Summer School in Statistics for Astronomers V June 1 - June 6, Regression. Mosuk Chow Statistics Department Penn State University.

Transcription

1 Summer School in Statistics for Astronomers V June 1 - June 6, 2009 Regression Mosuk Chow Statistics Department Penn State University. Adapted from notes prepared by RL Karandikar

2 Mean and variance Recall we had defined the Expectation or the mean of a random variable X as i µ = E(X) = x ip(x = x i ) xf (x)dx for discrete X for continuous X with density f (x) and the variance of a random variable X as i σ 2 = Var(X) = E(X 2 ) µ 2 = x2 i P(X = x i) µ 2 x2 f (x)dx µ 2

3 If X and Y are random variables with means µ X and µ Y, then the covariance of X and Y is defined by Cov (X, Y) = E (X µ X ) (Y µ Y ) = E(XY) µ X µ Y The correlation coefficient ρ (X, Y) of X and Y is defined by ρ(x, Y) = Cov (X, Y) Var (X) Var (Y)

4 Properties of mean and variance E (ax + b) = aex + b, Var (ax + b) = a 2 Var (X) E (ax + by + c) = aex + bey + c Var (ax + by + c) = a 2 Var (X) + b 2 Var (Y) + 2abCov (X, Y)

5 It may be shown that for any n n matrix A and n 1 vector b E (AY + b) = AEY + b, Cov (AY + b) = ACov (Y) A T. which is the basic result used in regression.

6 Hubble s data (1929) In 1929 Edwin Hubble investigated the relationship between distance and radial velocity of extra-galactic nebulae (celestial objects). It was hoped that some knowledge of this relationship might give clues as to the way the universe was formed and what may happen later. His findings revolutionized astronomy and are the source of much research today. Given here is the data which Hubble used for 24 nebulae.

7 X = Distance (in Megaparsecs) from earth Y = The recession velocity (in km/sec) X Y X Y X Y X Y lib.stat.cmu.edu/dasl/datafiles/hubble.html

8 From this data-set Hubble obtained the relation Recession Velocity = H 0 Distance where H 0 is Hubble s constant thought to be about 75 km/sec/mpc.

9 Back to Hubble s data Scatterplot of Recession Velocity vs Distance Recession Velocity (km/sec) Distance (megaparsecs)

10 The ML Method for Linear Regression Analysis Scatterplot data: (x 1, y 1 ),..., (x n, y n ) Basic assumption: The x i s are non-random measurements; the y i are observations on Y, a random variable Statistical model: Y i = α + βx i + ɛ i, i = 1,..., n Errors ɛ 1,..., ɛ n : a random sample from N(0, σ 2 ) Parameters: α, β, σ 2 Y i N(α + βx i, σ 2 ): The Y i s are independent The Y i are not identically distributed, because they have differing means

11 The likelihood function is the joint density function of the observed data, Y 1,..., Y n n [ L(α, β, σ 2 1 ) = exp (Y i α βx i ) 2 ] i=1 2πσ 2 2σ 2 n (Y = (2πσ 2 ) n/2 i α βx i ) 2 exp i=1 2σ 2 Use partial derivatives to maximize L over all α, β and σ 2 > 0 (Wise advice: Maximize ln L) The ML estimators are: n i=1 ˆβ = (x i x)(y i Ȳ) n i=1 (x i x) 2, ˆα = Ȳ ˆβ x and ˆσ 2 = 1 n n (Y i ˆα ˆβx i ) 2 i=1

12 Using this on Hubble s data we get ˆβ = , ˆα = where the intercept term is not significant. If we use regression through origin, the result is ˆβ = The result from the historical data set obtained by Hubble is very different from the current estimate of the Hubble s constant due to various issues of measurement errors and censoring, etc. Those issues will be discussed in a subsequent lecture.

13 How do we obtain the least squares estimates? Let Y i be the response for the i th data point and let x i be the p-dimensional (row vector) of the predictors for the ith data point, i = 1,, n. There are p-1 predictors. We assume that Y i = x i β + e i, where β, an unknown parameter, is a p 1 column vector, and ( e i N 1 0, σ 2 ), and the e i are independent. Note that σ 2 is another parameter for this model.

14 We further assume that the predictors are linearly independent. Thus we could have the second predictor be the square of the first predictor, the third one the cube of the first one, etc, so this model includes polynomial regression.

15 We often write this model in matrices. Let Y = Y 1. Y n, X = x 1. x n, e = e 1. e n so that Y and e are n 1 and X is n p. The assumed linear independence of the predictors implies that the columns of X are linearly independent and hence rank(x) = p. x i = (1, x i,1,..., x i,p 1 ).

16 The normal model can be stated more compactly as Note that: β = (β 0, β 1,..., β p 1 ) T. Y = Xβ + e, e N n ( 0, σ 2 I ) The input matrix X is of dimension n (p): 1 x 1,1 x 1,2... x 1,p 1 1 x 2,1 x 2,2... x 2,p x n,1 x n,2... x n,p 1

17 Therefore, using the formula for the multivariate normal density function, we see that the joint density of Y is f β,σ 2 (y) = (2π) n/2 σ 2 I 1/2 exp{ 1 2 (y Xβ)T ( σ 2 I ) 1 (y Xβ)} = (2π) n/2 ( σ 2) n/2 exp{ 1 2σ 2 y Xβ 2 }

18 Therefore the likelihood for this model is L Y ( β,σ 2 ) = (2π) n/2 ( σ 2) n/2 exp{ 1 2σ 2 Y Xβ 2 }

19 Estimation of β First, we note that the assumption on the X matrix implies that X T X is invertible. The ordinary least square (OLS) estimator of β is found by minimizing q (β) = (Y i x i β) 2 = Y Xβ 2 The formula for the OLS estimator of β is β = ( X T X ) 1 X T Y

20 Note that E β = ( X T X ) 1 X T EY = ( X T X ) 1 X T Xβ = β ) Cov ( β = ( X T X ) 1 X T σ 2 IX ( X T X ) 1 = σ 2 ( X T X ) 1 = σ 2 M Therefore β N p ( β, σ 2 M )

21 Properties of the OLS estimator 1. (Gauss-Markov) For the non-normal model the OLS estimator is the best linear unbiased estimator (BLUE), i.e., it has smaller variance than any other linear unbiased estimator. 2. For the normal model, the OLS is the best unbiased estimator i.e., has smaller variance than any other unbiased estimator 3. Typically, the OLS estimator is consistent, i.e. β β

22 The unbiased estimator of σ 2 In regression we typically estimate σ 2 by σ 2 = Y X β 2 / (n p) which is called the unbiased estimator of σ 2. we first state the distribution of σ 2. (n p) σ 2 χ 2 n p independently of β σ 2

23 Properties of σ 2 1. For the general model σ 2 is unbiased. 2. For the normal model σ 2 is the best unbiased estimator. 3. σ 2 is consistent.

24 Interval estimators and tests We first discuss inference about β i the ith component of β. Note that β i the ith component of the OLS estimator is the estimator of β i. Further ) Var ( βi = σ 2 M ii which implies that that the standard error of β i is σ βi = σ M ii Therefore we see that a 1 α confidence interval for β i is β i ( β i t α/2 n p σ βi, β i + t α/2 n p σ βi ).

25 To test the null hypothesis β i = c against one and two-sided alternatives we use the t-statistic t = β i c σ βi t n p.

26 Now consider inference for δ = a T β, let δ = a T ( β N1 δ, σ 2 a Ma ) therefore we see that δ is the estimator of δ, and ) Var ( δ = σ 2 a Ma so that the standard error of δ is = σ a σ δ Ma

27 Therefore the confidence interval for δ is δ ( δ t α/2 n p σ δ, δ + t α/2 n p σ δ) and the test statistic for testing δ = c is given by δ c σ δ t n p under the null hypothesis There are tests and confidence regions for vector generalizations of these procedures.

28 Let x 0 be a row vector of predictors for a new response Y 0. Let µ 0 = x 0 β = EY 0. µ 0 = x 0 β is the obvious estimator of µ0 and Var ( µ 0 ) = σ 2 x 0 Mx 0 σ µ0 = σ x 0 Mx 0 and therefore a confidence interval for µ 0 is µ 0 ( µ 0 t α/2 n p σ µ 0, µ 0 + t α/2 n p σ µ 0 )

29 A 1 α prediction interval for Y 0 is an interval such that P (a (Y) Y 0 b (Y)) = 1 α A 1 α prediction interval for Y 0 is Y 0 ( µ 0 t α/2 n p σ 2 + σ 2 µ 0, µ 0 + t α/2 n p σ 2 + σ 2 µ 0 ) The derivation of this interval is based on the fact that Var (Y 0 µ 0 ) = σ 2 + σ 2 µ 0

30 R 2 Let T 2 = ( Yi Y ) 2, S 2 = Y X β 2 be the numerators of the variance estimators for the regression model and the intercept only model. We think of these as measuring the variation under these two models. Then the coefficient of determination R 2 is defined by Note that R 2 = T2 S 2 T 2 0 R 2 1

31 Note that T 2 S 2 is the amount of variation in the intercept only model which has been explained by including the extra predictors of the regression model and R 2 is the proportion of the variation left in the intercept only model which has been explained by including the additional predictors.