CN114266191B - Multi-model space-time modeling method based on density peak clustering - Google Patents
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Abstract
The invention provides a multi-model space-time modeling method based on density peak clustering, which comprises the following steps: dividing the space-time variable into a plurality of local subspaces by using a DPC clustering algorithm, wherein each subspace represents the local space-time characteristic of the original system; learning a corresponding local space basis function by adopting a KL method; establishing all local time coefficient models through ELM, and reconstructing a local space-time model by using the obtained local space basis function and the corresponding time coefficient model; and calculating the weight of the corresponding sub-model through LASSO regression, and carrying out weighted summation on each local space-time model to obtain an integrated model which approximates to the original system. The beneficial effects provided by the invention are as follows: the invention establishes a data-driven distributed parameter system thermal process temperature prediction model, which can effectively solve the problem of difficult accurate modeling caused by complex system, and compared with other methods at present, the model established by the invention not only can improve modeling accuracy, but also is more suitable for a nonlinear distributed parameter system with a large range and multiple working conditions.
Description
Technical Field
The invention relates to the technical field of modeling and data prediction of industrial distribution parameter systems, in particular to a multi-model space-time modeling method based on density peak clustering.
Background
Thermal processes, convective diffusion reaction processes, chemical vapor deposition processes, etc. in the industrial field are nonlinear distributed parameter systems whose physical models are usually represented using partial differential equations, but in most cases, their precise partial differential equations are often difficult to determine due to the complexity of the system, resulting in difficulty in accurately modeling and predicting them. The space-time model based on data driving can describe the space-time dynamics of the distributed parameter system without priori knowledge, and therefore, the space-time model is widely applied to nonlinear distributed parameter system modeling.
In the existing data-driven industrial thermal process space-time separation modeling method, a space-time model is generally built only for the condition that a small working area or a single working condition exists, however, in reality, many complicated distribution parameter systems have the characteristics of large range and multiple working conditions, which results in that the whole process of the systems cannot be accurately simulated by the traditional single global space-time model, and how to build the accurate space-time model of the distribution parameter systems has a certain challenge.
Disclosure of Invention
In view of this, the present invention provides a multi-model spatio-temporal modeling method based on density peak clustering. Dividing a space-time variable into a plurality of local subspaces by using a DPC clustering algorithm, wherein each subspace represents the local space-time characteristic of an original system; secondly, learning a corresponding local space basis function by adopting a KL method; then, establishing all local time coefficient models through ELM, and reconstructing a local space-time model by using the obtained local space basis function and the corresponding time coefficient model; and finally, calculating the weight of the corresponding sub-model through LASSO regression, and carrying out weighted summation on each local space-time model to obtain an integrated model to approach the original system. Compared with other methods of space-time model constructed by global space basis functions, the DPC-based multi-model space-time separation method approximates an original system through a plurality of locally simplified space-time models, so that the established space-time model of the distributed parameter system is higher in precision. The method is suitable for a nonlinear distribution parameter system with a large range and multiple working conditions, and has the advantages of being simple in modeling, high in applicability, high in prediction accuracy and the like.
In order to achieve the above purpose, the invention provides a multi-model space-time modeling method based on density peak clustering, which comprises the following steps:
s101: collecting historical space-time data in the thermal process of a distributed parameter system by using a discrete control system as a data set; the dataset includes system input variables And temperature distribution data/>Wherein, N is the number of samples, m is the number of input variables, N is the number of output variables, R is the real number set;
S102: preprocessing the data set, and obtaining a new data set with a mean value of 0 and a variance of 1 after each variable is subjected to standardization processing, wherein the input variable of the preprocessed new data set is u epsilon R N×m, and the output variable is space-time variable data Y epsilon R N×n;
S103: dividing the preprocessed space-time variable data into a plurality of different subspaces { Y 1(Si,t),…,YK(Si, t) } by adopting a DPC clustering algorithm, wherein K is the number of subspaces, and the subspaces represent the local characteristics of the original system; the preprocessed space-time variable data is preprocessed temperature distribution data Y, which is defined as { Y (S i, t) |i=1,., L; t=1, 2,. -%, N; s epsilon omega, wherein L is the number of data points of the preprocessed space-time variable data in the space direction, S represents the position of the space-time data, S i represents the space position of the ith space-time data point, and omega is the coordinate space;
s104: learning corresponding local space basis functions for each subspace space-time data divided by DPC clustering algorithm by utilizing KL method
S105: projecting the divided subspace space-time variable data onto the local space basis function learned in step S104, thereby obtaining corresponding subspace low-dimensional time sequence data
S106: construction of unknown time sequence dynamic characteristics of each low-dimensional space by using ELM method, namely, each subspace input variable u k (t) and low-dimensional time sequence dataA model of the relationship between the two;
s107: predicting each subspace low-dimensional time sequence data output according to the relation model Outputting// >, the low-dimensional time sequence data of each subspaceWith spatial basis functions/>Each subspace local space-time prediction output/>, obtained by space-time synthesis
S108: calculating the weight of the corresponding local space-time model by using LASSO regression, and approximating the original system, namely the global space-time prediction output by using an integrated model obtained by carrying out weighted summation on each local space-time model
S109: and performing inverse normalization processing on the global space-time prediction output to obtain a temperature prediction value and evaluating the performance of the space-time model.
Further, in step S102, the normalization is specifically: wherein μ u、μY is the mean value of the raw data, σ u、σY is the standard deviation of the raw data.
Further, in step S103, the pre-processed space-time variable data is divided into a plurality of different subspaces by adopting DPC clustering algorithm, which specifically includes:
S201: the data matrix is { Y (S i, t) |i=1,..l; t=1, 2,. -%, N; s.epsilon.Ω }, the truncated distance d c is expressed as equation (1) according to the initialization parameter:
dc＝sda[round(n×(n-1)×p)] (1)
in the formula (1), n is the number of samples, round represents rounding, p is an adjusting parameter, and the value is 1% -2%;
s202: calculating the distance between any two data points in the data matrix to obtain a distance matrix;
s203: according to the cut-off distance d c, calculating the local density rho i of any data point according to any one of the formula (2) or the formula (3):
In the formulas (2) and (3), d ij＝dist(xi,xj is the Euclidean distance between the data points x i and x j;
S204: the distance δ i for any data point is calculated by equation (4):
s205: drawing a rho-delta decision graph by taking rho i as a horizontal axis and delta i as a vertical axis;
S206: points where ρ i and δ i are both relatively high are labeled as cluster centers, points where ρ i is relatively low but δ i is relatively high are labeled as noise points using the ρ - δ decision diagram;
S207: assigning the remaining points, wherein each remaining point is assigned to a cluster where a data point which is nearest neighbor and has a density greater than that of the remaining point is located;
S208: returning to the multi-model subspace { Y 1(Si,t),…YK(Si, t) }, K is the number of subspaces. Further, in step S104, the KL method is used to learn the corresponding local spatial basis function for each subspace space-time data divided by the DPC clustering algorithm, and the specific process is as follows:
s301: for each subspace Y k(Si, t), k=1, …, K is decoupled into a spatial basis function And corresponding time coefficient/>In the form of the inner product of (a) as in formula (5):
In the formula (5), the amino acid sequence of the compound, Is an infinite dimensional space basis function,/>Is an infinite dimensional time coefficient;
s302: the truncation of formula (5) is simplified to formula (6):
In the formula (6), the amino acid sequence of the compound, Is an n-order approximation of Y k (S, t);
S303: according to the spatial basis function By using a KL method, and designing a minimization objective function, wherein the minimization objective function is as shown in formula (7):
In the formula (7), i f (S, t) i= (f (S, t), f (S, t)) 1/2,
S304: solving the equation (7) and converting it into a problem of searching for the eigenvalue of the equation (8):
In the formula (8), R k(S,ζ)＝<Tk(S,t),Tk (ζ, t) > in the formula (4) is a covariance function; is a Lagrangian multiplier; ζ represents another arbitrary point different from S;
S305: will be The linear combination expressed as the spatio-temporal output is as in formula (9):
S306: substituting formula (9) into formula (8), and converting the eigenvalue problem formula (8) into a formula (10):
in the formula (10), the amino acid sequence of the compound, For the ith eigenvector,/>
S307: calculating a feature vector of the formula (10), and obtaining a space basis function formula (9) of the subspace;
Since C is a symmetric semi-definite matrix, the obtained characteristic functions are also orthogonal, and the characteristic values are ordered so as to meet the requirement The order n is calculated according to the following selection criteria:
In the formula (11), when the proportion eta exceeds 99.9%, the parameter n is determined by the formula (11), and the n-order space-time can be approximated to an original system;
S308: obtaining a local spatial basis function of the subspace by the above formulas (7) to (11) Further, in step S105, the spatial basis function is a unit orthogonal satisfying formula (12):
In the formula (12), the amino acid sequence of the compound, Is/>And/>Is an inner product of (2);
Obtaining corresponding subspace low-dimensional time sequence data The specific calculation is as shown in formula (13):
further, in step S106, the ELM method is used to construct the unknown time sequence dynamic characteristics of each low-dimensional space, namely each subspace input variable u k (t) and the low-dimensional time sequence data The relation model comprises the following specific steps:
S401: the ELM consists of an input layer, a hidden layer and an output layer, wherein the input layer is connected with the hidden layer, and the hidden layer is connected with the output layer through neurons; wherein, the single hidden layer neural network of the L hidden layer nodes can be expressed as follows:
In the formula (14), the amino acid sequence of the compound, Representing one of N arbitrary samples,/> G (·) is the activation function,/>To input weights,/>To output weight,/>Is the bias of the i-th hidden layer element,/>Expressed as W i k and/>Is an inner product of (2);
s402: ELM approximates any one continuous nonlinear objective function with zero error, i.e There is W i k,/>So that the following formula (15) holds:
S403: converting the N equations in equation (15) into a matrix product form as in equation (16):
Hk·βk＝Tk (16)
in formula (16), H k is the output of the hidden layer node, β k is the output weight, and T k is the desired output, which is expressed as formula (17):
S404: calculated to obtain Such that:
In equation (18), i=1, …, L, is equivalent to minimizing the loss function:
the solution of formula (19) is determined by inverting formula (20), namely:
in the formula (20), the amino acid sequence of the compound, Is the Moore-Penrose generalized inverse of matrix H k.
Further, in step S107, the low-dimensional time series data of each subspace is outputtedWith spatial basis functions/>The output of each subspace local space-time prediction obtained by space-time synthesis is specifically shown as a formula (21):
Further, in step S108, the weights of the corresponding sub-models are calculated by using LASSO regression, and the integrated model obtained by weighting and summing each local space-time model approximates the original system to obtain a global space-time prediction output, which specifically includes the steps of:
s501: the integration of each subspace local space-time model is described mathematically as in equation (22):
In the formula (22), K is the number of the local models, As the kth local model, W K＝[w1,w2,…,wK]T is the weight of the kth local model;
S502: the L1 regularized linear regression method, namely the LASSO algorithm, is adopted to estimate the weight W K, and the specific calculation is as shown in the formula (23):
s503: the global spatiotemporal predictive model outputs are combined by summing, with the specific calculation being as in equation (24):
in the formula (24), the amino acid sequence of the compound, Is a global spatiotemporal predictive output. In step S109, there are 3 model performance evaluation indexes, including: absolute error AE, time normalized absolute error TNAE, spatially normalized absolute error SNAE.
The beneficial effects provided by the invention are as follows: the method is suitable for a nonlinear distribution parameter system with a large range and multiple working conditions, and has the advantages of being simple in modeling, high in applicability, high in prediction accuracy and the like.
Drawings
FIG. 1 is a schematic diagram of a multi-model space-time modeling method based on density peak clustering;
FIG. 2 is a flow diagram of a multi-model spatio-temporal modeling method based on density peak clustering of the present invention;
FIG. 3 is a schematic illustration of a catalytic reactor rod process in accordance with an embodiment of the present invention;
FIG. 4 is a schematic diagram of a random input signal employed by an embodiment of the present invention;
FIG. 5 is a schematic diagram of output data collected in accordance with an embodiment of the present invention;
FIG. 6 is a graph of rho-delta decisions obtained in an embodiment of the present invention;
FIG. 7 is a schematic diagram of partitioning subspaces in accordance with an embodiment of the present invention;
FIG. 8 is a schematic diagram of the predicted output of the present invention;
FIG. 9 is a schematic diagram of the absolute error AE distribution of the present invention;
FIG. 10 is a SNAE comparison of the total spatial error information for each time instant of the respective model;
FIG. 11 is a comparison of TNAE of the total time error information for each sensor for each model.
Detailed Description
For the purpose of making the objects, technical solutions and advantages of the present invention more apparent, embodiments of the present invention will be further described with reference to the accompanying drawings.
The invention provides a multi-model space-time modeling method based on density peak clustering, and the basic idea is shown in FIG. 1. The following is explained with respect to fig. 1:
(1) In the whole industrial thermal process, an accurate partial differential equation between the input quantity u and the output quantity Y is difficult to obtain, and the application is established by a data-driven indirect mode;
(2) Dividing the preprocessed space-time variable data into a plurality of different subspaces { Y 1(Si,t),…,YK(Si, t) } by adopting a DPC clustering algorithm;
(3) Preliminarily representing the output Y of each subspace as the product of the infinite space basis function and the infinite time coefficient
(4) Reducing the dimension of an infinite space basis function and an infinite time coefficient of each subspace to a finite dimension by utilizing a KL method;
(5) An ELM method is adopted to establish the input quantity u k and the finite-dimensional time coefficient of each subspace A model of the relationship between the two;
(6) Space-time synthesis is carried out on the relationship model established by ELM and the space basis function, so that local space-time prediction output of each subspace can be obtained;
(7) And calculating the weight of the corresponding sub-model by using LASSO regression, and carrying out weighted summation on each local space-time model to obtain an integrated model which approximates to the original system, namely the global space-time prediction output.
Aiming at the modeling problem of the thermal process of a distributed parameter system, the method firstly adopts a DPC clustering algorithm to divide the preprocessed space-time variable data into a plurality of different subspaces; then, carrying out space-time separation on the space-time data by adopting a KL method to obtain a space basis function and a time coefficient; then, the ELM algorithm is used for approximating the unknown time sequence dynamic characteristics of the low-dimensional space, namely a relation model between the system input variable and the low-dimensional time sequence data, and space-time synthesis is carried out on each subspace; and finally, calculating the weight of the corresponding sub-model by using LASSO regression, carrying out weighted summation on each local space-time model to obtain an integrated model to approach the original system, and evaluating the performance of the built prediction model.
The flow of the technical scheme adopted by the invention is shown in figure 2, and the main steps are as follows:
s101: the discrete control system is used to collect the historical data of the distributed parameter system in the hot process, mainly including the system input variable Temperature distribution data/>Where N is the number of samples, m is the number of input variables, N is the number of output variables, and R is the set of real numbers.
S102: preprocessing the data set by adoptingNormalization was performed, where μ u、μY is the mean of the raw data and σ u、σY is the standard deviation of the raw data. After the standardization processing of each variable, a new data set with a mean value of 0 and a variance of 1 is obtained, the input variable is u epsilon R N×m, and the output variable is Y epsilon R N×n.
S103: dividing the preprocessed space-time variable data into a plurality of different subspaces { Y 1(Si,t),…,YK(Si, t) } by adopting a DPC clustering algorithm, wherein K is the number of the subspaces, and the data matrix is { Y (S i, t) |i=1, and the number of the subspaces is L; t=1, 2,. -%, N; s.epsilon.Ω }, the truncated distance d c is expressed as equation (1) according to the initialization parameter:
dc＝sda[round(n×(n-1)×p)] (1)
In the formula (1), n is the number of samples, round represents rounding, p is an adjusting parameter, and the size is between 1% and 2%.
And calculating the distance between any two data points to obtain a distance matrix.
From the cutoff distance d c, the local density ρ i of any data point is calculated by equation (2) or equation (3):
In the formulas (2) (3), d ij＝dist(xi,xj is the euclidean distance between data points x i and x j.
The distance δ i for any data point is calculated by equation (4):
A ρ - δ decision graph is drawn with ρ i as the horizontal axis and δ i as the vertical axis.
Points where ρ i and δ i are both relatively high are labeled as cluster centers, and points where ρ i is relatively low but δ i is relatively high are labeled as noise points using the ρ - δ decision diagram.
The remaining points are assigned, each remaining point being assigned to the cluster in which it is nearest neighbor and its dense data point is located.
Returning to the multi-model subspace { Y 1(Si,t),…YK(Si, t) }, K is the number of subspaces.
S104: learning corresponding local space basis functions for each subspace space-time data divided by DPC clustering algorithm by utilizing KL methodFor each subspace Y k(Si, t), k=1, …, K can be decoupled into a spatial basis function/>And corresponding time coefficient/>In the form of the inner product of (a) as in formula (5):
In the formula (5), the amino acid sequence of the compound, Is an infinite dimensional space basis function,/>Is an infinite dimensional time coefficient.
In practical application, the formula (5) can be truncated and simplified into the following formula (6):
In the formula (6), the amino acid sequence of the compound, Is an n-order approximation of Y k (S, t).
According to the spatial basis functionBy using a KL method, and designing a minimization objective function, wherein the minimization objective function is as shown in formula (7):
In the formula (7), i f (S, t) i= (f (S, t), f (S, t)) 1/2,
Solving equation (7) can be converted into a problem of finding eigenvalues of equation (8):
In the formula (8), R k(S,ζ)＝<Tk(S,t),Tk (ζ, t) > in the formula (4) is a covariance function; Is a Lagrangian multiplier; ζ represents any other point different from S.
Assume thatThe linear combination, which can be expressed as a spatio-temporal output, is as in equation (9):
further substituting formula (9) into formula (8), the eigenvalue problem formula (8) can be converted into the following formula (10):
in the formula (10), the amino acid sequence of the compound, For the ith eigenvector,/>
The feature vector is calculated and the spatial basis function of the subspace can be obtained from equation (9). Since C is a symmetric semi-definite matrix, the resulting eigenvalues are also orthogonal, assuming that the eigenvalues satisfyThen the order n may be calculated according to the following selection criteria:
In the formula (11), the parameter n is generally determined by the formula (11) when the proportion eta exceeds 99.9%, and represents that the n-order space-time can approximate the original system.
The partial space basis functions of the subspaces can be obtained by the above equations (7) to (11)
S105: low-order time sequence data of each subspaceCan be obtained by projecting onto the spatial basis functions obtained in the fourth step as follows:
in the formula, the angle brackets represent the inner product.
S106: construction of unknown time sequence dynamic characteristics of each low-dimensional space by using ELM method, namely, each subspace input variable u k (t) and low-dimensional time sequence dataA model of the relationship between the two. The ELM is composed of an input layer, a hidden layer and an output layer, wherein the input layer is connected with the hidden layer, and the hidden layer is connected with the output layer through neurons. Assume that there are N arbitrary samples/>Wherein the method comprises the steps ofThen for a single hidden layer neural network with L hidden layer nodes it can be expressed as follows:
In the formula (14), g (·) is an activation function, To input weights,/>In order to output the weight of the object,Is the bias of the i-th hidden layer element,/>Expressed as W i k and/>Is a product of the inner product of (a).
ELM approximates any one continuous nonlinear objective function with zero error, i.eThat is, W i k,/>So that the following formula (15) holds:
the N equations in equation (15) can be converted into a matrix product form as in equation (16):
Hk·βk＝Tk (16)
In formula (16), H k is the output of the hidden layer node, β k is the output weight, and T k is the desired output, which may be represented as formula (17):
In order to be able to train a single hidden layer neural network, it is desirable to have Such that:
in equation (18), i=1, …, L, which is equivalent to minimizing the loss function:
In the ELM parameter identification process, the input weight W i k and the hidden layer bias Are randomly generated, independent of training data, and once randomly determined, the output matrix H k is uniquely determined, with only the output weights β i being unknown. Its solution can be determined by inverting equation (20), namely:
in the formula (20), the amino acid sequence of the compound, Is the Moore-Penrose generalized inverse of matrix H k.
S107: predicting each subspace low-dimensional time sequence data output according to the relation modelOutputting// >, the low-dimensional time sequence data of each subspaceWith spatial basis functions/>Each subspace local space-time prediction output/>, obtained by space-time synthesisSpecifically, the formula (21):
S108: calculating the weight of the corresponding sub-model by using LASSO regression, and approximating the original system, namely the global space-time prediction output by using an integrated model obtained by carrying out weighted summation on each local space-time model The integration of each subspace local space-time model is described mathematically as in equation (22): /(I)
In the formula (22), K is the number of the local models,For the kth local model, W K＝[w1,w2,…,wK]T is the weight of the kth local model.
The L1 regularized linear regression method, namely the LASSO algorithm, is adopted to estimate the weight W K, the method can enable the characteristic weight which is partially learned to be 0, thereby achieving the purposes of sparsity and characteristic selection, and the problem that the correlation between each local space-time model is very large can be well solved, so as to obtain a steady weight, and the specific calculation is as shown in the formula (23):
The global spatiotemporal predictive model outputs may be combined by summing, specifically calculated as equation (24):
in the formula (24), the amino acid sequence of the compound, Is a global spatiotemporal predictive output.
S109: and carrying out inverse normalization processing on the predicted output value to obtain a temperature predicted value and evaluating the performance of the model.
The effectiveness of the invention is described below in connection with a specific catalytic reactor rod simulation process. The chemical reaction rod is a transport-diffusion reaction process in the chemical industry, belongs to a typical thermal process of a distribution parameter system, and is shown in a schematic view in fig. 3. Assuming that the density, specific heat capacity, thermal conductivity and temperature at both ends of the catalytic reaction rod are constant, the partial differential equation is expressed as follows:
Where T (x, T) is the space-time temperature distribution of the reaction rod, β r is the reaction heat, γ is the number of actuators, β u is the heat transfer system, u (T) is the input signal, and b (T) T is the input signal distribution.
Meets the boundary condition and initial condition of dirichlet:
The process parameters are typically set as follows:
The simulation experiment adopts random input signals, and the range of values of the random input signals is [ -3,5], as shown in figure 4. 20 sensors are used for collecting system output signals, and the 20 sensors are uniformly distributed on the horizontal direction of the catalytic reaction rod. Sampling interval Δt=0.01 s, sampling time t=40 s. And 4000 groups of simulation experiment data are acquired by adopting a finite difference method. Acquired output data T (S, T), as shown in fig. 5, with the 6 th and 11 th sensors for model verification use and the remaining 18 sensors for modeling training use. The first 3500 groups of simulation data are collected for training, and the last 500 groups of simulation data are used for verifying effects.
After data preprocessing, firstly, the space-time variable is divided into a plurality of local subspaces by using a DPC clustering algorithm, each subspace represents the local space-time characteristic of an original system, in the example, a rho-delta decision diagram is drawn in plane coordinates by using the DPC algorithm, as shown in fig. 6, as can be seen from the diagram, the three data points 44, 73 and 1463 have larger rho values and delta values at the same time, and are in accordance with the principle of DPC determining a clustering center, so that the clustering center is automatically confirmed, and the original space is divided into three subspaces, and the division result is shown in fig. 7. Then, the KL method is adopted to learn the corresponding local space basis functions, and the first three dominant basis functions in the example can obtain more than 99% of main dynamic characteristics of the system, so that three subspaces are obtained in total, and each subspace has three space basis functions. Finally, modeling a time coefficient in the dimension-reducing subspace by adopting ELM in the dimension-reducing subspace of the thermal process of the distributed parameter system, combining a space basis function with an ELM time number model to complete synthesis of a local space-time model, and carrying out local weighted summation on the local space-time model by using a LASSO method to obtain a final integrated model, so that the predicted future temperature distribution condition can be obtained.
In order to test the performance of the model, 500 test input signals are used for exciting the obtained integrated multi-model, the prediction output of the method is shown in fig. 8, the absolute error AE distribution is shown in fig. 9, and it can be seen from the graph that the prediction output of the established model to test data is basically the same as the actual output, and the absolute error is small, so that the multi-model modeling method has good precision in the classical catalytic rod reaction process.
To further verify the model performance, it was compared with two other modeling methods under the same experimental conditions. One is a traditional single model space-time model based on the KL method; the other method adopts a traditional K-means clustering algorithm to divide space-time variable data into different subspaces, and constructs a local space-time model based on the KL method for each subspace, and because the K-means clustering algorithm needs to specify the number of clustering centers in advance, the number of the clustering centers arranged in the space-time variable data is consistent with the number of the clustering centers determined by DPC, namely the number of the clustering centers K=3, and in addition, a PCR regression method is adopted when the K-means multimode integration is carried out. Fig. 10 and 11 show a comparison of SNAE representing total spatial error information at each instant in time with TNAE representing total temporal error information for each sensor, respectively. As can be seen from the figure, the proposed method is smaller in both SNAE and TNAE compared with the other two methods overall, which means that the error of the proposed method is smaller and the modeling accuracy is significantly improved.
The invention has the beneficial effects that: the method is suitable for a nonlinear distribution parameter system with a large range and multiple working conditions, and has the advantages of being simple in modeling, high in applicability, high in prediction accuracy and the like.
The above-described embodiments of the invention and features of the embodiments may be combined with each other without conflict.
The foregoing description of the preferred embodiments of the invention is not intended to limit the invention to the precise form disclosed, and any such modifications, equivalents, and alternatives falling within the spirit and scope of the invention are intended to be included within the scope of the invention.
Claims (9)
1. A multi-model space-time modeling method based on density peak clustering is characterized in that:
s101: collecting historical space-time data in the thermal process of a distributed parameter system by using a discrete control system as a data set; the dataset includes system input variables And temperature distribution data/>Wherein, N is the number of samples, m is the number of input variables, N is the number of output variables, R is the real number set;
S102: preprocessing the data set, and obtaining a new data set with a mean value of 0 and a variance of 1 after each variable is subjected to standardization processing, wherein the input variable of the preprocessed new data set is u epsilon R N×m, and the output variable is space-time variable data Y epsilon R N×n;
S103: dividing the preprocessed space-time variable data into a plurality of different subspaces { Y 1(Si,t),…,YK(Si, t) } by adopting a DPC clustering algorithm, wherein K is the number of subspaces, and the subspaces represent the local characteristics of the original system; the preprocessed space-time variable data is preprocessed temperature distribution data Y, which is defined as { Y (S i, t) |i=1,., L; t=1, 2,. -%, N; s epsilon omega, wherein L is the number of data points of the preprocessed space-time variable data in the space direction, S represents the position of the space-time data, S i represents the space position of the ith space-time data point, and omega is the coordinate space;
s104: learning corresponding local space basis functions for each subspace space-time data divided by DPC clustering algorithm by utilizing KL method
S105: projecting the divided subspace space-time variable data onto the local space basis function learned in step S104, thereby obtaining corresponding subspace low-dimensional time sequence data
S106: construction of unknown time sequence dynamic characteristics of each low-dimensional space by using ELM method, namely, each subspace input variable u k (t) and low-dimensional time sequence dataA model of the relationship between the two;
s107: predicting each subspace low-dimensional time sequence data output according to the relation model Outputting// >, the low-dimensional time sequence data of each subspaceWith spatial basis functions/>Each subspace local space-time prediction output/>, obtained by space-time synthesis
S108: calculating the weight of the corresponding local space-time model by using LASSO regression, and approximating the original system, namely the global space-time prediction output by using an integrated model obtained by carrying out weighted summation on each local space-time model
S109: and performing inverse normalization processing on the global space-time prediction output to obtain a temperature prediction value and evaluating the performance of the space-time model.
2. The multi-model space-time modeling method based on density peak clustering as claimed in claim 1, wherein: in step S102, the normalization specifically includes: wherein μ u、μY is the mean value of the raw data, σ u、σY is the standard deviation of the raw data.
3. The multi-model space-time modeling method based on density peak clustering as claimed in claim 1, wherein: in step S103, the pre-processed space-time variable data is divided into a plurality of different subspaces by adopting DPC clustering algorithm, and the specific process is as follows:
S201: the data matrix is { Y (S i, t) |i=1,..l; t=1, 2,. -%, N; s.epsilon.Ω }, the truncated distance d c is expressed as equation (1) according to the initialization parameter:
dc＝sda[round(n×(n-1)×p)] (1)
in the formula (1), n is the number of samples, round represents rounding, p is an adjusting parameter, and the value is 1% -2%;
s202: calculating the distance between any two data points in the data matrix to obtain a distance matrix;
s203: according to the cut-off distance d c, calculating the local density rho i of any data point according to any one of the formula (2) or the formula (3):
In the formulas (2) and (3), d ij＝dist(xi,xj is the Euclidean distance between the data points x i and x j;
S204: the distance δ i for any data point is calculated by equation (4):
s205: drawing a rho-delta decision graph by taking rho i as a horizontal axis and delta i as a vertical axis;
S206: points where ρ i and δ i are both relatively high are labeled as cluster centers, points where ρ i is relatively low but δ i is relatively high are labeled as noise points using the ρ - δ decision diagram;
S207: assigning the remaining points, wherein each remaining point is assigned to a cluster where a data point which is nearest neighbor and has a density greater than that of the remaining point is located;
s208: returning to the multi-model subspace { Y 1(Si,t),…YK(Si, t) }, K is the number of subspaces.
4. The multi-model space-time modeling method based on density peak clustering as claimed in claim 1, wherein: in step S104, the KL method is used to learn the corresponding local spatial basis function for each subspace space-time data divided by the DPC clustering algorithm, and the specific process is as follows:
S301: for each subspace Y k(Si, t0, k=1, …, K is decoupled into a spatial basis function And corresponding time coefficient/>In the form of the inner product of (a) as in formula (5):
In the formula (5), the amino acid sequence of the compound, Is an infinite dimensional space basis function,/>Is an infinite dimensional time coefficient;
s302: the truncation of formula (5) is simplified to formula (6):
In the formula (6), the amino acid sequence of the compound, Is an n-order approximation of Y k (S, t);
S303: according to the spatial basis function By using a KL method, and designing a minimization objective function, wherein the minimization objective function is as shown in formula (7):
In the formula (7), i f (S, t) i= (f (S, t), f (S, t)) 1/2,
S304: solving the equation (7) and converting it into a problem of searching for the eigenvalue of the equation (8):
In the formula (8), R k(S,ζ)＝<Tk(S,t),Tk (ζ, t) > in the formula (4) is a covariance function; is a Lagrangian multiplier; ζ represents another arbitrary point different from S;
S305: will be The linear combination expressed as the spatio-temporal output is as in formula (9):
S306: substituting formula (9) into formula (8), and converting the eigenvalue problem formula (8) into a formula (10):
in the formula (10), the amino acid sequence of the compound, For the ith eigenvector,/>
S307: calculating a feature vector of the formula (10), and obtaining a space basis function formula (9) of the subspace;
Since C is a symmetric semi-definite matrix, the obtained characteristic functions are also orthogonal, and the characteristic values are ordered so as to meet the requirement The order n is calculated according to the following selection criteria:
In the formula (11), when the proportion eta exceeds 99.9%, the parameter n is determined by the formula (11), and the n-order space-time can be approximated to an original system;
S308: obtaining a local spatial basis function of the subspace by the above formulas (7) to (11)
5. The multi-model space-time modeling method based on density peak clustering as claimed in claim 1, wherein: in step S105, the spatial basis function satisfies equation (12) for unit orthogonality:
In the formula (12), the amino acid sequence of the compound, Is/>And/>Is an inner product of (2);
Obtaining corresponding subspace low-dimensional time sequence data The specific calculation is as shown in formula (13):
6. The multi-model space-time modeling method based on density peak clustering as claimed in claim 1, wherein: in step S106, the ELM method is used to construct the unknown time sequence dynamic characteristics of each low-dimensional space, namely each subspace input variable u k (t) and the low-dimensional time sequence data The relation model comprises the following specific steps:
S401: the ELM consists of an input layer, a hidden layer and an output layer, wherein the input layer is connected with the hidden layer, and the hidden layer is connected with the output layer through neurons; wherein, the single hidden layer neural network of the L hidden layer nodes can be expressed as follows:
In the formula (14), the amino acid sequence of the compound, Representing one of N arbitrary samples,/> G (·) is the activation function,/>To input weights,/>To output weight,/>Is the bias of the i-th hidden layer element,/>Expressed as W i k and/>Is an inner product of (2);
s402: ELM approximates any one continuous nonlinear objective function with zero error, i.e W i k is present,So that the following formula (15) holds:
S403: converting the N equations in equation (15) into a matrix product form as in equation (16):
Hk·βk＝Tk (16)
in formula (16), H k is the output of the hidden layer node, β k is the output weight, and T k is the desired output, which is expressed as formula (17):
S404: calculated to obtain Such that:
In equation (18), i=1, …, L, is equivalent to minimizing the loss function:
the solution of formula (19) is determined by inverting formula (20), namely:
in the formula (20), the amino acid sequence of the compound, Is the Moore-Penrose generalized inverse of matrix H k.
7. The multi-model space-time modeling method based on density peak clustering as claimed in claim 1, wherein: in step S107, the low-dimensional time sequence data of each subspace is outputWith spatial basis functions/>The output of each subspace local space-time prediction obtained by space-time synthesis is specifically shown as a formula (21):
8. The multi-model space-time modeling method based on density peak clustering as claimed in claim 1, wherein: in step S108, the weight of the corresponding sub-model is calculated by using LASSO regression, and the integrated model obtained by weighting and summing each local space-time model approximates the original system to obtain the global space-time prediction output, which comprises the following specific steps:
s501: the integration of each subspace local space-time model is described mathematically as in equation (22):
In the formula (22), K is the number of the local models, As the kth local model, W K＝[w1,w2,…,wK]T is the weight of the kth local model;
S502: the L1 regularized linear regression method, namely the LASSO algorithm, is adopted to estimate the weight W K, and the specific calculation is as shown in the formula (23):
s503: the global spatiotemporal predictive model outputs are combined by summing, with the specific calculation being as in equation (24):
in the formula (24), the amino acid sequence of the compound, Is a global spatiotemporal predictive output.
9. The multi-model space-time modeling method based on density peak clustering as claimed in claim 1, wherein: in step S109, there are 3 model performance evaluation indexes, including: absolute error AE, total time error information TNAE, total spatial error information SNAE.
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