Statistics and data analyses

Transcription

1 Statistics and data analyses Designing experiments Measuring time Instrumental quality Precision Standard deviation depends on Number of measurements Detection quality Systematics and methology σ tot = σ stat +σ instr +σ syst b baryon lifetime (ps) : 1.593ps, stat = / , syst = FYS-KJM590 - Nuclear Measurement Methods and 1

2 Probability, discrete and continues Die with 6 sides. Random variable: x = 1,,3,4,5 or 6 P(x = 4) = % Probability density p. Probability to find persons with height 00-01cm: 01cm P(00 < h < 01) = p(h)dh 1% 00cm P(x) 0. p(h) [1/cm] 0.01 i=1 P(x i ) =1 x h(cm) cm 0cm p(h)dh =1 FYS-KJM590 - Nuclear Measurement Methods and

3 Expectation value (Probability density is from now on denoted P) E[x] =< x >= xp(x) dx E[ f (x)] =< f (x) >= f (x)p(x) dx Distribution parameters: Mean value µ = E[x] Standard deviation σ = E[(x µ) ] Skewness γ = µ 3 µ 3 / Second and third moment: µ = E[(x µ) ] = E[x ] E[x] = E[x ] µ = σ µ 3 = E[(x µ) 3 ] = E[x 3 ] 3E[x ]E[x] + E[x] 3 = E[x 3 ] 3E[x ]µ + µ 3 FYS-KJM590 - Nuclear Measurement Methods and 3

4 Probability distribution P(x) = x /8 0.5 σ=0.94 γ= x µ=.67 FYS-KJM590 - Nuclear Measurement Methods and 4

5 Covariance Expresses the linear correlation between two parameters: P(x, y) cov(x, y) = E[(x µ x )(y µ y )] Correlation coefficient : ρ = cov(x, y) σ x σ y If linear dependence, say y ax + b : ρ ±1 If not linear dependence, say y ax + b : ρ 0 FYS-KJM590 - Nuclear Measurement Methods and 5

6 Probability distributions in physics Surprisingly few theoretical distributions important for most physics: Binomial distribution Poisson distribution Gaussian distribution Chi-square distribution FYS-KJM590 - Nuclear Measurement Methods and 6

7 Binomial distribution (I) Bi = two: yes - no, 0-1, up - down Let r = success (yes, 0, up, etc.), N = trials and p = probability of success in a single trial: P(r) = µ = rp(r) r N! r!(n r)! pr (1 p) N r = Np σ = (r µ) P(r) = Np(1 p) r (a + b) N = [(1 p) + p] N =1 = P(r) Binomial expansion coefficients r FYS-KJM590 - Nuclear Measurement Methods and 7

8 Binomial distribution (II) What is the probability to draw 0, 1 and after 5 trials? 0 5! # 1 & # P(r = 0) = 1 1 & 0!(5 0)! $ 5' $ 5' 5! # 1 & P(r = ) =!(5 )! $ 5' 5 # 1 1 & $ 5' µ = =1 σ = 5 1 # & = 0.99 $ 5' 1 5! # 1 & # = 36% P(r =1) = 1 1 & 1!(5 1)! $ 5' $ 5' # = & $ 5' 50 = 51 # 51& 5 $ 5' =17% = 37% What is the probability to get 0, 1, and 3 coins after 3 trial? 3! # 1 P(r = 0) = % & ( 0!(3 0)! $ ' 3! # 1 P(r = ) = % & (!(3 )! $ ' 0 # 1 1 & $ ' # 1 1 & $ ' µ = =1.5 σ = = ! # 1 =1.5% P(r =1) = % & 1 # ( 1 1 & 1!(3 1)! $ ' $ ' 3! # 1 = 37.5% P(r = 3) = % & ( 3!(3 3)! $ ' # 1 1 & $ ' = 37.5% 3 3 =1.5% Large N and not too small p => Gaussian distribution If Np is not large => Poisson distribution FYS-KJM590 - Nuclear Measurement Methods and 8

9 Poisson distribution P(r) = µr e µ r! µ = rp(r) r σ = µ To be used when p -> 0 and N -> infinity, but pn remains finite. Typically for radioactivity and reaction rates. Case: 137 Cs source of 1 ug with 7 years half life: p = s 1 and N =10 15 µ = pn = s 1 FYS-KJM590 - Nuclear Measurement Methods and 9

10 Gaussian distribution The most used distribution in science! % 1 P(x) = σ π exp ' x µ &' σ µ = xp(x) dx σ = + + ( ) (x µ) P(x) dx FWHM =.35 σ ( * )* P(x) 68.3% 95.5% 99.7% P(x) P(x) σ FWHM 1 3σ x 1 3σ x 1 3σ x FYS-KJM590 - Nuclear Measurement Methods and 10

11 Chi-square distribution ( ) n x χ i µ i =, where x i are Gaussian distributed values around u i. i=1 σ i Thus, also χ will be distributed randomly, and given by # χ & P(χ $ ' ) = # % $ # Γ ν & $ ' (ν / ) 1exp χ Γ is the gamma function. µ = ν σ = ν & ( ', where integer ν is the degree of freedom and Useful for evaluating the goodness-of-fit FYS-KJM590 - Nuclear Measurement Methods and 11

12 How to find good values with a limited set of data? The measured values will approach the theoretically once for large samples : ˆ µ µ and σ ˆ = σ for n Poisson distribution : ˆ µ = 1 n n x i and ˆ i=1 σ = ˆ µ n Gaussian distribution : ˆ µ = 1 n n x i and ˆ i=1 σ = n i=1 (x i ˆ µ ) n FYS-KJM590 - Nuclear Measurement Methods and 1

13 Weighted mean Assuming Gaussian distribution, but x i is measured with different precision: n ˆ µ = w i x i with w i = σ = i=1 j 1 1/σ j 1/σ i 1/σ j j FYS-KJM590 - Nuclear Measurement Methods and 13

14 Propagation of errors Assume that we will estimate the errors of a quantity f depending on x and y: u = f (x, y) $ σ u = f ' & ) % x ( $ σ x + f ' & ) % y ( σ y + cov(x,y) f x Usually x and y are independent: $ σ u = f ' & ) % x ( $ σ x + f ' & ) % y ( σ y f y FYS-KJM590 - Nuclear Measurement Methods and 14

15 Examples Difference : u = x y σ u = ( 1) σ x + ( 1) σ y = σ x +σ y Ratio : u = x / y σ u = 1 % ' * σ & y x + x ( ' ) & y * ) Logarithm: u = x ln y σ u = ln y % ( ) σ x + ' x y & ( * ) σ y σ u u σ y = σ x x + σ y y FYS-KJM590 - Nuclear Measurement Methods and 15

16 Goodness of fit 30% may be outside theory with 1σ f(x) ν=14-4=10 3/10=30% ν=14-3=11 7/11=64% FYS-KJM590 - Nuclear Measurement Methods and x ν=14-=1 5/1=4% ν=14-1=13 8/13=6% 16

17 More accurate estimation χ = n i=1 % y exp i f teo (x i ;a 1,a,,a p ) ( ' * & σ i ) Reduced χ : χ ν = Degree of freedom ν = n - p χ n p χ ν 1 for a good fit If χ ν <<1 you may have too big error bars, 30% should lay outside If χ ν >>1 you may have a "wrong" theory or too small error bars 17 FYS-KJM590 - Nuclear Measurement Methods and

18 More accurate estimation If P(χ χ ( found)) > 5%, the fit can be accepted Assume χ =.08 for ν = 4 χ ν = 0.5 (very good) and P(χ >.08) 97.5acceptable!) Assume χ =15.6 for ν = 8 χ ν =1.96 (possible OK) and P(χ >15.6) 5just acceptable) P(χ > χ ( found)) > 5% χ FYS-KJM590 - Nuclear Measurement Methods and 18

19 Multi-parameter fit or P(x,y)=R(y-x)S(x) P has 00 points ρ has 70 parameters T has 70 parameters FYS-KJM590 - Nuclear Measurement Methods and 19

20 Estimate of error in parameter Find the σ that increases χ χ (a +σ) ν ν by 1: 1+ χ (a) ν χ ν σ Δ χ ν 1 0 a 7 ± σ FYS-KJM590 - Nuclear Measurement Methods and a 7 0