CN114611398A - Brain-like modular neural network-based soft measurement method for nitrogen oxides in urban solid waste incineration process - Google Patents

Brain-like modular neural network-based soft measurement method for nitrogen oxides in urban solid waste incineration process Download PDF

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CN114611398A
CN114611398A CN202210266639.9A CN202210266639A CN114611398A CN 114611398 A CN114611398 A CN 114611398A CN 202210266639 A CN202210266639 A CN 202210266639A CN 114611398 A CN114611398 A CN 114611398A
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蒙西
王岩
乔俊飞
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Abstract

The invention relates to a brain-like modular neural network-based soft measurement method for nitrogen oxides in urban solid waste incineration process, which realizes NO soft measurementXThe method for accurately acquiring the concentration in real time comprises the following steps: firstly, acquiring data; preprocessing the acquired data, and determining an input variable and an output variable of the model; then, establishing a soft measurement model by adopting a brain-like modular neural network; and finally, the test data is used as the input of the model, and the validity of the model is verified. The invention effectively realizes NOXThe real-time accurate detection of the concentration has important theoretical significance and application value.

Description

Brain-like modular neural network-based soft measurement method for nitrogen oxides in urban solid waste incineration process
Technical Field
The invention relates to Nitrogen Oxide (NO) in the process of burning urban solid wastesX) A soft measurement method; establishes NO based on Brain-like Modular Neural Network (BIMNN)XSoft measurement model, realizes to NOXAnd (4) accurately acquiring the concentration in real time. Not only belongs to the field of urban solid waste treatment, but also belongs to the field of intelligent modeling.
Background
With the rapid development of Chinese economy and the continuous acceleration of urbanization process, the urban solid waste yield increases day by dayCities face the crisis of "solid waste enclosure". The incineration treatment mode of solid wastes increasingly becomes the main mode of urban solid waste treatment in China. And NOXIs one of main pollutants generated in the process of incinerating urban solid wastes, and seriously influences the health of people and the quality of ecological environment. With the increasing requirements of environmental protection and treatment in China, the control of NOx emission becomes one of the key problems to be solved urgently in municipal solid waste incineration plants, and the real-time accurate detection of NOx is one of the important prerequisites for improving the denitration efficiency of the municipal solid waste incineration plants. Thus, realize NOXThe real-time accurate detection has important theoretical significance and application value.
Disclosure of Invention
The invention aims to provide a brain-like modular neural network-based urban solid waste incineration process NOXSoft measurement method, adopting brain-like modular neural network to establish NOXSoft measurement model, implementation to NOXAnd (4) accurately acquiring the concentration in real time.
The invention adopts the following technical scheme and implementation steps:
1. collecting data;
2. determining the input and output variables of the model: determining the input variable of the model by adopting a maximum correlation minimum redundancy (mRMR) algorithm, wherein the output variable of the model is NO at the current momentXConcentration;
the method for determining the input variables of the model by adopting the mRMR algorithm is as follows:
given two random variables a and b, the mutual information between the two random variables is calculated as follows:
Figure BDA0003552096710000011
wherein, I is mutual information between random variables a and b, and p (a) and p (b) are edge probability distribution of the random variables a and b respectively; p (a, b) is the joint distribution of random variables a and b;
firstly, based on mutual information, finding a feature subset S having the maximum correlation with a variable c to be measured:
Figure BDA0003552096710000021
wherein ,miThe method comprises the following steps of taking characteristic variables in a characteristic subset S, | S | is the number of the characteristic variables in the characteristic subset S, D is the correlation between the selected characteristic variables and a variable c to be measured, and if D is larger, the correlation between the selected characteristic variables and the variable c to be measured is higher;
considering that certain similarity exists among the selected feature variables, and the elimination of the 'redundant' feature does not affect the model performance, therefore, the redundancy among the features is calculated, and the 'mutually exclusive' feature is found:
Figure BDA0003552096710000022
wherein ,mi，njThe characteristic variables in the characteristic subset S are taken as R, the redundancy among the variables in the characteristic subset S is taken as R, and the smaller the R is, the lower the redundancy is;
in the mRMR algorithm application process, the maximum correlation index D and the minimum redundancy index R are generally unified into an evaluation function Φ ═ D-R, and then an optimal feature subset S is determined by finding the maximum value of the evaluation function:
maxΦ(D,R),Φ＝D-R (4)
3. designed for NOXA brain-like modular neural network model for soft measurement of concentration;
(1) task decomposition
In order to measure and evaluate the modularization degree of a network and simulate the modularization characteristic of a brain network, a modularization index (MQ) facing a modularization neural network is provided; the modularization index consists of the density in the module and the sparsity among the modules, wherein the density in the module is calculated as follows:
Figure BDA0003552096710000023
wherein ,JCIs the density in the moduleSet degree, P is the number of modules in the current network, NlNumber of samples assigned to the l-th module, xiTo input samples, hl and rlRespectively representing the position and the action range of the core node of the ith module;
the sparsity between the modules is calculated based on the Euclidean distance:
Figure BDA0003552096710000024
wherein ,JSD (h) is the degree of sparsity between modulesl,hs) Representing the distance between the core node of the l-th module and the core node of the s-th module, and comprehensively considering the density J in the moduleCAnd degree of sparsity J between modulesSThe modularization index measurement mode is provided as follows:
Figure BDA0003552096710000031
therefore, the larger the value of MQ is, the higher the modularization degree of the network is, so a brain-like modularization partition method is proposed, and the main idea is as follows: firstly, distributing training samples through core nodes to determine whether to distribute the training samples to a current existing module or a newly added module; then, a new core node of an existing module is determined by seeking the maximum "modularity" degree of the network, and therefore, the modular structure construction can be divided into two cases: adding new modules and updating existing modules;
adding new module
At the initial moment, the number of modules of the whole network is 0;
when the first data sample enters the network, setting the first data sample as a core node of a first submodule:
h1＝x1 (8)
Figure BDA0003552096710000032
Figure BDA0003552096710000033
wherein ,h1 and r1Respectively representing the location and range of action, x, of the first module core node1As input vector for the first training sample, dmaxFor training sample xiAnd xjThe maximum distance therebetween;
at time t, when the tth training sample enters the network, assuming that k modules exist, finding the core node closest to the sample:
Figure BDA0003552096710000034
wherein ,xtIs the input vector of the t-th training sample, hsDenotes the location of the core node of the s-th module, kminRepresenting distance training samples xtThe nearest core node;
if the t-th training sample is not in kminIn the action range of the core node, a new module is needed to learn the current sample, and the parameters of the core node corresponding to the new module are set as follows:
hk+1＝xt (12)
Figure BDA0003552096710000041
wherein ,hk+1，rk+1Position and range of action of core node corresponding to newly added module, xtIs the input vector for the t-th training sample,
Figure BDA0003552096710000042
the farthest distance from other core nodes to the core node of the newly added module;
② optimizing existing modules
Otherwise, the sample is considered to be classifiedIs kminIn the module, in order to make the network have the optimal modularization degree, according to the formula (7), modularization index values MQ of the whole network under the condition that the current sample and the original core node are respectively used as the core nodes are respectively calculatedtAnd
Figure BDA0003552096710000043
if it is
Figure BDA0003552096710000044
If the network modularization degree of the current input sample as the core node is considered to be higher, the sample is used for replacing the existing core node to become a new core node, and the initial parameter setting is as follows:
Figure BDA0003552096710000045
Figure BDA0003552096710000046
wherein ,
Figure BDA0003552096710000047
respectively as a new core node and an action range of the module; n is a radical ofkIs the number of samples allocated to the kth module;
if it is
Figure BDA0003552096710000048
The location of the current core node
Figure BDA0003552096710000049
Keeping unchanged, only adjusting the action range of the node
Figure BDA00035520967100000410
Namely:
Figure BDA00035520967100000411
wherein ,
Figure BDA00035520967100000412
is the original core node of the module;
after all training samples are compared, the samples are distributed to different sub-modules, a partition structure is formed, the modularization degree of the current network can be considered to be the maximum, and then a sub-network needs to be constructed according to the task of each sub-module;
(2) subnetwork structure design
Training data set
Figure BDA00035520967100000413
Is divided into M subsets;
wherein ,xi，yiRespectively the input variables and the output variables of the model,
Figure BDA00035520967100000414
representing the domain, V being the input vector xiN is the number of model input variables and output variable data;
an adaptive task-oriented radial basis function neural network (ATO-RBF) is adopted to construct a sub-network corresponding to each module, and the design of the sub-network comprises three parts: network structure growth, network structure pruning and network parameter adjustment;
(ii) network architecture growth
The center, radius and connection weight to the output layer of the node of the first hidden layer of the s-th module are based on the sample with the largest absolute output
Figure BDA0003552096710000051
Setting:
Figure BDA0003552096710000052
Figure BDA0003552096710000053
Figure BDA0003552096710000054
Figure BDA0003552096710000055
wherein ,
Figure BDA0003552096710000056
for the sample with the largest absolute output,
Figure BDA0003552096710000057
respectively representing the input variable and the output variable corresponding to the sample with the largest absolute output, rsIs the range of action of the s-th module, NsIs the number of samples assigned to the s-th module;
Figure BDA0003552096710000058
the center and the radius of a first hidden layer node of the s-th module and the connection weight to the output layer are respectively;
at time tsAt time, training error vector e (t)s) Obtained by the following formula:
Figure BDA0003552096710000059
Figure BDA00035520967100000510
wherein ,yfIs the desired output of the f-th sample,
Figure BDA00035520967100000511
is the f-th sample at time tsIs calculated by the following equation:
Figure BDA00035520967100000512
wherein H is the number of neurons in the hidden layer,
Figure BDA00035520967100000513
is the function of the jth hidden layer node of the s-th module,
Figure BDA00035520967100000514
for the jth hidden layer node of the s-th module to the output layer connection weight,
Figure BDA00035520967100000515
the center and radius of the jth hidden layer node of the s-th module;
finding the sample with the largest difference between the expected output and the network output
Figure BDA00035520967100000516
Figure BDA00035520967100000517
Then, adding an RBF neuron pair
Figure BDA00035520967100000518
Learning is carried out on each sample, and the initial parameters of the newly added neurons are as follows:
Figure BDA0003552096710000061
Figure BDA0003552096710000062
wherein
Figure BDA0003552096710000063
And
Figure BDA0003552096710000064
respectively representing the center of the new neuron of the s-th module and the connection weight to the output layer;
Figure BDA0003552096710000065
is a first
Figure BDA0003552096710000066
The input variable corresponding to each of the samples,
Figure BDA0003552096710000067
are respectively the first
Figure BDA0003552096710000068
Expected output sum corresponding to each sample at tsA network output value of a time;
when the following relation is satisfied, the influence of the existing neurons on the newly added neurons is small:
Figure BDA0003552096710000069
Figure BDA00035520967100000610
wherein
Figure BDA00035520967100000611
Is the center of the neuron closest to the newly added neuron;
from equations (27) and (28), the radius of the newly added neuron is set as:
Figure BDA00035520967100000612
when a neuron is newly added, network parameters are adjusted through a second-order learning algorithm; when reaching the preset maximum structure JmaxOr period of timeInspection training precision E0The network structure growth process ends; in the course of the experiment Jmax＝10，E0The training accuracy of the network is measured using Root Mean Square Error (RMSE) at 0.0001, and is calculated as follows:
Figure BDA00035520967100000613
wherein ,yi and yiThe expected output and the network output of the ith sample are respectively;
② trimming of network structure
In order to avoid redundancy of the network structure, it is proposed to measure the contribution value of hidden layer neurons based on the index of the connection weight:
Figure BDA00035520967100000614
wherein SI (j) is the contribution value of the jth hidden layer node,
Figure BDA00035520967100000615
is the weight from the jth hidden layer node of the s-th module to the output layer;
finding the hidden layer node with the smallest contribution value:
Figure BDA0003552096710000071
wherein ,
Figure BDA0003552096710000072
the hidden layer node with the minimum contribution value in the s-th module, and J is the number of the hidden layer nodes in the network;
therefore, deleting the hidden node with the smallest contribution, adjusting parameters through a second-order learning algorithm, and comparing the root mean square error value (RMSE _1) after the node is deleted with the root mean square error value (RMSE _0) when the node is not deleted, wherein the calculation formula of RMSE is shown as a formula (30); if RMSE _1 is less than or equal to RMSE _0, the selected nodes can be pruned under the condition of not sacrificing the network learning capability, and then the process is repeated, otherwise, the selected nodes cannot be deleted; at this time, the network structure trimming process is finished, and the sub-network construction is completed; adjusting parameters by using a second-order learning algorithm every time when the neurons are deleted;
adjusting network parameters
The second order learning algorithm is as follows:
θL+1＝θL-(QL+μLE)-1gL (33)
wherein θ refers to parameters to be adjusted, including center, radius and connection weight, Q is a Hessian-like matrix, μ is a learning coefficient (0.01 in the experiment), E is an identity matrix, g is a gradient vector, and L is an iteration step number (set to 50 in the experiment);
in order to reduce memory requirements and calculation time, the calculation of the Hessian-like matrix Q and the gradient vector g is converted into Hessian-like sub-matrix summation and gradient sub-vector summation:
Figure BDA0003552096710000073
Figure BDA0003552096710000074
qzis a Hessian-like sub-matrix, ηzThe gradient subvectors can be calculated by the following formula:
Figure BDA0003552096710000075
Figure BDA0003552096710000076
wherein ,ezIs the desired output y of the z-th samplezAnd network prediction output
Figure BDA0003552096710000077
Difference of jzFor the Jacobian vector, the calculation is as follows:
Figure BDA0003552096710000078
Figure BDA0003552096710000079
according to the chain derivation rule, each component in the jacobian vector in equation (39) is calculated as follows:
Figure BDA0003552096710000081
Figure BDA0003552096710000082
Figure BDA0003552096710000083
wherein
Figure BDA0003552096710000084
The center and radius of the jth neuron of the s-th module and the connection weight value x to the output layerzIs the input vector of the z-th training sample;
4、NOXsoft measurement of concentration;
taking test sample data as the input of the brain-like modular neural network, wherein the output of the model is NOXSoft measurements of concentration; and at time T, after the Tth test sample enters the BIMNN, searching a core node closest to the sample, and activating a sub-module to which the core node belongs:
Figure BDA0003552096710000085
wherein ,xTAs input vector for the Tth test sample, hsIs the core node of the s-th module, lactIs a distance of the Tth test sample xTThe nearest core node belongs to the sub-network, and A is the number of sub-network modules;
therefore, the actual output of BIMNN is the lactOutput of the sub-network:
Figure BDA0003552096710000086
wherein ,
Figure BDA0003552096710000087
for the actual output of the BIMNN at time T,
Figure BDA0003552096710000088
is the firstactThe actual output of the sub-network;
and (3) quantitatively evaluating the test precision by adopting a Root Mean Square Error (RMSE), an average percent error (MAPE) and a Correlation Coefficient (CC), wherein the RMSE is calculated according to a formula (30), and the MAPE and CC are calculated according to the following formulas:
Figure BDA0003552096710000089
Figure BDA00035520967100000810
in the formula ,yiAnd
Figure BDA00035520967100000811
expected output and net output, N, respectively, for the ith sampleTThe number of samples to be tested.
The invention has the following obvious advantages and beneficial effects:
1 the invention is based on the classGood nonlinear mapping capability and generalization capability of brain modular neural network, and establishes stable and effective NOXSoft concentration measurement model for NOXThe concentration is accurately obtained in real time, and NO is generated in the process of burning urban solid wastesXEmission control is of great significance.
Drawings
FIG. 1 is a flow chart of the present invention;
FIG. 2 is a diagram of a brain-like modular neural network architecture;
FIG. 3 is a diagram of the structure of an ATO-RBF neural network;
FIG. 4 is a graph of the training results of subnetwork 1;
FIG. 5 is a graph of the results of the training of subnetwork 2;
FIG. 6 is a graph of the results of the training of subnetwork 3;
FIG. 7 is a BIMNN soft measurement model test output diagram;
FIG. 8 is a BIMNN soft measurement test error graph.
Detailed Description
The invention uses training data to establish for NOXA brain-like modular neural network model for soft measurement of concentration; verifying NO output by brain-like modular neural network soft measurement model by using test data setXAccuracy of real-time concentration.
As an example, the effectiveness of the method provided by the invention is verified by using actual data from a solid waste incineration plant in a certain city of Beijing. After removing obvious abnormal data, 1000 groups of 96-dimensional experimental data are obtained. Based on the obtained data sample, adopting an mRMR algorithm to perform feature selection, selection and NOXThe 20 variables with larger correlation are used as input variables of the soft measurement model, and are shown in table 1 in detail.
TABLE 1
Figure BDA0003552096710000091
Figure BDA0003552096710000101
For 1000 groups of data after dimension reduction, 750 groups of data are used for establishing a soft measurement model, and the other 250 groups of data are used for testing the performance of the model;
(1) based on 750 groups of training data, adopting a brain-like modular partitioning method, dividing 750 groups of training data into three subsets, wherein the sample number of each subset is 215, 273 and 262 respectively; correspondingly, the BIMNN consists of 3 sub-networks, and the sub-networks are established by the sub-network construction method in step 3; fig. 4, 5, and 6 are graphs of training results of respective sub-networks, where X-axis: number of training samples, in units of units per sample, Y-axis: NOXConcentration in mg/Nm3；
(2) NO by brain-like modular neural network based on 250 sets of test dataXConcentration soft measurement, the test results are shown in fig. 7, X-axis: number of samples tested, in units of units per sample, Y-axis: NOXConcentration in mg/Nm3(ii) a Test error as shown in fig. 8, X-axis: number of training samples, in units of units per sample, Y-axis: NOXError in concentration measurement in mg/Nm3；
(3) The measurement accuracy is quantitatively evaluated by using the root mean square error RMSE, the average percentage error MAPE and the correlation coefficient CC, and the calculation result is that RMSE is 6.5031, MAPE is 3.8514% and CC is 0.9777.
It will be apparent to those skilled in the art that various modifications and variations can be made in the present invention without departing from the spirit or scope of the invention. It should be noted that the above-mentioned embodiments illustrate rather than limit the invention, and that modifications and variations can be made without departing from the spirit and scope of the invention as defined by the appended claims.

Claims (1)

1. A brain-like modular neural network-based soft measurement method for nitrogen oxides in an urban solid waste incineration process is characterized by comprising the following steps:
step 1, data acquisition;
step 2, determining model input and output variables;
model input variables passing maximum phaseDetermining by a minimum redundancy mRMR algorithm, wherein the output variable of the model is NO at the current momentXConcentration;
step 3, designing a brain-like modular neural network, and establishing a soft measurement model;
step 4, using the test data as the input of the model, wherein the output of the model is NO at the current momentXA concentration measurement;
in step 2, the input variable selection method based on the mRMR algorithm is as follows:
given two random variables a and b, the mutual information between the two random variables is calculated as follows:
Figure FDA0003552096700000011
wherein, I is mutual information between random variables a and b, and p (a) and p (b) are edge probability distribution of the random variables a and b respectively; p (a, b) is the joint distribution of random variables a and b;
firstly, based on mutual information, finding a feature subset S having the maximum correlation with a variable c to be measured:
Figure FDA0003552096700000012
wherein ,miThe method comprises the following steps of taking characteristic variables in a characteristic subset S, | S | is the number of the characteristic variables in the characteristic subset S, D is the correlation between the selected characteristic variables and a variable c to be measured, and if D is larger, the correlation between the selected characteristic variables and the variable c to be measured is higher;
considering that certain similarity exists among the selected characteristic variables, and the elimination of the 'redundant' characteristic does not influence the performance of the model; therefore, the redundancy between features is calculated, and the "mutually exclusive" features are found:
Figure FDA0003552096700000013
wherein ,mi，njThe characteristic variables in the characteristic subset S are taken as R, the redundancy among the variables in the characteristic subset S is taken as R, and the smaller the R is, the lower the redundancy is;
in the mRMR algorithm application process, the maximum correlation index D and the minimum redundancy index R are generally unified into an evaluation function Φ — D-R, and then the optimal feature subset S is determined by finding the maximum value of the evaluation function:
maxΦ(D,R),Φ＝D-R (4)
in step 3, the design method of the soft measurement model based on the brain-like modular neural network is as follows:
(1) task decomposition
In order to measure and evaluate the modularization degree of the network and simulate the modularization characteristic of the brain network, a modularization index MQ facing to a modularization neural network is provided, the modularization index MQ consists of the density degree in the modules and the sparsity degree among the modules, wherein the density degree in the modules is calculated as follows:
Figure FDA0003552096700000021
wherein ,JCFor the density within a module, P is the number of modules in the current network, NlNumber of samples assigned to the l-th module, xiTo input samples, hl and rlRespectively representing the position and the action range of the core node of the first module;
the sparsity between the modules is calculated based on the Euclidean distance:
Figure FDA0003552096700000022
wherein ,JSD (h) is the degree of sparsity between modulesl,hs) Representing the distance between the core node of the l-th module and the core node of the s-th module, and comprehensively considering the density J in the moduleCAnd degree of sparsity J between modulesSThe modularization index measurement mode is provided as follows:
Figure FDA0003552096700000023
therefore, the larger the value of MQ is, the higher the modularization degree of the network is, so a brain-like modularization partition method is proposed, and the main idea is as follows: firstly, distributing training samples through core nodes to determine whether to distribute the training samples to a current existing module or a newly added module; then, determining a new core node of the existing module by seeking the maximum modularization degree of the network; thus, modular structure construction can be divided into two cases: adding new modules and updating existing modules;
adding new module
At the initial moment, the number of modules of the whole network is 0;
when the first data sample enters the network, setting the first data sample as a core node of a first sub-module:
h1＝x1 (8)
Figure FDA0003552096700000024
Figure FDA0003552096700000031
wherein ,h1 and r1Respectively representing the location and range of action, x, of the first module core node1As input vector for the first training sample, dmaxFor training sample xiAnd xjThe maximum distance therebetween;
at time t, when the tth training sample enters the network, assuming that k modules exist, finding the core node closest to the sample:
Figure FDA0003552096700000032
wherein ,xtIs the input vector of the t-th training sample, hsDenotes the location of the core node of the s-th module, kminRepresenting distance training sample xtThe nearest core node;
if the t-th training sample xtIs not at kminIn the action range of the core node, a new module is needed to learn the current sample, and the parameters of the core node corresponding to the new module are set as follows:
hk+1＝xt (12)
Figure FDA0003552096700000033
wherein ,hk+1，rk+1Position and range of action of core node corresponding to newly added module, xtIs the input vector for the t-th training sample,
Figure FDA0003552096700000034
the farthest distance from other core nodes to the core node of the newly added module;
② optimizing existing modules
Otherwise, the sample is considered to be classified as kminIn the module, in order to make the network have the optimal modularization degree, according to the formula (7), modularization index values MQ of the whole network under the condition that the current sample and the original core node are respectively used as the core nodes are respectively calculatedtAnd
Figure FDA0003552096700000035
if it is
Figure FDA0003552096700000036
Then the network modularization degree of the current input sample selected as the core node is considered to be higher, and the sample is used for replacing the existing core nodeThe new core node, the initial parameters set as follows:
Figure FDA0003552096700000037
Figure FDA0003552096700000038
wherein ,
Figure FDA0003552096700000039
respectively as a new core node and an action range of the module; n is a radical ofkIs the number of samples assigned to the kth module;
if it is
Figure FDA00035520967000000310
The location of the current core node
Figure FDA00035520967000000311
Keeping unchanged, only adjusting the action range of the node
Figure FDA00035520967000000312
Namely:
Figure FDA00035520967000000313
wherein ,
Figure FDA00035520967000000314
is the original core node of the module;
after all training samples are compared, the samples are distributed to different sub-modules, a partition structure is formed, the modularization degree of the current network is considered to be the maximum, and then a sub-network needs to be constructed according to the task of each sub-module;
(2) subnetwork structure design
Training data set
Figure FDA0003552096700000041
Is divided into M subsets;
wherein ,xi，yiRespectively the input variables and the output variables of the model,
Figure FDA0003552096700000042
representing the domain, V being the input vector xiN is the number of model input variables and output variable data;
an adaptive task-oriented radial basis function neural network (ATO-RBF) is adopted to construct a sub-network corresponding to each module, and the design of the sub-network comprises three parts: network structure growth, network structure pruning and network parameter adjustment;
(ii) network architecture growth
The center, radius and connection weight to the output layer of the node of the first hidden layer of the s-th module are based on the sample with the largest absolute output
Figure FDA0003552096700000043
Setting:
Figure FDA0003552096700000044
Figure FDA0003552096700000045
Figure FDA0003552096700000046
Figure FDA0003552096700000047
wherein ,
Figure FDA0003552096700000048
for the sample with the largest absolute output,
Figure FDA0003552096700000049
respectively representing the input variable and the output variable corresponding to the sample with the largest absolute output, rsIs the range of action of the s-th module, NsIs the number of samples assigned to the s-th module;
Figure FDA00035520967000000410
the center and the radius of a first hidden layer node of the s-th module and the connection weight to the output layer are respectively;
at time tsAt time, training error vector e (t)s) Obtained by the following formula:
Figure FDA00035520967000000411
Figure FDA00035520967000000412
wherein ,yfIs the desired output of the f-th sample,
Figure FDA00035520967000000413
is the f-th sample at time tsIs calculated by the following equation:
Figure FDA00035520967000000414
wherein H is the number of neurons in the hidden layer,
Figure FDA00035520967000000415
is the function of the jth hidden layer node of the s-th module,
Figure FDA00035520967000000416
for the jth hidden layer node of the s-th module to the output layer connection weight,
Figure FDA00035520967000000417
the center and radius of the jth hidden layer node of the s-th module;
finding the sample with the largest difference between the expected output and the network output
Figure FDA0003552096700000051
Figure FDA0003552096700000052
Then, adding a new RBF neuron pair
Figure FDA0003552096700000053
Learning is carried out on each sample, and the initial parameters of the newly added neurons are as follows:
Figure FDA0003552096700000054
Figure FDA0003552096700000055
wherein
Figure FDA0003552096700000056
And
Figure FDA0003552096700000057
respectively representing the center of the new neuron of the s-th module and the connection weight to the output layer;
Figure FDA0003552096700000058
is a first
Figure FDA0003552096700000059
The input variable corresponding to each of the samples,
Figure FDA00035520967000000510
are respectively the first
Figure FDA00035520967000000511
Expected output sum corresponding to each sample at tsA network output value of a time;
when the following relation is satisfied, the influence of the existing neurons on the newly added neurons is small:
Figure FDA00035520967000000512
Figure FDA00035520967000000513
wherein
Figure FDA00035520967000000514
Is the center of the neuron closest to the newly added neuron;
from equations (27) and (28), the radius of the newly added neuron is set as:
Figure FDA00035520967000000515
when a neuron is newly added, network parameters are adjusted through a second-order learning algorithm; when reaching the preset maximum structure JmaxOr desired training accuracy E0The network structure growth process ends; in the course of the experiment Jmax＝10，E00.0001, adoptThe root mean square error RMSE is used to measure the training accuracy of the network and is calculated as follows:
Figure FDA00035520967000000516
wherein ,yi and yiThe expected output and the network output of the ith sample are respectively;
② trimming of network structure
In order to avoid redundancy of the network structure, it is proposed to measure the contribution value of hidden layer neurons based on the index of the connection weight:
Figure FDA00035520967000000517
wherein SI (j) is the contribution value of the jth hidden layer node,
Figure FDA00035520967000000518
is the weight from the jth hidden layer node of the s-th module to the output layer;
finding the hidden layer node with the smallest contribution value:
Figure FDA00035520967000000519
wherein ,
Figure FDA0003552096700000061
the hidden layer node with the minimum contribution value in the s-th module, and J is the number of the hidden layer nodes in the network;
therefore, the hidden node with the smallest contribution is deleted, parameters are adjusted through a second-order learning algorithm, the root mean square error value (RMSE _1) after the node is deleted is compared with the root mean square error value (RMSE _0) when the node is not deleted, and the calculating formula of RMSE is shown as a formula (30); if RMSE _1 is less than or equal to RMSE _0, the selected nodes can be pruned under the condition of not sacrificing the network learning capability, and then the process is repeated, otherwise, the selected nodes cannot be deleted; at this time, the network structure trimming process is finished, and the sub-network construction is completed; adjusting parameters by using a second-order learning algorithm every time when the neurons are deleted;
network parameter adjustment
The second order learning algorithm is as follows:
θL+1＝θL-(QL+μLE)-1gL (33)
wherein theta refers to parameters needing to be adjusted and comprises a center, a radius and a connection weight, Q is a Hessian-like matrix, mu is a learning coefficient, and 0.01 is taken; e is an identity matrix, g is a gradient vector, and L is an iteration step number set to be 50;
in order to reduce memory requirements and calculation time, the calculation of the Hessian-like matrix Q and the gradient vector g is converted into Hessian-like sub-matrix summation and gradient sub-vector summation:
Figure FDA0003552096700000062
Figure FDA0003552096700000063
qzis a Hessian-like sub-matrix, ηzThe gradient subvectors can be calculated by the following formula:
Figure FDA0003552096700000064
Figure FDA0003552096700000065
wherein ,ezIs the desired output y of the z-th samplezAnd network prediction output
Figure FDA0003552096700000066
Difference of jzFor the Jacobian vector, the calculation is as follows:
Figure FDA0003552096700000067
Figure FDA0003552096700000068
according to the chain derivation rule, each component in the jacobian vector in equation (39) is calculated as follows:
Figure FDA0003552096700000069
Figure FDA00035520967000000610
Figure FDA0003552096700000071
wherein
Figure FDA0003552096700000072
The center and radius of the jth neuron of the s-th module and the connection weight value x to the output layerzIs the input vector of the z-th training sample;
in step 4, the test data is used as the input of the model, and the output of the model is NO at the current momentXA concentration measurement;
and at time T, after the Tth test sample enters the BIMNN, searching a core node closest to the sample, and activating a sub-module to which the core node belongs:
Figure FDA0003552096700000073
wherein ,xTAs input vector for the Tth test sample, hsIs the core node of the s-th module, lactIs a distance of the Tth test sample xTThe nearest core node belongs to the sub-network, and A is the number of sub-network modules;
therefore, the actual output of BIMNN is the lactOutput of the sub-network:
Figure FDA0003552096700000074
wherein ,
Figure FDA0003552096700000075
for the actual output of the BIMNN at time T,
Figure FDA0003552096700000076
is the firstactThe actual output of the sub-network.
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CN112733876A (en) * 2020-10-28 2021-04-30 北京工业大学 Soft measurement method for nitrogen oxides in urban solid waste incineration process based on modular neural network
CN113077039A (en) * 2021-03-22 2021-07-06 北京工业大学 Task-driven RBF neural network-based water outlet total nitrogen TN soft measurement method
CN113780639A (en) * 2021-08-29 2021-12-10 北京工业大学 Urban solid waste incineration nitrogen oxide NOx emission prediction method based on multitask learning framework

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* Cited by examiner, † Cited by third party
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CN112733876A (en) * 2020-10-28 2021-04-30 北京工业大学 Soft measurement method for nitrogen oxides in urban solid waste incineration process based on modular neural network
CN113077039A (en) * 2021-03-22 2021-07-06 北京工业大学 Task-driven RBF neural network-based water outlet total nitrogen TN soft measurement method
CN113780639A (en) * 2021-08-29 2021-12-10 北京工业大学 Urban solid waste incineration nitrogen oxide NOx emission prediction method based on multitask learning framework

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