Multi-time-sequence collaborative optimization sample selection method for proxy model training
Through the multi-time series collaborative optimization sample selection method, the problem of the uncertainty of a single timing performance in the prior art ignores the mutual influence of multiple timing performances, achieving more accurate and efficient proxy model training, and improving the timing analysis effect of integrated circuit design.
Patent Information
- Application Number
- CN202510314700.6
- Authority / Receiving Office
- CN · China
- Patent Type
- Applications(China)
- Current Assignee / Owner
- Filing Date
- 2025-03-14
- Publication Date
- 2025-08-08
AI Technical Summary
The existing sample selection method only considers the uncertainty of a single timing performance and cannot effectively capture the mutual influence between multiple timing performances, resulting in the inability to fully capture the changes in circuit performance in statistical timing analysis.
The multi-time series collaborative optimization sample selection method is adopted to predict the uncertainty of multiple time series performance indicators by determining timing performance indicators and assigning weight factors. The uncertainty contribution is predicted using the weighting method, and the local maximum operation and probability-based selection strategy are used to select samples to ensure that the samples are distributed throughout the distribution and the effective samples are selected through the probability distribution function.
It improves the accuracy and efficiency of proxy model training, can more comprehensively consider the relationship between multiple timing performances, and improves the accuracy and reliability of timing analysis of integrated circuit design.
Smart Images
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Abstract
Description
Technical Field
[0001] The present invention relates to the field of integrated circuit design and provides a multi-time series collaborative optimization sample selection method for agent model training. Background Art
[0002] As semiconductor processes continue to advance and transistor dimensions shrink, process variations are increasingly impacting circuit performance. Statistical circuit analysis has become crucial for accurately evaluating circuit performance. In digital circuit timing analysis, the Liberty Variation Format (LVF) has become the industry-leading representation of timing distributions in cell libraries for 22nm and below processes. However, LVF characterization relies on Monte Carlo (MC) methods, which typically require extensive SPICE simulations to handle cells with process variation.
[0003] Existing sample selection methods primarily consider the uncertainty of individual timing performance, while ignoring the mutual impact of multiple timing performances. This results in an inability to fully capture circuit performance variations in statistical timing analysis (SSTA), thus affecting the accuracy of design signoff.
[0004] Traditional LVF library generation optimization techniques attempt to improve efficiency by employing machine learning for LVF data prediction. However, these methods are primarily effective at finding bounds on timing performance, rather than the overall timing distribution required for LVF. Furthermore, they do not consider the impact of uncertainty quantification (UQ). UQ analysis is an important practice that promotes a quantitative understanding of uncertainty in models. It provides information on predictive likelihood, inferring the highest probability outcomes, and calibrating models. Within the broader UQ community, there is potential to accelerate the estimation of distribution functions through active learning. Although they can handle arbitrary target distributions, this flexibility comes at a higher computational cost. This hinders their application to LVFs that focus on the first three statistical moments.
[0005] Furthermore, existing sample selection methods primarily consider the uncertainty of a single timing property, such as delay or transition time, while ignoring the interplay between multiple timing properties. For example, in digital circuits, delay and transition time are often interrelated, and the uncertainty of a single timing property can indirectly impact the other timing properties. Consequently, existing sample selection methods are unable to effectively capture this interplay, resulting in reduced accuracy in timing distribution estimation.
[0006] In summary, existing sample selection methods cannot effectively capture the mutual influence between multiple timing performances, resulting in the inability to fully capture the changes in circuit performance in statistical timing analysis. Summary of the Invention
[0007] The purpose of the present invention is to solve the problem that the existing sample selection method only considers the uncertainty of a single time series performance and cannot effectively capture the mutual influence between multiple time series performances.
[0008] In order to achieve the above-mentioned purpose, the present invention adopts the following technical means:
[0009] The present invention provides a multi-time series collaborative optimization sample selection method for agent model training, comprising the following steps:
[0010] Step 1: Determine the timing performance indicators and assign a weight factor w to each timing performance indicator. i ;
[0011] Step 1.1. List all the timing performance indicators that need to be considered, such as delay time, transition time, constraint time, etc.
[0012] Step 1.2: Assign a weight factor w to each timing performance indicator based on the requirements of timing analysis and the importance of each indicator. i ,The weight factor should reflect the relative importance of the indicator in the overall performance evaluation;
[0013] Step 2: Predict multiple time series performance indicators predicted by the agent model to obtain a predicted time series performance vector;
[0014] Step 3: Calculate the prediction uncertainty of each time series performance indicator to obtain the prediction uncertainty vector;
[0015] Step 4: Use a weighted approach to combine the prediction uncertainty contributions of multiple time series performance indicators into an overall acquisition score, where the weight factor of each time series performance indicator is set according to its importance in the time series analysis;
[0016] Step 5: Apply the local maximum operation to identify the samples with the local maximum acquisition score among the candidate samples, ensuring that the selected samples are distributed throughout the distribution, not just the tail;
[0017] Step 6: Use a probability-based selection strategy and use the normalized acquisition value as the probability of selecting a valid sample, so that even samples with low acquisition scores but located in the center of the distribution have a chance to be selected;
[0018] Step 7: Use the probability distribution function to select M valid samples. The probability of each sample being selected is proportional to its normalized acquisition value.
[0019] Step 8: Add the selected valid samples to the valid sample set, annotate them, and use them to train the proxy model for the next iteration.
[0020] In the above solution, the proxy model is used in step 2 to predict the timing performance of all samples, and the predicted timing performance vector Y = [y1, y2, ..., y n ], where y i represents the prediction time series performance of the i-th sample;
[0021] In the above scheme, step 3 specifically includes the following steps:
[0022] Step 3.1: Calculate the mean of the statistical moment of each timing performance indicator based on the predicted timing performance vector variance and skewness
[0023] Step 3.2, calculate the mean separately variance and skewness The expectation and variance of and Where E[*] represents expectation and V[*] represents variance;
[0024] Step 3.3: Use calculation tools to calculate the gradient and
[0025] , where x n The vector representing the nth sample;
[0026] Step 3.4: Express each sample x n Contribution to the uncertainty of the three statistical moments Var
[0027] Step 3.5, calculate the gradient of the objective function L(i) with respect to the sample, and get
[0028] Step 3.6: Integrate variational analysis to derive the total variance change Var[L (i) ];
[0029] Step 3.7: Obtain the collection score of each candidate sample for each time series performance, that is, obtain the contribution A of each sample to the uncertainty of the prediction result. (i) (x n ).
[0030] Furthermore, in step 4, for each sample x n , using a weighted approach to combine the prediction uncertainty contributions of multiple time series performance indicators into an overall acquisition score A(x n ):
[0031]
[0032] A (i) (x n ) The i-th time series performance index for candidate sample x n The forecast uncertainty contribution of w i is the weight factor of the i-th time series distribution.
[0033] Furthermore, step 5 includes the following steps:
[0034] Step 5.1: For each sample x n Define a local neighborhood by setting a fixed distance threshold;
[0035] Step 5.2: For each sample x n , find the maximum value of the score in the local neighborhood defined by it;
[0036] Step 5.4: Record the maximum acquisition score and its corresponding sample in each local neighborhood;
[0037] Step 5.5: Check the position of the sample with the identified local maximum acquisition score in the entire distribution. If the sample distribution is uneven, adjust the distance threshold in the definition of the local neighborhood in step 5.1 to promote a more uniform distribution.
[0038] Step 6: Use a probability-based selection strategy and use the normalized acquisition value as the probability of selecting a valid sample, so that even samples with low acquisition scores but located in the center of the distribution have a chance to be selected;
[0039] Step 6.1: Normalize the acquisition scores of all samples to obtain the normalized acquisition value of each sample:
[0040]
[0041] L1, L2, …, L M′ Indicates the index of the candidate sample.
[0042] The present invention also provides a multi-time series collaborative optimization sample selection device for agent model training, comprising the following modules:
[0043] Timing performance index determination module, used to determine the timing performance index and assign a weight factor w to each timing performance index i ;
[0044] A prediction uncertainty calculation module is used to predict multiple time series performance indicators predicted by the agent model to obtain a predicted time series performance vector, and calculate the prediction uncertainty of each time series performance indicator to obtain a prediction uncertainty vector;
[0045] A weighted uncertainty merging module for merging the prediction uncertainty contributions of multiple timing performance indicators into an overall acquisition score using a weighted approach, where the weight factor of each timing performance indicator is set according to its importance in timing analysis;
[0046] A local maximum operation module is used to apply the local maximum operation to identify the samples with the local maximum acquisition score among the candidate samples, ensuring that the selected samples are distributed in the entire distribution, not just the tail;
[0047] A probability selection strategy module is used to use a probability-based selection strategy, using the normalized acquisition value as the probability of selecting a valid sample, so that even samples with low acquisition scores but located in the center of the distribution have a chance to be selected;
[0048] A probability distribution function selection module is used to select M valid samples using a probability distribution function, where the probability of each sample being selected is proportional to its normalized acquisition value;
[0049] The valid sample set update module is used to add the selected valid samples to the valid sample set, annotate them, and use them to train the proxy model for the next iteration.
[0050] In the above device, the prediction uncertainty calculation module includes:
[0051] The statistical moment calculation unit calculates the mean of the statistical moment of each timing performance indicator based on the predicted timing performance vector variance and skewness
[0052] Statistical moment expectation and variance calculation unit, calculate the mean respectively variance and skewness The expectation and variance of and Where E[*] represents expectation and V[*] represents variance;
[0053] Gradient calculation unit, use calculation tools to calculate the gradient and , where x n The vector representing the nth sample;
[0054] Uncertainty contribution expression unit, expression of each sample x n Contribution to the uncertainty of the three statistical moments Var
[0055] The objective function gradient calculation unit calculates the gradient of the objective function L(i) with respect to the sample and obtains
[0056] Total variance change derivation unit, integrated variational analysis to derive the total variance change Var[L (i) ];
[0057] The acquisition score acquisition unit obtains the acquisition score of each candidate sample for each time series performance, that is, the contribution of each sample to the uncertainty of the prediction result A (i) (x n ).
[0058] In the above device, for each sample x in the weighted uncertainty merging module n , using a weighted approach to combine the prediction uncertainty contributions of multiple time series performance indicators into an overall acquisition score A(x n ):
[0059]
[0060] A (i) (x n ) The i-th time series performance index for candidate sample x n The forecast uncertainty contribution of w i is the weight factor of the i-th time series distribution.
[0061] In the above device, the local maximum operation module includes:
[0062] The local neighborhood defines the unit, for each sample x n Define a local neighborhood by setting a fixed distance threshold;
[0063] The local maximum acquisition score search unit, for each sample x n , find the maximum value of the score in the local neighborhood defined by it;
[0064] A local maximum acquisition score recording unit records the maximum acquisition score and its corresponding sample in each local neighborhood;
[0065] The sample distribution inspection and adjustment unit inspects the positions of the identified samples of the local maximum acquisition scores in the entire distribution, and if the samples are unevenly distributed, adjusts the distance threshold in the definition of the local neighborhood in the local neighborhood definition unit to promote a more even distribution.
[0066] In the above device, the probability selection strategy module normalizes the acquisition scores of all samples to obtain a normalized acquisition value for each sample:
[0067]
[0068] L1, L2, …, L M′ Indicates the index of the candidate sample.
[0069] Because the present invention provides a multi-timing collaborative optimization sample selection method, the present invention achieves the following technical effects in the field of integrated circuit design:
[0070] Comprehensive consideration of timing performance metrics: This approach identifies multiple timing performance metrics and assigns weighting factors to them, thereby more comprehensively considering all aspects of timing performance. This comprehensive consideration helps improve the accuracy of surrogate model training, especially when dealing with complex integrated circuit designs.
[0071] Accurate assessment of prediction uncertainty: By calculating the prediction uncertainty of each time series performance metric, this method can more accurately assess the uncertainty of different metrics, thereby improving the reliability of time series analysis.
[0072] Optimizing sample selection: This method applies a local maximum operation and a probability-based selection strategy to ensure that sample selection considers not only the performance of a single time series but also the interaction between multiple time series. This helps improve the representativeness of the samples, thereby improving the training efficiency and accuracy of the proxy model.
[0073] Improved training efficiency of proxy models: Through the above-mentioned optimized sample selection method, proxy models can more effectively capture the complexity of timing performance, thereby enhancing their application value in integrated circuit design.
[0074] Implementation of the device: A multi-time series collaborative optimization sample selection device is provided, which includes multiple modules, such as a time series performance index determination module, a prediction uncertainty calculation module, etc. These modules work together to ensure the efficiency and accuracy of sample selection.
[0075] Advantages over existing technologies: Compared with existing sample selection methods, the present invention is more comprehensive and detailed in processing multiple time series performance indicators. Through weighted processing and probabilistic selection strategies, the quality of sample selection and the training efficiency of the proxy model are improved. BRIEF DESCRIPTION OF THE DRAWINGS
[0076] Figure 1 A simplified flow chart of the present invention;
[0077] Figure 2 This is a simplified flowchart of the uncertainty vector prediction process of the present invention. DETAILED DESCRIPTION
[0078] The following is a detailed description of the embodiments of the present invention. Although the present invention will be described and illustrated in conjunction with certain specific embodiments, it should be noted that the present invention is not limited to these embodiments. On the contrary, modifications or equivalent substitutions of the present invention are intended to fall within the scope of the claims of the present invention.
[0079] In addition, in order to better illustrate the present invention, numerous specific details are given in the following detailed description. It will be understood by those skilled in the art that the present invention can also be implemented without these specific details.
[0080] The present invention provides a multi-time series collaborative optimization sample selection method for agent model training, comprising the following steps:
[0081] Step 1: Determine the timing performance indicators and assign a weight factor w to each timing performance indicator. i ;
[0082] Step 1.1. List all the timing performance indicators that need to be considered, such as delay time, transition time, constraint time, etc.
[0083] Step 1.2: Assign a weight factor w to each timing performance indicator based on the requirements of timing analysis and the importance of each indicator. i ,The weight factor should reflect the relative importance of the indicator in the overall performance evaluation;
[0084] Step 2: Predict multiple time series performance indicators predicted by the proxy model to obtain a predicted time series performance vector. Specifically, the proxy model is used to predict the time series performance of all samples to obtain a predicted time series performance vector Y = [y1, y2, ..., y n ], where y i represents the prediction time series performance of the i-th sample;
[0085] Step 3: Calculate the prediction uncertainty of each time series performance indicator to obtain the prediction uncertainty vector;
[0086] Step 3.1: Calculate the mean of the statistical moment of each timing performance indicator based on the predicted timing performance vector variance and skewness
[0087]
[0088] N: number of samples;
[0089] Indicates the ,th time series performance index in sample x n The predicted value at
[0090] Step 3.2, calculate the mean separately variance and skewness The expectation and variance of and Where E[*] represents expectation and V[*] represents variance;
[0091] calculate The expectation and variance of :
[0092] expect:
[0093] variance:
[0094] μ (i) (x n ) represents the predicted average value of the i-th time series performance of the n-th sample;
[0095] v (i) (x n ): represents the prediction variance of the i-th time series performance of the n-th sample;
[0096] calculate The expectation and variance of :
[0097] expect:
[0098] variance:
[0099]
[0100] calculate The expectation and variance of :
[0101] expect:
[0102] variance:
[0103] Step 3.3: Use calculation tools to calculate the gradient and where x n The vector representing the nth sample;
[0104] Step 3.4: Express each sample x n Contribution to the uncertainty of the three statistical moments
[0105]
[0106] Represents sample xn Contribution to *uncertainty;
[0107] Step 3.5, calculate the gradient of the objective function L(i) with respect to the sample, and get Calculate the objective function L (i) right Gradient:
[0108]
[0109] Where LVF represents the LVF distribution, D represents the distance measure between the two distributions, using the divergence measure, and δ represents a small perturbation value;
[0110] Step 3.6: Integrate variational analysis to derive the total variance change Var[L (i) ]:
[0111]
[0112] Step 3.7: Obtain the collection score of each candidate sample for each time series performance, that is, obtain the contribution A of each sample to the uncertainty of the prediction result. (i) (x n ):
[0113]
[0114] Step 4: Use a weighted approach to combine the prediction uncertainty contributions of multiple time series performance indicators into an overall acquisition score, where the weight factor of each time series performance indicator is set according to its importance in time series analysis. Specifically, for each sample x n , using a weighted approach to combine the prediction uncertainty contributions of multiple time series performance indicators into an overall acquisition score A(x n ):
[0115]
[0116] A (i) (x n ) The i-th time series performance index for candidate sample x n The forecast uncertainty contribution of w i is the weight factor of the i-th time series distribution.
[0117] Step 5: Apply the local maximum operation to identify the samples with the local maximum acquisition score among the candidate samples, ensuring that the selected samples are distributed throughout the distribution, not just the tail;
[0118] Step 5.1: For each sample x n Define a local neighborhood by setting a fixed distance threshold;
[0119] Step 5.2: For each sample x n , find the maximum value of the score in the local neighborhood defined by it;
[0120] Step 5.4: Record the maximum acquisition score and its corresponding sample in each local neighborhood;
[0121] Step 5.5: Check the position of the sample with the identified local maximum acquisition score in the entire distribution. If the sample distribution is uneven, adjust the distance threshold in the definition of the local neighborhood in step 5.1 to promote a more uniform distribution.
[0122] Step 6: Use a probability-based selection strategy and use the normalized acquisition value as the probability of selecting a valid sample. This ensures that even samples with low acquisition scores but located in the center of the distribution have a chance to be selected. Specifically, the acquisition scores of all samples are normalized to obtain the normalized acquisition value of each sample:
[0123]
[0124] L1, L2, …, L M ′ represents the index of the candidate sample.
[0125] Step 7: Use the probability distribution function to select M valid samples. The probability of each sample being selected is proportional to its normalized acquisition value.
[0126] Step 8: Add the selected valid samples to the valid sample set, annotate them using SPICE simulation, and use them to train the proxy model for the next iteration.
Claims
1. A multi-time series collaborative optimization sample selection method for agent model training, characterized in that: The following steps are involved: Step 1: Determine the timing performance indicators and assign a weight factor w to each timing performance indicator. i ; Step 2: Predict multiple time series performance indicators predicted by the agent model to obtain a predicted time series performance vector; Step 3: Calculate the prediction uncertainty of each time series performance indicator to obtain the prediction uncertainty vector; Step 4: Use a weighted approach to combine the prediction uncertainty contributions of multiple time series performance indicators into an overall acquisition score, where the weight factor of each time series performance indicator is set according to its importance in the time series analysis; Step 5: Apply the local maximum operation to identify the samples with the local maximum acquisition score among the candidate samples, ensuring that the selected samples are distributed throughout the distribution, not just the tail; Step 6: Use a probability-based selection strategy and use the normalized acquisition value as the probability of selecting a valid sample, so that even samples with low acquisition scores but located in the center of the distribution have a chance to be selected; Step 7: Use the probability distribution function to select M valid samples. The probability of each sample being selected is proportional to its normalized acquisition value. Step 8: Add the selected valid samples to the valid sample set, annotate them, and use them to train the proxy model for the next iteration.
2. The multi-time series collaborative optimization sample selection method for agent model training according to claim 1 is characterized in that: In step 2, the proxy model is used to predict the timing performance of all samples, and the predicted timing performance vector Y = [y1, y2, ..., y n ], where y i represents the prediction time series performance of the i-th sample.
3. The multi-time series collaborative optimization sample selection method for agent model training according to claim 1 is characterized in that: Step 3 includes the following steps: Step 3.1: Calculate the mean of the statistical moment of each timing performance indicator based on the predicted timing performance vector variance and skewness Step 3.2, calculate the mean separately variance and skewness The expectation and variance of and Where E[*] represents expectation and V[*] represents variance; Step 3.3: Use calculation tools to calculate the gradient and where x n The vector representing the nth sample; Step 3.4: Express each sample x n Contribution to the uncertainty of the three statistical moments Step 3.5, calculate the gradient of the objective function l(i) with respect to the sample, and get Step 3.6: Integrate variational analysis to derive the total variance change Var[L (i) ]; Step 3.7: Obtain the collection score of each candidate sample for each time series performance, that is, obtain the contribution A of each sample to the uncertainty of the prediction result. (i) (x n ).
4. The multi-time series collaborative optimization sample selection method for agent model training according to claim 1 is characterized in that: In step 4, for each sample x n , using a weighted approach to combine the prediction uncertainty contributions of multiple time series performance indicators into an overall acquisition score A(x n ): A (i) (x n ) The i-th time series performance index for candidate sample x n The forecast uncertainty contribution of w i is the weight factor of the i-th time series distribution.
5. The multi-time series collaborative optimization sample selection method for agent model training according to claim 1 is characterized in that: Step 5 includes the following steps: Step 5.1: For each sample x n Define a local neighborhood by setting a fixed distance threshold; Step 5.2: For each sample x n , find the maximum value of the score in the local neighborhood defined by it; Step 5.3: Record the maximum acquisition score and its corresponding sample in each local neighborhood; Step 5.
4. Check the location of the sample with the identified local maximum acquisition score in the entire distribution. If the sample distribution is uneven, adjust the distance threshold in the definition of the local neighborhood in step 5.1 to promote a more uniform distribution.
6. The multi-time series collaborative optimization sample selection method for agent model training according to claim 1 is characterized in that: In step 6, the acquisition scores of all samples are normalized to obtain the normalized acquisition value of each sample: L1,L2,…,L M′ Indicates the index of the candidate sample.
7. A multi-time series collaborative optimization sample selection device for agent model training, characterized in that: Includes the following modules: Timing performance index determination module, used to determine the timing performance index and assign a weight factor w to each timing performance index i ; A prediction uncertainty calculation module is used to predict multiple time series performance indicators predicted by the agent model to obtain a predicted time series performance vector, and calculate the prediction uncertainty of each time series performance indicator to obtain a prediction uncertainty vector; A weighted uncertainty merging module for merging the prediction uncertainty contributions of multiple timing performance indicators into an overall acquisition score using a weighted approach, where the weight factor of each timing performance indicator is set according to its importance in timing analysis; A local maximum operation module is used to apply the local maximum operation to identify the samples with the local maximum acquisition score among the candidate samples, ensuring that the selected samples are distributed in the entire distribution, not just the tail; A probability selection strategy module is used to use a probability-based selection strategy, using the normalized acquisition value as the probability of selecting a valid sample, so that even samples with low acquisition scores but located in the center of the distribution have a chance to be selected; A probability distribution function selection module is used to select M valid samples using a probability distribution function, where the probability of each sample being selected is proportional to its normalized acquisition value; The valid sample set update module is used to add the selected valid samples to the valid sample set, annotate them, and use them to train the proxy model for the next iteration.
8. The multi-time series collaborative optimization sample selection device for agent model training according to claim 7, characterized in that: The prediction uncertainty calculation module includes: The statistical moment calculation unit calculates the mean of the statistical moment of each timing performance indicator based on the predicted timing performance vector variance and skewness Statistical moment expectation and variance calculation unit, calculate the mean respectively variance and skewness The expectation and variance of and Where E[*] represents expectation and V[*] represents variance; Gradient calculation unit, use calculation tools to calculate the gradient and where x n The vector representing the nth sample; Uncertainty contribution expression unit, expression of each sample x n Contribution to the uncertainty of the three statistical moments The objective function gradient calculation unit calculates the gradient of the objective function L(i) with respect to the sample and obtains Total variance change derivation unit, integrated variational analysis to derive the total variance change Var[L (i) ]; The acquisition score acquisition unit obtains the acquisition score of each candidate sample for each time series performance, that is, the contribution of each sample to the uncertainty of the prediction result A (i) (x n ).
9. The multi-time series collaborative optimization sample selection method for agent model training according to claim 7, characterized in that: For each sample x in the weighted uncertainty merging module n , using a weighted approach to combine the prediction uncertainty contributions of multiple time series performance indicators into an overall acquisition score A(x n ): A (i) (x n ) The i-th time series performance index for candidate sample x n The forecast uncertainty contribution of w i is the weight factor of the i-th time series distribution.
10. The multi-time series collaborative optimization sample selection device for agent model training according to claim 7, characterized in that: The local maximum operation module includes: The local neighborhood defines the unit, for each sample x n Define a local neighborhood by setting a fixed distance threshold; The local maximum acquisition score search unit, for each sample x n , find the maximum value of the score in the local neighborhood defined by it; A local maximum acquisition score recording unit records the maximum acquisition score and its corresponding sample in each local neighborhood; The sample distribution inspection and adjustment unit inspects the positions of the identified samples of the local maximum acquisition scores in the entire distribution, and if the samples are unevenly distributed, adjusts the distance threshold in the definition of the local neighborhood in the local neighborhood definition unit to promote a more even distribution.
11. The multi-time series collaborative optimization sample selection device for agent model training according to claim 7, characterized in that: In the probability selection strategy module, the acquisition scores of all samples are normalized to obtain the normalized acquisition value of each sample: L1,L2,…,L M′ Indicates the index of the candidate sample.