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Abstract

This paper proposes a revised counter-propagation network (CPN) model by integrating rough set in structural damage detection, applicable for processing redundant and uncertain information as well as assessing structural health states. Firstly, rough set is used in the model to deal with a large volume of data; secondly, a revised training algorithm is developed to improve the capabilities of the CPN model; and lastly, the least input vectors are input to the revised CPN (RCPN) model, hence the rough set-based RCPN is proposed in the paper. To validate the model proposed, numerical experiments are conducted, and, as a result, six damage patterns have been successfully identified in a steel frame. The influence of measurement noise, the network models adopted, and the data preprocessing methods on damage identification is also discussed in the paper. The results show that the proposed model not only has good damage detection capability and noise tolerance, but also significantly reduces the data storage requirement and saves computing time.

1. Introduction

The Structural Health Monitoring (SHM) issue has attracted more and more attention, largely due to the rapid increase in the number of deteriorated and damaged structures, especially tall buildings, long span bridges, and large spatial structures [1, 2]. Great efforts have been made to develop SHM techniques and damage detection methods, some of which have been applied to practical engineering problems to some extent in the past two decades [3, 4]. Although the number of permanent structural health monitoring systems installed in various kinds of structures is increasing, it is still of an emergency and constitutes a big challenge as to how to deal effectively with the uncertainties inherent in the great volume of measured data from the health monitoring systems [5].

Recently, neural networks are becoming increasingly popular with data processing and damage detection due to their powerful nonlinear modeling capability and high fault tolerance [1, 6–11]. Although such methods are available and applicable for processing of uncertain data from SHM systems in some cases, these neural network models also have some disadvantages, for instance, local optimal solution and poor extrapolation for a BP network, complex network architecture for a fuzzy neural network structure, and difficult determination of the parameter σ of Gaussian kernel function in a probabilistic neural network [9]. The counter-propagation network (CPN) began to be adopted in pattern classification, function approximation, and statistical analysis not very long ago [10, 11], because it has a relatively simple network structure and does not have an error criterion for convergence owing to its combination of the Kohonen self-organizing map and Grossberg competitive learning network model [12, 13]. Nevertheless, the CPN model often requires higher data storage memory and more runtime than other neural network models [10, 11]. Consequently, for any CPN model which is applied to practical engineering problems, it is becoming increasingly urgent and vital to develop efficient learning algorithms and reduce the spatial dimensions of the measured data from the SHM systems of large and complex structures.

Numerous techniques have been developed to extract feature parameters and reduce the spatial dimensions of the great quantities of data, for example, principal component analysis (PCA) and kernel principal component analysis [3, 4, 14, 15]. These multivariate statistics-based methods are efficient for some problems that are involved with dimension reduction, but in other cases, they either cause key information loss or fail the precise physical interpretation of results. When this happens, intelligent information techniques, such as neural networks, fuzzy logic, data fusion, and rough set, are usually needed for solving uncertainty and imprecision as well as dealing with the large amount of data concerning feature extraction and damage detection [2, 8, 9]. Compared with multivariable statistical analysis methods, intelligent information techniques are considered to be more efficient in solving the above problems, because (1) they can treat both numerical data and linguistic data taken together in an uncertain and imprecise situation, (2) they can allow for nonlinear relationships in the analysis, and (3) they lead to more precise physical interpretation. As a consequence, there has been a rapid growth in interest in rough set theory and its applications recently [7–9, 16, 17]. Some scholars have tried to combine rough set with other intelligent methods in order to reduce uncertainty and spatial dimension of data, thereby to improve pattern recognition and damage diagnosis accuracy [7–9, 16, 17]. To date, in spite of great efforts on developing integrating methodologies, few of them seem to be successful in dealing with great quantities of noisy measured data.

In view of the abovementioned description, it is no denying that it is necessary and significant to develop efficient data processing and damage detection methods. In the following sections, the training algorithm of a CPN model is revised firstly, and then the revised CPN model and rough set are integrated to address the processing of great quantities of information with uncertainty, as well as to detect damages in a structure. Finally, a novel damage detection model is presented, which is especially suitable for structural damage detection with noisy data and/or a great volume of measured data. In the validation of the proposed model, both single- and multidamage patterns of a 7-story steel frame are detected, and its effects on the prformance of the model are investigated as well.

The rest of the paper is organized as follows. The principles of counter-propagation network are briefly reviewed in Section 2. In Section 3, we propose a revised counter-propagation network (CPN) model by integrating rough set for structural damage detection. In Section 4, six damage patterns from a steel frame are identified firstly, and then the effects of measurement noise, network models, and data preprocessing methods on damage detection results are also discussed, respectively. Finally, Section 5 draws some conclusions and remarks.

2. Counter-Propagation Network (CPN)

The counter-propagation neural network (CPN) has been described in great detail in relevant textbooks and articles [10, 11, 18]. Its applications in damage detection and fault diagnosis are also reported in the literature [14, 18–20]. Therefore, only a brief introduction of CPN is given in the paper as for how the CPN works.

In general, a CPN model consists of three layers (as shown in Figure 1), specifically, input layer, competitive layer, and output layer [10, 18, 19]. The first two layers constitute self-organising maps (SOMs). The SOM performs the mapping of input data into two-dimensional plane, and in most cases, neurons are ordered in a rectangular or hexagonal matrix. The last two layers form a Grossberg network [10, 11, 18], which is a general competitive network model.

Figure 1

Architecture of a CPN model.

From the input layer to the output layer, the winning neuron of competitive layer is obtained in accordance with the self-organization mapping (SOM) learning norm, and then the connecting weights between the input layer and the competitive layer are adjusted using the following learning rule:

\begin{matrix} W_{g} (new) = W_{g} (old) + α [A - W_{g} (old)], \end{matrix}

(1)

where

W_{g}

is the weight connecting the winning neuron g in the competitive layer to each neuron in the input layer; α is the learning rate; A is the input vector.

From the competitive layer to the output layer, the actual output value of each output neuron is attained in accordance with the learning rule of a general competitive network, and the connecting weights between them are adjusted according to the supervised learning algorithm. Then the weight vectors are modified by the following equation:

\begin{matrix} V_{g} (new) = V_{g} (old) + β b_{g} [C - V_{c} (old)], \end{matrix}

(2)

where

V_{g}

is the weight connecting the winning neuron g in the competitive layer to neurons in the output layer;

b_{g}

is the output value of neuron g, with a value of 1; C is the desired output vector corresponding to the input vector A; β is the learning rate.

CPN has been widely used in the fields of pattern recognition, decision-making optimization, and robot intelligence for its adaptability and robustness. In addition, CPN is a hybrid network, which combines unsupervised learning with supervised learning and thus enjoys the advantages of both teacher type and nonteacher type networks, such as simple structure, less iterations of data set, and no error criterion for convergence. However, the CPN model often requires much largerstorage memory and more computing time than other neural network models, and it might fail in its learning process when the same weight value is adjusted for samples with similar values. So we hereby propose a novel revised CPN (RCPN) to address the abovementioned problems, and more details can been seen in Section 3.4.

In recent years, rough set has attracted increasingly more attention and has been applied to structural health monitoring and damage detection owing to its capability of processing the data uncertainty and reducing the space dimension of data [21, 22]. To take advantage of both rough set and neural network methods, a novel damage detection method based on the CPN model combined with rough set is proposed in this paper, and a comparison of different methods' performance is also conducted.

3. Structural Damage Detection Methods

To improve the capabilities of damage detection methods in dealing with massive data redundancy and noise, rough set is used for reducing redundant data and a revised counter-propagation neural network (RCPN) is adopted for damage detection. Thereby, a 6-phase structural damage detection method based on rough set and RCPN is proposed, which involves data preprocessing, feature extraction, feature reduction, least input vector, RCPN, and results output as shown in Figure 2.

Figure 2

Structural damage detection method.

3.1. Data Preprocessing and Feature Extraction

On account of environment variations, measurement errors, the skills of erectors, and so forth, measurement noise and errors are inevitably introduced along with the true data acquired from multisensors in structural damage detection and health monitoring. As a consequence, the digitized raw data must be preprocessed to eliminate noise and errors using threshold, averaging, and image processing techniques before it goes through subsequent phases.

3.2. Feature Extraction

Feature extraction plays an important role in damage detection and structural health monitoring because it has a remarkable impact on processing efficiency and detection results. Frequency, mode shape, and the combined feature indices of structures are used to detect the damages of structures, because it is easy to extract this information from structural responses. In doing so, the normalized damage signature index (NDSI) and the normalized frequency change ratio (NFCR) are employed as features in this study, which are denoted as follows [7, 9]:

\begin{matrix} NFC R_{i} = \frac{FF C_{i}}{\sum_{j = 1}^{m} FF C_{j}}, \\ NDS I_{i} = \frac{DS I_{i} (k)}{\sum_{j = 1}^{n} | DS I_{j} (k) |}, \end{matrix}

(3)

where

FF C_{i}

(

FF C_{i} = (f_{u i} - f_{d i}) / f_{u i}

) and

DS I_{i}

(

DS I_{i} = ({Φ_{u i}} - {Φ_{d i}}) / (f_{u i}^{2} - f_{d i}^{2})

) are the fractional frequency change and damage signature index of the ith mode, respectively; m is the number of modes with measured frequencies; n is the total number of normalized damage indices;

f_{u i}

and

f_{d i}

are the frequencies of the ith mode of the structure in the undamaged and damaged states;

{Φ_{u i}}

and

{Φ_{d i}}

are the modal vector values of the ith mode in the undamaged and damaged states, respectively.

3.3. Feature Reduction and Least Input Vector

Rough set is employed to implement the feature reduction. Attribute discretization and reduction are two key aspects of the rough set theory. Attribute reduction mainly refers to the reduction of condition attributes in a decision table. When the rough set theory is employed to process data for a training sample, all data must be discretized before the reduction of condition attributes can be performed on the decision table created.

3.3.1. Discretization of Continuous Attribute Values

Rough set analysis is a symbolized analysis method where the continuous variable must be discretized. K-means are used to discretize condition attribute values.

K-means clustering is a partitioning method that treats observations in data as objects having locations and distances from each other. It partitions the objects into k mutually exclusive clusters so that objects within each cluster are as close to each other as possible and as far as possible from objects in other clusters. Each cluster is characterized by its centroid or centre point. The K-means function in MATLAB is employed to analyze condition attributes, and the function KMEANS is called to complete K-means clustering, using an iterative algorithm that assigns objects to clusters so that the sum of distances from each object to its cluster centroid, over all clusters, is a minimum.

3.3.2. Constructing the Decision Table

On the basis of the rough set theory, a two-dimensional table, known as a decision table, is created, where each row describes an object while each column describes an attribute of the object. Decision pertains mainly to decision attributes. Discretized features are condition attributes, and damage patterns are decision attributes.

3.3.3. Reduction of Attributes

Reduction of attributes is implemented by removing all redundant condition attributes, meanwhile keeping indispensable attributes in the decision table, which is governed by reduction rules. During the process of attribute reduction, there are numerous types of reduction rules [7, 8, 20], and the importance of attribute is selected as a reduction rule owing to its precision and convenience. The reduced decision table has the same function as the original decision table but contains fewer condition attributes. A computer program developed is finally run to achieve the reduction of condition attributes in this study [20].

3.3.4. Least Input Vectors

The least input vectors are created with the least condition attributes for both training samples and test samples. Then these feature parameters, which are mutually independent, are input to the input layer of a revised CPN for structural damage detection purposes.

3.4. Revised Counter-Propagation Network (RCPN)

Network structure and learning algorithm are two key issues to a neural network model. For the learning algorithm of a CPN model, all training samples are used to train the network, and the connecting weights are adjusted simultaneously during one training echo. The training phase will stop until the network output error is less than the allowable error or it achieves the maximum training echo. In doing so, the CPN model needs to adjust the connecting weights between the neurons for each sample in each training echo. This not only leads to adjusting the connecting weights every now and then in each iterative step, but also costs longer time in the training phase. Furthermore, the CPN model will not converge in some cases when two samples are very similar [19].

In order to solve the abovementioned problems, a revised CPN learning algorithm is addressed in this study. With the revised learning algorithm, a sample is used to train the CPN model firstly. Only when the network output error is less than the allowable error value or the network reaches the maximum iterative steps will the next sample be continued to train the CPN model. This process is repeated until all samples are done with the training, and the training phase is over finally. It is noted that the trained weights using certain sample will keep unchangeable and will not be adjusted in the training of the next sample [19].

Besides the learning algorithm, another important but difficult task is to determine the architecture of a CPN. Generally, the numbers of neurons in the input layer and output layer are in accordance with the numbers of least input vectors and pattern classes, respectively. The most difficult issue is to determine the appropriate neuron number of the competitive layer. However, there is no guideline in theory, so it is set by trial and error in this study.

After the architecture of the RCPN model is constructed, the above revised learning algorithm is used to train the RCPN model. A training sample is input to the network, and the weighted input summation of each neuron in the competitive layer is obtained. The winning neuron is the one with the maximum value. Then the connecting weights are trained in accordance with the abovementioned learning algorithm. The training is over when the network output error is up to the allowable error or the network achieves the maximum echo for all training samples.

3.5. Results Output

In the last phase, new test samples are input to the trained RCPN model, and the output of the competitive layer is obtained as follows:

\begin{matrix} b_{g} = \max_{j = 1,2, \dots, Q} (\sum_{i = 1}^{N} w_{j i} a_{j}), \end{matrix}

(4)

where

b_{g}

is the output of the competitive layer;

w_{j i}

is the connecting weight between the input layer and competitive layer;

a_{i}

is the test sample value; Q is the number of neurons in the competitive layer; N is the number of neurons in the input layer.

The output value of the winning neuron g is 1 in the competitive layer, and the output values of the others are 0. The output of neurons in the output layer is represented as follows:

\begin{matrix} c_{j} = V_{j g} b_{g}, \end{matrix}

(5)

where

c_{j}

is the output of neurons in the output layer, which indicates the pattern class;

V_{j g}

is the connecting weight between the competitive layer and output layer.

4. Simulation Examples

4.1. Structural Model and Damage Simulation

To check the applicability and performance of the proposed method, a numerical simulation study is carried out on examples with known values. A scaled 7-story shear-beam type building model with known damages is used to observe the performance of the damage detection method based on RCPN. For comparison purpose, a probabilistic neural network (PNN) model is also applied to detect the damages.

Details of a 7-story shear-beam type steel frame model are as follows: the stiffnesses of all stories are identical; that is, $k_{1} = \dots = k_{7} = 375$ KN/m. The masses of all stories are also the same, $m_{1} = \dots = m_{6} = 3.78$ kg, except for the 7th story where $m_{7} = 3.31 kg$ . According to the solution method of the eigenvalue problem, the natural frequencies and mode shapes of the healthy structural building model can be easily obtained as long as stiffness and mass matrices are known. Damage scenarios were simulated by reduction of the stiffness in each story. Two magnitudes of damage and four damage locations were used to validate the proposed approach. The damage magnitudes were classified quantitatively as small (4.1%) and large (16.7%), while the damage locations were denoted by their corresponding story numbers. Thus six damage patterns were simulated (as shown in Table 1) and are discussed further next.

Table 1

Damage extent and location of each damage pattern.

Pattern class	Small damage	Large damage
Pattern 1	4th story	—
Pattern 2	—	4th story
Pattern 3	6th story	4th story
Pattern 4	3rd and 6th stories	4th story
Pattern 5	3rd story	4th and 6th stories
Pattern 6	—	3rd, 4th, and 6th stories

The numerically simulated response is obtained for 1 s at time step of 0.002 s (500 steps) by using Newmark's constant-acceleration method, and random noise is added. As the natural frequencies and first mode shape could be measured easily and accurately, just as in the healthy building model, natural frequencies of the first four modes and the components of the first mode vector at 7 stories were calculated. In addition, as the noise is inevitable, each set of the analytical computed modal parameters for both healthy and damage scenarios were then added by a random sequence to simulate the measured data; that is,

\begin{matrix} y_{i} = y_{i}^{a} \times (1 + ε R), \end{matrix}

(6)

where

y_{i}

is the measured modal parameter polluted by noise (i.e., natural frequency or modal vector);

y_{i}^{a}

is the analytically computed modal parameter for healthy or a certain damage scenario; R is a normally distributed random variable with zero mean and a derivation of 1; ε is an index representing the noise level; here ε is set to 0.1%, 0.2%, and 0.3%, respectively.

For each noise level, 200 sets were randomly produced for each damage scenario, and thus there were 200 sets of measured data. In these data sets for each scenario, the first 100 sets were used to create training samples, and the others were used to create test samples. As there were totally six damage scenarios, thus a total of $100 \times 6 = 600$ measured data sets were created for training along with a total of 600 measured data sets for test.

4.2. Damage Detection

4.2.1. Damage Detection Model

In order to construct the rough set-based RCPN model (RSRCPN), some work had to be done in advance, such as extracting feature parameters, producing training and test samples, and determining the structure of the network model. Here the model was taken as an example to describe the process of constructing the network model when $ε = 0.1 %$ .

(1) Feature Extraction. NFCR and NDSI were taken as feature parameters in this example; thus 7 NDSIs for the components of the first mode vector at 7 stories and 4 NFCRs for the natural frequencies of first four modes were computed by (3). Six hundred training samples and another six hundred test samples different from the training samples were produced. Each feature parameter was regarded as a condition attribute, so there were $7 + 4 = 11$ condition attributes for each sample.

(2) Cluster Analysis. The function KMEANS in MATLAB was used to analyze cluster condition attributes. Herein 4 clusters were set to 4 by trial and error, which means that the feature parameter data would be classified into 4. This was done using two-phase iterative analysis so that the sum of the distances from each object to its cluster centroid, over all clusters, was minimized. Consequently attribute values could be replaced with the clusters to which that data is affiliated, and the feature parameters for all samples were discretized into discrete data represented with 1, 2, 3, and 4.

(3) Decision-Table Reduction. Discretized feature parameters were used to construct the decision table for damage detection, where condition attributes included NFCR1, NFCR2, NFCR3, NFCR4, and NDSI1, $NDSI 2, \dots, NDSI 7$ , and decision attributes consisted of the damage pattern that could be represented by a number ranging from 1 to 6. The rough set-based attribute reduction program [20] was run to reduce the decision table; accordingly, the core sets of feature parameters were NFCR1, NFCR2, NFCR3, NFCR4 and $NDSI 2, \dots, NDSI 7$ , which meant that the feature parameter NDSI1 was redundant and therefore eliminated.

(4) RCPN Calculation. In correspondence with the new set of feature parameters of the training and test samples that was produced by the reduction process, the least input vectors were reextracted and subsequently fed to the input layer of the RSRCPN. Thus the number of neurons in the input layer was equal to the number of least condition attributes; that is, $4 + 6 = 10$ . The number of neurons in the output layer was six, which meant one neuron corresponded to one damage pattern. The study [23] showed that the number of neurons in the competitive layer had little impact on a CPN model, when it was more than twice that of training samples. As the number of training samples was 600, the number of neurons in the competitive layer was set to 2000. Thus the structure of the rough set-based RCPN (RSRCPN) was represented by 10-2000-6.

After the RSRCPN was configured, the model was trained by the abovementioned revised learning algorithm, wherein both α and β were set to 0.1 in (1) and (2). Once the RSRCPN reached the allowable error or the maximum echo, the training was over.

(5) Results Output. Given a sample (or test sample) to the trained RSRCPN, the winning neuron g in the competitive layer and the output values corresponding to output neurons were obtained using (4) and (5), and the classification result was finally attained.

4.2.2. Damage Detection Results

Here the identification accuracy (IA) is defined as the ratio of the total number of correct identification samples to the number of total test samples. Table 2 and Figure 3 show the identification results of test samples.

Table 2

Identification results of test samples by using different models.

Method	ε (%)	The number of correct pattern identification						Average IA (%)
Method	ε (%)	Pattern 1	Pattern 2	Pattern 3	Pattern 4	Pattern 5	Pattern 6	Average IA (%)
RSRCPN	0.1	86	93	96	100	100	100	95.83
	0.2	90	81	83	90	100	100	90.67
	0.3	89	67	70	83	96	100	84.17

PCARCPN	0.1	84	97	97	100	100	100	96.33
	0.2	89	79	81	91	100	100	90.00
	0.3	88	66	69	86	97	100	84.33

RCPN	0.1	86	97	97	100	100	100	96.67
	0.2	89	79	81	91	100	100	90.00
	0.3	87	66	69	85	97	100	84.00

RSPNN [7]	0.1	35	95	98	98	100	100	88.00
	0.2	79	50	80	97	100	100	84.33
	0.3	86	44	55	86	99	100	78.33

PNN	0.1	35	95	99	100	100	100	88.17
	0.2	79	53	81	97	100	100	85.00
	0.3	84	43	57	88	99	100	78.50

Figure 3

Identification results using RSRCPN.

It is found that the average IA of RSRCPN is 95.83% for all patterns when $ε = 0.1 %$ . Even in pattern 1, the IA value is the lowest and amounts to 86%. Some samples from pattern 1 are misclassified into other patterns (as shown in Figure 3(a)); that is, 6, 4, 3, and 1 samples belonging to pattern 1 are classified into patterns 2, 3, and 4, respectively. The reason for this is that Pattern 1 has only a small damage extent on the 4th story, whilst patterns 2, 3, and 4 all have large damage extents on the 4th story, and apart from that, pattern 3 has one more small extent damage on the 6th story, and Pattern 4 has additional small extent damages on both the 3rd and 6th stories. Because of the similarity in location and damage extent, the changes in modal parameters for patterns 1, 2, 3, and 4 also look similar, as shown in Figure 4(a). Apparently, small extent damages result in slight changes of modal parameters. Most of the projection points of pattern 1 are located nearby the point clusters of patterns 2, 3, and 4 (see Figure 4), as a result of which they are misidentified as patterns 2, 3, and 4, hence a low IA for pattern 1. Although some samples belonging to patterns are misclassified into other patterns, such as pattern 2, IA for pattern 2 is not affected since IA is defined as the ratio of the number of correctly identified samples (matching the predefined pattern 2) to the number of total test samples. It is why the model proposed still enjoys high identification accuracy greater than 93% for pattern 2.

Figure 4

The projection diagram of features for each pattern.

Except for patterns 1 and 2, the IA values of other patterns are over 96%; even the IA values of patterns 4, 5, 6 are all up to 100%. As patterns 1, 2, and 3 have larger extent of damage, the samples can be correctly identified from the above three patterns (as shown in Table 2 and Figure 3). It is also observed that the projection points of feature data from patterns 4, 5, and 6 are clearly classified into three different clusters as shown in Figure 4(a); thus none of samples are misclassified for these three patterns. The results indicate that the proposed model has a good damage detection capability.

4.3. Comparison and Discussion

In an attempt to prove the applicability and efficiency, we investigated the effects of measurement noise, network models, and data preprocessing methods on the performance of the RSRCPN model.

4.3.1. Noise Level

Just as $ε = 0.1 %$ , the RSRCPN models with other noises, that is, $ε = 0.2 %$ and 0.3%, were constructed, and both structures of RSRCPN models were also 10-2000-6. Then both models were trained and stopped just as the case when $ε = 0.1 %$ . The identification results are also shown in Table 2 and Figure 3. Some phenomena are found and deductions are made herein.

First, the average IA value decreases with the increase of noise level, just as expected. As shown in Table 2, the average IA is 95.83% when $ε = 0.1 %$ and still as high as 90.67% when $ε = 0.2 %$ . Even when the noise level is up to 0.3%, the average IA is 84.17%. This indicates that the proposed model has good damage detection capability, noise tolerance, and robustness.

Second, it is remarkable that the IA value of pattern 1 does not change or slightly increase with the increase of noise level, contrary to common knowledge. There are three reasons for this [7]: (1) the changes in modal parameters induced by noise are comparatively more significant than those by small extent damage; (2) the number of projection points of pattern 1 located nearby the point clusters of patterns 2, 3, and 4 decreases with noise level (as shown in Figure 4); consequently, the number of samples identified correctly as pattern 1 increases with noise level (as shown in Figure 3 and Table 2); and (3) the neural networks have good tolerance, noise-resistance, and self-learning capabilities. Furthermore, neural network models have better pattern recognition and generality capabilities in noisy environment than in noise-free environment to some extent.

4.3.2. Data Processing Method

We also studied the effects of data processing methods on damage detection using the RCPN model, and the same method as that used in RSRCPN was adopted to construct new classifiers with different data processing methods. Equation (3) were employed to extract features; 4 NFCRs and 7 NDSIs were chosen as inputs. The extracted feature data were directly input to the RCPN model, while the extracted feature data were processed with the principal component analysis and then input to the proposed RCPN model so as to identify damage. It is noted that the components whose contribution ratio is less than 2% are removed. The two models were represented with RCPN and PCARCPN, respectively; thus a comparison study of different models' performance was conducted. The results are listed in Tables 2 and 3, respectively.

Table 3

Runtime and memory usage comparison of RCPN using different models.

Models	ε (%)	Training phase		Test phase
Models	ε (%)	Runtime (s)	Memory usage (kB)	Runtime (s)	Memory usage (kB)
RSRCPN	0.1	585.58	664	1.37	384
	0.2	584.16	756	1.38	480
	0.3	589.81	797	1.30	576

PCARCPN	0.1	579.39	544	1.29	356
	0.2	581.94	668	1.33	412
	0.3	584.11	712	1.28	474

RCPN	0.1	612.13	5688	1.46	506
	0.2	611.80	5736	1.44	688
	0.3	612.67	5976	1.42	753

In contrast with the RCPN, the average IA changes from 96.67% to 95.83% with a decrease of 0.84% using RSRCPN when $ε = 0.1 %$ , and the average IA value rises from 90% to 90.67% with an increase of 0.67% when $ε = 0.2 %$ , whilst the average IA is from 84.0% to 84.17% with an increase of only 0.17% when $ε = 0.3 %$ . The results show that the proposed model has good damage detection capability and noise tolerance.

Except the identification accuracy, a comparison of different identification models is made in the runtime and memory usage (as shown in Table 3). It is shown that both RSRCPN and PCARCPN models use less time and memory usage than the RCPN model in both training and test phases. For example, the training time of RSRCPN decreases from 612.13 seconds with RCPN to 585.58 seconds, a decrease of 27 seconds when $ε = 0.1 %$ , while the training time of PCARCPN decreases from 612.13 seconds with RCPN to 579.39 seconds, a decrease of 32 seconds; with regard to the memory usage, in the training phase, the memory usage decreases from 5688 kB with RCPN to 664 kB with RSRCPN when $ε = 0.1 %$ , while the memory usage decreases from 5688 kB with RCPN to 544 kB with PCARCPN. The same phenomenon can be found in the test stage.

All in all, these facts have proven that the proposed model cannot only have good damage detection capability and noise tolerance, but also reduces spatial data dimensions and the memory requirements for data storage without sacrificing identification accuracy.

4.3.3. Network Model

In order to illuminate the advantages of RSRCPN model, the probabilistic neural network (PNN) based on rough set (RSPNN) [9], RCPN, and PNN were employed to identify the above structural model and damage patterns. These models were constructed in a similar way, in which the number of neurons in the input layer was equal to the number of feature vectors, the number of neurons in the output layer was six, each of which corresponded to a damage pattern, the number of training samples was 600, and $σ = 0.25$ . It is noted that the data used by the RSPNN model were preprocessed using rough set, while the data used by other two models were not preprocessed. The identification results are also shown in Table 2.

As shown in Table 2, the neural network models have a great impact on the results of damage identification. The IA of RCPN is apparently higher than that of PNN at each noise level. For example, when $ε = 0.1 %$ , 0.2%, and 0.3%, the average IA values of RCPN are higher than PNN by 8.5%, 5.00%, and 5.50%, respectively. The largest difference in IA is seen in pattern 1 when $ε = 0.1 %$ , and the IA of RCPN is 51% greater than PNN. When $ε = 0.2 %$ , the IA of RCPN in pattern 2 was greater than PNN by 26%. Similarly, the difference between the RSRCPN and the RSPNN is similar to the case of RCPN and PNN. In a word, the RCPN has better damage detection capability than that of PNN.

In summary, rough set or principal component analysis can be used to reduce the redundant information, the spatial dimension, and data storage space. This not only saves the running time, but also improves the damage detection capabilities and robustness. This implies that the proposed model is feasible and effective in structural damage detection.

5. Conclusions

This paper presents a revised CPN model by integrating rough set for damage detection, which is suitable for dealing with redundant and uncertain information in structural damage detection. Furthermore, a numerical example has proved the effectiveness of the proposed method. The results show that excellent performances are achieved with this model in comparison with different network models and data processing methods. That is, the presented method not only has good damage detection and noise-resistant capabilities, but also significantly reduces the memory requirements for data storage and saves runtime as a consequence of the rough set technique and RCPN model processing.

The study shows that the integration of rough set processing and a revised CPN has great potential in the field of structural damage detection. However, it is noted that the proposed model has been proved feasible and efficient only in the case of numerical examples. Its actual performance in experimental work needs to be studied in the coming investigations.

Footnotes

Acknowledgments

The work is supported by the National Natural Science Foundation of China (nos. 51278127 and 50878057), the National 12th Five-Year Research Program of China (Grant no. 2012BAJ14B05), and Research Plan for Minjiang Scholar of Fujian Province, China. Many thanks are given to Mr. Jie Lin for his simulation and analysis.

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A Revised Counter-Propagation Network Model Integrating Rough Set for Structural Damage Detection

Abstract

1. Introduction

2. Counter-Propagation Network (CPN)

3. Structural Damage Detection Methods

3.1. Data Preprocessing and Feature Extraction

3.2. Feature Extraction

3.3. Feature Reduction and Least Input Vector

3.3.1. Discretization of Continuous Attribute Values

3.3.2. Constructing the Decision Table

3.3.3. Reduction of Attributes

3.3.4. Least Input Vectors

3.4. Revised Counter-Propagation Network (RCPN)

3.5. Results Output

4. Simulation Examples

4.1. Structural Model and Damage Simulation

4.2. Damage Detection

4.2.1. Damage Detection Model

4.2.2. Damage Detection Results

4.3. Comparison and Discussion

4.3.1. Noise Level

4.3.2. Data Processing Method

4.3.3. Network Model

5. Conclusions

Footnotes

Acknowledgments

References