CN112100574A - Resampling-based AAKR model uncertainty calculation method and system - Google Patents

Resampling-based AAKR model uncertainty calculation method and system Download PDF

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CN112100574A
CN112100574A CN202010852271.5A CN202010852271A CN112100574A CN 112100574 A CN112100574 A CN 112100574A CN 202010852271 A CN202010852271 A CN 202010852271A CN 112100574 A CN112100574 A CN 112100574A
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成玮
张乐
陈雪峰
李芸
周光辉
高琳
邢继
堵树宏
孙涛
徐钊
于方小稚
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Xian Jiaotong University
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Abstract

The invention discloses a resampling-based method and a resampling-based system for calculating uncertainty of an AAKR model, wherein a sensor historical state data set is divided into a training data set and a testing data set, the training data set is denoised by a wavelet denoising method, noise variance is calculated, data accuracy is improved, then the sensor historical state data is randomly selected and replaced to obtain a new training data set sample, model prediction variances of a plurality of model prediction values can be obtained by optimizing an AAKR model architecture and the change among the plurality of model prediction values, and the mean square error between the prediction values and the testing value is calculated by resampling training data; the model deviation is calculated by combining the variance of the prototype model to form a 95% uncertainty value, an empirical distribution model is not required to perform modeling calculation on the noise estimation value, the resampling process is simplified, the calculation efficiency is improved, the reliability of the confidence interval deviation is guaranteed by combining the Jackknife method, and the estimation efficiency is improved on the basis of keeping the convergence performance.

Description

Resampling-based AAKR model uncertainty calculation method and system
Technical Field
The invention relates to a quantification method of AAKR model uncertainty, in particular to a resampling-based AAKR model uncertainty calculation method and system.
Background
An online condition monitoring system for critical equipment of a nuclear power plant helps to reduce the risk of catastrophic failure and reduce the excess costs incurred by unnecessary periodic maintenance. The state monitoring method based on the empirical model does not depend on deep understanding of a fault mechanism model, whether the equipment is abnormal or not is judged from historical operation data and operation experience of the equipment, and the method is widely applied along with rapid development of the Internet of things and big data technology. However, when the empirical model is used for monitoring nuclear power key instruments and equipment, the problem of uncertainty affecting the stability of the model is involved, the uncertainty of the empirical model must be estimated, and meanwhile, the accurate estimation of an uncertainty interval can effectively reduce false alarm rate and false alarm rate of the equipment, so that the economic loss caused by equipment shutdown is reduced. At present, uncertain analysis and research on model regression values are few, and the traditional Monte Carlo uncertainty determination method uses probability distribution simulation noise to obtain sampling data, needs total distributed prior knowledge and enough large sample data, is low in efficiency and high in economic cost, and cannot effectively ensure prediction accuracy of the state of a key equipment sensor.
Disclosure of Invention
The invention aims to provide a resampling-based method and a resampling-based system for calculating the uncertainty of an AAKR model, so as to overcome the defects of the prior art.
In order to achieve the purpose, the invention adopts the following technical scheme:
a resampling-based AAKR model uncertainty calculation method comprises the following steps:
step 1), dividing a sensor historical state data set into a training data set and a testing data set;
step 2), denoising the training data set by a wavelet denoising method and calculating a noise variance;
step 3), resampling the training data sets for multiple times by a Bootstrap method, obtaining a group of new training data sets after resampling for each time, establishing a plurality of new models according to the groups of new training data sets after sampling, obtaining a plurality of model predicted values according to the plurality of new models, and calculating the change among the plurality of model predicted values to obtain the model prediction variance of the plurality of model predicted values;
step 4), calculating the mean square error between the model predicted value and the test observation value;
step 5), calculating according to the noise variance, the model prediction variance and the mean square error to obtain a model deviation;
and 6) estimating according to the model deviation and the model variance to obtain the Monte Carlo uncertainty estimated value which is 2 times of the square value of the sum of the square of the model deviation and the model variance.
Furthermore, the historical sensor data are loaded, the historical sensor data are detected and abnormal values are corrected, and the historical sensor state data set is divided into a training data set and a testing data set.
Furthermore, a wavelet denoising method is used for denoising the training data set,
Figure BDA0002645133380000021
wherein the content of the first and second substances,iis the ith training observation X in the training datasetiEstimating the noise of (2);
Figure BDA0002645133380000022
is the ith training observation X in the training datasetiAn estimated value of the true value of (d); the variance of the i variable noise in the training dataset is:
Figure BDA0002645133380000023
ntrnis the number of training observations;
Figure BDA0002645133380000024
is the expected value of the noise estimate;
Figure BDA0002645133380000025
is the training data set noise variance.
Further, the variance of the model prediction values is calculated by using the following formula:
Figure BDA0002645133380000031
wherein the content of the first and second substances,
Figure BDA0002645133380000032
Figure BDA0002645133380000033
variance of ith observation for jth variable; to obtain ntstAnd f, estimating the variance in x p dimension, arranging the variance of each p variable in ascending order, and selecting the maximum value of the 95 th percentile to conservatively estimate the single-point variance.
Furthermore, each new model established by the resampling training data set can give a model predicted value, namely
Figure BDA0002645133380000034
Calculating the mean square error MSE between the predicted value of the new model and the test observation value:
Figure BDA0002645133380000035
wherein Xtst,iAnd
Figure BDA0002645133380000036
respectively, the test observed value and the model predicted value of the ith new model. The dimension of the MSE is 1 xp, and N predictors yield N MSEs of dimension 1 xp.
Further, the model bias is:
Figure BDA0002645133380000037
further, according to the Monte Carlo uncertainty estimation value, a confidence interval and a prediction interval corresponding to the 95% confidence level are calculated, and the Jackknife deviation estimation method is used for correcting the deviation of the Confidence Interval (CI) predicted by the AAKR model and calculating the prediction interval.
Further, according to the Monte Carlo uncertainty estimation value, a confidence interval and a prediction interval corresponding to the 95% confidence level are calculated, and the Jackknife deviation estimation method is used for correcting the deviation of the Confidence Interval (CI) predicted by the AAKR model and calculating the prediction interval.
Further, the general equation for the confidence interval is:
Figure BDA0002645133380000038
wherein the content of the first and second substances,
Figure BDA0002645133380000039
if the model is an estimate of the expected θ of the predicted value of the model, the deviation is as follows:
Figure BDA0002645133380000041
model prediction value
Figure BDA0002645133380000042
Is ntst×pA time state sequence is maintained;
Figure BDA0002645133380000043
the estimated value obtained by removing the i (i ═ 1, 2.., N) th predicted value is averaged to obtain the average value
Figure BDA0002645133380000044
The Jackknife bias is then estimated as:
Figure BDA0002645133380000045
thereby obtaining
Figure BDA0002645133380000046
Deviation correction estimator:
Figure BDA0002645133380000047
therefore, the confidence interval after rectification is as follows:
Figure BDA0002645133380000048
the prediction intervals for the 95% confidence levels were:
Figure BDA0002645133380000049
a resampling-based AAKR model uncertainty calculation system comprises a data acquisition module, a data denoising module and a data processing module;
the data acquisition module is used for acquiring a sensor historical state data set, dividing the acquired data set into a training data set and a testing data set, and transmitting the training data set to the data denoising module;
the data denoising module denoises a received training data set, calculates a noise variance, transmits the noise variance to the data denoising module, and transmits denoised training data to the data processing module;
the data processing module conducts repeated resampling on the training data set through the data acquisition module, a group of new training data sets are obtained after each repeated resampling, a plurality of new models are built according to the groups of new training data sets after sampling, a plurality of model predicted values are obtained according to the plurality of new models in a predicting mode, and the model prediction variance of the plurality of model predicted values can be obtained by calculating the change among the plurality of model predicted values; simultaneously calculating the mean square error between the model predicted value and the test observed value; finally, calculating according to the noise variance, the model prediction variance and the mean square error to obtain a model deviation; and estimating by using the model deviation and the model variance, and outputting a Monte Carlo uncertainty estimation value which is a value 2 times of the square value of the sum of the model deviation and the model variance.
Compared with the prior art, the invention has the following beneficial technical effects:
the invention relates to a resampling-based AAKR model uncertainty calculation method, which comprises the steps of dividing a sensor historical state data set into a training data set and a testing data set, denoising the training data set and calculating noise variance through a wavelet denoising method, improving data accuracy, randomly selecting and replacing sensor historical state data to obtain a new training data set sample, obtaining model prediction variances of a plurality of model prediction values by optimizing AAKR model architecture and changes among a plurality of model prediction values, developing and testing a plurality of prototype models by resampling training data through the prototype models and the testing data, and calculating mean square error between the prediction values and the testing values; the model deviation is calculated by combining the variance of a prototype model to form a 95% uncertainty value, an empirical distribution model is not required to perform modeling calculation on a noise estimation value, the resampling process is simplified, the calculation efficiency is improved, the reliability of the uncertainty interval deviation is guaranteed by combining a Jackknife method, the estimation efficiency is improved on the basis of keeping the convergence performance, a set of reliable, efficient and complete method flow is provided for uncertainty estimation of the empirical model of the nuclear power plant key equipment, and the method has important engineering application value for improving the accuracy of state prediction of the key equipment sensor.
Furthermore, by calculating a confidence interval and a prediction interval of a 95% confidence level and correcting the distribution deviation of the confidence interval by using a Jackknife deviation estimation method, the analysis process is simplified, the deviation of the confidence interval is reduced, and the estimation efficiency is improved on the basis of keeping the convergence performance.
The resampling-based AAKR model uncertainty calculation system is simple in structure, training is carried out by acquiring a sensor historical state normal data set, and the analysis process is simplified by the Bootstrap resampling-based experience model uncertainty calculation method, and the reliability of the system is ensured by reducing confidence interval deviation by combining with a Jackknife method.
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FIG. 1 is a schematic diagram of a flow of calculating uncertainty of an AAKR model according to an embodiment of the present invention.
FIG. 2 is a graph illustrating an uncertainty convergence analysis in an embodiment of the present invention.
FIG. 3 is a diagram of confidence interval before rectification of the AAKR model in the embodiment of the present invention.
FIG. 4 is a diagram of confidence intervals after deviation rectification of the AAKR model in the embodiment of the present invention.
FIG. 5 is a diagram illustrating the prediction window of the AAKR model according to an embodiment of the present invention.
Detailed Description
The invention is described in further detail below with reference to the accompanying drawings:
as shown in fig. 1, a resampling-based method for calculating uncertainty of AAKR model includes the following steps:
step 1), dividing a sensor historical state data set into a training data set and a testing data set;
specifically, the historical sensor data is loaded, the historical sensor data is detected and abnormal values are corrected, and the data set is divided into a training data set and a testing data set.
Step 2), denoising the training data set by a wavelet denoising method and calculating a noise variance;
specifically, a wavelet denoising method is used for denoising the training data set,
Figure BDA0002645133380000061
wherein the content of the first and second substances,iis the ith training observation X in the training datasetiEstimating the noise of (2);
Figure BDA0002645133380000062
is the ith training observation X in the training datasetiThe estimated value of the true value is a result obtained by performing wavelet denoising on the training observed value; the variance of the i variable noise in the training dataset is:
Figure BDA0002645133380000063
ntrnis the number of training observations;
Figure BDA0002645133380000064
is an expected value of the noise estimate, which is zero or near zero for drift-free data;
Figure BDA0002645133380000065
is the training data set noise variance.
Random error predicted by a variance measurement model of variable noise;
step 3), resampling the training data set for N times by a Bootstrap method (namely, taking a training test observation value to replace the value of the training data set at the current position) to obtain N groups of resampled training data sets, establishing N new models according to the N groups of resampled training data sets, predicting and obtaining N model predicted values according to the N new models, and calculating the change among the N model predicted values to obtain the model prediction variance of the N model predicted values;
the method specifically comprises the following steps:
performing Bootstrap resampling on a group of training data sets for N times, establishing a new model through each obtained new sampling data set, namely N new models, and estimating the change between model prediction values from the N new models;
wherein XiRepresenting a state of the system, XjRepresenting a time state sequence of a monitored variable.
Figure BDA0002645133380000071
Bootstrap resampling: for monitoring variable XjWith replaced random samples ntrnObtaining Bootstrap resampling sample X of X*。
Figure BDA0002645133380000072
And adopting LHS resampling technology: applying wavelet denoising method to training data to obtain 'true' value, subtracting the 'true' value from original training data to obtain estimated value of noise; modeling the noise probability distribution as a normal distribution by splitting the distribution into ntrnNon-overlapping compartments (also known as "bins"), each bin having
Figure BDA0002645133380000073
The probability of (d); random values are chosen from each bin at equal frequencies and the final noise distribution is uniformly sampled to construct the prototype training set.
The AAKR model established by the resampling training data set can give a test observation value XtstModel predictive value of
Figure BDA0002645133380000081
Model prediction value
Figure BDA0002645133380000082
Is to the test observed value XtstEvaluation of (2) test the observed value XtstContaining ntst(ii) an observed value;
Figure BDA0002645133380000083
wherein the content of the first and second substances,
Figure BDA0002645133380000084
is a test observation XtstPrediction of the kth prototype model, here
Figure BDA0002645133380000085
To represent
Figure BDA0002645133380000086
An ith observation for a jth variable; the predicted value of the ith observation of the jth variable is expected to be the average of the predicted values of the N new models, i.e.:
Figure BDA0002645133380000087
namely XtstThe predicted value of (d) is expressed as:
Figure BDA0002645133380000088
Figure BDA0002645133380000089
likewise, variance of ith observation for jth variable:
Figure BDA00026451333800000810
i.e. the model prediction variance can be written as:
Figure BDA0002645133380000091
after simplification: the variance of the model predictions, which is the change between the N model predictions, is calculated using the following equation:
Figure BDA0002645133380000092
to obtain ntstEstimating the variance in x p dimension, arranging the variance of each p variable in ascending order, and selecting the maximum value of the 95 th percentile to conservatively estimate the single-point variance;
the model prediction variance is defined as the expectation of the squared difference of the parameter and its expected value, so the model prediction variance can also be expressed as:
Figure BDA0002645133380000093
step 4), calculating the Mean Square Error (MSE) between the predicted values of the N models and the test observation value;
each new model established by the resampling training data set can give a model predicted value, namely
Figure BDA0002645133380000094
Calculating the mean square error MSE between the predicted value of the new model and the test observation value:
Figure BDA0002645133380000095
wherein Xtst,iAnd
Figure BDA0002645133380000096
respectively, the test observed value and the model predicted value of the ith new model. The dimension of the MSE is 1 x p, N predicted values can generate N MSEs with the dimension of 1 x p, and for p variables, the maximum value of the 95 th percentile is taken as a single-point estimation value of the MSE.
Step 5), calculating according to the noise variance, the model prediction variance and the mean square error to obtain a model deviation;
the deviation measures any systematic error.
The prediction performance of the model is quantified by Mean Square Error (MSE), and the MSE is calculated according to the prediction value of the model:
Figure BDA0002645133380000101
wherein the content of the first and second substances,
Figure BDA0002645133380000102
is a measure of the error that can be approximated,irrirreducible error.
E[irr]＝0，
Figure BDA0002645133380000103
The reducible error is the model prediction value
Figure BDA0002645133380000104
And test observation XtstTrue model M (X)tst) The square of the distance between; irreducible errors are differences between the true parameter values and the measured parameter values, caused by random processes and measurement noise, and are called irreducible errors because they cannot be modeled deterministically.
Figure BDA0002645133380000105
The error of treaty explains how the model adequately represents the true model M (X)tst) Depending only on the selected model architecture, training procedure and data set. The error may be further decomposed into a bias component and a variance component.
Figure BDA0002645133380000106
The model deviation is defined as the systematic error of the model, which is the difference between the expected predicted value and the true target value of the model, and can be expressed as:
Figure BDA0002645133380000107
total uncertainty
Figure BDA0002645133380000108
Is a combination of model bias, model prediction variance, and irreducible error:
Figure BDA0002645133380000111
and total uncertainty
Figure BDA0002645133380000112
Quantized by MSE, having a value of MSE, i.e.
Figure BDA0002645133380000113
The model bias is constant for each variable and can be approximated by its expected value. Then, if the expectation of the squared deviation is negative (i.e., the sum of the model predicted variance and the estimated noise variance is greater than MSE), it is set to zero;
the final available model bias is specifically as follows:
Figure BDA0002645133380000114
step 6), estimating according to the model deviation and the model variance to obtain the estimated Monte Carlo uncertainty value
Figure BDA0002645133380000115
And convergence with the number of prototype models, the results are shown in fig. 2.
Example (b):
according to the Monte Carlo uncertainty estimation value, a confidence interval and a prediction interval corresponding to the 95% confidence level are calculated, the Jackknife deviation estimation method is used for correcting the deviation of the Confidence Interval (CI) predicted by the AAKR model and calculating the prediction interval, and a diagram of the confidence interval before correction of the AAKR model is shown in figure 3.
The general equation for the confidence interval is given by the following equation:
Figure BDA0002645133380000116
wherein the content of the first and second substances,
Figure BDA0002645133380000117
if the model is an estimate of the expected θ of the predicted value of the model, the deviation is as follows:
Figure BDA0002645133380000118
model prediction value
Figure BDA0002645133380000119
Is ntstA time state sequence of dimension x p,
Figure BDA00026451333800001110
the estimated value obtained by removing the i (i ═ 1, 2.., N) th predicted value is averaged to obtain the average value
Figure BDA00026451333800001111
The Jackknife bias is then estimated as:
Figure BDA0002645133380000121
thereby obtaining
Figure BDA0002645133380000122
Deviation correction estimator
Figure BDA0002645133380000123
So the CI after rectification is:
Figure BDA0002645133380000124
CI does not contain noise term
Figure BDA0002645133380000125
Only the uncertainty in the model prediction is estimated, and the natural change of the modeling value is not considered, wherein the confidence interval after rectification of the AAKR model is shown in FIG. 4;
PI at 95% confidence level is expressed as:
Figure BDA0002645133380000126
since the PI contains a noise variance term, by definition, CI, it is a more conservative estimate of the model uncertainty; the prediction intervals of the AAKR model are shown in FIG. 5
The invention relates to a Bootstrap resampling-based self-association kernel regression model (AAKR) uncertainty calculation method, which randomly selects and replaces sensor historical state data to obtain a Bootstrap sample so as to optimize an AAKR model architecture and determine uncertainty of a current predicted value, and comprises the following steps: loading historical data, detecting and correcting abnormal values, and dividing the data into training and testing data sets; utilizing Bootstrap to resample training data, developing and testing a plurality of prototype models, obtaining a predicted value through the prototype models and the test data, and calculating a Mean Square Error (MSE) between the predicted value and a test observation value; denoising the training data by using a wavelet denoising method to estimate a noise variance; the model deviation is calculated by combining the variance of the prototype model to form 95% uncertainty value estimation, the confidence interval and the prediction interval of 95% confidence level are calculated, and the distribution deviation of the confidence interval is corrected by using a Jackknife deviation estimation method.

Claims (10)

1. A resampling-based AAKR model uncertainty calculation method is characterized by comprising the following steps:
step 1), dividing a sensor historical state data set into a training data set and a testing data set;
step 2), denoising the training data set by a wavelet denoising method and calculating a noise variance;
step 3), resampling the training data sets for multiple times by a Bootstrap method, obtaining a group of new training data sets after resampling for each time, establishing a plurality of new models according to the groups of new training data sets after sampling, obtaining a plurality of model predicted values according to the plurality of new models, and calculating the change among the plurality of model predicted values to obtain the model prediction variance of the plurality of model predicted values;
step 4), calculating the mean square error between the model predicted value and the test observation value;
step 5), calculating according to the noise variance, the model prediction variance and the mean square error to obtain a model deviation;
and 6) estimating according to the model deviation and the model variance to obtain the Monte Carlo uncertainty estimated value which is 2 times of the square value of the sum of the square of the model deviation and the model variance.
2. The method for calculating the uncertainty of the AAKR model based on resampling as claimed in claim 1, wherein the historical sensor data is loaded, the historical sensor data is detected and abnormal values are corrected, and the corrected data set is divided into a training data set and a testing data set.
3. The method of claim 1, wherein the training data set is denoised by a wavelet denoising method,
Figure FDA0002645133370000011
wherein the content of the first and second substances,iis the ith training observation X in the training datasetiEstimating the noise of (2);
Figure FDA0002645133370000012
is the ith training observation X in the training datasetiAn estimated value of the true value of (d); trainingThe variance of the i variable noise in the dataset is:
Figure FDA0002645133370000013
ntrnis the number of training observations;
Figure FDA0002645133370000021
is the expected value of the noise estimate;
Figure FDA0002645133370000022
is the training data set noise variance.
4. The method of claim 3, wherein the model prediction variance is calculated as a change between model predictions using the following equation:
Figure FDA0002645133370000023
wherein the content of the first and second substances,
Figure FDA0002645133370000024
Figure FDA0002645133370000025
variance of ith observation for jth variable; to obtain ntstAnd f, estimating the variance in x p dimension, arranging the variance of each p variable in ascending order, and selecting the maximum value of the 95 th percentile to conservatively estimate the single-point variance.
5. The method of claim 4, wherein the new model created for each resampled training data set provides a model prediction value (AAKR model uncertainty) for each resampled training data set
Figure FDA0002645133370000026
Calculating the mean square error MSE between the predicted value of the new model and the test observation value:
Figure FDA0002645133370000027
wherein Xtst,iAnd
Figure FDA0002645133370000028
respectively testing observed values and model predicted values of the ith new model; the dimension of the MSE is 1 xp, and N predictors yield N MSEs of dimension 1 xp.
6. The resampling-based AAKR model uncertainty calculation method according to claim 5, wherein the model bias is:
Figure FDA0002645133370000029
7. the method for calculating the uncertainty of the AAKR model based on resampling of claim 1, wherein the confidence interval and the prediction interval corresponding to 95% confidence level are calculated according to the estimated Monte Carlo uncertainty, and the Jackknife deviation estimation method is used to correct the Confidence Interval (CI) predicted by the AAKR model and calculate the prediction interval.
8. The method for calculating the uncertainty of the AAKR model based on resampling of claim 7, wherein the confidence interval and the prediction interval corresponding to 95% confidence level are calculated according to the estimated Monte Carlo uncertainty, and the Jackknife deviation estimation method is used to correct the Confidence Interval (CI) predicted by the AAKR model and calculate the prediction interval.
9. The resampling-based AAKR model uncertainty computation method according to claim 8, wherein the general equation for confidence interval is:
Figure FDA0002645133370000031
wherein the content of the first and second substances,
Figure FDA0002645133370000032
if the model is an estimate of the expected θ of the predicted value of the model, the deviation is as follows:
Figure FDA0002645133370000033
model prediction value
Figure FDA0002645133370000034
Is ntstX p-dimensional time state sequence;
Figure FDA0002645133370000035
the estimated value obtained by removing the i (i ═ 1, 2.., N) th predicted value is averaged to obtain the average value
Figure FDA0002645133370000036
The Jackknife bias is then estimated as:
Figure FDA0002645133370000037
thereby obtaining
Figure FDA0002645133370000038
Deviation correction estimator:
Figure FDA0002645133370000039
therefore, the confidence interval after rectification is as follows:
Figure FDA00026451333700000310
the prediction intervals for the 95% confidence levels were:
Figure FDA00026451333700000311
10. a resampling-based AAKR model uncertainty calculation system is characterized by comprising a data acquisition module, a data denoising module and a data processing module;
the data acquisition module is used for acquiring a sensor historical state data set, dividing the acquired data set into a training data set and a testing data set, and transmitting the training data set to the data denoising module;
the data denoising module denoises a received training data set, calculates a noise variance, transmits the noise variance to the data denoising module, and transmits denoised training data to the data processing module;
the data processing module conducts repeated resampling on the training data set through the data acquisition module, a group of new training data sets are obtained after each repeated resampling, a plurality of new models are built according to the groups of new training data sets after sampling, a plurality of model predicted values are obtained according to the plurality of new models in a predicting mode, and the model prediction variance of the plurality of model predicted values can be obtained by calculating the change among the plurality of model predicted values; simultaneously calculating the mean square error between the model predicted value and the test observed value; finally, calculating according to the noise variance, the model prediction variance and the mean square error to obtain a model deviation; and estimating by using the model deviation and the model variance, and outputting a Monte Carlo uncertainty estimation value which is a value 2 times of the square value of the sum of the model deviation and the model variance.
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CN112926656A (en) * 2021-02-25 2021-06-08 西安交通大学 Method, system and equipment for predicting state of circulating water pump of nuclear power plant
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CN114996830A (en) * 2022-08-03 2022-09-02 华中科技大学 Visual safety assessment method and equipment for shield tunnel to pass through existing tunnel
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