STA Object Data Analysis - A List of Projects. January 18, 2018

Transcription

1 STA 6557 Jauary 8, 208 Object Data Aalysis - A List of Projects. Schoeberg Mea glaucomatous shape chages of the Optic Nerve Head regio i aimal models 2. Aalysis of VW- Kedall ati-mea shapes with a applicatio to medical imagig - Yufa Wag 3. Distributios of Trajectories of Objects i the Solar System. See Ch. data ad ( of tras-neptuia objects) 4. VW regressio for projective shapes of shells of Acrosterigma magum mollusk i the N Florida coastal waters. 5. 3D Projective Shape Aalysis i Face Recogitio from Digital Camera Images - Seug-Hee Choi. 6. Topological Data Aalysis 7. Procipal Compoets o Object Spaces

2 2 REVIEW ( BUILDING THE NECESSARY MATH STAT LANGUAGE ) We begi by reviewig a list of simple defiitios ad otatios for otios that are at the basis of statistical iferece, such as sample space, evets, σ-algebra, probability measure probability space radom variable radom vector mea vector, covariace matrix characteristic fuctio 2. The Multivariate Normal Distributio REMARK 2.. The multivariate ormal distributios play a cetral role i Large Sample Theory, the key to Noparametric Data Aalysis (see Bhattacharya ad Rao (976) or Tsybakov(2009)).. A r.vec. X : Ω R p has a multivariate ormal distributio if a R p the r.v. a T X has a ormal distributio. Assume X is as i ad E(X) = µ, Cov(X) = Σ, ( otatio X N p (µ, Σ) ). The characteristic fuctio of X is φ X (t) = e itt µ 2 tt Σt. If X N p (µ, Σ), where Σ is positive defiite the the joit p.d.f. of X is give ϕ X (x) = (2π) p/2 Σ /2 exp( 2 (x µ)t Σ (x µ)) () 2

3 If Z = (Z,..., Z p ) N p (0, I p ), the the margials Z,..., Z p are idepedet. X N p (µ, Σ), where Σ is positive defiite the (X µ) T Σ (X µ) has a χ 2 p distributio. 2.2 Covergece i Distributio ad The Cetral Limit Theorem. The mai referece for this sectio is Ferguso (996). 2.3 Covergece i distributio ( weak covergece ) DEFINITION 2.. A sequece of radom vectors Y, N, coverges i distributio (or i law) to a radom vector Y, if the sequece of joit c.d.f. s F Y (y) coverges to F Y (y) at all vectors y where F Y is cotiuous. Notatio Y d Y, or Y L Y. The followig is a combiatio of the Helly-Bray theorem with the cotiuity theorem, ad is take straight from Ferguso (996), icludig the proof. THEOREM 2.. Assume X, N, is a sequece of radom vectors i R p. The followig are equivalet: a. X L X. b. Ef(X ) Ef(X) = R p fdp X as for all real-valued cotiuous fuctios f o R p, with compact support. c. X coverges weakly to X, that is Ef(X ) Ef(X) as (2) for all bouded real-valued cotiuous fuctios f o R p. d. Ef(X ) Ef(X) as for all measurable fuctios f o R p, such that X takes values o the set of poits of cotiuity of f with probability. e. The sequece of characteristic fuctios of X coverges to the characteristic fuctio of X. 3

4 2.4 The Cetral Limit Theorem This sectio is from Bhattacharya et al.(206)(see STA5334). The most importat covergece theorem i law is the followig theorem, abbreviated as CLT. THEOREM 2.2 (Cetral Limit Theorem). If Y is a sequece of i.i.d. radom vectors i R p with commo mea vector zero ad a fiite commo covariace matrix Σ, the 2 (Y + + Y ) L N p (0, Σ). (3) that For a proof of the CLT, we cosider the sequece X = Ȳ = 2 (Y + + Y ). Note φ X (t) = E(exp(i 2 t T X j )) = (E(exp(i 2 t T X ))) = (φ X ( 2 t)) j= (4) Let σ 2 (t) = E((t T X ) 2 ). From equatio (4) it follows that lim φ X (t) = exp( σ2 (t) ). (5) 2 The right had side of equatio (5) is φ U (), where U N (0, σ 2 (t)). Fially ote that if X N d (0, Σ), the t T X N(0, σ 2 (t)), ad from Theorem?? it follows that X L X, doe. We will sometimes use the alterative otatio Φ 0,Σ for N p (0, Σ), ad deote the correspodig distributio fuctio by Φ 0,Σ (x), ad, wheever Σ is positive defiite, the joit desity fuctio by φ 0,Σ (x) = (2π)p Σ e 2 xt Σ x. The CLT says that if Σ is positive defiite, the P( 2 (Y + + Y ) x) Φ 0,Σ (x) = (2π)p Σ... e 2 yt Σ y dy (6) y x 4

5 for all x. A old result of Polya implies that actually the covergece i (6) is uiform over all x: sup P( 2 (Y + + Y ) x) Φ 0,Σ (x) 0 as. (7) x A immediate applicatio of the CLT is that the r th sample momet (X r + + X r )/ of a uivariate distributio is asymptotically ormal if EX 2r <. (Here r is a positive iteger). The italicized statemet meas: ( ) Xi r EX r L N(0, σ 2 ), (8) i= where σ 2 = var(x r ) = EX 2r (EX r ) 2. REMARK 2.2. The LLN shows i particular that for a sequece of i.i.d. radom variables Z, i= Z i µ (= EZ ), i.e., the differece betwee the two sides i goes to zero as. Uder the additioal assumptio EZ 2 <, Chebyshev s iequality stregthes this approximatio by showig that the differece is of the order of 2 i probability: i= This meas that, give ε < 0 there exists A > 0 such that P( Z i µ = O p ( ). (9) Z i µ > A ) < ε (0) i= for all sufficietly large. The CLT is of course a more precise statemet tha (9). I particular, it provides a computatio of the left side of (0) as ( 2Φ 0, A ) + δ = 2 A/σ e y2 /2 dy + δ, () σ 2π where δ 0 as (uiformly w.r.t. A).] A useful extesio of the CLT i the uivariate case is 5

6 THEOREM 2.3. (Lideberg s CLT). Let X, X 2,... be idepedet radom variables with distributio fuctios F, F 2,..., respectively, such that EX = µ ad Var X = σ 2 <, with at least oe σ > 0. Let S = X + + X ad s = Var(S ) = σ σ. 2 The the ormalized partial sums S ES s coverge i distributio to a radom variable with ormal distributio N(0, ) (i.e. the ormal covergece holds,) if the followig Lideberg coditio is satisfied: ε > 0, lim (x µ s 2 k ) 2 df k (x) = 0. x µ k >εs k= PROBLEM 2.. Assume we have k sequeces U (i) P a i ( i k) ad a fuctio g of k vector variables cotiuous at (a, a 2,..., a k ). Show that g(u (),..., U (k) ) coverges i probability to (a, a 2,..., a k ). PROBLEM 2.2. Prove that the sample covariace matrix is a cosistet estimator of the covariace matrix. PROBLEM 2.3. Let X ( ) be a sequece of i.i.d. radom variables, ad assume that the ifimum ad supremum of values of X are m ad M, respectively. That is, P(m X M) =, P(X < a) > 0 a > m, P(X > b) > 0 b < M (Here X real-valued, but m ad/or M may be ifiite). Prove that max{x,..., X } P M ad mi{x,..., X } P m. PROBLEM 2.4. Prove that if Y coverges i distributio to a costat c (i.e., the distributio of Y coverges to the Dirac measure δ {c} ), the Y P c. PROBLEM 2.5. Let P ( ), P be probability measures o (R, B ) such that P coverges weakly to P. (a) Give a example to show that P (B) eed ot coverge to P(B) for all Borel sets B. (b) Give a example to show that the distributio fuctio F of P may ot coverge to the distributio fuctio F of P at every poit x. 6