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Segregating Solar Features by Temperature
Nathan Stein
California-Boston-Smithsonian AstroStatistics Collab oration

February 8, 2011


Big Picture

Solar Dynamics Observatory generates up to 1.4 terabytes/day Atmospheric Imaging Assembly: four-telescope array on the SDO satellite High-resolution (4096 в 4096) images of the corona in 7 extreme ultraviolet filters every 10 seconds More than anyone could examine by eye Need fast methods for processing data


Statistical Model

Observe photon counts {Yib } in pixel i through filter b i = unknown amount of plasma in pixel i
ij

= proportion of plasma in temperature bin j (

j



ij

= 1)

i ij = DEM(log Tj ) b = known exposure time bj = known response function Likelihood: Y
ib

Poisson i b
j i

bj

ij

Goal: identify regions with similar


Statistical Model

i is a nuisance parameter
b

Y

ib

= Ni Pois

i

j

Mj

ij

, with Mj =

b

b bj

Distribution of Ni depends on i and i , whereas distribution of YiB Yi1 ,..., Ni Ni (conditional on Ni ) only depends on
i i

(1)

Cluster pixels with similar proportions, ignore totals N


Clustering probability vectors

How to cluster vectors of probabilities or proportions? Squared Hellinger distance between p and q is d2 (p, q ) = H 1 2 ( pb - q b )
b 2

=1-
b

pb q

b

Modify k-means clustering to use Hellinger distance:
"h-means clustering"


Clustering probability vectors

Observations p1 , . . . , pn Cluster assignments c1 , . . . , c Cluster centers q 1 , . . . , q
k n

1. Given cluster centers, set ci = arg minj d2 (pi , q j ) H 2. Given cluster assignments, set q j = arg minq
i:ci =j

d2 (pi , q ) H

(2)

The minimization in (2) has an analytic solution: ( i:ci =j pib )2 qj b = pib )2 b( i:ci =j


Clustering AIA data

Cluster the vectors (yi1 /ni , . . . , y

iB

/ni ) for i = 1, . . . , n

For illustration, examine a coarsened (256 в 256) set of images, with 3 clusters


Clustering AIA data


Clusters in log Y space


Clusters in Y /n space


Distribution of pixels in each cluster


Distribution of pixels in each cluster


Simulated Data
Simulated Temperature Distributions

4

5

6 logT

7

8

9


Simulated Data


Simulated Data


Simulated Data


Simulated Data


Simulated Data: Results


Simulated Data: Results


Simulated Data: Results


Simulated Data: Results

How to make images of results more meaningful? (Cluster label is arbitrary number) Assign a level lj to each cluster Many choices for quantitatively meaningful lj 's For example: lj =
i:ci =j

yi,

171

i:ci =j

ni


Full Resolution Images

4096 в 4096 pixels в 6 bands k = 64 + 1 clusters (1 extra for pixels with zero counts in all 6 images) Two observations:
1. Octob er 2, 2010, 05:57 2. Octob er 2, 2010, 18:43


05:57


18:43


Next Steps

Model-based clustering to compare multiple images, to identify regions that are thermally similar in different observations DEM reconstruction: what is the state of the art?


Acknowledgements

Vinay Kashyap Xiao-Li Meng