Nonparametric Confidence Regions for Veronese-Whitney Means and Antimeans on Planar Kendall Shape Spaces
NNonparametric Confidence Regions for Veronese-Whitney Meansand Antimeans on Planar Kendall Shape Spaces
Yunfan Wang and Vic PatrangenaruOctober 10, 2018
To date, Object Data Analysis (ODA) is the most inclusive type of data analysis, as far as sample metric spaces areconcerned. Early examples of object spaces were spaces of directions (see Watson(1983) [21]), direct similarity shapespaces (see Kendall(1984)[10]), axial spaces (see Beran and Fisher(1998)[2], Fisher et al.(1996) [7]), Stiefel manifolds(see Hendriks and Landsman(1998)[9]). In the infinite dimensional case, ODA leads to a nonlinear extension offunctional data analysis (see Patrangenaru and Ellingson (2015)[15]).Fr´echet (1948)[8] noticed that for higher complexity data, such as the shape of a random contour, numbers orvectors do not provide a meaningful representation. To investigate these kind of data he introduced the notion of elements , which are nowadays called objects; as an example he mentioned that “the shape of an egg randomly takenfrom a basket of eggs” may be viewed as a random object. Fr´echet’s visionary concepts, were nevertheless hard tohandle computationally during his time. It took many decades, until such data became the bread and butter of moderndata analysis. Nowadays, various types of shapes of configurations extracted from digital images are representedas points on projective shape spaces (see Mardia and Patrangenaru (2005)[14], Patrangenaru et al.(2010)[16]), onaffine shape spaces(see Patrangenaru and Mardia(2003)[17], Sugathadasa(2006) [20]), or on Kendall shape spaces(see Kendall(1984) [10], Dryden and Mardia(2016)[6]). To analyze the mean and variance of the random object X on1 a r X i v : . [ s t a t . O T ] O c t smooth object space M with a metric ρ , Fr´echet defined what we call now the Fr´echet function given by(1.1) F ( p ) = E ( ρ ( p, x )) , and if ( M , ρ ) is complete, the minimizers of the Fr´echet function form the Fr´echet mean set . In general, if ρ = ρ g isthe geodesic distance associated with a Riemannian structure g on a manifold M , there are no necessary and sufficientconditions for the existence of a unique minimizer of F in (1.1) (see eg Patrangenaru and Ellingson (2015)[15], ch.4),therefore, with the possible exception of complete flat Riemannian manifolds, it is advisable to consider only the casewhen ρ is the “chord” distance on M induced by the Euclidean distance in R N via an embedding j : M → R N , andthe Fr´echet function becomes(1.2) F ( p ) = (cid:90) M (cid:107) j ( x ) − j ( p ) (cid:107) Q ( dx ) , where Q = P X is the probability measure on M , associated with X. Also, given X , . . . , X n i.i.d.r.o.’s from Q ,their extrinsic sample mean (set) is the extrinsic mean (set) of the empirical distribution ˆ Q n = n (cid:80) ni =1 δ X i (see egPatrangenaru and Ellingson(2015)[15], chapter 4).In this paper we will assume in addition that ( M , ρ ) is a compact metric space, therefore the Fr´echet function isbounded, and its extreme values are attained two set of points on M . It makes sense to also consider as locationparameter for X, the extrinsic antimean set , set of maximizers of the Fr´echet function in (1.2) (see eg Patrangenaru,Guo and Yao (2016)[18]). In case the extrinsic antimean set has one point only, that point is called extrinsic antimean of X, and is labeled αµ j,E ( Q ) , or simply αµ E , when j and Q are known.In this paper after a brief revision of Veronese-Whitney means (VW means) in Section 2, which are extrinsic meanson real and complex projective spaces, relative to the Veronese-Whitney embeddings, we give two examples of sampleVW means computations on Kendall shape spaces. In Section 3 we derive large sample and pivotal nonparametricbootstrap confidence regions for VW antimeans, using VW anticovariance matrices, and their sample counterparts.2 VW antimeans on C P k − Planar direct similarity shapes of k -ads ( set of k labeled points at least two of which are distinct) in the Euclideanspace, were introduced by D. G. Kendall (1984)[10], who showed that in the 2D case, these shapes can be representedas points on a complex projective space C P k − . A standard shape analysis method, due to Kent(1992)[11], consistsin using the so called Veronese-Whitney (VW) embedding of C P k − in the space of ( k − × ( k − self adjointcomplex matrices, to represent shape data in an Euclidean space. This VW embedding j : C P k − → S ( k − , C ) ,where S ( k − , C ) is the space of ( k − × ( k − Hermitian matrices, is given by(2.1) j ([ z ]) = zz ∗ , z ∗ z = 1 . This embedding is a SU ( k − equivariant embedding, where SU ( k − is the special unitary group ( k − × ( k − matrices of determinant 1, since j ([ Az ]) = Aj ([ z ]) A ∗ , , ∀ A ∈ SU ( k − . The corresponding extrinsic mean (set) ofa random shape X on C P k − is called the VW mean (set) (See Patrangenaru and Ellingson (2015), ch. 3 [15]), andthe
VW mean , when it exists, and is labeled µ V W ( X ) , µ V W or simply µ E . The corresponding extrinsic antimean (set)of a random shape X , is called the VW antimean (set) and is labeled αµ V W ( X ) , αµ V W or αµ E . We have the following theorem for VW antimeans associated with the embedding (2.1).
THEOREM 2.1.
Let Q be a probability distribution on C P k − and let { [ Z r ] , (cid:107) Z r (cid:107) = 1 r =1 ,...,n } be i.i.d.r.o.’s from Q . ( a ) Q is VW nonfocal iff λ , the smallest eigenvalue of E [ Z Z ∗ ] is simple and in this case αµ E Q = [ ν ] , where ν is an eigenvector of E [ Z Z ∗ ] corresponding to λ , with (cid:107) ν (cid:107) = 1 . ( b ) The sample VW antimean αX E = [ m ] , where m is an eigenvector of norm 1 of J = n (cid:80) ni =1 z i z ∗ i , (cid:107) z i (cid:107) = 1 , i = 1 , . . . , n , corresponding to the smallest eigenvalueof J , provided this eigenvalue has multiplicity one. Proof. (a). The squared distance on the space S ( k − , C ) of Hermitian matrices is d ( A, B ) =
T r (( A − B )( A − B ) ∗ ) = T r (( A − B ) ) . A random object X = [ U ] on C P k − , with U ∗ U = 1 , has Fr´echet function(2.2) F X ([ u ]) = E ( T r ( U U ∗ − uu ∗ ) ) , u ∗ u = 1 . The matrix A = E ( T r ( U U ∗ ) is positive semidefinite, having the eigenvalues λ ≥ λ ≥ . . . ≥ , λ k − ≥ , andcan be represented as A = B Λ B ∗ , where B ∈ SU ( k − and Λ =
Diagonal ( λ ≥ λ ≥ . . . ≥ , λ k − ) . From32.2), we get F X ([ u ]) = T r ( A ) + 1 − T r ( u ∗ Au ) , thus F X is maximized iff [ u ] → T r ( u ∗ Au ) is minimized, or [ v ] → T r ( v ∗ Λ v is minimized, where v = Bu.
Note that v ∗ v = 1 , and if v T = ( v . . . v k − ) , then T r ( v ∗ Λ v ) = (cid:80) k − a =1 λ a | v a | ≥ λ k − = F X ([ u k − ]) , where u k − is an eigenvector of A corresponding to the eigenvalue λ k − . Part(b) follows, by taking the empirical distribution, with a matrix corresponding to the population expectation A in Part(a) being given by(2.3) J = ˆ A := n − n (cid:88) r =1 Z r Z ∗ r . We ran a simulation using an example of the VW embedding of a complex projective space ( a Kendall shape space)to compare VW means and VW antimeans for a data set of landmark configuration. In this context we ran a nonpara-metric bootstrap for sample VW means and sample VW antimeans. The objective of our simulations was to see if thebootstrap distributions of the sample VW means (respectively sample VW antimeans) is concentrated or not. For thissimulations, the data represents coordinates of k = 11 landmarks, and it has N = 100 observations. The data aredisplayed in figure 1. Note that the corresponding shape variable is valued in C P (real dimension = 18).Figure 1: Simulated centered and scaled landmarkconfigurations - affine coordinate representation Figure 2: Simulated and Location removed landmarkconfigurationsOur study is on Kendall shape spaces, slightly more general than just using Bookstein coordinates (see Bookstein41997)[5]) on this shape manifold. A useful tool for “removing location” of a k -ad, is the multiplication by a Helmertsub-matrix H, consisting in the last ( k − × k rows of a Helmert matrix. The full Helmert matrix HF, commonlyused in Statistics, is a square k × k orthogonal matrix with its first row equal to / √ k Tk , having the remainingrows orthogonal to the first row, with an increasing number of nonzero entries, as in (2.4) . We drop the first rowof HF so that the resulting matrix H does not depend on the original location of the configuration (see Dryden andMardia(2016)[6]). The j th row of the Helmert sub-matrix H is given by(2.4) ( h j , · · · , h j , − jh j , , · · · , , h j = { j ( j + 1) } − / . To compute the sample VW mean or the sample VW antimean, we multiply by the Helmert sub-matrix H to“remove location” of the original data. The Helmerized data after having removed location, is displayed in Figure 2.The figure 3 is a representative (icon) of the sample VW mean of the coordinates of landmarks of the mean shape afterremoving location. One may notice that the configurations in Figure 2 and Figure 3 look fairly similar, and the iconof the VW mean configuration is close, up to a rotation and scaling, to the icons of the sampled configurations. WeFigure 3: Icon of Sample VW mean shape of simu-lated landmark configurations Figure 4: Distribution of sample VW means for boot-strap resamplescomputed the nonpivotal bootstrap distribution of the sample VW means in MATLAB, that we ran for 500 randomresamples with repetition. An icon of the spherical representation of the bootstrap distribution of the sample VW meansis displayed in Figure 3. Note that the distribution of sample VW means bootstrap resamples is very concentrated5round the sample VW means. As for the sample VW antimean shape, its representative is shown in Figure 5. Therelative location of the landmarks in the icon of the sample VW antimean shape should look very different, whencompared with the original landmark configuration, after the registration process, and indeed it does (see Figure 5).Figure 5: Icon of sample VW antimean shape of sim-ulated landmark data Figure 6: Distribution of icons of sample VW an-timeans for bootstrap resamples.We computed the nonpivotal bootstrap sample VW antimeans distribution using MATLAB, that we ran on 500random resamples. Coordinates of the bootstrap distribution of the icons of the sample VW antimeans are displayedin Figure 6. Note that the distribution of the landmark configuration for icons of the bootstrap sample VW antimeansare not too concentrated around the sample VW antimean; nonetheless they are similarly positioned.From Theorem 2.1, in our simulation example, we know that the sample VW antimean is represented by aneigenvector of norm 1 of J = n (cid:80) ni =1 z i z ∗ i , (cid:107) z i (cid:107) = 1 , i = 1 , . . . , n , corresponding to the smallest eigenvalue of J,where [ z i ] ∈ C P are obtained by applying the submatrix of the last 10 rows of the Helmert matrix (see Mardia et al[13], p. 461) to the centered normalized data point x i + y i ∈ C . The smallest eigevalue of J is very close to zero,since data is fairly concentrated, explaining the pattern in Figure 6.6 .2 Application We are interested to determine how concentrated is the bootstrap distribution of the sample VW antimeans around thesample VW antimean, in the case of shapes of landmark configurations extracted from medical imaging outputs. Ourdata consists of shapes for a group of eighth midface anatomical landmarks labeled X-rays of skulls of eight year oldand fourteen year-old North American children(72 boys and 52 girls), known as the
University School data . The dataset represents coordinates of landmarks, whose names and position on the skull are given in Bookstein ((1997)[5]). InBhattacharya and Patrangenaru(2005) [4] only part of this data set ( boys only) was used. The registered coordinatesare displayed in Figure 7. The shape variable is valued in a Kendall space of planar octads, C P ( real dimension = 12). Figure 7: The coordinates of mid-sagital landmarkconfigurations of midfaces of children skulls Figure 8: The coordinates of configurations in Figure7 after location was removedIn our application, the data is registered using a Helmert sub-matrix H in equation (2.4), with k = 8 , and isdisplayed in Figure 8. In Figure 9 is displayed an icon of the sample VW mean of the Helmertized data, in a sphericalrepresentation. Like with the simulated data, one may notice that with VW mean icon, has a fairly close shape to theshapes of sampled configuration.Next, we computed the nonparametric bootstrap distribution of the sample VW means shapes in MATLAB, thatwe ran for 500 random resamples. An icon of the Helmertized spherical representation of the bootstrap distribution of7igure 9: Icon of Helmertized sample VW meanshape of midface cranial landmark configurations Figure 10: Distribution of icons of Helmertized boot-strap sample VW meansthe sample VW means is displayed in Figure 10. Note that the bootstrap distribution of the sample VW means is veryconcentrated around the sample VW means, as theoretically predicted.As for the sample VW antimean, its representative is shown in Figure 11. Since the sample VW antimean ison average far from the shape data, it is not surprising that the relative location of the landmarks in the sample VWantimean icon looks quite different from the one in the configurations in the original shape data.We computed the nonparametric bootstrap distribution using MATLAB, that we ran again for 500 random resam-ples. A spherical representation of the bootstrap distribution of the sample VW antimeans in Helmetrized coordinates isdisplayed in Figure 12. Here again, the icons of configurations for bootstrap distribution of the sample VW antimeanshave a similar look with the one of the VW sample antimean, however is that concentrated around the registered iconof the sample VW antimean, partially due to computational rounding errors for eigenvectors associated with the small-est eigenvalue of J ∗ . The standard affine embedding: C k − → C P k − is ( z , · · · , z k − ) → [ z : · · · : z k − : 1] ,leads to the notion of affine coordinates of a projective point(2.5) p = [ z : · · · : z k − ] , z k − (cid:54) = 0 ( w , w , · · · , w k − ) = ( z z k − , · · · , z k − z k − ) . Using simultaneous complex confidence intervals (See Bhattacharya and Patrangenaru 2005 [4]) for the affine coordi-nates of the VW antimean, we obtain the following results: w : [-0.1804 - 0.1808i 0.0549 + 0.1365i], w : [0.4913- 0.2301i 0.6136 - 0.0747i], w : [0.4455 + 0.0385i 0.5885 + 0.2288i], w : [0.1344 - 0.1923i 0.2346 - 0.0748i], w :[0.2376 - 0.4823i 0.5682 - 0.1533i], w : [-0.2752 + 0.2558i 0.1936 + 0.8011i]. In this section we will discuss the asymptotic distribution of sample antimeans in axial data analysis and in planarshape analysis, after a review of a Central Limit Theorem for extrinsic sample antimeans.9 .1 Central Limit Theorem for Extrinsic Sample Antimeans
In preparation, we are an using the large sample distribution for extrinsic sample antimeans given in Patrangenaru etal (2016 [18]).Assume j is an embedding of a d -dimensional manifold M such that j ( M ) is closed in R k , and Q = P X is a αj -nonfocal probability measure on M such that j ( Q ) has finite moments of order 2. Let µ and Σ be the mean andcovariance matrix of j ( Q ) regarded as a probability measure on R k . Let F be the set of αj -focal points of j ( M ) , andlet P F,j : F c → j ( M ) be the farthest projection on j ( M ) . P F,j is differentiable at µ and has the differentiabilityclass of j ( M ) around any αj nonfocal point.A local frame field p → ( e ( p ) , . . . , e k ( p )) , defined on an open neighborhood U ⊆ R k is adapted to the embedding j if it is an orhonormal frame field and ∀ x ∈ j − ( U ) , e r ( j ( x )) = d x j ( f r ( x )) , r = 1 , . . . , d, where ( f , . . . , f d ) is alocal frame field on M . Let e , . . . , e k be the canonical basis of R k and assume ( e ( p ) , . . . , e k ( p )) is an adapted frame field around P F,j ( µ ) = j ( αµ E ) . Then d µ P F,j ( e b ) ∈ T P F,j ( µ ) j ( M ) is a linear combination of e ( P F,j ( µ )) , . . . , e d ( P F,j ( µ )) :(3.1) d µ P F,j ( e b ) = d (cid:88) a =1 ( d µ P F,j ( e b )) · e a ( P F,j ( µ )) e a ( P F,j ( µ )) By the delta method, n / ( P F,j ( j ( X )) − P F,j ( µ )) converges weakly to N k (0 k , α Σ µ ) , where j ( X ) = n (cid:80) ni =1 j ( X i ) and(3.2) α Σ µ = [ d (cid:88) a =1 d µ P F,j ( e b ) · e a ( P F,j ( µ )) e a ( P F,j ( µ ))] b =1 ,...,k × Σ[ d (cid:88) a =1 d µ P F,j ( e b ) · e a ( P F,j ( µ )) e a ( P F,j ( µ ))] Tb =1 ,...,k here Σ is the covariance matrix of j ( X ) w.r.t the canonical basis e , . . . , e k .The asymptotic distribution N k (0 k , α Σ µ ) is degenerate and the support of this distribution is on T P F,j j ( M ) , sincethe range of d µ P F,j is T P F,j ( µ ) j ( M ) . Note that d µ P F,j ( e b ) · e a ( P F,j ( µ )) = 0 for a = d + 1 , . . . , k .The tangential component tan ( v ) of v ∈ R k , w.r.t the basis e a ( P F,j ( µ )) ∈ T P F,j ( µ ) j ( M ) , a = 1 , . . . , d is givenby(3.3) tan ( v ) = [ e ( P F,j ( µ )) T v, . . . , e d ( P F,j ( µ )) T v ] T ( d αµ E j ) − ( tan ( P F,j (( j ( X ))) − P F,j ( µ ))) = (cid:80) da =1 X aj f a has the following covariancematrix w.r.t the basis f ( αµ E ) , . . . , f d ( αµ E ) :(3.4) α Σ j,E = e a ( P F,j ( µ )) T α Σ µ e b ( P F,j ( µ )) ≤ a,b ≤ d = [Σ d µ P F,j ( e b ) · e a ( P F,j ( µ ))] a =1 ,...,d Σ [Σ d µ P F,j ( e b ) · e a ( P F,j ( µ ))] Ta =1 ,...,d which is the anticovariance matrix of the random object X . Similarly, given i.i.d.r.o.’s X , · · · , X n from Q , we definethe sample anticovariance matrix aS j,E,n as the anticovariance matrix associated with the empirical distribution ˆ Q n . R P N − and C P k − We first consider the case when M = R P N − , the real projective space which can be identified with the sphere S N − = { x ∈ R N |(cid:107) x (cid:107) = 1 } with antipodal points identified(see Mardia and Jupp (2009)[12] ). Here the points in R N are regarded as N × vectors. R P N − can be identified with the quotient space S N − / { x, − x } ; it is a compacthomogeneous space, with the group SO ( N ) acting transitively on ( R P N − , ρ ), where the distance ρ on R P N − is induced by the chord distance on the space S ( N, R ) of symmetric N × N and the embedding j that is compatiblewith two transitive group actions of SO ( N ) on R P N − , respectively on j ( R P N − ) , that is(3.5) j ( T · [ x ]) = T ⊗ j ([ x ]) , ∀ T ∈ SO ( N ) , ∀ [ x ] ∈ R P N − where T · [ x ] = [ T x ] and T ⊗ A is given in (3.7) below.Such an embedding is said to be equivariant (See Kent 1992 [11]). The equivariant embedding of R P N − that wasused so far in the axial data analysis literature is the Veronese Whitney (VW) embedding j : R P N − → S + ( N, R ) ,that associates to an axis the matrix of the orthogonal projection on this axis (See Patrangenaru and Ellingson 2015[15], chapter 3)(3.6) j ([ x ]) = xx T , (cid:107) x (cid:107) = 1 Here S + ( N, R ) is the set of nonnegative definite symmetric N × N matrices, and in this case(3.7) T ⊗ A = T AT T , ∀ T ∈ SO ( N ) , ∀ A ∈ S + ( N, R ) EFINITION 3.1.
A random object [ X ] = Y on R P N − is α VW-nonfocal if it is αj -nonfocal w.r.t. the VW embed-ding in (3.6). Then we have the following proposition from Patrangenaru et al (2016) [19].
PROPOSITION 3.1.
A random object [ X ] = Y on R P N − , X T X = 1 is α VW-nonfocal iff the smallest eigenvalueof E ( XX T ) is positive and has multiplicity 1. Now we consider the anticovariance on R P N − . PROPOSITION 3.2.
Assume [ X r ], X Tr X r = 1 , r = 1 , . . . , n is a random sample from a αj -nonfocal probabilitymeasure Q on R P N − . And λ a , a = 1 , . . . , N , are eigenvalues of K := n (cid:80) nr =1 X r X Tr in increasing order and m a , a = 1 , . . . , N , are corresponding linearly independent unit eigenvectors. Then the sample VW anticovariancematrix aS j,E,n is given by (3.8) ( aS j,E,n ) ab = n − ( λ a − λ ) − ( λ b − λ ) − × (cid:88) r ( m a · X r )( m b · X r )( m · X r ) The proof is along the lines of a similar result from sample VW covariance on R P N − (see Bhattacharya andPatrangenaru (2003)[3]).As the embedding j is equivariant, w.l.o.g. we may assume that the farthest projection of sample mean j ( a ¯ X j,E ) = P F,j ( j ( X )) is a diagonal matrix, a ¯ X j,E = [ m ] = [ e ] and the other unit eigenvectors of j ( X ) = D are m a = e a , ∀ a = 2 , . . . , N . Based on this description of tangent space at [ x ] we evaluate d D P F,j . T [ x ] R P N − , one can selectthe orthonormal frame e a ( P F,j ( D )) = d [ e ] j ( e a ) in T P F,j ( D ) j ( R P N − ) . Note that S ( N, R ) has the orthobasis F ba , b ≤ a , where, for a < b , the matrix F ba has the positions ( a, b ) , ( b, a ) that are equal to − / , and all other entriesequal to zero. We note that also F ba = j ([ e a ]) . d D P F,j ( F ba ) = 0 , ∀ a, b, ≤ b < a ≤ N and d D P F,j ( F b ) = ( λ b − λ ) − e a ( P F,j ( D )) ; If λ a , a = 1 , . . . , N arethe eigenvalues of D in their increasing order. Then from this equation it follows that, if j ( X ) is a diagonal matrix D ,then the entry ( aS j,E,n ) ab is given by. ( aS j,E,n ) ab = n − ( λ b − λ ) − ( λ a − λ ) − n (cid:88) r =1 X ar X br ( X r ) j ( X ) to be a diagonal matrix and m a = e a Let T ([ ν ]) = n (cid:107) aS − E,n tan ( P F,j ( j ( X )) − P F,j ( E ( j ( Q )))) (cid:107) (see formula (27) from Patrangenaru et al 2016[18]). We can derive now the following theorem. Note that αµ j,E = [ ν ] , where ( ν a ) , a = 1 , . . . , N , are uniteigenvectors of E ( XX t ) = E ( j ( Q )) corresponding to eigenvalues in their increasing order. THEOREM 3.1.
Assume j is the Veronese-Whitney embedding of R P N − and let [ X r ], X Tr X r = 1 , r = 1 , . . . , n be a random sample from an αj -nonfocal distribution Q . Then T ([ ν ]) is given by T ([ ν ]) = nν t [( ν a ) a =2 ,...,N ] aS − E,n [( ν a ) a =2 ,...,N ] t ν, and, asymptotically, T ([ ν ]) has a χ N − distribution. Proof: Let ν , . . . , ν N be the orthobasis of the tangent space T [ ν ] R P N − . Based on that the VW embedding j isisometric and using the method of moving frame(See Bhattacharya and Patrangenaru 2005 [4]). Let e a ( P F,j ( ν E,j )) =( d [ ν ] j )( ν a ) be the first elements of the adapted moving frame. Then the a th tangential component of P F,j (( j ( X ))) − P F,j ( ν ) w.r.t. this basis of T P F,j ( E ( j ( Q ))) j ( R P N − ) equals up to a sign the a th component of m − ν w.r.t. theorthobasis ν , . . . , ν N in T [ ν ] R P N − , namely ν ta m .We say that a random object on C P k − is α VW-nonfocal if it is αj -nonfocal w.r.t. the embedding in (2.1)Similarly with Proposition 3.1, we get the following proposition. PROPOSITION 3.3.
The random object Y = [ Z ] on C P k − is α VW-nonfocal iff the smallest eigenvalue of E ZZ ∗ Z ∗ Z is positive and has multiplicity 1. Similar asymptotic results can be obtained for the large sample distribution of VW means of planar shapes, asfollowing. Recall that the planar shapes space M = Σ k of an ordered set of k points in C at least two of whichare distinct, can be identified in different ways with the complex projective space C P k − (see Bhattacharya andPatrangenaru (2003) [3] ,and Balan and Patrangenaru(2005) [1]). Here we regard C P k − as a set of equivalence classes C P k − = S k − /S where S k − is the space of complex vectors in C k − of norm 1, and the equivalence relation on S k − is by multiplication with scalars in S . The action of S on S k − is by multiplication of complex vectors withscalars in S (complex numbers of modulus 1). A complex vector z = ( z , z , . . . , z k − ) of norm 1 corresponding to13 given configuration of k landmarks, with the identification described in Bhattacharya and Patrangenaru ((2003) [3]),can be displayed in the Euclidean plane (complex line) with the superscripts as labels.A random variable X = [ Z ] , (cid:107) Z (cid:107) = 1 , valued in C P k − is αj -nonfocal if the smallest eigenvalue of E [ ZZ ∗ ] issimple, and then the VW-antimean of X is αµ j,E = [ ν ] , where ν ∈ C k − , (cid:107) ν (cid:107) = 1 , is an eigenvector correspondingto this eigenvalue. The sample VW-antimean α [ z ] j,E of a random sample [ z r ] = [( z r , · · · , z k − r )] , (cid:107) z r (cid:107) = 1 , r =1 , · · · , n, from such a nonfocal distribution exists with probability converging to 1 as n → ∞ , and is the same as thatgiven by(3.9) a [ z ] j,E = [ m ] , where m is the unit eigenvector of(3.10) K := 1 n n (cid:88) r =1 z r z ∗ r . corresponding to the smallest eigenvalue. PROPOSITION 3.4.
Assume X r = [ Z r ] , Z Tr Z r = 1 , r = 1 , . . . , n is a random sample from a αj -nonfocal prob-ability measure Q with a nondegenerate αj -extrinsic anticovariance matrix on C P k − . λ a , a = 2 , · · · , k − areeigenvalues of K in (3.10) in their increasing order and m a , a = 2 , · · · , k − are corresponding linearly independentunit eigenvectors. Then the VW-extrinsic sample anticovariance matrix aS E,n as a complex matrix has the entries (3.11) ( aS E,n ) ab = n − ( λ a − λ ) − ( λ b − λ ) − × n (cid:88) r =1 ( m a · Z r )( m b · Z r ) ∗ | m · Z r | The proof is based on the equivariance of the VW embedding. The embedding j : C P k − → S ( k − , C ) , wherethe action SU ( k − is a non-negative semi defined self-adjoint complex matrices(see Bhattacharya and Patrange-naru(2003) [3]). First we need to assume that K := n (cid:80) nr =1 z r z ∗ r is a diagonal matrix, the smallest eigenvaluecorresponding complex eigenvector of norm 1 of K is a simple root of the characteristic polynomial over C , with m = e . The tangent space T [ m ] j ( C P k − ) has an orthobasis m (cid:48) a = ie a , a = 2 , · · · , k − , where m a = e a are eigenvectorcorresponding to the largest eigenvalue. Here we define a path η z ( t ) = [ cos tm + sin tz ] , where z is orthogonal to14 ∈ C k − . T P F,j ( K ) j ( C P k − ) is generated by the vectors tangent to such paths η z ( t ) at t = 0 . Such a vector, hasthe form zm ∗ + m z ∗ , as a matrix in S ( k − , C ) .Thus we take z = m a , a = 2 , · · · , k − , or z = im a , a = 2 , · · · , k − based on the eigenvectors of K areorthogonal w.r.t. the complex scalar product. We norm these vectors to have unit length to obtain the orthonormalframe. e a ( P F,j ( K )) = d [ m ] j ( m a ) = 2 − / ( m a m ∗ + m m ∗ a ) ,e (cid:48) a ( P F,j ( K )) = d [ m ] j ( m a ) = i − / ( m a m ∗ + m m ∗ a ) . As we assume K is diagonal. In this case m a = e a , e a ( P F,j ( K )) = 2 − / E a and e (cid:48) a ( P F,j ( K )) = 2 − / F a ,where E ba has the positions ( a, b ) and ( b, a ) that are equal to 1 and all other entries zero, and F ba has all the positions ( a, b ) and ( b, a ) that are equal to i , respectively − i and other entries zero. That we have d K P F,j ( E ba ) = d K P F,j ( F ba ) =0 , ∀ < a ≤ b ≤ k − , and d K P F,j ( E a ) = ( λ a − λ ) − e a ( P F,j ( K )) ,d K P F,j ( F a ) = ( λ a − λ ) − e (cid:48) a ( P F,j ( K )) . We evaluate the extrinsic sample anticovariance matrix aS E,n in formula (25) in Patrangenaru et al (2016 [18]) usingthe real scalar product in S ( k − , C ) , namely, U · V = ReT r ( U V ∗ ) . Note that, d K P F,j ( E b ) · e a ( P F,j ( K )) = ( λ a − λ ) − δ ba ,d K P F,j ( E b ) · e (cid:48) a ( P F,j ( K )) = 0 and d K P F,j ( F b ) · e (cid:48) a ( P F,j ( K )) T = ( λ a − λ ) − δ ba ,d K P F,j ( F b ) · e a ( P F,j ( K )) = 0 Thus we may regard aS E,n as a complex matrix noting that in this case we get(3.12) ( aS E,n ) ab = n − ( λ a − λ ) − ( λ b − λ ) − n (cid:88) r =1 ( e a · Z r )( e b · Z r ) ∗ | m · Z r | K is diagonal. The general case follows by equivariance.Next we consider the statistic T ( a ¯ X E , αµ E ) = n (cid:107) ( aS E,n ) − / tan ( P F,j ( ¯ j ( X )) − P F,j ( µ j ( X ) ) (cid:107) given in Patrangenaru et al 2016 [18], in the our context of i.i.d.r.o objects on a complex projective space to get: THEOREM 3.2.
Let X r = [ Z r ] , Z Tr Z r = 1 , r = 1 , · · · , n, be a random sample from a VW- α nonfocal probabilitymeasure Q on C P k − . Then the random variable given by (3.13) T ([ m ] , [ ν ]) = n [( m · ν a ) a =2 , ··· ,k − ]( aS E,n ) − [( m · ν a ) a =2 , ··· ,k − ] ∗ has asymptotically a χ k − distribution. Proof . Since the VW embedding j is by definition isometric, and ( ν , · · · , ν k − , ν ∗ , · · · ν ∗ k − ) is an orthogonalbasis in the tangent space T [ ν ] C P k − , the first elements of the adapted orthogonal moving frame are e a ( P j ( µ )) =( d [ ν ] j )( ν a ) e ∗ a ( P j ( µ )) = ( d [ ν ] j )( ν ∗ a ) . Then the a th tangential component of P F,j ( j ([ m ])) − P F,j ( µ j ( X ) ) w.r.t. thisbasis of T P j ( µ ) C P k − equals up to a sign to the component of m − ν w.r.t. the orthobasis ν , · · · , ν k − in T [ ν ] C P k − ,which is ν ta m ; and the a ∗ th tangential components are given by ν ∗ at m , and together(in complex multiplication) theyyield the complex vector [( m · ν a ) a =2 , ··· ,k − ] . The result follows by taking [ m ] = P F,j ( ¯ j ( X )) = j ( a ¯ X E ) . We may derive the following large sample confidence regions for the VW-antimean shape
COROLLARY 3.1.
Assume X r = [ Z r ] , Z Tr Z r = 1 , r = 1 , · · · , n, is a random sample from a αj -nonfocalprobability measure Q on C P k − . An asymptotic (1 − β ) − confidence region for αµ jE ( Q ) = [ ν ] is given by R β ( X ) = { [ ν ] : T ([ m ] , [ ν ]) ≤ χ k − ,β } , where T ([ m ] , [ ν ]) is given in (3.13) . If Q has a nonzero absolutely continuouscomponent w.r.t. the volume measure on C P k − , then the coverage error of R α ( X ) is of order O ( n − ) . When the sample size is small, the coverage error could be quite large, and a bootstrap analogue of Theorem 3.2is preferred.
THEOREM 3.3.
Let X r = [ Z r ] , Z Tr Z r = 1 , r = 1 , · · · , n, be a random sample from a αV W -nonfocal distribution Q on C P k − , such that X r has a nonzero absolutely continuous component w.r.t. the volume measure on C P k − . f j is the VW embedding, and the restriction of the covariance matrix of j ( X ) to T [ ν ] j ( C P k − ) is nondegenerate,where αµ E ( Q ) = [ ν ] be the extrinsic antimean of Q . For a bootstrap resample { X ∗ r } r =1 . ··· ,n from the given sample,consider the matrix K ∗ := n − (cid:80) Z ∗ r Z ∗ r ∗ . Let ( m ∗ a ) a =1 , ··· ,k − be the unit complex eigenvectors, corresponding tothe eigenvalues ( m ∗ a ) a =1 , ··· ,k − in increasing order. Let ( aS E,n ) ∗ be the matrix obtained from aS E,n by substitutingall the entries with ∗ -entires. Then the bootstrap distribution function of (3.14) T ([ m ] ∗ , [ m ]) = n [( m ∗ · m a ) a =2 , ··· ,k − ]( aS ∗ E,n ) − [( m ∗ · m a ) a =2 , ··· ,k − ] ∗ approximates the true distribution function of T ([ m ] , [ ν ]) given in Theorem 3.2 with an error of order O P ( n − ) . References [1] V. Balan, and V. Patrangenaru (2005).
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