International Core Journal of Engineering 2020-26 | Page 26
X ˆ E
(10) calculated evidence and each proposition to obtain the
evidence BPA.
Where, X̂ is the estimation matrix of X, D is the number
of variables after dimension reduction, and T D is the score
matrix of I u D , which represents the projection of matrix X
on load vector p i , and the combination of load vectors 2) ICA detection statistics and BPA: Provide I 2 and
SPE as evidence. I 2 explains the influence of the main
independent component space on the data, SPE explains the
influence of residual space on the data.
X
T D P D T E
obtains load matrix P D KJ u D . The corresponding
The corresponding statistic I 2 and SPE are shown in
Equation (15) and (16).
statistic T 2 and SPE expressions are Equation (11) and (12).
T 2
SPE
T D / D 1 T D T (11)
E T E (12)
I 2
SPE
E T E
S d T S d
(15)
X X ˆ X X ˆ
T
(16)
Where, / D is the diagonal matrix of the principal
element after dimension reduction. Where S d is an independent component matrix obtained
using maximum likelihood estimation
After calculating the statistic as evidence, it is necessary
to assign the BPA. Before that, we need to understand the
concept of control limits. If the control limit is exceeded, the
variable is considered abnormal. The control limit for
statistic T 2 is calculated as Equation (13). The control limit of I 2 and SPE can be obtained by the
kernel density estimation method, in which the kernel
function selects the Gaussian kernel function, shown in
Equation (17).
T T 2
D I 2 1
I I D
F D , I D ; T
f ˆ x
(13)
C. Multiple evidence fusion and decision making
Using the method described in Fig. 1, the statistical
evidence obtained through PCA and ICA was subjected to
multiple evidence fusions that accounted for conflicting
evidence. In terms of decision rules, the Pignistic probability
method is the most suitable among the five decision
methods mentioned above, the paper adopts B&P method
base on Pignistic probability. The final reliability of the
hypothesis is related to each element in the proposition, and
is distributed according to s i shown in Equation (18).
The control limit for statistic SPE is calculated as
Equation (14).
§ v · 2
g F 2 h ¨
¸ F 2 P 2
© 2 P ¹ v , T
(14)
Where P and v is mean and variance of data,
g
v / (2 P ) is weighted value, h
freedom, F
2
2 P 2 / v , T
(17)
Where n is the number of samples; h is the bandwidth,
which is obtained by cross-validation. The size of the
confidence level is the percentage of the kernel function.
Where, F D , I D ; T refers to the critical value of F
distribution of significant level T and degree of freedom D
and I D .
SPE T
2
1 n 1
° § x x i · ½
°
exp ® ¨
¦
¸ ¾
nh i 1 2 S
h
4
©
¹
°
°
¯
¿
2 P 2 / v is degree of
is the critical value of the F 2 distribution
with a significant level of T and a degree of freedom of
2 P 2 / v .
s i
The confidence level is different, the statistic control
limits will have different values, and each evidence will be
associated with multiple variables. Therefore, the statistic in
the data corresponding to each variable can be calculated
first. The value is equal to the control limit and the
corresponding confidence level is calculated, and the
confidence level can be used as the basis for calculating the
BPA. It is easy to see that if the statistic of a variable A
differs from the evidence value, the calculated confidence
level will be lower, indicating that the lower the correlation
between the evidence and the variable A, the more support
the proposition A will be small and vice versa. Since the
BPA needs to satisfy Equation (3), it is necessary to
standardize the confidence level 1 T n , i between each of the
Bel A i
, A M 2 4 , A i A M
1 Pls A i (18)
§ s i
¨ ¨
A i A M © ¦ s j (19)
PrBP A i
¦
·
¸ ¸ m A M
¹
The calculated result can be used as the final reliability,
and the B&P method comprehensively considers Bel(•) and
Pls(•), and the performance and efficiency can reach a good
level, which can effectively avoid the decision risk and
obtain accurate conclusions.
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