Ginseng growth age rapid identification method based on terahertz time-domain spectroscopy technology

Abstract

The invention discloses a rapid ginseng growth age identification method based on terahertz time-domain spectroscopy, and relates to the technical field of spectrum detection.The method comprises the steps that a ginseng sample needing growth age identification is prepared; analyzing the ginseng sample by using a terahertz time-domain spectrometer; collecting terahertz spectrum data of the ginseng sample in the analysis process; carrying out standardization processing on the collected terahertz spectrum data; performing grouping identification on the ginseng samples according to the terahertz spectrum data; optimizing parameters of the clustering algorithm by adopting an algorithm optimizer; and evaluating the recognition result of the clustering algorithm through the evaluation index. According to the method, the terahertz time-domain spectroscopy technology is combined with multivariate clustering analysis, the growth age limit of the ginseng can be judged according to the spectrum absorption condition of the ginseng, and automatic identification of the age limit of the ginseng without labeling is realized.

Description

TECHNICAL FIELD

[0001] The present application relates to the technical field of spectral detection, in particular to a ginseng growth age rapid identification method based on terahertz time-domain spectroscopy. BACKGROUND

[0002] With the progress of science and technology, terahertz time-domain spectroscopy (THz-TDS) as a new emerging spectral analysis method, gradually shows its unique advantages in the fields of material science, life science and so on. Especially in the quality control and identification of traditional Chinese medicinal materials, THz-TDS technology is widely concerned due to its non-contact, high sensitivity and rapid detection characteristics. Ginseng as a traditional Chinese medicinal material, its growth age directly affects its efficacy and market value. Therefore, accurately identifying ginseng of different growth ages is of great significance to guarantee the quality of traditional Chinese medicine and improve market competitiveness.

[0003] However, the current identification of ginseng age mainly relies on experience-based sensory judgment, chemical analysis or optical component detection methods, such as high performance liquid chromatography, mass spectrometry and other technologies. These methods not only have a complicated process and a long detection period, but also have a certain destructive effect on samples, which is difficult to meet the urgent needs of modern traditional Chinese medicine industry for rapid and non-destructive detection. In addition, the existing methods have certain limitations in processing multi-dimensional complex data, and the accuracy and stability of identification still need to be improved.

[0004] Therefore, the present application provides a ginseng growth age rapid identification method based on terahertz time-domain spectroscopy, which combines terahertz time-domain spectroscopy technology with multivariate cluster analysis to realize rapid and accurate identification of ginseng of different growth ages. SUMMARY

[0005] The present application provides a ginseng growth age rapid identification method based on terahertz time-domain spectroscopy, which uses a method combining terahertz time-domain spectroscopy technology with multivariate cluster analysis to solve the technical problem of material and labor waste caused by relying on manual identification and personnel level in the existing identification method.

[0006] To solve the above technical problems, the present application provides the following technical solutions: The present application provides a ginseng growth age rapid identification method based on terahertz time-domain spectroscopy, which includes: S1 preparing ginseng samples that need to be identified for growth age; S2 analyzing the ginseng samples by using a terahertz time-domain spectrometer; S3 collecting terahertz spectrum data of the ginseng samples in the analysis process; S4 performing standardization processing on the collected terahertz spectrum data; S5 performing cluster identification on the ginseng samples according to the terahertz spectrum data; S6 uses an algorithm optimizer to optimize the parameters of the clustering algorithm; S7 evaluates the identification result of the clustering algorithm through an evaluation index.

[0007] The technical solution provided by the present application has at least the following beneficial effects: The method combines terahertz time-domain spectroscopy technology with multivariate cluster analysis, can determine the growth years of ginseng according to the absorption of the spectrum, realizes automatic identification of ginseng years without labeling, provides an efficient, non-destructive and unsupervised new path for quality control of traditional Chinese medicinal materials, and has the potential for popularization in other complex medicinal material identification scenarios. BRIEF DESCRIPTION OF DRAWINGS

[0008] In order to more clearly illustrate the technical solutions in the embodiments of the present application, the drawings needed in the embodiment description will be briefly introduced. Obviously, the drawings in the following description are only some embodiments of the present application, and other drawings can be obtained by those skilled in the art without creative labor.

[0009] Figure 1 is a flowchart of the ginseng growth year rapid identification method based on terahertz time-domain spectroscopy technology provided by the embodiments of the present application; Figure 2 is a flowchart of the mean shift algorithm provided by the embodiments of the present application; Figure 3 is a flowchart of the SABO algorithm provided by the embodiments of the present application. DETAILED DESCRIPTION

[0010] In order to make the purpose, technical solutions and advantages of the present application clearer, the embodiments of the present application will be further described in detail below with reference to the drawings.

[0011] EMBODIMENT A ginseng growth year rapid identification method based on terahertz time-domain spectroscopy technology.

[0012] Please refer to Figure 1 is a flowchart of the ginseng growth year rapid identification method based on terahertz time-domain spectroscopy technology provided by the embodiments of the present application.

[0013] S1 prepares ginseng samples that need to be identified for growth years; The preparation process includes crushing, drying and tabletting, the crushing process is used to crush the ginseng samples to particles with a diameter of less than 200 mesh, the drying process is used to keep the moisture in the ginseng particles below 5%, and the tabletting process is used to compress the ginseng particles into tablets for easy detection; The thickness, diameter and weight of the tablets are 1mm, 12mm and 200mg, respectively.

[0014] It should be noted that the specific preparation process of the tablet is: the main root part of ginseng is selected, crushed and then passed through a 200 mesh sieve to obtain a ginseng granule sample, the sample is dried in a vacuum drying oven at a temperature of 50°C for 1h, and then the powder is pressed into a circular tablet under a pressure of 12MPa for 1min using a press.

[0015] S2 analyzes the ginseng sample using a terahertz time domain spectrometer; The environmental conditions during the analysis include relative humidity, ambient temperature, and detection space. The relative humidity condition is maintained within 1%RH, the ambient temperature condition is maintained at 18±2°C, and the detection space condition is maintained in a sealed environment.

[0016] It should be noted that the terahertz time domain spectrometer is placed in a sealed test box during the detection process, and dry nitrogen is continuously introduced, the relative humidity is controlled at 1%RH, and the test environment temperature is maintained at 18°C. The time domain spectrum signal of the ginseng tablet sample is collected by the terahertz time domain spectrometer.

[0017] S3 collects the terahertz spectrum data of the ginseng sample during the analysis; The terahertz spectrum data takes the time domain signal of the ginseng sample as the sample signal and the time domain signal of dry nitrogen as the reference signal. By using fast Fourier transform to convert the time domain to the frequency domain, two frequency domain signals are obtained; The terahertz spectrum data is obtained by comparing the amplitude ratio and phase difference of the two frequency domain signals to obtain the refractive index and absorption coefficient of the sample at different frequencies, and by calculating the absorbance spectrum of the sample from the refractive index and absorption coefficient; The terahertz spectrum data calculation process is as follows:

[0018]

[0019] In the formula, T represents the transmittance of the ginseng sample, i.e. the ratio between the refractive index and the absorption coefficient; T represents the representation of the time domain signal of the ginseng sample in the frequency domain form; T represents the representation of the time domain signal of dry nitrogen in the frequency domain form; T represents the absorbance spectrum, i.e. the terahertz spectrum data.

[0020] It should be noted that the frequency range of the terahertz spectrum data needs to be collected within 0.06-1.8THz.

[0021] S4 standardizes the collected terahertz spectrum data; The standardization processing includes spectrum pretreatment and feature dimension reduction, the spectrum pretreatment is used to solve the problems of dielectric interference, surface scattering and baseline drift in the terahertz spectrum, and the feature dimension reduction is used to reduce the dimension of the pretreated data; The spectrum pretreatment process is: based on a dynamic high-order nonlinear fitting mechanism, a dynamic reference is constructed by using a moving local average spectrum, a third-order and above polynomial regression fitting residual is obtained, and local outliers are iteratively removed in the fitting process, and a hybrid residual fitting strategy is used to correct the original absorbance curve, and compensate for the noise of moisture, particle distribution and nonlinear scattering; It should be noted that the key absorption peaks in the range of 0.3-1.8 THz need to be retained in the pretreatment process.

[0022] The feature dimension reduction adopts principal component analysis and uniform manifold approximation mapping methods to reduce the dimension of the pretreated data.

[0023] It should be noted that the specific implementation process of the principal component analysis method is: the pretreated data is standardized, so that each feature has a mean of 0 and a variance of 1, and the covariance matrix of the standardized data is calculated to represent the linear relationship between the features. At this time, the covariance matrix is decomposed to obtain the eigenvalues and the corresponding eigenvectors, the eigenvalues are arranged in descending order, the first k (k represents the dimension after dimension reduction) eigenvectors are selected to form a projection matrix, and the original data is projected onto the selected principal component direction to obtain the data after dimension reduction; the specific implementation process of the uniform manifold approximation mapping method is: a weighted graph is constructed, wherein the nodes represent the samples and the edges represent the adjacent relationship, a low-dimensional space is constructed, and a point set is randomly initialized. The position of the low-dimensional point is optimized by minimizing the cross entropy between the high-dimensional graph and the low-dimensional graph, and the gradient descent method is used to minimize the loss function for iteration. After the iteration is completed, the data representation in the low-dimensional space is obtained.

[0024] Please refer to Figure 2 is the flowchart of the mean shift algorithm provided by the embodiment of the application.

[0025] S5, according to the terahertz spectrum data, carries out cluster identification on the ginseng samples; The cluster identification process is realized by using a mean shift clustering algorithm, the mean shift clustering algorithm adjusts a dynamic access matrix and a density-driven center merging strategy, and introduces a double convergence criterion to avoid pseudo-convergence and overfitting small cluster phenomenon; The merging strategy includes initialization, iterative clustering, convergence judgment, center merging and label assignment. The specific process of the initialization step is: inputting a data set, an initial width r and a convergence threshold value epsilon, initializing an access count matrix and a clustering center matrix, and marking all data points as unvisited state. The specific process of the iterative clustering step is: calculating the Euclidean distance of each unvisited point with all points, screening the neighborhood points with a distance less than r2, taking the number of neighborhood points as the current local density, then updating the cluster center to the mean of the neighborhood points, and generating a drift vector; The specific process of the convergence judgment step is: using the norm of the drift vector and the density change rate as two conditions to judge whether the clustering converges; The specific process of the center merging and label assignment step is: the merging rule is that if the distance between the new center and the existing center is less than r2, then merge to the cluster with higher density, and determine the final category according to the center with the most visited data points.

[0026] It should be noted that after determining the final category, the cluster center and sample label are returned, and the low-density noise cluster is removed.

[0027] The generation process of the drift vector is as follows:

[0028] In the formula, represents the displacement of the cluster center update, that is, the drift vector; represents the number of neighborhood points of the current unvisited point; represents the geometric center position of a single cluster; represents the sample point in the data space.

[0029] Please refer to Figure 3 is the flowchart of the SABO algorithm provided by the embodiment of the application.

[0030] S6 uses an algorithm optimizer to optimize the parameters of the clustering algorithm; The algorithm optimizer is based on the SABO algorithm, balances the global exploration and local development ability through a differential perturbation mechanism and an adaptive search strategy; It should be noted that the SABO algorithm is used to optimize the clustering radius and the convergence threshold Two key parameters in the clustering algorithm, which can reduce the performance fluctuation problem of the clustering algorithm in ginseng THz spectrum data due to parameter sensitivity.

[0031] The differential perturbation mechanism sorts and classifies the fitness of the population individuals, and uses different perturbation mechanisms according to the grouping situation. The grouping includes an elite group, a medium group and a lag group. The individuals in the elite group use small amplitude Gaussian perturbation, the individuals in the medium group use cross perturbation to the elite individuals, and the individuals in the lag group use Levy flight to reset; It should be noted that the elite group individuals impose a Gaussian distribution random disturbance with a mean of 0 and a standard deviation of 2% of the parameter value on the parameter value, and only a part of the parameters are randomly selected for disturbance each time; the medium group individuals move in the direction of the randomly selected elite individual parameter value from the current individual parameter value, and the moving step is a random value between 0% and 20% of the distance; the lag group individuals use Levy flight to generate a random step, and the parameter α of the Levy flight is 1.6, and the disturbance amplitude is 30% of the parameter space range.

[0032] The adaptive search strategy calculates the population diversity index and the convergence degree index, and judges whether the current is in the exploration stage or the development stage based on the population diversity threshold and the convergence degree threshold, the search strategy in the exploration stage is to increase the crossover probability, the mutation strength and the random initialization probability, and the search strategy in the development stage is to reduce the crossover probability and the mutation strength and increase the local search operation.

[0033] It should be noted that the population diversity index is obtained by calculating the average Euclidean distance between all individuals in the population, and the convergence degree index is obtained by calculating the improvement rate of the optimal fitness. When the population diversity is lower than the preset threshold, or the optimal solution has no significant improvement for continuous generations, or it is in the early 40% of the iteration, it is determined to be in the exploration stage; when the population diversity is moderate, or there is continuous slight improvement in the last few generations, or it is in the later 60% of the iteration, it is determined to be in the development stage.

[0034] S7 evaluates the identification result of the clustering algorithm through the evaluation index; The evaluation index includes the adjusted Rand index and the classification accuracy, the adjusted Rand index is used for evaluation from the perspective of clustering consistency, and the classification accuracy is used for evaluation from the perspective of sample discrimination accuracy; The identification result is determined by assigning a weight coefficient to the evaluation index, and the final evaluation result is determined by calculating the weighted comprehensive score.

[0035] It should be noted that for the application scenario with high clustering quality requirement, the adjusted Rand index weight is set to be greater than the classification accuracy weight, and the recommended adjusted Rand index weight is 0.7 and the classification accuracy weight is 0.3; for the application scenario with high classification accuracy requirement, the adjusted Rand index weight is set to be less than the classification accuracy weight, and the recommended adjusted Rand index weight is 0.3 and the classification accuracy weight is 0.7.

[0036] The calculation process of the adjusted Rand index is as follows:

[0037] In the formula, represents the adjusted Rand index result; represents the number of overlapping sample pairs between the real and predicted clusters; represents the number of combinations of sample pairs in the real cluster. denotes the number of combinations of sample pairs in the predicted cluster; denotes the number of combinations of total sample pairs.

[0038] It should be noted that the range of the Landau index is [-1, 1], wherein 1 represents complete consistency, 0 represents a result close to randomness, and a negative value represents a result worse than randomness.

[0039] The calculation process of the classification accuracy is specifically as follows:

[0040] In the formula, denotes the classification accuracy result; denotes the total number of samples; denotes the true label of the i th sample; denotes the predicted label of the i th sample; denotes an indicator function, which is 1 when the prediction is correct, and 0 otherwise.

[0041] In addition, it should be noted that the present application can be provided as a method, device or computer program product. Therefore, the embodiments of the present application can adopt a completely hardware embodiment, a completely software embodiment or an embodiment combining software and hardware aspects. Moreover, the embodiments of the present application can adopt the form of a computer program product implemented on one or more computer usable storage media containing computer usable program code.

[0042] The embodiments of the present application are described with reference to flowcharts and / or block diagrams according to the method, terminal device (system) and computer program product of the embodiments of the present application. It should be understood that each flow and / or block in the flowcharts and / or block diagrams, and the combination of the flows and / or blocks in the flowcharts and / or block diagrams can be realized by computer program instructions. These computer program instructions can be provided to a general-purpose computer, embedded processor or other programmable data processing terminal device to produce a machine, so that the instructions executed by the computer or other programmable data processing terminal device produce a device for implementing the functions specified in the flowcharts and / or block diagrams. Figure 1 The functions specified in one or more flows and / or blocks Figure 1 The device for implementing the functions specified in one or more flows and / or blocks.

[0043] These computer program instructions can also be stored in a computer readable storage medium capable of guiding a computer or other programmable data processing terminal device to work in a specific manner, so that the instructions stored in the computer readable storage medium produce a product including instruction devices, which implement the functions specified in the flowcharts and / or block diagrams. Figure 1 The functions specified in one or more flows and / or blocks Figure 1The functions specified in one or more boxes. These computer program instructions may also be loaded onto a computer or other programmable data processing terminal equipment to cause a series of operational steps to be performed on the computer or other programmable terminal equipment to produce a computer-implemented process, thereby providing instructions that execute on the computer or other programmable terminal equipment for implementing the process. Figure 1 One or more processes and / or boxes Figure 1 The steps of the function specified in one or more boxes.

[0044] It should also be noted that, in this document, relational terms such as "first" and "second" are used only to distinguish one entity or operation from another, and do not necessarily require or imply any such actual relationship or order between these entities or operations. The terms "comprising," "including," or any other variations thereof are intended to cover non-exclusive inclusion, such that a process, method, article, or terminal device that comprises a list of elements includes not only those elements but also other elements not expressly listed, or elements inherent to such a process, method, article, or terminal device. Without further limitation, an element defined by the phrase "comprising one..." does not exclude the presence of other identical elements in the process, method, article, or terminal device that includes said element.

[0045] Finally, it should be noted that the above description represents a preferred embodiment of the present invention. It should be pointed out that although preferred embodiments have been described, those skilled in the art, once they understand the basic inventive concept of the present invention, can make various improvements and modifications without departing from the principles described herein. These improvements and modifications should also be considered within the scope of protection of the present invention. Therefore, the appended claims are intended to be interpreted as including both the preferred embodiments and all changes and modifications falling within the scope of the embodiments of the present invention.

Claims

1. A rapid identification method for the growth years of ginseng based on terahertz time-domain spectroscopy, characterized in that, include: S1 preparation requires identification of ginseng growth years; S2 used a terahertz time-domain spectrometer to analyze ginseng samples; S3 collects terahertz spectral data of ginseng samples during the analysis process; S4 performs standardization processing on the acquired terahertz spectral data; S5 performs clustering and identification of ginseng samples based on terahertz spectral data; S6 uses an algorithm optimizer to optimize the parameters of the clustering algorithm; S7 evaluates the clustering algorithm's identification results using evaluation metrics.

2. The method for rapid identification of ginseng growth years based on terahertz time-domain spectroscopy as described in claim 1, characterized in that, S1 prepares ginseng samples that require identification of their growth period, wherein: The preparation process includes pulverizing, drying, and tableting. The pulverizing process is used to pulverize the ginseng sample into particles with a diameter of less than 200 mesh. The drying process is used to keep the moisture content of the ginseng particles below 5%. The tableting process is used to compress the ginseng particles into tablets that are easy to test. The tablet has a thickness of 1 mm, a diameter of 12 mm, and a weight of 200 mg.

3. The method for rapid identification of ginseng growth years based on terahertz time-domain spectroscopy as described in claim 1, characterized in that, The S2 method utilizes a terahertz time-domain spectrometer to analyze ginseng samples, wherein: The environmental conditions during the analysis process include relative humidity, ambient temperature, and detection space. The relative humidity is maintained below 1%RH, the ambient temperature is maintained at 18±2℃, and the detection space is maintained in a sealed environment.

4. The method for rapid identification of ginseng growth years based on terahertz time-domain spectroscopy as described in claim 1, characterized in that, S3 collects terahertz spectral data of ginseng samples during the analysis process, wherein: The terahertz spectral data uses the time-domain signal of the ginseng sample as the sample signal and the time-domain signal of dry nitrogen as the reference signal. By applying a fast Fourier transform to the sample signal and the reference signal, the time domain is converted to the frequency domain, resulting in two frequency domain signals. The terahertz spectral data is obtained by comparing the amplitude ratio and phase difference of two frequency domain signals to obtain the refractive index and absorption coefficient of the sample at different frequencies, and the absorbance spectrum of the sample is calculated by using the refractive index and absorption coefficient. The specific calculation process for the terahertz spectral data is as follows: ； ； In the formula, This represents the transmittance of a ginseng sample, which is the ratio between the refractive index and the absorption coefficient. The representation of the time-domain signal of a ginseng sample in the frequency domain; The time-domain signal of dry nitrogen is represented in the frequency domain. This represents the absorbance spectrum, i.e., terahertz spectral data.

5. The method for rapid identification of ginseng growth years based on terahertz time-domain spectroscopy as described in claim 1, characterized in that, S4 performs standardization processing on the acquired terahertz spectral data, wherein: The standardization process includes spectral preprocessing and feature reduction. The spectral preprocessing is used to address the problems of dielectric interference, surface scattering, and baseline drift in terahertz spectroscopy, while the feature reduction is used to reduce the dimensionality of the preprocessed data. The spectral preprocessing process is as follows: Based on the dynamic high-order nonlinear fitting mechanism, a dynamic reference is constructed using the moving local average spectrum. The residuals are fitted by third-order or higher polynomial regression. Local outliers are iteratively removed during the fitting process. At the same time, a hybrid residual fitting strategy is used to correct the original absorption curve and compensate for noise from moisture, particle distribution, and nonlinear scattering. The feature dimensionality reduction employs two methods: principal component analysis and unified manifold approximation mapping, to reduce the dimensionality of the preprocessed data.

6. The method for rapid identification of ginseng growth years based on terahertz time-domain spectroscopy as described in claim 1, characterized in that, S5 performs clustering and identification of ginseng samples based on terahertz spectral data, wherein: The clustering identification process is implemented using the mean-drift clustering algorithm. The mean-drift clustering algorithm avoids pseudo-convergence and overfitting small clusters by adopting a dynamic access matrix adjustment and density-driven center merging strategy, and introducing a dual convergence criterion. The steps of the merging strategy include initialization, iterative clustering, convergence judgment, center merging, and label assignment. The specific process of the initialization step is as follows: input the dataset, initial width r, convergence threshold ε, initialize the access count matrix and cluster center matrix, and mark all data points as unaccessed; The specific process of the iterative clustering step is as follows: calculate the Euclidean distance between each unvisited point and all points, filter out the neighboring points with a distance less than r2, and use the number of neighboring points as the current local density. Then update the cluster center to the mean of the neighboring points and generate a drift vector. The specific process of the convergence judgment step is as follows: the norm of the drift vector and the density change rate are used as two conditions to determine whether the clustering has converged. The specific process of the center merging and label allocation steps is as follows: the merging rule is that if the distance between the new center and the existing center is less than r2, then it is merged into a cluster with higher density, and the final category is determined according to the center with the most data point visits.

7. The specific process of the iterative clustering step as described in claim 6 is as follows: calculate the Euclidean distance between each unvisited point and all other points, filter out neighboring points with a distance less than r², and use the number of neighboring points as the current local density; then update the mean of the cluster center to its neighboring points, and generate a drift vector, characterized in that... in: The process of generating the drift vector is as follows: ； In the formula, This represents the displacement amount of the updated cluster centers, i.e., the drift vector; This indicates the number of neighboring points of the currently unvisited point; Indicates the geometric center location of a single cluster; This represents a sample point in the data space.

8. The method for rapid identification of ginseng growth years based on terahertz time-domain spectroscopy as described in claim 1, characterized in that, S6 employs an algorithm optimizer to optimize the parameters of the clustering algorithm, wherein: The algorithm optimizer is based on the SABO algorithm and balances global exploration and local development capabilities through a differentiated perturbation mechanism and an adaptive search strategy. The differentiated perturbation mechanism sorts and classifies individuals based on their fitness, and adopts different perturbation mechanisms according to the grouping. The grouping includes an elite group, a middle group, and a lagging group. Individuals in the elite group are subjected to small-amplitude Gaussian perturbation, individuals in the middle group are subjected to cross perturbation towards the elite individuals, and individuals in the lagging group are reset using the Levy flight method. The adaptive search strategy calculates population diversity and convergence indices, and determines whether the current stage is exploration or development based on population diversity and convergence thresholds. The search strategy in the exploration stage is to increase crossover probability, mutation intensity, and random initialization probability. The search strategy in the development stage is to decrease crossover probability and mutation intensity, and increase local search operations.

9. The method for rapid identification of ginseng growth years based on terahertz time-domain spectroscopy as described in claim 1, characterized in that, S7 evaluates the clustering algorithm's identification results using evaluation metrics, wherein: The evaluation metrics include the adjusted RAND index and classification accuracy. The adjusted RAND index is used to evaluate from the perspective of cluster consistency, and the classification accuracy is used to evaluate from the perspective of sample discrimination accuracy. The identification results are used to assign weight coefficients to the evaluation indicators and to determine the final evaluation result by calculating a weighted comprehensive score.

10. The evaluation indicators as described in claim 9 include the adjusted RAND index and classification accuracy, characterized in that, in: The specific calculation process for adjusting the RAND index is as follows: ； In the formula, This indicates an adjustment to the RAND Corporation index results; This represents the number of overlap pairs between the actual and predicted clusters; This represents the number of combinations of sample pairs within a real cluster; This represents the number of combinations of sample pairs within the predicted cluster; This represents the total number of combinations of sample pairs; The calculation process for the classification accuracy is as follows: ； In the formula, This indicates the classification accuracy result; Indicates the total number of samples; This represents the true label of the i-th sample; This represents the predicted label of the i-th sample; This indicates an indicator function, with a value of 1 when the prediction is correct and 0 otherwise.

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