Intrusion Monitoring Based on High Dimensional Random Matrix by Using Ultra-Weak Fiber Bragg Grating Array

Abstract

In order to ensure that a perimeter security system can work effectively, a convenient and effective event detection algorithm has an important engineering significance. Given the above background, in this paper, we propose a high reliability intrusion event recognition method and vibration sensing system, based on ultra-weak fiber Bragg grating array, by using high dimensional random matrix. We obtain a high sensitivity optical interference signal by constructing a patch-matched optical interference system, then compose the demodulated interference signal into a high-dimensional random matrix. The statistical characteristics of the matrix for the Marcenko-Pastur (M-P) law and ring law are used to confirm the presence of intrusion events efficiently, which can reflect the limit spectrum distribution of the high-dimensional random matrix; meanwhile, the abnormal state quantity and moment are obtained. Further, the average spectral radius value is used to judge the fault cause. Field experimental results show that the proposed method can effectively obtain the correct monitoring data for the sensor array. By comparing the monitoring results of normal operation and crusher operation, we can detect the intrusion event in 4.5 s, and the accuracy rate can reach more than 90%, which verifies that the proposed high-dimensional random matrix analysis method can work properly, proving a practical engineering application prospect.

1. Introduction

With the rapid increase in monitoring range and distance of perimeter security systems, the complexity and maintenance difficulty of the monitoring system are challenged [1]. We all hope locate sudden intrusion events accurately in real time, so a monitoring system with high reliability and low false alarm rate is the trend in current development [2]. The key for a perimeter security system based on fiber-optic sensing technology is to measure vibration and environmental noise [3]. By establishing the relationship between distributed fiber-optic sensors and various characteristic parameters of the external environment, real-time monitoring and management of warning areas are finally realized [4,5].

General sensing optical cables rely on armored optical fibers to form high-strength composite cables. When there are different types of intrusion events around the optical cables, the Rayleigh scattered light intensity and power formed by the overall communication transmission link are extremely low. At the same time, the armored structure will also reduce the sensitivity of internal optical fibers for external vibration and shock, so it is of great engineering significance for a security system to realize high-sensitivity monitoring [6]. In order to improve the sensitivity, Rayleigh scattering perimeter security early warning technology based on phase sensitive optical time-domain reflectometry (Φ-OTDR) has been widely studied [7], and has made effective progress in multiple directions. However, compared with traditional security prediction, this technology can effectively improve the sensitivity of event monitoring, but also increase the false alarm rate of the system, so that the monitoring technology has to be improved in the terms of the classification and recognition algorithm [8]. Inspired by the above techniques, researchers combined ultra-weak fiber Bragg grating (uwFBG) with coherence detection technology [9,10], to improve the monitoring effect of intrusion events. Owing to the fact that uwFBG is written by the laser in one exposure, the reflectivity is usually less than −30 dB. The lower reflectivity enables the array to prepare thousands of sensors based on time division multiplexing, so the monitoring distance is longer. Combined with coherent monitoring technology, the external signal can be detected in the form of optical phase change. Therefore, the range and sensitivity of intrusion event monitoring are improved. However, the amount of sensing data based on time division multiplexing is extremely large, which increases the difficulty of optical signal data processing. Therefore, the monitoring, classification, recognition and judgment of different types of intrusion events become some of the difficulties hindering engineering applications.

To determine whether an intrusion event has occurred, researchers set a simple threshold for the signal at the beginning and use wavelet decomposition to realize detection [11]. In order to eliminate the influence of interference events further, they also study the pattern recognition algorithm [12]. In recent years, feature extraction algorithms based on big data analysis have been vigorously developed, but the algorithm has high requirements on the number of samples, and it is difficult to achieve fast event judgment. The theory of high-dimensional random matrix can map data of different states and study corresponding characteristics and state distribution from a mathematical perspective [13,14]; moreover, signal characteristics are analyzed according to feature root distribution and matrix correlation distribution function [15]. By detecting the feature vector of abnormal elements in the matrix, the abnormal state quantity and abnormal moment can also be analyzed [16,17]. The above advantages make the high-dimensional random matrix widely studied in anomaly detection. However, its construction requires the use of correlation analysis to extract the state quantity of the representation data, which has high requirements on the input data [18]. Although uwFBG has better signal-to-noise ratio (SNR) in interference signal acquisition [19], highly sensitive signals also increase the false alarm rate of security monitoring systems, so an intrusion event decision method with high reliability and SNR is significative [20]. In order to achieve the reliability of the FBG-based sensor system, Zhang [21] proposed a model reconstruction soft computing recognition algorithm based on genetic algorithm-support vector regression, with complicated calculations. An applied BP neural network for the FBG calibration method makes sense, with considerable application prospects in FBG, but this algorithm cannot distinguish the effect of different events on the grating [22]. In order to further classify phase modulated perimeter security events, Shi et.al. [23] proposed an interferometric method optical fiber perimeter security system based on multi-domain feature fusion and support vector machines (SVM). This is a method with high accuracy, but requires statistics of nine-dimensional eigenvalues, and the calculation is complicated. In 2015, high-dimensional random matrix was first introduced into the field of power systems to analyze big data [24]. The application of a stochastic matrix theory to the identification of power grid weak links can avoid the errors brought about by the modeling process, the calculation is simple, and online application can be realized [25].

Based on the above analysis, we first attempt to introduce high-dimensional random theory into the classification and recognition application of fiber optic sensing signals, in order to solve the problem of intrusion event judgment. We intend to use the non-equilibrium interferometer to construct the interference measurement system of uwFBGs, so as to realize the high sensitivity measurement of vibration signals, and then form the collected interference signals into a high dimensional random matrix. After performing the above work, it is possible to recognize intrusion events based on dual criteria, based on the comparison of the coincidence degree of the eigenvalue probability, density function with the Marcenko-Pastur (M-P) law curve, and the distribution characteristics of the singular value points on the complex plane. The cause is identified by the mean spectral radius (MSR), which will provide an effective technical decision for perimeter security.

2. Principle

Figure 1 shows the sensing schematic diagram of the distributed perimeter security system based on a uwFBG array. The continuous light emitted from a narrow linewidth laser is modulated into pulse light by an acousto-optic modulator (AOM), whereafter it is injected into circulator-1 (CIR-1) followed by amplification by an erbium-doped fiber amplifier (EDFA). Then, the pulse enters the uwFBG array, and the echo pulse train reflected by the uwFBGs is transmitted to CIR-2. The arm length difference of the nonequilibrium interferometer constructed by Faraday rotating mirrors (FRMs) is the same as the spacing of uwFBGs. Therefore, optical path compensation can be realized in the interferometer by successive reflected pulses, thus completing the pulse interference. Finally, the interference pulse is converted by photoelectric detectors (PDs) for data processing.

The proposed system constitutes a distributed perimeter security system array based on uwFBGs, since a non-equilibrium interferometer is used to achieve optical path matching, the sensing unit is the fiber between adjacent gratings, and the grating becomes a reflection mirror of weak reflectivity. When a single uwFBG reflects pulsed light, the echo light can be expressed as enhanced Rayleigh scattering. When the laser outputs continuous light at t₀, the amplitude of backward Rayleigh scattered light detected by the receiving end can be expressed as [26].

$$e(t)={\displaystyle \sum _{i=1}^N{a}_i}\exp \left(-\alpha \frac{v{\tau }_i}{n_f}\right)e^{\mathrm{j}2\pi f\left[t-{\tau }_i\right]}\mathrm{rect}\left[\frac{t-{\tau }_i}{W}\right]$$

(1)

The term rect [(t − τ_i)/W] accounts for the change in the reflecting volume seen as the pulse propagates. a_i is the amplitude of the scattered pulse light generated by the incident light at the i-scattering center; W is pulse width; τ_i is the time delay of the i-scattered pulse light in a single pulse period; N is the number of scattering centers; α is the fiber attenuation constant; v is the speed of light in a vacuum; f the frequency of quasi-monochromatic pulsed light; n_f is the refractive index of an optical fiber. Therefore, the interference power I(t) associated with the reflection wave e(t) is given by [27]. Through the non-equilibrium interferometer, double beam interference is formed, and the interference result can be expressed as

$$\begin{array}{ll}I& =2r{E}_0^2\cos \left[2\pi \left(f_1-f_2\right)t+\frac{4\pi n}{\lambda }L+\frac{4\pi n}{\lambda }\Delta L\right]\\ & =2r{E}_0^2\cos \left(2\pi \Delta ft+{\phi }_{12}+\Delta \phi \right)\end{array}$$

(2)

where E₀ is the electric field of the incident light and r is the reflectance of uwFBG. Δf is the frequency difference between two optical signals, because two beams are generated from the same light source in the uwFBG sensing system, with the same frequency, Δf = f₁ − f₂ = 0. φ₁₂ is the initial phase of the interference pulse, Δφ is the sensor phase change caused by the external environment, and the vibration phase signal can be demodulated by arctangent demodulation algorithm [28]. When we monitor n uwFBG sensors at sampling point ti, the demodulated result of n-sensor is expressed as x_nti.

We obtain the original sensor data by using the uwFBG array, then transmit the data collected at different locations to the processor. The spatial characteristics are related to the data at different locations simultaneously, and the time characteristics will be formed by the constantly updated data over time. By combining time and space data, we can generate a high dimensional random matrix.

When choosing measuring sensors n among demodulation data, within a certain period, a single measuring point has k state quantity, so there will be a total of N = n × k state quantity. By processing and analyzing the measurement data of all sensors, a time vector can be formed at the sampling point t_i, expressed as

$$x(ti)={\left[x_{1ti},x_{2ti},\cdots ,x_{nti}\right]}^T$$

(3)

We assume that the state quantity is N and the sampling point is T, then a high-dimensional matrix X of size N × T is formed, as

$$X_{N\times T}=\left[\begin{array}{cccc}{X}_{1t1}& X_{1t2}& \dots & X_{1T}\\ X_{2t1}& X_{2t2}& \dots & X_{2T}\\ ⋮& \dots & \dots & \dots \\ X_{Nt1}& X_{Nt2}& \dots & X_{NT}\end{array}\right]$$

(4)

Within the matrix, the same sampling time is arranged on the same column; similar to this, the sampling location is arranged on the row, which shows the corresponding spatial and temporal characteristics of the arranged data. Generally speaking, there are only small measurement errors, small perturbations and white noise in the measured data under normal circumstances, which meet the conditions of independence and of the same distribution, and satisfy the M-P law and the ring law. It is worth noting that the M-P law reflects the limit spectrum distribution law of the high-dimensional random matrix, and when the monitoring system detects the intrusion signal, the distribution law of the matrix limit spectrum of the fiber sensing signal will change. The intrusion event will cause abnormal phase variation generated by the distributed perimeter security system. The randomness of the system will be destroyed and, under these circumstances, the elements in the matrix will no longer satisfy the independent distribution, the M-P law, nor the ring law.

The M-P law can characterize the asymptotic state of a singular value of the random matrix. After the random matrix is constructed by using the sensing signal, the empirical spectral density of the covariance matrix of the sample will converge to meet the M-P law, and the density function can be expressed as

$$f_{\mathrm{ESD}}\left({\lambda }_s\right)=\left\{\begin{array}{l}\frac{\lambda E}{2\pi \lambda c{\sigma }^2}{\left(b-\lambda \right)}^{1/2}{\left(\lambda -a\right)}^{1/2},\left[a,b\right]\\ 0,otherwise\end{array}\right.$$

(5)

where $a={\sigma }^2{\left(1-c^{1/2}\right)}^2$; $b={\sigma }^2{\left(1+c^{1/2}\right)}^2$; λ is the eigenvalue; c = N/T, it is the scale coefficient, and E and σ are the expectation and variance of the matrix.

After the standardization of the high-dimensional random matrix, its empirical spectral density has a convergence limit, which can be expressed as

$$f_{\mathrm{ESD}}\left({\lambda }_z\right)=\left\{\begin{array}{l}\frac{1}{\lambda cL}{\left|\lambda \right|}^{2/\left(L-2\right)},{\left(1-c\right)}^{L/2}\le \left|\lambda \right|\le 1\\ 0,otherwise\end{array}\right.$$

(6)

where L is a standard number of the Hermitian matrix for independence. We can see that the eigenvalues of the matrix after normalization are distributed in the middle of two rings on the complex plane. On the complex plane of eigenvalues, the radius of the inner ring is (1 − c)^L/2, and the radius of the outer ring is 1. When N and T in the matrix tend to infinity, the proportionality coefficient c is in the interval of (0, 1).

In order to further determine the distribution characteristics of the eigenvalues of the random matrix, MSR is used to calculate the linear eigenvalues. The MSR can be expressed as

$$r_{\mathrm{MSR}}=\frac{{\displaystyle \sum _{i=1}^N\left|{\lambda }_i\right|}}{N},i=1,2,\dots ,N$$

(7)

where λ_i is the eigenvalue of the matrix which, when less than the radius of the inner ring, indicates an intrusion event in the system, and according to the different eigenvalue can distinguish different event types.

3. Simulation Comparisons

The M-P theorem describes the state of change of singular values of random matrices. For a general random matrix of N × T, E = 0, and variance σ² < ∞, is independent and identically distributed. When N and T approach infinity, c = N/T∈(0, 1), then the empirical spectral density of the covariance matrix will converge to the M-P law.

We use the demodulation signal in the case of general background noise as the data for matrix analysis, then analysis the distribution results of the M-P rate and ring rate under normal operation by simulation calculation. Figure 2 shows the probability density results of eigenvalues of different matrices. It can be seen that the probability difference of different eigenvalues is small, and the overall distribution is relatively uniform, displaying certain rules. Figure 3 shows the distribution results of real and imaginary parts of matrix eigenvalues (Ei). When the system runs normally, data is distributed between the outer ring (Or) and the inner ring (Ir). Due to systematic errors and other factors, about 5% of the data is distributed on the edge of the Ir and outside the Or. Because of the small amount of abnormal data, it cannot affect the judgment result [29].

4. Experimental Setup and Results

In order to construct a long distance distributed perimeter security system, we used a chirped uwFBG to prepare sensor monitoring array. Figure 4 shows the spectrum of the uwFBG array. It can be seen from the spectrum diagram that the full line length is 1.5 km, and the adjacent uwFBG spacing is 5 m, with consistent spectra. Meanwhile, the central wavelength of the chirped uwFBG array is about 1553.0 nm, with full width at half maximum (FWHM) of 1.5 nm, and the reflectivity is about −50 dB through the ordinate, which is several orders of magnitude lower than that of the traditional FBG. This proves that the uwFBG array can realize long distance regional warning based on time division multiplexing technology.

Figure 5 shows the demodulation results of four sensors in the monitoring array based on uwFBG. Four sensors are selected in the front, back and in the middle of the array, respectively, for the time domain test of 800 Hz vibration signals. The signals have good time-domain characteristics, and basically the same demodulation amplitude with high SNR. The above calibration results show that the uwFBG intrusion monitoring system can obtain complete demodulation data, which will ensure that it can correctly construct a high-dimensional random matrix.

The purpose of monitoring a single frequency signal is to verify the accuracy of the system we have built, so as to pave the way for the subsequent classification and recognition of a high-dimensional random matrix. A fiber-optic sensing system essentially realizes a high sensitivity signal based on the influence of external signals on optical phase changes; it has nothing to do with the periodicity and randomness of signals, but is related to the optical pulse frequency of the laser. The higher the pulse frequency, the larger the signal frequency that can be demodulated. Therefore, random signals can be demodulated according to the algorithm process mentioned in this paper.

To further study the results of intrusion detection under actual conditions, we use the vibration motor to create the construction state of the crusher outdoors, and carry out intrusion event monitoring through the recognition algorithm proposed in this paper. Figure 6 shows the M-P law and kernel density estimation (KDE) results of the characteristic values of the crusher during construction near the monitoring array. KDE can illustrate the spatial agglomeration of analysis targets.

In order to accurately estimate the distribution of probability density function, we calculate the KDE of signal synchronously, so as to compare the M-P law results of measured data, in order to ensure the accuracy of intrusion event judgment. In Figure 6, the upper curve is the measured result of M-P law, and the lower curve is the estimated result of kernel density. Vibration events lead to an increase in the variation of phase inside the sensor array. Due to the destruction of the randomness of the modulation data, there is a large separation of the double curve at this time, which proves that the M-P rate can identify the existence of vibration events in the warning area.

Figure 7 shows the calculation result of the ring law of the crusher during construction. It can be seen that the result is obviously different from the simulation result shown in Figure 3, and more than 95% of the overall Ei of the random matrix are distributed in the Ir, indicating that the intrusion event changes the distribution of the ring law of the random matrix.

To judge the specific time of the intrusion event, we calculate the average spectral radius of the eigenvalues of the random matrix. Figure 8a,b show the results of the mean spectral radius of the crusher and excavator during construction. The upper curve is the radius of the inner ring, the lower curve is MSR, and the system calculation period is 10 samples/s. We know that, starting from around 4.5 s, the MSR value of the matrix fluctuate sharply around 0.2 and 0.3, and the value is less than the Ir radius. The fault category can be roughly judged by the average spectral radius.

In order to verify the classification effect of the proposed algorithm, the accuracy of classification results is also compared between our proposed algorithm and the multi-domain feature fusion and SVM algorithm in reference [23]. The three event types are no-intrusion state, crusher working state and excavator working state. Each group of data is collected repeatedly, and 100 groups of data are collected altogether. In the data classification algorithm, each group of algorithms uses the same parameter and decision threshold, and the classification results of different algorithms are shown in

Table 1. Comparison results of different classification methods’ accuracy.

Method	No-Intrusion	Crusher	Excavator
Ref. [23]	100	89	87
Proposed method	100	91	95

. It can be seen that both algorithms have no false alarm when there is no intrusion state. However, in the working state of crusher and excavator, under the joint judgment condition of M-P law, ring law and MSR of eigenvalues of high-dimensional random matrix, the classification recognition effect of our algorithm has certain advantages, and the accuracy rate can reach more than 90%.

The above experimental scenarios prove that the proposed algorithm can analyze the interference of monitoring systems from the characteristics of matrix eigenvalues. However, for different intrusion monitoring systems, different monitoring purposes determine the classification standards of events. Therefore, targeted classification and long-term monitoring of intrusion events is an effective means to reduce the false alarm rate. Besides, it is fully proved that the proposed method can effectively extract the high-dimensional random matrix features of different intrusion events, which is expected to be further applied in pipeline monitoring, regional security, cultural relics protection and other scenarios.

5. Conclusions

In this paper, we use a high dimensional random matrix to realize data driven analysis of a uwFBG vibration sensing array. According to the different sags of MSR values of the matrix composed of different data, the fault types are identified. The effectiveness and engineering significance of the method are proved by actual measurement. On the other hand, the MSR results obtained by different intrusion events are also different. By further analyzing the statistical characteristics of the random matrix, the types of intrusion events can be distinguished more efficiently, and the classification and recognition can be realized to predict intrusion events more accurately.