Nonparametric statistics on manifolds with applications to shape spaces

IMS Collectio ns Pushing the Limits of Con temp orary Statist ics: Contributions in Honor of Jay an ta K. Ghosh V ol. 3 ( 2008) 282–301 c  Institute of Mathe matical Statistics , 2008 DOI: 10.1214/ 07492170 80000002 00 Nonparametric statistic s on ma nifolds with applications to shap e space s ∗ Abhishek Bhattac hary a 1 and Rabi Bhatta c hary a 1 University of A rizona Abstract: This ar ticle presen ts certain recen t methodologies and some new results for the statistical analysis of probability distributions on man ifolds. An imp ortan t example considered in some detail here is the 2- D shape space of k-ads, comprising all configurations of k planar landmarks ( k > 2)-mo dulo translation, scaling and r otation. Con ten ts 1 Int ro duction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282 2 F r ´ echet mean and v a r iation on metric spaces . . . . . . . . . . . . . . . . 286 2.1 F r´ ec het mean . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 86 2.2 F r´ ec het v ariation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 7 3 Extrinsic mean and v ariation . . . . . . . . . . . . . . . . . . . . . . . . . 29 0 3.1 Asymptotic distribution of the sample extr insic mean . . . . . . . . . 290 3.2 Application to the planar shap e spa c e o f k-ads . . . . . . . . . . . . 291 3.3 Asymptotic distribution of mean shap e . . . . . . . . . . . . . . . . . 292 3.4 Two sample testing problems on Σ k 2 . . . . . . . . . . . . . . . . . . 294 4 Int rinsic mean and v ariation . . . . . . . . . . . . . . . . . . . . . . . . . . 296 4.1 Asymptotic distribution of the sample intrinsic mean . . . . . . . . . 2 97 5 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 98 Ac knowledgmen ts. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 99 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 300 1. Introduction The statistical analys is of shap e distributio ns based on random samples is imp orta nt in many areas such as mor phometrics (discrimination and classification o f biolog- ical shap es), medical dia g nostics (detection of change or deformation of shap es in some o r gans due to some disease, for example) and mac hine vision (e.g., digital recording and a nalysis based on planar views o f 3- D ob jects). Among the pioneers on foundational studies leading to such applications, we men tion Kendall [ 20 ] (also see Kendall et al. [ 21 ]) and Boo kstein [ 9 ]. The g e ometries of the spa ces are those of differentiable ma nifolds often with appro priate Riemannian structures. ∗ Supported by NSF Grant DMS 04-06-143. 1 Departmen t of Mathematics, Universit y of Arizona, T ucson, AZ 85721, USA, e-mail : abhishek @math.ar izona.edu ; rabi @math.ar izona.ed u AMS 2000 subje c t classific ations: Pr imary 62 G20; secondary 62E20, 62 H35. Keywor ds and phr ases: extrinsic and intrinsic means and v ariations, Kendall’s shap e spaces, t w o-sample nonparametric tests. 282 Nonp ar ametric statist ics on manifolds 283 Our go al in this article is to esta blish some general principles for nonparametr ic statistical analy s is on s uch manifolds and a pply tho se to some shap e spa ces, es- pec ially K endall’s t w o-dimensional shap e space Σ k 2 of the so- c alled k-ads, i.e., the space o f configurations of k points on the plane (not all iden tical), identified mo d- ulo size and under Euclidean motions of translation a nd rotation. Tw o sample tests for the co mpa rison of b oth extrins ic and intrinsic F r ´ echet mean shap es and mean v aria tions of tw o distributions on Σ k 2 are pr ovided. As fa r as we know the explicit computations of these tests are new. In the case of the intrinsic mean and v ariation, the usual suppor t cr iterion (see, e.g., Le [ 24 ] a nd Bhattacharya and Patrangenaru [ 6 , 7 , 8 ]) is significantly relaxe d, ther eby substantially enha ncing the applica bilit y of the tests. F or recent r e s ults on statistical analysis of 3-D shap es, which w e do not consider here, we r efer to Dryden et al.[ 11 ] and Bandulasir i et al. [ 2 ]. Sometimes the sample sizes in s hap e a nalysis are only mo de r ately la r ge. Under such circumstances, one may more effectively use Effron’s b o o tstrap metho ds (Ef- fron [ 14 ]), who se supe r iority ov er the clas s ical CL T-based confidence reg ions and tests may b e established via higher or der a symptotics (see, e.g., Babu and Singh [ 1 ], B hattachary a and Qumsiyeh [ 5 ], Bhattacharya and Ghosh [ 4 ], Gho sh [ 16 ], Hall [ 17 ]). W e nex t tur n to the sp ecific ex ample of main interest to us, namely , Σ k 2 . F or purp oses of medical diagno stics, classification o f biolog ical sp ecies, etc., o ne may use exp er t help to choose a suitable o rdered set of k p oints or la ndmarks in the plane, or a k-ad , z = { ( x j , y j ) , 1 ≤ j ≤ k } , on a tw o-dimensional image of an ob ject under considera tion. One assumes that not all k po ints are the same, and k > 2. Kendall’s shap e space Σ k 2 comprises the equiv alence classes of all such k- ads under tr anslation, ro tation and sca ling. F or a g iven k-a d z , the effect of tr a nslation is r emov ed b y consider ing z − h z i where h z i is the vector whose elements are all equa l to the mean lo cation of the k-ad, namely , (1 /k ) P k j =1 ( x j , y j ). The tra nslated k-ads then lie in the (2 k − 2)-dimensiona l hyperplane H of ( ℜ 2 ) k ≈ ℜ 2 k , given by H = { ( x j , y j ) 1 ≤ j ≤ k : X x j = 0 , X y j = 0 } , and they comprise all of H except the origin. The effect of sca le, or length, is remov ed b y dividing z − h z i by k z − h z ik where k . k is the usua l Euclidean norm in ( ℜ 2 ) k , k ( u j , v j ) 1 ≤ j ≤ k k = [ X ( u 2 j + v 2 j )] 1 / 2 . The resulting tra nsformed k-ad w = ( z − h z i ) / k z − h z ik is called the pr eshap e o f the k-ad z . The set of presha pes is then natura lly identified with the unit sphere in H , which is basic a lly the same as the unit sphere S 2 k − 3 in ℜ 2 k − 2 . Fina lly , the shap e [ z ] of a k-a d z is given by the orbit of w = ( u j , v j ) ′ 1 ≤ j ≤ k under rotation, namely , (1.1) [ z ] = "  cos θ − s in θ sin θ cos θ   u j v j  1 ≤ j ≤ k # , − π < θ ≤ π . Thu s Σ k 2 is a quotient sp ac e of S 2 k − 3 , namely , S 2 k − 3 /S 1 , and it has dimens io n 2 k − 4. W e will use a mathema tically more conv enien t w a y of descr ibing Σ k 2 as achieved by viewing a k-ad a s an element of C k , na mely , z = ( x j + iy j ) 1 ≤ j ≤ k . The n h z i is 284 A. Bhattacharya and R. Bhattacharya the c omplex k -v ector whose elements are all e q ual to (1 / k ) P k j =1 ( x j + iy j ). The translated k-a d then lie s in the complex ( k − 1)-dimensional hyper plane H k − 1 = { ( a j ) 1 ≤ j ≤ k ∈ C k : k X j =1 a j = 0 } . The norm k z − h z ik has the sa me v alue as b e fore. But the r o tation by an angle θ of w = ( z − h z i ) / k z − h z ik may now be ex pressed as e iθ w . F or a system of co ordinate neighborho ods, or parametriz a tion, o f this spheric al r epr esentation o f Σ k 2 as a quotient spac e o f S 2 k − 3 , see Gallot et a l. ([ 15 ], pp. 3 2 , 34). Another parametriza tion of Σ k 2 , compatible with the ab ove, is o btained b y view- ing the sha pe o f a k-ad z ≡ ( x j + iy j ) 1 ≤ j ≤ k as the orbit { z 0 ( z − h z i ) : z 0 ∈ C \ { 0 }} . Note that z 0 = λ e iθ for λ = | z 0 | and some θ ∈ ( − π, π ], so that the o rbit, namely , a complex line through the o rigin in H k − 1 , is independent of b oth scale and ro tation and, therefor e, a representation of the sha pe o f z . Thus Σ k 2 is (isomorphic to) the space of all co mplex lines through the or igin in C k − 1 , the c omplex pr oje ctive sp ac e C P k − 2 , a familiar and imp ortant example in differ e n tial geometr y . F or a sy stem of co ordinate neig hborho o ds for Σ k 2 viewed as C P k − 2 , see Gallot et al. ([ 15 ], pp. 9, 10, 64, 65 ). W e next co nsider an extrinsic distanc e on Σ k 2 corres p o nding to a sp ecia l embed- ding, namely , the V er onese-Whitney emb e dding φ E of Σ k 2 int o the space S ( k , C ) of k × k co mplex Hermitian matrices : (1.2) φ E ([ z ]) = ww ∗ where w = ( z − h z i ) / k z − h z ik is the pr eshap e of z . Here w is re g arded as a column vector of k complex num b ers , w = ( w 1 , w 2 , . . . , w k ) ′ , and w ∗ is the transp ose o f its complex conjuga te. Obs e rve that the right side of (1.2 ) is constant on the orbit { e iθ w : − π < θ ≤ π } o f the presha p e w and is, therefore, a function of the shap e [ z ] o f the k-ad. Also, this function is o ne-to-one on Σ k 2 int o S ( k , C ). The v ector space S ( k, C ), with the real scaler field ℜ , has dimension k 2 . This is because a k × k Hermitian matrix is sp ecified by k r eal n um ber s on the diagona l and  k 2  complex nu m ber s (i.e., 2  k 2  real num ber s) as low er-right off-diagonal elements. On S ( k , C ) define the nor m k . k and distance d by k A k 2 = T race AA ∗ = T rac e A 2 , d 2 ( A, B ) = k A − B k 2 = T race( A − B ) 2 . (1.3) Note that this is the sa me as the Euclidean norm and dista nce in ℜ 2 k 2 . The induced distance ρ E on Σ k 2 is then given by ρ 2 E ([ z ] , [ w ]) = d 2 ( φ E ([ z ]) , φ E ([ w ])) = T rac e( uu ∗ − vv ∗ ) 2 = k X j =1 | u j | 2 + k X j =1 | v j | 2 − k X j =1 k X j ′ =1 ( u j ¯ u j ′ v j ′ ¯ v j + v j ¯ v j ′ u j ′ ¯ u j ) = 2 − 2 | u ∗ v | 2 (1.4) where u a nd v a re the preshap es of [ z ] a nd [ w ] resp ectively . The distance ρ E is known as the full Pr o crustes distanc e (Kendall [ 22 ], Kent [ 2 3 ] a nd Dryden and Mardia [ 12 ]). Nonp ar ametric statist ics on manifolds 285 Let X j , 1 ≤ j ≤ n , b e i.i.d. k-ads s uch that their shap es [ X j ], 1 ≤ j ≤ n , hav e the common distribution Q . Let ˜ µ denote the Euclidean mea n of ˜ Q = Q ◦ φ − 1 E viewed as a proba bilit y measure on S ( k, C ). Let ˜ M = φ E (Σ k 2 ), and denote the Euclidea n pro- jection of ˜ µ on ˜ M by P ˜ µ ≡ P ˜ M ˜ µ . The extrinsic me an of Q is then µ E = φ − 1 ( P ˜ µ ). It minimizes the F r´ echet function (2.1) with resp ect to the distance ρ E . Similar ly , for the sample extrinsic mean, ca lculate P ˜ X where ˜ X = (1 /n ) P n j =1 φ E ([ X j ]) is a co ordinate-w is e av erage of the matrix element s W j W ∗ j and W j is the preshap e of X j (1 ≤ j ≤ n ). The asymptotic distribution of √ n ( P ˜ X − P ˜ µ ) is given b y that of its pro jection on the ta ngent spa ce T P ˜ µ ˜ M at P ˜ µ , since its pro jection on the com- plement of T P ˜ µ ˜ M is negligible. F or computation of this pro jection, one c hoo ses a suitable ortho normal basis of S ( k , C ) (considered as a single orthonormal frame for its c onstant tange nt spaces), a nd calculates the differential of the pro jection map P = P ˜ M : S ( k, C ) → ˜ M in terms of these co or dinates. One thus arrives at a non- singular (2 k − 4)- dimens ional Normal distribution in the limit (see Sections 3.1 – 3.4 for details). T ur ning to the intrinsic me an o n a Riemannian manifold M , with geo desic distance d g , the first pro blem to resolve is its existence as the unique minimizer o f the F r ´ echet function R d 2 g ( p, m ) Q ( dm ). Here a result of Karchar [ 19 ] o n the existence of a unique minimizer is grea tly improv ed by a result of K endall [ 22 ], whic h allows the r adius r of a geo des ic ba ll B ( p, r ) containing the supp ort o f Q to b e twice as lar g e as re q uired by Karchar [ 1 9 ] (Pr op osition 4.1 ). On such a ball, the map φ = ex p − 1 p (the inv erse of the exp onential map at p ), is a diffeomorphism o nt o its image in the tange nt spac e T p M a t p . Using the co ordinates o f the vector spa ce T p M , c a lled normal c o or di nates , one arr ives at a central limit theor e m for the sample intrinsic mean µ nI (Theorem 4.2 ), following Bhattacharya and Patrang enaru [ 8 ]. Note that, with the (non- Euclidean) distance on T p M induced by φ from the geo desic distance d g on M , the imag e µ n = φ ( µ nI ) of µ nI is the minimizer of the F r´ echet function F n ( x ) ≡ Z d 2 g ( φ − 1 x, φ − 1 y ) ˜ Q n ( dy ) where ˜ Q n = Q n ◦ φ − 1 , Q n = (1 /n ) P n j =1 δ [ X j ] . Thus µ n is a M-estimator in the Euclidean space T q M . The a ssumptions in Theorem 4.2 guara n tee that this M- estimator is asymptotically Ga ussian a r ound µ = φ ( µ I ). The as ymptotic distribu- tion of the test s tatistic (4.5) follows from this. The computation of the test statistic (4.5) is genera lly mo re inv olv ed than that used for comparing extr ins ic means (s e e, e.g., (3.17 ) for the ca se M = Σ k 2 ). This inv o lves, in particular , the metric tensor of M to co mpute geo desic s and norma l co ordinates. W e refer to [ 3 ] for the asymptotic theory for in trinsic means, with explicit co mputations o f par ameters esp ecially for the planer sha p e space o f k- ads. Ho w ever in Section 5 of the pr esent article, w e display numerical v a lues of the intrinsic t w o-sample test s tatistics, along with the cor resp onding p-v alues, in t wo examples . It may be noted that for highly co ncent rated data in each of these examples, the extrinsic and intrinsic distances are close and hence the extr ins ic and int rinsic test statistics hav e virtually the sa me v alues. The minim um v a lue attained by the F r´ echet function is ca lle d the F r ´ echet vari- ation of Q and it is a mea s ure of sprea d of the distribution Q . The sample F r´ echet v aria tion is a co nsistent estimator of the F r´ echet v a riation of Q as pr ov ed in Prop o- sition 2.4 . If the F r ´ echet mean exists, we derive the as ymptotic distribution of the sample F r´ echet v ar iation in Theo rem 2.5 . This can b e used to construct a no npara- metric test sta tistic to co mpare the sprea d of tw o p opulatio ns on M . W e compute 286 A. Bhattacharya and R. Bhattacharya nu merical v alues o f the test statistic, along with the p-v a lues for M = Σ k 2 in Sec- tion 5 . F or highly concentrated data a s in the exa mples consider ed in Section 5 , the F r´ echet v ariations of the dis tr ibutions are very sma ll. Then the mean compa rison is usually sufficient to discrimina te b etw e e n the p o pula tions and the v ar iations show no significa nt difference. W e conclude this s ection with tw o brief r emarks. Fir st, the main o b jectiv e of inference in the tw o-sample pr oblem on Σ k 2 is to discriminate b etw een t w o differe n t distributions on it. It turns out, in most pr a ctical pr o blems that arise, that the means and v ariations (extr ins ic o r intrinsic) are ge nerally adequate for this dis- crimination. Mor e elab or ate pro cedures s uch as nonpara metric de ns it y es timation suffer from the “curse of dimensionality” on this commonly high-dimensiona l space. One can, how ever, do such densit y estimation on a tangent space (e.g., on T µ I M , via the inv erse exp onential map exp − 1 µ I ), as in the Euclide a n case. Excepting fo r the computation in normal co ordina tes, this pres e n ts no nov elt y . Secondly , in ex a mples with real data sets that we hav e studied (e.g., thos e in Section 5 ), the p-v alues of the nonparametr ic tw o-sample tes ts for comparing means, developed in this ar ti- cle, a re alwa ys muc h smaller (often b y an o rder of ma g nitude or more) than thos e based on existing, mostly par a metric, tests in the literature (see Dryden and Mar - dia [ 12 ]). This seems to indicate that the tes ts prop os e d her e may b e more p ow erful than thos e that hav e been used in the past, for many da ta s ets that arise in prac- tice. This pe r haps also po int s to the inadeq ua cy o f pa rametric mo dels of shap es po pularly used in the literature in capturing cer tain impo rtant shap e features. 2. F r ´ echet mean and v ariation on me tric spaces Let ( M , ρ ) b e a metric spa c e , ρ b eing the distance o n M . F or a g iven proba bilit y measure Q on (the Borel sigma-field of ) M , define the F r´ echet function o f Q as (2.1) F ( p ) = Z M ρ 2 ( p, x ) Q ( dx ) , p ∈ M . 2.1. F r ´ echet me an Definition 2.1. Supp ose F ( p ) < ∞ for some p ∈ M . Then the s e t of all p for which F ( p ) is the minimum v alue of F on M is ca lled the F r ´ echet me an set of Q , de no ted by C Q . If this set is a singleton, sa y { µ F } , then µ F is called the F r ´ echet me an of Q . If X 1 , X 2 , . . . , X n are indep endent a nd iden tically distributed (i.i.d.) with common distribution Q , and Q n . = (1 /n ) P n j =1 δ X j is the corres po nding empirica l distribution, then the F r´ echet mea n set o f Q n is ca lle d the sample F r´ echet me an set , denoted by C Q n . If this set is a singleton, say { µ F n } , then µ F n is called the sample F r ´ echet me an . The following result has been prov ed in Theorem 2.1, Bhattachary a and Pa- trangenar u [ 7 ]. Prop ositi o n 2. 1. Supp ose every close d and b oun de d subset of M is c omp act. If the F r ´ echet function F ( p ) of Q is finite for some p , then C Q is nonempty and c omp act. The next res ult es tablishes the strong consis tency o f the sample F r ´ echet mean. F o r a pro of, s e e Theorem 2.3, Bhattacharya and Patrangenaru [ 7 ]. Nonp ar ametric statist ics on manifolds 287 Prop ositi o n 2.2 . Assu me (i) t hat every close d b ounde d subset of M is c omp act, and (ii) F is finite on M . Then given any ǫ > 0 , ther e exists an int e ger value d r andom variable N = N ( ω , ǫ ) and a P -nul l set A ( ω , ǫ ) such that (2.2) C Q n ⊂ C ǫ Q ≡ { p ∈ M : ρ ( p, C Q ) < ǫ } , ∀ n ≥ N outside of A ( ω , ǫ ) . In p articular, if C Q = { µ F } , t hen every me asur able sele ction µ F n fr om C Q n is a str ongly c onsistent estimator of µ F . Remark 2.1 . It is known that a connected Riemannian manifold M which is complete (in its geo desic dis tance) satisfie s the top ologica l hypothesis of Prop osi- tions 2.1 and 2.2 : every clo sed b o unded subset of M is compact (see Theorem 2.8 , Do Ca rmo [ 10 ], pp. 1 46–1 47). W e will inv estigate conditions for the existence o f the F r´ echet mean of Q (as a unique minimizer of the F r´ echet function F of Q ) in the subsequent s ections. Remark 2. 2 . One can show that the r everse of (2.2), that is, “ C Q ⊂ C ǫ Q n ∀ n ≥ N ( ω , ǫ )” do es no t hold in general. See, for example, Bha ttachary a and Patrangenaru ([ 7 ], Remark 2 .6). Next we consider the a symptotic distribution of µ F n . F or Theor em 2.3 , we assume M to b e a differentiable manifold of dimension d . Le t ρ b e a distance metrizing the top ology of M . F or a pro o f of the following result, see Theore m 2.1 , Bhattacharya and Patrangenar u [ 8 ]. Theorem 2.3. Supp ose the fol lowing assumptions hold: (i) Q has supp ort in a s ingle c o or di nate p atch, ( U, φ ) , φ : U − → ℜ d smo oth. L et Y j = φ ( X j ) , j = 1 , . . . , n . (ii) The F r´ echet me an µ F of Q is u nique. (iii) ∀ x , y 7→ h ( x, y ) = ρ 2 ( φ − 1 x, φ − 1 y ) is twic e c ontinuously differ entiable in a neighb orho o d of φ ( µ F ) = µ . (iv) E (D r h ( Y 1 , µ )) 2 < ∞ ∀ r . (v) E ( sup | u − v | ≤ ǫ | D s D r h ( Y 1 , v ) − D s D r h ( Y 1 , u ) | ) → 0 as ǫ → 0 ∀ r, s . (vi) Λ = E (D s D r h ( Y 1 , µ )) is n onsingular. (vii) Σ = Cov (D h ( Y 1 , µ )) is nonsingular. L et µ F n b e a me asur able sele ction fr om the F r ´ echet sample me an set, and write µ n = φ ( µ F n ) . Then under the assumptions (i) – (vii) , (2.3) √ n ( µ n − µ ) L − → N (0 , Λ − 1 Σ(Λ ′ ) − 1 ) . 2.2. F r ´ echet variati on Definition 2. 2. The F r ´ echet variation V of Q is the minimum v alue atta ine d by the F r´ ech et function F defined by (2.1) on M . Similarly the minimu m v a lue attained by the sample F r ´ echet function , (2.4) F n ( p ) = 1 n n X j =1 ρ 2 ( X j , p ) is called the sample F r´ echet variation a nd denoted by V n . 288 A. Bhattacharya and R. Bhattacharya F r om Pr op osition 2.1 it follows tha t if the F r´ echet function F ( p ) is finite for some p , then V is finite a nd equals F ( p ) for all p in the F r´ echet mea n se t C Q . Similar ly the sample v a riation V n is the v alue of F n on the sample F r ´ echet mean set C Q n . The following result establishes the stro ng co nsistency of V n as an estimator of V . Prop ositi o n 2.4. Supp ose every close d and b ounde d subset of M is c omp act, and F is finite on M . Then V n is a st r ongly c onsist ent estimator of V . Pr o of. In view of Pro po sition 2.2 , for any ǫ > 0, there e xists N = N ( ω , ǫ ) such tha t (2.5) | V n − V | = | inf p ∈ M F n ( p ) − inf p ∈ M F ( p ) | ≤ sup p ∈ C ǫ Q | F n ( p ) − F ( p ) | for all n ≥ N almost surely . F r om the pro o f of Theor em 2.3 in Bhattacharya and Patrangenaru [ 7 ], it follows tha t for any co mpact set K ⊂ M , sup p ∈ K | F n ( p ) − F ( p ) | − → 0 a.s. as n → ∞ . Since C ǫ Q is compact, it follows from (2.5) that | V n − V | − → 0 a.s. as n → ∞ . Remark 2. 3. The sample v ariation is a consistent estimator of the p opulation v aria tion even when the F r´ echet function F of Q do es not hav e a unique minimizer. Next we derive the asymptotic distribution o f V n when there is a unique p opu- lation F r´ echet mean. Theorem 2 .5. L et M b e a differ entiable manifo ld. Using the notation of The o- r em 2.3 , un der assumptions (i) – (vii) and assuming E ( ρ 4 ( X 1 , µ F )) < ∞ , one has (2.6) √ n ( V n − V ) L − → N  0 , v ar( ρ 2 ( X 1 , µ F ))  . Pr o of. Let F ( x ) = Z ρ 2 ( φ − 1 ( x ) , m ) Q ( dm ) , F n ( x ) = 1 n n X j =1 ρ 2 ( φ − 1 ( x ) , X j ) . Let µ F n be a measur able selection from the sample mea n set and µ n = φ ( µ F n ). Then √ n ( V n − V ) = √ n ( F n ( µ n ) − F ( µ )) = √ n ( F n ( µ n ) − F n ( µ )) + √ n ( F n ( µ ) − F ( µ )) , (2.7) √ n ( F n ( µ n ) − F n ( µ )) = 1 √ n n X j =1 d X r =1 ( µ n − µ ) r D r h ( Y j , µ ) + 1 2 √ n n X j =1 d X r =1 d X s =1 ( µ n − µ ) r ( µ n − µ ) s D s D r h ( Y j , µ ∗ n ) (2.8) for so me µ ∗ n in the line segment joining µ and µ n . By a ssumption (v) of Theo rem 2.3 and b eca use √ n ( µ n − µ ) is asymptotically normal, the seco nd term on the r ight of Nonp ar ametric statist ics on manifolds 289 (2.8) conv erges to 0 in pr obability . Also (1 /n ) P n j =1 D h ( Y j , µ ) → E ( D h ( Y 1 , µ )) = 0, so that the first term on the right o f (2.8 ) converges to 0 in probability . Hence (2 .7) bec omes √ n ( V n − V ) = √ n ( F n ( µ ) − F ( µ )) + o P (1) = 1 √ n n X j =1  ρ 2 ( X j , µ F ) − E ρ 2 ( X 1 , µ F )  + o P (1) . (2.9) By the CL T for the i.i.d. seq uence { ρ 2 ( X j , µ F ) } , (2.9) conv erges in law to N (0 , v ar( ρ 2 ( X 1 , µ F )). Remark 2.4. Theor e m 2 .5 require s the p opulation mean to e x ist for the sample v aria tion to be a symptotically Normal. It may b e shown by examples that it fails to give the co rrect distribution if there is not a unique mean. Theorem 2.5 ca n b e used to constr uct a nonpara metric test for testing whether t wo p opula tions ha v e the same sprea d. Supp ose Q 1 and Q 2 are tw o pro bability distributions with unique F r´ echet means µ 1 F and µ 2 F and F r´ echet v ariations V 1 and V 2 , resp ectively . W e hav e i.i.d. s a mples X 1 , X 2 , . . . , X n and Y 1 , Y 2 , . . . , Y m from Q 1 and Q 2 , r esp ectively . L e t µ F n and µ F m denote the sa mple mea ns, V n and V m denote the sa mple v aria tio ns. Then the null hypo thesis is H 0 : V 1 = V 2 = V . Under H 0 , from (2.6), √ n ( V n − V ) L − → N (0 , σ 2 1 ) , (2.10) √ m ( V m − V ) L − → N (0 , σ 2 2 ) , (2.11) where σ 2 1 = v ar ( ρ 2 ( X 1 , µ 1 F )) , σ 2 2 = v ar( ρ 2 ( Y 1 , µ 2 F )) . Suppo se n/ ( m + n ) → p , m/ ( m + n ) → q , for some p, q > 0; p + q = 1. Then from (2.10) and (2.1 1), √ n + m ( V n − V m ) L − → N (0 ,  σ 2 1 p + σ 2 2 q  ) , (2.12) V n − V m q s 2 1 n + s 2 2 m L − → N (0 , 1) , (2.13) where s 2 1 = (1 / n ) P n j =1 ( ρ 2 ( X j , µ F n ) − V n ) 2 and s 2 2 = (1 /m ) P m j =1 ( ρ 2 ( Y j , µ F m ) − V m ) 2 are the s ample estimates of σ 2 1 and σ 2 2 , resp ectively . Hence the test statistic used is (2.14) T nm = V n − V m q s 2 1 n + s 2 2 m . F o r a test of size α , we reject H 0 if | T nm | > Z 1 − ( α/ 2) where Z 1 − ( α/ 2) is the (1 − ( α/ 2)) th quantile of N (0 , 1). F r om now o n, unles s o therwise stated, w e as sume that ( M , g ) is a d -dimensional connected complete Riemannian manifold, g b e ing the Riemannian metric tenso r on M . W e shall co me acro s s different no tions of mea ns a nd v a riations dep ending o n the distance chosen on M . W e b egin with the extrinsic distanc e in the next section. 290 A. Bhattacharya and R. Bhattacharya 3. Extrins ic mean and v ariation Let φ : M → ℜ k be an em bedding of M in to ℜ k , and let ˜ M = φ ( M ) ⊂ ℜ k . Define the distance o n M a s: ρ ( x, y ) = k φ ( x ) − φ ( y ) k , wher e k · k denotes Euclidean norm ( k u k 2 = P k i =1 u i 2 , u = ( u 1 , u 2 , . . . , u k ) ′ ). This is called the ext rins ic distanc e on M . Assume that ˜ M is a closed subset of ℜ k . The n for every u ∈ ℜ k there exis ts a compact set of po int s in ˜ M whose dista nce fro m u is the smallest among a ll p oints in ˜ M . W e will deno te this set by P u ≡ P ˜ M u = { x ∈ ˜ M : k x − u k ≤ k y − u k ∀ y ∈ ˜ M } . If this set is a singleton, u is said to b e a nonfo c al p oint o f ℜ k (with resp ect to ˜ M ); otherwise it is said to be a fo c al p oint of ℜ k . Definition 3.1. Let ( M , ρ ), φ be as a bove. Let Q b e a pro bability measure on M with finite F r´ echet function. The F r ´ echet mean (set) of Q is called the ex trinsic me an (set) o f Q , and the F r´ echet v ariation of Q is called its ext rinsic variation . If X j ( j = 1 , . . . , n ) are iid obs e rv atio ns from Q , and Q n = 1 n P n j =1 δ X j is the empirical distribution, then the F r´ echet mean(set) o f Q n is called the ext rins ic sample m e an (set) a nd the F r ´ echet v a riation of Q n is called the extrinsic sample variation . Let ˜ Q and ˜ Q n be the ima ges of Q and Q n , resp ectively , o n ℜ k under φ : ˜ Q = Q ◦ φ − 1 , ˜ Q n = Q n ◦ φ − 1 . The next result gives us a way to calculate the extr insic mean a nd establishes the consistency of the sample mean as an estimator of the po pulation mean if that exists. F or a pro of s ee P rop osition 3.1 in Bhatta charya a nd Patrangenaru [ 7 ]. Prop ositi o n 3.1. (a ) If ˜ µ = R R k u ˜ Q ( du ) is the me an of ˜ Q , then the ext rinsic me an set of Q is given by φ − 1 ( P ˜ µ ) . (b) If ˜ µ is a nonfo c al p oint of ℜ k (r elative to ˜ M ), then the ex trinsic sample me an µ nE (any me asur able sele ction fr om t he extrinsic me an set of Q n ) is a s t r ongly c onsist ent estimator of the ex t rinsic me an µ E = φ − 1 ( P ˜ µ ) . 3.1. Asymptotic distribution of the sample extrinsic me an W e ca n use Theorem 2.3 to get the a symptotic distribution of the sa mple extrinsic mean. Howev er, e x pressions for the parameters Λ and Σ are not easy to g e t. Here we devise ano ther w ay to de r ive the asy mptotic distribution. W e a ssume that the mean ˜ µ o f ˜ Q is a nonfo ca l p oint, so that the pro jection P ˜ µ of ˜ µ o n φ ( M ) is unique, and the extrinsic mea n of Q is µ E = φ − 1 ( P ˜ µ ). Let ˜ X = (1 /n ) P n j =1 ˜ X j denote the sample mea n of ˜ X j = φ ( X j ). The extrinsic sample mean set is C Q n = φ − 1 ( P ˜ X ), where P ˜ X is the set of pr o jection of ˜ X on φ ( M ). In a neighborho o d of a nonfo cal po int such as ˜ µ , P ( . ) is smo oth. So we ca n write (3.1) √ n [ P ( ˜ X ) − P ( ˜ µ )] = √ n (d ˜ µ P )( ˜ X − ˜ µ ) + o P (1) = (d ˜ µ P )( √ n ( ˜ X − ˜ µ )) + o P (1) where d ˜ µ P is the differential (map) o f P ( · ) , which takes vectors in the tang ent space of ℜ k at ˜ µ to tangent vectors of φ ( M ) at P ( ˜ µ ). Hence the left side is asympto tica lly normal. Nonp ar ametric statist ics on manifolds 291 F o r the case o f r egular submanifolds em bedded in an Euc lidea n space by the inclusion map, a similar a symptotic distr ibution and a tw o-sample test w ere co n- structed indepe ndently b y Hendricks and Landsman [ 18 ] and, for more general manifolds, by Patrang enaru [ 26 ] and B hattachary a and Patrangenar u [ 8 ]. 3.2. Applic ation to the planar shap e sp ac e of k-ads Consider a set of k p oints on the plane, e.g., k lo cations o n a skull pro jected on a plane, not a ll p o ints b eing the s ame. W e will ass ume k > 2 and refer to such a se t as a k-ad (or a set of k landmarks ). F or conv enience we will denote a k - ad by k complex nu m ber s ( z j = x j + iy j , 1 ≤ j ≤ k ), i.e., we w ill re present k-ads on a complex plane. By the shap e o f a k-ad z = ( z 1 , z 2 , . . . , z k ), we mean the equiv alence cla ss, or orbit of z under translation, ro tation and scaling. T o remove trans lation, one may subs tr act h z i ≡ ( h z i , h z i , . . . , h z i ) ( h z i = (1 /k ) P k j =1 z j ) from z to get z − h z i . Rotation of the k-ad by an angle θ and scaling (b y a factor r > 0) are achiev ed by mult iplying z − h z i by the complex num ber λ = r ex p iθ . Hence one may repres e nt the s hap e of the k-ad as the complex line pa ssing thro ugh z − h z i , namely , { λ ( z − h z i ) : λ ∈ C \ { 0 }} . Thu s the space of k-ads is the set of all complex lines on the (complex ( k − 1 )- dimensional) hyperpla ne, H k − 1 = { w ∈ C k \ { 0 } : P k 1 w j = 0 } . Therefore the shap e spa ce Σ k 2 of planer k- ads has the structure of the c omplex pr oje ctive sp ac e C P k − 2 : the space of a ll complex line s throug h the o rigin in C k − 1 . As in the case of C P k − 2 , it is conv enien t to represent the elemen t of Σ k 2 corres p o nding to a k-ad z by the curve γ ( z ) = [ z ] = { e iθ (( z − h z i ) / k z − h z ik ) : 0 ≤ θ < 2 π } on the unit sphere in H k − 1 ≈ C k − 1 . If we denote by u the quantit y ( z − h z i ) / k z − h z ik , c a lled the pr eshap e of the shap e of z , then another representation of Σ k 2 is v ia the V er onese–Whitney emb e ddi ng φ int o the spac e S ( k , C ) of all k × k complex Hermitian matrices. S ( k , C ) is viewed as a (real) vector space with res pec t to the scaler field ℜ . The embedding φ is given by φ : Σ k 2 → S ( k , C ) , φ ([ z ]) = u u ∗ ( u = ( u 1 , . . . , u k ) ′ ∈ H k − 1 , k u k = 1) = (( u i ¯ u j )) 1 ≤ i,j ≤ k . (3.2) The sha pe of z , [ z ] = { e iθ u : 0 ≤ θ < 2 π } is the o rbit of the vector u under ro tation. Note that if v 1 , v 2 ∈ [ z ], then φ ([ v 1 ]) = φ ([ v 2 ]) = φ (( z − h z i ) / k z − h z ik ). Define the extrinsic distanc e ρ on Σ k 2 by that induced fro m this embedding, namely , (3.3) ρ 2 ([ z ] , [ w ]) = k uu ∗ − v v ∗ k 2 , u . = z − h z i k z − h z ik , v . = w − h w i k w − h w ik where for arbitra ry k × k complex ma tr ices A, B, (3.4) k A − B k 2 = X j,j ′ | a j j ′ − b j j ′ k 2 = T race( A − B )( A − B ) ∗ is just the squared euclidean dista nce b etw een A a nd B regar ded as elements of C k 2 (or, ℜ 2 k 2 ). Since the matr ices uu ∗ , v v ∗ in (3.2) ar e Her mitian, one notes that the image φ (Σ k 2 ) of Σ k 2 is a closed subset of C k 2 and the “co njugate-transp o s e” symbo l * may b e dr o pp e d from (3.4) in computing dis tances in φ (Σ k 2 ). 292 A. Bhattacharya and R. Bhattacharya Let Q b e a pro ba bilit y measure o n the s hap e space Σ k 2 , le t [ X 1 ] , [ X 2 ] , . . . , [ X n ] be an i.i.d. sample from Q and let ˜ µ deno te the mea n vector of ˜ Q . = Q ◦ φ − 1 , regar ded as a pro bability measure on C k 2 (or, ℜ 2 k 2 ). Note that ˜ µ b elongs to the conv ex hull of ˜ M = φ (Σ k 2 ) and in particular , is an element of H k − 1 . Let T b e a (complex) or thogonal k × k matrix such that T ˜ µT ∗ = D = Diag ( λ 1 , λ 2 , . . . , λ k ), where λ 1 ≤ λ 2 ≤ · · · ≤ λ k are the eigenv alues of ˜ µ . Then, writing v = T u with u as in (3.3), k uu ∗ − ˜ µ k 2 = k v v ∗ − D k 2 = k X j =1 ( | v j | 2 − λ j ) 2 + X j 6 = j ′ | v j v j ′ | 2 = X λ j 2 + k X j =1 | v j | 4 − 2 k X j =1 λ j | v j | 2 + k X j =1 | v j | 2 . k X j ′ =1 | v j ′ | 2 − k X j =1 | v j | 4 = X λ j 2 + 1 − 2 k X j =1 λ j | v j | 2 (3.5) which is minimized (on φ (Σ k 2 )) by taking v = e k = (0 , . . . , 0 , 1) ′ , i.e., u = T ∗ e k , a unit eigenvector having the larg est e ig env alue λ k of ˜ µ . It follows that the extrinsic mean µ E , say , of Q is unique if and o nly if the eigenspac e for the larges t eigenv alue of ˜ µ is (co mplex) one-dimensional, and then µ E = [ µ ], µ ( 6 = 0) ∈ the eigenspace of the large s t eigenv alue of ˜ µ . F r om (3.5), the extrinsic v aria tion o f Q has the expre ssion V = E k X 1 X ∗ 1 − µµ ∗ k 2 = E k X 1 X ∗ 1 − ˜ µ k 2 + k ˜ µ − µµ ∗ k 2 = 2(1 − λ k ) (3.6) Therefore, we ha ve the following consequence of Prop os ition 2.4 and Pro p osition 3.1 . Corollary 3.2. L et µ n denote an eigenve ctor of (1 /n ) P n j =1 X j X j ∗ having the lar gest eigenvalue λ kn . (a ) If the lar gest eigenvalue λ k of ˜ µ is simple, then the extrinsic sample me an [ µ n ] is a st ro ngly c onsistent estimator of the extrinsic me an [ µ ] . (b) The sample ext rinsic variation, V n = 2 (1 − λ kn ) is a st r ongly c onsistent estimator of the extrinsic variation, V = 2(1 − λ k ) . The distance ρ o n Σ k 2 in (3.3) ca n be expressed as (3.7) ρ 2 ([ z ] , [ w ]) ≡ k uu ∗ − v v ∗ k 2 = 2(1 − | u ∗ v | 2 ) . This is the so-called ful l Pr o crustes distanc e for Σ k 2 . See Kent [ 23 ], Dry den and Mardia [ 12 ] and Kendall et a l. [ 21 ]. 3.3. Asymptotic distribution of me an shap e T o get the asymptotic distributio n of the sample extrinsic mean sha p e using (3.1 ), we e m bed M = Σ k 2 int o S ( k , C ), the spa ce of a ll k × k co mplex self-adjoint matrices, via the ma p φ in (3 .2 ). W e co nsider S ( k , C ) as a linear subspace of C k 2 (ov er ℜ ) and as such a regula r submanifold of C k 2 embedded b y the inclusion map, and inheriting the metr ic tenso r: h A, B i = Re  T r ace( A ¯ B ′ )  . Nonp ar ametric statist ics on manifolds 293 The (rea l) dimensio n of S ( k , C ) is k 2 . An orthonorma l bas is for S ( k , C ) is given by { v a b : 1 ≤ a ≤ b ≤ k } a nd { w a b : 1 ≤ a < b ≤ k } , defined as v a b = ( 1 √ 2 ( e a e t b + e b e t a ) , a < b e a e t a , a = b w a b = + i √ 2 ( e a e t b − e b e t a ) , a < b. where { e a : 1 ≤ a ≤ k } is the standard canonica l ba s is for ℜ k . W e a lso take { v a b : 1 ≤ a ≤ b ≤ k } and { w a b : 1 ≤ a < b ≤ k } as the (co nstant) orthogo nal fra me for S ( k , C ). F or any U ∈ O ( k ) ( U U ∗ = U ∗ U = I ), { U v a b U ∗ : 1 ≤ a ≤ b ≤ k } , { U w a b U ∗ : 1 ≤ a < b ≤ k } is also an orthog onal frame for S ( k, C ). Assume that the mean ˜ µ of ˜ Q has its larges t eigenv alue simple. T o apply (3.1 ), we vie w d ˜ µ P : S ( k , C ) → T P ( ˜ µ ) φ (Σ k 2 ). Cho o se U ∈ O ( k ) such that U ∗ ˜ µU = D ≡ Diag( λ 1 , . . . , λ k ), λ 1 ≤ · · · ≤ λ k − 1 < λ k being the eigenv alues of ˜ µ . Cho ose the basis fr a me { U v a b U ∗ , U w a b U ∗ } for S ( k , C ). Then one ca n show that d ˜ µ P ( U v a b U ∗ ) = ( 0 , if 1 ≤ a ≤ b < k , a = b = k , ( λ k − λ a ) − 1 U v a k U ∗ , if 1 ≤ a < k , b = k . d ˜ µ P ( U w a b U ∗ ) = ( 0 , if 1 ≤ a < b < k , ( λ k − λ a ) − 1 U w a k U ∗ , if 1 ≤ a < k , b = k . (3.8) W r ite √ n ( ¯ ˜ X − ˜ µ ) = X X 1 ≤ a ≤ b ≤ k h √ n ( ¯ ˜ X − ˜ µ ) , U v a b U ∗ i U v a b U ∗ + X X 1 ≤ a 0 ; p + q = 1. Then √ n + m ( ¯ T ( µ ) − ¯ S ( µ )) L − → N 2 k − 4 (0 , 1 p Σ 1 ( µ ) + 1 q Σ 2 ( µ )) . Thu s assuming Σ 1 ( µ ), Σ 2 ( µ ) and hence 1 p Σ 1 ( µ ) + 1 q Σ 2 ( µ ) to b e nons ingular, (3.16) ( n + m )( ¯ T ( µ ) − ¯ S ( µ )) ′ ( 1 p Σ 1 ( µ ) + 1 q Σ 2 ( µ )) − 1 ( ¯ T ( µ ) − ¯ S ( µ )) L − → X 2 2 k − 4 . Note that the nonsingularity assumption for Σ 1 ( µ ) and Σ 2 ( µ ) are satisfied if, for example, Q 1 and Q 2 hav e nonzer o absolutely contin uous co mpo nents with resp ect to the volume measure on Σ k 2 (iden tified with the Riemannian manifold C P k − 2 ). W e can choose µ to b e a ny p ositive linear co m bination of µ 1 and µ 2 . Then under H 0 , µ will hav e the same pr o jection o n φ (Σ k 2 ) as µ 1 and µ 2 . W e may take µ = pµ 1 + qµ 2 . In practice, since µ 1 and µ 2 are unknown, so is µ . Then we may estimate µ by the po oled sample mean ˆ µ = ( n ¯ X + m ¯ Y ) / ( m + n ), Σ 1 ( µ ) and Σ 2 ( µ ) by their sample estimates ˆ Σ 1 ( ˆ µ ) and ˆ Σ 2 ( ˆ µ ), where ˆ Σ 1 ( µ ) = 1 n T ( µ ) T ( µ ) ′ − ¯ T ( µ ) ¯ T ( µ ) ′ , ˆ Σ 2 ( µ ) = 1 m S ( µ ) S ( µ ) ′ − ¯ S ( µ ) ¯ S ( µ ) ′ . Then the tw o-sample test statistic in (3.16) can b e e s timated by (3.17) T nm = ( ¯ T ( ˆ µ ) − ¯ S ( ˆ µ )) ′ ( 1 n ˆ Σ 1 ( ˆ µ ) + 1 m ˆ Σ 2 ( ˆ µ )) − 1 ( ¯ T ( ˆ µ ) − ¯ S ( ˆ µ )) . Given level α , we reject H 0 if (3.18) T nm > X 2 2 k − 4 (1 − α ) . The expr ession for T nm depe nds on the sp ectrum of ˆ µ through the o rbit [ U k ( ˆ µ )] and the subspa c e spanned b y { U 2 ( ˆ µ ) , . . . , U k − 1 ( ˆ µ ) } . If the p opulation mean exists, [ U k ( ˆ µ )] is a consistent estimator of [ U k ( µ )] and by p erturbation theory (see Dunfor d and Schw a rtz [ 13 ], p. 598), the pro jection on Span { U 2 ( ˆ µ ) , . . . , U k − 1 ( ˆ µ ) } c onv erges to that on Span { U 2 ( µ ) , . . . , U k − 1 ( µ ) } . Thus from (3.16) a nd (3 .17), T nm has an asymptotic X 2 2 k − 4 distribution. Hence the test in (3.18) has asymptotic level α . T o test if the p opulations hav e the sa me sprea d around their r esp ective means, we use the test statistic in (2.14 ), which is (3.19) T nm = 2 λ km − λ kn q s 2 1 n + s 2 2 m , where λ kn and λ km are the lar g est eigenv alues of ¯ X and ¯ Y , resp ectively . Under H 0 , T nm has asymptotic Normal distribution. 296 A. Bhattacharya and R. Bhattacharya 4. Intrinsic mean and v ariation Let ( M , g ) be a d-dimensional connected complete Riemannian manifold, g b eing the Riemannian metric on M . Let the distance ρ = d g be the g e o desic dis ta nce under g . Let Q b e a probability distr ibution on M with finite F r´ echet function, (4.1) F ( p ) = Z M d 2 g ( p, m ) Q ( dm ) , p ∈ M . Definition 4.1. The F r´ echet mean (set) of Q under the distance d g is called its intrinsic me an (set). The F r´ echet v ariation o f Q under d g is called its intrinsic variation . Let X 1 , X 2 , . . . , X n be i.i.d. observ ations on M with common distribution Q . The sa mple F r ´ echet mean (set) is calle d the sample intrinsic me an (set) and the sample F r´ echet v ariation is called the sample intrinsic variation . Let us define a few technical terms rela ted to Riemannian manifolds which we will use extensively in the subsequent sections. F or details on Riemannian Manifolds, see DoCar mo [ 10 ], Gallot et al. [ 1 5 ] or Lee [ 25 ]. 1. Ge o desic : These a re curves γ o n the manifold with zero a cceleration. They a re lo- cally length minimizing curves. F or example, cons ider great circle s on the sphere or straig ht lines in ℜ d . 2. Exp onent ial map : F or p ∈ M , v ∈ T p M , w e define exp p v = γ (1), wher e γ is a geo desic with γ (0) = p and ˙ γ (0) = v . 3. Cut lo cu s : Let γ b e a unit spee d ge o desic s ta rting at p, γ (0) = p . Let t 0 be the supremum o f all t for which γ is leng th minimizing o n [0 , t ]. Then γ ( t 0 ) is called the cut point of p along γ . The cut lo cus of p , C ( p ), is the set of all cut p oints of p along a ll geo desics. F or example, C ( p ) = {− p } on S d . 4. Convex b al l : A ball B is c alled convex if, for any p, q ∈ B , a unique geo des ic fr o m p to q lies in B , which is also the shor test geo desic from p to q . F or example, any ball of radius π / 2 or les s in S d is conv ex. 5. Se ct ional Curvature : Recall the notio n of Gauss ian curv a tur e o f tw o dimensional surfaces. On a Riemannian manifold M , ch o ose a pa ir of linearly independent vectors u, v ∈ T p M . A t w o dimensiona l submanifold of M is swept out by the set of all geo desics star ting at p and with initial velocities lying in the t w o- dimensional section π s panned b e u, v . The curv ature of this submanifold is called the se ctional curv ature at p of the section π . In all subse q uent sectio ns, we assume that M ha s a ll sectio nal c ur v ature s b ounded ab ov e by some C ≥ 0. The next res ult, due to Kendall [ 22 ], gives a sufficient c o ndition for the exis tence of a unique lo ca l minim um of F in a g e o desic ball of rea sonably wide radius. Prop ositi o n 4.1. If the supp ort of Q is c ontaine d in B ( p, r ) with r < π / (2 √ C ) and B ( p, r ) ∩ C ( p ) = φ , then the F r´ echet fun ct ion F of Q has a unique lo c al minimum in B ( p, r ) . Recall that (Karchar [ 19 ]; see also Theorem 2.1 in Bhattacharya a nd Patrange- naru [ 7 ]) if Q ( C ( p )) = 0 ∀ p ∈ M , then every lo cal minimum µ of F sa tis fie s (4.2) Z T µ M v ˜ Q ( dv ) = 0 where ˜ Q is the imag e of Q under the map e xp − 1 µ on M \ C ( µ ). Nonp ar ametric statist ics on manifolds 297 4.1. Asymptotic distribution of the sample intrinsic me an One can use Theo rem 2.3 to get the as ymptotic distribution of the sa mple intrinsic mean. F or that we need to v erify ass umptions (i) to (vii). The next result gives sufficient co nditio ns for those assumptions to hold. Theorem 4.2. Supp ose the supp ort of Q is c ontaine d in a ge o desic b al l B ( p, r ) with c enter p and r adius r as in Pr op osition 4.1 . L et φ = exp − 1 p : B ( p, r ) − → T p M ( ≈ ℜ d ) . Define h ( x, y ) = d 2 g ( φ − 1 x, φ − 1 y ); x, y ∈ ℜ d . L et ((D r h )) d r =1 and ((D r D s h )) d r,s =1 b e t he matric es of first and se c ond or der derivatives of y 7→ h ( x, y ) . L et ˜ X j = φ ( X j )( j = 1 , . . . , n ), X 1 , . . . , X n b eing i.i.d. observations fr om Q . L et µ = φ ( µ I ) , µ I b eing t he p oint of lo c al minimum of F in B ( p, r ) . L et µ n = φ ( µ nI ) , µ nI b e- ing the p oint of lo c al minimum of F n in B ( p, r ) . Define Λ = E((D r D s h ( ˜ X 1 , µ ))) d r,s =1 , Σ = Cov((D r h ( ˜ X 1 , µ ))) d r =1 . If Λ and Σ ar e nonsingular, then (4.3) √ n ( µ n − µ ) L − → N (0 , Λ − 1 ΣΛ − 1 ) . Pr o of. When Q is considered a s a probability mea sure on the co mpact ba ll B ( p, r ) (as the underlying metric space), µ n is a consistent e stimator of µ , by P rop osi- tion 2 .2 . In view of Pr op osition 4 .1 , as in the pro o f of Theorem 2 .3 in Bhattacharya and Patrengenaru [ 8 ], Assumptions (i)–(vii) o f Theorem 2.3 are verified. Remark 4.1. The no nsingularity of Σ in Theorem 4 .2 is a mild conditio n which holds in particular if Q has a dens it y (comp onent) with res pe c t to the v olume measure. The nonsingularity of Λ is a more delicate ma tter in ge neral, inv olving a detailed analysis inv olving curv a tur e and Jacobi fields. These matter s ar e considered in detail in B hattachary a and Bhattacharya [ 3 ]. Remark 4.2. Under the hypothesis of Prop osition 4.1 (and Theorem 4 .2 ), the po int of lo cal minimum µ I of F in B ( p, r ) may not b e the globa l minimizer of F on M . How ever, if one r e s tricts attention to the clos ed ba ll B ( p, r ) a s the under lying metric s pace o f interest, this po int of lo cal minim um is the intrinsic mea n (on B ( p, r )). The adv an tage of Theorem 4 .2 over the ear lier res ult Theor em 2.3 in Bhattacharya and Patrengenaru [ 8 ] is that here one a llows a m uc h wider supp ort of Q , namely , the radius r her e is twice as large a s that allow ed in the earlier re s ult. This is par ticularly impo rtant in tw o-sample problems as well as in problems of classification inv olving several p opulations. Also from a statistical p oint o f view, the mean shap e is p er haps b etter repres ented if defined as the F r´ ech et mean over B ( p, r ) tha n ov er the whole of M , since Q ( M \ B ( p, r ) ) = 0 and since B ( p, r ) is a connected Riemannian manifold inheriting the metric o f M . Theorem 4.2 ca n b e use d to construct a n asymptotic 1 − α confidence set for µ I which is given b y (4.4) { µ I : n ( µ n − µ ) t ( ˆ Λ − 1 ˆ Σ ˆ Λ − 1 ) − 1 ( µ n − µ ) ≤ X 2 d (1 − α ) } where ( ˆ Σ , ˆ Λ) are consistent sample estimates of (Σ , Λ) and X 2 d (1 − α ) is the upp er (1 − α ) th quantile of the c hi-squared distribution with d degr ees of freedom. Also we ca n p erform a nonpar ametric test to test if tw o distr ibutions Q 1 and Q 2 hav e the same intrinsic mean µ I . Let µ = φ ( µ I ). Let X 1 , . . . , X n and Y 1 , . . . , Y m be i.i.d. observ ations from Q 1 and Q 2 , r e sp ectively . Let Q n and Q m be the empir ical distributions and µ n 1 and µ m 2 be the corres po nding s ample mean co ordinates. W e wan t to tes t H 0 : µ 1 I = µ 2 I = µ I , say , a g ainst H 1 : µ 1 I 6 = µ 2 I , wher e µ 1 I and µ 2 I 298 A. Bhattacharya and R. Bhattacharya are the tr ue intrinsic means of Q 1 and Q 2 , r e s pe ctively . T hen the test statistic used is T nm = ( n + m )( µ n 1 − µ m 2 ) ′ ˆ Σ − 1 ( µ n 1 − µ m 2 ) , (4.5) ˆ Σ = ( m + n )  1 n ˆ Λ − 1 1 ˆ Σ 1 ˆ Λ − 1 1 + 1 m ˆ Λ − 1 2 ˆ Σ 2 ˆ Λ − 1 2  , (4.6) (Λ 1 , Σ 1 ) a nd (Λ 2 , Σ 2 ) b eing the parameters in the asymptotic distr ibution o f √ n ( µ n 1 − µ ) and √ m ( µ m 2 − µ ), resp ectively , as defined in Theo rem 4.2. ( ˆ Λ 1 , ˆ Σ 1 ) and ( ˆ Λ 2 , ˆ Σ 2 ) are c onsistent sample estimates. In case n, m → ∞ such that n/ ( m + n ) → θ , 0 < θ < 1, then under the h ypo thesis of Theor em 4.2, assuming H 0 to b e true, (4.8) √ n + m ( µ n 1 − µ m 2 ) L − → N d (0 , 1 θ Λ − 1 1 Σ 1 Λ − 1 1 + 1 1 − θ Λ − 1 2 Σ 2 Λ − 1 2 ) . So T nm L − → X 2 d . W e reject H 0 at asymptotic level 1 − α if T nm > X 2 d (1 − α ). W e conclude with the test for the equality o f intrinsic v aria tions V 1 , V 2 of Q 1 and Q 2 . Under the h ypo thesis of Theorem 4.2, the test for H 0 : V 1 = V 2 , aga inst H 1 : V 1 6 = V 2 , is provided by the a symptotically Normal sta tistic T nm in (2.14), as describ ed at the end of Section 2 . 5. Exampl es In this section, we record the res ults of t w o-sample tests in tw o examples. Example 1 (Sc hizophrenic Children) . In this example from Bo o k stein [ 9 ], 13 la nd- marks a r e recor ded on a midsagittal tw o-dimensional slice from a Magnetic Res - onance brain scan of each of 14 sc hizophrenic children and 14 no rmal children. Figures 1 (a), (b) show the preshap es of the landmarks for the pa tient and normal samples along with the respec tive sample extrinsic mean pr eshap es. The s a mple preshap es ar e rotated appropr iately as to minimize their Euclidean distance from the mean preshap e. Figur e 2 shows the pre shap es of the normal and the patient sample extrins ic means along with the p o oled s ample mean. The v alues of the t w o-sample test statistics (3.17), (4.5 ) for testing equality of the mean sha pes , along with the p-v a lues are as follows. Extrinsic: T nm = 95 . 547 6, p-v alue = P ( X 2 22 > 95 . 547 6) = 3 . 8 × 10 − 11 . Int rinsic: T nm = 95 . 458 7, p-v alue = P ( X 2 22 > 95 . 4 587) = 3 . 97 × 10 − 11 . The extr insic sample v ariations for pa tient a nd normal samples are 0 . 010 7 and 0 . 0093 , re s pe c tively . The v alue of the tw o-sample test statistic (3.19) for testing equality of extrins ic v a riations is 0 . 94 61, a nd the p-v alue is 0 . 3 441. The v alue of the likelihoo d ratio test statistic, using the so-c alled offset normal shap e distribut ion (Dryden and Mardia [ 12 ], pp. 1 45–1 46) is − 2 log Λ = 43 . 124, p-v alue = P ( X 2 22 > 43 . 124 ) = 0 . 005. The corresp onding v alues of Goo da ll’s F- s tatistic and Bo okstein’s Monte Carlo test (Dry den and Mar dia [ 12 ], pp. 145–1 46) a re F 22 , 572 = 1 . 89 , p-v alue = P ( F 22 , 572 > 1 . 89 ) = 0 . 01 . The p-v alue for Bo o kstein’s test = 0 . 0 4. Example 2 (Gorilla Skulls) . T o test the difference in the sha pe s of skulls of male and female g orillas, eight landma r ks ar e chosen on the midline plane of the skulls o f 29 male and 30 female gorillas . W e use the data o f O’Higgins a nd Dryden repr o duced in Dryden and Mardia ([ 12 ], pp. 317 –318 ). The statistics (3.17) and (4.5) y ield the following v alues: Nonp ar ametric statist ics on manifolds 299 (a) (b) Fig 1 . (a) and (b) show 13 landmarks f or 14 normal and 14 schizophr enic p atients, resp e ct ively, along with t he me an shap es, * c orr esp ond to the me an landmarks; 1c shows the sample extrinsic me ans for the 2 gr oup s along with the p o ole d sample me an. Extrinsic: T nm = 392 . 6, p-v a lue = P ( X 2 12 > 392 . 6) < 10 − 16 . Int rinsic: T nm = 391 . 63 , p-v alue = P ( X 2 12 > 391 . 63) < 1 0 − 16 . The extrinsic sa mple v ariations for male and female samples are 0 . 0 0 5 and 0 . 0 038, resp ectively . The v alue of the tw o-sample tes t statistic (3.19 ) for tes ting equa lity o f extrinsic v aria tions is 0 . 92 3, and the p-v alue is 0 . 356. A parametric F-test (Dryden and Mardia [ 12 ], p. 154) yields F = 26 . 4 7, p-v alue = P ( F 12 , 46 > 2 6 . 47) = 0 . 0001. A par ametric (Normal) mo del for Bo okstein co o rdinates leads to the Hotelling’s T 2 test (Dryden and Mardia [ 12 ], pp. 170 –172 ) yields the p-v a lue 0 . 0001 . Ac kno wle dgment s. The autho r s g reatly appre c iate the kind and helpful sug - gestions by the e ditors and an a nonymous refer ee which led to a substantial im- prov emen t in exp os ition. 300 A. Bhattacharya and R. Bhattacharya Fig 2 . 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