Statistical Analysis for Hull Stress Monitoring Network Signal with Nested Array

Abstract

The hull stress monitoring system is to measure and display the ship motions and real-time stresses by strain gauge and accelerometer sensor networks. The statistical parameters such as standard deviation of measurements through hull stress sensor network are important for analysis of hull structure status. Due to the large amount of measurement data, it is difficult to acquire the standard deviation directly. Nested array is a sparse sampling algorithm which can keep the statistical property of the original data. This paper presents an algorithm for standard deviation computed of hull stress data based on nested array. From the experimental results, it can be seen that the provided algorithm can achieve higher accurate distribution of standard deviation with much less samples. This proves that the nested array sampling could be used in statistical computing for hull stress data.

1. Introduction

The hull stress monitoring system is intended to accomplish the following goals: firstly, it is a system that measures and displays the ship motions and real-time stresses directly by strain gauge or indirectly by monitoring pressures sensor networks. And the mariners can determine the onset and severity of hull structural response to the sea to initiate ship handling changes to mitigate dangerous stress levels and other hazards. Secondly, the hull stress monitoring system can be extended by measuring, recording, and analyzing hull stresses in conjunction with other ship motions, navigation, and performance data. The extended benefits include fatigue assessment, decision rules, and guidance to assist the mariner in mitigating current dangers and quantifying design constraints for future ships [1–3].

In the extended goal of hull stress monitoring system, the following statistical parameters should be calculated for each of the selected sensor data: standard deviation, skewness, and kurtosis.

The statistical parameters are acquired based on thirty minutes or more continuously through hull stress monitoring networks. The data is big and the calculation of standard deviation, skewness, and kurtosis is tremendous. Furthermore, it is not conducive to acquire the statistical parameters in real time.

Stress sensors on the ship will collect a large amount of data; if processing the data one by one, it will produce great workload. This paper adopts a new method called nested sparse sampling, which can collect less data to obtain second-order statistical estimator.

Nested sparse sampling is a nonuniform sampling, which overcomes Nyquist sampling requirement, using two different samplers in each period. That is why we call it “sparse” sampling compared with traditional Nyquist sampling. Although the signal is sampled sparsely and nonuniformly; the samples in the time domain are arbitrarily sparse, which could keep the statistical property of the original signal; this sampling algorithms could be used to estimate second-order statistics to highly reduce the transmission cost of big data [4, 5].

This paper will use the nested array algorithm to achieve the second-order moment statistics estimation algorithm through the sparse sampling to analyze the statistical parameters acquired from hull stress monitoring system, and the structure of this paper is as follows: first, we introduce the principle of nested sparse sampling including theoretical analysis and formula derivation. In Section 2, we give a brief description of hull stress monitoring sensor networks. At last, we show the results of nested sparse sampling, analyze the data, and make a conclusion.

2. Statistical Parameters Acquired Algorithm Based on Nested Array

The concept of nesting is shown to be easily extensible to multiple stages and the structure of the optimally nested array is found analytically. Hence, it can resolve significantly more sources than the actual number of physical sensors. We will demonstrate that using only second-order statistics of the sensor data. The nested array is an effective approach to array processing with enhanced degrees of freedom [6, 7]. The time domain autocorrelation could also be obtained from sparse sampling with nested sampling structure [8]. And the autocorrelation of the samples can be computed, although the samples from this nested sparse sampling are sparse and nonuniform. The nested sampling structure is obtained by systematically nesting two or more uniform linear arrays; here we use the simplest form, two levels of sampling density, with level-1 samples at the $N_{\mathrm{1}}$ locations and level-2 samples at the $N_{\mathrm{2}}$ locations. Figure 1 shows an example of periodic sparse sampling using nested sampling structure with $N_1=\mathrm{3}$ and $N_2=\mathrm{6}$ .

Figure 1

Nested sampling structure with $N_1=3$ and   $N_2=6$ .

The sampling points are given by $1\le s\le N_1$ , for level 1, and $(N_1+1)m$ , $1\le m\le N_2$ , for level 2.

The sampling points are in the following range:

$$\begin{array}{cc}& \mathrm{1},\mathrm{2,3},\mathrm{4},\left(N_{\mathrm{1}}+\mathrm{1}\right),\mathrm{2}\left(N_{\mathrm{1}}+\mathrm{1}\right),\dots ,N_{\mathrm{2}}\left(N_{\mathrm{1}}+\mathrm{1}\right).\end{array}$$

(1)

For example, considering the example in Figure 1, where $\mathrm{1}\le m\le \mathrm{6}$ and $\mathrm{1}\le s\le \mathrm{3}$ , the sampling points will achieve these values $\mathrm{1,2},\mathrm{3,4},\mathrm{8,12,16,20,24}$ . Beside these integers, difference 0 is also missing, for the reason that m and s are nonzero. When using nested array structure, with sparse samples, we could obtain the degrees of freedom as

$$\begin{array}{cc}& 2[(N_1+1)N_2-1]+1=2(N_1+1)N_2-1.\end{array}$$

(2)

Using the above principle, a sparse sampling using nested sampling structure could be achieved as shown in Figure 1. There are two levels of sampler, with $N_1$ level-1 samples and $N_2$ level-2 samples in each period, with period $(N_1+1)N_2$ . It is obvious that nested sampling is nonuniform and the samples obtained are very sparse compared with traditional Nyquist sampling. Hence, there are totally $N_1+N_2$ samples in $(N_1+1)N_{2}T$ seconds [5]. The average sampling rate is

$$\begin{array}{cc}& f_{s,\text{nested}}=\frac{N_1+N_2}{(N_1+1)N_{2}T}\approx \frac{1}{N_{1}T}+\frac{1}{N_{2}T}<\frac{1}{T}.\end{array}$$

(3)

We can see that although the signal is sampled sparsely and nonuniformly, $1\le s\le N_1$ and $(N_1+1)m$ , $1\le m\le N_2$ , for each period, the autocorrelation $R_c(\tau )$ of the original signal $x_c(t)$ could be estimated; that is, the second-order statistical property of the signal is kept for nested sparse sampling [6].

The autocorrelation samples for all k could be estimated [5] by averaging the products $x(n_1)*x(n_2)$ , over L periods:

$$\begin{array}{cc}& R\left(k\right)=\frac{\mathrm{1}}{L}{{\displaystyle \sum }}_{l=\mathrm{0}}^{L-\mathrm{1}}x\left(n\right)*x\left(n-k\right).\end{array}$$

(4)

In this paper, the standard deviation of data is needed for analysis. According to the principle of probability and statistics, we can derive the relationship between the correlation function and its standard deviation as follows:

$$\begin{array}{cc}& \sigma \left(x\right)=\sqrt{R\left(x\right)-{\stackrel{-}{x}}^{\mathrm{2}}},\end{array}$$

(5)

in which $\stackrel{-}{x}$ is the average value of 2400 random sampling values and $R(x)$ is the autocorrelation values of new sampling values after the sparse sampling.

Consequently, the nested sparse array sampling can calculate second-order statistics moment and obtain the standard deviation for further research.

3. Hull Stress Monitoring Sensor Networks and Experiments Design

3.1. Hull Stress Monitoring Sensor Networks

Prolonging the ship lifetime and reducing maintenance costs with the hull stress monitoring sensor network, the monitoring system can provide real-time monitoring of the forces exerted on the ship's hull. It provides immediate feedback to the crew if hull stress or fatigue levels exceed SAFT thresholds and reduces maintenance costs by avoiding or reducing the risk of structural damage to the vessel. The standard components of a hull stress monitoring sensor network include four strain sensors to capture the global loading level and one bow accelerometer to capture slamming events. Figure 2 showed the schematic of hull stress monitoring sensor network. Statistical calculation of sensor response, such as standard deviation, skewness, and kurtosis, can assist the mariner to assess the current dangers and quantify design constraints for future ships.

Figure 2

The schematic of hull stress monitoring sensor network.

3.2. Experimental Design

In this paper, according to the simulation of a stress sensor wave load response, we analyze the statistical standard deviation parameters based on sparse nested array sampling algorithm presented in this paper.

The distribution of the ship stress wave load usually should be subjected to the two parameters of Weibull distribution. The probability density function and distribution function of Weibull distribution are as follows:

$$\begin{array}{cc}& f_S\left(S\right)=\frac{\xi }{\alpha }{\left(\frac{S}{\alpha }\right)}^{\xi -\mathrm{1}}\exp \left[-{\left(\frac{\xi }{\alpha }\right)}^{\xi }\right],\end{array}$$

(6)

$\begin{array}{cc}& F_S\left(S\right)=\mathrm{1}-\exp \left[-{\left(\frac{\xi }{\alpha }\right)}^{\xi }\right],\end{array}$

(7)

where ξ is called the shape parameter, the general values of which are between 0.7 to 1.3, which is related to the marine environment, the response characteristics, and the type of structure, and α is called the scale parameter. If the shape parameter ξ is known, the scale parameter can be represented by the stress range which corresponds to an exceedance probability, which is obtained by long-term analysis of the recovery period of fatigue load.

Monte Carlo method is usually used to construct random numbers obeying a certain distribution. Following the principle of the method, let the random variable y be continuous distribution and the distribution function $F(x)$ ; by stochastic simulation theory, random variables are

$$\begin{array}{cc}& R=F\left(y\right),\end{array}$$

(8)

where R is uniformly distributed random variables on $[0,1]$ . To obtain the inverse function of distribution function $F(x)$ , consider

$$\begin{array}{cc}& y=F^{-\mathrm{1}}\left(R\right).\end{array}$$

(9)

When random sampling R is between $[\mathrm{0,1}]$ , the random distribution sequence of y can be gotten. After parameters ξ and α are determined, according to (6), we can generate the random load history of the ship which obeys the Weibull distribution.

This paper generated 960000 sampling-points signal of hull stress wave load by Monte Carlo method, which obey the Weibull distribution with the shape parameter $\xi =\mathrm{0.98}$ and the scale parameter $\alpha =\mathrm{6.02}$ . The waveform is shown in Figure 3.

Figure 3

Waveform of 960000 sampling points.

3.3. Results and Analysis

This paper simulates the real-time sampling process with a single sampling data length of 2400. So it can obtain 400 standard deviations by calculating directly due to the total length of 96000. At the same time, it calculates the standard deviation with (5) based on the nested array algorithm, which selects $N_1$ , $N_2$ for 3,5; 3,6; 4,6; and 5,8. To acquire the distribution of standard deviations, a histogram of all standard deviation was established. The number of standard deviations within each interval preset was counted. And the histogram is shown in Figure 4.

Figure 4

The histogram of standard deviations based on all algorithms.

According to Figure 4, we can see that the distribution of standard deviations based on all algorithms concentrated between 6 and 8. The interval of the histogram is just the uniform distribution of numbers. The vertical axis represents the number of numerical distribution in a certain interval.

In Table 1, it describes the two largest numbers of numerical distribution and its proportion.

Table 1

The number distribution of variance.

	5–5.5	5.5–6	6–6.5	Proportion	6.5–7	Proportion	7–7.5	Total
Normal	0	1	254	63.5%	145	36.25%	0	400
(3, 5)	5	50	182	45.5%	133	33.25%	30	400
(3, 6)	3	61	155	38.75%	141	35.25%	40	400
(4, 6)	6	63	162	40.5%	129	32.25%	40	400
(5, 8)	9	57	156	39%	124	31%	54	400

According to Table 1, the proportion of the main standard deviation values is 99.75% which were computed directly with standard deviation formula. And the proportion is 78.75%, 74%, 72.75%, and 70% which were computed based on nested array samples with different $N_1$ and $N_2$ for 3,5; 3,6; 4,6; and 5,8. Meanwhile, the sparse proportions are 53.3%, 50%, 41.7%, and 32.5%. Obviously, we can get two conclusions. Firstly, the higher the sparse proportion samples used to acquire standard deviation were, the higher the accurate distribution of standard deviation we could get. Secondly, when the sparse proportion was 53.3%, the relative error of standard deviation distribution was $(99.75\%\hspace{0.17em}-\hspace{0.17em}78.75\%)/99.75\%=21.1\%$ . In other words, we could get higher accurate distribution with much less samples.

4. Conclusions

Nested array sparse sampling is a nonuniform sampling which overcomes Nyquist sampling requirement and could keep the statistical property of the original signal. The hull structure status could be learned through the statistical analysis of hull stress sensor network measurement data. This paper addressed a new algorithm to acquire the standard deviation through sensor network data based on nested array.

It demonstrated that it can achieve better statistical results with much less samples with the nested array sparse sampling. Furthermore, it can be studied that the third statistical moment of the sensor network data will be acquired based on the nested array.