Anomaly identification method for structural monitoring data considering spatial-temporal correlation

Abstract

The present invention belongs to the technical field of health monitoring for civil structures, and an anomaly identification method considering spatial-temporal correlation is proposed for structural monitoring data. First, define current and past observation vectors for the monitoring data and pre-whiten them; second, establish a statistical correlation model for the pre-whitened current and past observation vectors to simultaneously consider the spatial-temporal correlation in the monitoring data; then, divide the model into two parts, i.e., the system-related and system-unrelated parts, and define two corresponding statistics; finally, determine the corresponding control limits of the statistics, and it can be decided that there is anomaly in the monitoring data when each of the statistics exceeds its corresponding control limit.

Claims

1. An anomaly identification method for structural monitoring data considering spatial-temporal correlation, wherein, steps of which are as follows: Step 1: Monitoring data preprocessing (1) Define current and past observation vectors for a normal monitoring data:
y.sup.c(t)=y(t)
y.sup.p(t)=[y.sup.T(t1),y.sup.T(t2), . . . ,y.sup.T(tl)].sup.T where y(t).sup.m represents the sample at time t in the normal monitoring data, and m represents the number of measured variables; y.sup.c (t) and y.sup.p(t) represent the current and past observation vectors defined at time t, respectively; l represents the time-lag; (2) Pre-whiten the current observation vector y.sup.c (t) and past observation vector y.sup.p(t):
{tilde over (y)}.sup.c(t)=R.sup.cy.sup.c(t)
{tilde over (y)}.sup.p(t)=R.sup.py.sup.p(t) where R.sup.c and R.sup.p represent pre-whitening matrices corresponding to y.sup.c (t) and y.sup.P (t), respectively; {tilde over (y)}.sup.c(t) and {tilde over (y)}.sup.p(t) represent the pre-whitened current and past observation vectors, respectively; Step 2: Spatial-temporal correlation modeling (3) Establish a spatial-temporal correlation model for the normal monitoring data, that is, establish a statistical correlation model between {tilde over (y)}.sup.c(t) and {tilde over (y)}.sup.p(t) as follows:
({tilde over (C)}.sub.pp.sup.1{tilde over (C)}.sub.pc{tilde over (C)}.sub.cc.sup.1{tilde over (C)}.sub.cp)=.sup.2
({tilde over (C)}.sub.cc.sup.1{tilde over (C)}.sub.cp{tilde over (C)}.sub.pp.sup.1{tilde over (C)}.sub.pc)=.sup.2 where {tilde over (C)}.sub.pp=E{{tilde over (y)}.sup.p{tilde over (y)}.sup.pT} and {tilde over (C)}.sub.cc=E{{tilde over (y)}.sup.c{tilde over (y)}.sup.cT} represent auto-covariance matrices of {tilde over (y)}.sup.p(t) and {tilde over (y)}.sup.c(t), respectively; {tilde over (C)}.sub.pc=E{{tilde over (y)}.sup.p{tilde over (y)}.sup.cT} and {tilde over (C)}.sub.cp=E{{tilde over (y)}.sup.c{tilde over (y)}.sup.pT} represent cross-covariance matrices of {tilde over (y)}.sup.p(t) and {tilde over (y)}.sup.c(t), respectively; (4) Since {tilde over (y)}.sup.p(t) and {tilde over (y)}.sup.c(t) are pre-whitened data, {tilde over (C)}.sub.pp and {tilde over (C)}.sub.cc are both identity matrices; by additionally considering {tilde over (C)}.sub.pc.sup.T={tilde over (C)}.sub.cp and {tilde over (C)}.sub.cp.sup.T={tilde over (C)}.sub.pc, the statistical correlation model between {tilde over (y)}.sup.p(t) and {tilde over (y)}.sup.c(t) can be further simplified as follows:
({tilde over (C)}.sub.pc{tilde over (C)}.sub.pc.sup.T)=.sup.2
({tilde over (C)}.sub.cp{tilde over (C)}.sub.cp.sup.T)=.sup.2 (5) The solution of the above statistical correlation model can be obtained by a following singular value decomposition:
{tilde over (C)}.sub.pc=E{{tilde over (y)}.sup.p{tilde over (y)}.sup.cT}=.sup.T where =[.sub.1,.sub.2 . . . ,.sub.ml].sup.mlml and =[.sub.1, .sub.2 . . . , .sub.m].sup.mm represent matrices consisting of all left and right singular vectors, respectively; .sup.mlm represents a singular value matrix, in which m non-zero singular values are correlation coefficients between {tilde over (y)}.sup.p(t) and {tilde over (y)}.sup.c(t); (6) Define a projection of {tilde over (y)}.sup.p(t) on , termed as z (t), which can be obtained by:
Z(t)=.sup.T{tilde over (y)}.sup.p(t)=.sup.TR.sup.py.sup.p(t)=Qy.sup.p(t) where Q=.sup.TR.sup.p; Step 3: Define statistics (7) Since there are only m non-zero correlation coefficients, variables in z(t) can be divided into two parts:
z.sub.s(t)=Q.sub.sy.sup.p(t)
z.sub.n(t)=Q.sub.ny.sup.p(t) where z.sub.s(t) and z.sub.n(t) represent system-related and system-unrelated parts of z(t), respectively; Q.sub.s and Q.sub.n represent first m rows and last m (l1) rows of Q, respectively; (8) To identify anomalies in monitoring data, two statistics can be defined for z.sub.s(t) and z.sub.n(t):
H.sub.s.sup.2=z.sub.s.sup.Tz.sub.s=y.sup.pT(Q.sub.s.sup.TQ.sub.s)y.sup.p
H.sub.n.sup.2=z.sub.n.sup.Tz.sub.n=y.sup.pT(Q.sub.n.sup.TQ.sub.n)y.sup.p For newly acquired monitoring data, the past observation vector y.sup.p is constructed firstly; two corresponding statistics H.sup.2.sub.s and H.sup.2.sub.n are then calculated, respectively; it can be decided that there exist anomalies in the monitoring data when each of the statistics exceeds its corresponding control limit; Step 4: Determine control limits (9) If the monitoring data is Gaussian distributed, the two corresponding statistics H.sup.2.sub.s and H.sup.2.sub.n theoretically follow a F-distribution, and the theoretical values of the control limits are determined as: $H_{s,\mathrm{li}m}^s\left(\right)\frac{m\left(m^2{l}^2-1\right)}{ml\left(ml-m\right)}{F}_{m,ml-m}\left(\right)$ $H_{n,lim}^2\left(\right)\frac{m\left(l-1\right)\left(m^2{l}^2-1\right)}{m^{2}l}{F}_{ml-m,m}\left(\right)$ where H.sup.2.sub.s,lim and H.sup.2.sub.n,lim represent the control limits of the two corresponding statistics H.sup.2.sub.s and H.sup.2.sub.n, respectively; represents a significance level, it is generally set to 0.01; (10) If the monitoring data is not Gaussian distributed, probability density distributions of the two corresponding statistics H.sub.s.sup.2 and H.sub.n.sup.2 can be separately estimated by other methods, and then the control limits are determined according to a given significance level.

Description

DETAILED DESCRIPTION

(1) Take a two-span highway bridge model, with a length of 5.5 m and a width of 1.8 m, as an example. A finite element model is built to simulate structural responses, and the responses at 16 finite element nodes are acquired as monitoring data. There are two datasets generated: the training dataset and the testing dataset; the training dataset consists of normal monitoring data, and part of the testing dataset is used to simulate abnormal monitoring data; both datasets last for 80 s and the sampling frequency is 256 Hz. The key of the present invention lies in the spatial-temporal correlation modeling process for the structural monitoring data, as shown in the following schematic:

(2) ##STR00001##

(3) (1) Construct the current observation vector y.sup.c(t) and the past observation vector y.sup.p(t) for each data point in the training dataset; then pre-whiten all current and past observation vectors (i.e., y.sup.c(t) and y.sup.p(t)) to obtain the whitening matrices (i.e., R.sup.c and R.sup.p) and the pre-whitened data (i.e., {tilde over (y)}.sup.c(t) and {tilde over (y)}.sup.p(t)).

(4) (2) Establish spatial-temporal correlation model for the training dataset, that is, build a statistical correlation model for {tilde over (y)}.sup.c(t) and {tilde over (y)}.sup.p(t) to obtain the model parameters Q=.sup.TR.sup.p and ; since there are only 16 non-zero correlation coefficients in , the first 16 rows of the matrix Q are used to construct Q.sub.s and the others are used to construct Q.sub.n.

(5) (3) Determine the control limits of the statistics, i.e., H.sub.s,lim.sup.2 and H.sub.n,lim.sup.2; after new monitoring data is acquired, the past observation vector is first constructed, and then the two statistics, i.e., H.sub.s.sup.2=y.sup.pT(Q.sub.s.sup.TQ.sub.s)y.sup.p and H.sub.n.sup.2=y.sup.pT(Q.sub.n.sup.TQ.sub.n)y.sup.p, are calculated; it can be decided that there exist anomalies in the monitoring data when each of the two statistics exceeds its corresponding control limit.

(6) (4) Simulate abnormal monitoring data in the testing dataset, that is, the monitoring data of sensor 3 gains anomaly during time 4080 s; identify anomalies in the monitoring data using the two proposed statistics H.sub.s.sup.2 and H.sub.n.sup.2, results show that both H.sub.s.sup.2 and H.sub.n.sup.2 can successfully identify anomalies in the monitoring data.