Information Technology Reference
In-Depth Information
precision. The result of interpolation is represented as a product df=
Ψ
g of SVD
(principal components) to SVD-transformed RBF weights g=V
T
w.
Finally one has f(x)=<y>+df(x), computational cost of interpolation is reduced to O(m
Nmod), plus once-charged O(m Nexp
2
) cost of SVD. This method is convenient when
interpolation should be performed many times (>>Nexp), e.g. for interactive
exploration of database.
More generally, for representation of bulky data one can use clustering techniques
[14]. They also decompose bulky data over a few basis vectors (modes) and
accelerate linear algebra operations with them.
modes
Ψ
=U
Λ
3
Reliability Analysis
The purpose of reliability analysis is an estimation of confidence limits (CL) for
simulation results: P(y<CL)=C, where P is probability measure and C is a user
specified confidence level. For example, median corresponds to 50% CL, i.e.
P(y<med)=0.5; while 68% CL corresponds to confidence interval [CLmin,CLmax),
where P(y<CLmin)=0.16, P(y≥CLmax)=1-P(y<CLmax)=0.16; etc. Several methods
for solution of this task are available.
3.1
First Order Reliability Method (FORM)
FORM is applicable for linear mapping f(x) and normal distribution of simulation
parameters:
(x)~exp(-(x-x
0
)
T
cov
x
-1
(x-x
0
)/2).
(8)
ρ
Here x
0
=<x> is mean value of x and
(cov
x
)
ij
=<(x-x
0
)
i
(x-x
0
)
j
>
(9)
is covariance matrix of x. In this case y is also normally distributed, with mean value
(10)
y
0
=<y>=med(y)=f(x
0
)
and covariance matrix
cov
y
= J cov
x
J
T
,
(11)
where J
ij
=
x
j
is Jacoby matrix of f(x), called also sensitivity matrix. The diagonal
part of cov
y
gives standard deviations
∂
f
i
/
∂
2
σ
directly defining CL(y), e.g.
y
CL
min/max
(68%)=<y>±
σ
y
,
(12)
CL
min/max
(99.7%)=<y>±3
σ
y
.
(13)
In particular, when simulation parameters are independent random values, cov
x
becomes diagonal: cov
x
=diag(
σ
x
2
), and
(14)
2
x
j
)
2
2
σ
=
j=1..n
(
∂
f
i
/
∂
σ
.
yi
xj
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