new results in the geometry of random fields, with applications to cmb and galaxy density keith...
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New results in the geometry of random fields, with applications
to CMB and galaxy density
Keith Worsley, Hugo Drouin, Pauline Klein
Department of Mathematics and Statistics, McConnell Brain Imaging Centre, Montreal Neurological Institute,
McGill University
Milky way
-5 -4 -3 -2 -1 0 1 2 3 4 5-100
-80
-60
-40
-20
0
20
40
60
80
100CfA red shift survey, FWHM=13.3
Gaussian threshold
Eul
er C
hara
cter
istic
(E
C)
"Bubble"topology
"Sponge"topology
"Meat ball" topology
CfARandomExpected
The 2dF Galaxy Redshift Survey: spectra and redshiftsMon. Not. R. Astron. Soc. 328, 1039–1063 (2001)
Milky way
0
500
1000First scan of fMRI data
-5
0
5
T statistic for hot - warm effect
0 100 200 300
870880890 hot
restwarm
Highly significant effect, T=6.59
0 100 200 300
800
820hotrestwarm
No significant effect, T=-0.74
0 100 200 300
790800810
Drift
Time, seconds
fMRI data: 120 scans, 3 scans each of hot, rest, warm, rest, hot, rest, …
T = (hot – warm effect) / S.d. ~ t110 if no effect
Gaussianized χ2 field?
-5 -4 -3 -2 -1 0 1 2 3 4 5-30
-20
-10
0
10
20
30
CfAGaussianChisq, 5 dfChisq, 4 dfChisq, 3 df
Gaussian threshold
EC
-5 -4 -3 -2 -1 0 1 2 3 4 5-30
-20
-10
0
10
20
30
CfAGaussianT, 10 dfT, 6 dfT, 4 df
Gaussianized T field?
Gaussian threshold
EC
EC for WMAP, unsmoothed, FWHM = 1.6o
-5 -4 -3 -2 -1 0 1 2 3 4 5-2500
-2000
-1500
-1000
-500
0
500
1000
1500
2000
2500
Gaussian threshold
EC
ObservedExpected
-8 -7 -6 -5 -4-10
0
10
4 5 6 7 80
10
20
-5 -4 -3 -2 -1 0 1 2 3 4 5-200
-150
-100
-50
0
50
100
150
Observed - Expected+1 sd (~ \sqrt Expected)-1 sd (~ \sqrt Expected)
Observed – Expected EC, with Poisson sd in tails
EC
Gaussian threshold
EC for WMAP smoothed to FWHMo (expected)no smoothing
scale space
-5 0 5
-2000
0
2000 1.6o
EC
-5 0 5-1000
0
10002.7o
-5 0 5-500
0
500
4.1o
-5 0 5-200
0
200
6.2o
EC
-5 0 5-100
0
1008.7o
-5 0 5-50
0
5012.1o
-5 0 5-50
0
50
16.1o
EC
threshold-5 0 5
-20
0
20
20.5o
threshold-5 0 5
-1000
0
1000
20001.6o 20.5o
threshold
Simulation (expected)no smoothing
scale space
-5 0 5
-2000
0
20001.7o
EC
-5 0 5-2000
0
2000
2o
-5 0 5-1000
0
1000
2.7o
-5 0 5-500
0
5003.5o
EC
-5 0 5
-200
0
200 4.8o
-5 0 5
-100
0
100 6.6o
-5 0 5-100
0
100
9.1o
EC
threshold-5 0 5
-50
0
50
12.7o
threshold-5 0 5
-1000
0
1000
2000
1.7o 12.7o
threshold
Scale space WMAP data
-5 -4 -3 -2 -1 0 1 2 3 4 5-1000
-500
0
500
1000
1500
2000
1.6o 20.5o
Gaussian threshold
EC
Observed
Expected
-4.5 -4 -3.5-10
0
10
3.5 4 4.5 50
10
20