自适应稀疏主成分分析提升过程监控与故障诊断
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自适应稀疏主成分分析（Adaptive Sparse Principal Component Analysis, ASPCA）是一种创新的统计方法，旨在提升过程监控和故障隔离的性能。传统的主要成分分析（Principal Component Analysis, PCA）在工业流程监控中广泛应用，它通过将数据转换为一组新的线性组合，即主成分（PCs），来发现数据中的潜在结构和异常。然而，PCA的一个主要局限是缺乏物理解释性，因为每个主成分是由所有变量的线性组合构成的，这使得故障检测变得复杂，尤其是当过程参数随时间变化或存在噪声干扰时。
ASPCA针对这一问题提出了改进。它引入了自适应性元素，允许模型能够动态适应数据的变化，提高了对过程动态特性的捕捉能力。此外，ASPCA还强调了稀疏性，即只保留少数重要的特征或变量，这有助于减少冗余信息，增强信号与背景噪声之间的对比，从而提高故障检测的精度和效率。通过这种方式，ASPCA不仅保留了PCA的主要优点——数据降维和异常检测，还克服了其固有的不足，提供了更具有实际意义的主成分解释。
在研究论文中，作者Kangling Li、Zhengshun Fei、Boxuan Yue、Jun Liang等人合作，他们构建了一个理论框架，并可能通过实验验证了ASPCA在实时过程监控和故障识别场景中的有效性。他们可能讨论了ASPCA的算法设计、实施步骤、优化方法以及如何处理非平稳数据和异常情况。文章历史显示，该研究在2014年12月首次提交，经过修订于2015年6月接受，最终于同年6月在线发布。
关键词包括：主成分分析、自适应性、稀疏性、过程监控、故障隔离。这些关键词揭示了本文的核心关注点，即如何利用ASPCA技术在不断变化的工业环境中实现更为精确和智能的过程监控和故障诊断。
自适应稀疏主成分分析作为一种前沿的统计工具，对于提高复杂工业系统的实时监控能力和故障隔离效果具有显著的优势，为工业自动化领域的研究人员和工程师提供了一种有力的数据分析手段。
different sub-structures. In summary, these local sub-structures cover
the whole process and reflect the natural dynamics of process.
Note that the presentence of jp
i;k
p
i; j
s
z
τ
jk
j and ‖p
i
‖
2
2
= 1 renders the
ASPCA optimization problem non-convex. An algorithm for solving
ASPCA is developed in Algorithm 1.
The online procedure is to adaptively update the ASPCA optimiza-
tion problem with the current measurement z
τ
. The process is
remodeled when any new measurement is available, resulting in large
computation and storage requirements. For simplicity, we partition Z
into B segments Z =[Z
1
, Z
2
, …, Z
B
]
T
, and each seg ment represents an
operating cond ition. The process is remodeled with each seg ment
Z
b
(b =1,2,…, B) for complying with changes in operating conditions
or proces s c haracteristics. Arrange varia nce of each PC λ
i
τ
= p
i
τT
Σ
X
p
i
τ
(i =1,2,…, m) in a descend order into λ
1
τ
=[λ
1
τ
, λ
2
τ
, ⋯, λ
m
τ
]
(λ
1
τ
≥ λ
2
τ
≥⋯≥λ
m
τ
) with corresponding loadings P
τ
=[p
1
τ
, p
2
τ
, ⋯, p
m
τ
].
Then, the matrix X is decomposed into the score matrix T
τ
¼ X
^
P
τ
and
the residual matrix E
τ
¼ XðI−
^
P
τ
^
P
T
τ
Þ, which is given by X ¼ T
τ
^
P
T
τ
þ E
τ
,
where
^
P
τ
¼½p
τ
1
; p
τ
2
; …; p
τ
l
is the loading matrix and Σ
T
τ
¼
^
P
τ
Σ
X
^
P
T
τ
is
the covariance matrix. The number of PCs (l) is selected as the one min-
imizing the following BIC-type criterion.
BIC
l
¼ − log
X
l
i¼1
λ
τ
i
!
þ l
log n
n
l ¼ 1; 2; …; m−1ðÞð3Þ
where the first term decreases and the second term increases with l,
and, theoretically, BIC has a minimum solution. The BIC criterion is
established based on the idea that the score matrix contains mostly var-
iance information and the residual matrix contains noise.
Algorithm 1. (Iterative interior point algorithm for solving ASPCA)
(i) For each sparse loading vector p
i
τ
, without loss of generality,
we individually perform the following steps in the order of i =
1, 2, …, m.
(ii) Start the algorithm by setting the ith PCA loading vector p
i
⁎
as the
initial solution of the ith sparse loading vector, p
i
(0)
= p
i
⁎
.
(iii) For any a ≥ 1 based on the last so lution p
i
τ(a − 1)
, the original
ASPCA optimization problem is revised as the following optimi-
zation problem, where y
i
=[y
i,1
, y
i,2
, ⋯, y
i,m
] denotes the absolute
value terms of p
i
.
p
τ;
i
¼ min
p
i
;t
i
‐p
i
T
Σ
X
p
i
þ β
X
m
j¼1
X
m
k¼1
y
i;k
y
i; j
s
z
τ
jk
s:t:
p
T
i
p
τ a−1ðÞ
i
¼ 1
p
T
i
p
τ
j
¼ 0 j ¼ 1; 2; …; i−1ðÞ
−y
i;k
≤p
i;k
≤y
i;k
k ¼ 1; 2; …; mðÞ
8
>
<
>
:
(iv) Solve the above optimization problem using the interior point
method (available in Matlab). Then, obtain p
i
τ(a)
by normalizing
p
i
τ,∗
: p
i
τ(a)
= p
i
τ,∗
/‖p
i
τ,∗
‖.
(v) Set a = a + 1, and then repeat Steps iii–iv until convergenc e,
‖p
i
τ(a)
− p
i
τ(a − 1)
‖
1
≤ ε,whereε denotes the convergence thresh-
old (take ε = 0.05 in this work).
2.2. Process monitoring based on ASPCA
Based on the ASPCA model built in the above section, the first l PCs
span the PC subspace and the last m–l PCs construct the residual sub-
space. Thus, each measurement is identified by its score Mahalanobis
distance in the PC subspace and the model error in the residual sub-
space. Then, two monitoring statistics, Quasi- T
2
(QT
2
) in the PC sub-
space and SPE in residual subsp ace, are respectively defined and
compared to their corresponding confidence limits as follows.
QT
2
¼ t
τT
∑
−1
T
τ
t
τ
¼ z
T
τ
^
P
τ
∑
−1
T
τ
^
P
T
τ
z
τ
≤ QT
2
lim
ð4Þ
SPE ¼ e
τT
e
τ
¼ z
T
τ
I−
^
P
τ
^
P
T
τ
z
τ
≤ SPE
lim
ð5Þ
where t
τ
¼
^
P
T
τ
z
τ
contains the scores and e
τ
¼ðI−
^
P
τ
^
P
T
τ
Þz
τ
contains the
residuals of z
τ
. Note that if β = 0, ASPCA reduces to PCA. In PCA moni-
toring, PCs are independent with each other and then Σ
T
τ
is diagonal.
In this case, monitoring statistic QT
2
equates to T
2
. However, in ASPCA
monitoring, the extracted PCs are often not independent and barely
conform to a specific distribution, hence, the confidence limit cannot
be determined directly from a particular approximate distribution. An
alternative approach to determine the confidence limit of QT
2
(QT
lim
2
)
is to use kernel density estimation (KDE) [22,23]. A univariate kernel es-
timator is used in this work, and it is defined as
^
fx; dðÞ¼
1
nd
X
n
i¼1
K
x−x
i
d
hi
ð6Þ
where x is the data point under consideration; x
i
is an observation value
from the data set; d is the window width or the smoothing parameter;
n is the number of observations; and K is the kernel function, which
determines the shape of the smooth curve and satisfies the following
conditions:
KxðÞ≥0
Z
þ∞
−∞
KxðÞdx ¼ 1
8
<
:
ð7Þ
In this work, a Gaussian function is chosen for K. Thus, we can deter-
mine the confidence limit with 95% or 99% area coverage of de nsity
function. It should be po inted out that the confidence limit of SPE
(SPE
lim
) can also be determined by KDE. Consequently, a fault is report-
ed to be detected if any one of monitoring statistics QT
2
and SPE violates
its corresponding confidence limit.
2.3. Fault isolation based on ASPCA and dominant PCs
After a fault is detected, it is crucial to root the cause of the out-of-
control status. With the assumption that variables associated with the
fault likely exhibit large contributions, reconstruction based contribu-
tion (RBC) [24] is popularly adopted to isolate faulty variables. Despite
contribution plot approach requires no prior fault information, it leads
to obscure diagnosis because faulty variables can increase contributions
of unaffected variables. To reduce this “smearing” effect, a fault isolation
scheme with dominant PCs is presented in this section.
2.3.1. Dominant principal components
PCs are not of the same sensitivity to a fault. For a specific fault, usually
only several PCs dominate. An indefinite mapping of fault information re-
sults in the relevant information loss and poor monitoring performance.
Therefore, it is of much importance for fault isolation to select fault-
dominant PCs and concentrate the fault information on the selected dom-
inant PCs. The algorithm for selecting dominant PCs proceeds as follows.
Algorithm 2. (Iterative algorithm for selecting dominant PCs)
(i) Start the algorithm by initializing dominant PCs set DT = ∅.
(ii) Select the jth PC with the largest amount of contribution as an-
other new dominant PC. The contribution of each PC i ∉ DT can
be defined by
CT
i
¼ φ
QT
2
t
τ
DT
−φ
QT
2
t
τ
Θ
i
ð8Þ
where, Θ
i
= DT ∪ {i}, reconstructed index φ
QT
2
ðt
τ
a
Þ¼t
τ
a
T
Σ
−1
T
τ;a
t
τ
a
and t
a
τ
, T
τ,a
are obtained by removing all corresponding PCs in the
set a from t
τ
and T
τ
.Theoptimalj value is selected as the one maxi-
mizing the above contribution equation, so j ¼ max
i
CT
i
.
428 K. Liu et al. / Chemometrics and Intelligent Laboratory Systems 146 (2015) 426–436
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