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THE FULL RANK CASE FOR A LINEARISABLE MODEL

by

Nicoleta Breaz, Daniel Breaz

Abstract. We consider a model which can be reduced to a linear one by substitution. For this model, we obtain a full rank case theorem for uniquely fitting written in terms of initial matrix of sample data.

Definition 1

Let be Y a variable which depends on influence of some factors expressed by other p

variables X₁,X₂,...,X_p. The regression is a search method for dependence of variable Y on variables X₁,X₂,...,X_p and consist in determination of a functional connection f

such that

(

)

+ε

= f X X Xp

Y ₁, ₂,..., (1)

where

ε

is a random term (error) which include all factors that can not be

quantificated by f and which satisfies the conditions: a) E

( )

ε =0

b)Var

( )

ε has a small value

Formula (1) with conditions a) and b) is called regressional model, variable Y is called the endogene variable and variables X1,X2,...,Xp are called the exogene variables.

Definition 2

The next regression is called a parametric regression

(

X X Xp

) (

f X X Xp p

)

f ₁, ₂,..., = ₁, ₂,..., ;α₁,α₂,...,α

Otherwise the regression is called a nonparametric regression. The regression bellow is called a linear regression

(

)

∑

= = p

k

k k p

p X

X X X f

1 2

1 2

1, ,..., ;α ,α ,...,α α

Remark 3If function f from regressional models is linear with respect to the parameters α1,α2,...,αp, that is

(

)

(

p

)

p

k k k p

p X X X

X X X

f , ,..., ; , ,..., ₁, ₂,...,

1 2

1 2

1

∑

= = α ϕ α

α α

than regression can be reduced to linear one.

Definition 4

It is called the linear regressional model between variable Y and variables

p

X X

∑

₌ + = p

k

k kX

Y

1

ε

α (2)

Remark 5.

The liniar regression problem consists in study of variable Y behavior whit respect to the factors

X

₁

,

X

₂

,...,

X

_p the study made by “ evaluation” of regressional parameters

α

1

,

α

2

,...,

α

p and random term

ε

.

Let be considered a sample of n data

   

 

   

  =

n

y y y

y

M

2 1

(

)

n p

x x

x

x x

x

x x

x

x x x

np n

n

p p

p >>

   

 

   

  =

= ,

... ... ... ... ...

... ...

,...,

2 1

2 22

21

1 12

11

1

Than one can make the problem of evaluations for regressional parameters

(

)

∈

= p

T α α α

α ₁, ₂,..., p and for error term =

(

n

)

∈

T ε ε ε

ε ₁, ₂,..., n , from these data. From this point of views the fitting of theoretic model can offering solutions. Matriceal the model (2) can be written in form

ε α + =x

y (2’)

and represent the linear regressional theoretical model.

By fitting this models using a condition of minim results the fitted model

e xa

y= + (2’’)

where aT ∈p,eT ∈n.

It is desirable that residues e1,e2,...,ento be minimal. Then can be realised using the least squares criteria.

Definition 6.

It is called the least squares fitting, the fitting which corresponds to the solutions (a,e) of the system y= xa+e, which minimise the expression

∑

₌

= n

k k T

e e e

1 2

Theorem 7.(full rank case)

If rang

( )

x = pthen the fitting solution by least squares criteria is uniquely given by

( )

x x x y a= T −1 T Remark 8.

In this paper we consider the next model

ε α

α

α + + + +

= x x x x p− xp−xp

where =

(

n

)

∈ T

y y y

y ₁, ₂,..., n, =

(

1, 2,..., p₋1

)

∈

T α α α

α p-1

,

(

)

∈

=

n

T

ε

ε

ε

ε

₁

,

₂

,...,

n and x is the sample data/sample variables matrix

p n

M

x∈ _. ,

(

)

   

 

   

  = =

− − −

np np n

n

p p

p p

p

x x x

x

x x x

x

x x x

x

x x x x

1 2

1

2 1 2 22

21

1 1 1 12

11

2 1

...

... ... ... ... ...

... ...

,..., ,

With substitutions xixi+1= zi,∀i=1,p−1, we obtain the new matrix,

1 . − ∈Mnp

z ,

(

)

   

 

   

 

⋅ ⋅

⋅

⋅ ⋅

⋅ ⋅ ⋅

⋅ = =

− − −

−

np np n

n n n

p p

p p

p

x x x

x x x

x x x

x x x

x x x

x x x

z z z z

1 3

2 2 1

2 1 2 23

22 22 21

1 1 1 13

12 12 11

1 2

1

...

... ...

... ...

... ...

,..., ,

and the linear model y=α₁z₁+α₂z₂ +...+α_p−1z_p−1+ε, which after the least squares

fitting becomes y =a₁z₁+a₂z₂ +...+a_p−1z_p−1+e, where aT =

(

a₁,a₂,...,a_p−1

)

∈p-1,

(

)

∈

= n

T

e e e

e ₁, ₂,..., n. The fitting uniquely solutions results from

( )

z = p−1

rang (see theorem 7).

The purpose of our paper is to give an analogouse theorem based on initial sample variables.

Theorem 9.

If rang

( )

x , = p p∈

{ }

2,3 , and if the sample data are not nulls than uniquely exist the

least squares fitting solution a

( )

xTx x*Ty

1 * *

−

= where x* =

(

x1∗x2,...,xp−1∗xp

)

and

j i x

x ∗ is the natural product between the vectors

x

i and

x

j that is the vector which can be obtained by multiplications one components of the two vectors.

Proof

Case 2p= :

The sample matrix is x∈M_n,2, x=

(

x₁,x₂

)

,

   

 

   

  =

2 1

22 21

12 11

... ...

n

n x

x x x

x x

x and the model

ε

α +

= 1x1x2

y .

Because the sample data are not nulls it results rang

( )

z =1, where

   

 

   

 

⋅ ⋅ ⋅ =

2 1

22 21

12 11

...

n

n x

x x x

x x

z .

One can observe that is sufficient that for a single unit of sample, data must be differed from zero. According to theorem 7 if rang

( )

z =1 then uniquely exist

( )

z z z y

( )

x x x y a

a T T T *T

1 * * 1

1

− −

= =

= where x∗ =

(

x1∗x2

)

∈Mn,1.

Case

p

=

3

:

The sample matrix is x∈Mn_,3, x=

(

x1,x2,x3

)

,

   

 

   

  =

3 2 1

23 22 21

13 12 11

... ... ...

n n

n x x

x

x x x

x x x

x

and the model

ε α

α + +

= 1x1x2 2x2x3

y . We use the substitutions x₁x₂ = z₁,x₂x₃ = z₂ and we obtain

ε α

α + +

= 1z1 2z2

y . If rang

( )

x =3 then results that at least one minor of three order

is differed from zero and let be this one (without restrict the generality)

33 32 31

23 22 21

13 12 11

x x x

x x x

x x x

d = .

If d₃ ≠0, developing by the second column results that at least one of the three minor from the development is not null and let be, by example, this one

0

23 21

13 11

2 = ≠

x x

x x

d . On the other way we calculate a minor of two order from z, by

example

(

)

2 22 12

21 13 23 11 22 12 22 21 13 12 23 22 12 11 21 12 22 11 22 21

12 11 2

d x x

x x x x x x x x x x x x x x z z z z z z

z z d

z ⋅ =

= −

= −

= −

= =

Because from the hypothesis, the sample data are not nulls and rang

( )

x =3, 0

2 ≠

d then results that ₂ ≠0

z

d , so rang

( )

z =2. According to theorem 7 it results

that uniquely exists the solution a

( )

zTz zTy

( )

xTx x*Ty

1 * *

1 −

− =

= where

(

x1 x2,x2 x3

)

x_∗ = ∗ ∗ .

A weak condition, namely rang

( )

x = p−1,p∈

{ }

2,3 is not sufficient because this is not implied that rang

( )

z = p−1. However, it can be given an intermediary condition between rang

( )

x = p−1 and rang

( )

x = p.

Theorem 11.

If in sample data matrix there exists a minor of second order differed from zero, at

least, which not contains elements from second column, then a

( )

xTx x*Ty

1 * *

−

= with

(

x1 x2,x2 x3

)

x_∗ = ∗ ∗ .

Proof

By example _{11 23 13 21} 0

23 21

13 11

2 = =x x −x x ≠

x x

x x

d and

(

_{11 23 13 21}

)

0

22 12 23 22 22 21

13 12 12 11

2 = =x x x x −x x ≠

x x x x

x x x x d

z

Remark 12.

These theorems can not be generalised for any

p

. A similar theorem with 11 can be given in genarall case if we define the pseudominor in follow sense.

Definition 13.

Let be x∈M_n,p. We call the pseudo-minor of p−1 order from matrix x, formed by the first p−1rows of x, the expression

( )

(

( ) ( ) ( )

)

(

( ) ( ) ( )

)

( )

( ) ( )

∑

∏

∏

∑

− −

∈

−

= +

−

=

∈ − − + + − − +

    

  

⋅ ⋅

− =

= ⋅

⋅ ⋅

⋅ ⋅ ⋅ −

=

1 1

1

1 1 1

1

1 1 1 1

2 2 1 1 1 1 1 2

2 1 1

1

... ...

1

p p

S

p

i i i p

i i i sign

S

p p p

p sign

f

x x

x x

x x

x x d

σ σ σ

σ

σ σ σ σ σ σ σ

σ

Remark 14.

In calculus of d we apply the ordinary formulla for a determinant of

p

−

1

order only that the elements from products appearing in determinant are product of elements which are from minor of

p

−

1

obtained by elimination of the first column and from minor of

p

−

1

order obtained by elimination of last column.

(

)(

) (

)(

)

(

)(

) (

)(

) (

)(

)

(

33 12 21

)(

34 13 22

)

33 24 12 32 23 11 32 23 14 31 22 13 24

13 32 23 12 31

33 22 14 32 21 13 34

23 12 33 22 11

34 33 32 31

24 23 22 21

14 13 12 11

3

x x x x x x

x x x x x x x

x x x x x x

x x x x x

x x x x x x x

x x x x x

x x x x

x x x x

x x x x

df

−

− −

− +

+ +

By example, d₃f is the pseudo-minor of p−1 order from matrix x∈Mn_,p

(

p=4

)

, formed with the first three rows of this matrix.

Theorem 15.

If in sample data matrix there exist at least one pseudo-minor of p−1order different

from zero then exists uniquely fitting solution a=

( )

x_*Tx_* −1x_*Ty with

(

x x x x xp xp

)

x_∗ = ₁∗ ₂, ₂∗ ₃,..., ₋₁∗ .

Proof

Let be the pseudo-minor of p−1 order, different from zero, formed with the first

1

−

p rows, without restrict the generality

(

d_pf₋₁ ≠0

)

.

With substitutions xjxj₊₁ =zj,∀j=1,p−1, we obtain matrix

( )

1

1 1

1 , ₊

− ≤ ≤≤

≤ = ⋅

= ij ij ij

p j

n i

ij z x x

z z

We calculate the minor of p−1 from

z

formed with the first p−1 rows:

( )

(

( ) ( ) ( )

)

( )

1

(

( ) ( )

)

(

( ) ( )

)

...

(

( ) ( )

)

0 ...

1

1 1 1 1 1 1 1

2 2 2 2 1 1 1 1 1

1 1 2

2 1 1

1 1

≠ = ⋅

⋅ ⋅ ⋅

⋅ −

=

= ⋅

⋅ ⋅ −

=

−

∈ + + − − − − +

∈ − −

∑

∑

− −

f p S

p p p p sign

S

p p sign

z

d x

x x

x x

x

z z

z d

p p

σ σ σ σ σ σ σ

σ

σ σ σ σ

σ

So rang z= p−1 and a=

( )

zTz −1zTy=

( )

x_*Tx_* −1x_*Ty.

REFERENCES

[1] Stapleton J.H.-Linear Statistical Models, A Willey-Interscience Publication, Series in Probability and Statistics, New York,1995.

[2] Saporta G.-Probabilités, analyse des données et statistique, Editions Technip, Paris, 1990.

Authors:

Nicoleta Breaz- „1 Decembrie 1918” University of Alba Iulia, str. N. Iorga, No. 13, Departament of Mathematics and Computer Science, email: nbreaz@lmm.uab.ro