1

Sample variance fitting

𝑥2 𝑥3𝑥4 𝑥5

𝑥1

𝑥

𝑦 (𝑥 )

𝜒2 (⋆ )=𝛿 𝑦1

2 (⋆ )𝑠𝑚1

2 (⋆ )+𝛿 𝑦2

2 (⋆ )𝑠𝑚 2

2 (⋆ )+⋯+

𝛿 𝑦𝑀2 (⋆ )

𝑠𝑚𝑀

2 (⋆ )

min𝑃𝐴𝑅𝐴𝑀𝑆

𝜒2 (⋆ )∼𝑀−𝑁 𝑃𝐴𝑅𝐴𝑀

𝑥2 𝑥3𝑥4 𝑥5

𝑥1

2

Comparing data to a priori distributions

𝑥

𝑦 (𝑥 )

𝜒2 (⋆ )𝑥2 𝑥3𝑥4 𝑥5

𝑥1

3

Comparing data to a priori distributions

𝑥

𝑦 (𝑥 )

𝛿 𝑦5 (⋆ )

𝛿 𝑦4 (⋆ )

𝛿 𝑦3 (⋆ )𝛿 𝑦2 (⋆ )

𝛿 𝑦1 (⋆ )

𝜎 1 𝜎 2

𝜎 3 𝜎 4𝜎 5

|𝛿 𝑦 𝑖 (⋆ )|∼𝜎 𝑖

𝑃 (𝛿 𝑦1 (⋆ ) , 𝛿𝑦 2 (⋆ ) ,⋯ )=𝑃1 (𝛿𝑦1 (⋆ ) ) ∙𝑃2 (𝛿 𝑦 2 (⋆ ) )⋯

⟨ 𝜒2 ⟩=𝑀

¿ 1(2𝜋 )𝑀 /2𝜎1𝜎2⋯

exp [− 12 ( 𝛿 𝑦12 (⋆ )𝜎12 +

𝛿𝑦 22 (⋆ )𝜎22 +⋯+

𝛿𝑦𝑀2 (⋆ )𝜎𝑀2 )]

4

Sample variance fitting

𝑥

𝑦 (𝑥 )

𝑥2 𝑥3𝑥4 𝑥5

𝑥1

𝑥

𝑦 (𝑥 )2) Vertical offset y0

Example parameters1) Horizontal stretch x1/2

5

Sample variance fitting

𝑥

𝑦 (𝑥 )

𝑥2 𝑥3𝑥4 𝑥5

𝑥1

𝑥

𝑦 (𝑥 )

𝑃 (𝛿 𝑦1 (⋆ ) , 𝛿𝑦 2 (⋆ ) ,⋯ )∼ 1

(2𝜋 )𝑀 /2𝑠𝑚1𝑠𝑚2

⋯exp [− 12 (𝛿 𝑦1

2 (⋆ )𝑠𝑚1

2 (⋆ )+𝛿 𝑦2

2 (⋆ )𝑠𝑚2

2 (⋆ )+⋯+

𝛿 𝑦𝑀2 (⋆ )

𝑠𝑚𝑀

2 (⋆ ) )]

Example parameters

Adjust parameters to maximize “probability”, i.e. minimize

2) Vertical offset y0

1) Horizontal stretch x1/2

Adjust parameters to maximize “probability”, i.e. minimize

Adjust parameters to maximize “probability”, i.e. minimize

6

Sample variance fitting

𝑥

𝑦 (𝑥 )

𝑃 (𝛿 𝑦1 (⋆ ) , 𝛿𝑦 2 (⋆ ) ,⋯ )∼ 1

(2𝜋 )𝑀 /2𝑠𝑚1𝑠𝑚2

⋯exp [− 12 (𝛿 𝑦1

2 (⋆ )𝑠𝑚1

2 (⋆ )+𝛿 𝑦2

2 (⋆ )𝑠𝑚2

2 (⋆ )+⋯+

𝛿 𝑦𝑀2 (⋆ )

𝑠𝑚𝑀

2 (⋆ ) )]

𝑥2 𝑥3𝑥4 𝑥5

𝑥1

𝑥

𝑦 (𝑥 )

Example parameters

2) Vertical offset y0

1) Horizontal stretch x1/2

𝑥

𝑦 (𝑥 )Big dysc2 big

𝑥

𝑦 (𝑥 )Big dysc2 big

𝑥

𝑦 (𝑥 )Small dysc2 small

7

Sample variance fitting

𝑥

𝑦 (𝑥 )

𝑥2 𝑥3𝑥4 𝑥5

𝑥1

𝑥

𝑦 (𝑥 )

Adjust parameters to maximize “probability”, i.e. minimize

min𝑃𝐴𝑅𝐴𝑀𝑆

𝜒2 (⋆ )

𝑥1 /2 (⋆ )±𝑠𝑥1/2 (⋆ )𝑦 0 (⋆ )± 𝑠𝑦0 (⋆ )

𝑃 (𝛿 𝑦1 (⋆ ) , 𝛿𝑦 2 (⋆ ) ,⋯ )∼ 1

(2𝜋 )𝑀 /2𝑠𝑚1𝑠𝑚2

⋯exp [− 12 (𝛿 𝑦1

2 (⋆ )𝑠𝑚1

2 (⋆ )+𝛿 𝑦2

2 (⋆ )𝑠𝑚2

2 (⋆ )+⋯+

𝛿 𝑦𝑀2 (⋆ )

𝑠𝑚𝑀

2 (⋆ ) )]

𝑠𝑥1 /2(⋆ )=√ [𝑥1/2 (⋆+∆⋆1 )−𝑥1/2 (⋆ ) ]2+⋯

8

Sample variance fitting

𝑥2 𝑥3𝑥4 𝑥5

𝑥1

𝑥

𝑦 (𝑥 )

Adjust parameters to maximize “probability”, i.e. minimize

𝑥1 /2 (⋆ )±𝑠𝑥1/2 (⋆ )𝑦 0 (⋆ )± 𝑠𝑦0 (⋆ )

IF the fitting curve can be adjusted to be “correct,”

min𝑃𝐴𝑅𝐴𝑀𝑆

𝜒 2 (⋆ )  

𝜈  ∼1

⟨ min𝑃𝐴𝑅𝐴𝑀𝑆

𝜒 2 (⋆ )   ⟩⋆=𝑀 −𝑁 𝑃𝐴𝑅𝐴𝑀

min𝑃𝐴𝑅𝐴𝑀𝑆

𝜒2 (∎ ) min𝑃𝐴𝑅𝐴𝑀𝑆

𝜒2 (Δ )  

𝜈

min𝑃𝐴𝑅𝐴𝑀𝑆

𝜒2 (⋆ )

𝑃 (𝛿 𝑦1 (⋆ ) , 𝛿𝑦 2 (⋆ ) ,⋯ )∼ 1

(2𝜋 )𝑀 /2𝑠𝑚1𝑠𝑚2

⋯exp [− 12 (𝛿 𝑦1

2 (⋆ )𝑠𝑚1

2 (⋆ )+𝛿 𝑦2

2 (⋆ )𝑠𝑚2

2 (⋆ )+⋯+

𝛿 𝑦𝑀2 (⋆ )

𝑠𝑚𝑀

2 (⋆ ) )]

9

Sample variance fitting

𝑃 (𝛿 𝑦1 (⋆ ) , 𝛿𝑦 2 (⋆ ) ,⋯ )∼ 1

(2𝜋 )𝑀 /2𝑠𝑚1𝑠𝑚2

⋯exp [− 12 (𝛿 𝑦1

2 (⋆ )𝑠𝑚1

2 (⋆ )+𝛿 𝑦2

2 (⋆ )𝑠𝑚2

2 (⋆ )+⋯+

𝛿 𝑦𝑀2 (⋆ )

𝑠𝑚𝑀

2 (⋆ ) )]

𝑥2 𝑥3𝑥4 𝑥5

𝑥1

𝑥

𝑦 (𝑥 )

min𝑃𝐴𝑅𝐴𝑀𝑆

𝜒2 (⋆ )

𝑥1 /2 (⋆ )±𝑠𝑥1/2 (⋆ )𝑦 0 (⋆ )± 𝑠𝑦0 (⋆ )

1) Measure individual samples to construct sample means and standard errors at various x

2) Justify fitting function and parameters.

min𝑃𝐴𝑅𝐴𝑀𝑆

𝜒 2 (⋆ )  

𝜈∼13) Is ?

4) Do normalized residuals look plausibly like random noise?

5) IF pass QC, report

𝑥

𝛿 𝑦 𝑖 (⋆ )𝑠𝑚𝑖

1

-1

0

( )

min𝑃𝐴𝑅𝐴𝑀𝑆

𝜒2 (⋆ )

𝜒2 (⋆ )

𝑥

𝑦 (𝑥 )Big dysc2 big

𝑥

𝑦 (𝑥 )Big dysc2 big

𝑥

𝑦 (𝑥 )Small dysc2 small