Issue 58

J. Wang et alii, Frattura ed Integrità Strutturale, 58 (2021) 114-127; DOI: 10.3221/IGF-ESIS.59.09

n

n

(3)

   ( )] i i

E



[ E w y x

w E y x

[ ( )] y x

[ ( )]

i

i

0





i

i

1

1

where, w i are the weight coefficients and therefore the unbiased condition followed is as follows:

n

   1 i i w

1

(4)

(ii) Minimizing error estimation (optimal results) The estimated variance of the unknown point x 0 can be obtained as

n



2

2 ( )) ]

 ( ) ( )) ] 2





 E y x y x [( ˆ ( )





[( E w y x y x

E

i

i

0

0

0



i

1

(5)

   1 1 n n i j

n





 ( ( ), ( )) 2

 w C y x y x C y x y x ( ( ), ( )) ( ( ), ( ))

w w C y x y x

i

j

i

j

i

i

0

0

0



i

1

where, C denotes the matrix of the covariance, while the correlation vector R (x i , x j ), based on the unbiased results, can convert Eqn. (5) into the following form:

n

n n

   0 ( , ) i i w R x x

 2 E



 w w R x x R x x , 0 0 ( , ) ( , ) i j i j

   2

2

R x x

C x x

( , )

( , )

(6)

i

j

i

j



  j

i

i

1

1 1

The minimization of estimated variance of Eqn. (6), at the unbiased condition, is an extreme-value problem with constraints, while the solution can be obtained based on the Lagrange Multiplier Method, which means:

n

        2 1 2 ( E i i w

(7)

L w

( , )

1)

i

where, Φ is the vector of Lagrange Multiplier, the partial derivatives of w i and Φ to L are set to 0, deriving:

 L w R x x R x x w L w             2 i     1 2( 1) 0 n j i j i j n i

 ( , ) 2 ( , ) 2 0   

0

(8)



i

1

after adjustment, the form is:



n

        1 j j n i w

  

w R x x

R x x

( , )

( , )

i

j

i

0

(9)

1



i

1

The weight coefficient w i and Lagrange coefficient Φ can be obtained via solving linear Eqn. (9), while the evaluated value of ŷ ( x 0 ) at unknown point x 0 can be derived, based on Eqn. (2). Eqn. (9) can be written as a matrix:

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