A spline aproximation of the factors path in MDFrez
Nicoleta Breaz , Daniel Breaz: A spline aproximation of the factors path in MDF
A SPLINE APROXIMATION OF THE FACTORS PATH IN MDF
by
Nicoleta Breaz , Daniel Breaz
Abstract: It is known that the factorial index obtained by the factors path method (MDF)
depend on the path. Beginning with some particular cases of the factors path we will propose in
this paper a spline factors path.
The statistical index is an indicator which measures the variation of a variable,
variation that can be registered both in space and time. If the variable Z is depending
on other variables
X 1 , X 2 ,..., X n , Z = f ( X 1 , X 2 ,..., X n ) we could speak about a
single index of the integral variation and n indices of the factorial variations that are
generated by the explicative variables X i .The variable which we are refering to in this
paper ,is the value of a panel with n goods Z = ∑ pi qi where p = ( p1 , p 2 ,..., p n ) and
n
q = (q1 , q 2 ,..., q n ) are the prices and quantities vectors coresponding to those n goods.
In this case, we speak about an index of the prices which is actually a factorial index
of Z with respect to p. This index expresses the partial variation of the panel value,
variation measured from the base situation j to the observed situation k.The cause of
this partial variation of the value is in this case, the variation of prices.
i =1
There are many methods of calculus for the factorial indices and implicitly for
prices indices.
Here we make a short presentation of four of these methods.
Laspeyres method
I zk // pj (•, L ) =
∑ pi (k )qi ( j )
m
i =1
m
∑ pi ( j )qi ( j )
i =1
I zk //qj (L,•) =
∑ pi ( j )qi (k )
m
i =1
m
∑ pi ( j )qi ( j )
i =1
35
Nicoleta Breaz , Daniel Breaz: A spline aproximation of the factors path in MDF
∑ pi (k )qi (k )
Paasche method
I zk // pj (•, P ) =
m
i =1
m
∑ pi ( j )qi (k )
I zk //qj (P,•) =
i =1
∑ pi (k )qi (k )
m
i =1
m
∑ pi (k )qi ( j )
.
i =1
Edgeworth method
qi ( j ) + qi (k )
2
I zk // pj (•, E ) = im=1
q ( j ) + q (k )
∑ pi ( j ) i 2 i
i =1
∑ pi (k )
m
and simillary I zk //qj (E ,•) .
Factors path method(MDF)
Definition 1.
A factorial index of a variable z = z (t ) = f (x1 (t ), x 2 (t ),...x m (t ))
to X i factor and ( j, k ) time interval, given by the formula
Ι kz // xji = exp
f x´ ( x1 , x2 ,...xm )
∫ f (x , x ,...x ) dxi
(Pj Pk )
1
2
m
(
with respect
(1)
)
is called the MDF index.
In the formula (1) Pj , Pk represents the segment of arc i.e. a part of the factors path,
which links the points Pj , Pk , in R m space, Pj (x1 ( j )...x m ( j )) , Pk (x1 (k )...x m (k )) that
symbolize the base and the observed situation. The path is given by the equations
x1 = x1 (t ),...x m = x m (t ) where t is a time parameter.
Definition 2.
An index given by one of next formula (2) with (2’) or (2’’)is called the prices
index MDF.
Ι kz // pj = ∏ Ι kz // pj i
m
(2)
i =1
Ι kz // pj i = exp ∫M
z ′pi
jMk
z
dp i = exp ∫M
qi
jMk
∑ pi qi
m
i =1
36
dp i = exp ∫t 1
t
0
q1 (t ) pi′ (t )
∑ qi (t ) pi (t )
m
i =1
dt
(2’)
Nicoleta Breaz , Daniel Breaz: A spline aproximation of the factors path in MDF
Ι kz // pj
= exp ∫t
∑
m
t1
0
i =1
m
q i (t ) p1′ (t )
∑ qi (t ) pi (t )
dt
(2’’)
i =1
Remark 3.
Contrary to the first methods which don’t take into account the real data of
prices and quantities in j and k situations, the MDF method realizes all these. The only
inconvenient of this method is that the index depends on the path described by the
price and quantity factors between the situations nominated by Pj and Pk . This fact,
although well sustained by the economic reality, makes impossible the calculus of
index whenever the path is unknown.
We present below some particular paths of factors.
The linear path of the factors
In particular cases of a linear path given by the equations
i = 1, m
xi (t ) = xi ( j ) + t [xi (k ) − xi ( j )] = xi ( j ) + t∆xi
t ∈ [0,1]
the index (1) becomes
1 f ′ (t )∆xi
Ι kz // xji = exp ∫0 x
(3)
dt
f (t )
For an index of prices we use the equations
pi (t ) = pi ( j ) + t ( pi (k ) − pi ( j )) = pi ( j ) + t∆pi
with i = 1, m , t ∈ [0,1]
qi (t ) = qi ( j ) + t (qi (k ) − qi ( j )) = qi ( j ) + t∆qi
This situation can be represented in a space of 2m dimension:
q
Pk
C
Pj
p
So the index (2) becomes
37
Nicoleta Breaz , Daniel Breaz: A spline aproximation of the factors path in MDF
Ι kz // pj i = exp ∫0
1
[qi ( j ) + t∆qi ]∆pi
dt
∑ [qi ( j ) + t∆qi ]( pi ( j ) + t∆pi )
i =1
or
Ι kz // pj = exp ∫0
1
(4)
m
∑ [qi ( j ) + t∆qi (t )]∆pi
m
i =1
∑ [qi ( j ) + t∆qi ]( pi ( j ) + t∆pi )
m
(4’)
dt
i =1
Examples of polygonal paths
Proposition 4.
The Laspeyres index Ι kz // pj (•, L ) is MDF index on ∆ 1 path,(figure no.1 )
q
P1 ( p( j ), q( j ))
P2 ( p(k ), q( j ))
P3 ( p(k ), q(k ))
∆1 = P1 P2 ∪ P2 P3
P2
P1
P3
p
Fig. 1
Proof: We have Ι kz // pj (MDF , ∆ 1 ) = Ι z / p (MDF , P1 P2 ) ⋅ Ι z / p (MDF , P2 P3 )
The path P1 P2 is given by the equation qi = qi ( j ) , pi = pi ( j ) + t∆pi , t ∈ [0,1] so the
index Ι z / p (MDF , P1 P2 ) becomes
Ι z / p (MDF , P1 P2 ) = exp ∫0
1
∑ qi ( j )∆pi
m
i =1
dt = exp ∫0
∑ qi ( j )[ p i ( j ) + t∆p i ]
m
i =1
38
1
∑ qi ( j )∆p i
m
i =1
dt
∑ (q i ( j ) pi ( j ) + tq i ( j )∆p i )
m
i =1
=
Nicoleta Breaz , Daniel Breaz: A spline aproximation of the factors path in MDF
= exp ln ∑ (qi ( j ) pi ( j ) + tqi ( j )∆pi ) =
1
m
i =1
=Ι
∑ (qi ( j ) pi ( j ) + qi ( j )∆pi ) ∑ qi ( j ) pi (k )
m
i =1
0
(•, L )
∑ qi ( j ) pi ( j )
m
=
i =1
m
i =1
m
∑ qi ( j ) pi ( j )
=
i =1
qi = qi ( j ) + t∆qi
so the index
The path P2 P3 is given by the equation
pi = pi (k ) ⇒ ∆pi = 0
Ι z / p (MDF , P2 P3 ) becomes
k/ j
z/ p
Finally, we have Ι kz // pj (MDF , ∆1 ) = Ι z / p (MDF , P1 P2 ) = Ι kz // pj (•, L ) .
Proposition 5.
The Paasche index Ι kz // pj (•, P ) is MDF index on ∆ 2 path,(figure no.2 )
q
P1 ( p( j ), q( j ))
P2 ( p( j ), q(k ))
P3 ( p(k ), q(k ))
∆ 2 = P1 P2 ∪ P2 P3
P1
P2
P3
p
Fig. 2
Proof: We have Ι kz // pj (MDF , ∆ 2 ) = Ι z / p (MDF , P1 P2 ) ⋅ Ι z / p (MDF , P2 P3 )
Similary with the previous proposition we proove Ι z / p (MDF , P1 P2 ) = 1.
The path P2 P3 is given by the equation qi = qi (k ), pi = pi ( j ) + t∆pi , t ∈ [0,1] so the
index Ι z / p (MDF , P2 P3 ) becomes
Ι z / p (MDF , P2 P3 ) =
∑ pi (k )qi (k )
m
i =1
m
∑ pi ( j )qi (k )
i =1
= Ι kz // pj (•, P )
Proposition 6.
The Edgeworth index Ι kz // pj (•, E ) is MDF index on ∆ 3 path,(figure no.3 )
39
Nicoleta Breaz , Daniel Breaz: A spline aproximation of the factors path in MDF
P1 ( p( j ), q( j ))
q( j ) + q(k )
P2 p( j ),
2
q ( j ) + q (k )
P3 p (k ),
2
P4 ( p(k ), q(k ))
∆ 3 = P1 P2 ∪ P2 P3 ∪ P3 P4
P1
q
P3
P2
P4
p
Fig. 3
Proof:
The path P2P3 is given by the equations
qi ( j ) + qi (k )
and pi = pi ( j ) + t∆pi , t ∈ [0,1]
2
so the index Ι z / p (MDF , P2 P3 ) becomes
qi =
Ιz / p (MDF,
m q ( j) + q (k )
m q ( j ) + q (k )
i
i
i
i
∆pi
∆pi
∑
∑
1
1
2
2
i =1
i =1
P2P3 ) = exp∫0 m
dt = exp∫0 m
dt =
qi ( j) + qi (k)
qi ( j) + qi (k)
qi ( j) + qi (k)
⋅ pi ( j) + t∆pi
⋅ ( pi ( j) + t∆pi )
∑
∑
qi ( j) + qi (k)
pi (k)
q ( j) + qi (k)
q ( j) + qi (k)
2
⋅ pi ( j) + t∆pi i
= Ιkz // pj (•, E)
exp ln ∑ i
= mi=1
(
)
(
)
+
q
j
q
k
2
2
i=1
0
∑ i 2 i ⋅ pi ( j)
i =1
Finally we have:
Ιkz // pj (MDF, ∆3 ) = Ι z / p (MDF, P1P2 ) ⋅ Ι z / p (MDF, P2P3 ) ⋅ Ι z / p (MDF, P3P4 ) =
i =1
2
∑
i =1
2
2
m
1
m
= 1⋅ Ι z / p (⋅ E) ⋅1 = Ιkz // pj (⋅ E)
Corollary 7. The MDF price index on the path which is presented in the figure no.4 is
a product of Laspeyres indices.
q
Mj
M1
M2
M3
Mk
p
Fig. 4
40
Nicoleta Breaz , Daniel Breaz: A spline aproximation of the factors path in MDF
(
)
Ιkz // pj (MDF) = Ιz / p MDF, MjM1 ⋅ Ιz / p(MDF, M1M2 ) ⋅ Ιz / p(MDF, M2M3) ⋅ Ιz / p(MDF, M3Mk ) = ∏Ιz / p(⋅ L)
The problem is to determine the path that aproximate the most the empiric path
of the
prices and quantities between j and k
The case of a spline path
Remark 8.
At the beginning ,we define the factorial indices of prices MDF with respect to a
family
of
parabolas.
We
consider
the
parametric
equations
2
2
qi = ai + bi t + ci t ,
pi = ai′ + bi′t + ci′t , t ∈ [0,1] (5)
where qi (0) = qi ( j ), qi (1) = qi (k ), pi (0) = pi ( j ), pi (1) = pi (k ) that become
q i = qi ( j ) + ∆qi t + ci t 2
t ∈ [0,1] , ci ci′ ∈ R
p i = pi ( j ) + ∆pi t + ci' t 2
If ci = ci′ = 0 we have the factorial index of prices on the linear path .Otherwise we
obtain
Ι kz // pj = exp ∫0
1
∑ (qi ( j ) + ∆qi t + ci t 2 )⋅ (∆pi + 2tci′ )
n
∑ (qi ( j ) + ∆qi t + ci t
i =1
n
i =1
2
)⋅ (p ( j ) + ∆p t + c′t )
dt
(6)
2
i
i
i
where the choice of constants ci , ci′ , n .could be made taking into account the new
information for the situation t ∈ [ j , k ] (moment of time).
The formula (6) can be generated when in (5) we have the polynoms of m degree, so
the index will depend on a number of 2n(m-1) constants.
Remark 9.
We suppose that we know the empiric data
p( j ), q( j ), p(l ), q(l ), p(k ), q(k ), l ∈ [ j , k ] and we want to calculate the
index Ι kz // pj (MDF ) .
a)If we consider that between every two points the path is linear then from the point
( p( j ), q( j )) to the point ( p(k ), q(k )) the path is a polygonal curve, so:
Ι kz // pj (MDF ) = Ι lz// jp (MDF − liniar ) ⋅ Ι kz // lp (MDF − liniar )
qi = ai + bi t + ci t 2
,
b)The parametric equations
pi = ai′ + bi′t + ci′t 2
t ∈ [0,2]
qi (0) = qi ( j )
qi (1) = qi (l )
qi (2) = qi (k )
pi (0 ) = pi ( j )
pi (1) = pi (l )
pi (2) = pi (k )
41
i = 1, n having the conditions
Nicoleta Breaz , Daniel Breaz: A spline aproximation of the factors path in MDF
become qi = qi ( j ) +
4∆qi
− ∆qi
∆q
− 2∆qi
t+ i
t 2 and similarly for pi.
2
2
So, one could observe that in both cases, although the index measures the variation
from the moment j to moment k , it also depends of the intermediar moment l.
Therefore, we calculate Ι kz // pj (MDF , ϕ ) where ϕ is the curve which links the points
l/ j
k/ j
k/ j
l/ j
j,k,l.
Remark 10.
If we generalize the remark (9) we obtain that the index MDF for the points j,k
and for the empiric path between j and k which passed by the intermediar points
coresponding to m moments of time t1 , t 2 ,....t m−1 and t 0 = j , t m = k can be
calculate either as a product of m linear path indices or by reducing the problem to
one of Lagrange
polynomial interpolation on the nodes (0,...,m)
with
pi (s ) = pi (t s ), s = 0, m , i = 1, n .Thus it can be determined a number of 2n Lagrange
polynoms of m degrees at the most qi = ( p m )i , pi = ( p ′m )i , i = 1, n .
The choosing one method in spite of the other is a meter of comparison of the errors
resulted in each case.
Remark 11.
For a better comparison of errors and taking into account that the polygonal
line is the graphic representation of a particular spline function we define the factorial
index of prices attached to Mj,Mk points and the linear path M j , M t1 , M t 2 ....M t m , M k
by using the spline polynomial interpolation problem.
We use the next notions and results from the spline functions theory, the interpolation
properties of these functions being the reason to recomand a spline path in MDF:
Definition 12. Let ∆ : −∞ < x1 < x 2 < ... < x N < ∞, x0 = −∞, x N +1 = ∞ .The function
s : R → R is called m degrees spline with nodes x1 < x 2 < ... < x N if
a)
s / Ι i ∈ Pm (Ι i ), Ι i = (xi , xi +1 ), i = 0.N
s ∈ c m −1 (R )
Definition 13.The function define in (12) which satisfies s (t i ) = y i , i = 1, N + m + 1 is
b)
called the interpolation polynomial spline function attached to the vector {y i }i =1
m + N +1
and
the points t1 < t 2 < ... < t m+ N +1 .
Proposition 14. Any polynomial spline function of m degrees can be expressed
uniquely s ( x ) = p0 ( x ) + ∑ ci ( x − xi )+ , p 0 ∈ Pm .
N
i =1
m
42
Nicoleta Breaz , Daniel Breaz: A spline aproximation of the factors path in MDF
Proposition 15. Let m > 0, t1 < t 2 < ... < t m + N +1 , the real numbers y i , i = 1, m + N + 1 ,
and the division ∆ , with the nodes x1 < x 2 < ... < x N which satisfy the relations
t i < xi < t i + m+1 , i = 1, N .
Then, a unique polynomial spline function of m degree exists so that
s (t i ) = y i , i = 1, m + N + 1 .
Proposition 16. Any of odd degrees with the nodes ∆ : a = x1 < x 2 < ... < x N = b, can
be written as s 2 m −1 (x ) = p 2 m −1 (x ) + ∑ ci (x − xi )
N
2 m −1
with
p 2 m −1 ∈ P2 m−1 , ci ∈ R and s 2 m −1 ∈ c 2 m −2 [a, b] .
Definition 17. The polynomial spline function of 2m-1 degree which satisfy the
conditions s 2 m−1 (xi ) = yi , ∀ i = 1, N
i =1
+
s2( mj )−1 (x1 ) = s2( mj )−1 (xN ) = 0, j = m, m + 1, ...2m - 1. (s ∈ Pm −1 x < x1 şi x > x n ) is called
the natural spline function.
Theorem 18. Any natural spline function can be written uniquely
s(x ) =
∑ ai x i + ∑ ci (x − xi )+
m −1
N
i =0
i =1
2 m −1
, where ∑ ci xir = 0, r = 0, 1, ...m − 1
N
i =0
Theorem 19. Let m ≥ 1 , N ≥ m and y j , j = 1, N . Then only a natural spline function
( )
exists, so that s x j = y j , j = 1, N .
Remark 20.
In the case when the degree of polynomial spline function is three then we
speak about cubic spline function.
Remark 21.
With respect to the index problem we use the empiric data for m time
moments and we interpolate them by 2m spline functions pi and qi.
Remark 22.
Let the time moments j and k and N-2 intermediar moments j 2 ,... j N −1 for
which we know the prices and quantities.
We consider a parameter t ∈ [a, b] , qi = qi (t ) , pi = pi (t ) so that qi (t1 ) = qi ( j ) ,
qi (t 2 ) = qi ( j 2 ), ...qi (t N −1 ) = qi ( j N −1 ), qi (t N ) = qi (k ) and similar for p , ∀ i = 1, n ,
t s being the division points ∆ : a = t1 < t 2 < ... < t N = b .
a)To determine the cubic spline function qi that satisfies the interpolation conditions
qi (t s ) = qi ( j s ), s = 1, N we must have with respect to proposition (4) a number of
N+2 conditions because there are N+2 coefficients (the interpolation conditions are
imposed on nodes).
There are more ways to determine the coefficients :
43
Nicoleta Breaz , Daniel Breaz: A spline aproximation of the factors path in MDF
-To the interpolations conditions (in number of N) we add qi′′(t1 ) = 0 , qi′′(t N ) = 0
and thus unique natural cubic spline functions results.
-If we consider two nodes of the function we have four coefficients and after the
use of this N conditions both in nodes and in points t 2 ,...t N −1 , a Lagrange interpolation
polynom it results.
-At last, if we consider N-2 nodes x1 < x 2 < ... < x N − 2 between the points t1 ,...t N it
results a function with N-2 coefficient defined on the interval (x1 , x 2 ),...(x N −3 , x N − 2 ) .
The coefficients result from N-2 interpolation conditions in nodes and the two
conditions in t j1 and t j2 that are not nodes.
∑ c ij (t − t j )
-Generally speaking, we can determine such an interpolation cubic spline given by
qi (t ) = ∑ bli (t − t1 ) +
2
l
N −1
3
l =o
+
j =1
l
N −1
+
2
+
j =1
pi (t ) = ∑ bli′ (t − t1 ) +
l =o
nodes and qi , pi ∈ C 2 [a, b] .
∑ c ij (t − t j )
′
3
, i = 1, n .
+
where qi and pi are polynoms of three degree at the most in the interval generated by
We have Ι kz // pj (MDF , ϕ ) = exp ∫ϕ
∑ qi dp
i
∑ pi qi
where ϕ will be the spline curve that
passes through j,k and through the intermediar points, being in fact the graphic
representations of interpolation spline functions.
∫ϕ = ∫t
tN
1
= ∫t 2 + ∫t 3 +... + ∫t N
t
t
t
1
2
N −1
In each case of these integrals the spline parametrization becomes a polynomial
parametrization.
In the third case mentioned above, when only x1 < x 2 < ... < x N − 2 nodes exist we
have
∫ϕ = ∫t
tN
1
= ∫x 2 + ∫x 3 +... + ∫x N − 2 .
x
x
1
2
x
N −3
b)In a similar way we use a spline function of m degree.
If interpolation is not made in nodes but in points t1 < t 2 < ... < t m + N +1 and we know
( )
( )
the values qi t j , pi t j , j = 1, m + N + 1 we will consider, for example, the quantity
qi as a spline polynomial function of m degree with nodes x1 < x 2 < ... < x N choosed
so that t l < xl < t l + m +1 , l = 1, N .
According to proposition (15) there is only a spline polynomial function of m degree
which solves the interpolation conditions in points t j , j = 1, m + N + 1 and it has the
44
Nicoleta Breaz , Daniel Breaz: A spline aproximation of the factors path in MDF
form q i (t ) = hi (t ) + ∑ cli (t − t l )+ , where hi ∈ Pm and similary
N
m
l =1
p i (t ) = hi′ (t ) + ∑ cl (t − t l )+
i′
N
l =1
m
Then the index MDF leads to an integral of type
∫ϕ = ∫t
t m + N +1
1
= ∫t 1 + ∫x 2 +... + ∫x N + ∫xm + N +1
x
x
1
1
x
t
N −1
N
It is more practically if we use the spline natural functions. In this case we consider N
data regarding to p and q as interpolation conditions on nodes t1 < t 2 < ... < t N .
In respect with proposition (19) if m ≥ 1, N ≥ m and we have interpolation
conditions in nodes there is only an interpolation spline functions. Taking to account
the proposition (18) theorem we can write
q i (t ) =
pi (t ) =
∑ ali t l + ∑ c ij (t − t j )+
m −1
N
l =0
j =1
2 m −1
(
∑ ali 't l + ∑ c ij' t − t j
m −1
N
l =0
j =1
)
2 m −1
+
where the coefficients will be determined from the interpolation conditions( N
conditions for every qi and pi ) plus the conditions
∑ c ij t rj
N
j =1
=0
, r = 0,1,...m − 1 ,
∑ c ij't rj
N
j =1
= o, i = 1, n
qi( j ) (t1 ) = qi( j ) (t N ) = 0 , ∀j = m, m + 1, 2m − 1
pi( j ) (t1 ) = pi( j ) (t N ) = 0
Remark 23.
If we take into account that the empiric data ( pi , qi ), i = 1, n obtained for N different
moments of time comprise errors then we speak rather of a numerical adjustment.
Analogous, we have the spline functions q = f ( p ) respectively regression model of
spline type q = f ( p ) + ε .
REFERENCES
[1] A. Vogt, J.Barta,- The Making of Tests for Index Number-Mathematical Methods
of Descriptive Statistics, Physica-Verlag, Heidelberg 1997.
[2] Ghe. Micula- Funcţii spline şi aplicaţii, Editura Tehnică, Bucureşti 1978.
45
Nicoleta Breaz , Daniel Breaz: A spline aproximation of the factors path in MDF
Authors:
Nicoleta Breaz - „ 1 Decembrie 1918” University of Alba Iulia, str. N.Iorga, No.13,
Departament of Mathematics and Computer Science, nbreaz@lmm.uab.ro
Daniel Breaz - „ 1 Decembrie 1918” University of Alba Iulia, str. N.Iorga, No.13,
Departament of Mathematics and Computer Science, dbreaz@lmm.uab.ro
46
A SPLINE APROXIMATION OF THE FACTORS PATH IN MDF
by
Nicoleta Breaz , Daniel Breaz
Abstract: It is known that the factorial index obtained by the factors path method (MDF)
depend on the path. Beginning with some particular cases of the factors path we will propose in
this paper a spline factors path.
The statistical index is an indicator which measures the variation of a variable,
variation that can be registered both in space and time. If the variable Z is depending
on other variables
X 1 , X 2 ,..., X n , Z = f ( X 1 , X 2 ,..., X n ) we could speak about a
single index of the integral variation and n indices of the factorial variations that are
generated by the explicative variables X i .The variable which we are refering to in this
paper ,is the value of a panel with n goods Z = ∑ pi qi where p = ( p1 , p 2 ,..., p n ) and
n
q = (q1 , q 2 ,..., q n ) are the prices and quantities vectors coresponding to those n goods.
In this case, we speak about an index of the prices which is actually a factorial index
of Z with respect to p. This index expresses the partial variation of the panel value,
variation measured from the base situation j to the observed situation k.The cause of
this partial variation of the value is in this case, the variation of prices.
i =1
There are many methods of calculus for the factorial indices and implicitly for
prices indices.
Here we make a short presentation of four of these methods.
Laspeyres method
I zk // pj (•, L ) =
∑ pi (k )qi ( j )
m
i =1
m
∑ pi ( j )qi ( j )
i =1
I zk //qj (L,•) =
∑ pi ( j )qi (k )
m
i =1
m
∑ pi ( j )qi ( j )
i =1
35
Nicoleta Breaz , Daniel Breaz: A spline aproximation of the factors path in MDF
∑ pi (k )qi (k )
Paasche method
I zk // pj (•, P ) =
m
i =1
m
∑ pi ( j )qi (k )
I zk //qj (P,•) =
i =1
∑ pi (k )qi (k )
m
i =1
m
∑ pi (k )qi ( j )
.
i =1
Edgeworth method
qi ( j ) + qi (k )
2
I zk // pj (•, E ) = im=1
q ( j ) + q (k )
∑ pi ( j ) i 2 i
i =1
∑ pi (k )
m
and simillary I zk //qj (E ,•) .
Factors path method(MDF)
Definition 1.
A factorial index of a variable z = z (t ) = f (x1 (t ), x 2 (t ),...x m (t ))
to X i factor and ( j, k ) time interval, given by the formula
Ι kz // xji = exp
f x´ ( x1 , x2 ,...xm )
∫ f (x , x ,...x ) dxi
(Pj Pk )
1
2
m
(
with respect
(1)
)
is called the MDF index.
In the formula (1) Pj , Pk represents the segment of arc i.e. a part of the factors path,
which links the points Pj , Pk , in R m space, Pj (x1 ( j )...x m ( j )) , Pk (x1 (k )...x m (k )) that
symbolize the base and the observed situation. The path is given by the equations
x1 = x1 (t ),...x m = x m (t ) where t is a time parameter.
Definition 2.
An index given by one of next formula (2) with (2’) or (2’’)is called the prices
index MDF.
Ι kz // pj = ∏ Ι kz // pj i
m
(2)
i =1
Ι kz // pj i = exp ∫M
z ′pi
jMk
z
dp i = exp ∫M
qi
jMk
∑ pi qi
m
i =1
36
dp i = exp ∫t 1
t
0
q1 (t ) pi′ (t )
∑ qi (t ) pi (t )
m
i =1
dt
(2’)
Nicoleta Breaz , Daniel Breaz: A spline aproximation of the factors path in MDF
Ι kz // pj
= exp ∫t
∑
m
t1
0
i =1
m
q i (t ) p1′ (t )
∑ qi (t ) pi (t )
dt
(2’’)
i =1
Remark 3.
Contrary to the first methods which don’t take into account the real data of
prices and quantities in j and k situations, the MDF method realizes all these. The only
inconvenient of this method is that the index depends on the path described by the
price and quantity factors between the situations nominated by Pj and Pk . This fact,
although well sustained by the economic reality, makes impossible the calculus of
index whenever the path is unknown.
We present below some particular paths of factors.
The linear path of the factors
In particular cases of a linear path given by the equations
i = 1, m
xi (t ) = xi ( j ) + t [xi (k ) − xi ( j )] = xi ( j ) + t∆xi
t ∈ [0,1]
the index (1) becomes
1 f ′ (t )∆xi
Ι kz // xji = exp ∫0 x
(3)
dt
f (t )
For an index of prices we use the equations
pi (t ) = pi ( j ) + t ( pi (k ) − pi ( j )) = pi ( j ) + t∆pi
with i = 1, m , t ∈ [0,1]
qi (t ) = qi ( j ) + t (qi (k ) − qi ( j )) = qi ( j ) + t∆qi
This situation can be represented in a space of 2m dimension:
q
Pk
C
Pj
p
So the index (2) becomes
37
Nicoleta Breaz , Daniel Breaz: A spline aproximation of the factors path in MDF
Ι kz // pj i = exp ∫0
1
[qi ( j ) + t∆qi ]∆pi
dt
∑ [qi ( j ) + t∆qi ]( pi ( j ) + t∆pi )
i =1
or
Ι kz // pj = exp ∫0
1
(4)
m
∑ [qi ( j ) + t∆qi (t )]∆pi
m
i =1
∑ [qi ( j ) + t∆qi ]( pi ( j ) + t∆pi )
m
(4’)
dt
i =1
Examples of polygonal paths
Proposition 4.
The Laspeyres index Ι kz // pj (•, L ) is MDF index on ∆ 1 path,(figure no.1 )
q
P1 ( p( j ), q( j ))
P2 ( p(k ), q( j ))
P3 ( p(k ), q(k ))
∆1 = P1 P2 ∪ P2 P3
P2
P1
P3
p
Fig. 1
Proof: We have Ι kz // pj (MDF , ∆ 1 ) = Ι z / p (MDF , P1 P2 ) ⋅ Ι z / p (MDF , P2 P3 )
The path P1 P2 is given by the equation qi = qi ( j ) , pi = pi ( j ) + t∆pi , t ∈ [0,1] so the
index Ι z / p (MDF , P1 P2 ) becomes
Ι z / p (MDF , P1 P2 ) = exp ∫0
1
∑ qi ( j )∆pi
m
i =1
dt = exp ∫0
∑ qi ( j )[ p i ( j ) + t∆p i ]
m
i =1
38
1
∑ qi ( j )∆p i
m
i =1
dt
∑ (q i ( j ) pi ( j ) + tq i ( j )∆p i )
m
i =1
=
Nicoleta Breaz , Daniel Breaz: A spline aproximation of the factors path in MDF
= exp ln ∑ (qi ( j ) pi ( j ) + tqi ( j )∆pi ) =
1
m
i =1
=Ι
∑ (qi ( j ) pi ( j ) + qi ( j )∆pi ) ∑ qi ( j ) pi (k )
m
i =1
0
(•, L )
∑ qi ( j ) pi ( j )
m
=
i =1
m
i =1
m
∑ qi ( j ) pi ( j )
=
i =1
qi = qi ( j ) + t∆qi
so the index
The path P2 P3 is given by the equation
pi = pi (k ) ⇒ ∆pi = 0
Ι z / p (MDF , P2 P3 ) becomes
k/ j
z/ p
Finally, we have Ι kz // pj (MDF , ∆1 ) = Ι z / p (MDF , P1 P2 ) = Ι kz // pj (•, L ) .
Proposition 5.
The Paasche index Ι kz // pj (•, P ) is MDF index on ∆ 2 path,(figure no.2 )
q
P1 ( p( j ), q( j ))
P2 ( p( j ), q(k ))
P3 ( p(k ), q(k ))
∆ 2 = P1 P2 ∪ P2 P3
P1
P2
P3
p
Fig. 2
Proof: We have Ι kz // pj (MDF , ∆ 2 ) = Ι z / p (MDF , P1 P2 ) ⋅ Ι z / p (MDF , P2 P3 )
Similary with the previous proposition we proove Ι z / p (MDF , P1 P2 ) = 1.
The path P2 P3 is given by the equation qi = qi (k ), pi = pi ( j ) + t∆pi , t ∈ [0,1] so the
index Ι z / p (MDF , P2 P3 ) becomes
Ι z / p (MDF , P2 P3 ) =
∑ pi (k )qi (k )
m
i =1
m
∑ pi ( j )qi (k )
i =1
= Ι kz // pj (•, P )
Proposition 6.
The Edgeworth index Ι kz // pj (•, E ) is MDF index on ∆ 3 path,(figure no.3 )
39
Nicoleta Breaz , Daniel Breaz: A spline aproximation of the factors path in MDF
P1 ( p( j ), q( j ))
q( j ) + q(k )
P2 p( j ),
2
q ( j ) + q (k )
P3 p (k ),
2
P4 ( p(k ), q(k ))
∆ 3 = P1 P2 ∪ P2 P3 ∪ P3 P4
P1
q
P3
P2
P4
p
Fig. 3
Proof:
The path P2P3 is given by the equations
qi ( j ) + qi (k )
and pi = pi ( j ) + t∆pi , t ∈ [0,1]
2
so the index Ι z / p (MDF , P2 P3 ) becomes
qi =
Ιz / p (MDF,
m q ( j) + q (k )
m q ( j ) + q (k )
i
i
i
i
∆pi
∆pi
∑
∑
1
1
2
2
i =1
i =1
P2P3 ) = exp∫0 m
dt = exp∫0 m
dt =
qi ( j) + qi (k)
qi ( j) + qi (k)
qi ( j) + qi (k)
⋅ pi ( j) + t∆pi
⋅ ( pi ( j) + t∆pi )
∑
∑
qi ( j) + qi (k)
pi (k)
q ( j) + qi (k)
q ( j) + qi (k)
2
⋅ pi ( j) + t∆pi i
= Ιkz // pj (•, E)
exp ln ∑ i
= mi=1
(
)
(
)
+
q
j
q
k
2
2
i=1
0
∑ i 2 i ⋅ pi ( j)
i =1
Finally we have:
Ιkz // pj (MDF, ∆3 ) = Ι z / p (MDF, P1P2 ) ⋅ Ι z / p (MDF, P2P3 ) ⋅ Ι z / p (MDF, P3P4 ) =
i =1
2
∑
i =1
2
2
m
1
m
= 1⋅ Ι z / p (⋅ E) ⋅1 = Ιkz // pj (⋅ E)
Corollary 7. The MDF price index on the path which is presented in the figure no.4 is
a product of Laspeyres indices.
q
Mj
M1
M2
M3
Mk
p
Fig. 4
40
Nicoleta Breaz , Daniel Breaz: A spline aproximation of the factors path in MDF
(
)
Ιkz // pj (MDF) = Ιz / p MDF, MjM1 ⋅ Ιz / p(MDF, M1M2 ) ⋅ Ιz / p(MDF, M2M3) ⋅ Ιz / p(MDF, M3Mk ) = ∏Ιz / p(⋅ L)
The problem is to determine the path that aproximate the most the empiric path
of the
prices and quantities between j and k
The case of a spline path
Remark 8.
At the beginning ,we define the factorial indices of prices MDF with respect to a
family
of
parabolas.
We
consider
the
parametric
equations
2
2
qi = ai + bi t + ci t ,
pi = ai′ + bi′t + ci′t , t ∈ [0,1] (5)
where qi (0) = qi ( j ), qi (1) = qi (k ), pi (0) = pi ( j ), pi (1) = pi (k ) that become
q i = qi ( j ) + ∆qi t + ci t 2
t ∈ [0,1] , ci ci′ ∈ R
p i = pi ( j ) + ∆pi t + ci' t 2
If ci = ci′ = 0 we have the factorial index of prices on the linear path .Otherwise we
obtain
Ι kz // pj = exp ∫0
1
∑ (qi ( j ) + ∆qi t + ci t 2 )⋅ (∆pi + 2tci′ )
n
∑ (qi ( j ) + ∆qi t + ci t
i =1
n
i =1
2
)⋅ (p ( j ) + ∆p t + c′t )
dt
(6)
2
i
i
i
where the choice of constants ci , ci′ , n .could be made taking into account the new
information for the situation t ∈ [ j , k ] (moment of time).
The formula (6) can be generated when in (5) we have the polynoms of m degree, so
the index will depend on a number of 2n(m-1) constants.
Remark 9.
We suppose that we know the empiric data
p( j ), q( j ), p(l ), q(l ), p(k ), q(k ), l ∈ [ j , k ] and we want to calculate the
index Ι kz // pj (MDF ) .
a)If we consider that between every two points the path is linear then from the point
( p( j ), q( j )) to the point ( p(k ), q(k )) the path is a polygonal curve, so:
Ι kz // pj (MDF ) = Ι lz// jp (MDF − liniar ) ⋅ Ι kz // lp (MDF − liniar )
qi = ai + bi t + ci t 2
,
b)The parametric equations
pi = ai′ + bi′t + ci′t 2
t ∈ [0,2]
qi (0) = qi ( j )
qi (1) = qi (l )
qi (2) = qi (k )
pi (0 ) = pi ( j )
pi (1) = pi (l )
pi (2) = pi (k )
41
i = 1, n having the conditions
Nicoleta Breaz , Daniel Breaz: A spline aproximation of the factors path in MDF
become qi = qi ( j ) +
4∆qi
− ∆qi
∆q
− 2∆qi
t+ i
t 2 and similarly for pi.
2
2
So, one could observe that in both cases, although the index measures the variation
from the moment j to moment k , it also depends of the intermediar moment l.
Therefore, we calculate Ι kz // pj (MDF , ϕ ) where ϕ is the curve which links the points
l/ j
k/ j
k/ j
l/ j
j,k,l.
Remark 10.
If we generalize the remark (9) we obtain that the index MDF for the points j,k
and for the empiric path between j and k which passed by the intermediar points
coresponding to m moments of time t1 , t 2 ,....t m−1 and t 0 = j , t m = k can be
calculate either as a product of m linear path indices or by reducing the problem to
one of Lagrange
polynomial interpolation on the nodes (0,...,m)
with
pi (s ) = pi (t s ), s = 0, m , i = 1, n .Thus it can be determined a number of 2n Lagrange
polynoms of m degrees at the most qi = ( p m )i , pi = ( p ′m )i , i = 1, n .
The choosing one method in spite of the other is a meter of comparison of the errors
resulted in each case.
Remark 11.
For a better comparison of errors and taking into account that the polygonal
line is the graphic representation of a particular spline function we define the factorial
index of prices attached to Mj,Mk points and the linear path M j , M t1 , M t 2 ....M t m , M k
by using the spline polynomial interpolation problem.
We use the next notions and results from the spline functions theory, the interpolation
properties of these functions being the reason to recomand a spline path in MDF:
Definition 12. Let ∆ : −∞ < x1 < x 2 < ... < x N < ∞, x0 = −∞, x N +1 = ∞ .The function
s : R → R is called m degrees spline with nodes x1 < x 2 < ... < x N if
a)
s / Ι i ∈ Pm (Ι i ), Ι i = (xi , xi +1 ), i = 0.N
s ∈ c m −1 (R )
Definition 13.The function define in (12) which satisfies s (t i ) = y i , i = 1, N + m + 1 is
b)
called the interpolation polynomial spline function attached to the vector {y i }i =1
m + N +1
and
the points t1 < t 2 < ... < t m+ N +1 .
Proposition 14. Any polynomial spline function of m degrees can be expressed
uniquely s ( x ) = p0 ( x ) + ∑ ci ( x − xi )+ , p 0 ∈ Pm .
N
i =1
m
42
Nicoleta Breaz , Daniel Breaz: A spline aproximation of the factors path in MDF
Proposition 15. Let m > 0, t1 < t 2 < ... < t m + N +1 , the real numbers y i , i = 1, m + N + 1 ,
and the division ∆ , with the nodes x1 < x 2 < ... < x N which satisfy the relations
t i < xi < t i + m+1 , i = 1, N .
Then, a unique polynomial spline function of m degree exists so that
s (t i ) = y i , i = 1, m + N + 1 .
Proposition 16. Any of odd degrees with the nodes ∆ : a = x1 < x 2 < ... < x N = b, can
be written as s 2 m −1 (x ) = p 2 m −1 (x ) + ∑ ci (x − xi )
N
2 m −1
with
p 2 m −1 ∈ P2 m−1 , ci ∈ R and s 2 m −1 ∈ c 2 m −2 [a, b] .
Definition 17. The polynomial spline function of 2m-1 degree which satisfy the
conditions s 2 m−1 (xi ) = yi , ∀ i = 1, N
i =1
+
s2( mj )−1 (x1 ) = s2( mj )−1 (xN ) = 0, j = m, m + 1, ...2m - 1. (s ∈ Pm −1 x < x1 şi x > x n ) is called
the natural spline function.
Theorem 18. Any natural spline function can be written uniquely
s(x ) =
∑ ai x i + ∑ ci (x − xi )+
m −1
N
i =0
i =1
2 m −1
, where ∑ ci xir = 0, r = 0, 1, ...m − 1
N
i =0
Theorem 19. Let m ≥ 1 , N ≥ m and y j , j = 1, N . Then only a natural spline function
( )
exists, so that s x j = y j , j = 1, N .
Remark 20.
In the case when the degree of polynomial spline function is three then we
speak about cubic spline function.
Remark 21.
With respect to the index problem we use the empiric data for m time
moments and we interpolate them by 2m spline functions pi and qi.
Remark 22.
Let the time moments j and k and N-2 intermediar moments j 2 ,... j N −1 for
which we know the prices and quantities.
We consider a parameter t ∈ [a, b] , qi = qi (t ) , pi = pi (t ) so that qi (t1 ) = qi ( j ) ,
qi (t 2 ) = qi ( j 2 ), ...qi (t N −1 ) = qi ( j N −1 ), qi (t N ) = qi (k ) and similar for p , ∀ i = 1, n ,
t s being the division points ∆ : a = t1 < t 2 < ... < t N = b .
a)To determine the cubic spline function qi that satisfies the interpolation conditions
qi (t s ) = qi ( j s ), s = 1, N we must have with respect to proposition (4) a number of
N+2 conditions because there are N+2 coefficients (the interpolation conditions are
imposed on nodes).
There are more ways to determine the coefficients :
43
Nicoleta Breaz , Daniel Breaz: A spline aproximation of the factors path in MDF
-To the interpolations conditions (in number of N) we add qi′′(t1 ) = 0 , qi′′(t N ) = 0
and thus unique natural cubic spline functions results.
-If we consider two nodes of the function we have four coefficients and after the
use of this N conditions both in nodes and in points t 2 ,...t N −1 , a Lagrange interpolation
polynom it results.
-At last, if we consider N-2 nodes x1 < x 2 < ... < x N − 2 between the points t1 ,...t N it
results a function with N-2 coefficient defined on the interval (x1 , x 2 ),...(x N −3 , x N − 2 ) .
The coefficients result from N-2 interpolation conditions in nodes and the two
conditions in t j1 and t j2 that are not nodes.
∑ c ij (t − t j )
-Generally speaking, we can determine such an interpolation cubic spline given by
qi (t ) = ∑ bli (t − t1 ) +
2
l
N −1
3
l =o
+
j =1
l
N −1
+
2
+
j =1
pi (t ) = ∑ bli′ (t − t1 ) +
l =o
nodes and qi , pi ∈ C 2 [a, b] .
∑ c ij (t − t j )
′
3
, i = 1, n .
+
where qi and pi are polynoms of three degree at the most in the interval generated by
We have Ι kz // pj (MDF , ϕ ) = exp ∫ϕ
∑ qi dp
i
∑ pi qi
where ϕ will be the spline curve that
passes through j,k and through the intermediar points, being in fact the graphic
representations of interpolation spline functions.
∫ϕ = ∫t
tN
1
= ∫t 2 + ∫t 3 +... + ∫t N
t
t
t
1
2
N −1
In each case of these integrals the spline parametrization becomes a polynomial
parametrization.
In the third case mentioned above, when only x1 < x 2 < ... < x N − 2 nodes exist we
have
∫ϕ = ∫t
tN
1
= ∫x 2 + ∫x 3 +... + ∫x N − 2 .
x
x
1
2
x
N −3
b)In a similar way we use a spline function of m degree.
If interpolation is not made in nodes but in points t1 < t 2 < ... < t m + N +1 and we know
( )
( )
the values qi t j , pi t j , j = 1, m + N + 1 we will consider, for example, the quantity
qi as a spline polynomial function of m degree with nodes x1 < x 2 < ... < x N choosed
so that t l < xl < t l + m +1 , l = 1, N .
According to proposition (15) there is only a spline polynomial function of m degree
which solves the interpolation conditions in points t j , j = 1, m + N + 1 and it has the
44
Nicoleta Breaz , Daniel Breaz: A spline aproximation of the factors path in MDF
form q i (t ) = hi (t ) + ∑ cli (t − t l )+ , where hi ∈ Pm and similary
N
m
l =1
p i (t ) = hi′ (t ) + ∑ cl (t − t l )+
i′
N
l =1
m
Then the index MDF leads to an integral of type
∫ϕ = ∫t
t m + N +1
1
= ∫t 1 + ∫x 2 +... + ∫x N + ∫xm + N +1
x
x
1
1
x
t
N −1
N
It is more practically if we use the spline natural functions. In this case we consider N
data regarding to p and q as interpolation conditions on nodes t1 < t 2 < ... < t N .
In respect with proposition (19) if m ≥ 1, N ≥ m and we have interpolation
conditions in nodes there is only an interpolation spline functions. Taking to account
the proposition (18) theorem we can write
q i (t ) =
pi (t ) =
∑ ali t l + ∑ c ij (t − t j )+
m −1
N
l =0
j =1
2 m −1
(
∑ ali 't l + ∑ c ij' t − t j
m −1
N
l =0
j =1
)
2 m −1
+
where the coefficients will be determined from the interpolation conditions( N
conditions for every qi and pi ) plus the conditions
∑ c ij t rj
N
j =1
=0
, r = 0,1,...m − 1 ,
∑ c ij't rj
N
j =1
= o, i = 1, n
qi( j ) (t1 ) = qi( j ) (t N ) = 0 , ∀j = m, m + 1, 2m − 1
pi( j ) (t1 ) = pi( j ) (t N ) = 0
Remark 23.
If we take into account that the empiric data ( pi , qi ), i = 1, n obtained for N different
moments of time comprise errors then we speak rather of a numerical adjustment.
Analogous, we have the spline functions q = f ( p ) respectively regression model of
spline type q = f ( p ) + ε .
REFERENCES
[1] A. Vogt, J.Barta,- The Making of Tests for Index Number-Mathematical Methods
of Descriptive Statistics, Physica-Verlag, Heidelberg 1997.
[2] Ghe. Micula- Funcţii spline şi aplicaţii, Editura Tehnică, Bucureşti 1978.
45
Nicoleta Breaz , Daniel Breaz: A spline aproximation of the factors path in MDF
Authors:
Nicoleta Breaz - „ 1 Decembrie 1918” University of Alba Iulia, str. N.Iorga, No.13,
Departament of Mathematics and Computer Science, nbreaz@lmm.uab.ro
Daniel Breaz - „ 1 Decembrie 1918” University of Alba Iulia, str. N.Iorga, No.13,
Departament of Mathematics and Computer Science, dbreaz@lmm.uab.ro
46