On The Symmetrical Property Of Procrustes Measure Of Distance

J

,

....

ISSN 1311-8080

. \

0

International Journal of Pure and

Applied Mathematics

Vol. 99, No. 3, March 2015

Contents

K. Kannan: Amenable action and invariant mean

C . Promsakon: The partial order on category of semigroups and Endo-Cayley digraphs

K . Paykan, A. Moussavi: A generalization of bear rings

S. Noeiaghdam: Numerical solution of n-th order Fredholm integro-differential equations by integral mean value theorem method

D.W. Yoon: On translation surfaces with zero Gaussian curvature in H2 x R

D. Hac helfi, Y . Las kri , M.L. Sahari: Trust region with nonlinear conjugate gradient method

T. Bakhtiar, Siswadi: On the symmetrical property of Procrustes measure of distance

S.B. Belhaouari: The density of the process of coalescing random walks

M.A. Alqudah: Construction of Tchebyshev-ll weighted orthogonal polynomials on triangular

M.K. Bansal, R . Jain: Application of generalized differential transform method to fractional order Riccati differential equation and numerical result

S. Acharjee: Fixed point theorem in structure fuzzy metric space

H. Rahimpoor, A. Ebadian, M.E. Gordji, A . Zohri:

Common fixed point theorems in modular metric spaces

227 - 244 245 - 256

257 - 276

277 - 288

289 - 298

299 - 314

315 - 324

325 - 342

343 - 354

355 - 366

367 - 372

373 - 383

..,

.

ｾ＠

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International Journal of Pure and Applied Mathematics

Volume 99 No. 3 2015, 315-324

ISSN: 1311-8080 (printed version): ISS!': 1314-3395 (on-line version) url: http://www.ijpam.eu

doi: http://dx.doi.ol"g/10 l2732/ijpam.,·99i3.i

ON THE SYMMETRICAL PROPERTY OF PROCRUSTES MEASURE OF DISTANCE

Toni

Bakhtiar1

S,

Siswadi2

L 2Department of ).lathematics

Bogor Agricultural University

Dramaga Campus, Bogor. 16680. INDONESIA

ｾ＠

ijpam.eu

Abstract: ｾｉ｣｡ｳｵｲ･ｭ･ｮｴ＠ the degree of differe11c:e bctwl'C'n two matric:<>s by

using Procrustes analysis is preceded by a series of Euclidean si 111ila rit.v

trans-formations namely translation , rotation. and dilation, performed i11 rc:;pcctccl

order. for gai11i11g maximal matching. It b easy. by a counter cxainplc. to show

that Procrustes measure ､ｯ＼Ｎｾ＠ not obey the symmetrical property, something

s hould he owned by any distnne<.' function. In thi:-. paper we analytically proved

that normalization uvc>r co11figuratio11 mat ri<"es as au ct<l<lit io11al t ra11sfonnatio11

r<'::.ults in the satisfaction of the sy111metrical property hy Procrustc•:-. a11alysis .

\Ve also proved that normalizatio11 can be undertaken prior to or aftl'r rotatio11

to preserve symmetrical property. ).forc'Ovcr, we proved that Procru:-.t<'s

mea-sure can be expressed in term of sint:,'Ular value.'> of the matrix. We here very

much exploited the characteristic of full ｾｩＱＱｩＮＬＧＱ Ｑｬ ｡ｲ＠ ,·al11<' ｣ｬ｣｣ｯＱＱＱｰｯｾｩｴｩｯＱＱ＠ under

similarity trausformations.

AMS Subject Classification: OOA35. 62H20

Key Words: procrustc.·s atmlysis, simi larit y trn11sforuint io11. si11guh1r ntltH•

cleco111po:-.ition. sy111111rtrical property

1. Procrus tes Measure

316

T.

Bakhtiar. Siswadi

and scaling e ffects are filtered out from a n object. Two con figura tion

matri-｣ｾ＠ as representation of objects ha,·e the sam e s h ape if they can be tra nslat ed.

ro t ated, and dilated to each other so that they match exactly [3]. Procrus t es

analysis refers to a technique of comparing objects with different s hapes and

producing a m easure of the match. It eliminates possible ｩｮ ｣ｯ ｭｾｮ ･ｵｳｵｲ｡｢ ｩｬﾭ

ity or variables wi t hin the individual data sets and size difference:> between

data sets by employing data and configuration scalings i11 calculating the

dis-tance. rcspcd i\'cly. Thus, a series of Euclidean s imilarity tn.msfonnations. i.e ..

trans lation- rotation-dilation, is carried-out to b est match up to their m<:Lximal

agreement. Init ially. the difference d between mat rice:;; X

=

(l' 1j) and Y = (Yii)

in Rmxn can be quantified by

m 11

d( X , Y) =

L

L (.r,

_{1 -} y,J)2 .= tr (X - } ')'( X - Y). (1)

1=1 j=l

\Ve mean by translation a moving ｰｲｯ｣｣ｳｾ＠ o f all (•lements of m a trix within

fixed distance and into the sam e direction with res pt•(:t to its ce11t roid. \Ye can

minimize the distance between X and }r a ft er translation by coi11ciding their

centroids at the origin, without loss of generalit y. ｔｨｵｾＮ＠ matrices X a11d } ·

after optimal translation arc gi\·cn by

XT

=

X -

Cx.

)'1-

= }' -

C\ .

(2)

where C'x := ｾ

Ｑ

Ｑ

ＱＱＱ

ＱｾＬｘ＠ a ud C )· := ［ｨｬｭ ｬ ｾＬＬｙ＠ arc, ｲ｣ｳｰ･｣ｴｩｮ ｾ ｬｹＮ＠ cent roids of.\'

a 11d Y , with 1111 is them x 1 v<X'tor ofo11c:,.

fl otntiou is a procCfiS o f moving all clem ent s or matrix 1111cler the tix('d

rottttio11 a 11gll' without any changes in the poi11t-to-ce11troid clistanc<'s. R otatio n

011 YT over XT is carried-out by pos t-multiplyi11g Y:r by a rot at ion 111at rix

Q.

The minimum di:-;tancc after rotntiou is o btained by srlcct iug

Q

= VU'. wh<'re

UE \I' is the' full :-;ingular value decomposilio11 ( fSVD ) of

X4-}'i" [2. 5J.

It is

ohvions t hnt Q is orthogonal, i.e ..

QQ'

= Q' Q

=

I .

Dilation is a dat a scaling which :-;trl'lchcs or s hrinkng<.'!-i the point-t o-cC'ntroicl

distn11c<' inn configuration by multiplication of a fi x<'d scaling factor. Dilatiou

011 }i-Q owr .\'T is undertaken hy 111nltipl.\'i11g }1·Q ｨｾ ﾷ＠ a scalar r E IR. when·

tr

x.;.

FrQ · · · · I 1· I ' ' I } . Q ft 1·1 .

,. ｴｲＨｾＩﾷｲ＠ 1111111m1z111g t 1c c 1stancc Jetwccn · ' T nuc T a er< 1 at1011.

S i11nilt ｡ｮｃＧｯ ｮ ｳ ｬｾ ﾷＮ＠ it is pron'<l in

11 J

that a scric:-. o f si111ilarit \' t rnnsformations i11

the ord c•r of t ranslation-rotatiou-dilation prO\'ide'!'i t lw ｬｯｷｃＧｾｉ＠ ｰｯｾｾｩｨｬ＼Ｇ＠ cl ｩｾｴ＠ amT

ON THE SV1Il\1ETRICAL PROPERTY OF ...

317

Theorem 1. Give11 two matrices X aud Y in R111

x".

_{tlie Procrustes}

measure p bct\\·ecn X and Y after optimal translatio11-rotation-dilntio11 process is pro\'idcd by

( , , 1,.) = . , ,, , , _ tr2 Xl-YTQ

PA. t1 ."'T""T }"''}'

tr T T (-1)

P roo f For complet e proof, see

!I].

2. Singular Value Decomposition

Ju this section we provide a s hort review on the :;;ing ular value <lecornpositio11.

which could be de:>cribed as the most useful tool i11 matrix algebra because th i:-.

factorization ca11 be applied to a11y real or complex matrix. sq uan• or rectangu-la r.

Theorem 2

([-tj).

If A is an m x ll matrix of ra1Jk r :=; 111 i11{111.11}, t/1£'11 t/Jcrc exists 01t /Jogomi/ m x rn nwtrix U (rcspcctfrcly 111Jitmy). ur t l wgo1wl 11 x"

1w1trix ll · Ｈ Ｑ ﾷｾｰ･｣ｴｩ｜Ｇｃｬｹ＠ 1111itl1ry}, am/ m x 11 1JWtrix .S'

=-

(s,J) s11d1 tlwt

A = US !\''.

(5)

Supp ost• t hat <'igcnvalue:-; of AA' can be urckrecl a:-.

>.,

2

..\2

2 · · ·

ｾ＠

>.r

>

>-r+I

- · · ·

=

>.,,,

-

0 with t he <'Orrcspo11di11g l'igl'H\'l'Ctors

11

₁ (i

ｾ＠

1, .... r.1·,..

I. ... ,m) and those of A'A arc>.,

2

>.2

2 · · · 2

>.,.

>

>.,.+J

- · · ·

=

>.,,

-

0 wi th

thl• corrcspo11d i11g cigcn vC'ctors 11•, (i = l. .... r. r

+

I , .. .. 11 ). To co111p ly wit It

(5). it is rcquirl'cl thnt. ci t hc•r

Au·; A'1t,

u, · "- - - or w, ·.:....

-.c.; 11 811 (6)

11111st IH' sath;fic•d. with s., =

4;.

The Su is call<·d t he' si11g1dar nil11e

or

A.

which is d<'fii1Pcl to h<' t hC' positi\'C' :squnr<' root oft h<' rn11k<'d <'ig<•nvalt 1t• of AA'

or A'A ( no t e thnt AA' a nd A'A pus:-.es the SHiil<' sl'I of positin• 1•igt>m·al11<•s).

Tltc11 it ca11 hl' written that

318 T. Bakhtiar. Siswadi

where ilr = diag(Siii ... , Srr) is an r x r diagonal matrix and 0 is a matrix

of zeros with compatible dimension. Note that we baYe S in the form of (8)

wheu r

<

m and r

<

n.

If

r

=

m

=

n then it reduces to S

=

ｾｲﾷ＠

If

r

=

m

<

n then S = (6. 0). and if r

=

n

<

m then S

=

(6. O)'. Since

u, (i = 1. ... , r) and

u.·

₁

(i

= 1. ....

r)

are. respectively.

･ｩｧ･ｵｮｾ｣ｴｯｲｳ＠

correspond

to positive cigc1walucs of

AA'

and

A' A.

theu

U

and

n·

should be constructed

by orthonormal columns. In fact. if the number of orthonormal columns is

less t h an 111 or 11 then we should expand this set to orthononual basb.

If

zero

eigenvalues are included a long with their corresponding eigenn,'Ctors. then we

have an FSVD of matrix

A

as given by Theorem 2.

As an illustrative example of FSVD let consider a -1

x

3 matrix A of rank 2:

(3 -1]

A=

ｾ＠

ｾ＠

.

ｾ ｬ ｡ｴｲｩｸ＠

A' A posscsse:> eigenvalues .A1 = 12 and

>.2

= 10 with corrcspo11cli11g

eigenvectors construct lhe followi n g H' matrix:

ff =

[l/

.J2.

- 1/

v12l

1/.J'i. l/v'2

j'

\Yhilc. matrix

.4A'

has cige11rnlues A1 = 12.

>.2

= 10. and >.:1 - 0. According to

(6). we obtaiu

(

1]

1 1 '

'Ill=

JG

2

(-2]

1 0 .

IL2 =

j5

5

13y expanding {u₁• u2} t o an ortltouormal basis in Ra and hy cxploitiug

Grn111-Schmidt orthonormalizntion procedure

[-tJ

we get

1 -5 .

( 1 ]

U3

=

J3(}

2

Thus, the corrc::.pon<ling cigcuYcctors of AA' form tlw followi11g C.: matrix:

u

=

l/./G

o

-5/-ISO .

[

1/./6 - 2//5 1/J30]

2/./G 1//5 2/JJf>

Obviously. W<' ran avoid ba.'iiS cxpm1sion and Grn111-Scl1111idt orthonorurnlization

ｨｾﾷ＠

initially starting with AA'. 13ut in

ｴｨｩｾ＠

ｷ｡ｾ ﾷ＠

wt-haw to

ｾｯｨｔ＠

th<' t hi rel or<l<•r

... ;_,,

ON T HE SYMMETRICAL PROPERTY OF ...

319

F\trthcr, it is easy to verify that both U and W arc orthonormal, i.e .. UU' =

U'U = [3 and TVl·V' = H"W = /2. Since m

=

4, n = 3. and r = 2. then the

matrix Sis provided

by

S=

[

vTI

ｾ＠

v;;o .

0 ]

Alternatively. the FSVD wit h economic ::.ize. i.e .. reduced order FSVD. can be

obtained by removing columns a n d rows relate to zero eigenvalues. i.e ..

-

｛ｬＯｾ＠

-

2

/J

rm

0

]

[l/-/2

-l/V2]'

A -

ＲＯｾ＠

I/y'6

0

0

v1Q

1/-/2 1/ -/2

J

l/Vs

3 . Symme trical P rop erty

A distance is

a

fu 11ction which associates to any pair of vector:, a n.•al

nonucga-tivc n umber. The notion of distance is essential becam;e many statistical

tech-nique::;. such as principal component

｡ｮ｡ｬｹｾ｢Ｎ＠

mult idime11sio11al scaling analysis.

and cor respondcu ce analysis, arc equiva lent to t he analysis of a sp0ci fic: dista11cC'

table. A distance function

ｰｯＺＺｩｍＮＡｓｓｃｾ＠

the followi ng prop<?rtics

HJ.

Defi11itio11 3. For any vectors .r. y. m1cl ::: i11 V(.'Ctor spncC' §. a funct ion d

is a distanct• function 011 S if the following prop<'rti<'S Hr<' satisfi<-<l:

1. d(x,

y)

2".

0,

2. d(:i:.y)

=

0 if and only if .r

=

!J. 3. d(:r. y) = d(y, :r),

·L d(.r. z) ｾ＠ d(.r, y)

+

d(y, z).

The concept of distance over VC'ctors might lie adopted for cak11lati11g tlte

d<'gr<'<' of diff<'r<'11<T• of matri<·<·s of tit<' SHiil<' di111C'11sion . For i11i:;tm1ce. to

｡ｵｾﾷ＠

matrices X = (:r,j) a11d Y .;... (y,j) i11

lll"'"'".

we• 111ay d<>fi11<'cl th<' cliff<'rC'W'<'

bct \vecu t h C'111 a . .., (1). as

｣ｭｨｲ｡ｾｸﾷ｣ｬ＠

h.'· Procrusl<•:-. measure witlii11 t

ranslation-rotalion-dilation

ｰｲｯ｣･ｾＮＮＬＮ＠

How('\·C'r. it is l'a:-;y to sho"' that Procrustes lll<'a:-.11rc•

do<'s not comply wit It distann• propc·rti0s provided i11 Dd1nit ion 3.

By

taki11g

two diff<'rf'nt matri<"<'S

[

:3

o']

.\ =

ｾ＠ ｾ＠

.

[

j

r.]

)' =

ｾ＠ ｾ＠

.

320 T. Bakhtiar. Siswadi

from (4) we have p(X, Y) = 0. which re,·eals a violation of the second property.

This fact, however, provides another kind of idiosyncrasies of matrix algebra.

in addition to the well-known fact that

AB

= 0 can happen although neither

A

nor

B

is itself

a

zero matrix. Another pair of matrices

[

-3 OJ

x

=

ｾ＠ ｾ＠

.

y

= _[3 2 5 3

l

.

4 -5

(11)

produces µ(X, Y)

=

3.t.67 and p(}·. X) - -!7.13. which show::; tlu\t in general

Procruste:; measure disobeys the third property. i.e .. the symmetrical property. This paper deals with the embedding of the symmetrical property of Pro-crustes measure. Its primary objecti,·e i::; to ascertain that an additional

trarn;-formation, namely a normalization. can set the symmetrical property i11to

Pro-cru::;tes m easure.

In

fact. normalization is already considered

by

::\IATLAn®

built-i11 functio11 pro crust es (X, Y). Pre-scaling to normal matric(>s 1-trl' abo

employed b.r

13]

iu calculating Procrustc::; dbtancc by using shape anulysi:;

ap-proach. \\'e here provide the proof of the !>ymrnctrical properly and discuss

the effect of n ormalization place in orde r of thc transformation ::.t'C)uencL'S. \Ye greatly exploit the characteristic of FSVD under similarity transformations.

Let given two data matrices

X

and } · with the same number of objects. If

the number of variables of both matrices is not the same. we may place 11 1111mber

of columns of zeros either in the last or ＱＱＱＱｾ ﾷ Ｌ｜ｨ｣ｲ･＠ such that th(• di11w11sio11 of

matrices arc the same

IGJ.

Thus. without loss of generality W(' assume that

X

aud Y <UC in the snnw space. i.e ..

X. }'

E JR111_x11 •

In thb analysis, ,..,·c mean by normali:wt ion making the nor111 of 111111 rix oil<'

by dividing it by its FrobC'nius norm. \\'c i11troclucc a nonnalization proc-edun.·

following a translation. Hence, th<· 11on11aliwtion o\·N

X1

and

l'T

prodncc-s

RT =

ax,..

f'1 = M;..

( 12)

where

1 1 1 1

(I :3)

Cl:- = .

,, .__

-llKrllF

Jtr ｘｾｘｔ＠

.- 1111-llF -

JtrY.p;-'

with

XT

and

>1-

arc given by (2). [11 {13) we• deuotc by

II· II,.-

fh(• Frohenius

norm of a matrix. hence Ｑ Ｑ Ｍｾｔｬｬｆ＠

=

lif:r llF

= I.

Theore1n 4. GfrC'11 t\m 1m1trkes X 1111</ } • i11 IR"''°'. tlw Pmcrnst<'s

mmsurc' JJ bct\1·('('11 .\" all(/}" after optinwl

trn11.-;/;itio11-11ornw/ix;itio11-rot:rtio11-dilatiu11 prncc;.;s olJ<'ys t/I(' sym111<' tric:1/ JH"OfJ<'rty rwd is f>W\·ic/NI 11.' ·

p(X. r·) =

J,p· .

.Y) _{I -}

(t.

a,.)'

(I .I )

ON THE snn.tETRICAL PROPERTY OF ...

321

where r and <1ii

(i

= l, ... , r) respectfrely arc the rank ruid singular ,•alucs of

X1-YT

or

Y.f.XT

with

XT

and YT are given in

(12).

Proof. By performing optimal rotation over XT and

YT

and subsequently

optimal dilation. then by considering (1) we obtain the Procrustes measure of

X and Y as follows

p(X. Y) =inf d(XT.CYTQi) = inftr(XT - cfi-Q1)'(XT - cfi-Qi) ,

c c

where Q1 = ｜ＧＱｕｾ＠ and U1E1 V{ is the FSVD of X1-f'T· \Ye can expand the above

equation to get a quadratic form in c:

v(X, Y) = inf(tr

x.;. ...

tT - 2ctr(X4-f'TQi)

+

c2 tr Y.f.YT ). ( 15)

c

By optimally selecting

tr(X1-f:rQ1) c

-- trf'.p;. {16)

a11cl sub::.tituting it back into quadratic form in (15) alongside with (12) ( 13)

we have

v(X,Y) = 1-tr2

(X4-YTQ

1 )

=

1-

tr2E1. ( 17)

Note that the last <'Xpr<•:-.sio11 is obtained by remembering that

QI

=

Vi

ｵｾ＠ and

Xl-YT

=

U₁E₁

V{.

\Ye now consider the opposite direction, where the matrix X

ｾｨｯｵｬ､＠ be· fitt<'cl by similarity transformations for obtainiug the bc:-.l matching

to the observed matrix

Y .

In other words. we aim to calculate the Procru:-.tl's

111cH.">ttrl' JJ(Y • .\' ). By following the similar tcd111iquc, we may shortly arriw at

11(> '. X)

=

I - tr2(Y.fXTQ2) = 1 - tr2 E2. {18)

where Q2 = ｜ＱＲｕｾ＠ and ｕＲｅＲ｜ｉｾ＠ is the FSVD of Y.fXT. Since for auy matrix A.

the S\K'<'lrnm of .4'

A

is the sn.mc <'I.'> that of

AA'.

tlien

r

trE1 = trE2

=La,,.

1- l

where• r and a₁₁ (i

=

1. .. . r) respl'cli,·el.'· ar<' llH• rank aud singular ,·nines

of

s.;.

f;.

or f:f.XT. ｔｨｵｾＮ＠ Wt' pron'd the symmetrical prope rty of Prun11:-.I ('S

322 _{T. Bakhtiar. Siswadi}

Remark 5. Procrustes measure p in (14) is basically calculated by using

( 1). Since tr A' A

2:

0 for auy matrix A, then it is gunrantet.>d that the

nonnega-tivity property of a distance function is satisfied. i.e .. p(X. Y)

2:

0. Furthermore.

suppose that there exists matrices X and }' such that p(X. Y)

>

l. Then we

hm·c 1 -

ｯ］ｾ］ｉ＠

0'11 )

2

>

1. Or equivaJent}y

Ｈｌｾ］ｬ＠

u,,)2

<

0. which is a

contradic-tion . It must be the case that p(X. Y ) $ I. ａｾ＠ a consequence of th(':)e facts. we

conclude 0 $ p(X. Y) $ I for any matrices X and l · of the ｾ ｡ｭ ｣＠ dimension.

R('Sult presented in Theorem -1 is in agreement with that of

l3J.

Remark 6. Note that the optimal scaling parameter c in (16) ·can further

be written as

r

c= L aii· r= l

Remark 7. Suppose that we have the following FSVD: Xl-l"T = U3E3

\':f.

Together with (17) we obtain

X1'f'T

=

U1E1V{ _{<=> abX1-Yr = U1E1} \ '{

Y' }'

[ ! E1 , .,

· T T - 1-b 1·

(I (19)

<=>

from which W(.' conclude

u.

= ｵｾｪﾷ＠

\'i

= i ;j. E1 "'- o/JEa. a11cl furtlwnnurc \\'(.'

hnw• t lac relation ｢｣ｴｷｴＮｾｮ＠ singular values

o,,

=

aM,,.

wlwrc

6;

₁ i:-. the si11b11ilar

mluc of XJ.. Yr . Tims we can al tern at i,·l'ly l'l'\\Tite ( I .J) a_..,

2 p(X. }') = p(}'. X) = l -

n

2

t/

1

(t

<>,,)

1- 1

Remark 8. 13y (H). if we lim·t• p(X. }.) - 0 llH·11 it 111c•a11:-. that .\and

} · lmv<' <'Xac-t ly tltc :-;wnc shape. Convcrsl'ly . .\ a1HI } • with 11( X. } ·) = l han·

<•11tird.v diff<'n·11t shapc. In this r<'gmd we· mil 11 a clissi111ilarity 111('as111·l'. If

we would likl· to liavc Procrustc:-; lllL•;u,urc fl with ｯｰｰｯｾｩｬ｣＠ i11t erpr<'tatio11. tlic·n

we· d<'tillP p(X. Y) :=I -11(X. } ')

=

0:::·""

₁ a,,)2. It 111ea11:-; that the biggl'r the'

mC'ffidC'11t p , th<.• closer the similarity of 111<1trircs X a11cl }'. Thus we• <'all 11 a

si111il11rity HINlSt1rc' which tall conrluct ru; a good11<'ss-of-fit of tllC' hc·:-.t 111atd1i11g.

4. Whe n Should We Normalize?

In the prcvio11:-. ｾ｣ｲｴｩｯｮ＠ we have ;rnalyzC'd the inrorporat ing 11un11alizntion as

an additional similarity transformation into tlw proct•<l11n•. It i:-. :-.how11 that

ON THE SYMMETRICAL PROPERTY OF ...

323

by normalizing the matrices after translation, the symmetrical property is as-serted by Procrustes measure. In this section we examine the situation where normalization is performed sequentially different. As an illustrative example.

Jet consider again matrices X and Y in (11). Table below provides Procrustes

measure under different sequence of transformations. e.g .. we mean by T-N-R-D the degree o f difference between matrices is calculated after translation {T), normalization (N), rotation (R), and dilation (D). respectively.

No Transformations p(X, Y) p(Y, X)

1 N-T-R-D 0.5174 0.5355

2 T-N-R-D 0.8125 0.8125

3

T-R-N-D 0.8125 0.8125

4 T-R-D-N 1.1340 1.1340

It is shown that normalizing prior to rotation (T-N-R-D) or after rota-tion (T-R-N-D) provides a consistent result and guarantees the symmetrical property. It can generally be explained as follows. After translation a nd

ro-tation we have XT and

YTQ3,

where

Q3

= V3U£ and U3E3 V3 is the FSVD

of Xl-YT. Normalizing both matrices provides XT

=

aXT as in (12) and

ZT

:=

ＱＱ

ｾＯｑｱｾ

ｉｆ＠

= bYTQ3. The procrustcs measure after optimal dilatio11 is

achieved by selecting c = tr

ｾｲＺｔ＠

with

lr T T

p(X, Y) = 1 - tr2 Xl-ZT = 1 - tr2(abE3) = 1 - tr2 E1, {20)

where the last expression is came from (19). Equation (20), however , proves that

Procrustes measures under T-N-R-D and T-R- N-D are the same. ｏ｢ｶｩｯｵｳＡｾ ＧＮ＠

we can a lso prove the symmetrical property similarly.

From the table we can also see that when normalization is applied in the first stngc. i.e .. we consider N-T-R-D. then the symmetrical property is generally not satisfied. even though this sequence may provide a srnalll'r mcru;urc. While, when we put normalization in the last stage, i.e., we consider T-fl-D-N, then we get a diffen•nt magnitude of measure with the symmetri<'al property convinced.

However, the measure is 11ot always staying in

!O, 1J

interval, somethi ng which

is not expected for a goodness of fit coefficient.. l n fact. Procrustes measure under T-R-D-N scheme is given by

T. Bakhtiar, Siswadi

5. Conclusion

this paper we have shortly reviewed the optimal similarity transformations of

Procrustes analysis. It has been shown that

by

performing translation, rotation,

and dilation consecutively over configuration matrices. then their differenC"e is

minimized. However, by simple example it has also been re\'ealed that

Pro-crustes measure violates the symmetrical property. To overcome the problem,

we

propose the inclusion of normalization, that is a di\'ision by its Frobenius

norm, as an additional similarity transformation. We analytically proved that

Procrustes measure can assert the symmetrical property

as

long

as

normal-ization procedure is accomplished prior to or after rotation.

By

this we also

guarantee that the measure belongs to the interval of

[O, l]. In

particular. we

have shown that the goodness-of-fit between X and Y can be expressed as the

squared sum of singular values of

X-'rYT.

R eferen ces

11)

T. Bakhtiar, Siswadi, Orthogonal procrustcs analysis: its transformation

arrangement a11d minimal distance,

Int.

J. App. Math. Stat., 20, No. ｾＱＱＱ＠

(2011), Hi-24, doi: 10.12988/ams.

[2}

J.M.F. ten Berge, Orthogornil procrnst<'S rotation for two or more matrices.

Psychometrika, 42, No. 2 ( 1977). 2G7- 27G, doi: 10.1007 /Df'0229-1053.

l3J

I.L. Dryden, K.V. ｾｬ｡ｲ､ｩ｡Ｌ＠ Sta'1st1cal Shape Analysis, John Wiley

&

Sons,

New York (1998).

l4J

J.R. Schott, Matrix Analysis for Statistics, John Wiley & Sons, New Jersey

(2005).

!SJ

R. Sibson, Studies in the robustness of multidimensional scaling:

pro-crustcs statistics, J.

R. Statist. Soc. B,

40 , No. 2 {1978), 234-238.

[6]

Siswadi,

T.

Dakhtiar.

It

ｾｬ｡ｨ｡ｲｳｩ Ｌ＠ Procrustes analysis a nd the good

ncss-of-fit of biplots: some thoughts and findings. App. Math. Sci .. 6. No. 72

(2012). 3579-3590. doi: 10.12!:>88/am:-..

I

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D.H. Ackeley, G.E. Hilton, T.J. Sejnovski, A learning algorithm for Bolzmann machine.

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