Error Bars in both X and Y

Transcription

1 Error Bars n both X and Y Wrong ways to ft a lne : 1. y(x) a x +b (σ x 0). x(y) c y + d (σ y 0) 3. splt dfference between 1 and. Example: Prmordal He abundance: Extrapolate ft lne to [ O / H ] 0. [ He / H ] Correct method s to mnmse : χ (a, b) ( Y (a X + b) ) σ (Y )+ a σ (X ) Let s see why. [ O / H ]

2 Vector Space Perspectve data ponts, M parameters. X Model µ(α) defnes a parametersed M-dmensonal surface n the -dmensonal data space. µ ( α) χ (α) squared dstance from the observed data to the model surface. Best-ft model s the one closest to the data. For lnear models (scalng patterns), the model surface s a flat M-dmensonal hyper-plane.

3 Revew: Vector Spaces Vectors have a drecton and a length. Addton of vectors gves another vector. Scalng a vector stretches ts length. Dot product: a a - b a b a b cosθ θ "angle" between vectors a, b. θ b Length of a vector: a a a (dstance from base to tp) Dstance between vectors: a b

4 Ortho-normal Bass Vectors Ortho-normal bass vectors e : e e j δ j 1 j 0 j Any vector a s a lnear combnaton of the bass vectors e, wth scale factors a Example: a a e ( a e ) e y 3 x a. e x 4 a e 3 e e 1 1 e y ex y a. e y 3 x

5 Data Space s a Vector Space data ponts defne a vector n -dmensonal data space : x {,x,...,x } e 1 + x e x e bass vectors: e 1 {1,0,...,0} e {0,1,...,0} x x... e {0,0,...,1} e Bass s ortho-normal f: e e j δ j e1 Bass vector e selects data pont x : x e x Data pont x s the projecton of data vector x along the bass vector e.

6 on-orthogonal Bass Vectors x In the non-orthogonal case, e 1 e cosθ 0 x x Two ways to measure coordnates: Contravarant coordnates (ndex hgh): x project parallel to bass vectors: e θ e 1 x e 1 +x e +...+x e Covarant coordnates (ndex low): x project perpendcular to bass vectors. + x cosθ x x + cosθ x Metrc tensor: Dot product: j g j x j g j e e j x 1 cosθ cosθ 1 x x y x y j e e j x y j g j x y x j y j, j, j j

7 Metrc for non-orthonormal Bass Vectors x x x e θ g j e e j e 1 e 1 e 1 e cosθ e 1 e cosθ e Metrc s symmetrc: g j g j. Off-dagonal terms vansh f the bass vectors are orthogonal. Dagonal terms defne the lengths of the bass vectors.

8 Data sets and Functons as Vector Spaces A data set, X, 1,...,, s also an -component vector ( X 1, X,..., X ), one dmenson for each data pont. The data vector s a sngle pont n the -dmensonal data space. A functon, f( t ), s a vector n an nfnte-dmensonal vector space, one dmenson for each value of t. The dot product between functons depends on a weghtng functon w( t ): f, g f (t) g(t) w(t) dt Weghtng functon Each weghtng functon w( t ) gves a dfferent dot product, a dfferent dstance measure, a dfferent vector space. Whch w( t ) to use for data analyss?

9 χ as (dstance) n functon space The (absolute value) of a functon f( t ) : f f, f f (t) w(t) dt The (dstance) between f( t ) and g( t ) : f g f g, f g ( f (t) g(t) ) w(t) dt A dataset ( X +/- σ ) at t t defnes a specfc weghtng functon: δ(t t w(t) ) σ Wth ths w( t ), the (dstance) from data X( t ) to model µ( t ) s χ : X µ X µ(t ) σ χ. Each dataset defnes ts own weghtng functon.

10 The Data-Space Metrc: σ s the unt of dstance. χ s (dstance) Defne the data-space dot product wth nverse-varance weghts: w 1 a b a b w σ a b a b σ. a b σ Then, the (dstance) between data x and parametersed model µ(α) s: X χ " $ # X µ (α) σ % ' & X µ(α). µ ( α)

11 Optmal Scalng n Vector Space otaton Mnmse χ -> pck model closest to the data. Scalng a pattern: µ( α ) α P : X µ (α) αp The pattern P s a vector n data space. The model α P s a lne n data space, multples of P. The best ft s the pont along the lne closest to the data X ˆα X P /σ P /σ X P P P X " µ( ˆα) ˆα P X P % $ 'P X e P # P P & Unt vector along P : e P P P ( ) e P α 1 P α 1 α 0 α α 3 ˆα P

12 Stretchng the Bass Vectors Usng the vector notaton, ˆα P X P P j j X P j g j P P j g j So the e bass vectors are orthogonal but not unt length, gven the data-space metrc X P σ ( P ) σ g j e e j 1 σ δ j.e. σ s the natural unt of dstance on the th axs of data space! We can stretch axs by factor σ to defne a new set of ortho-normal bass vectors b : b 1 {σ 1,0,...,0} b σ e b b j δ j e 1 {1,0,...,0} e {0,1,...,0}... e {0,0,...,1} b {0,σ,...,0}... b {0,0,...,σ }

13 Stretch bass vectors to make χ ellpses become crcles Old bass vectors: x x e g j e e j δ j σ x e x χ contours are ellpses Orthogonal, but not normalsed. Stretched bass vectors are orthonormal: e 1 b x /σ x χ contours are crcles b σ e g j b b j δ j b 1 x x, b b x σ b b b 1 /σ 1

14 Error Bars n both X and Y Wrong ways to ft a lne : 1. y(x) a x +b (σ x 0). x(y) c y + d (σ y 0) 3. splt dfference between 1 and. Example: Prmordal He abundance: [ He / H ] Extrapolate ft lne to [ O / H ] 0. Key concept: X +/- σ X and Y +/- σ Y are ndependent dmensons of the -dmensonal data space. [ O / H ]

15 Lne Ft wth error bars n both X and Y Data: X ±σ X Y ±σ Y Model: y a x + b Δy Δx For σ X σ Y, where s the pont of closest approach? ot obvous. L y a x + b Δy Y - (a X + b) Δx X - (Y b) / a Horzontal stretch by factor σ Y / σ X makes the probablty cloud round. Also changes the slope: a > a Δ x σ Y Δx a Δy σ X Δ x σ X a tanθ σ Y Closest approach at R Δy cosθ R Δy R σ Y Δy cos θ cos θ + sn θ 1 1+ tan θ σ Y σ Y + a σ X Δy σ Y θ R Δy R Δx θ Δy σ Y + a σ X y a x + b Crcle radus s σ Y σ X

16 Defnng χ for errors n both X and Y Horzontal stretch makes probablty cloud round. Crcle radus s σ Y σ X. Dstance R at closest approach s : R σ Y Δy σ Y + a σ X Δy R Δx y a x + b Crcle radus s σ Y σ X ote: eed a dfferent stretch for each data pont. Total (dstance) n the - dmensonal data space: χ ε(y ) + ε( X ) σ (Y ) σ ( X ) R σ (Y ) 1 ( Y (a X + b) ) σ (Y )+ a σ (X ) ε(y ) +ε( X ) σ (Y ) ε(y) ε(x ) R