統計的データ解析

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ゆあのじま
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2 ds45 xspec qdp guplot oocalc

4 (Error) gg (Radom Error)(Systematc Error)

5 x, x,, x

6 (

7 x, x,..., x x = s x x µ = lm = σ µ x x = lm ( x ) = σ ( ) =

8 - x = js j ( ) = j= ( j) x x + xj x + xj j x + xj = ( x x j) = ( ( x x ) ( x j x ) ) = ( ) + ( ) ( )( ) = s s ( x x x j x x x x j x ) = ( x x ) ( ) = j= ( + ( x j x) ( x x)( x j x) ) = ( ) ( ) = ( x x) ( ) ( x x) ( x x) ( x j x) = = = j=

9 x f( u, v,...) σ x = lm ( x x) = x x x x ( u u) + ( v v) + L u v σ σ σ = x = x x lm ( u u) + ( v v) + L u v x x x x lm ( u u) + ( v v) + ( u u)( v v) + L = u v u v u = lm ( u u), σv = lm ( v v) = = lm [ ( u u)( v v) ] (covarace) uv = x x x u + v + L+ uv σ σ σ σ u v u,v u,v x x + L u v

10 σ lm ( )( ) uv = [ u u v v ] x u v L uv x x x x σ σ + σ + + σ + L u v u v u σ uv x x x u v σ σ + σ + L u v

11 x= u+ v x= u v σ = σ + σ x u v x= uv σ σ σx = σuv + σvu = u v + u v u v

12 (s) r(c/s), b(c/s

13 (Error)(Ucertaty) x, x,..., x x = σ σ σ σ x x x = = = N / σ

14 (Maxmum Lkelhood Method) x, c,..., x µ σgauss) x x + dx dq = Pdx x µ P exp σ π σ µ µ ' µ '

15 µ ' σ ' = σ x x µ ' P ( µ ') = exp σ π σ x, x,..., x = P( µ ') = P( µ ') x µ ' = exp σ π = σ P( µ ') µ ' µ

16 ( ') ' ' 0 ' ' P X x X x dx d x x µ µ σ µ µ σ µ = = = = = = = = (mea) (average)

17 xσ x µ ' P( µ ') = exp = σ π σ µ ' = x µ ' x µ ' x σ = 0 µ ' = = = = d ( / ) dµ ' σ σ (/ σ ) µ ' σ (/ ) µ ' = σ

18 Coservatve x

19 . x,x,,x. FWHM0eV 0eV 0eV 3. u,v=f(u,v)u,v

21 xp( x) µ = xp( x) dx ( x ) σ = µ P( x) dx xx, x,... P, P,... µ = = xp = ( x ) = σ µ P x,x,,x

22 ! x PB (;, x p) = p ( p) ( x)! x! x = p = p( p) = ( p) σ µ p << x µ e Px (; µ ) = x! x = µ = σ µ µ x Posso Dstrbuto x µ=

23 !! PB (;, x p) = p ( p) = p ( p) ( p) ( x)! x! x!( x)! µ = p σ µ µ / p µ µ ( p) = lm ( p) = = e p 0 p 0 e lm lm p 0 = p( p) = ( p) x µ PB (; x p, ) = Pp ( x; µ ) e x! x x x x µ p <<! x (for x<< ) ( x)! x ( p) + px µ

24 ( τ xp(;,) x t τ t t+ dt dt P(0; t, τ) = P(0; t, τ) P(0; t+ dt, τ) dp(0; t, τ) τ t / τ P(0; t, τ) = e τ t / τ e τ x dtdt x dpxt (;,) τ = e x! t / τ x = txt0 t x dt τ t / τ e t Pxt (;,) τ = x! τ t = µ t τ x µ µ Pp (; x µ ) e x!

25 Posso Dstrbuto µ= µ0 µσ =µ x

26 exp(-t/tau) MBq55Fe55Fe

27 P G ( x µ ) ( x; µσ, ) = exp( ) πσ σ Bevgto &Robso

28 t φt µ σx x t = ( x µ )/ s / φ = 0 s

29 χ χ 0xχ = x χ νχ ν / χ / ν / ν ( χ ) = {( χ ) e }/ Γ( ν /) E ( ) = V( ) = χ ν χ ν ( x µ ) µ σ χ σ ( x x) σ χ

32 (Ft) X X

33 ( xyxy yx () = ax+ b ab, a, b y () x = a x+ b yy ( x) σ X

34 ( yp y y0 ( x) P = exp σ σ π y y y0 ( x) Pa ( 0, b0) = P = exp = = σ π = σ ab, y y yx ( ) Pab (,) = exp σ π = = σ Pa (, b) 0 0 Pab (,) (,) ab( a, b) 0 0

35 ( χ y( x) y ax b y = σ σ = = Pab (,) χ χ = 0, χ = 0 χ ab, xy x y a = σ σ σ σ x y x x y b = σ σ σ σ x x = σ σ σ χ χ = y ax b a, b) σ = ( ) ( x, y ) x, ax + b) ( a, b) a= ( xy x y ) b= x y x x y ( ) = x ( x)

37 (Goodess of Ft) t y y( x) = σ ( ) y y x y ax b χ = σ = σ m( ma, b - m ab, χ νχ ( χ / ν) reduced ch-square ν

38 educed- Pν Data Reducto ad Error Aalyss for the Physcal Sceces, Bevgto & Robso

39 χ Data Reducto ad Error Aalyss for the Physcal Sceces, Bevgto & Robso reduced ch-squaredch-squared

41 σ σ a = = a σ = y σ = = b x b σ = y σ

42 yx () y y( x) χ = σ m m χ ν χ χ a χ χ m χ aa aa a = χm + + χm a aχm a aχm + a+

43 a,a 68% =(a,a )68% =.3 Numercal Recpes C,

44 y χ χ χ

45 ( x, y ),...,( x y ) y,..., y, σ,..., σ f( x; a,..., a ) p ( a,..., a ) Pa (,..., a ) = exp ( y ) f( x; a,..., ap) p = σ σ π y f( x; a,..., ap) p expχ χ χ = σ Pa (,..., a ) ( aˆ,..., aˆ ) y ˆ ˆ f( x; a,..., ap) m = m = σ χ χ χ ( ) ( ( ;,..., )) p f x a a a a aˆ aˆ χ = ;,..., p,..., p (,..., p) χ ( ) y f x a a y p p ( ( ˆ ˆ f x; a,..., ap) ) = j= χ δ = a A p Pa (,..., a ) = Fa (,..., a) exp - p χ ( ˆ ) ( a ˆ j aj) p p j= δ δ j π j m f x a,..., a a,..., a ; p p j σ j p A a a j j j χ χ χ χ χ χ χ χ m p

46 χ y y( x) ( y( )) ( y( )) = = σ = = = y() 0

47 Grd Search Gradet Search Expaso Method best f best f Gradet-Expaso algorthm (Marquardt method) Data Reducto ad Error Aalyss for the Physcal Sceces, Bevgto & Robso

48 K K Data Reducto ad Error Aalyss for the Physcal Sceces, Bevgto & Robso

49 t P = Ap(;) t τ = Ae pt (;) τ = e τt t+ dt t / τ t / τ A τ At τ d, d t, ta t t t t / τ t Pdt = A e dt = N N L() τ = P = Ae N = = t / τ τ

50 N L() τ = P = Ae = = t / τ t M() τ = l L() τ = la τ t N = 0, t = A = / τm( τ) = t Nlτ τ dm() τ N = t 0 / = τ = t N dτ τ τ t, t = A = / M t / t / e τ τ dt = e t [ t t ] / τ ( τ) = l L= l dm() τ dτ τ = 0 τ = t / N t τ

51 Data Reducto ad Error Aalyss for the Physcal Sceces, Bevgto & Robso

52 K M=/

53 (x) F-test

55 x H(0.5 H'0.5 H H' P P(a a

56 σ ->t ->t F

57 x f(t) α -α α/ α t µ σ µ σ x µ )/ s / φ = tα α µ α x )/ s / α µ α x s / x + s / 00 (- α ) µ xα s / µ x + α s /

58 -σ<x-µ<σ68.3% -σ<x-µ<σ95.5% -3σ<x-µ<3σ99.7% -.96σ<x-µ<.96σ 95% -.58σ<x-µ<.58σ 99%

59 x,y r N xy x y ( ) ( ) N x x N y y / /

61 Data Reducto ad Error Aalyss for the Physcal Sceces, Bevgto & Robso

カイ二乗フィット検定、パラメータの誤差

カイ二乗フィット検定、パラメータの誤差統計的データ解析 008 008.. 林田清 ( 大阪大学大学院理学研究科 ) 問題 C (, ) ( x xˆ) ( y yˆ) σ x πσ σ y y Pabx (, ;,,, ) ˆ y σx σ y = dx exp exp πσx ただし xy ˆ ˆ はyˆ = axˆ+ bであらわされる直線モデル上の点 ( ˆ) ( ˆ ) ( ) x x y ax b y ax b Pabx (,