394
P. Z. Korondi et al.
Assuming the following form for the RSM function:
ˆ
y = ˆ
f (x, α),
(12.9)
where ˆ
f (x, α) is the RSM function, x = [x 1 , . . . , x m ] is the vector of independent
input variables, and α = [α 1 , . . . , α p ] is the vector of model parameters to be
adjusted, the LSM problem can be formulated as follows:
min
α
n
i=1
(y i − ˆ
f (x i , α))
2 ,
(12.10)
where n is the number of observations and y i are the actual responses. The sum in
Eq. (12.10) corresponds to the RSS index.
The parameters of the RSM can be determined according to the general minimum
condition equation. The RSS index will reach its minimum where its first derivative
equals zero,
∂RSS
∂α j
= −2
n
i=1
(y i − ˆ
f (x i , α))
∂ ˆ
f (x i , α)
∂α j
= 0, j = 1, . . . , p.
(12.11)
Further details on LSM can be found in Cavazzuti’s book [3].
In case of a Linear Least Squares Problem (Linear LSM) the regression model
can be written as,
ˆ
y = Xα,
(12.12)
where X is a matrix of functions containing the basis functions that depend only
on input variables. The model is linear because it is a linear combination of the
basis functions: it is linear in the model coefficients, while the basis functions can
be nonlinear.
By introducing the linear model of Eq. (12.12) in Eq. (12.10), the Linear Least
Squares Problem is formulated as,
min
α
n
i=1
(y i − Xα)
2 .
(12.13)
According to Eq. (12.11) the model parameters can be obtained from the following
equation:
∂
∂α
(y − Xα)
T (y − Xα) = −2X
T y + 2X
T Xα = 0.
(12.14)
The coefficients of the RSM can be given in closed form,
P. Z. Korondi et al.
Assuming the following form for the RSM function:
ˆ
y = ˆ
f (x, α),
(12.9)
where ˆ
f (x, α) is the RSM function, x = [x 1 , . . . , x m ] is the vector of independent
input variables, and α = [α 1 , . . . , α p ] is the vector of model parameters to be
adjusted, the LSM problem can be formulated as follows:
min
α
n
i=1
(y i − ˆ
f (x i , α))
2 ,
(12.10)
where n is the number of observations and y i are the actual responses. The sum in
Eq. (12.10) corresponds to the RSS index.
The parameters of the RSM can be determined according to the general minimum
condition equation. The RSS index will reach its minimum where its first derivative
equals zero,
∂RSS
∂α j
= −2
n
i=1
(y i − ˆ
f (x i , α))
∂ ˆ
f (x i , α)
∂α j
= 0, j = 1, . . . , p.
(12.11)
Further details on LSM can be found in Cavazzuti’s book [3].
In case of a Linear Least Squares Problem (Linear LSM) the regression model
can be written as,
ˆ
y = Xα,
(12.12)
where X is a matrix of functions containing the basis functions that depend only
on input variables. The model is linear because it is a linear combination of the
basis functions: it is linear in the model coefficients, while the basis functions can
be nonlinear.
By introducing the linear model of Eq. (12.12) in Eq. (12.10), the Linear Least
Squares Problem is formulated as,
min
α
n
i=1
(y i − Xα)
2 .
(12.13)
According to Eq. (12.11) the model parameters can be obtained from the following
equation:
∂
∂α
(y − Xα)
T (y − Xα) = −2X
T y + 2X
T Xα = 0.
(12.14)
The coefficients of the RSM can be given in closed form,