sample variance fitting
Post on 06-Feb-2016
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1
Sample variance fitting
π₯2 π₯3π₯4 π₯5
π₯1
π₯
π¦ (π₯ )
π2 (β )=πΏ π¦1
2 (β )π π1
2 (β )+πΏ π¦2
2 (β )π π 2
2 (β )+β―+
πΏ π¦π2 (β )
π ππ
2 (β )
minππ΄π π΄ππ
π2 (β )βΌπβπ ππ΄π π΄π
π₯2 π₯3π₯4 π₯5
π₯1
2
Comparing data to a priori distributions
π₯
π¦ (π₯ )
π2 (β )π₯2 π₯3π₯4 π₯5
π₯1
3
Comparing data to a priori distributions
π₯
π¦ (π₯ )
πΏ π¦5 (β )
πΏ π¦4 (β )
πΏ π¦3 (β )πΏ π¦2 (β )
πΏ π¦1 (β )
π 1 π 2
π 3 π 4π 5
|πΏ π¦ π (β )|βΌπ π
π (πΏ π¦1 (β ) , πΏπ¦ 2 (β ) ,β― )=π1 (πΏπ¦1 (β ) ) βπ2 (πΏ π¦ 2 (β ) )β―
β¨ π2 β©=π
ΒΏ 1(2π )π /2π1π2β―
exp [β 12 ( πΏ π¦12 (β )π12 +
πΏπ¦ 22 (β )π22 +β―+
πΏπ¦π2 (β )ππ2 )]
4
Sample variance fitting
π₯
π¦ (π₯ )
π₯2 π₯3π₯4 π₯5
π₯1
π₯
π¦ (π₯ )2) Vertical offset y0
Example parameters1) Horizontal stretch x1/2
5
Sample variance fitting
π₯
π¦ (π₯ )
π₯2 π₯3π₯4 π₯5
π₯1
π₯
π¦ (π₯ )
π (πΏ π¦1 (β ) , πΏπ¦ 2 (β ) ,β― )βΌ 1
(2π )π /2π π1π π2
β―exp [β 12 (πΏ π¦1
2 (β )π π1
2 (β )+πΏ π¦2
2 (β )π π2
2 (β )+β―+
πΏ π¦π2 (β )
π ππ
2 (β ) )]
Example parameters
Adjust parameters to maximize βprobabilityβ, i.e. minimize
2) Vertical offset y0
1) Horizontal stretch x1/2
Adjust parameters to maximize βprobabilityβ, i.e. minimize
Adjust parameters to maximize βprobabilityβ, i.e. minimize
6
Sample variance fitting
π₯
π¦ (π₯ )
π (πΏ π¦1 (β ) , πΏπ¦ 2 (β ) ,β― )βΌ 1
(2π )π /2π π1π π2
β―exp [β 12 (πΏ π¦1
2 (β )π π1
2 (β )+πΏ π¦2
2 (β )π π2
2 (β )+β―+
πΏ π¦π2 (β )
π ππ
2 (β ) )]
π₯2 π₯3π₯4 π₯5
π₯1
π₯
π¦ (π₯ )
Example parameters
2) Vertical offset y0
1) Horizontal stretch x1/2
π₯
π¦ (π₯ )Big dysc2 big
π₯
π¦ (π₯ )Big dysc2 big
π₯
π¦ (π₯ )Small dysc2 small
7
Sample variance fitting
π₯
π¦ (π₯ )
π₯2 π₯3π₯4 π₯5
π₯1
π₯
π¦ (π₯ )
Adjust parameters to maximize βprobabilityβ, i.e. minimize
minππ΄π π΄ππ
π2 (β )
π₯1 /2 (β )Β±π π₯1/2 (β )π¦ 0 (β )Β± π π¦0 (β )
π (πΏ π¦1 (β ) , πΏπ¦ 2 (β ) ,β― )βΌ 1
(2π )π /2π π1π π2
β―exp [β 12 (πΏ π¦1
2 (β )π π1
2 (β )+πΏ π¦2
2 (β )π π2
2 (β )+β―+
πΏ π¦π2 (β )
π ππ
2 (β ) )]
π π₯1 /2(β )=β [π₯1/2 (β+ββ1 )βπ₯1/2 (β ) ]2+β―
8
Sample variance fitting
π₯2 π₯3π₯4 π₯5
π₯1
π₯
π¦ (π₯ )
Adjust parameters to maximize βprobabilityβ, i.e. minimize
π₯1 /2 (β )Β±π π₯1/2 (β )π¦ 0 (β )Β± π π¦0 (β )
IF the fitting curve can be adjusted to be βcorrect,β
minππ΄π π΄ππ
π 2 (β )
π βΌ1
β¨ minππ΄π π΄ππ
π 2 (β ) β©β=π βπ ππ΄π π΄π
minππ΄π π΄ππ
π2 (β ) minππ΄π π΄ππ
π2 (Ξ )
π
minππ΄π π΄ππ
π2 (β )
π (πΏ π¦1 (β ) , πΏπ¦ 2 (β ) ,β― )βΌ 1
(2π )π /2π π1π π2
β―exp [β 12 (πΏ π¦1
2 (β )π π1
2 (β )+πΏ π¦2
2 (β )π π2
2 (β )+β―+
πΏ π¦π2 (β )
π ππ
2 (β ) )]
9
Sample variance fitting
π (πΏ π¦1 (β ) , πΏπ¦ 2 (β ) ,β― )βΌ 1
(2π )π /2π π1π π2
β―exp [β 12 (πΏ π¦1
2 (β )π π1
2 (β )+πΏ π¦2
2 (β )π π2
2 (β )+β―+
πΏ π¦π2 (β )
π ππ
2 (β ) )]
π₯2 π₯3π₯4 π₯5
π₯1
π₯
π¦ (π₯ )
minππ΄π π΄ππ
π2 (β )
π₯1 /2 (β )Β±π π₯1/2 (β )π¦ 0 (β )Β± π π¦0 (β )
1) Measure individual samples to construct sample means and standard errors at various x
2) Justify fitting function and parameters.
minππ΄π π΄ππ
π 2 (β )
πβΌ13) Is ?
4) Do normalized residuals look plausibly like random noise?
5) IF pass QC, report
π₯
πΏ π¦ π (β )π ππ
1
-1
0
( )
minππ΄π π΄ππ
π2 (β )
π2 (β )
π₯
π¦ (π₯ )Big dysc2 big
π₯
π¦ (π₯ )Big dysc2 big
π₯
π¦ (π₯ )Small dysc2 small
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