AST 418/518 Instrumentation and Statistics

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1 AST 418/518 Instrumentation and Statistics Cass Website: Cass Texts: Practica Statistics for Astronomers, J.V. Wa, and C.R. Jenkins, Second Edition. Measuring the Universe, G.H. Rieke 1

2 Statistics and Instrumentation 2

3 Statistica Methods for Astronomy Probabiity (Lecture 1) Statistics (Lecture 2) Read: W&J Ch Read: W&J Ch. 3 Why do we need statistics? Usefu Statistics Definitions Error Anaysis Probabiity distributions Error Propagation Binomia Distribution Least Squares Poisson Distribution chi-squared Gaussian Distribution Significance Bayes Theorem Comparison Statistics Centra Limit theorem 3

4 Practica Statistics Lecture 3 (Aug. 30) Read: W&J Ch Correation - Hypothesis Testing Lecture 4 (Sep. 1) - Principe Component Anaysis Lecture 5 (Sep. 6): Read: W&J Ch. 6 - Parameter Estimation - Bayesian Anaysis - Rejecting Outiers - Bootstrap + Jack-knife Lecture 6 (Sep. 8) Read: W&J Ch. 7 - Random Numbers - Monte Caro Modeing Lecture 7 (Sep. 13): - Markov Chain MC Lecture 8 (Sep. 15): Read: W&J Ch. 9 - Fourier Techniques - Fitering 4 - Uneveny Samped Data

5 What use are statistica methods? Hep you make a decision! Is a signa in a set of observations meaningfu? Do the data fit our mode of the phenomenon under study? Simuate Observations Pan size of sampe, etc. What woud happen if we repeated the observations? Compare different observations Are two sets of data consistent with each other? Are the observations truy independent? 5

6 Usefu References Data Reduction and Error Anaysis, Bevington and Robinson Good basic introduction to statistics Practica Statistics for Astronomers, Wa and Jenkins Numerica Recipes, Press et a. The bibe for numerica computation. Understanding Data Better with Bayesian and Goba Statistica Methods, Press, 1996 (on astro-ph) 6

7 Process of Decision Making Ask a Question Take Data Reduce Data Derive Statistics describing data Refect on what is needed Probabiity Distribution Error Anaysis Does the Statistic answer your question? No Hypothesis Testing Yes Simuation Pubish! 7

8 Some definitions Statistic a number or set of numbers that describe a set of data. Probabiity distribution the reative chances for different outcomes for your set of data. Sampe distribution Set of data that aow us to estimate usefu vaues of the object under study. Parent distribution Presumed probabiity distribution of the data that one woud measure if an infinite data set were acquired. Mean - the 1st moment of a distribution, which gives information about the most ikey vaue one wi observe. Variance the 2 nd moment of a distribution, which gives information about the range of vaues one wi observe. 8

9 Typica Statistics If we have n data points, we typicay want to know things ike the vaue of a data point, or how much the variation in the data is: Mean m = X j x j n s 2 = X j Variance (x j m) 2 n 1 More on this next ecture. 9

10 Probabiity Distributions 10

11 Assumptions Principe of Indifference: The system under study has known reative chances of arriving at a particuar state. The state of the system is arrived at by random chance. Independence: For two observations, the resut of one outcome is not infuenced by the other -> P(A and B) = P(A)P(B) 11

12 Probabiity Distributions Probabiity distributions (P(x)) can be any strange function (or non-anaytica curve) that can be imagined as ong as prob(a <x<b)= Z 1 1 P (x)dx =1 Z b a P (x)dx P(x) is a singe, non-negative vaue for a rea x. 12

13 Mean and Variance of Probabiity Distributions Mean Variance Discrete: n j P (n j ) (n j ) 2 P (n j ) µ 2 Continuous: µ = X j µ = Z xp (x)dx 2 = 2 = X j Z (n j µ) 2 P (n j )= X j (x µ) 2 P (x)dx = Z x 2 P (x)dx µ 2 2 = hx 2 i hxi 2 13

14 The Binomia distribution You are observing something that has a probabiity, p, of occurring in a singe observation. You observe it M times. Want chance of obtaining n successes. For one, particuar sequence of observations the probabiity is: P 1 (n) =p n (1 p) M n There are many sequences which yied n successes: M! P (n) = n!(m n)! pn (1 p) M n M = p n (1 p) M n n Mean Variance Mp Mp(1-p) 14 Often said M choose n

15 The Poisson Distribution Consider the binomia case where p 0, but Mp µ. The binomia distribution, then becomes: P (n) =µ n e µ n! Mean Mp=µ Variance Mp(1-p)~Mp=µ 15

16 Gaussian Distribution The imiting case of the Poisson distribution for arge µ is the Gaussian, or norma distribution P (x)dx = 1 p 2 e (x µ) dx Mean Variance µ σ 2 Large µ Poisson distributions are Gaussian with σ2 =µ. In genera, Gaussian distributions can have unreated mean and variance vaues. 16

17 Gaussian Distribution The Gaussian distribution is often used (sometimes incorrecty) to express confidence. P ( x <µ+ ) > 0.68 P ( x <µ+2 ) > 0.95 P ( x <µ+3 ) >

18 Mean and Variance of Distributions Distribution Mean Variance Binomia Mp Mp(1-p) Poisson µ µ Gaussian µ σ 2 Uniform [a,b) (a+b)/2 (b-a)/12 18

19 Frequentist vs. Bayesian Statistics 19

20 Two approaches to discussing the probem: Knowing the distribution aows us to predict what we wi observe. Frequentist We often know what we have observed and want to determine what that tes us about the distribution. Bayesian 20

21 Frequentist Approach I hypothesize that there are an equa number of red and white bas in a box. I see I have drawn 6 red bas out of 10 tota trias. Prediction A box with equa number of bas wi have a mean of 5 red bas with a standard deviation of 1.6. Based on this I cannot reject my origina hypothesis. 21

22 Bayesian Approach I hypothesize that there are an equa number of red and white bas in a box. I see I have drawn 6 red bas out of 10 tota trias. Odds on what is in the box. There is a 24% chance that my hypothesis is correct. 22

23 Approaches to Statistics Frequentist approaches wi cacuate statistics that a given distribution woud have produced, and confirms or rejects a hypothesis. These are computationay easy, but often sove the inverse of the probem we want. Locked into a distribution (typicay Gaussian) Bayesian approaches use both the data and any prior information to deveop a posterior distribution. Aows cacuation of parameter uncertainty more directy. More easiy incorporates outside information. 23

24 Conditiona Probabiity If two events, A and B, are reated, then if we know B the probabiity of A happening is: Reversing the events, we get: P (A B) = P (A and B) P (B) P (B A) = P (BandA) P (A) P(B A) shoud be read as probabiity of B given A Now, P(A and B) = P (B and A) which gives us the important equaity: P (B A) = P (A B) P (B) P (A) This is Bayes Formua. 24

25 Bayes Theorem Bayes formua is used to merge data with prior information. P (B A) = P (A B) P (B) P (A) A is typicay the data, B the statistic we want to know. P(B) is the prior information we may know about the experiment. P(data) is just a normaization constant P (B data) P (data B) P (B) 25

26 Exampe of Bayes Theorem A game show host invites you to choose one of three doors for a chance to win a car (behind one) or a goat (behind the other two). After you choose a door (say, door 1), the host opens another door (say, door 3) to revea a goat. Shoud you switch your choice? 26

27 Using Bayes' theorem Assume we are ooking for faint companions, and expect them to be around 1% of the stars we observe. From putting in fake companions we know that we can detect objects in the data 90% of the time. From the same tests, we know that we see fase panets 3% of the observations. What is the probabiity that an object we see is actuay a panet? P (panet + det.) = P (panet) =0.01 P (nopanet) =0.99 P (+det. panet) =0.9 P ( det. panet) =0.1 P (+det. nopanet) =0.03 P (+det panet)p (panet) P (+det) P (+det.) =P (+det panet)p (panet)+p (+det nopanet)p (nopanet) P (panet + det.) = =

28 Genera Bayesian Guidance Focuses on probabiity rather than accept/reject. Bayesian approaches aow you to cacuate probabiities the parameters have a range of vaues in a more straightforward way. A common concern about Bayesian statistics is that it is subjective. This is not necessariy a probem. Bayesian techniques are generay more computationay intensive, but this is rarey a drawback for modern computers. 28

29 Why are Gaussian statistics so pervasive? Even an unusua probabiity distribution wi converge to a Gaussian distribution, for a arge enough number, N, of sampings. Referred to as the Centra Limit Theorem From statisticaengineering.com 29

30 Odd Distributions This works for any unusua distributions that an individua random number may be drawn from: 30

31 Thursday: Read Wa and Jenkins Ch

AST 418/518 Instrumentation and Statistics

AST 418/518 Instrumentation and Statistics Class Website: http://ircamera.as.arizona.edu/astr_518 Class Texts: Practical Statistics for Astronomers, J.V. Wall, and C.R. Jenkins Measuring the Universe,