kobe university repository : kernel · terms of variations of between (shortly, b) and within...
TRANSCRIPT
-
Kobe University Repository : Kernel
タイトルTit le
多面断データの図示表現に関する研究(その2)(GraphicalRepresentat ion of Mult i Cross-sect ional Date : Part II)
著者Author(s) 磯貝, 恭史
掲載誌・巻号・ページCitat ion
神戸大学大学院海事科学研究科紀要 = Review of the Faculty ofMarit ime Sciences, Kobe University,06:7-15
刊行日Issue date 2009-07
資源タイプResource Type Departmental Bullet in Paper / 紀要論文
版区分Resource Version publisher
権利Rights
DOI
JaLCDOI 10.24546/81001459
URL http://www.lib.kobe-u.ac.jp/handle_kernel/81001459
PDF issue: 2021-06-09
-
������������� ���
�������� !"#$%&��'(��)
Graphical Representation of Multi Cross-sectional Data: Part II
Kobe University
����
Takafumi ISOGAI
���������� ���
Abstract
Three-way (or more generally multiway) data analysis has been introduced as the extension of
two-way data analysis to higher-order datasets in Tucker [5]. There are a large number of varieties
of three-way datasets. Especially, when two of three modes (or ways) in a three-way dataset
describe n objects and p-dimensional observational characteristics, and the other mode describes g
time samples, this n p g× × dataset X has a form of 3-dimensional array and is called “multi
cross-sectional data”. Now we focus on 2-dimensional graphical methods for multi cross-sectional
data to pursue behaviors of n objects in time. A multi cross-sectional dataset X is approximated in
terms of variations of between (shortly, B) and within (shortly, W) groups specified by g time
samples. Using the above-mentioned approximation for X , a 2-dimensional B-W scatter plot has
been introduced in Isogai & Matsuura [1] to perform an exploratory data analysis for X . In this
paper we propose a new type of B-W scatter plot, called a “directional” B-W scatter plot, to extract
clear patterns with respect to behaviors of n objects in time.
(Received March 28, 2009)
�. ��
p������������ ( )p
xxx ,,,21
L=′x ��������� x�� n������
��� g���� �!"#�� � k�$�������
( ) ( ) ( ) ( )
( ) ( )gkxxxk
p
kkk
,,2,1,,,21
LL ==
′
��x
%&�� n�����'(��)*�+, ( )1,2, ,i i n= L �-���. i��� k�$������
�
( ) ( ) ( ) ( )
( ) ( )1 2, , , 1, 2, , ; 1, 2, ,k k k k
i i i ipx x x i n k g
′
= = =x L L L��
%&��n���������k�$�������/*)0��12345( )k
X �����( )k
X �
( )
( )
( )
( ) ( ) ( )
( ) ( ) ( )
( ) ( ) ( )
( )
1 11 12 1
( ) 21 22 22
1 2
1,2, ,
k k k k
p
k k kk
k p
k k kk
n n npn
x x x
x x x
i g
x x x
′⎛ ⎞ ⎛ ⎞
⎜ ⎟ ⎜ ⎟
′⎜ ⎟ ⎜ ⎟
= = =⎜ ⎟ ⎜ ⎟
⎜ ⎟ ⎜ ⎟
⎜ ⎟ ⎜ ⎟′⎜ ⎟⎝ ⎠⎝ ⎠
x
x
X
x
L
LL
M M MM
L
��
���
-
������������� ���
%!6"#�� ( )( )
1,2, ,k
k g=X L �7�/*)123�85�
( )
( ) ( )1,2, , ; 1, 2, , ; 1, 2, ,k
ijx i n j p k g= = = =X L L L�� �
����9��:�X� gpn ×× �; gpn ×× � 3 Tucker����
B-W {1��yz�©ª�K���0���Tucker �.«{1�NTucker [6]O�YM#�0�
%f��Tucker�.«{1���;
-
������������� ���
Å���|¶ )45�����Æ2ÇÈvÉ�Ê�ËÌ%
YX
~~
− �½Í
�Îs�Y~
�º*�^_B�°±²²³�Ïй�Îs���SÑ�X~
�½0ÒÓÔ��Y~
�
º*�^_%f��Tucker�.«{1��ÕÖ������Kiers [3] ?£× Tucker[5]�¾¿�9��
���� B-W ���
B-W {1���Tucker �.«{1��?Ø�°±²²³��ÙD@©�ÙÚ@©�&�±²�(Û
�-��9��£K�!6"#��9%�|¶45N
H ��
ngngNHICHH ⊗+⊗=
�²¹���)Ü �⊗�45�ÝÞ2ß%f��
1
g g g g
g
′= −H I 1 1 �
1
n n n
n
′=C 1 1 �
1
n n n n
n
′= −H I 1 1 >
%f�N)Ü � ( )1,1, ,1 : 1g
g′
= ⋅ ⋅ ⋅ ×1 ���� ( )1,1, ,1 : 1n
n
′= ⋅ ⋅ ⋅ ×1 ���O�9��:�
( ) ( )
( ) ( )�������� +=
⊗′+⊗′=′ XHIXXCHXXHXngngN
�K����à�áâ�ãä���)*��
WBN
PPH += > ( )ngWngB
HIPCHP ⊗=⊗= ��
�å�B
P �ÙD@©��W
P �ÙÚ@©�`�g�45�K����9#�J��¦�
2
2
(1) ,
(2) ,
(3)
B B B B
W W W W
B W
′= =
′= =
=
P P P P
P P P P
P P O
��
��
��
�æ*#��
B-W{1���}��ßç45
XPXXPXWB
′′���
�#è#�·¸��²¹��(Û�°±²�º*�9����
> ½�éê(Jollife [2]¾¿)�ë�ìs���B-W{1���Tucker�.«{1��?��º*
£b���Y~
��ÙD�ÙÚ@©�í�î��ï² �B BB
=Y P Y ð �W WW
=Y P Y �²¹ ��
WWBB
YPYPX −−
~
�½Í
�Îs� �B
Y � �W
Y �º*�^_� �!6"#��9#£��
22 2
B W B WB W B B W W
− − = − + −X P Y P Y P X P Y P X P Y%
���ÙD�ËÌ%X�½0£ÓÔ��45B
Y ��ÙÚ�ËÌ%X�½0£ÓÔ��45W
Y
���
-
������������� ���
Bº*#��
> ½ÍÐ�Îs�45BB
P Y �WW
P Y �º*���ÃÄ�£b����Àk�-��45�ñA��
( ) , ( ) ( ), ( ) , ( ) ( )B WB B W W
rank m rank s m rank q rank h q= = < = =
P X U V U U I V V I
����99%�45B
U �B
V �.«5�.ó5À%�ô:g �õ± )45�(1)
B
U �(1)
B
V �å
:�B
Δ �ö÷±²�.«±²�.ó±²À%ô:g ��(1)
1 2( , , , )
B B B
B s
diag δ δ δΔ = ⋅⋅ ⋅ �åØM�
2
BB B
−P X P Y �½ÍÐ�Îs�BB
P Y ��
(1) (1) (1)
BB B B B
′= ΔP Y U V > N
2
BB B
−P X P Y �½ÍÐ ( ) ( ) ( )2 2 2
1 2
B B B
s s m
δ δ δ+ +
= + + ⋅⋅ ⋅ + O
%Îs#��Ïø��45W
U �W
V �.«5�.ù5À%�ô:g �õ± )45�(1)
W
U �(1)
W
V
�å:�W
Δ �ö÷±²�.«±²�.ù±²À%ô:g ��(1)
1 2( , , , )
W W W
W h
diag δ δ δΔ = ⋅⋅ ⋅ �åØ
M�2
WW W
−P X P Y �½ÍÐ�Îs�WW
P Y ��
(1) (1) (1)
WW W W W
′= ΔP Y U V N
2
WW W
−P X P Y �½ÍÐ ( ) ( ) ( )2 2 2
1 2
W W W
h h qδ δ δ
+ +
= + + ⋅⋅ ⋅ + O
%Îs#��
> B-W{1��¹BB
P Y �WW
P Y �-�)�:� �X�ö��ú¤û�í�üÎý�
( ) ( )
( ) ( )
2 22
1 1
2 2 2
1 1
1
s hB W
i jB WB W i j
m qB W
i ji j
δ δ
δ δ
= =
= =
+− −
− =
+
∑ ∑
∑ ∑
X P Y P Y
X
%
%
%Îs#��f����ÏÐ&þ� �
> > > > > > >
( )
( )
( )
( ) ⎪⎭
⎪
⎬
⎫
⎪⎩
⎪
⎨
⎧
′
−
−
′
+
⎪⎭
⎪
⎬
⎫
⎪⎩
⎪
⎨
⎧
′
−
−
′
=
−−
−
XPX
YPXP
X
XPX
XPX
YPXP
X
XPX
X
YPYPX
W
WWWW
B
BBBB
WWBB
tr
1
~
tr
tr
1
~
tr
~
~
1
2
2
2
2
2
2
%Îs#��
����
-
������������� ���
���> B-W ����B-W scatter plotB-W ���
� B-W�(B-W)��¹(1) (1) (1)
BB B B B
′= ΔP Y U V �?Ø�
(1)
B
U �.«51
B
u �(1)
B
Δ �.«ö
÷±²1
B
δ �ß1 1
B B
δ u ��� x ��� �¹(1) (1) (1)
WW W W W
′= ΔP Y U V �?Ø�
(1)
W
U �.«51
W
u �
(1)
W
Δ �.«ö÷±²1
W
δ �ß1 1
W W
δ u �� y ��� ��¢���} B-W ��4b©ª����!6��B
P X�.«°±²�CD�HIBþ#��
!6��W
P X�.«°±²���CD�����Ø�c��:@©Bþ#�9���
� ����#�£K�������32�����²��4b)*�HI�¤����9��$
���� � B�B-W��E¼�%f�)*�� B-W�����
¥�*�i��$EB�±"#�����b��½���7��%:��9�B»���
K��������)*���0bl ËE� B-W �����Bf���¼�
B� � B-W ���ÙD@©�õ±��B
P X�ÙÚ@©�õ±��W
P X��!Û�p¶�
f�HI�J��qg ���î���-��¢�������%f��B
P X����
"Ìf�HI�CD%f��W
P X����"Ìf�HI���C#%�¢�������$�
�F¡%f��F¡�HI�%EW�£K�¤¥�Îs#�êR0f� �b%�êR0f
��tuvwx[1]%Îs)&%��'Ç2�()*�+,#�Ç2�±-�F¡�-�����
> 9%�12345X�õ±�.%f�/C%���45 ( )( )
1,2, ,k
k g=X L �ö �����
C����� ( 1,2, , )n
k k g=1 L � ��#�À�*)CD��� t�
1 1
2 2
: 1
n
n
n
n
ng
g g
⎛ ⎞ ⎛ ⎞
⎜ ⎟ ⎜ ⎟
⎜ ⎟ ⎜ ⎟= = ⊗ ×
⎜ ⎟ ⎜ ⎟
⎜ ⎟ ⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠
1
1
t 1
1
M M����
����
-
������������� ���
%!6���À)�/C%���45 ( )( )
1,2, ,k
k g=X L �¢�������$��F¡�HIB
U#���� ��F¡HI�«�¢À%�0A�-��& )0�� ( )( )
1,2, ,k
i
r i n= L %í�
N«�½1�F¡�¢�½2�F¡�Îs�U���O�99%�33v123B45���:��
F¡�p��U�~6Ð%å:ìs�0�����"�� ( )( )
1,2, ,k
i
r i n= L ��.������
� ( )( ) ( ) ( ) ( )
1 2, , , ' ( 1,2, , )
k k k k
n
r r r k g= =r L L %& �( )
( 1,2, , )k
k g=r L �æ*)F¡���r�
(1)
(2)
( )
: 1
g
ng
⎛ ⎞
⎜ ⎟
⎜ ⎟= ×⎜ ⎟
⎜ ⎟⎜ ⎟
⎝ ⎠
r
r
r
r
M���
%!6���
99%�B
P X�5����« > > > �SÑ� ( )1B
′= =P Xa z a a �å��� ′t z�½��rs��
^_(II) W
P X�5����« > > > �SÑ� ( )1W
′= =P Xb u b b �å��� ′r u�½��rs��
î��^_�¹� ^_(I)�¹�����9:�Á!;A��£��
> > > > > > > > > B
B
′
=
′
XP t
a
XP t
> ^_(II)�¹�Ïø������9:�Á!;A��£��
> > > > > > > > > W
W
′
=
′
XP r
b
XP r
> � B-W���^_(I)�(II)�¹�-����� z��u����¢������
}
-
������������� ���
> ^_ ( )I′>
N
H X�5����« > > > > > > �SÑ� ( )1N
′= =H Xa z a a �å��� ′t z�½��rs��
> ^_ ( )II′
N
H X�5����« > > > > > > �SÑ� ( )1N
′= =H Xb u b b �å��� ′r u�½��rs��
� B-W �B^_ ( )I′� ( )II
′�¹�Îs�9���� B-W ��1237��
N
H X�HI�qg �����bËÌ%»?�½��JK����rs#��
���� ���
� � B-W ��£Â#)123�- �Q£b�`�@�123�ABCDEª
p(FAO)�yF��123�|�FGHÝ�I2N«J«KL)� kcalO�p��0�%�FGHÝ�
I2�õ±�������««M$N�SÑ�p=11Of��#��N��O�?£×%PQ�R
S��T��UV�W�X��Y��Z[��\]?£×]^_�`a�%f��9�����««
M$�����Kb�cde`fg�@h�ij��)*��1948+� 1974+��Ø��«k=�
�DN�SÑ�g=10O�lK�Kb�mnoAN}kÞAO�123B/*#�ApqB4S#
)Nr|[4]¾¿O�1970+sLC��KbAt�cde`fg�mnBuv"#��)�
r|[4]�ÕÖ²³NX�ö��°±²²³��·32²³�O�£�7Z�wx� 9�Z[%��}kÞA�mnoA�bÑ��Ð�J�·¸3�����Kb�«ÞA�m
noAN�SÑ�n=20O%¹³�4b�VWX123X� 20 11 10n p g× × = × × �õ±%f��
Àk� X�ö��}
-
������������� ���
������������� ������� ������� ������� �����
����
��
�
��
�
����
������������� ������� ������� ������� �����
����
��
�
��
�
����
��
��
��
��
��
��
�
����
�� �������� �������� ������������ ���������� ������ ��� !���"��� #�� ��$��% &�����"��' &���(���) *����+�� ��,��� +���(���� -�����"��� -���,��� .�����"��� �/�0�����"��� ��/�,�� 1���������% 2/�"����' 2/��3�����"��) 4����"�5���"� ��� 42�
«�X�ö��°±²�NßüÎý¯63.3%O
������������� ������� ������ �����
���
��
��
�
��
�
���
������������� ������� ������ �����
���
��
��
�
��
�
���
�� ��������� �������� ����������� ��������� ��� ��!��� "���#���$ %��!��&��� '�����#��( '���)���� *����+��!��,��� +���)���� -�����#��� -���,�� .����#�� ��/�0�����#��� ��/�,��$ 1��� ����� 2/�#����( 2/��3�����#��� 4����#�5�� #!��� 42�
}�X�ö�� B-W�NßüÎý¯48.2%O
½°��X�ö��� B-W ��;�Îs���� ���=��D�CDFG�
½0Ò&�£b���±! ��º*�C���cde`�²HÝ�I2��SÑ««M$
�B_Ù�e`�£�U#)²³ÞÇ2�����=��D´���A�F¡HI�-�)�½
1�²e`³ÞÇ2�A��U«��½2�²e`³ÞÇ2�A��U}k� �«µ
Q%}kÞA�U�Îs�ÏF¡�A�p ��U�~6Ð�-�)�9��:�ßüÎý�
44.1%N�z7.7%�z36.4%O%fK)�
;�����=��D�CDFG�HIB�¢�&þ"#���9��¤Ë �?�¶� B-W
�� ·Ïø�r¸�U�B�"̹���33Ç�©:%f��0b»����i�
����
-
������������� ���
¬��nA�º:292��È�cde`õ{�»�o%A����%f��� B-W �
��B-W��p¯�����32�����£�é)T���eBf��¼S#��
������������� ������� ������ �����
�
��
��
�
��
�
�
������������� ������� ������ �����
�
��
��
�
��
�
�
�� ��������� �������� ����������� ��������� ��� ��!��� "���#���$ %��!��&��� '�����#��( '���)���� *����+��!��,��� +���)���� -�����#��� -���,�� .����#�� ��/�0�����#��� ��/�,��$ 1��� ����� 2/�#����( 2/��3�����#��� 4����#�5�� #!��� 42�
;�X�ö��� B-W�NßüÎý¯44.1%O
���
> 9�Z[%��3