Identification
• Expansion:
–
–
–
–
Domains sorted by homogeneity
Expand by considering all adjacent grid cells
Expand by adding grid cell with max
After each expansion check if merging is possible
33
δ-MAPS:
Domain Identification
• Expansion:
–
–
–
–
Domains sorted by homogeneity
Expand by considering all adjacent grid cells
Expand by adding grid cell with max
After each expansion check if merging is possible
34
δ-MAPS:
Domain Identification
• Expansion:
–
–
–
–
Domains sorted by homogeneity
Expand by considering all adjacent grid cells
Expand by adding grid cell with max
After each expansion check if merging is possible
35
δ-MAPS:
Domain Identification
• Expansion:
–
–
–
–
Domains sorted by homogeneity
Expand by considering all adjacent grid cells
Expand by adding grid cell with max
After each expansion check if merging is possible
36
δ-MAPS:
Domain Identification
• Expansion:
–
–
–
–
Domains sorted by homogeneity
Expand by considering all adjacent grid cells
Expand by adding grid cell with max
After each expansion check if merging is possible
37
δ-MAPS:
Domain Identification
• Termination:
– No further merging or expansion is possible
38
Outline
•
•
•
•
•
Related Work
–
–
–
Methods for Analyzing Spatio-Temporal Data
Synthetic Data
Method Limitations
–
–
–
–
Method Overview
Domain Identification
Network Inference
Application on Synthetic Data
δ-MAPS
Applications in Climate Science
Applications in Neuroscience
Conclusions & Future Work
39
Network Inference
Prior Work
• Functional components might be correlated at a non-zero lag
– Compute Pearson correlation for a range of lags [-τmax ≤ τ ≤ τmax]
• Testing for significant correlations1
– T-test given significance level α
• Ignores autocorrelation structure
– Multiple testing problem
• Need to control for the number of false positives
• Selecting the appropriate lag2
– Consecutive lags produce almost maximal correlations
– Point estimates are not robust
•
•
1: A new dynamical mechanism for major climate shifts (Tsonis et al., GRL, 2007)
2: Network inference with confidence from multivariate time series (Kramer et al., Phys. Rev. E, 2009)
40
δ-MAPS:
Domain Signal
• Domain Level Signal XA(t)
– Application Specific
41
δ-MAPS:
Test for Statistical Significance
• Domains might by correlated at a non-zero lag τ
• For each pair of domains Α,Β:
– Compute Pearson rA,B(τ) correlation for a range of lags [-τmax ≤ τ ≤ τmax]
• Statistical significance
– Uncorrelated signals can produce spurious correlations if they have a
strong autocorrelation structure
• Bartlett’s formula
– Estimates the variance of rAB(τ)
– Null hypothesis: XA(t), XB(t) uncorrelated
– E[rAB(τ)] = 0 and
–
~ N(0,1)
42
δ-MAPS:
Multiple Testing Problem
• Multiple testing
– For N domains and a max lag τmax
–
– N = 100, τmax = 5, α = 1%, roughly 550 false positives
• False Discovery Rate (FDR)
– Select false discovery rate q
• q: Controls expected fraction of
false positives
– Sort the M p-values in ascending
order pi-1 < pm < pm+1
– Keep the first m < M p-values
• pm < qm/M
• pm: m’th lowest p-value
43
δ-MAPS:
Lag Inference
• Domains are connected if there exists at least one
significant correlation
• What is the appropriate lag?
• Proposed approach
– Associate a range of lags
– That produce significant correlations
– Located within one standard deviation from the max. absolute
correlation
44
δ-MAPS:
Edge Direction
• Edge direction
– Lag range positive: A -> B (A precedes B)
– Lag range negative: A <-B (A succeeds B)
– Lag range includes zero: Bi-directed edge
45
δ-MAPS:
Edge Weight
• Edge weight:
– Covariance between the domain signals
–
maximum correlation in absolute sense
– Edge weight captures the magnitude of the signal
of the two domains
– Weights can be positive or negative
• Final Network
– Directed, weighted graph
46
δ-MAPS:
Domain Strength
• Domain strength:
– Sum of the absolute weights of the edges
of a domain
47
Outline
•
•
•
•
•
Related Work
–
–
–
Methods for Analyzing Spatio-Temporal Data
Synthetic Data
Method Limitations
–
–
–
–
Method Overview
Domain Identification
Network Inference
Application on Synthetic Data
δ-MAPS
Applications in Climate Science
Applications in Neuroscience
Conclusions & Future Work
48
Application on Synthetic Data
• Revisiting the synthetic data set example
49
Application on Synthetic Data
•
•
δ-MAPS parameters: K=4, δ = 0.55, q = 10%, τmax = 20
More than one seeds in the core, seeds in the overlapping regions
•
•
•
Identified domains: Subset of ground truth
Correctly identifies overlaps
Network
– Eventually merged to a single domain
–
–
–
Correctly identifies all three edges and their polarity
Lag ranges always include the correct value
Hierarchy of edge weights is preserved
50
Outline
•
•
•
•
•
Related Work
–
–
–
Methods for Analyzing Spatio-Temporal Data
Synthetic Data
Method Limitations
–
–
–
–
Method Overview
Domain Identification
Network Inference
Application on Synthetic Data
δ-MAPS
Applications in Climate Science
Applications in Neuroscience
Conclusions & Future Work
51
Applications in Climate:
Climate Modes of Variability
• Recurring patterns with
identifiable
characteristics and
specific regional effects
• Represent the state of
the climate
• An example:
– El Niño Southern
Oscillation (ENSO)
52
Applications in Climate:
Teleconnections
• Teleconnections
– Climate anomalies related
to each other at large
distances
• An example:
– Variations in temperature
in tropical Pacific (
)
– Cause
• Rainfall in remote places
of the world (
)
• Temperature increase in
others (
)
53
Applications in Climate:
Data Description
• Data:
– Monthly averages of sea surface temperature (SST) from
HadISST.
– Period: 1956-2005 (50 years, 600 months)
• Preprocessing:
– Removal of seasonal cycle
– Removal of linear trends (Theil-Sen estimator)
– Transform to zero-mean
• δ-MAPS parameters
– Neighborhood size K = 4 grid cells
– δ = 0.37
– False discovery rate q = 3%
• 30 edges in network (no more than 1 false positive)
– Max lag τmax = 12 months
54
Applications in Climate:
The Climate Network
• From 6000 grid cells to 18 domains
• 35% of the grid cells do not belong to a domain
• Largest domain ENSO
55
Applications in Climate:
The Climate Network
•
•
•
Strongest domain: ENSO
–
Hierarchy in terms of strength in the three ocean basins
–
Indian Ocean, Horse-shoe pattern, North Atlantic
–
Domain C
ENSO teleconnections
ENSO predecessors
–
•
Improved El Nino forecasting by cooperativity detection, Ludescher et a.l, PNAS 2013)
•
See also: Are Atlantic Ninos enhancing Pacific ENSO events in recent decades (Rodriguez-Fonseca, GRL 2009)
Domain Q (South Atlantic), precedes all other domains in the climate network
56
Applications in Climate:
Structural Balance
• Network decomposed to 5 weakly connected
components
• Network is structurally balanced
– Partitioned into two groups of domains
57
Outline
•
•
•
•
•
Related Work
–
–
–
Methods for Analyzing Spatio-Temporal Data
Synthetic Data
Method Limitations
–
–
–
–
Method Overview
Domain Identification
Network Inference
Application on Synthetic Data
δ-MAPS
Applications in Climate Science
Applications in Neuroscience
Conclusions & Future Work
58
Applications in Neuroscience:
Introduction
• Functional Magnetic Resonance Imaging (fMRI)
– Blood Oxygen – Level Dependent (BOLD) signal
– Measures changes in the level of oxygen
concertation in the brain
• The brain as a functional network
–
–
–
–
Spatially distributed regions
Each having their own task and function
Continuously interacting with each other
How is functional connectivity altered
due to neurodegenerative diseases?
• Resting State fMRI
– Subject scanned while at rest
– During rest the functional network is not idle
– Resting State Networks: Strongly functionally linked subnetworks
between different brain regions
59
Applications in Neuroscience:
Data Description
• Data
– HCP, Cortical resting state fMRI
– Single subject, 2 scans, ≈ 15 minutes per scan
– Time resolution 0.72 seconds
• Data Preprocessing
–
–
–
–
–
–
HCP ‘’fix-extended’’ minimal preprocessing pipeline
Correction for B0 distortions
Head motion correction
Registration to structural image
Masking of non-brain voxels
Removal of physiological artifacts…
•
Resting-state Fmri in the human connectome project (Smith et al., Neuroimage, 2013)
– Bandpass-filtering 0.01-0.08Hz
60
Applications in Neuroscience:
Network Properties
• δ-MAPS parameters:
– Neighborhood size K = 6
– δ = 0.37
– false discovery rate q = 10-4
• (expected 1 out of 10k edges false positive),
– τmax = 3 (2.2 seconds)
• Majority of domains are small (95% < 250 voxels)
• Polarity of edges changes across scans (time
varying?)
• Degree and size of a domain are positively
correlated
• Networks are assortative
61
Applications in Neuroscience:
Resting State Networks
• Resting state networks: Regions highly interconnected to
each other during rest
• Community detection (OSLOM*):
– Network communities correspond to well known resting state
networks
– Consistent between the two scans
–
* Finding statistically significant communities in networks (Lancichinetti et al., PLoS One, 2011)
62
Applications in Neuroscience:
K-Core decomposition
•
A process that iteratively removes nodes based on their degree
•
After the removal of the 14th core (16th for scan-2) density of the
network increases by a factor of 2
Reveals a few domains, densely interconnected to each other
•
– After the extraction of the k’th core all nodes have degree > k
– Backbone of the functional brain network
•
Rich-club organization of the human connectome (van den Heuvel and Sporns, Journal of
neuroscience, 2011).
63
Outline
•
•
•
•
•
Related Work
–
–
–
Methods for Analyzing Spatio-Temporal Data
Synthetic Data
Method Limitations
–
–
–
–
Method Overview
Domain Identification
Network Inference
Application on Synthetic Data
δ-MAPS
Applications in Climate Science
Applications in Neuroscience
Conclusions & Future Work
64
• δ-MAPS
Conclusions
– Bridging overlapping community detection and spatial
clustering
– Identifies the functional components of a spatiotemporal system
– Used to study their possible lagged and weighted
interactions
– Validated against synthetic data
– Overpowers traditional dimensionality
reduction/network based methods
• Applied to climate data
– Successfully uncovering known modes of variability and
teleconnections
• Applications in neuroscience
– Successfully uncovers well known resting state
networks at a single subject analysis
– Identifies the backbone of the brain network
65
Future Work
• Climate networks over time
– Investigate trajectories of the functional components as
reflected by their size/strength
• Climate models and controlled perturbation experiments
– How do they propagate in the climate network scale?
• Effective connectivity
– Application of probabilistic graphical models to remove noncausal edges
• Dynamic networks using contextual time series detection
– Automatic identification of changes between two time series
• Structural-functional networks
– Combine functional connectivity with structural connectivity
• Extension to other spatio-temporal data
– Species migration patterns, seismic data
66
Publications – Posters - Talks
•
Book chapters
–
•
A. Bracco, R.K. Archibald, C. Dovrolis, I. Fountalis, H. Luo and J.D Neelin. The parameter
optimization problem in state-of-the-art climate models and network analysis for systematic data
mining in model intercomparison projects. CISMCoursesandLectures:
TheFluidDynamicsofClimate,Springer Ed. 201
Journal papers
–
–
I. Fountalis, A. Bracco, C. Dovrolis. Spatio temporal network analysis for studying climate patterns.
Climate Dynamics 42 3-4 (2014) 879-899
I. Fountalis, A. Bracco, C. Dovrolis. ENSO in CMIP5 simulations: Network connectivity from the
recent past to the twenty-third century. Climate Dyncamics (2014)
•
Under Review
•
Poster abstracts
– I. Fountalis, A.Bracco, B. Dilkina, C. Dovrolis, S. Keilholz. δ-MAPS: From spatio-temporal
data to a weighted and lagged network between functional domains (Submitted KDD 2016)
– C. Dovrolis, I. Fountalis, B. Dilkina, S. Keilholz. From fMRI data to a weighted network
between functional domains (Submitted PRNI 2016)
–
–
•
I. Fountalis, A. Bracco, C. Dovrolis. A network based analysis of CMIP5 historical experiments.
Climate Informatics (2013). Poster abstract.
I. Fountalis, C. Dovrolis, A. Bracco. A network based methodology for the study of climate
teleconnections. Second workshop on understanding climate change from Data (2012). Poster
abstract.
Invited talks
–
–
Evaluation of climate models using network analysis. Complenets 2012.
Validation of the CMIP5 models using network analysis. Conference on artificial intelligence 67
applications to environmental sciences (2012)
Thank you!
68
Backup Slides
69
Network Based Methods
• Related work
– Step 1: grid cells -> nodes
• Do not correspond to functionally distinct
units
70
Network Based Methods
• Related work
– Step 1: grid cells -> nodes
• Do not correspond to functionally distinct
units
– Step 2: Compute pairwise correlations
between all pairs of nodes
Pair-wise correlations
71
Network Based Methods
• Related work
– Step 1: grid cells -> nodes
• Do not correspond to functionally distinct
units
– Step 2: Compute pairwise correlations
between all pairs of nodes
– Step 3: Threshold to obtain network
• Fixed threshold approach
– “Community structure and dynamics in climate networks,
Tsonis et al., 2011”
• Fixed density approach
– “The backbone of the climate network, Donges et al., 2009”
• Not clear which edges to prune
72
Network Based Methods
• Related work
– Step 1: grid cells -> nodes
• Do not correspond to functionally distinct
units
– Step 2: Compute pairwise correlations between
all pairs of nodes
– Step 3: Threshold to obtain network
• Fixed threshold approach
–
“Community structure and dynamics in climate networks, Tsonis et al.,
2011”
• Fixed density approach
–
“The backbone of the climate network, Donges et al., 2009”
• Not clear which edges to prune
– Step 4: Final network
• Binary (ignores magnitude and sign of correlations)
–
“Simple models of human brain functional networks, Vėrtes et al., 2011”
–
“Improved El Niňo forecasting by cooperativity detection, Ludescher et
al., 2013”
• Weighted (typically only + correlations are
considered)
73
Heuristic to Infer δ
• δ determines the minimum degree of homogeneity
of a domain
• δ heuristic
– Start with a random sample of pairs of grid cells
– Calculate their zero-lag correlation
– Infer the significant correlations for a given
significance level α
– δ depends on:
• Significance level α (input to domain identification)
• Underlying correlation distribution
• Underlying autocorrelation structure
• Intuition
– A domain is a spatially contiguous set of grid cells
– Mean pair-wise correlation should be higher than mean
correlation of randomly picked grid cells
74
Dimensionality Reduction:
Community Detection
•
OSLOM
– Automatically identifies hierarchical structure of communities
– Accounts for overlaps between communities
– INPUT: Pruned cell-level network
– Can not distinguish between positive and negative correlations
– Cannot infer actual connectivity
– Communities are not guaranteed to be spatially contiguous
–
Finding statistically significant communities in networks (Lancichinetti et al., PLoS One, 2011)
75
Applications in Climate:
Lag-consistent Triangles
• Lag-consistent triangle: Nodes can be placed in a
consistent temporal order
• All triangles are consistent with one exception (C,D,G)
– However, an alternate path exists
76
Applications in Climate:
EOFs, Communities & Spatial
Clustering
•
EOFs:
•
Communities:
•
Spatial Clustering:
– ENSO dominates, teleconnections to South Atlantic are lost
– Grouping spatially disjoint modes of variability together
– More areas. No Overlaps.
– Spatial extent of an area constrained because no overlaps are allowed
77
Applications in Neuroscience:
Data Representation
• Volumetric representation
– Grid cell (voxel): isomorphic cube
– Confounds arise from the variability of the
convolutions of the human brain
– Neighboring voxels might belong to different
sulci/gyri
– Functional organization of the cortex is largely two78
dimensional
Applications in Neuroscience:
Data Representation
• Surface based registration
– Individual volumetric data are projected to
a surface mesh
79