CN113255215A - Lithium battery health state estimation method based on voltage segments - Google Patents

Lithium battery health state estimation method based on voltage segments Download PDF

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CN113255215A
CN113255215A CN202110546304.8A CN202110546304A CN113255215A CN 113255215 A CN113255215 A CN 113255215A CN 202110546304 A CN202110546304 A CN 202110546304A CN 113255215 A CN113255215 A CN 113255215A
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lithium battery
health state
voltage
value
kernel function
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CN113255215B (en
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孟锦豪
黄焕炀
蔡磊
刘平
刘天琪
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Sichuan University
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    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06F—ELECTRIC DIGITAL DATA PROCESSING
    • G06F30/00—Computer-aided design [CAD]
    • G06F30/20—Design optimisation, verification or simulation
    • G06F30/27—Design optimisation, verification or simulation using machine learning, e.g. artificial intelligence, neural networks, support vector machines [SVM] or training a model
    • G—PHYSICS
    • G01—MEASURING; TESTING
    • G01R—MEASURING ELECTRIC VARIABLES; MEASURING MAGNETIC VARIABLES
    • G01R31/00—Arrangements for testing electric properties; Arrangements for locating electric faults; Arrangements for electrical testing characterised by what is being tested not provided for elsewhere
    • G01R31/36—Arrangements for testing, measuring or monitoring the electrical condition of accumulators or electric batteries, e.g. capacity or state of charge [SoC]
    • G01R31/367—Software therefor, e.g. for battery testing using modelling or look-up tables
    • G—PHYSICS
    • G01—MEASURING; TESTING
    • G01R—MEASURING ELECTRIC VARIABLES; MEASURING MAGNETIC VARIABLES
    • G01R31/00—Arrangements for testing electric properties; Arrangements for locating electric faults; Arrangements for electrical testing characterised by what is being tested not provided for elsewhere
    • G01R31/36—Arrangements for testing, measuring or monitoring the electrical condition of accumulators or electric batteries, e.g. capacity or state of charge [SoC]
    • G01R31/392—Determining battery ageing or deterioration, e.g. state of health
    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06F—ELECTRIC DIGITAL DATA PROCESSING
    • G06F2111/00—Details relating to CAD techniques
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    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06F—ELECTRIC DIGITAL DATA PROCESSING
    • G06F2119/00—Details relating to the type or aim of the analysis or the optimisation
    • G06F2119/04—Ageing analysis or optimisation against ageing

Abstract

The invention discloses a lithium battery health state estimation method based on voltage segments, which can accurately predict the health state of a retired power lithium battery. According to the method, the empirical model and the data driving model are combined, the mode that the estimated empirical model is converted into the data driving model kernel function is realized by depending on the charge-discharge cycle times of the lithium battery, the battery electrochemical characteristics of the empirical model are integrated into the data driving model, and the estimation accuracy of the health state of the lithium battery is improved.

Description

Lithium battery health state estimation method based on voltage segments
Technical Field
The invention belongs to the technical field of lithium batteries, and particularly relates to a lithium battery health state estimation method based on voltage segments.
Background
The power lithium battery has the advantages of long service life, high energy density, no pollution, low self-discharge rate and the like, and is widely used in the current electric transportation and energy storage fields. However, each field has strict requirements on the state of health of the power battery in use. How to accurately evaluate the health state of the power battery and timely retire the aged power battery has important significance for maintaining the normal work of equipment and recovering the power battery in a gradient manner.
A large number of off-line tests are carried out to establish an empirical model of the power lithium battery, and the method is a widely proposed method for estimating the health state of the power lithium battery. However, the testing process of this kind of method depends on a lot of off-line tests, and the electrochemical characteristics of different types of batteries have certain differences, and even the same type of battery may have performance deviation due to different specific design methods of the battery system. And then the factory inconsistency of the lithium battery and the deviation of the aging test process caused by the manufacturing process are comprehensively considered, the accuracy of the empirical model is limited by the determined battery type, and the effectiveness of different batteries is still further discussed.
The lithium battery experience model summarized by offline test is a widely researched health state estimation method of a power lithium battery, and the experience model contains electrochemical characteristic information of the lithium battery and can reflect an aging track of the lithium battery. The estimation of the state of health of the battery by the empirical model is mainly based on the number of charge and discharge cycles of the battery. However, even for the same type of lithium battery, the aging condition of the lithium battery is different according to different use scenes; moreover, for the lithium battery needing to be retired, the charging and discharging times are difficult to quantify. Therefore, the estimation accuracy of the empirical model for the health state of the lithium battery in an actual scene is low.
The data driving model based on the data is suitable for various use scenes of the battery, and the health state of the lithium battery can be accurately predicted by the data driving model through testing the characteristic information of the lithium battery on site. However, since the data-driven model only depends on the characteristics of the data information, the model is built without considering the characteristics of the battery itself, and the prediction capability of the data model is limited.
In the data-driven model, the kernel function plays a decisive role in the model accuracy. In order to further improve the estimation accuracy, the invention provides a method for integrating an empirical model and a data driving model, the battery electrochemical characteristics of the empirical model are integrated into the data driving model by converting the empirical model which depends on the lithium battery charge-discharge cycle number to realize estimation into a data driving model kernel function, and the characteristics of the two models are integrated to realize the combination of the two models, so that the accurate estimation of the health state of the lithium battery is facilitated.
Disclosure of Invention
Aiming at the defects in the prior art, the lithium battery health state estimation method based on the voltage segment provided by the invention solves the problem that the estimation precision of the lithium battery health state in the prior art is low.
In order to achieve the purpose of the invention, the invention adopts the technical scheme that: a lithium battery health state estimation method based on voltage segments comprises the following steps:
s1, intercepting a voltage segment from a charging voltage curve of the lithium battery, and acquiring aging characteristics of the lithium battery according to the voltage segment;
s2, converting the empirical model into a kernel function based on the number of charge and discharge cycles of the lithium battery;
s3, collecting health state data of lithium battery aging characteristics as training data;
s4, introducing the kernel function into the Gaussian process regression model, and training the Gaussian process regression model by adopting the lithium battery aging characteristics and the corresponding health state data;
and S5, inputting the aging characteristics of the lithium battery with the service life to be estimated into the trained Gaussian process regression model to obtain the estimation result of the health state of the lithium battery.
Further, the step S1 is specifically:
s1.1, setting a voltage starting point UstartAnd a segment time length l cut off on the voltage curve of the constant-current charging processThe starting point is UstartAnd a voltage segment u of length l;
wherein u ═ u1,u2,...,ul]，u1,u2,...,ulAll represent voltage points, u, in a voltage segment1＝Ustart；
S1.2, giving the number of main components, and performing main component analysis on the voltage section u to obtain the aging characteristic of the lithium battery;
s1.3, collecting the aging characteristics of a plurality of lithium batteries according to the method of the steps S1.1-S1.2;
s1.4, collecting the lithium battery health state value corresponding to each aging characteristic, and taking out a group of data as a test sample, and taking the other data as training samples;
s1.5, training a Gaussian process regression model by adopting a gradient descent method according to the training samples;
s1.6, inputting the aging characteristics of the test sample into a Gaussian process regression model to obtain an estimated value of the health state of the lithium battery;
s1.7, comparing the lithium battery health state estimated value with the lithium battery health state value of the test sample to obtain prediction precision;
s1.8, according to the method of the steps S1.4-S1.7, each group of data is used as a test sample, a plurality of prediction precisions are obtained, and an average value is taken as a final prediction precision;
s1.9, resetting a voltage starting point UstartAnd the segment time length l, and obtaining the final prediction precision according to the method of the steps S1.1-S1.8;
and S1.10, judging whether the final prediction precision obtained at the next time is greater than that of the previous time, if so, taking the parameter of the Gaussian process regression model at the moment as an initial parameter, and storing the battery aging characteristic and the corresponding lithium battery health state value, otherwise, returning to the step S1.9.
Further, the step S1.2 is specifically:
s1.21, giving the number N of principal components, and performing N-dimensional linear transformation on the voltage segment u, wherein the N-dimensional linear transformation specifically comprises the following steps:
Figure BDA0003073628940000031
Figure BDA0003073628940000041
Figure BDA0003073628940000042
wherein x isiThe principal component after the ith dimensionality reduction, i.e., the ith aging characteristic,
Figure BDA0003073628940000043
denotes a unit vector, T denotes transpose, i and j denote different aging characteristics, i 1,2,.. and N, j 1, 2.. and N,
Figure BDA0003073628940000044
denotes that the voltage segment u ═ u1,u2,...,ul]Dimensionality reduction to the ith principal component xiThe corresponding weight vector.
S1.22, obtaining the aging characteristics of the N batteries according to the method described in step S1.21, and obtaining an aging characteristic of x ═ x1,x2,...,xN]。
Further, the step S2 is specifically:
s2.1, establishing a relation between the health state of the lithium battery and the number of charge-discharge cycles based on the number of charge-discharge cycles of the lithium battery as follows:
Figure BDA0003073628940000045
wherein f (·) is a nonlinear equation with two independent variables, S represents the capacity of the lithium battery, C represents the charge-discharge cycle number of the lithium battery, and d represents the differential;
s2.2, carrying out Taylor expansion on two independent variables in the relational expression in the step S2.1, specifically:
Figure BDA0003073628940000046
wherein, a1And a2Respectively representing attenuation factors and fatigue damage accumulation factors;
and S2.3, solving the step S2.2 according to the condition that when the lithium battery is not used, C is 0 and S is 100% to obtain an empirical model as follows:
S＝k1C+k2eαC+1-k2
wherein k is1Denotes a first unknown variable, k2Representing a second unknown variable, and alpha representing a third unknown variable;
s2.4. adopting square | | · | | of Euclidean distance in low-dimensional space2Replacing the charging and discharging cycle number C, converting the empirical model into a kernel function, specifically:
Figure BDA0003073628940000051
where k (-) represents the kernel function, i.e., the distance of two sets of features in the high-dimensional space; | l | · | represents the euclidean distance between two observation points, x and x' represent two different sets of features, respectively, θ1Denotes a first hyperparameter, θ2Denotes a second hyperparameter, θ3Represents a third hyperparameter;
s2.5, collecting the cycle number and the health state value of the lithium battery, importing the cycle number and the health state value into an empirical model, and acquiring parameter values of the empirical model by adopting a least square fitting method, wherein the method specifically comprises the following steps:
Figure BDA0003073628940000052
where δ (-) denotes the sum of the squared errors of the predicted and true values under the corresponding parameters, Si'Represents the real value of the health state of the lithium battery in the ith' cycle,
Figure BDA0003073628940000053
representing the predicted value of the health state of the lithium battery in the ith' cycle under the corresponding parameters;
and S2.6, acquiring an initial value of the hyperparameter in the kernel function according to the parameter value of the empirical model and the relation between the empirical model and the kernel function.
Further, the step S4 is specifically:
s4.1, introducing a kernel function into a Gaussian process regression model;
and S4.2, optimizing the hyper-parameters of the kernel function by adopting a negative logarithm maximum likelihood estimation function according to the aging characteristics of the lithium battery and the corresponding health state data thereof, and obtaining an optimized Gaussian process regression model.
Further, the negative log-maximum likelihood estimation function L (θ) in step S4.2 is specifically:
Figure BDA0003073628940000054
Figure BDA0003073628940000055
wherein theta represents a set of hyper-parameters to be optimized in the kernel function,
Figure BDA0003073628940000056
representing a noise term, InRepresenting a diagonal matrix; tr (-) represents the trace of the matrix, y represents the output value of the regression model of the Gaussian process, and K represents an intermediate parameter.
The invention has the beneficial effects that:
(1) the invention provides a lithium battery health state estimation method based on voltage segments, which can accurately predict the health state of a retired power lithium battery.
(2) According to the method, the empirical model and the data driving model are combined, the mode that the estimated empirical model is converted into the data driving model kernel function is realized by depending on the charge-discharge cycle times of the lithium battery, the battery electrochemical characteristics of the empirical model are integrated into the data driving model, and the estimation accuracy of the health state of the lithium battery is improved.
(3) The invention has low complexity, improves the working efficiency and has wide application prospect.
Drawings
Fig. 1 is a flowchart of a lithium battery health state estimation method based on voltage segments according to the present invention.
Detailed Description
The following description of the embodiments of the present invention is provided to facilitate the understanding of the present invention by those skilled in the art, but it should be understood that the present invention is not limited to the scope of the embodiments, and it will be apparent to those skilled in the art that various changes may be made without departing from the spirit and scope of the invention as defined and defined in the appended claims, and all matters produced by the invention using the inventive concept are protected.
Embodiments of the present invention will be described in detail below with reference to the accompanying drawings.
As shown in fig. 1, a method for estimating the state of health of a lithium battery based on voltage segments includes the following steps:
s1, intercepting a voltage segment from a charging voltage curve of the lithium battery, and acquiring aging characteristics of the lithium battery according to the voltage segment;
s2, converting the empirical model into a kernel function based on the number of charge and discharge cycles of the lithium battery;
s3, collecting health state data of lithium battery aging characteristics as training data;
s4, introducing the kernel function into the Gaussian process regression model, and training the Gaussian process regression model by adopting the lithium battery aging characteristics and the corresponding health state data;
and S5, inputting the aging characteristics of the lithium battery with the service life to be estimated into the trained Gaussian process regression model to obtain the estimation result of the health state of the lithium battery.
The step S1 specifically includes:
s1.1, setting a voltage starting point UstartAnd a segment time length l during constant current chargingThe interception starting point on the voltage curve is UstartAnd a voltage segment u of length l;
wherein u ═ u1,u2,...,ul]，u1,u2,...,ulAll represent voltage points, u, in a voltage segment1＝Ustart；
S1.2, giving the number of main components, and performing main component analysis on the voltage section u to obtain the aging characteristic of the lithium battery;
s1.3, collecting the aging characteristics of a plurality of lithium batteries according to the method of the steps S1.1-S1.2;
s1.4, collecting the lithium battery health state value corresponding to each aging characteristic, and taking out a group of data as a test sample, and taking the other data as training samples;
s1.5, training a Gaussian process regression model by adopting a gradient descent method according to the training samples;
s1.6, inputting the aging characteristics of the test sample into a Gaussian process regression model to obtain an estimated value of the health state of the lithium battery;
s1.7, comparing the lithium battery health state estimated value with the lithium battery health state value of the test sample to obtain prediction precision;
s1.8, according to the method of the steps S1.4-S1.7, each group of data is used as a test sample, a plurality of prediction precisions are obtained, and an average value is taken as a final prediction precision;
s1.9, resetting a voltage starting point UstartAnd the segment time length l, and obtaining the final prediction precision according to the method of the steps S1.1-S1.8;
and S1.10, judging whether the final prediction precision obtained at the next time is greater than that of the previous time, if so, taking the parameter of the Gaussian process regression model at the moment as an initial parameter, and storing the battery aging characteristic and the corresponding lithium battery health state value, otherwise, returning to the step S1.9.
The step S1.2 is specifically as follows:
s1.21, giving the number N of principal components, and performing N-dimensional linear transformation on the voltage segment u, wherein the N-dimensional linear transformation specifically comprises the following steps:
Figure BDA0003073628940000081
Figure BDA0003073628940000082
Figure BDA0003073628940000083
wherein x isiThe principal component after the ith dimensionality reduction, i.e., the ith aging characteristic,
Figure BDA0003073628940000084
denotes a unit vector, T denotes transpose, i and j denote different aging characteristics, i 1,2,.. and N, j 1, 2.. and N,
Figure BDA0003073628940000085
denotes that the voltage segment u ═ u1,u2,...,ul]Dimensionality reduction to the ith principal component xiThe corresponding weight vector.
S1.22, obtaining the aging characteristics of the N batteries according to the method described in step S1.21, and obtaining an aging characteristic of x ═ x1,x2,...,xN]。
The step S2 specifically includes:
s2.1, establishing a relation between the health state of the lithium battery and the number of charge-discharge cycles based on the number of charge-discharge cycles of the lithium battery as follows:
Figure BDA0003073628940000086
wherein f (·) is a nonlinear equation with two independent variables, S represents the capacity of the lithium battery, C represents the charge-discharge cycle number of the lithium battery, and d represents the differential;
s2.2, carrying out Taylor expansion on two independent variables in the relational expression in the step S2.1, specifically:
Figure BDA0003073628940000087
wherein, a1And a2Respectively representing attenuation factors and fatigue damage accumulation factors;
and S2.3, solving the step S2.2 according to the condition that when the lithium battery is not used, C is 0 and S is 100% to obtain an empirical model as follows:
S＝k1C+k2eαC+1-k2
wherein k is1Denotes a first unknown variable, k2Representing a second unknown variable, and alpha representing a third unknown variable;
s2.4. adopting square | | · | | of Euclidean distance in low-dimensional space2Replacing the charging and discharging cycle number C, converting the empirical model into a kernel function, specifically:
Figure BDA0003073628940000091
where k (-) represents the kernel function, i.e., the distance of two sets of features in the high-dimensional space; | l | · | represents the euclidean distance between two observation points, x and x' represent two different sets of features, respectively, θ1Denotes a first hyperparameter, θ2Denotes a second hyperparameter, θ3Represents a third hyperparameter;
s2.5, collecting the cycle number and the health state value of the lithium battery, importing the cycle number and the health state value into an empirical model, and acquiring parameter values of the empirical model by adopting a least square fitting method, wherein the method specifically comprises the following steps:
Figure BDA0003073628940000092
where δ (-) denotes the sum of the squared errors of the predicted and true values under the corresponding parameters, Si'Represents the real value of the health state of the lithium battery in the ith' cycle,
Figure BDA0003073628940000093
representing the predicted value of the health state of the lithium battery in the ith' cycle under the corresponding parameters;
and S2.6, acquiring an initial value of the hyperparameter in the kernel function according to the parameter value of the empirical model and the relation between the empirical model and the kernel function.
The step S4 specifically includes:
s4.1, introducing a kernel function into a Gaussian process regression model;
and S4.2, optimizing the hyper-parameters of the kernel function by adopting a negative logarithm maximum likelihood estimation function according to the aging characteristics of the lithium battery and the corresponding health state data thereof, and obtaining an optimized Gaussian process regression model.
The negative log-maximum likelihood estimation function L (θ) in step S4.2 is specifically:
Figure BDA0003073628940000094
Figure BDA0003073628940000101
wherein theta represents a set of hyper-parameters to be optimized in the kernel function,
Figure BDA0003073628940000102
representing a noise term, InRepresenting a diagonal matrix; tr (-) represents the trace of the matrix, y represents the output value of the regression model of the Gaussian process, and K represents an intermediate parameter.

Claims (6)

1. A lithium battery health state estimation method based on voltage segments is characterized by comprising the following steps:
s1, intercepting a voltage segment from a charging voltage curve of the lithium battery, and acquiring aging characteristics of the lithium battery according to the voltage segment;
s2, converting the empirical model into a kernel function based on the number of charge and discharge cycles of the lithium battery;
s3, collecting health state data of lithium battery aging characteristics as training data;
s4, introducing the kernel function into the Gaussian process regression model, and training the Gaussian process regression model by adopting the lithium battery aging characteristics and the corresponding health state data;
and S5, inputting the aging characteristics of the lithium battery with the service life to be estimated into the trained Gaussian process regression model to obtain the estimation result of the health state of the lithium battery.
2. The method for estimating the state of health of a lithium battery based on voltage slices as claimed in claim 1, wherein the step S1 specifically comprises:
s1.1, setting a voltage starting point UstartAnd the segment time length l is obtained by intercepting a starting point U on a voltage curve in the constant current charging processstartAnd a voltage segment u of length l;
wherein u ═ u1,u2,...,ul]，u1,u2,...,ulAll represent voltage points, u, in a voltage segment1＝Ustart；
S1.2, giving the number of main components, and performing main component analysis on the voltage section u to obtain the aging characteristic of the lithium battery;
s1.3, collecting the aging characteristics of a plurality of lithium batteries according to the method of the steps S1.1-S1.2;
s1.4, collecting the lithium battery health state value corresponding to each aging characteristic, and taking out a group of data as a test sample, and taking the other data as training samples;
s1.5, training a Gaussian process regression model by adopting a gradient descent method according to the training samples;
s1.6, inputting the aging characteristics of the test sample into a Gaussian process regression model to obtain an estimated value of the health state of the lithium battery;
s1.7, comparing the lithium battery health state estimated value with the lithium battery health state value of the test sample to obtain prediction precision;
s1.8, according to the method of the steps S1.4-S1.7, each group of data is used as a test sample, a plurality of prediction precisions are obtained, and an average value is taken as a final prediction precision;
s1.9, resetting a voltage starting point UstartAnd the segment time length l, and obtaining the final prediction precision according to the method of the steps S1.1-S1.8;
and S1.10, judging whether the final prediction precision obtained at the next time is greater than that of the previous time, if so, taking the parameter of the Gaussian process regression model at the moment as an initial parameter, and storing the battery aging characteristic and the corresponding lithium battery health state value, otherwise, returning to the step S1.9.
3. The method for estimating the state of health of a lithium battery based on voltage slices as claimed in claim 2, wherein the step S1.2 is specifically as follows:
s1.21, giving the number N of principal components, and performing N-dimensional linear transformation on the voltage segment u, wherein the N-dimensional linear transformation specifically comprises the following steps:
Figure FDA0003073628930000021
Figure FDA0003073628930000022
Figure FDA0003073628930000023
wherein x isiThe principal component after the ith dimensionality reduction, i.e., the ith aging characteristic,
Figure FDA0003073628930000024
denotes a unit vector, T denotes transpose, i and j denote different aging characteristics, i 1,2,.. and N, j 1, 2.. and N,
Figure FDA0003073628930000025
denotes that the voltage segment u ═ u1,u2,...,ul]Dimensionality reduction to the ith principal component xiCorrespond toThe weight vector of (2). .
S1.22, obtaining the aging characteristics of the N batteries according to the method described in step S1.21, and obtaining an aging characteristic of x ═ x1,x2,...,xN]。
4. The method for estimating the state of health of a lithium battery based on voltage slices as claimed in claim 1, wherein the step S2 specifically comprises:
s2.1, establishing a relation between the health state of the lithium battery and the number of charge-discharge cycles based on the number of charge-discharge cycles of the lithium battery as follows:
Figure FDA0003073628930000031
wherein f (·) is a nonlinear equation with two independent variables, S represents the capacity of the lithium battery, C represents the charge-discharge cycle number of the lithium battery, and d represents the differential;
s2.2, carrying out Taylor expansion on two independent variables in the relational expression in the step S2.1, specifically:
Figure FDA0003073628930000032
wherein, a1And a2Respectively representing attenuation factors and fatigue damage accumulation factors;
and S2.3, solving the step S2.2 according to the condition that when the lithium battery is not used, C is 0 and S is 100% to obtain an empirical model as follows:
S＝k1C+k2eαC+1-k2
wherein k is1Denotes a first unknown variable, k2Representing a second unknown variable, and alpha representing a third unknown variable;
s2.4. adopting square | | · | | of Euclidean distance in low-dimensional space2Replacing the charging and discharging cycle number C, converting the empirical model into a kernel function, specifically:
Figure FDA0003073628930000033
where k (-) represents the kernel function, i.e., the distance of two sets of features in the high-dimensional space; | l | · | represents the euclidean distance between two observation points, x and x' represent two different sets of features, respectively, θ1Denotes a first hyperparameter, θ2Denotes a second hyperparameter, θ3Represents a third hyperparameter;
s2.5, collecting the cycle number and the health state value of the lithium battery, importing the cycle number and the health state value into an empirical model, and acquiring parameter values of the empirical model by adopting a least square fitting method, wherein the method specifically comprises the following steps:
Figure FDA0003073628930000034
where δ (-) denotes the sum of the squared errors of the predicted and true values under the corresponding parameters, Si'Represents the real value of the health state of the lithium battery in the ith' cycle,
Figure FDA0003073628930000041
representing the predicted value of the health state of the lithium battery in the ith' cycle under the corresponding parameters;
and S2.6, acquiring an initial value of the hyperparameter in the kernel function according to the parameter value of the empirical model and the relation between the empirical model and the kernel function.
5. The method for estimating the state of health of a lithium battery based on voltage slices as claimed in claim 4, wherein the step S4 is specifically as follows:
s4.1, introducing a kernel function into a Gaussian process regression model;
and S4.2, optimizing the hyper-parameters of the kernel function by adopting a negative logarithm maximum likelihood estimation function according to the aging characteristics of the lithium battery and the corresponding health state data thereof, and obtaining an optimized Gaussian process regression model.
6. The method for estimating the state of health of a lithium battery based on voltage slices as claimed in claim 5, wherein the negative log-maximum likelihood estimation function L (θ) in the step S4.2 is specifically:
Figure FDA0003073628930000042
Figure FDA0003073628930000043
wherein theta represents a set of hyper-parameters to be optimized in the kernel function,
Figure FDA0003073628930000044
representing a noise term, InRepresenting a diagonal matrix; tr (-) represents the trace of the matrix, y represents the output value of the regression model of the Gaussian process, and K represents an intermediate parameter.
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