Correlation Matrices in Problems of Identification of
Seismic Stability and Technical Condition of HighRise Buildings and Building Structures
Naila Musaeva1, Elchin Aliyev2, Ulkar Sattarova3, Narmin Rzayeva4
1,3
Azerbaijan University of Architecture and Construction, Baku, Azerbaijan
2,4
Cybernetics Institute of ANAS, Baku, Azerbaijan
1
[email protected], [email protected], [email protected], [email protected]
Abstract— It is shown that correlation matrices and normalized
correlation matrices are applied in solving of static and dynamic
identification problems for diagnostics and prediction of
technical condition and seismic stability of high-rise buildings
and building structures. Specific characteristics and properties of
those matrices are determined.
Keywords— correlation matrices; normalized correlation matrices;
static and dynamic identification;, diagnostics; prediction; seismic
stability; high-rise buildings; buildings structures
I.
INTRODUCTION
It is known that a high-rise building consists of a large
number of structural elements, any of which can trigger a train
of events resulting in destruction of the whole building.
Specialists therefore must be capable of performing precise
diagnostics, predicting deformation and vibration of a building
or other building structure and forecasting and promptly
determining degree of damage. Only maximally quick reliable
threat identification and immediate alarming will allow
carrying out of evacuation and preventing negative effects
caused by gas leakage, fire situations, troubles with electrical
equipment, including elevators, etc.
Constant monitoring of all structural elements, meanwhile,
is first of all impossible due to technical and economic reasons;
on the other hand, it does not produce the desired effect.
Thereby, methods of indirect measurements in hard-to-reach
spots, modeling and analysis methods, methods of empirical
dependences, methods of one-dimensional, two-dimensional
and complex multidimensional models are applied to solving of
this problem. It allows one to carry out diagnostics of the
technical condition of a high-rise building or building structure,
predict a failure, determine the moment, at which mechanical
deformations exceed threshold values, and detect the beginning
of a rescue mission.
On the other hand, most accurate picture of the current
condition of a high-rise building or other building structure,
such as its wear rate, presence of hidden faults, etc., is obtained
after dynamic tests. A full-scale monitoring of technical
condition of a high-rise building therefore requires information
on its dynamic characteristics and changes. In reality, dynamic
probing and early diagnostics of deformation condition of
bearing structures are as a rule based on analysis of change in
transfer functions built for building sectors of different height.
This method is applicable to high-rise building of different
architecture, and transfer functions in such cases are built for
different sections of the building and constructions.
Transfer function of a building section is understood as
correlation of components of power spectra of registered
signals in two spots of the building, specifically, at the point of
dynamic effect set, for instance, as a broadband impulse from
an inelastic impact, and at the point, where this effect is
registered after having passed through the part of the building
under consideration. Such transfer function characterizes mode
of deformation of constructions exactly in the part of the
building, which the broadband impulse passed through.
Change in transfer function, in particular, change in values
of force coefficients for different frequencies, indicates a
change in the mode of deformation in this part of the building,
which allows localizing such a change within the number of
floors of the building between neighboring measure points.
This, diagnostics and prediction of technical condition and
seismic stability of a high-rise building or a building structure
require solving of static and dynamic identification problems.
The paper considers specifics of the mathematical apparatus
applied to solving of those problems.
II.
PROBLEM STATEMENT
Solving of static identification problems by probabilisticstatistical methods is known to be reduced to numerical solving
of correlation matrix equations.

R


X X

0  C  R 0 
 
XY
where
 R   0 R   0
X1 X 2
 X1 X1


R
0
R



 0 
  X 2 X1
 0 
X2 X2
Xi X j
 


 R   0 R   0
Xn X2
 X n X1

R   0  R 
XX
 R

X1 X n
 R

X2 Xn
0
0  i, j  1, n 


 R   0
Xn Xn




R

XY
0   RX Y 0 RX Y 0  RX Y 0
X Y


0 
R




1
i


2

T

n

T
C  c1 c2  cn  
Meanwhile, general dynamic identification problem is
reduced to calculation of impulse transition function k  or
transfer function W   as a result of certain kind of effect on
the input and measuring the system response. Statistical
methods of determining of transfer function or impulse
transition function are based on the integral equation

where

C is column vector of coefficients; R 0

0
0
Xi X
is matrix of
j
estimates of auto- and cross-correlation functions
R
0 ,
  0 of centered input useful
at time shift
j

signals X i it   X i it   mX ; R 
i

XiY
0
is column matrix
of estimates of cross-correlation functions R 

Xi Y
shift

Xi Xi
0

Xi X
R
  0 between centered input useful signal
0
XY
at time

X i it  and
mY are
mathematical expectations of signals X i it  and Y it 
and
respectively;
R

XX
N

Assume that the construction object is in the normal state,
and useful signals X t  , Y t  are received from vibration,
motion, tilt, deformation and other sensors. Then solving of
statistical dynamic of the construction object can be reduced to
solving of a system of equations, which looks as follows in
matrix representation:

XY
N

XX


mX ; R
 i t  ,  t  , which are imposed
X i t  , i  1, n , correlation matrix
However, due to noises
on useful input signals
equation is transformed to the following form



R   0 B   R   0 
g
  is square symmetric matrix of auto-correlation
 
XY

X t  = X t  -
  is column vector of cross-correlation functions
between the input X t  and output Y t  = Y t  = mY ,


mX ,
mY are mathematical expectations of X t  , Y t  respectively;

W   is column vector of impulse transition function [3].


Correlation matrices R     , R     and column
XY
XX

vector of impulse transition functions W   in this case have
the following:
where
 R   0  R   0 
g1 g 2
 g1 g1

R   0  R   0   Rg g 0  Rg g 0 
2 2
gg
gi g j
 2 1

 
 R   0  R   0 
gn g2
 g n g 1

R   0   R 
g
N  1t 
t , 2t, ,
 
functions of dimension N  N of input signal
k 1
gg

   R   W      0,

where R 
   1  X kt Y k   t  

  of signal
XY

k 1
N



X t  at the input of the construction object and crosscorrelation function R     between the output Y t  and the
XX
R
   k  d        
XX
XY
   1  X kt  X k   t  

 
XX
which allows one, using correlation function R 

R
N
0
input, to determine impulse transition function.

output signal Y it   Y it   mY ; m X i
R       R


 0   R 
gi 


g1 
0
R
 R

g1 g n
 R

0

0  i, j  1, n 

R
R

XX


g2
0
R

X X
0
t 
R
R
R
T

 R   0 
gn


R

   R 0
XY


 
XY

X X
N  1t 
X X


X X



 R   0 
gn gn

g2 gn

X X
  
t 
0

N  2t 
X X
R
R

XY

t 
N  1t 

R  N  2 t  
X X

R

X X






R

X X
0
N  1t 
XY
 R

T


T
W    W 0 W t   W  N  1t  


R   0 

g1 g1


Dg1 

R   0


g 2 g1
r  0   

D
g
gg
2   D  g1 



R   0 

g n g1

 D g n   Dg1 


When corresponding noises  t ,  t are imposed on
useful signals X t  , Y t  , correlation matrices take the
following form:

R     
R    N  1t 
R   0
R   t 

R   t 
R   0 
 R    N  2 t  
gg
gg
gg
gg
gg

R 
gg
N  1t 

R 
gg
N  2t 
gg
gg



R   0 
gg
T

R       R   0 R   t   R   N  1t  
g
g
g
 g

Thus, dynamic identification problem, as well as static
identification problem, is reduced to solving of a symmetric
system of linear algebraic equations by means of correlation
matrices. Let us therefore consider properties and specifics of
correlation matrices in more detail.
III.
As is known, a correlation matrix is a matrix composed of
estimates of auto- and cross-correlation functions.
In static identification problem, system n  n of
correlation functions calculated at time shift   0 and placed
in a rectangular table with n rows and n columns
 
Rg g
 R   0  R   0  R   0 
g1 g 2
g1 g3
 g1 g1
 Rg g 0  Rg g 0  Rg g 0 
2 2
2 3
0   2 1




 R   0  R   0  R   0 
gn g2
g n g3
 gn g1
 R

g1 g n
 R

0

0


 
 R   0 
gn gn

g2 gn
0
D g1   D g 2 
R   0
D g 2 

R   0 

Dg n   D g1 

g2 g2
 r

g1 g n
 r


g1 g n

gn g2
0


Dg1   Dg n  
R   0 

g2 gn

D g 2   D g n  


R   0  
gn gn

D g n 

R

0

0 


 
 r  0 
gn gn

g2 gn

variances of parameters.
In dynamic identification problem, system n  n of
correlation functions calculated at all time shifts  and placed
in a rectangular table with n rows and n columns is called
correlation dynamic matrix:
R    N  1t 
gg

 R    N  2 t 
gg






R   0 
gg

R   0 
R   t 

gg
gg

R   0 
 Rg g t 
gg




 R    N  1t  R    N  2 t 
gg
 gg

R    
gg

Normalized correlation dynamic matrix is a matrix
composed of normalized estimates of auto- and crosscorrelation functions at all time shifts  , i.e. of elements of
the above mentioned matrix are divided by variance of
corresponding parameter:
R   0 

r     
gg
gg
Dg 
R   t 

gg
Dg 
gg

Dg 


R    N  2 t 
gg

Dg 
Dg 
R    N  2 t  
gg
Dg 

R   0
gg
Dg 
r  t 
 r   N  1t 
r  t 
r  0 
 r   N  2 t 


gg
gg
gg
gg
gg

r   N  1t  r   N  2 t  
gg
gg
r  0 
gg


Dg 
R   0 
Dg 

R    N  1t 
gg
R    N  1t 
R   t 
gg
is called a correlation static matrix. rows and columns of the
above mentioned matrix are called series of correlation matrix.
Normalized correlation static matrix is a matrix composed
of normalized estimates of auto- and cross-correlation
functions at time shift   0 , i.e. of elements of the above
mentioned matrix are divided by corresponding values of
variances of parameters:

g1 g 2
Dgi  , D g j  - i  1, 2,, n; j  1, 2,, n are
where
TECHNOLOGY OF BUILDING CORRELATION MATRICES
AND THEIR PROPERTIES IN PROBLEMS OF STATIC AND DYNAMIC
IDENTIFICATION OF TECHNICAL CONDITION AND SEISMIC
STABILITY OF HIGH-RISE BUILDINGS AND BUILDING
STRUCTURES

 r  0  r  0 
g1 g 2
 g1 g1
rg g 0  rg g 0 
2 2
 2 1

 
r  0  r  0 
gn g2
 g n g1
R
gg
where Dgi  D g j  i  1,2,, n
of parameters.

 
r  0 
gg
j  1,2,, n are variances



Correlation functions R gi g j 0 , i  1, 2,, n; j  1, 2,, n
and R g g     0,1, 2,, N  composing above mentioned
matrices respectively are called elements of correlation
matrices.

Elements of normalized correlation matrices above
mentioned respectively r  0  , is auto-correlation function,
r

gi g j
 
gg
gg
R
 

 
0
0
R 

det R   0   g2 g1
gg

R

g n g1
0
In solving of static identification problems, when there is
no correlation between parameters gi and g j , correlation
R

g1 g1
0
0 
0 
 R   0 
 g1 g1

R   0  0 
0 

 0
g2 g2
 R   0   
 
gg







 0
0
0  R   0 
gn gn


and is called a diagonal matrix. Diagonal normalized
correlation matrix will have the following form in solving of
static identification problems:
0
Dg 2 
R

g2 g2

0
R
R

gn g2

g1 g3
R

g 2 g3

0
0

0

det r  0  
gg
R

g1 g 2
D g1 
R   0 
R

g n g3
0
0
D g1   D g 2 
R   0 
g 2 g1
g2 g2
Dg 2 

R   0 
D  g 2   D g1 

R   0 
D g n   D g1 
 R

g1 g n
 R

g2 gn

0
0

 R

gn gn
0
gn g2
D  g n   D g1 
R

g1 g n
0

D g1   D g n 
R   0 

Dg 2   Dg n 

R   0 


g2 gn
gn gn
Dg n 
 
In solving of dynamic identification problems, determinant



of correlation matrix R    has the following form:
gg

det R     
 
R   0 
R   t 
R   t 
R   0 


gg
gg
gg
gg
gg
R    N  1t 
gg
 R    N  2 t  .

gg

R    N  1t  R    N  2 t  
gg






0


 
R   0 

gn gn


Dg n  

g2 g2
0
g n g1

R
0
0
gg
matrix has the following form:
0

g1 g 2
In solving of static identification problems, determinant of

normalized correlation matrix r  0 has the following form:
Such a matrix, where n  n , is called a square matrix of
n order. In particular, correlation matrix of 1 n type is
called a row vector, and correlation matrix of n  1 type is
called a column vector.
 R   0 
 g1 g1
 Dg1 



r  0    0
gg
 


 0

R


gg

g1 g1
For correlation above mentioned matrices abridged
gg
is linked to
 n, n
gi g j
form:
is cross-correlation function. Here, the first index i



notations
are
often
used:
R   0   R   0
gg
g
g
 i j 



,
,
i  1, 2,, n; j  1, 2,, n R    R     0,1, 2,, N  .
0

a determinant. In solving of static identification problems,

determinant of correlation matrix R   0  has the following
gi gi
represents the number of element row, and the second j is the
number of its column.


Square correlation matrix R   0   R 

gg

R   0 
gg
In solving of dynamic identification problems, determinant

of normalized correlation matrix r    has the following
gg
0
form:
R   0

gg

det r    
gg
D g 
R   t 
gg
D g 

R   N  1t 
gg
D g 
R   N  1t 
R   t 
gg
D g 
R   0
gg


D g 


R   N  2t 
gg

D g 
gg
Dg 
R    N  2t 
gg
Dg 

R   0
gg
Dg 
Matrix is a rectangular array of numbers, and its determinant is
a number determined from known rules, that is:

det R        1  R
gg
where
R

g1 g i1


sum
0, R
is


g 2 g i2


g1 gi1
applied
0,, R
to


g n g in
and therefore contains
n
0, R


g 2 g i2
all
0
0,, R


g n g in
0 

possible permutations
of elements 1, 2, , n
  0 , if
summands, with
  1 , if permutation is odd.

By norm of correlation matrix Rgg (0)  Rg g j (0) is
i

meant a real number Rgg (0) , complying with the following
permutation is even and


d)

By
norm

of
normalized

i

a)
r gg 0  0 , and
r gg 0  0 only when
r gg 0  0;
  r gg 0   r gg 0



R gg 0   0 only when




matrix
complying with the following conditions:
b)
R gg 0  0 , and
correlation



rgg (0)  rg g j (0) is meant a real number rgg (0) ,
conditions:
a)

R gg  B  R gg    B ;


(
a
number) and,

in particular,  r gg 0   r gg 0  ;

R gg 0  0;

  R gg 0   R gg 0 (   a

b)
c)




number) and,

d)
in particular,  R gg 0   R gg 0  ;
By

c)

R gg 0  B  R gg 0  B ;



 R gg 0 B  R gg 0  B ;




By norm of correlation matrix Rgg (  )  [ Rgg (  )] is

meant a real number Rgg (  ) , complying with the following

R gg    0 , and

R gg    0 only when
correlation
matrix

r gg    0 only when
r gg    0;
  r gg     r gg   (   a





number) and,

in particular,  r gg    r gg   ;
  R gg     R gg   (   a number) and,

r gg    B  r gg    B ;





r gg    0 , and


in particular,  R gg    R gg   ;
c)
normalized

c)

b)
of

a)

R gg    0;
norm
complying with the following conditions:
b)
conditions:



 r gg 0 B  r gg 0  B ;





rgg (  )  [rgg (   it )] is meant a real number rgg (  ) ,
d)
a)

r gg 0  B  r gg 0  B ;

R gg    B  R gg    B ;
d)





r

B

r gg    B ;
gg




Rgg 0 , Rgg  , rgg 0 , rgg   and B are matrices, for
which corresponding operation make sense. For square matrix
in particular, we have


 R gg 0



P
P


  R gg 0 


R gg 0 
3.



 R gg  


R gg   


  R gg   



r gg 0 


  r gg 0  


r gg   
IV.
P
P


  r gg   



and normalized correlation matrices


r gg 0  rgg 0ij ,

are considered:
i
m

R gg 0, R gg   and normalized

 Rgg 0 Rgg 0  E 
R  R  
 Rgg   Rgg    E 
1
gg
gg
1
1
gg
1

i

R gg    max  Rgg  ij ( l - norm);
j
i

r gg 0  max  rgg 0ij ( l - norm);
i
rgg    max  rgg  ij ( l - norm);
i
r  r  
 rgg   rgg    E 
gg
1
gg
j
R gg 0  max  Rgg 0ij ( l - norm);
 rgg 0 rgg 0  E 
1
r gg    max  rgg  ij (т- norm);
i
r 0r 0
gg
j
j
gg
j
i
j
INVERSE CORRELATION MATRIX
R 0R 0

l
( k - norm)
i, j
Then we will have the following by definition:
r gg 0  max  rgg 0ij (т- norm);
j
2
ij
r gg 0, r gg   , let us denote inverse
1
1
1
1
matrices by R gg 0, R gg  , rgg 0, rgg   respectively.

l
gg
j
i
l
 r  


l
( k - norm)
correlation matrices
R gg    max  Rgg  ij (т- norm);
2.
2
ij


m
gg
For correlation matrices
R gg 0  max  Rgg 0ij (т- norm);
m
( k - norm)
Inverse correlation matrix of the given one is a correlation
matrix, which, being multiplied by the given correlation matrix
both from the right and from the left, gives an identity
correlation matrix.
r gg    rgg  ij , basically three easily calculated norms
m
 r 0


1.
2
ij
i, j

For correlation matrices R gg 0  Rgg 0ij  , R gg    Rgg  ij 

gg
P
P
where p is a natural number.

( k - norm);
i, j
k


 r gg  


2
ij
 R  
k
k


 r gg 0 


gg
i, j
k
P
P
 R 0
where
gg
1
1
E is identity matrix.
Determination of inverse correlation matrix for the given
one is called correlation matrix inversion.
A correlation matrix is called nonsingular, if its determinant
is different from zero.
Otherwise, a correlation matrix is called singular. Any
nonsingular correlation matrix has an inverse matrix.
Let us build so-called augmented (or union) matrices for

matrices



R gg 0, R gg   and r gg 0, r gg   :
 RAg1g1 0  RAg2 g1 0
 RA 0  RA 0 

g1g 2
g2 g2
RA gg 0   
 


 RAg1g n 0 RAg2 g n 0
where
 RAgn g 1 0 
 RAg n g2 0 


 

 RAgn g n 0
RAg1g1 0 

R gg





RA     
RA   0 
gg
 RA   N  1t 
gg

 RA    N  2 t  
gg






RA   0
gg

RA   t 
gg
RA   t 
RA   0 
gg
gg




 RA    N  1t  RA    N  2t 
gg
 gg
gg

RA gg   are algebraic complements (minors with
signs) of corresponding elements Rgg   ;
where
 ra g1g1 0  ra g 2 g1 0 
ra 0  ra 0 

gg
g2 g2
r a gg 0    1 2
 


ra g1g n 0  ra g 2 g n 0 
 ra g n g 1 0 
 ra g n g 2 0 


 

 ra g n g n 0 
where ra g n g n 0  are algebraic complements (minors with
signs) of corresponding elements rg n g n 0  ;

r a     
ra   0 
ra   t 
 ra    N  1t 
ra   t 
ra   0 
 ra    N  2 t  


gg
gg
gg
gg
gg
gg
1


det R gg 0 
RAg1g2 0  RAg 2 g 2 0   RAgn g 2 0  





RAg1gn 0 RAg2 g n 0   RAgn g n 0 
R Agg  
Let us divide all elements of correlation matrix

by the value of determinant, i.e. by
det R gg   :
RA   0 
RA   t 

gg
gg

RA
RA   0 
  t 

1
1
gg
gg
R gg   




det R gg   
 RA    N  1t  RA    N  2 t 
gg
 gg

RA    N  1t 
gg

 RA    N  2 t 
gg






RA   0 
gg


Let us divide all elements of normalized correlation matrix

r agg 0 by the value of determinant, i.e. by det r gg 0 :

 ra g1g1 0  ra g2 g1 0 
ra 0  ra 0 
1
g2 g2
1
 g1g 2
r gg 0  

 

det r gg 0 
ra g1gn 0  ra g2 g n 0 

 ra gn g 1 0 
 ra g n g2 0  

 

 ra gn gn 0
Let us divide all elements of normalized correlation matrix


r agg   by the value of determinant, i.e. by det r gg   :
ra   0 
ra   t 

gg
gg

ra   0 

 ra g g t 
1
1
gg
r gg   




det r gg   
ra    N  1t  ra    N  2 t 
gg
 gg
 ra    N  1t 
gg

 ra    N  2 t 
gg






ra   0 
gg

gg

ra    N  1t  ra    N  2 t  
gg
0 

RAgi g j 0 are algebraic complements (minors with
signs) of corresponding elements RAg g 0 , i, j  1, 2,, n ;
i j
1
RAg2 g1 0   RAgn g 1 0
gg


ra   0 
gg
where ra g n gn   are algebraic complements (minors with
signs) of corresponding elements rg n g n   .
It should be noted that algebraic complements of row
elements are placed into corresponding columns, i.e.
transposition is performed.
Those are inverse correlation matrices

R gg
1

R gg
1
0
,

   r gg 1 0  r gg 1   
Thus, reliable prediction of technical condition of a highrise building or building structure requires application of the
correlation matrices described above, which makes it possible
to solve problems of control, monitoring, identification,
predication, diagnostics and detection of malfunction at early
stages.

Let us divide all elements of correlation matrix

by the value of determinant, i.e. by
det R gg 0 :
R Agg 0
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