Structural wind-induced response deep learning modeling-oriented unsupervised sample reduction method
By combining discrete wavelet transform and deep autoencoder with Bootstrap and k-means clustering, the adaptiveness problem of sample selection in deep learning of structural wind response is solved, achieving effective reduction of sample size and preservation of model performance.
Patent Information
- Application Number
- CN202511681829.7
- Authority / Receiving Office
- CN · China
- Patent Type
- Applications(China)
- Current Assignee / Owner
- Filing Date
- 2025-11-17
- Publication Date
- 2026-02-10
AI Technical Summary
In existing technologies for deep learning modeling of structural wind-induced responses, random sampling methods lead to the loss of key features or the omission of important samples, making it difficult to meet the needs of engineering modeling and high-precision prediction. Furthermore, existing sample reduction methods rely on researchers' experience and lack adaptability.
Low-frequency features of the samples are extracted by discrete wavelet transform, a deep autoencoder model is designed for feature compression, the sample size is determined by combining the Bootstrap method and the bisection method, and the sample is clustered and hierarchically using the k-means clustering algorithm to construct the reduced sample set.
It effectively reduces the number of training samples, lowers the training cost of deep learning models, and ensures minimal or no loss in model performance. It adapts to different confidence coefficients and tolerance error limits, and improves the adaptability and accuracy of sample selection.
Smart Images
Figure CN121502357A_ABST
Abstract
Description
Technical Field
[0001] This invention belongs to the field of structural health monitoring technology and relates to an unsupervised sample reduction method for deep learning modeling of structural wind-induced response. Background Technology
[0002] In the fields of machine learning and data mining, the size of the training samples directly affects the training efficiency and computational resource consumption of the model. To reduce training time and computational costs, sample reduction strategies have gradually become a research focus.
[0003] Early studies often employed simple random sampling methods, randomly selecting a subset from the original training set for model training. This method is simple to implement, computationally efficient, and widely applicable, suitable for scenarios with large datasets and tight modeling timelines. However, random sampling does not consider the characteristics of sample distribution and their contribution to model performance, easily leading to the loss of key features or the omission of important samples, thus causing a decline in model performance and making it difficult to meet the needs of engineering modeling and high-precision prediction tasks. With the maturity of supervised learning techniques, researchers have gradually combined sample reduction with the structural information of classification tasks to improve data utilization efficiency. Classification models rely on labeled data for training, and the geometric distribution of their decision boundaries becomes an important basis for selecting representative samples. Sample selection methods based on boundary analysis are one of the mainstream strategies. By retaining samples close to the decision boundary, the model strengthens its learning of boundary information and reduces reliance on redundant samples. For example, sample selection methods based on support vector machines utilize the discriminative properties of support vectors to retain samples that have a significant impact on the classification results and remove samples far from the boundary. Clustering methods are also commonly used for sample selection. By dividing the data into several clusters and combining the distance between cluster centers and categories, the most representative samples are selected from each cluster, especially retaining data points near the boundaries of different categories. This balances the representativeness and diversity of the samples, improving training efficiency while maintaining classification performance. Meanwhile, Active Learning, as a dynamic sample selection technique, has also been widely studied. This method uses a small number of labeled samples for initial model training, and then selects the most valuable samples from the unlabeled data according to certain criteria to request labeling, thereby reducing manual costs and improving learning efficiency. Although Active Learning does not directly reduce the total number of samples, it complements sample reduction strategies by optimizing the sample selection order and labeling resource allocation.
[0004] In the field of earthquake engineering and seismic motion modeling, with the rapid expansion of global seismic motion databases, the number of available records has surged. How to select representative seismic motion samples from these databases to balance modeling accuracy and computational efficiency has become a key research focus. Existing methods mostly employ sample reduction techniques based on clustering analysis, such as Yuchen Liao, Ruiyang Zhang, Gang Wu, Hao Sun. A frequency-based ground motion clustering approach for data-driven surrogate modeling of bridges. Journal of Engineering Mechanics-ASCE, 2023, 149(9): 04023069. and Chunxiao Ning, Yazhou Xie. Convolutional variational autoencoder for ground motion classification and generationtoward efficient seismic fragility assessment. Computer-Aided Civil and Infrastructure Engineering, 2023, 39(2): 165-185. These methods divide seismic motions into several categories based on parameters such as response spectrum shape, duration, and peak ground acceleration, and then select representative samples from each category to construct a reduced set. This method effectively reduces the computational complexity and resource consumption of response spectrum analysis, nonlinear dynamic time history analysis, and seismic motion-based machine learning model training while preserving the integrity of structural response information. It should be noted that the sample size selected for this method is entirely determined by the researchers' experience. Summary of the Invention
[0005] This invention provides an unsupervised sample reduction method and apparatus for deep learning modeling of structural wind-induced response, which can effectively reduce the number of training samples. While reducing the training cost of deep learning models, it ensures minimal or no performance loss.
[0006] The technical solution of the present invention:
[0007] An unsupervised sample reduction method for deep learning modeling of structural wind-induced response includes:
[0008] (1) Deploy a health monitoring system on the target structure to obtain monitoring data of the target structure; divide the monitoring data of the target structure into a sample set according to a fixed time length; the monitoring data includes wind speed, wind direction, strain and displacement, and the wind speed and wind direction correspond one-to-one with the strain and displacement at the same time.
[0009] Since wind speed and wind direction data have a one-to-one correspondence with strain and displacement data over time, the number of samples can be reduced by selecting any one of the wind speed, wind direction, strain, or displacement data for processing.
[0010] (2) Using the wavelet approximation method of Discrete Wavelet Transform (DWT), the wind speed, wind direction, strain, or displacement in each sample are subjected to discrete wavelet transform to obtain the wavelet approximation coefficients characterizing their low-frequency features; the wavelet basis function is the Daubechies wavelet basis. After discrete wavelet transform, the wind speed, wind direction, strain, or displacement are expressed as:
[0011]
[0012] In the formula, This represents wind speed, wind direction, strain, or displacement data; m and n represent wavelet scaling parameters and wavelet displacement parameters, respectively. and These represent the approximation coefficient and the detail coefficient, respectively. and Representing scaling functions and wavelet functions;
[0013] In the above formula, the first term is the approximation component, reflecting the low-frequency characteristics of the data; the second term is the detail component, reflecting the high-frequency variation characteristics and noise of the data; by ignoring the detail component, dimensionality reduction of the data is achieved, and wavelet approximation coefficients representing the time-domain and frequency-domain characteristics of the data are extracted. At this time, the wind speed, wind direction, strain, or displacement data are expressed as follows:
[0014]
[0015] (3) Design and train a deep autoencoder model, and extract low-dimensional features from the wavelet approximation coefficients of each group of samples based on the deep autoencoder model structure;
[0016] The deep autoencoder model consists of an encoder and a decoder. The encoder and decoder are structurally symmetrical and are both composed of multiple fully connected neural networks. The number of units in the initial layer neurons of the encoder is consistent with the dimension of the input data. The number of units in each subsequent layer is halved until the number of units in the final layer neurons is consistent with the preset dimension reduction dimension, thus achieving stepwise feature compression of the input data. The number of units in each layer neurons of the decoder increases exponentially layer by layer, thus achieving the restoration and reconstruction of low-dimensional features. The activation function of each layer is the Rectified Linear Unit (ReLU), and the loss function is the Mean Squared Error (MSE), which is used to optimize the error between the reconstructed output and the input data.
[0017] Wavelet approximation coefficients are used as input data for the deep autoencoder model. The network is trained to learn and reconstruct the features of the input data and extract the low-dimensional features of the samples.
[0018] (4) Under the preset confidence coefficient and allowable error limit, the Bootstrap method is used to select different numbers of samples and calculate the mean of low-dimensional features. This mean is then compared with the mean of low-dimensional features of all samples to determine whether the sampling accuracy requirement is met. An iterative search is performed using the bisection method to determine the minimum number of samples that meet the accuracy requirement, which is then used as the final sample size. The relationship between sampling accuracy, confidence coefficient, and allowable error limit is expressed as follows:
[0019]
[0020] In the formula, P represents the sampling precision. and denoted by and , respectively, the mean of the low-dimensional features under the given sample size and the mean of the low-dimensional features of all samples; r represents the tolerance limit; 1-α represents the confidence coefficient;
[0021] The process for determining the sample size is as follows:
[0022] The low-dimensional features of all samples are represented as follows Where N represents the number of samples, F represents the dimension of the extracted low-dimensional features; set the confidence coefficient 1-α, the tolerance limit r, and the number of samplings M for the Bootstrap method;
[0023] Calculate the mean of the low-dimensional features of all samples. The upper bound of the binary search for initializing the sample size. and lower limit ;
[0024] Pick and The integer value of the mean is used as the candidate sample size. In candidate sample size Next, perform M Bootstrap samplings on all samples, calculate the mean of the low-dimensional features of the samples in each Bootstrap sampling, and obtain the probability distribution of the mean of the low-dimensional features of all Bootstrap samples.
[0025] Calculate the confidence coefficient 1-α and the candidate sample size. At that time, the mean of the low-dimensional features of the sample sampled by Bootstrap Calculate the error limit e. ;
[0026] Determine whether the error limit e meets the allowable error limit r requirement. If the condition is not met, then increase the lower bound of the binary search. Repeat the above process; if satisfied, further determine the upper limit of the binary search. and lower limit Does it meet the requirements? If the conditions are met, then output the candidate sample size. As the final determined sample size If the condition is not met, then reduce the upper limit of the binary search. Repeat the above process; continue repeating this process until all judgment conditions are met, thus obtaining the final determined sample size. ;
[0027] (5) Based on the k-means clustering algorithm, cluster analysis is performed on all samples in combination with low-dimensional features; the elbow method and the silhouette coefficient method are used to comprehensively evaluate the k-means clustering results, determine the optimal number of clusters, improve the stability and rationality of the clustering results, and realize the clustering stratification of all samples;
[0028] A stratified sampling strategy with equal sample size is adopted, in which the same number of samples are drawn in each cluster stratum to ensure the representativeness and balance of the samples, and a reduced sample set is constructed.
[0029] The beneficial effects of this invention are:
[0030] (1) This invention provides an unsupervised sample reduction method for deep learning modeling of structural wind-induced response. Monitoring data is divided into segments based on a fixed time length to prepare a sample set; discrete wavelet transform is performed on the samples to obtain wavelet approximation coefficients; an autoencoder is designed and trained, and low-dimensional features of the samples are extracted based on the encoder; based on the low-dimensional features of the samples, under specified confidence coefficients and tolerance limits, the sample size is determined using the Bootstrap and bisection methods, while meeting sampling accuracy requirements; the samples are clustered hierarchically using the k-means clustering algorithm and low-dimensional features; samples are selected using a hierarchical sampling method with equal sample size allocation to construct the reduced sample set, thereby effectively reducing the number of training samples. This invention effectively solves the key problems of "sample selection quantity" and "sample selection method" in redundant sample reduction, achieving effective reduction of the number of training samples, reducing the training cost of deep learning models while ensuring minimal or no loss in model performance.
[0031] (2) This invention provides a sample size determination algorithm based on the Bootstrap method and the bisection method. This algorithm does not rely on distribution assumptions and can adaptively determine the sample size according to the sample characteristics under given confidence coefficients and tolerance limits (such as a confidence coefficient of 0.95 and a tolerance limit of 0.05), while meeting the sampling accuracy requirements.
[0032] (3) The present invention is simple to implement and highly efficient. It can provide effective guidance for the efficient reduction of redundant samples in time series data regression tasks. It has good application prospects, wide applicability and high practical value. Attached Figure Description
[0033] Figure 1 This is a flowchart illustrating an embodiment of the unsupervised sample reduction method for deep learning modeling of structural wind-induced response provided by the present invention.
[0034] Figure 2 A flowchart illustrating an embodiment of the sample size determination algorithm provided by the present invention;
[0035] Figure 3 A schematic diagram of the structure of an embodiment of the self-encoder provided by the present invention;
[0036] Figure 4 This is a schematic diagram of the structure of one embodiment of the LSTM model provided by the present invention. Detailed Implementation
[0037] The specific embodiments of the present invention will be further described below with reference to the accompanying drawings and technical solutions.
[0038] Please see Figures 1 to 4The example is a 220kV transmission tower located in the southeastern coastal region of China, with a total height of 34m. A structural health monitoring system is installed on the tower; detailed structural information can be found in "Fig. 5" of "Zhang Wen-Sheng, Fu Xing, Li Hong-Nan, Zhu Deng-Jie. Wind-induced fragility analysis of a transmission tower based on multi-source monitoring data and deep learning methods. Journal of Wind Engineering and Industrial Aerodynamics, 2024, 252: 105834." In one embodiment of the invention, unsupervised efficient sample reduction methods for time series data regression tasks can all be implemented using self-written programs or related software. This embodiment uses the widely used software MATLAB as an example to demonstrate the method's application.
[0039] See Figure 1 An embodiment of the present invention provides an unsupervised sample reduction method for deep learning modeling of structural wind-induced response. The method includes steps S101 to S105, each step of which is as follows:
[0040] Step S101: Divide the monitoring data into fixed time periods to form a sample set.
[0041] In this embodiment of the invention, a health monitoring system is deployed on the target structure to acquire monitoring data of the target structure. The monitoring data includes wind speed, wind direction and strain, with wind speed and wind direction corresponding one-to-one with strain at the same time. The sampling frequency is 5Hz. The monitoring data of the target structure is divided into a sample set with a time interval of 10 minutes. The sample set includes 4912 samples.
[0042] Step S102: Perform discrete wavelet transform on each group of samples and extract the wavelet approximation coefficients of the samples.
[0043] In this embodiment of the invention, to further improve the sample reduction efficiency, the wind speed and wind direction data in each sample group are downsampled to 1Hz, and discrete wavelet transform is performed on each to obtain wavelet approximation coefficients characterizing their low-frequency features. In this embodiment of the invention, the Daubechies wavelet basis function is selected. After discrete wavelet transform, the wind speed or wind direction is expressed as:
[0044]
[0045] In the formula, This represents wind speed or wind direction data; m and n represent wavelet scaling parameters and wavelet displacement parameters, respectively. and These represent the approximation coefficient and the detail coefficient, respectively. and This represents the scaling function and the wavelet function.
[0046] In the above formula, the first term is the approximation component, reflecting the low-frequency characteristics of the data; the second term is the detail component, reflecting the high-frequency variation characteristics and noise of the data; by ignoring the detail component, dimensionality reduction of the data is achieved, and wavelet approximation coefficients representing the time-domain and frequency-domain characteristics of the data are extracted. At this time, the wind speed or wind direction is expressed as:
[0047]
[0048] Step S103: Design and train a deep autoencoder to extract low-dimensional features of sample wavelet coefficients based on the encoder.
[0049] In this embodiment of the invention, two deep autoencoders with the same architecture are trained using the wavelet approximation coefficients of wind speed and wind direction in each set of samples, such as... Figure 3 As shown, the low-dimensional features of the wavelet approximation coefficients are further extracted, and the two features are concatenated to form the low-dimensional features of the sample.
[0050] The deep autoencoder model consists of an encoder and a decoder. The encoder and decoder are structurally symmetrical, both composed of multiple fully connected neural networks. The initial number of neurons in the encoder's network layers matches the dimension of the input data. Subsequently, the number of neurons in each network layer is halved layer by layer until the final network layer's number of neurons matches the preset dimensionality reduction dimension, achieving step-by-step feature compression of the input data. The number of neurons in each layer of the decoder increases exponentially layer by layer, achieving the restoration and reconstruction of low-dimensional features. The activation function for each network layer is the Rectified Linear Unit (ReLU). The number of neurons in each fully connected neural network layer of the deep autoencoder is 320, 80, 20, 80, and 320, respectively. Model training uses the Adaptive Moment Estimation (Adam) algorithm, with the Mean Square Error (MSE) loss function, a mini-batch size of 50, and a maximum number of iterations of 200. The initial learning rate is 0.001, and a segmented learning rate adjustment strategy is adopted, with a learning rate decrease cycle of 30 and a decrease factor of 0.5.
[0051] Wavelet approximation coefficients are used as input data for the deep autoencoder model. The network is trained to learn and reconstruct the features of the input data and extract the low-dimensional features of the samples.
[0052] Step S104: Under the set confidence coefficient and tolerance limit, the Bootstrap method is used to select different numbers of samples and calculate the mean of low-dimensional features. The mean of low-dimensional features of all samples is compared with the mean of low-dimensional features of all samples to determine whether the sampling accuracy requirement is met. The minimum number of samples that meet the accuracy requirement is determined by the bisection method iteratively, which is used as the sample size.
[0053] In this embodiment of the invention, based on the "3σ principle" of the normal distribution, the confidence coefficients are set to 0.95 and 0.99 respectively, and the tolerance limit is set to 0.05. The sampling precision is expressed as:
[0054]
[0055] In the formula, P represents the sampling precision. and denoted by and , respectively, the mean of the low-dimensional features under the given sample size and the mean of the low-dimensional features of all samples; r represents the tolerance limit; 1-α represents the confidence coefficient;
[0056] By using different combinations of confidence coefficients and tolerance limits, according to Figure 2 Algorithm flow, determining sample size under different combinations.
[0057] Step S105: Perform stratified clustering of all samples based on the k-means clustering algorithm and the low-dimensional features of the samples; select samples using a stratified sampling method with equal sample size allocation to construct a reduced sample set.
[0058] In this embodiment of the invention, all samples are clustered into hierarchical groups based on the k-means clustering algorithm and combined with the low-dimensional features of the samples; the elbow method and the silhouette coefficient method are used to comprehensively evaluate the k-means clustering results and determine the optimal number of clusters.
[0059] A stratified sampling strategy with equal sample size is adopted, in which the same number of samples are drawn in each cluster stratum to ensure the representativeness and balance of the samples, and a reduced sample set is constructed.
[0060] As an example of an embodiment of the present invention, the following is adopted: Figure 4The Long Short-Term Memory (LSTM) model shown is used to model the wind-induced response of a transmission tower. The surrogate model takes wind speed and direction data as input and outputs strain responses at four locations. The input and output have the same length. The LSTM model constructed in this embodiment consists of one LSTM layer, one fully connected layer, and one forget layer. The LSTM layer contains 200 hidden nodes, and the fully connected layer contains 100 hidden nodes. The probability of the forget layer is 0.1. The model is trained using the Adam algorithm with a mini-batch size of 50 and a maximum of 120 iterations. The initial learning rate is 0.005, and a segmented learning rate adjustment strategy is used, with a learning rate decrease period of 30 and a decrease factor of 0.5.
[0061] The performance of the LSTM model trained on the reduced sample set is compared with that trained on the full sample set. Table 1 summarizes the performance degradation of the LSTM model trained on different reduced sample sets, using the performance of the model trained on the full sample set as a benchmark.
[0062]
[0063] Note: The numerical values in parentheses are explained as follows: (1) Number of samples in the sample set: the percentage of the number of samples in the reduced sample set relative to the total number of samples in the sample set; (2) Model performance: the performance change of the LSTM model trained on the reduced sample set relative to the model trained on the total number of samples. Negative values indicate the percentage decrease in performance, and positive values indicate the percentage increase in performance.
[0064] This invention provides an unsupervised sample reduction method for deep learning modeling of structural wind-induced response.
[0065] With a confidence level of 0.95 and an error tolerance of 0.05, this invention significantly reduces the number of samples in the sample set by about 45%, while keeping the model performance degradation within 1%, demonstrating good sample reduction effect and performance preservation ability.
[0066] Under the conditions of a confidence level of 0.99 and an error tolerance of 0.05, although the number of samples in the sample set is reduced by only about 5%, the present invention effectively removes noisy samples, improving the model performance by 0.38%, further verifying that the proposed method has the ability to optimize data quality while ensuring model accuracy.
[0067] The specific embodiments described above further illustrate the purpose, technical solution, and beneficial effects of the present invention. It should be understood that the above descriptions are merely specific embodiments of the present invention and are not intended to limit the scope of protection of the present invention. In particular, it should be noted that any modifications, equivalent substitutions, improvements, etc., made within the spirit and principles of the present invention should be included within the scope of protection of the present invention for those skilled in the art.
Claims
1. An unsupervised sample reduction method for deep learning modeling of structural wind-induced response, characterized in that, include: (1) Deploy a health monitoring system on the target structure to obtain monitoring data of the target structure; The monitoring data of the target structure is divided into sample sets according to a fixed time length; the monitoring data includes wind speed, wind direction, strain and displacement, and the wind speed and wind direction correspond one-to-one with the strain and displacement at the same time. (2) Using the wavelet approximation method of discrete wavelet transform, the wind speed, wind direction, strain, or displacement in each sample are subjected to discrete wavelet transform to obtain the wavelet approximation coefficients that characterize their low-frequency features; the wavelet basis function is the Daubechies wavelet basis. After discrete wavelet transform, the wind speed, wind direction, strain, or displacement data are expressed as follows: ； In the formula, This represents wind speed, wind direction, strain, or displacement data; m and n represent wavelet scaling parameters and wavelet displacement parameters, respectively. and These represent the approximation coefficient and the detail coefficient, respectively. and Representing scaling functions and wavelet functions; In the above formula, the first term is the approximation component, reflecting the low-frequency characteristics of the data; the second term is the detail component, reflecting the high-frequency variation characteristics and noise of the data; by ignoring the detail component, dimensionality reduction of the data is achieved, and wavelet approximation coefficients representing the time-domain and frequency-domain characteristics of the data are extracted. At this time, the wind speed, wind direction, strain, or displacement data are expressed as follows: ； (3) Design and train a deep autoencoder model, and extract low-dimensional features from the wavelet approximation coefficients of each group of samples based on the deep autoencoder model structure; The deep autoencoder model consists of an encoder and a decoder. The encoder and decoder are structurally symmetrical and are both composed of multiple fully connected neural networks. The number of units in the initial layer neurons of the encoder is consistent with the dimension of the input data. The number of units in each subsequent layer neurons is halved until the number of units in the final layer neurons is consistent with the preset dimension reduction dimension, thereby achieving progressive feature compression of the input data. The number of units in each layer neurons of the decoder increases exponentially layer by layer, thereby achieving the restoration and reconstruction of low-dimensional features. The activation function of each layer adopts the linear rectified function, and the loss function adopts the mean squared error to optimize the error between the reconstructed output and the input data. Wavelet approximation coefficients are used as input data for the deep autoencoder model. The network is trained to learn and reconstruct the features of the input data and output the results. The encoder of the deep autoencoder model is used to compress the features of the input data to obtain the low-dimensional features of the samples. (4) Under the preset confidence coefficient and allowable error limit, the Bootstrap method is used to select different numbers of samples and calculate the mean of low-dimensional features. This mean is then compared with the mean of low-dimensional features of all samples to determine whether the sampling accuracy requirement is met. An iterative search is performed using the bisection method to determine the minimum number of samples that meet the accuracy requirement, which is then used as the final sample size. The relationship between sampling accuracy, confidence coefficient, and allowable error limit is expressed as follows: ； In the formula, P represents the sampling precision. and denoted by and , respectively, the mean of the low-dimensional features under the given sample size and the mean of the low-dimensional features of all samples; r represents the tolerance limit; 1-α represents the confidence coefficient; The process for determining the sample size is as follows: The low-dimensional features of all samples are represented as follows Where N represents the number of samples, F represents the dimension of the extracted low-dimensional features; set the confidence coefficient 1-α, the tolerance limit r, and the number of samplings M for the Bootstrap method; Calculate the mean of the low-dimensional features of all samples. The upper bound of the binary search for initializing the sample size. and lower limit ; Pick and The integer value of the mean is used as the candidate sample size. In candidate sample size Next, perform M Bootstrap samplings on all samples, calculate the mean of the low-dimensional features of the samples in each Bootstrap sampling, and obtain the probability distribution of the mean of the low-dimensional features of all Bootstrap samples. Calculate the confidence coefficient 1-α and the candidate sample size. At that time, the mean of the low-dimensional features of the sample sampled by Bootstrap Calculate the error limit e. ; Determine whether the error limit e meets the allowable error limit r requirement. If the condition is not met, then increase the lower bound of the binary search. Repeat the above process; if satisfied, further determine the upper limit of the binary search. and lower limit Does it meet the requirements? If the conditions are met, then output the candidate sample size. As the final determined sample size If the condition is not met, then reduce the upper limit of the binary search. Repeat the above process; continue repeating this process until all judgment conditions are met, thus obtaining the final determined sample size. ; (5) Based on the k-means clustering algorithm, cluster analysis is performed on all samples in combination with low-dimensional features; the elbow method and the silhouette coefficient method are used to comprehensively evaluate the k-means clustering results, determine the optimal number of clusters, improve the stability and rationality of the clustering results, and realize the clustering stratification of all samples; A stratified sampling strategy with equal sample size is adopted, in which the same number of samples are drawn in each cluster stratum to ensure the representativeness and balance of the samples, and a reduced sample set is constructed.