CN116595862B - Self-adaptive modeling method based on Gaussian process regression - Google Patents
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Abstract
The invention discloses a self-adaptive modeling method based on Gaussian process regression, which uses extreme points and inflection points of the last sampling sample Gaussian Process Regression (GPR) modeling as reference points to determine sample points of the next sampling, and verifies and finely adjusts a proxy model through repeated sampling processes and modeling process iteration to obtain a final prediction proxy model. Fine but important local features (such as narrow peaks or nulls) can be captured in detail, thereby greatly reducing the time and hardware costs required for high-precision simulation models.
Description
Technical Field
The invention belongs to the technical field of model building methods, and relates to a self-adaptive modeling method based on Gaussian process regression.
Background
Gaussian process regression (Gaussian Process Regression, GPR) is a typical covariance function-based prediction algorithm, has high nonlinear expression capability and good generalization performance, and is widely applied to aspects such as performance analysis and pattern recognition. For example, jacobs et al model the quasi-periodic radar cross section (Radar Cross Section, RCS) response of a missile using GPR based on a local periodic covariance function followed by GPR based on a spectral mixture covariance function; ding et al propose a single-station RCS shape-dependent variation law combining GPR and Bayesian committee machine analysis of three-dimensional objects; bilal et al analyzed the aperiodic RCS response of an aircraft parameterized model using GPR. They can effectively mine the relationship between sparsely distributed samples to obtain the complete RCS response of the target, significantly reducing the expensive time and hardware costs required for full wave simulation. However, conventional GPR-based modeling techniques all employ a single sampling strategy (e.g., a uniform sampling strategy and a uniform random sampling strategy), which makes it difficult to capture narrow peaks or zeros contained in the RCS response when the samples are sparse, thereby reducing modeling accuracy.
Disclosure of Invention
The invention aims to provide a self-adaptive modeling method based on Gaussian process regression, which solves the problem of low modeling precision in the prior art.
The technical scheme adopted by the invention is that the self-adaptive modeling method based on Gaussian process regression comprises the following steps:
step 1, acquiring an RCS frequency domain original sample set, wherein the original sample set comprises frequency and RCS, and training a GPR model by using the original sample set to obtain a proxy model of RCS response;
step 2, carrying out first-order derivation on the proxy model to obtain the frequency corresponding to the extreme point; calculating a curvature at each extreme point frequency, a minimum frequency separation of each extreme point frequency from the original sample set frequency; calculating sampling probability vectors of the extreme point frequency according to the curvature and the minimum frequency interval;
step 3, determining whether a new sampling frequency exists according to the sampling probability vector; if not, carrying out the next step; if yes, obtaining RCS response under the newly increased sampling frequency through a simulation method, updating an original data set, and calculating the approximate precision of the RCS response of each frequency in the updated original data set by the agent model and the simulation method; if the approximation precision is smaller than a preset threshold value, performing the next step; otherwise, adjusting parameters of the proxy model, training the proxy model by using the updated original data set, and returning to the step 2;
step 4, performing second order derivation on the agent model obtained in the step to obtain the frequency corresponding to the inflection point; calculating the minimum frequency interval between each inflection point frequency and the updated original sample set frequency; calculating sampling probability vectors of the extreme point frequency according to the minimum frequency interval;
step 5, substituting the sampling probability vector obtained in the step 4 into the step 3;
and 6, after the sampling is finished, adjusting parameters of the proxy model, and training the proxy model by using the updated original data set in the step 5 to obtain a final proxy model.
The invention is also characterized in that:
the step 1 specifically comprises the following steps: obtaining RCS frequency domain original sample set D (f,σ) ＝{(f i ,σ i ) I=1,.. i 、σ i Training a GPR model by using an original sample set to obtain a proxy model m (f) for the frequency and RCS of the ith sample respectively;
in the above, the weightWherein v, l 1 、l 2 P are the hyper-parameters to be solved, σ= [ σ (f) 1 ),...,σ(f n )] T Is a vector containing all RCS values at the sampling frequency, I is an n×n-dimensional identity matrix, k= (K) ij ) Is an n x n-dimensional covariance matrix estimated at all sampling frequencies, i.e., K ij ＝C(f i ,f j ),i,j＝1,...,n。
The step 2 specifically comprises the following steps:
step 2.1, let the first derivative m' (f) =0 of the proxy model to obtain the frequency corresponding to the extreme point
Step 2.2, calculating the frequency at the ith extreme pointCurvature at, i-th extreme point frequency and minimum frequency in the original sample setInterval->
Step 2.3, pairNormalization is carried out:
step 2.4, willAs the probability of the ith extreme point frequency to be sampled, the sampling probability vector of all extreme point frequencies is +.>
And step 3, determining whether a new sampling frequency exists or not according to the sampling probability vector by adopting a Monte Carlo method.
In the step 3, the approximate precision of the RCS response of each frequency in the updated original data set obtained by the agent model and the simulation method is calculated, specifically: computing agent model and simulation method in frequencyThe approximation accuracy of the RCS response of (2) is the absolute percentage error, and the calculation formula is:
the step 4 specifically comprises the following steps:
step 4.1, let the second derivative m "(f) =0 of the trimmed proxy model to obtain the frequency at the extreme point
Step 4.2, calculating the minimum frequency interval at each extreme point frequencyAnd normalize it:
step 4.3, regarding the normalized result as the ith inflection point frequency to be sampledProbability of (2), i.e.)>Obtaining the sampling probability vector of all inflection frequencies +.>
The beneficial effects of the invention are as follows: compared with the one-step sampling modeling, the adaptive modeling method based on Gaussian process regression can capture fine but important local features (such as narrow peaks or zero points), can reduce root mean square error, enables frequency domain RCS characteristic analysis of a target to be more efficient and accurate, and further improves modeling accuracy; the adaptive GPR proxy model building technique finds high representative samples to avoid redundant samples, which can further reduce the computational effort.
Drawings
FIG. 1 is a flow chart of the adaptive modeling method of the present invention based on Gaussian process regression;
FIG. 2 is a graph of a scaling model of an embodiment of the adaptive modeling method of the present invention based on Gaussian process regression;
FIG. 3 is a graph of performance trends of conventional GPR modeling techniques to reconstruct the RCS frequency response.
Detailed Description
The invention will be described in detail below with reference to the drawings and the detailed description.
The adaptive modeling method based on Gaussian process regression, as shown in FIG. 1, comprises the following steps:
step 1, acquiring an RCS frequency domain original sample set, wherein the original sample set comprises frequency and RCS response, and training a GPR model by using the original sample set to obtain a proxy model;
specifically, an RCS frequency domain original sample set D is obtained (f,σ) ＝{(f i ,σ i ) I=1,.. i 、σ i Parameters of the GPR model, including initial learning rate, maximum number of iterations, initial number of samples, and error threshold, were set to 0.1, 1000, 26, and 1% for the frequency of the i-th sample, RCS, respectively. Training a GPR model by using an original sample set to obtain a proxy model m (f);
in the above, the weightWherein v, l 1 、l 2 P are the hyper-parameters to be solved, σ= [ σ (f) 1 ),...,σ(f n )] T Is a vector containing all RCS values at the sampling frequency, I is an n×n-dimensional identity matrix, k= (K) ij ) Is an n x n-dimensional covariance matrix estimated at all sampling frequencies, i.e., K ij ＝C(f i ,f j ),i,j＝1,...,n。
Step 2, carrying out first-order derivation on the agent model m (f) to obtain the frequency corresponding to the extreme point; calculating the curvature at the ith extreme point frequency, and the minimum frequency interval between the ith extreme point frequency and the frequency in the original sample set; calculating sampling probability vectors of the extreme point frequency according to the curvature and the minimum frequency interval;
step 2.1, let the first derivative m' (f) =0 of the proxy model to obtain the frequency corresponding to the extreme point
Step 2.2, calculating the frequency at the ith extreme pointCurvature at, minimum frequency interval in the original sample set +.>
Step 2.3, pairNormalization is carried out:
step 2.4, willAs the probability of the ith extreme point frequency to be sampled, the sampling probability vector of all extreme point frequencies is +.>
And 3.1, determining whether a new sampling frequency exists or not according to the sampling probability vector by adopting a Monte Carlo method. Tool withGenerating random vectorsWherein element->Uniformly distributed over the interval [0,1 ], will ]>And->For comparison, if->The elements in (a) are greater than->Corresponding elements in the two vectors, the corresponding frequency is the newly added sampling frequency, and the comparison of all elements in the two vectors is completed to determine all sampling frequencies to be newly added +.>
Step 3.2, if not, carrying out the next step; if yes, obtaining RCS response under the newly increased sampling frequency through a full-wave simulation method, and updating the original data set, namely adding the newly increased sampling frequency and the corresponding RCS response to the original data set to obtain a sampling data set; calculating the approximate precision of the RCS response of each frequency in the sampling data set by the agent model and the simulation method; if the approximation precision is smaller than a preset threshold e, performing the next step; otherwise, adjusting parameters of the proxy model (taking a smaller learning rate of 0.0001), training the proxy model by using the updated original data set, and returning to the step 2; the approximation accuracy is absolute percentage error, and the calculation formula is:
step 4, performing second order derivation on the agent model obtained in the step to obtain the frequency corresponding to the inflection point; calculating the minimum frequency interval between each inflection point frequency and the updated original sample set frequency; calculating sampling probability vectors of the extreme point frequency according to the minimum frequency interval;
step 4.1, let the second derivative m "(f) =0 of the trimmed proxy model to obtain the frequency at the extreme point
Step 4.2, calculating the minimum frequency interval at each extreme point frequencyAnd normalize it:
step 4.3, regarding the normalized result as the ith inflection point frequency to be sampledProbability of (2), i.e.)>Obtaining the sampling probability vector of all inflection frequencies +.>
Step 5, substituting the sampling probability vector obtained in the step 4 into the step 3;
and 5.1, determining whether a newly increased sampling frequency exists or not by adopting a Monte Carlo method according to the sampling probability vector obtained in the step 4. Specifically, a random vector is generatedWherein element->Uniformly distributed over the interval [0,1 ], will ]>And->Comparing, determining the final frequency to be sampled +.>
Step 5.2, if not, carrying out the next step; if yes, obtaining RCS response under the newly increased sampling frequency through a full-wave simulation method, and updating the original data set updated in the step 3, namely adding the newly increased sampling frequency and the corresponding RCS response to the original data set updated in the step 3 to obtain a new sampling data set; calculating the approximate precision of the RCS response of each frequency in the new sampling data set by the agent model and the simulation method; if the approximation precision is smaller than a preset threshold e, performing the next step; otherwise, adjusting parameters of the proxy model (taking a smaller learning rate of 0.0001), training the proxy model with the new sampling data set, and then returning to step 2; the approximation accuracy is absolute percentage error, and the calculation formula is:
and 6, after the sampling is finished, adjusting parameters of the proxy model, and training the proxy model by using the updated original data set in the step 5 to obtain a final proxy model.
Through the mode, compared with the self-adaptive modeling method based on Gaussian process regression, the self-adaptive modeling method based on Gaussian process regression can capture fine but important local features (such as narrow peaks or zero points) more, can reduce root mean square error, and enables frequency domain RCS characteristic analysis of a target to be more efficient and accurate; the adaptive GPR proxy model building technique finds high representative samples to avoid redundant samples, which can further reduce the computational effort.
Examples
To verify that the present invention improves accuracy and efficiency over conventional methods, we consider the RCS frequency response of the warhead model as an example and compare to conventional single-step sampling modeling, with the warhead targets shown in FIG. 2. The variation trend of the RCS of the target reconstructed under 60 samples based on the GPR modeling technology relative to the frequency is shown in fig. 3, and compared with the result obtained by the moment simulation, the general trend of the RCS obtained by the conventional technology along with the frequency is basically consistent, but the RCS does not perform well in some positions with more severe variation. The invention can capture fine but important local features and reconstruct the relationship between the RCS and the frequency of the target more accurately. The invention introduces the approximate precision of the Root Mean Square Error (RMSE) quantitative modeling, and under the condition of the same sample number, the RMSE of the traditional technology is 0.6378dBsm respectively, and the RMSE of the invention is 0.3322dBsm. Namely, under the condition of the same sample number, the invention reduces the traditional modeling error by 47.91 percent. The self-adaptive GPR proxy model building technology provided by the invention searches for a high-representative sample to avoid redundant samples, so that the calculation workload can be further reduced.
Claims (3)
1. The self-adaptive modeling method based on Gaussian process regression is characterized by comprising the following steps of:
step 1, acquiring an RCS frequency domain original sample set, wherein the original sample set comprises frequency and RCS, the original sample set is utilized to train a GPR model, the RCS is a quasi-periodic radar cross section, the GPR model is a Gaussian process regression model, and a proxy model of RCS response is obtained;
the step 1 specifically comprises the following steps: obtaining RCS frequency domain original sample set D (f,σ) ＝{(f i ,σ i ) I=1,.. i Sum sigma i Training a GPR model by using the original sample set to obtain a proxy model m (f) for the frequency and RCS of the ith sample respectively;
in the above, the weightWherein v, l 1 、l 2 P are the hyper-parameters to be solved, σ= [ σ (f) 1 ),...,σ(f n )] T Is a vector containing all RCS values at the sampling frequency, I is an n×n-dimensional identity matrix, k= (K) ij ) Is an n x n-dimensional covariance matrix estimated at all sampling frequencies, i.e., K ij ＝C(f i ,f j ),i,j＝1,...,n；
Step 2, carrying out first-order derivation on the proxy model to obtain the frequency corresponding to the extreme point; calculating a curvature at each extreme point frequency, a minimum frequency separation of each extreme point frequency from the original sample set frequency; calculating a sampling probability vector of the extreme point frequency according to the curvature and the minimum frequency interval;
step 2.1, let the first derivative m' (f) =0 of the proxy model to obtain the frequency corresponding to the extreme point
Step 2.2, calculating the frequency at the ith extreme pointCurvature of the region->And the i-th extreme point frequency is +.f. from the minimum frequency interval in the original sample set>
Step 2.3, pairNormalizing to obtain->
Step 2.4, willAs the probability of the ith extreme point frequency to be sampled, the sampling probability vector of all extreme point frequencies is +.>
Step 3, determining whether a new sampling frequency exists according to the sampling probability vector; if not, carrying out the next step; if yes, obtaining RCS response under the newly increased sampling frequency through a simulation method, updating an original data set, and calculating the approximate precision of the RCS response of each frequency in the updated original data set by the agent model and the simulation method; if the approximation precision is smaller than a preset threshold value, performing the next step; otherwise, adjusting parameters of the proxy model, training the proxy model by using the updated original data set, and returning to the step 2;
step 4, performing second order derivation on the agent model obtained in the step 3 to obtain the frequency corresponding to the inflection point; calculating the minimum frequency interval between each inflection point frequency and the updated original sample set frequency; calculating sampling probability vectors of the extreme point frequency according to the minimum frequency interval;
the step 4 specifically comprises the following steps:
step 4.1, let the second derivative m' (f) =0 of the proxy model obtained in step 3, obtain the frequency at the extreme point
Step 4.2, calculating the minimum frequency interval at each extreme point frequencyAnd normalize it:
step 4.3, regarding the normalized result as the ith inflection point frequency to be sampledProbability of (2), i.e.)>Obtaining the sampling probability vector of all inflection frequencies +.>
Step 5, substituting the sampling probability vector obtained in the step 4 into the step 3;
and 6, after the sampling is finished, adjusting parameters of the proxy model, and training the proxy model by using the updated original data set in the step 5 to obtain a final proxy model.
2. The adaptive modeling method based on gaussian process regression according to claim 1, wherein the monte carlo method is adopted in step 3 to determine whether there is a new sampling frequency according to the sampling probability vector.
3. The adaptive modeling method based on gaussian process regression according to claim 1, wherein the approximate accuracy of the RCS response of each frequency in the updated raw data set obtained by the calculation proxy model and the simulation method in step 3 is specifically: computing agent model and simulation method in frequencyThe approximation accuracy of the RCS response of (2) is the absolute percentage error, and the calculation formula is:
in the method, in the process of the invention,represents 1 st of the newly added sampling frequencies, < >>Represents the first' in the newly added sampling frequency, ">Indicating frequency->Lower target RCS response->Indicating frequency->The target RCS response below.
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