Download Associating earth-orbiting objects detected by astronomical telescopes

Outline
Introduction
Solution Approach - Cluster Analysis
Implementation and Results
Conclusions
Future Work
Associating earth-orbiting objects detected by
astronomical telescopes
Haseena Ahmed Prince Chidyagwai Kun Gou Yun Liu
Timur Milgrom Vincent Quenneville-Bélair
Mentor: Dr. Gary B. Green (The Aerospace Corporation)
August 17, 2007
Team 3
Associating Earth-Orbiting Objects
Outline
Introduction
Solution Approach - Cluster Analysis
Implementation and Results
Conclusions
Future Work
Introduction
Problem Statement
Streak Modeling
Solution Approach - Cluster Analysis
Agglomerative Hierarchical Clustering
k-means Clustering
Comparison
Implementation and Results
Conclusions
Future Work
Team 3
Associating Earth-Orbiting Objects
Outline
Introduction
Solution Approach - Cluster Analysis
Implementation and Results
Conclusions
Future Work
Problem Statement
Streak Modeling
Problem Statement
Satellites make streaks in telescope images
I Input:
1. Streak data
2. Orbit data
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Objective: Identify streaks made by the same object
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Process:
Take the image and find the streak (astronomers)
Estimate the orbit of the object (orbit analysts)
Cluster streaks (large cardinality problem - our task)
Team 3
Associating Earth-Orbiting Objects
Outline
Introduction
Solution Approach - Cluster Analysis
Implementation and Results
Conclusions
Future Work
Problem Statement
Streak Modeling
Streak Modeling
Streaks can be modeled in two spaces:
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Image space: A vector in R3 as a result of processing streak
points
points in a streak
{RAi , DEi , ti }#of
→ {RA, DE , α}
i=1
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Orbit space: A vector in R6 as a result of orbit estimation
points in a streak
→ {a, e, i, Ω, ωp , M}
{RAi , DEi , ti }#of
i=1
Team 3
Associating Earth-Orbiting Objects
Outline
Introduction
Solution Approach - Cluster Analysis
Implementation and Results
Conclusions
Future Work
Agglomerative Hierarchical Clustering
k-means Clustering
Comparison
Clustering
• Similarity and dissimilarity measures
• Depends mainly on the data set available
Team 3
Associating Earth-Orbiting Objects
Outline
Introduction
Solution Approach - Cluster Analysis
Implementation and Results
Conclusions
Future Work
Agglomerative Hierarchical Clustering
k-means Clustering
Comparison
Clustering
• Similarity and dissimilarity measures
• Depends mainly on the data set available
• Two commonly used methods of clustering
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Hierarchical clustering
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Tree structure
Agglomerative
Team 3
Associating Earth-Orbiting Objects
Outline
Introduction
Solution Approach - Cluster Analysis
Implementation and Results
Conclusions
Future Work
Agglomerative Hierarchical Clustering
k-means Clustering
Comparison
Clustering
• Similarity and dissimilarity measures
• Depends mainly on the data set available
• Two commonly used methods of clustering
I
Hierarchical clustering
I
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Tree structure
Agglomerative
Partitional clustering
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One level partitioning
k-means
Team 3
Associating Earth-Orbiting Objects
Outline
Introduction
Solution Approach - Cluster Analysis
Implementation and Results
Conclusions
Future Work
Agglomerative Hierarchical Clustering
k-means Clustering
Comparison
Agglomerative Hierarchical Clustering
Given a set of points to be clustered in 2D as in the figure
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We need to specify: distance measure, type of linkage
Team 3
Associating Earth-Orbiting Objects
Outline
Introduction
Solution Approach - Cluster Analysis
Implementation and Results
Conclusions
Future Work
Agglomerative Hierarchical Clustering
k-means Clustering
Comparison
Agglomerative Hierarchical Clustering
Compute the proximity matrix as in table
Team 3
Associating Earth-Orbiting Objects
Outline
Introduction
Solution Approach - Cluster Analysis
Implementation and Results
Conclusions
Future Work
Agglomerative Hierarchical Clustering
k-means Clustering
Comparison
Agglomerative Hierarchical Clustering
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Cluster points 3 and 6
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Obtain new proximity matrix by calculating the distance
between the new cluster {3, 6} and other points
dist({3, 6} , {1}) = min (dist(3, 1), dist(6, 1))
= min (0.22, 0.23) = 0.22
Team 3
Associating Earth-Orbiting Objects
Outline
Introduction
Solution Approach - Cluster Analysis
Implementation and Results
Conclusions
Future Work
Agglomerative Hierarchical Clustering
k-means Clustering
Comparison
Agglomerative Hierarchical Clustering
Dendogram representation can be given by figure
Team 3
Associating Earth-Orbiting Objects
Outline
Introduction
Solution Approach - Cluster Analysis
Implementation and Results
Conclusions
Future Work
Agglomerative Hierarchical Clustering
k-means Clustering
Comparison
k-means Clustering
Algorithm:
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Select k points as initial centroids
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repeat
Form k clusters by assigning each point to closest centroid.
Recompute the centroid of each cluster.
until Centroids do not change.
Team 3
Associating Earth-Orbiting Objects
Outline
Introduction
Solution Approach - Cluster Analysis
Implementation and Results
Conclusions
Future Work
Agglomerative Hierarchical Clustering
k-means Clustering
Comparison
Comparision - Agglomerative vs k-means
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Agglomerative
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Complexity is O(n2 ) in memory and O(n2 log n) in CPU time
Local optimal clustering
All merges are final
Team 3
Associating Earth-Orbiting Objects
Outline
Introduction
Solution Approach - Cluster Analysis
Implementation and Results
Conclusions
Future Work
Agglomerative Hierarchical Clustering
k-means Clustering
Comparison
Comparision - Agglomerative vs k-means
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Agglomerative
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Complexity is O(n2 ) in memory and O(n2 log n) in CPU time
Local optimal clustering
All merges are final
k-means
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Complexity is O(n) in memory space and CPU time
Number of clusters k needs to be known a-priori
Initialization of centers of clusters
Local optimal clustering
Team 3
Associating Earth-Orbiting Objects
Outline
Introduction
Solution Approach - Cluster Analysis
Implementation and Results
Conclusions
Future Work
Implementation
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Representations in orbit space
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Kepler (Orbit space)
Equinoctial elements
Cartesian ellipse
MATLAB
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Linkage
Distance function
Team 3
Associating Earth-Orbiting Objects
Outline
Introduction
Solution Approach - Cluster Analysis
Implementation and Results
Conclusions
Future Work
Time comparision
Satellites
6
32
74
137
Streaks
96
861
2191
4086
Kepler Time
.05
3.85
56.45
423.13
Ellipse Time
.06
4.48
61.7
443.17
Table: Computational time (seconds)
Team 3
Associating Earth-Orbiting Objects
Outline
Introduction
Solution Approach - Cluster Analysis
Implementation and Results
Conclusions
Future Work
Silhouette
For unperturbed data
Team 3
Associating Earth-Orbiting Objects
Outline
Introduction
Solution Approach - Cluster Analysis
Implementation and Results
Conclusions
Future Work
Silhouette
For perturbed data
Team 3
Associating Earth-Orbiting Objects
Outline
Introduction
Solution Approach - Cluster Analysis
Implementation and Results
Conclusions
Future Work
Distance Measure Comparison
Satellites
6
36
74
137
Euclidean
63
612
1563
3107
Weighted
7
99
273
764
Cosine
644
617
1537
3098
Table: Performance of norms (# clusters)
Team 3
Associating Earth-Orbiting Objects
Outline
Introduction
Solution Approach - Cluster Analysis
Implementation and Results
Conclusions
Future Work
Linkage Function Comparison
Satellites
6
36
74
137
Single
7
99
273
764
Average
13
86
260
520
Centroid
13
82
240
472
Table: Performance of linkage (# clusters)
Team 3
Associating Earth-Orbiting Objects
Outline
Introduction
Solution Approach - Cluster Analysis
Implementation and Results
Conclusions
Future Work
Effect of Variation in Cut-off
Satellites
6
36
36
74
137
Found
6
32
33
57
133
Cut-off
1.154
1.1546
1.1547
1.1546331
1.1546
Silhouette
0.70
0.70
0.79
0.48
0.47
Table: Effect of cut-off on silhouette (a, e weighted with 0.1)
Team 3
Associating Earth-Orbiting Objects
Outline
Introduction
Solution Approach - Cluster Analysis
Implementation and Results
Conclusions
Future Work
Large data clustering
Sectioning method tested on
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Number of streaks = 4400
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Actual number of satellites = 137
Sections
Time
Found
1
356
137
2
143
116
4
56
126
8
12
143
Table: Effective grouping
Team 3
Associating Earth-Orbiting Objects
Outline
Introduction
Solution Approach - Cluster Analysis
Implementation and Results
Conclusions
Future Work
Conclusions
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Weighted norm is effective
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Linkage function is inconclusive
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Cut-off is sensitive
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Sectional method is promising
Team 3
Associating Earth-Orbiting Objects
Outline
Introduction
Solution Approach - Cluster Analysis
Implementation and Results
Conclusions
Future Work
Future Work
Improving clustering
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Develop theory for choosing weights
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Develop theory for choosing cutoff
Improving sectioning method
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Optimal grouping
Team 3
Associating Earth-Orbiting Objects
Outline
Introduction
Solution Approach - Cluster Analysis
Implementation and Results
Conclusions
Future Work
Questions?
Team 3
Associating Earth-Orbiting Objects