Plant carotenoid determination method based on hyperspectrum and machine learning

Abstract

The invention belongs to the technical field of hyperspectral imaging and machine learning, and discloses a plant carotenoid rapid determination method based on hyperspectrum and machine learning. The measuring system comprises a hyperspectral imaging device, a spectrum preprocessing module, a characteristic wavelength extraction module, a machine learning modeling module and a carotenoid content prediction module which are sequentially connected through a data processing unit. According to the method, spectral data of a plant sample is obtained in combination with a hyperspectral imaging technology, key wavelengths are screened through spectrum preprocessing and a feature extraction algorithm, and a carotenoid content prediction model is established by using machine learning models such as random forest regression, partial least squares regression or a gradient elevator. According to the method, rapid and nondestructive detection of the plant carotenoid can be realized without a chemical reagent, the determination precision and efficiency are improved, and the defects of complex operation, long time consumption and strong destructiveness of a traditional detection method are overcome.

Description

Technical Field

[0001] The present invention belongs to the technical field of agricultural monitoring, and particularly relates to a method for estimating carotenoid content by machine learning and hyperspectral features. Background Technique

[0002] Carotenoids: A general term for a class of important natural pigments, belonging to compounds. Yellow, orange-red or red pigments that are commonly present in animals, higher plants, fungi, algae, and bacteria. The content of carotenoids can be used as a biochemical index for plants' response to drought stress. Under drought conditions, plants will undergo a series of physiological and biochemical changes, including a decrease in photosynthesis efficiency, the accumulation of reactive oxygen species, and the activation of the antioxidant system. Carotenoids play an important role in the light protection mechanism of plants. They can quench excess excitation energy and prevent light damage in photosynthesis. Therefore, by monitoring the change in carotenoid content, the drought stress state of plants can be indirectly evaluated.

[0003] Currently, many scholars, based on hyperspectral technology, combine different spectral processing methods with different machine learning prediction models to establish inversion models for monitoring carotenoid content in different regions.

[0004] Spectral data preprocessing algorithms: The carotenoid content in plants is affected by various factors. Its spectral signal is obtained by comprehensively superimposing the reflectance of plant leaves, stems, and other parts. The spectral information often contains feedback information of various factors such as moisture, nitrogen content, and chlorophyll. These factors will cause the plant spectrum to be non-specific, thus affecting the accuracy and precision of measuring carotenoid content through spectral data. In addition, due to the influence of factors such as the acquisition environment and instrument noise, the hyperspectral data obtained in the laboratory often contains noise. To improve the reliability and precision of carotenoid content prediction, scholars at home and abroad have carried out a large number of studies on spectral preprocessing and denoising methods. Research shows that preprocessing methods such as spectral denoising, spectral transformation methods, and characteristic band selection can effectively improve the accuracy of spectral data. Commonly used denoising methods include the moving average method, Savitzky-Golay smoothing method, etc.; spectral transformation methods include differential transformation, wavelet transformation, etc.; feature selection methods include principal component analysis, correlation analysis, etc.

[0005] Inversion model: Since the birth of hyperspectral remote sensing technology, a large number of experts and scholars at home and abroad have conducted in-depth research on the modeling and inversion methods for the content of carotenoids in plants. With the continuous development of technology, the modeling methods have been gradually improved, and the model accuracy has been continuously enhanced. Currently, the commonly used modeling and inversion methods include multiple stepwise regression, partial least squares regression, support vector machine, backpropagation neural network, random forest, convolutional neural network, etc. These studies indicate that near-infrared spectroscopy technology can significantly improve the efficiency of measuring carotenoid content in plants while maintaining high accuracy.

[0006] Although the existing near-infrared spectroscopy technology has made certain progress in measuring the carotenoid content in plants, there are still the following drawbacks:

[0007] Spectral interference problem: Hyperspectral data is usually interfered by other components such as water, chlorophyll, and nitrogen, resulting in the spectral signal of carotenoids being mixed. This interference reduces the prediction accuracy of carotenoid content.

[0008] Spectral noise: During the actual acquisition process, due to factors such as the noise of the instrument itself and environmental conditions, the spectral data may contain noise, which in turn affects the stability and accuracy of the model.

[0009] High data dimension: Hyperspectral data has a high dimension and complex feature information, which poses higher requirements for data processing and feature selection. The high dimension may lead to an increase in redundant information, affecting the training efficiency and prediction performance of the model.

[0010] Sample variability: The spectral characteristics of carotenoids in different plant species and at different growth stages vary greatly, resulting in the model being difficult to generalize to new plants or environments under certain conditions, affecting its wide application.

[0011] Complexity of preprocessing and modeling: There is still a certain degree of complexity in the preprocessing of hyperspectral data and the construction of the model. In particular, how to effectively remove noise, select characteristic bands, and select appropriate modeling methods remains a challenge in the research.

[0012] To solve the above problems, the present invention proposes an inversion method for the content of carotenoids in plants based on near-infrared hyperspectrum. Summary of the Invention

[0013] The purpose of the present invention is to propose an inversion method for the content of plant carotenoids based on hyperspectral technology to solve the problems raised in the background technology:

[0014] The existing hyperspectral technology still has problems such as complex data preprocessing, difficult feature selection, and insufficient model generalization ability in measuring the carotenoid content in plants.

[0015] To achieve the above object, the present invention adopts the following technical solutions:

[0016] A method for inverting carotenoid content based on near-infrared hyperspectral includes the following steps:

[0017] S1: Set the collection points and collection schemes of plant samples, and collect leaf samples of different types of plants.

[0018] S2: Conduct biochemical experiments on the collected plant leaf samples to obtain the measured values of carotenoids; meanwhile, collect the spectral data of plant leaves through a hyperspectral camera, and perform preprocessing, noise removal, and characteristic wavelength selection on the data.

[0019] S3: Optimize the random forest model based on the Bayesian optimization algorithm, establish a prediction model for the carotenoid content in plant leaves, and evaluate and optimize the model.

[0020] S4: Conduct data tests based on the optimized prediction model for plant carotenoid content, and evaluate the prediction accuracy and stability of the model in different plant samples.

[0021] Preferably, the types of plant samples collected in S1 include chrysanthemum, carrot, and spinach

[0022] Preferably, in S2, the carotenoid spectral reflectance is measured based on a hyperspectral camera with a spectral range of 900 nanometers to 1700 nanometers.

[0023] Preferably, in the preprocessing step of S2, conduct biochemical experiments on the collected plant samples to obtain the measured values of carotenoids; denoise the original spectrum based on the Savitzky-Golay smoothing method, and further preprocess the spectral data of plant leaves by using several differential transformation methods.

[0024] Preferably, in the preprocessing step of S2, conduct biochemical experiments on the sampled plant samples to obtain the measured values of carotenoids; denoise the original spectrum based on the Savitzky-Golay smoothing method, and further preprocess the plant spectral data by using first-order differential and second-order differential.

[0025] Preferably, the characteristic wavelength selection process in S2 is based on the competitive adaptive reweighted sampling method to find the optimal variable subset of spectral data, specifically as follows: The competitive adaptive reweighted sampling method calculates the regression coefficients of each wavelength variable through stepwise regression, selects the wavelength most relevant to the carotenoid content as the characteristic band; through multiple iterations of screening, finally obtains the variable subset with the minimum cross-validation error as the optimal characteristic band.

[0026] Preferably, in step S3, a prediction model of carotenoid content is established by simulating the non-linear relationship between carotenoid content and plant growth conditions, environmental factors, and other related biological characteristics based on a machine learning algorithm; wherein the structure and parameters of the machine learning algorithm are optimized based on the Bayesian algorithm; the specific Bayesian algorithm is as follows:

[0027] The Bayesian optimization algorithm starts the optimization process by initializing the prior distribution in the search space, combining the performance metrics of historical sampling points, and establishing a probability model of the objective function through a Gaussian process, as follows:

[0028] Initialization: First, select some initial points for sampling, usually randomly or based on prior knowledge. Set the initial points as {x1, x2, …, x n}, and obtain the objective value y i = f(x i )

[0029] The value corresponding to the objective function f(x) here is the actually measured response value.

[0030] Fitting the surrogate model: Bayesian optimization usually uses a Gaussian process as the surrogate model for estimation.

[0031] The objective function f(x). Gaussian process modeling assumes that the objective function is a function obtained from a sample drawn from a Gaussian process distribution. The Gaussian process is defined by the following formula:

[0032] f(x) ∼ GP(μ(x), k(x, x′))

[0033] where μ(x) is the mean function, usually assumed to be 0 or some prior estimate. k(x, x′) is the covariance function, used to measure the similarity between different points. After obtaining the values of the objective function, we use Bayesian inference methods (such as maximum a posteriori estimation) to update the model and obtain p(f(x) | {x i , y i}), that is, the conditional probability distribution of the objective function value under the existing data.

[0034] Optimizing the acquisition function: The core of Bayesian optimization lies in selecting the next sampling point through the acquisition function. Commonly used acquisition functions include expected improvement, probability of improvement, and upper confidence bound. Let the acquisition function be, and its goal is to select the point that maximizes this function, that is:

[0035]

[0036] The forms of different acquisition functions are different. The following are the expressions of several common acquisition functions:

[0037] Desired improvement: EI(x) = E[max(f(x + ) - f(x), 0)] where f(x + ) is the historical optimal value and f(x) is the predicted value at the current point. The expectation is an estimate of the potential improvement of the objective function.

[0038] Probability of improvement: where Φ is the cumulative distribution function of the standard normal distribution and σ(x) is the predicted standard deviation at the current point x.

[0039] Upper confidence bound: UCB(x) = μ(x) + κσ(x)f(x) where μ(x) is the mean prediction of the Gaussian process and σ(x) is a hyperparameter used to control the balance between exploration and exploitation. At each step of optimization, the point xn+1 that maximizes the acquisition function is selected as the next sampling point.

[0040] Update the model: After each sampling, new data (x n+1 , f(x n+1 ) is obtained. These new data points are added to the training set and the Gaussian process model is updated. The updated model will predict the objective function more accurately.

[0041] Repeat iteration: Repeat the steps of optimizing the acquisition function and updating the model until the termination condition is met, such as reaching the maximum number of samplings or finding a satisfactory solution.

[0042] Compared with the prior art, the present invention provides a method for inverting carotenoid content based on near-infrared hyperspectral, having the following beneficial effects:

[0043] The present invention optimally selects the near-infrared spectral range, precisely focuses on the key characteristic bands closely related to the changes in plant carotenoids, effectively reduces redundant information, improves the signal-to-noise ratio, and thus enhances the accuracy and reliability of the determination. The differential transformation method is used to preprocess the spectral data, which can effectively eliminate noise and interference factors, enhance the key details in the spectral signal, improve the data quality, and lay a solid foundation for subsequent analysis. Combining with an efficient feature selection algorithm, such as competitive adaptive reweighted sampling, the most representative characteristic bands are selected from a large amount of spectral data to enhance the generalization ability of the model. Further, a variety of machine learning algorithms are used and combined with a parameter optimization strategy to optimize the model structure, making it applicable to different plant samples and improving the accuracy and stability of carotenoid content prediction. Brief description of the drawings

[0044] Figure 1 is the flowchart of the method mentioned in Embodiment 1 of the present invention;

[0045] Figure 2 is the schematic diagram of the spectral device acquisition device mentioned in Embodiment 1 of the present invention;

[0046] Figure 3 This is the flowchart of the Bayesian optimization algorithm mentioned in Embodiment 1 of the present invention.

[0047] Figure 4 This is the spectrogram and the corresponding preprocessed graph mentioned in Embodiment 1 of the present invention. Detailed implementation manners

[0048] Next, the technical solutions in the embodiments of the present invention will be clearly and completely described in conjunction with the accompanying drawings in the embodiments of the present invention. Obviously, the described embodiments are only a part of the embodiments of the present invention, rather than all the embodiments.

[0049] The present invention optimally selects key bands for near-infrared spectral data, accurately locks the characteristic regions closely related to the change of carotenoid content in plants, reduces the interference of irrelevant information, improves the signal-to-noise ratio, and thus enhances the accuracy and stability of measurement. By using differential transformation preprocessing technology, noise and interference factors are effectively removed, the subtle changes in spectral signals are highlighted, and the data quality is improved, laying a solid foundation for subsequent analysis. Combining with the efficient feature selection algorithm - competitive adaptive reweighted sampling algorithm, the most representative characteristic wavelengths are extracted from large-scale spectral data, enhancing the prediction ability of the model. In addition, by combining multiple machine learning algorithms and parameter optimization methods, the model structure and hyperparameter configuration are optimized to make it applicable to different plant samples, improving the accuracy and generalization ability of carotenoid content prediction. The specific contents are as follows.

[0050] Embodiment 1:

[0051] Please refer to Figures 1-3 , a method for inverting carotenoid content based on near-infrared hyperspectrum of the present invention, includes the following steps:

[0052] S1: Determine the plant sampling scheme and implement the collection. Specifically as follows:

[0053] A total of 277 chrysanthemums were selected in the experiment, of which 222 were used to train the model and 55 were used to test the model. Every two weeks, 36 seedlings were randomly selected for hyperspectral data collection and used to measure carotenoids.

[0054] S2: Conduct biochemical experiments on the collected plant samples to measure the actual content of carotenoids; at the same time, use a hyperspectral camera to collect the spectral data of the plant samples, and perform data preprocessing and characteristic wavelength selection processing. Specifically as follows:

[0055] The carotenoids in chrysanthemums were determined by spectrophotometry. The collected chrysanthemum petal samples were shredded and weighed (0.03 g - 0.055 g), and then placed into 2-ml centrifuge tubes numbered from 1 to 100. 1600 μl of 95% ethanol solution was added to all samples in sequence, and they were placed in an incubator at 37 °C for 24 hours, with the whole process carried out in the dark. After standing, all centrifuge tubes were centrifuged at 12,000 rpm for 10 minutes, and 2 μl of the supernatant was taken and added to the microplate. The microplate was placed in a microplate reader, and the absorbance was measured at 665 nm, 649 nm, and 470 nm respectively to remove systematic errors. Finally, the carotenoid content was calculated from the absorbance values at the three wavelengths, and the carotenoid concentration of each sample was calculated based on the formula. The formula is as follows

[0056] Ca = 13.95xA 665 -6.68xA 669 #

[0057]

[0058] Substitute Cx into the pigment concentration in the formula to calculate the carotenoid value of each label.

[0059] Process of hyperspectral acquisition: The hyperspectral data was acquired using a Dualix Spectral Imaging hyperspectral camera. The spectral range of the near-infrared hyperspectral imager is from 870 to 1,720 nm, with a spatial resolution of 640 pixels, 512 bands, and a spectral resolution of 5 nm. The spectral equipment acquisition device is as Figure 2 shown. The size of the lifting platform is 300 mm × 300 mm, and the allowable lifting range is between 90 and 370 mm. To ensure high-quality hyperspectral images of the samples, the conveyor belt was set to move at a speed of 0.6 cm / s for 25 cm. The distance from the sample to the lens was maintained at 30 cm, the angle between the light source and the horizontal plane was set at 60 degrees, and the exposure time was 7 ms. To avoid the influence caused by uneven light source intensity distribution and dark current during the image acquisition process, a white reference image (W) was obtained from the white reference plate, and a dark reference image (D) was obtained by completely closing the lens of the camera with its opaque cover. Then, based on ENVI5.3 software, first, chrysanthemum leaves were selected as the region of interest (ROI) to extract spectral data. The specific steps were as follows: The hyperspectral data of the chrysanthemum samples were corrected using the hyperspectral data of a white board with a reflectivity of 93% to obtain the corrected hyperspectral data of the chrysanthemums. The "ROL Tool" was used on the corrected hyperspectral data to select the entire individual of the chrysanthemum in the hyperspectral data, and this region was the region of interest. The selected region of interest was used to obtain the average spectrum of the whole chrysanthemum through the "Stats" option, and this spectrum was the hyperspectral reflectance data of this chrysanthemum.

[0060] The preprocessing of plant spectral data can reduce the influence of factors such as noise, stray signals, and non-linear spectral effects in plant spectral data, and improve the accuracy and reliability of the data. This preprocessing method can perform non-destructive analysis on plant samples, and at the same time can provide more efficient, accurate, and comprehensive data information for plant analysis, thus providing important support and guarantee for plant science research. In this embodiment, the Savitzky-Golay smoothing method is first used to denoise the original spectrum, and then a variety of differential transformation methods are further used to preprocess the plant spectral data to further improve the correlation between the reflectance data and the plant indicators. Two differential transformation methods are adopted, including the first-order differential and the second-order differential. Among them, log is based on e, representing the spectral data after competitive adaptive reweighted sampling smoothing, and x' and x'' represent taking the first-order differential and the second-order differential respectively.

[0061] The competitive adaptive reweighted sampling method has significant advantages in dealing with high-dimensional data and complex problems. In the process of feature selection and sample resampling, the competitive adaptive reweighted sampling method can better capture the internal structure of the data and the relationship between features. By automatically adjusting the sample weights, the competitive adaptive reweighted sampling method can effectively handle unbalanced or noisy data sets, and at the same time quickly converge to the global optimal solution. While retaining the scale of the smaller sample set, it maximally retains the information of the original data, thereby improving the prediction performance and generalization ability of the model.

[0062] The calculation process of using the competitive adaptive reweighted sampling method to find the optimal variable subset of spectral data contains n loop processes. In each loop, the competitive adaptive reweighted sampling method first calculates the correlation of each wavelength variable, and performs feature selection through competitive adaptive reweighted sampling. Through the competition and adaptive adjustment of variables, the competitive adaptive reweighted sampling method can effectively screen out the characteristic bands with strong correlation with the target variable. After multiple loop screenings, the final variable subset is obtained, and the final variable subset is regressed through the partial least squares regression model to obtain its root mean square error of cross-validation. Finally, the subset with the minimum root mean square error of cross-validation value is selected as the optimal variable subset, that is, the optimal characteristic band.

[0063] S3: Establish a prediction model for carotenoid content based on the Bayesian algorithm and random forest and evaluate and optimize the model; specifically as follows:

[0064] The structure and parameters of the random forest model are optimized based on the Bayesian algorithm; the specific Bayesian algorithm is as follows:

[0065] The Bayesian optimization algorithm starts the optimization process by initializing the prior distribution in the search space, combining the performance metrics of historical sampling points, and establishing a probability model of the objective function through a Gaussian process, as follows: Initialization: First, select some initial points for sampling, usually randomly or based on prior knowledge. Set the initial points as {x1, x2, …, x n}, and obtain the objective values y i = f(x i ) through the objective function f(x). The value corresponding to the objective function f(x) here is the actually measured response value.

[0066] Fitting the surrogate model: Bayesian optimization usually uses a Gaussian process as a surrogate model to estimate the objective function f(x). The Gaussian process modeling assumes that the objective function is a function whose samples are drawn from a Gaussian process distribution. The Gaussian process is defined by the following formula:

[0067] f(x) ∼ GP(μ(x), k(x, x′))

[0068] where μ(x) is the mean function, usually assumed to be 0 or some prior estimate. k(x, x′) is the covariance function, which is used to measure the similarity between different points. After obtaining the values of the objective function, we use Bayesian inference methods (such as maximum a posteriori estimation) to update the model and obtain p(f(x) | {x i , y i}), that is, the conditional probability distribution of the objective function value under the existing data.

[0069] Optimizing the acquisition function: The core of Bayesian optimization lies in selecting the next sampling point through the acquisition function. Commonly used acquisition functions include expected improvement, probability of improvement, and upper confidence bound. Let the acquisition function be, and its goal is to select the point that maximizes this function, that is:

[0070]

[0071] The forms of different acquisition functions are different. The following are the expressions of several common acquisition functions:

[0072] Expected improvement: EI(x) = E[max(f(x + ) - f(x), 0)] where f(x + ) is the historical optimal value, f(x) is the predicted value of the current point, and the expectation is an estimate of the potential improvement of the objective function.

[0073] Probability of improvement: where Φ is the cumulative distribution function of the standard normal distribution, and σ(x) is the predicted standard deviation at the current point x.

[0074] Upper Confidence Bound: UCB(x) = μ(x) + κσ(x)f(x), where μ(x) is the mean prediction of the Gaussian process, and σ(x) is a hyperparameter used to control the balance between exploration and exploitation. At each optimization step, the point xn+1 that maximizes the acquisition function is selected as the next sampling point.

[0075] Update the model: After each sampling, new data (x n+1 , f(x n+1 ) is obtained. These new data points are added to the training set, and the Gaussian process model is updated. The updated model will predict the objective function more accurately.

[0076] Repeat the iteration: Repeat steps 3 and 4 until the termination condition is met, such as reaching the maximum number of samplings or finding a satisfactory solution.

[0077] Since Bayesian optimization has significant advantages in optimizing model parameters, especially when finding suitable hyperparameters, it can continuously update the understanding of the unknown function through a probabilistic model to find the optimal solution. Therefore, in this embodiment, the Bayesian optimization method is used to optimize the parameters of the random forest model (the number of generated trees and the maximum depth of the tree). First, set the initial exploration range of Bayesian optimization and configure relevant parameters (such as sampling strategy, convergence criterion, balance between exploration and exploitation, etc.).

[0078] Bayesian optimization predicts the behavior of the objective function by constructing

[0079] a surrogate model (such as Gaussian process regression) and selects the next sampling point by maximizing the acquisition function. The iterative process continues until the maximum number of iterations or the convergence condition is reached. Finally, the model after Bayesian optimization is evaluated and optimized based on indicators such as Pearson correlation coefficient and root mean square error, and the best parameter combination is selected to construct the optimal Bayesian optimization random forest model.

[0080] S4: Conduct data testing based on the optimized prediction model of carotenoid content.

[0081] Table 1 Influence of different preprocessing methods on carotenoid prediction

[0082]

[0083] Table 2 Prediction results of different models for carotenoid

[0084]

[0085] The above are only the preferred specific embodiments of the present invention, but the protection scope of the present invention is not limited thereto. Any person skilled in the art within the technical scope disclosed by the present invention, according to the technical solution and inventive concept of the present invention, making equivalent substitutions or changes should be covered within the protection scope of the present invention.

Claims

1. A method for detecting carotenoid content in plants based on hyperspectral and machine learning, characterized in that: The following steps are involved: S1: Set up plant sampling points and collection plans to collect plant samples of different types; S2: Conduct biochemical experiments on the sampled plant samples to obtain the measured values ​​of plant carotenoids; also collect spectral data of plant samples through a hyperspectral camera, and perform preprocessing and characteristic wavelength selection processing; S3: Establish a prediction model for plant carotenoid content based on Bayesian algorithm and random forest model and evaluate and optimize the model; S4: Data testing based on the optimized prediction model of plant carotenoid content.

2. The method for inverting carotenoid content in plants based on near infrared spectroscopy according to claim 1, characterized in that: The plant samples collected in S1 include chrysanthemum, carrot and spinach.

3. The method for inverting carotenoid content in plants based on near-infrared hyperspectroscopy according to claim 1, characterized in that: In S2, the spectral reflectance of plants is measured by collecting data using a hyperspectral camera with a spectral range of 900 nanometers to 1700 nanometers.

4. The method for inverting carotenoid content in plants based on near-infrared hyperspectroscopy according to claim 1, characterized in that: In the preprocessing step of S2, biochemical experiments are carried out on the sampled plant samples to obtain the measured values ​​of carotenoids in the plants; the original spectrum is denoised based on the Savitzky-Golay-smoothing method, and the plant spectrum data is further preprocessed using several differential transformation methods.

5. The method for inverting carotenoid content in plants based on near infrared spectroscopy according to claim 4, characterized in that: The differential transformation method includes first-order differential and second-order differential.

6. The method for inverting carotenoid content in plants based on near-infrared hyperspectroscopy according to claim 1, characterized in that: The characteristic wavelength selection process in S2 is based on the competitive adaptive reweighted sampling method to find the optimal variable subset of spectral data, which is as follows: the competitive adaptive reweighted sampling method calculates the regression coefficient of each wavelength variable by stepwise regression, and selects the wavelength most relevant to the carotenoid content as the characteristic band; through multiple iterative screening, finally obtains the variable subset with the minimum cross-validation error as the optimal characteristic band.

7. Preferably, in S3, a nonlinear relationship between carotenoid content and plant growth conditions, environmental factors and other relevant biological characteristics is simulated based on a machine learning algorithm to build a prediction model for carotenoid content; wherein the structure and parameters of the machine learning algorithm are optimized based on a Bayesian algorithm; the Bayesian algorithm is specifically as follows: the Bayesian optimization algorithm initializes the prior distribution in the search space, combines the performance indicators of the historical sampling points, and establishes a probability model of the objective function through a Gaussian process to start the optimization process, as follows: Initialization: First, select some initial points for sampling, usually randomly or based on prior knowledge. Set the initial points to {x1,x2,…,x n }, and obtain the target value y through the objective function f(x) i =f(x i ). The value corresponding to the objective function f(x) here is the actual measured response value. Fitting a surrogate model: Bayesian optimization often uses a Gaussian process as a surrogate model to estimate the target function f(x). Gaussian process modeling assumes that the target function is a function of a sample obtained from a Gaussian process distribution. The Gaussian process is defined by the following formula: f(x)~GP(μ(x),k(x,x′)) where μ(x) is the mean function, which is usually assumed to be zero or some a priori estimate. k(x,x ′ ) is the covariance function, which is used to measure the similarity between different points. After obtaining the value of the objective function, we use Bayesian inference methods (such as maximum a posteriori estimation) to update the model and obtain p(f(x)|{x i ,y i }), which is the conditional probability distribution of the objective function value under the existing data. Optimizing the acquisition function: The core of Bayesian optimization is to select the next sampling point through the acquisition function. Commonly used acquisition functions include expected improvement, probability improvement, and upper confidence bound. Let the acquisition function be its goal to select the point that can maximize the function, that is: Different acquisition functions have different forms. The following are expressions of several common acquisition functions: Expected improvement: EI(x) = E[max(f(x + )-f(x),0)]where f(x + ) is the historical optimal value, f(x) is the predicted value at the current point, and expectation is an estimate of the potential improvement of the objective function. Probability Improvement: Where Φ is the cumulative distribution function of the standard normal distribution and σ(x) is the predicted standard deviation at the current point x. Upper confidence bound: UCB(x) = μ(x) + κσ(x)f(x) where μ(x) is the mean prediction of the Gaussian process and σ(x) is a hyperparameter used to control the balance between exploration and exploitation. In each optimization step, the point xn+1 that maximizes the acquisition function is selected as the next sampling point. Update model: After each sampling, new data (x n+1 ,f(x n+1 ), add these new data points to the training set, and update the Gaussian process model. The updated model will predict the target function more accurately. Iteration: Repeat the steps of optimizing the acquisition function and updating the model until the termination condition is met, such as reaching the maximum number of sampling times or finding a satisfactory solution.

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