Journal of Computations & Modelling, vol.3, no.4, 2013, 165-193
ISSN: 1792-7625 (print), 1792-8850 (online)
Scienpress Ltd, 2013
Analysis of the Preference Shift of
Customer Brand Selection
and Its Matrix Structure
-Expansion to the second order lag
Kazuhiro Takeyasu 1
Abstract
It is often observed that consumers select the upper class brand when they buy the
next time. Suppose that the former buying data and the current buying data are
gathered. Also suppose that the upper brand is located upper in the variable array.
Then the transition matrix becomes an upper triangle matrix under the supposition
that the former buying variables are set input and the current buying variables are
set output. The goods of the same brand group would compose the Block Matrix
in the transition matrix. Condensing the variables of the same brand group into
one, analysis becomes easier to handle and the transition of Brand Selection can
1 Tokoha University. E-mail: takeyasu@fj.tokoha-u.ac.jp
Article Info: Received : August 18, 2013. Revised : September 29, 2013.
Published online : December 1, 2013.
166
Analysis of the Preference Shift of Customer Brand Selection…
be easily grasped. In this paper, equation using transition matrix stated by the
Block Matrix is expanded to the second order lag and the method of condensing
the variable stated above is also applied to this new model. Planners for products
need to know its brand position whether their brand is upper or lower than other
products. Matrix structure makes it possible to ascertain this by calculating
consumers’ activities for brand selection. Thus, this proposed approach makes it
possible to execute an effective marketing plan and/or establishing new brand.
Mathematics Subject Classification: 03C85, 15A23, 15A24
Keywords: brand selection, matrix structure, brand position, automobile
industry, brand group
1
Introduction
It is often observed that consumers select the upper class brand when they
buy the next time. Focusing the transition matrix structure of brand selection, their
activities may be analyzed. In the past, there are many researches about brand
selection [1-5]. But there are few papers concerning the analysis of the transition
matrix structure of brand selection. In this paper, we make analysis of the
preference shift of customer brand selection. The goods of the same brand group
would compose the Block Matrix in the transition matrix. Condensing the
variables of the same brand group into one, analysis becomes easier to handle and
Kazuhiro Takeyasu
167
the transition of Brand Selection can be easily grasped. Then we confirm them by
the questionnaire investigation for automobile purchasing case. If we can identify
the feature of the matrix structure of brand selection, it can be utilized for the
marketing strategy.
Suppose that the former buying data and the current buying data are
gathered. Also suppose that the upper brand is located upper in the variable array.
Then the transition matrix becomes an upper triangular matrix under the
supposition that the former buying variables are set input and the current buying
variables are set output. If the top brand were selected from the lower brand in
jumping way, corresponding part in the upper triangular matrix would be 0. These
are verified by the numerical examples with simple models.
Planners for products need to know its brand position whether their brand is
upper or lower than other products. Matrix structure makes it possible to ascertain
this by calculating consumers’ activities for brand selection. Thus, this proposed
approach makes it possible to execute an effective marketing plan and/or
establishing new brand.
A quantitative analysis concerning brand selection has been executed by [4,
5]. [5] examined purchasing process by Markov Transition Probability with the
input of advertising expense. [4] made analysis by the Brand Selection Probability
model using logistics distribution.
The goods of the same brand group would compose the Block Matrix in the
transition matrix. Condensing the variables of the same brand group into one,
168
Analysis of the Preference Shift of Customer Brand Selection…
analysis becomes easier to handle and the transition of Brand Selection can be
easily grasped. In this paper, equation using transition matrix stated by the Block
Matrix is expanded to the second order lag and the method of condensing the
variable stated above is also applied to this new model. Such research cannot be
found as long as searched.
The rest of the paper is organized as follows. Matrix structure is clarified for
the selection of brand in section 2. A block matrix structure is analyzed when
brands are handled in a group in section 3. Expansion of the block matrix structure
to the second order lag is executed in section 4. A block matrix structure when
condensing the variables of the same brand group is analyzed in section 5. Its
expansion to the second order lag in stated in section 6. Numerical calculation is
executed in section 7. Section 8 is a summary.
2
Brand Selection and Its Matrix Structure
2.1 Upper Shift of Brand Selection
It is often observed that consumers select the upper class brand when they
buy the next time. Now, suppose that x is the most upper class brand, y is the
second upper brand, and z is the lowest brand. Consumer’s behavior of selecting
brand would be z → y , y → x , z → x etc. x → z might be few. Suppose that x
is the current buying variable, and xb is the previous buying variable. Shift to x
is executed from xb , y b , or z b .
Therefore, x is stated in the following equation.
Kazuhiro Takeyasu
169
x = a11 xb + a12 y b + a13 z b
Similarly,
y = a 22 y b + a 23 z b
And
z = a33 z b
These are re-written as follows.
 x   a11
  
 y =  0
 z  0
  
a13  xb 
 
a 23  y b 
a33  z b 
a12
a 22
0
(1)
Set :
 x
 
X =  y ,
z
 
 a11

A= 0

 0
a12
a 22
0
a13 

a 23  ,
a33 
 xb 
 
X b =  yb 
z 
 b
then, X is represented as follows.
X = AX b
(2)
Here,
X ∈ R 3 , A ∈ R 3×3 , X b ∈ R 3 ,
A is an upper triangular matrix.
To examine this, generating the following data, which are all consisted by the
upper brand shift data,
170
Analysis of the Preference Shift of Customer Brand Selection…
1
 
X = 0
0
 
i
 0
 
X = 1
 0
 
i
b
i = 1,
1
 
0
0
 
0
 
1
0
 
…
1
 
0
0
 
0
 
0
1
 
…
2,
(3)
(4)
N
…
parameter can be estimated using least square method.
Suppose
X i = AX ib + ε i
(5)
and
N
J = ∑ ε iT ε i → Min
(6)
i =1
 which is an estimated value of A is obtained as follows.
N
N
ˆ =  X i X iT  X i X iT 
A
∑ b  ∑
b
b
i =1
 i =1

−1
(7)
In the data group of the upper shift brand, estimated value  should be an upper
triangular matrix. If the following data, that have the lower shift brand, are added
only a few in equation (3) and (4),
0
 
X = 1 ,
0
 
i
1
 
X =  0
 0
 
i
b
 would contain minute items in the lower part triangle.
Kazuhiro Takeyasu
171
2.2 Sorting Brand Ranking by Re-arranging Row
In a general data, variables may not be in order as x, y, z . In that case, large
and small value lie scattered in  . But re-arranging this, we can set in order by
shifting row. The large value parts are gathered in an upper triangular matrix, and
the small value parts are gathered in a lower triangular matrix.
Â
 x   ○ ○ ○
 

 y   ε ○ ○
 z   ε ε ○
 

Â
 z   ε ε ○


 x   ○ ○ ○

 
 y   ε ○ ○
Shifting row 
(8)
2.3 In the case that brand selection shifts in jump
It is often observed that some consumers select the most upper class brand
from the most lower class brand and skip selecting the middle class brand. We
suppose v, w, x, y, z brands (suppose they are laid from the upper position to the
lower position as v > w > x > y > z ).
In the above case, the selection shifts would be
v←z
v← y
Suppose there is no shift from z to y , corresponding part of the transition
matrix is 0 (i.e. a 45 =0). Similarly, if there is no shift from z to y , from z to
w , from y to x , form y to w , from x to w , then the matrix structure
would be as follows.
172
Analysis of the Preference Shift of Customer Brand Selection…
 v   a11
  
 w  0
 x = 0
  
 y  0
z  0
  
3
a12
a13
a14
a 22
0
0
0
a33
0
0
0
a 44
0
0
0
a15  vb 
 
0  wb 
0  xb 
 
0  y b 
a55  z b 
(9)
Block Matrix structure in Brand Groups
Next, we examine the case in brand groups. Matrices are composed by
Block Matrix.
(1)
Brand shift group － in the case of two groups
Suppose brand selection shifts from Corolla class to Mark II class in car. In
this case, it does not matter which company’s car they choose. Thus, selection of
cars are executed in a group and brand shift is considered to be done from group
to group. Suppose brand groups at time n are as follows.
X consists of p varieties of goods, and Y consists of q varieties of
goods.
 x1n 
 n
x 
Xn =  2 
  
 xn 
 p ,
 y1n 
 n
y 
Yn =  2 
  
 n
 yq 
 X n   A 11 , A 12  X n −1 
  = 


 Yn   0, A 22  Yn −1 
Here,
(10)
Kazuhiro Takeyasu
173
X n ∈ R p (n = 1,2, ) ,
Yn ∈ R q (n = 1,2, ) ,
A 11 ∈ R p× p ,
A 12 ∈ R p×q ,
A 22 ∈ R q×q
Make one more step of shift, then we obtain following equation.
 X n   A 11 2 , A 11 A 12 + A 12 A 22  X n − 2 


 = 

2

0,
A 22  Yn − 2 
 Yn  
(11)
Make one more step of shift again, then we obtain following equation.
 X n   A 11 3 , A 11 2 A 12 + A 11 A 12 A 22 + A 12 A 22 2  X n − 3 


  = 
3

A 22  Yn − 3 
 Yn   0,
(12)
Similarly,
 Xn   A114 , A113 A12 + A112 A12 A 22 + A11 A12 A 22 2 + A12 A 223   Xn − 4 

 =

A 22 4   Yn − 4 
 Yn   0,
(13)
 X n   A 11 5 , A 11 4 A 12 + A 11 3 A 12 A 22 + A 11 2 A 12 A 22 2 + A 11 A 12 A 22 3 + A 12 A 22 4  X n −5 


  = 
5

A 22  Yn −5 
 Yn   0,
(14)
Finally, we get generalized equation for s -step shift as follows.
s −1
 s
s −1
s −k
k −1
s −1 
 X n   A 11 , A 11 A 12 + ∑ A 11 A 12 A 22 + A 12 A 22  X n − s 

  = 
k =2

s  Yn − s 
 Yn   0,
A 22 

(15)
If we replace n − s → n, n → n + s in equation (15), we can make s -step
forecast.
(2)
Brand shift group － in the case of three groups
Suppose brand selection is executed in the same group or to the upper group,
and also suppose that brand position is x > y > z ( x is upper position). Then
brand selection transition matrix would be expressed as
174
Analysis of the Preference Shift of Customer Brand Selection…
 X n   A 11 , A 12 , A 13  X n −1 
  


 Yn  =  0, A 22 , A 23  Yn −1 
 Z   0,
0, A 33  Z n −1 
 n 
(16)
Where
 y1n 
 n
y 
Yn =  2 
  
 yn 
 q ,
 x1n 
 n
x 
Xn =  2 
  
 xn 
 p ,
 zn 
 1n 
z 
Zn =  2 
  
 zn 
 r
Here,
Xn ∈ R p
(n = 1,2,) ,
A 11 ∈ R p× p ,
Yn ∈ R q
A 12 ∈ R p×q ,
(n = 1,2,) ,
A 13 ∈ R p×r ,
Zn ∈ R r
A 22 ∈ R q×q ,
(n = 1,2,) ,
A 23 ∈ R q×r ,
A 33 ∈ R r×r
These are re-stated as
Wn = AWn −1
where,
 Xn 
 
Wn =  Yn 
Z 
 n ,
 A 11 , A 12 , A 13 


A =  0, A 22 , A 23 
 0,
0, A 33  ,

(17)
Wn −1
 X n −1 


=  Yn −1 


 Z n −1 
Hereinafter, we shift steps as is done in previous section.
In the general description, we state as
Wn = A (s) Wn −s
Here,
(18)
Kazuhiro Takeyasu
A (s)
 A 11 (s) , A 12 (s) , A 13 (s) 


(s)
(s)
0, A 22 , A 23 
=

(s) 
0,
0, A 33 

,
175
Wn −s
 X n −s 


=  Yn −s 
Z 
 n −s 
From definition,
A (1) = A
(19)
In the case s = 2 , we obtain
A ( 2)
 A 11 , A 12 , A 13  A 11 , A 12 , A 13 



=  0, A 22 , A 23  0, A 22 , A 23 
 0,
0, A 33 
0, A 33  0,

2
 A 11
, A 11 A 12 + A 12 A 22 , A 11 A 13 + A 12 A 23 + A 13 A 33 


=  0,
A 22 A 23 + A 23 A 33 
A 222 ,
 0,
0,
A 233 

(20)
Next, in the case s = 3 , we obtain
 Α 113 , Α 112 Α 12 + Α 11 Α 12 Α 22 + Α 12 Α 222 , Α 112 Α 13 + Α 11 Α 12 Α 23 + Α 11 Α 13 Α 33 + Α 12 Α 22 Α 23 + Α 12 Α 23 Α 33 + Α 13 Α 233 


A =  0,
Α 322 ,
Α 222 Α 23 + Α 22 Α 23 Α 33 + Α 23 Α 233 
 0,
0,
Α 333 

( 3)
(21)
In the case s = 4 , equations become wide-spread, so we express each Block
Matrix as follows.
(4 )
4
A 11
= A 11
( 4)
2
2
3
A 12
= A 11
A 12 + A 11
A 12 A 22 + A 11 A 12 A 22
+ A 12 A 322
2
2
2
( 4)
3
A 13 A 33 + A 11 A 12 A 22 A 23 + A 11 A 12 A 23 A 33 + A 11 A 13 A 33
A 13
A 13 + A 11
A 12 A 23 + A 11
= A 11
2
2
A 23 + A 12 A 22 A 23 A 33 + A 12 A 23 A 33
+ A 12 A 22
+ A 13 A 333
A (224 ) = A 422
A (234) = A 322 A 23 + A 222 A 23 A 33 + A 22 A 23 A 233 + A 23 A 333
A (334) = A 433
In the case s = 5 , we obtain the following equations similarly.
(22)
176
Analysis of the Preference Shift of Customer Brand Selection…
(5)
5
A 11
= A 11
(5)
4
3
2
A 12
= A 11
A 12 + A 11
A 12 A 22 + A 11
A 12 A 222 + A 11 A 12 A 322 + A 12 A 422
(5)
4
3
3
2
2
2
A 13
= A 11
A 13 + A 11
A 12 A 23 + A 11
A 13 A 33 + A 11
A 12 A 22 A 23 + A 11
A 12 A 23 A 33 + A 11
A 13 A 233
+ A 11 A 12 A 222 A 23 + A 11 A 12 A 22 A 23 A 33 + A 11 A 12 A 23 A 233 + A 11 A 13 A 333
+ A 12 A A 23 + A 12 A A 23 A 33 + A 12 A 22 A 23 A
3
22
2
22
2
33
+ A 12 A 23 A
3
33
+ A 13 A
(23)
4
33
A (235) = A 422 A 23 + A 322 A 23 A 33 + A 222 A 23 A 233 + A 22 A 23 A 333 + A 23 A 433
A (335) = A 533
In the case s = 6 , we obtain
(6)
6
A 11
= A 11
(6)
5
4
4
3
3
3
A 13
= A 11
A 13 + A 11
A 12 A 23 + A 11
A 13 A 33 + A 11
A 12 A 22 A 23 + A 11
A 12 A 23 A 33 + A 11
A 13 A 233
2
2
2
2
+ A 11
A 12 A 222 A 23 + A 11
A 12 A 22 A 23 A 33 + A 11
A 12 A 23 A 233 + A 11
A 13 A 333
(24)
+ A 11 A 12 A 322 A 23 + A 11 A 12 A 222 A 23 A 33 + A 11 A 12 A 22 A 23 A 233 + A 11 A 12 A 23 A 333 + A 11 A 13 A 433
2
4
+ A 12 A 422 A 23 + A 12 A 322 A 23 A 33 + A 12 A 222 A 23 A 33
+ A 12 A 22 A 23 A 333 + A 12 A 23 A 33
+ A 13 A 533
We get generalized equations for s -step shift as follows.
(s)
s
A 11
= A 11
s −1
(s)
k −1
s −1
s −1
s −k
+ A 12 A 22
= A 11
A 12 A 22
A 12 + ∑ A 11
A 12
k =2
2

 j +1 j +1− k
 s −3  S − 2 − j 
(s)
s −1
s −2 
k −1 
j +1
A 13
= A 11
A 13 + A 11
 ∑ A 1(k +1) A (k +1)3  + ∑  A 11
A 12  ∑ A 22 A 23 A 33  + A 13 A 33 
 k =1
 j =1 
 k =1



A
(s)
22
=A
s
22
s
s −k
A (23s ) = ∑ Α 22
Α 23 Α k33−1
K −1
s
A (33s ) = A 33
Expressing them in matrix, it follows.
(25)
Kazuhiro Takeyasu
177
s −1
2
 s
 

 s −3  s − 2 − j   j +1 j +1− k
 A , A s −1A + Α s − k Α Α k −1 + Α Α s −1 , A s −1A + A s − 2  A A
A12  ∑ A 22 A 23 A 33k −1  + A13 A 33j +1  

11
11 12 ∑ 11
12 22
12 22
11 13
11 ∑ 1(k + 1) (K + 1)3  + ∑  A11

k =2
 k =1
 j =1 

  k =1
 


s
s
s
−
k
k
(S )
A =  0,
A 22 ,
A 22 A 23 A 33−1 
∑


k =1
 0,
0,
A 33s 






(26)
Generalizing them to m groups, they are expressed as
 X n(1)   A 11
 ( 2)  
 X n   A 21
  = 

 
 X (m)   A
 n   m1
X n(1) ∈ R k1 ,
4
A 12
A 22

A m2
X n( 2 ) ∈ R k2 ,  , X n( m ) ∈ R k m ,
 A 1m  X n(1−)1 


 A 2m  X n( 2−)1 
   


 A mm  X n( m−1) 
A ij ∈ R
ki ×k j
(27)
(i = 1,, m )( j = 1,, m )
Expansion of the Block Matrix structure to the second
order lag
Expansion of the above stated Block Matrix model to the second order lag is
executed in the following method. Here we take three groups case.
Generating Eq.(16) and Eq.(18),we state the model as follows. Here we set P=3.
 X n   Α, B, C  X n −1 

  

 Yn  =  D, E, F  Yn −1 
 Z   G , H, J  Z 
 n 
 n −1 
Where
(28)
178
Analysis of the Preference Shift of Customer Brand Selection…
 x1n 
 y1n 
 z1n 
 
 
 
X n =  x2n , Yn =  y2n , Z n =  z2n 
 n
 n
 n
 x3 
 y3 
 z3 
 
 
 
(29)
Here,
Xn ∈=
R 3 ( n 1, 2,) ,
Yn ∈ =
R 3 ( n 1, 2,) ,
{A, B,C, D, E, F,G, H, J} ∈ R
3×3
Zn ∈ =
R 3 ( n 1, 2,) ,
. These are re-stated as
Wn = PWn −1
(30)
X 
 n
Wn =  Yn 
Z 
 n
(31)
 A , B, C 


P =  D, E, F 
 G , H, J 


(32)
Wn −1
 X n −1 


=  Yn −1 
Z 
 n −1 
(33)
if N amount of data exist, we can derive the following the equation similarly as
Eq.(5),
Wni = PWni −1 + ε in (i = 1,2,, N )
(34)
And
N
J n = ∑ ε iTn ε in → Min
i =1
∧
P which is an estimated value of P is obtained as follows.
(35)
Kazuhiro Takeyasu
179

 N
 N
P =  ∑ Wni WniT−1  ∑ Wni −1 WniT−1 

 i =1
 i =1
∧
−1
(36)
Now, we expand Eq.(34) to the second order lag model as follows.
Wni = P1 Wni −1 + P2 Wni − 2 + ε in
(37)
 A 2 , B 2 , C2 
 A 1 , B 1 , C1 




P1 =  D1 , E1 , F1 , P2 =  D 2 , E 2 , F2 
G , H , J 
G , H , J 
2
2 
1 
1
 2
 1
(38)
Here
It we set
P = (P1 , P2 )
(39)
∧
then P can be estimated as follows.
T
T
 N
 Wti−1   N  Wti−1  Wti−1  

i




P = ∑ Wt
i 
i  
 i =1
 W i   ∑

 t − 2   i =1  Wt − 2  Wt − 2  

We further develop this equation as follows.
−1
(40)
180
Analysis of the Preference Shift of Customer Brand Selection…
P = ( P1 , P2 )
 A1 , B1 , C1 , A 2 , B 2 , C2 


=  D1 , E1 , F1 , D2 , E2 , F2 
G , H , J , G , H , J 
1
1
2
2
2 
 1
N
 N
i
iT
i
iT 
W
W
,
 ∑ t −1 t −1 ∑ Wt −1Wt − 2 
N
N

 i 1 =i 1
=

=  ∑ Wti WtiT−1 , ∑ Wti WtiT− 2   N
N


=
 i 1 =i 1

i
iT
i
iT
 ∑ Wt − 2 Wt −1 , ∑ Wt − 2 Wt − 2 
=
 i 1 =i 1

i
i
  xt 

 xt 
N 
 N  i  iT iT iT


=  ∑  y t  ( xt −1 , y t −1 , z t −1 ) , ∑  y it  ( xiTt − 2 , y iTt − 2 , z iTt − 2 ) 
i 1 i 
=
 i 1 =

 i
 zt 
  zt 

−1
  xit −1 

 xit −1 
N
N
  i  iT iT iT
 i  iT iT iT 
 ∑  y t −1  ( xt −1 , y t −1 , z t −1 ) , ∑  y t −1  ( xt − 2 , y t − 2 , z t − 2 ) 
i 1 i
 i 1=

 i 

z
1
t
−
 


 z t −1 
×

i
i
 xt − 2 
 N  xt − 2 

N 
 iT iT iT 
i
  y i  ( xiT , y iT , z iT ) ,
 t − 2  t −1 t −1 t −1
 y t − 2  ( xt − 2 , y t − 2 , z t − 2 ) 
∑
∑
1
i 1=
i
 i 
  z it − 2 


 z t −2 
 

−1
N
N
N
N
N
 N i iT
i iT
i iT
i iT 
i iT
i iT
x
x
,
x
y
,
,
,
,
x
x
x
y
x
z
 ∑ t t −1 ∑ t t −1 ∑ t t −1 ∑ t t − 2 ∑ t t − 2 ∑ xt z t − 2 
=i 1 =i 1 =i 1=i 1 =i 1 =i 1


N
N
N
N
N
 N i iT
i iT
i iT
i iT 
i iT
i iT
=  ∑ y t xt −1 , ∑ y t y t −1 , ∑ y t z t −1 , ∑ y t xt − 2 , ∑ y t y t − 2 , ∑ y t z t − 2  ,
=
 i 1 =i 1 =i 1 =i 1 =i 1 =i 1

N
N
N
N
N
 N i iT
i iT
i iT
i iT
i iT
i iT 
 ∑ z t xt −1 , ∑ z t y t −1 , ∑ z t z t −1 , ∑ z t xt − 2 , ∑ z t y t − 2 , ∑ z t z t − 2 
=
 i 1 =i 1 =i 1 =i 1 =i 1 =i 1

Kazuhiro Takeyasu
 N i iT
 ∑ xt −1xt −1 ,
 i =1
 N i iT
 ∑ y t −1xt −1 ,
 i N=1
 z i xiT ,
t −1 t −1
∑
i =1
× N
 xi xiT ,
t − 2 t −1
∑
i =1
 N
 ∑ y it − 2 xiTt−1 ,
 i =1
 N i iT
 ∑ z t − 2 xt −1 ,
 i =1
181
N
N
N
N
∑ xit −1y iTt−1,
∑ xit −1z iTt−1,
∑ xit −1xiTt− 2 ,
∑ xit −1y iTt− 2 ,
∑y
iT
i
t −1 t −1
∑y
i
iT
t −1 t −1
∑y
i
iT
t −1 t − 2
y
,
∑z
i
iT
t −1 t −1
∑z
i
iT
t −1 t −1
∑z
i
iT
t −1 t − 2
,
i =1
N
i =1
N
i =1
y ,
y ,
N
i =1
N
i =1
N
i =1
z ,
z ,
i =1
N
∑y
i
iT
t −1 t − 2
,
∑z
i
iT
t −1 t − 2
,
i =1
N
i =1
N
x
x
∑x
i =1
N
i =1
N
i =1
N
y
N
∑ xit − 2y iTt−1,
∑ xit − 2z iTt−1,
∑ xit − 2xiTt− 2 ,
∑x
i
t −2
y iTt− 2 ,
∑y
i
t −2
y iTt−1 ,
∑y
i
iT
t − 2 t −1
∑y
i
iT
t −2 t −2
,
∑y
i
t −2
y iTt− 2 ,
∑z
i
t −2
y iTt−1 ,
∑z
i
iT
t − 2 t −1
∑z
i
iT
t −2 t −2
,
∑z
i
t −2
y iTt− 2 ,
i =1
N
i =1
N
i =1
i =1
N
i =1
N
i =1
z ,
z ,
i =1
N
i =1
N
i =1
x
x
i =1
N
i =1
N
i =1


i =1

N

i
iT
y t −1z t − 2 
∑
i =1

N
i
iT
z t −1z t − 2 
∑

i =1
 (41)
N
i
iT 
xt − 2 z t − 2
∑

i =1

N
i
iT

y
z
∑
t −2 t −2

i =1
N

z it − 2 z iTt− 2 
∑
i =1

N
i
iT
t −1 t − 2
z
We set this as:
 K1

P =  K4
K
 7
K2
K5
K 3 L1
K 6 L4
L2
L5
K8
K 9 L7
L8
 M1

M
L3   4
M
L6   7
Q
L9   1
 Q4

 Q7
M2
M3
N1
N2
M5
M8
Q2
Q5
M6
M9
Q3
Q6
N4
N7
R1
R4
N5
N8
R2
R5
Q8
Q9
R7
R8
N3 

N6 
N9 

R3 
R6 

R 9 
−1
(42)
Then when all consist of the same level shifts or the upper level shifts (suppose
𝐗 > 𝐘 > 𝐙),
𝑲4 , 𝑲7 , 𝑲8 , 𝑳4 , 𝑳7 , 𝑳8 , 𝑴2 , 𝑴3 , 𝑴6 , 𝑵4 , 𝑵7 , 𝑵8 , 𝑹2 , 𝑹3 , 𝑹4
are all 0.
As
𝑴4 = 𝑴𝑇2 , 𝑴7 = 𝑴𝑇3 , 𝑴8 = 𝑴𝑇6 , 𝑹6 = 𝑹𝑇2 , 𝑹7 = 𝑹𝑇3 , 𝑹8 = 𝑹𝑇6 , 𝑸2 = 𝑵𝑇4 , 𝑸3
= 𝑵𝑇7 , 𝑸6 = 𝑵𝑇8 ,
therefore they are all 0.
𝑴1 , 𝑴5 , 𝑴9 , 𝑹1 , 𝑹5 , 𝑹9 become diagonal Matrices.
182
Analysis of the Preference Shift of Customer Brand Selection…
Using a symbol “∗” as a diagonal matrix, P becomes as follows by using the
relation stated above.
0,
0, N1 , N 2 , N 3 
 ∗,


∗,
0,
0,
0, N 5 , N 6 

 K1 , K 2 , K 3 , L1 , L 2 , L3 
∗,
0,
0,
0, N 9 

  0,
P =  0, K 5 , K 6 , 0, L5 , L 6  

∗,
Q1 , 0
0,
0,
0 

 0,

0, K 9 , 0, 0, L9 

 Q 4 , Q5 , 0,
0,
∗,
0 


0,
∗ 
 Q 7 , Q8 , Q9 , 0,
5
−1
(43)
Matrix Structure When Condensing the Variables of the
Same Class
Suppose the customer select Bister from Corolla (Bister is an upper class
brand automobile than Corolla.) when he/she buy next time. In that case, there are
such brand automobiles as bluebird, Gallant sigma and 117 coupes for the
corresponding same brand class group with Bister in other companies.
Someone may select another automobile from the same brand class group.
There is also the case that the consumer select another company’s automobile of
the same brand class group when he/she buy next time.
Matrix structure would be, then, as follows.
Suppose w, x, y in the same brand class group in the example of 2.3. If there
exist following shifts:
Kazuhiro Takeyasu
183
y, x, w, v ← z b
x, w ← y b
y, w ← xb
x, y ← wb
v ← yb
v ← xb
v ← wb
y ← yb
y ← xb
y ← wb
then, transition equation is expressed as follows.
 v   a11
  
 w  0
 x = 0
  
 y  0
z  0
  
a12
a 22
a32
a 42
0
a13
a 23
a33
a 43
0
a14
a 24
a34
a 44
0
a15  vb 
 
a 25  wb 
a35  xb 
 
a 45  y b 
a55  z b 
(44)
Expressing these in Block Matrix form, it becomes as follows.
 a11

X= 0

 0
A 12
A 22
0
a15 

Α 23  X b
a55 
(45)
Where,
X ∈ R 5 , A 12 ∈ R 1×3 , A 22 ∈ R 3×3 , A 23 ∈ R 3×1 , X b ∈ R 5
As w, x, y are in the same class brand group, condensing these variables into one,
and expressing it as w , then Eq.(10) becomes as follows.
184
Analysis of the Preference Shift of Customer Brand Selection…
 v   a11
  
w =  0
z  0
  
a12
a 22
0
a15  vb 
 
a 23  wb 
a55  z 
(46)
(∀ j )
(47)
As aij satisfies the following equation :
5
∑a
i =1
ij
=1
Condensed version of Block Matrix A 22 are as follows, where sum of each item
of Block matrix is taken and they are divided by the number of variables.
a 22 =
1 4 4
∑∑ aij
3 i =2 j =2
(48)
a12 =
1 4
∑ a1 j
3 j =2
(49)
4
a 23 = ∑ ai 5
(50)
i =2
Generalizing this, it becomes as follows.
 X1   A 11
  
 X2   0
  = 
  
X   0
 r 
A 12  A 1r  X1,b 


A 21  A 2 r  X 2,b 

   


0  A rr  X r ,b 
(51)
where
X1 ∈ R p1 , , X r ∈ R pr , A ij ∈ R
pi × p j
i 1, , r ) , ( =
j 1, , r )
(=
When the variables of each Block Matrix are condensed into one, transition
matrix is expressed as follows.
Kazuhiro Takeyasu
185
 x1   a11 , a12 ,  a1r  x1,b 

  

 x2   0, a22 ,  a2 r  x2,b 
  = 

   

  

 x   0,
 x 
0
,

a
rr  r , b 
 r 
(52)
Where
aij =
1
pj
pi
pj
∑∑ a
k =1 l =1
ij
kl
(i = 1,, r ), ( j = i,, r )
(53)
Here, a klij (k = 1,  , Pi ), (l = 1,  , Pi ) means the item of A ij .
Taking these operations, the variables of the same brand class group are
condensed into one and the transition condition among brand class can be grasped
easily and clearly. Judgment when and where to put the new brand becomes easy
and may be executed properly.
6
Expansion to the Second Order Lag
Here we take up Eq. (37), (38), (39) and expand the method stated in section
5.
When the variables of each matrix are condensed into one, transition matrix is
expressed as follows in the same way stated in section 5.
 x n   a1 , b1 , c1  x n −1   a 2 , b2 , c 2  x n − 2 


 

  
 y n  =  d1 , e1 , f1  y n −1  +  d 2 , e2 , f 2  y n − 2 
 z   g , h , j  z   g , h , j  z 
1
1  n −1 
2
2  n − 2 
 n  1
 2
Where
(54)
186
=
a1
Analysis of the Preference Shift of Customer Brand Selection…
1 3 3 1
1 3 3 1
=
a
,
b
bij
∑∑ ij 1 3=∑∑
3=i 1 =j 1
i 1 =j 1

=
a2
1 3 3 2
1 3 3 2
=
a
,
b
bij
∑∑ ij 2 3=∑∑
3=i 1 =j 1
i 1 =j 1

Here,
aij1 , bij1 ,, aij2 , bij2 , are items of each Block Matrix
A1 , B1 ,, A 2 , B 2 ,,
respectively and a1 , b1 , a2 , b2 , are the condensed variables of each Block
Matrix A 1 , B1 ,  , A 2 , B 2  .
7
Numerical Example
We consider the case that brand selection shifts to the same class or upper
classes. As above-referenced, transition matrix must be an upper triangular matrix.
Suppose following events occur.
(1)
(2)
(3)
(4)
(5)
(6)
(7)
(8)
(9)
𝑋𝑡−2 𝑡𝑜 𝑋𝑡−1
𝐿1
𝐿1
𝐿1
𝐿2
𝐿2
𝐿2
𝐿3
𝐿3
𝐿2
𝐿2
𝐿1
𝐿1
𝐿1
𝐿1
𝐿1
𝐿1
𝐿2
𝐿2
2 𝑒𝑣𝑒𝑛𝑡𝑠
1 𝑒𝑣𝑒𝑛𝑡
3 𝑒𝑣𝑒𝑛𝑡𝑠
1 𝑒𝑣𝑒𝑛𝑡
2 𝑒𝑣𝑒𝑛𝑡𝑠
1 𝑒𝑣𝑒𝑛𝑡
2 𝑒𝑣𝑒𝑛𝑡𝑠
3 𝑒𝑣𝑒𝑛𝑡𝑠
1 𝑒𝑣𝑒𝑛𝑡
𝑋𝑡−1 𝑡𝑜 𝑋𝑡
𝐿1
𝐿1
𝐿2
𝐿2
𝐿2
𝐿3
𝐿3
𝐿3
𝐿2
𝐿2
𝐿1
𝑀1
𝐿1
𝑀2
𝐿1
𝐿2
𝑀3
𝑀1
2 𝑒𝑣𝑒𝑛𝑡𝑠
1 𝑒𝑣𝑒𝑛𝑡
3 𝑒𝑣𝑒𝑛𝑡𝑠
1 𝑒𝑣𝑒𝑛𝑡
2 𝑒𝑣𝑒𝑛𝑡𝑠
1 𝑒𝑣𝑒𝑛𝑡
2 𝑒𝑣𝑒𝑛𝑡𝑠
3 𝑒𝑣𝑒𝑛𝑡𝑠
1 𝑒𝑣𝑒𝑛𝑡
Kazuhiro Takeyasu
(10)
(11)
(12)
(13)
(14)
(15)
(16)
(17)
(18)
(19)
(20)
(21)
(22)
(23)
(24)
(25)
(26)
(27)
(28)
(29)
(30)
(31)
(32)
(33)
(34)
(35)
(36)
(37)
(38)
(39)
(40)
𝑋𝑡−2
𝐿2
𝐿2
𝐿3
𝐿3
𝐿3
𝐿1
𝐿1
𝐿1
𝐿2
𝐿2
𝐿2
𝑀1
𝑀1
𝑀1
𝑀2
𝑀2
𝑀3
𝑀1
𝑀1
𝑀1
𝑀1
𝑡𝑜 𝑋𝑡−1
𝐿2
𝐿2
𝐿3
𝐿3
𝐿3
𝐿2
𝐿2
𝐿3
𝐿3
𝐿3
𝐿3
𝑀1
𝑀2
𝑀2
𝑀2
𝑀3
𝑀3
𝑀1
𝑀1
𝑀1
𝑀2
𝑋𝑡−2 𝑡𝑜 𝑋𝑡−1
𝑀1
𝑀2
𝑀1
𝑀2
𝑀1
𝑀3
𝑀1
𝑀3
𝑀2
𝑀2
𝑀2
𝑀2
𝑀2
𝑀2
𝑀2
𝑀2
𝑀2
𝑀3
𝑀3
𝑀3
187
1 𝑒𝑣𝑒𝑛𝑡
2 𝑒𝑣𝑒𝑛𝑡𝑠
1 𝑒𝑣𝑒𝑛𝑡
3 𝑒𝑣𝑒𝑛𝑡𝑠
2 𝑒𝑣𝑒𝑛𝑡𝑠
1 𝑒𝑣𝑒𝑛𝑡
1 𝑒𝑣𝑒𝑛𝑡
1 𝑒𝑣𝑒𝑛𝑡
𝑋𝑡−1 𝑡𝑜 𝑋𝑡
𝐿2
𝑀2
𝐿2
𝑀3
𝐿3
𝑀1
𝐿3
𝑀2
𝐿3
𝑀3
𝐿2
𝑀1
𝐿2
𝑀2
𝐿3
𝑀3
1 𝑒𝑣𝑒𝑛𝑡
2 𝑒𝑣𝑒𝑛𝑡𝑠
1 𝑒𝑣𝑒𝑛𝑡
3 𝑒𝑣𝑒𝑛𝑡𝑠
2 𝑒𝑣𝑒𝑛𝑡𝑠
1 𝑒𝑣𝑒𝑛𝑡
1 𝑒𝑣𝑒𝑛𝑡
1 𝑒𝑣𝑒𝑛𝑡
1 𝑒𝑣𝑒𝑛𝑡
2 𝑒𝑣𝑒𝑛𝑡𝑠
1 𝑒𝑣𝑒𝑛𝑡
1 𝑒𝑣𝑒𝑛𝑡
2 𝑒𝑣𝑒𝑛𝑡𝑠
2 𝑒𝑣𝑒𝑛𝑡𝑠
3 𝑒𝑣𝑒𝑛𝑡𝑠
𝑋𝑡−1 𝑡𝑜 𝑋𝑡
𝑀2
𝑈2
𝑀2
𝑈3
𝑀3
𝑈1
𝑀3
𝑈2
𝑀2
𝑈1
𝑀2
𝑈2
𝑀2
𝑈3
1 𝑒𝑣𝑒𝑛𝑡
2 𝑒𝑣𝑒𝑛𝑡𝑠
1 𝑒𝑣𝑒𝑛𝑡
1 𝑒𝑣𝑒𝑛𝑡
2 𝑒𝑣𝑒𝑛𝑡𝑠
2 𝑒𝑣𝑒𝑛𝑡𝑠
3 𝑒𝑣𝑒𝑛𝑡𝑠
2 𝑒𝑣𝑒𝑛𝑡𝑠
2 𝑒𝑣𝑒𝑛𝑡𝑠
1 𝑒𝑣𝑒𝑛𝑡
3 𝑒𝑣𝑒𝑛𝑡𝑠
3 𝑒𝑣𝑒𝑛𝑡𝑠
2 𝑒𝑣𝑒𝑛𝑡𝑠
1 𝑒𝑣𝑒𝑛𝑡
1 𝑒𝑣𝑒𝑛𝑡
2 𝑒𝑣𝑒𝑛𝑡𝑠
2 𝑒𝑣𝑒𝑛𝑡𝑠
1 𝑒𝑣𝑒𝑛𝑡
3 𝑒𝑣𝑒𝑛𝑡𝑠
2 𝑒𝑣𝑒𝑛𝑡𝑠
2 𝑒𝑣𝑒𝑛𝑡𝑠
1 𝑒𝑣𝑒𝑛𝑡
2 𝑒𝑣𝑒𝑛𝑡𝑠
𝐿3
𝐿3
𝐿3
𝑀1
𝑀2
𝑀2
𝑀2
𝑀3
𝑀3
𝑀1
𝑀1
𝑀1
𝑀2
𝑀1
𝑀2
𝑀3
𝑀1
𝑀2
𝑀3
𝑀2
𝑀3
𝑀3
𝑈1
𝑈2
𝑈3
𝑈1
𝑀3
𝑀3
𝑀3
𝑈1
𝑈2
𝑈3
2 𝑒𝑣𝑒𝑛𝑡𝑠
2 𝑒𝑣𝑒𝑛𝑡𝑠
1 𝑒𝑣𝑒𝑛𝑡
3 𝑒𝑣𝑒𝑛𝑡𝑠
3 𝑒𝑣𝑒𝑛𝑡𝑠
2 𝑒𝑣𝑒𝑛𝑡𝑠
1 𝑒𝑣𝑒𝑛𝑡
1 𝑒𝑣𝑒𝑛𝑡
2 𝑒𝑣𝑒𝑛𝑡𝑠
2 𝑒𝑣𝑒𝑛𝑡𝑠
1 𝑒𝑣𝑒𝑛𝑡
3 𝑒𝑣𝑒𝑛𝑡𝑠
2 𝑒𝑣𝑒𝑛𝑡𝑠
2 𝑒𝑣𝑒𝑛𝑡𝑠
1 𝑒𝑣𝑒𝑛𝑡
2 𝑒𝑣𝑒𝑛𝑡𝑠
188
(41)
(42)
(43)
(44)
(45)
(46)
(47)
(48)
(49)
(50)
(51)
(52)
(53)
(54)
(55)
(56)
(57)
(58)
(59)
(60)
(61)
(62)
(63)
(64)
(65)
(66)
(67)
(68)
(69)
(70)
(71)
(72)
Analysis of the Preference Shift of Customer Brand Selection…
𝑀3
𝑀3
𝑀3
𝑈1
𝑈1
𝑈1
𝑈2
𝑈2
𝑈2
𝑈3
𝑈3
𝑈3
𝑀3
𝑀3
𝑀3
𝑀3
𝑀2
𝐿3
𝐿3
𝐿3
𝑀3
𝑀3
𝑀3
𝑈1
𝑈1
𝑈1
𝑈2
𝑈2
𝑈2
𝑈3
𝑈3
𝑈3
𝑀3
𝑀3
𝑀2
𝑀2
𝑀2
𝐿3
𝐿2
𝐿1
1 𝑒𝑣𝑒𝑛𝑡
2 𝑒𝑣𝑒𝑛𝑡𝑠
2 𝑒𝑣𝑒𝑛𝑡𝑠
1 𝑒𝑣𝑒𝑛𝑡
1 𝑒𝑣𝑒𝑛𝑡
2 𝑒𝑣𝑒𝑛𝑡𝑠
1 𝑒𝑣𝑒𝑛𝑡
2 𝑒𝑣𝑒𝑛𝑡𝑠
2 𝑒𝑣𝑒𝑛𝑡𝑠
3 𝑒𝑣𝑒𝑛𝑡𝑠
2 𝑒𝑣𝑒𝑛𝑡𝑠
1 𝑒𝑣𝑒𝑛𝑡
1 𝑒𝑣𝑒𝑛𝑡
2 𝑒𝑣𝑒𝑛𝑡𝑠
1 𝑒𝑣𝑒𝑛𝑡
1 𝑒𝑣𝑒𝑛𝑡
2 𝑒𝑣𝑒𝑛𝑡𝑠
3 𝑒𝑣𝑒𝑛𝑡𝑠
2 𝑒𝑣𝑒𝑛𝑡𝑠
1 𝑒𝑣𝑒𝑛𝑡
𝑀3
𝑀3
𝑀3
𝑈1
𝑈1
𝑈1
𝑈2
𝑈2
𝑈2
𝑈3
𝑈3
𝑈3
𝑀3
𝑀3
𝑀2
𝑀2
𝑀2
𝐿3
𝐿2
𝐿1
𝐿1
𝐿1
𝐿1
𝐿1
𝐿1
𝐿2
𝐿2
𝐿2
𝐿2
𝐿2
𝐿1
𝑀1
𝑈1
𝑈2
𝑈3
𝑈1
𝑈2
𝑈3
𝑈2
𝑈3
𝑈1
𝑈3
𝑈2
𝑈1
𝑀2
𝑀1
𝑀2
𝑀1
𝑀1
𝐿2
𝐿1
𝐿1
𝐿1
𝐿2
𝐿3
𝑀1
𝑀2
𝐿2
𝐿3
𝐿1
𝑀1
𝑈1
𝑈2
𝑀1
1 𝑒𝑣𝑒𝑛𝑡
2 𝑒𝑣𝑒𝑛𝑡𝑠
2 𝑒𝑣𝑒𝑛𝑡𝑠
1 𝑒𝑣𝑒𝑛𝑡
1 𝑒𝑣𝑒𝑛𝑡
2 𝑒𝑣𝑒𝑛𝑡𝑠
1 𝑒𝑣𝑒𝑛𝑡
2 𝑒𝑣𝑒𝑛𝑡𝑠
2 𝑒𝑣𝑒𝑛𝑡𝑠
3 𝑒𝑣𝑒𝑛𝑡𝑠
2 𝑒𝑣𝑒𝑛𝑡𝑠
1 𝑒𝑣𝑒𝑛𝑡
1 𝑒𝑣𝑒𝑛𝑡
2 𝑒𝑣𝑒𝑛𝑡𝑠
1 𝑒𝑣𝑒𝑛𝑡
1 𝑒𝑣𝑒𝑛𝑡
2 𝑒𝑣𝑒𝑛𝑡𝑠
3 𝑒𝑣𝑒𝑛𝑡𝑠
2 𝑒𝑣𝑒𝑛𝑡𝑠
1 𝑒𝑣𝑒𝑛𝑡
2 𝑒𝑣𝑒𝑛𝑡𝑠
2 𝑒𝑣𝑒𝑛𝑡𝑠
1 𝑒𝑣𝑒𝑛𝑡
1 𝑒𝑣𝑒𝑛𝑡
3 𝑒𝑣𝑒𝑛𝑡𝑠
2 𝑒𝑣𝑒𝑛𝑡𝑠
1 𝑒𝑣𝑒𝑛𝑡
2 𝑒𝑣𝑒𝑛𝑡𝑠
1 𝑒𝑣𝑒𝑛𝑡
1 𝑒𝑣𝑒𝑛𝑡
3 𝑒𝑣𝑒𝑛𝑡𝑠
2 𝑒𝑣𝑒𝑛𝑡𝑠
Kazuhiro Takeyasu
(73)
(74)
(75)
(76)
(77)
(78)
(79)
(80)
(81)
(82)
(83)
(84)
(85)
(86)
(87)
(88)
(89)
(90)
189
𝑋𝑡−1 𝑡𝑜 𝑋𝑡
𝐿3
𝑀2
𝑀3
𝑀1
𝑀1
𝑀2
𝑀3
𝑈1
𝑈2
𝑈3
𝑈1
𝑈1
𝑈3
𝑈3
𝑈2
𝑈2
𝐿3
𝑀2
𝐿3
𝑀2
𝑀3
𝑀2
𝑈1
𝑈3
𝑈2
𝑈1
𝑈2
𝑈3
𝑈2
𝑈3
𝑈2
𝑈1
𝑈1
𝑈3
𝐿1
𝐿1
2 𝑒𝑣𝑒𝑛𝑡𝑠
1 𝑒𝑣𝑒𝑛𝑡
1 𝑒𝑣𝑒𝑛𝑡
3 𝑒𝑣𝑒𝑛𝑡𝑠
3 𝑒𝑣𝑒𝑛𝑡𝑠
2 𝑒𝑣𝑒𝑛𝑡𝑠
1 𝑒𝑣𝑒𝑛𝑡
1 𝑒𝑣𝑒𝑛𝑡
2 𝑒𝑣𝑒𝑛𝑡𝑠
2 𝑒𝑣𝑒𝑛𝑡𝑠
1 𝑒𝑣𝑒𝑛𝑡
2 𝑒𝑣𝑒𝑛𝑡𝑠
2 𝑒𝑣𝑒𝑛𝑡𝑠
1 𝑒𝑣𝑒𝑛𝑡
1 𝑒𝑣𝑒𝑛𝑡
2 𝑒𝑣𝑒𝑛𝑡𝑠
3 𝑒𝑣𝑒𝑛𝑡𝑠
1 𝑒𝑣𝑒𝑛𝑡
𝑿𝑡−2
𝑿𝑡−1
𝑿𝑡
Vector � 𝒀𝑡−2 � , � 𝒀𝑡−1 � , � 𝒀𝑡 � in these cases are expressed as follows.
𝒁𝑡−2
𝒁𝑡−1
𝒁𝑡
We show (1) and (2) cases for example.
190
Analysis of the Preference Shift of Customer Brand Selection…
𝑋𝑡−2




 Xt − 2 






 Yt − 2 
(1)
=








 Zt − 2 




(2)
𝑋𝑡−1
0


 


0
 Xt −1 
0


 


0


 0=
 ,  Yt −1 
 


0


1


 


0
 Zt −1 
0


 



 0
  

 Xt − 2   0 
 0

  

 0

 Y  =  0 ,
 t −2   

 0
  

 1

 Zt − 2   0 
  


 0
𝑋𝑡
0
 
 
 
0
 Xt 
0
 
 
 
0
 
=
0  ,  Yt 
 
 
0
 
1
 
 
 
0
 Zt 
0
 
 
 

 0

  
 Xt −1   0 

 0

  

 0
 Y  =  0 ,
 t −1   
 0

  


 0
 Z t −1   1 

  

 0
0
 
0
0
 
0
0
 
0
1
 
0
0
 
  0
   
 Xt   0 
  0
   
  0
 Y  = 0
 t  
  0
   
  0
 Zt   1 
   
  0
Substituting these to equation (40), we obtain the following estimated Matrix.
2

2
4

0
P =  0
0

0
0

0
3 2 5 8 6 0 1 0 1 2 1 5 4 1 0 0 0

3 4 1 4 6 0 3 0 1 1 2 3 3 2 0 0 0
4 5 3 7 4 0 0 0 2 2 3 5 5 2 0 0 0

0 0 5 0 0 2 3 3 0 0 0 3 0 3 2 3 1
0 0 3 5 0 5 2 5 0 0 0 3 1 2 3 3 3 
0 0 0 2 4 3 2 4 0 0 0 2 1 2 4 3 2

0 0 0 0 0 5 4 3 0 0 0 0 0 0 2 0 3
0 0 0 0 0 2 5 3 0 0 0 0 0 0 1 2 3

0 0 0 0 0 1 4 3 0 0 0 0 0 0 0 3 1 
Kazuhiro Takeyasu
191
8 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 


 0 10 0 0 0 0 0 0 0 0 5 0 0 0 0 0 0 0 
 0 0 11 0 0 0 0 0 0 0 0 6 0 0 0 0 0 0 


 0 0 0 17 0 0 0 0 0 0 0 0 9 0 0 0 0 0 
 0 0 0 0 26 0 0 0 0 0 0 0 10 10 2 0 0 0 


 0 0 0 0 0 20 0 0 0 0 0 0 2 6 10 0 0 0 


 0 0 0 0 0 0 21 0 0 0 0 0 0 0 0 8 0 1 
 0 0 0 0 0 0 0 21 0 0 0 0 0 0 0 3 9 2 


 0 0 0 0 0 0 0 0 21 0 0 0 0 0 0 1 5 10 
×
4 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0


0 5 0 0 0 0 0 0 0 0 5 0 0 0 0 0 0 0 
0 0 6 0 0 0 0 0 0 0 0 6 0 0 0 0 0 0 


 0 0 0 9 10 2 0 0 0 0 0 0 21 0 0 0 0 0 


 0 0 0 0 10 6 0 0 0 0 0 0 0 16 0 0 0 0 
 0 0 0 0 2 10 0 0 0 0 0 0 0 0 12 0 0 0 


 0 0 0 0 0 0 8 3 1 0 0 0 0 0 0 12 0 0 
 0 0 0 0 0 0 0 9 5 0 0 0 0 0 0 0 14 0 


0
0
0
0
0
0
1
2
10
0
0
0
0
0
0
0
0
13


0.751
0.912
 0.250 0.200 0.200 0.522

0.282
0.521
 0.250 0.400 0.400 0.104
 0.500 0.400 0.400 0.130
0.174
0.130

0
0
0.151 − 0.304 − 0.497
 0
=  0
0
0
0.184
0.186 − 0.107
 0
0
0
− 0.007 0.125
0.277

0
0
0
0
0
 0
 0
0
0
0
0
0
 0
0
0
0
0
0

0.016 0.087
0.047 0.262
0
0
0.062 0.086
0.191 − 0.001
0.009 − 0.063
0.289 0.340
0.120 0.275
0.111 0.228
0.040
0.120
0
0.132
0.143
0.066
0.231
0.126
0.215
0 0.200 − 0.033 − 0.430 − 0.561 − 0.802
0 − 0.200 − 0.067 − 0.085 − 0.184 − 0.314
0
0
0.100
0.087
0.155
0.029
0
0
0
0.270
0.376
0.715
0
0
0
− 0.014 − 0.013 0.225
0
0
0
0.012 − 0.119 − 0.085
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
− 0.036
− 0.107
0
0.093
0.111
0.337
− 0.130
− 0.076
− 0.149
− 0.070
− 0.211
0
0.112
0.164
0.231
− 0.301
− 0.079
− 0.009
−1
− 0.045 

− 0.136 
0 

− 0.042 
0.106 
0.112 

− 0.021 
0.082 

− 0.132 
The Block Matrices make upper triangular matrix as is supposed. We can
confirm that 𝑲4 , 𝑲7 , 𝑲8 , 𝑳4 , 𝑳7 , 𝑳8 , 𝑴2 , 𝑴3 , 𝑴4 , 𝑴6 , 𝑴7 , 𝑴8 , 𝑵4 , 𝑵7 , 𝑵8 , 𝑸2 , 𝑸3 ,
𝑸6 , 𝑹2 , 𝑹3 , 𝑹4 , 𝑹6 , 𝑹7 , 𝑹8 are all 0. 𝑴1 , 𝑴5 , 𝑴9 , 𝑹1 , 𝑹5 , 𝑹9 become diagonal
Matrices as we have assumed.
192
Analysis of the Preference Shift of Customer Brand Selection…
Condensing the variables into one as stated in section 6, we obtain the
following equation.
 x n   1 1.175 0.190  x n −1   0 − 0.702 − 0.202  x n −1 
  

 


0.408  y n −1 
 y n  =  0 0.003 0.208  y n −1  +  0 0.456
 z  0
0
0.645  z n −1   0
0
− 0.272  z n −1 
 n 
This is by far a simple one compared with the result obtained so far.
The variables of the same brand class are condensed into one.
The matrix of P1 and P2 are 9 × 9 respectively, but the condensed versions
become 3× 3 Matrix. Therefore the brand transition condition among brand class
can be grasped easily and clearly.
8
Conclusion
It is often observed that consumers select the upper class brand when they
buy the next time. Suppose that the former buying data and the current buying
data are gathered. Also suppose that the upper brand is located upper in the
variable array. Then the transition matrix become an upper triangle matrix under
the supposition that former buying variables are set input and current buying
variables are set output. If the top brands are selected from the lower brand in
jumping way, corresponding part in an upper triangle matrix would be 0. The
goods of the same brand group would compose the Block Matrix in the transition
matrix. Condensing the variables of the same brand group into one, analysis
Kazuhiro Takeyasu
193
becomes easier to handle and the transition of Brand Selection can be easily
grasped.
In this paper, equation using transition matrix stated by the Block Matrix is
expanded to the second order lag and the method of condensing the variable stated
above is also applied to this new model. Unless planner for products does not
notice its brand position whether it is upper or lower than other products, matrix
structure make it possible to identify those by calculating consumers’ activities for
brand selection. Thus, this proposed approach enables to make effective marketing
plan and/or establishing new brand. Various fields should be examined hereafter.
References
[1] Aker, D.A., Management Brands Equity, Simon & Schuster, USA, 1991.
[2] Katahira, H., Marketing Science (In Japanese), Tokyo University Press, 1987.
[3] Katahira, H.,Y, Sugita, Current movement of Marketing Science (In
Japanese), Operations Research, 4, (1994), 178-188.
[4] Takahashi, Y. and T. Takahashi, Building Brand Selection Model Considering
Consumers Royalty to Brand (In Japanese), Japan Industrial Management
Association, 53(5), (2002), 342-347.
[5] Yamanaka, H., Quantitative Research Concerning Advertising and Brand
Shift (In Japanese), Marketing Science, Chikra-Shobo Publishing, 1982.