CN112016839A

CN112016839A - Flood disaster prediction and early warning method based on QR-BC-ELM

Info

Publication number: CN112016839A
Application number: CN202010899796.4A
Authority: CN
Inventors: 刘扬; 刘雪梅; 吴慧欣; 杨礼波; 闫新庆; 刘明堂
Original assignee: North China University of Water Resources and Electric Power
Current assignee: North China University of Water Resources and Electric Power
Priority date: 2020-08-31
Filing date: 2020-08-31
Publication date: 2020-12-01
Anticipated expiration: 2040-08-31
Also published as: CN112016839B

Abstract

The invention discloses a flood disaster prediction and early warning method based on QR-BC-ELM, which relates to the technical field of flood early warning, considers various flood causes, establishes a flood model based on an extreme learning machine and a geographic information system so as to simulate a flood prone area of a yellow river basin and verify the efficiency and precision advantages of the extreme learning machine relative to an artificial neural network. The learning speed of the improved extreme learning model provided by the invention is 32 times that of the artificial neural network and 1.2 times that of the traditional extreme learning model. Moreover, the noise processing capacity of the orthogonal triangular decomposition extreme learning model and the full-rank decomposition extreme learning model is greatly superior to that of an artificial neural network, and the BC-ELM and the QR-ELM have great advantages in the aspects of prediction precision and prediction efficiency, and are more suitable for selecting a flood forecasting model.

Description

Flood disaster prediction and early warning method based on QR-BC-ELM

Technical Field

The invention relates to the technical field of flood early warning, in particular to a flood disaster prediction early warning method based on QR-BC-ELM.

Background

Nowadays, flood disaster management has advanced to the intelligent era, and especially the development of data science provides experts with timely, accurate and rich disaster information and decision support. Under the current form, a flood model taking an intelligent algorithm as a leading part has made a great progress, and schemes like cluster analysis, collaborative filtering, regression prediction and the like are widely applied to the flood model. The neural network-dominated intelligent algorithm is widely applied to hydrologic prediction, and has scientificity, informativeness, timeliness and accuracy compared with the traditional scheme. The ideas of cluster analysis, collaborative filtering and the like are more to break through the problems of high memory consumption and complex calculation under high-latitude large data, and scientific support is provided for feature extraction and risk level determination in flood forecasting. With the development of intelligent algorithms, the deep learning era is developed nowadays, and the deep learning with the extremely strong expressive force is a very wide application tool in the flood model, so that the flood model has more expressive force, and the precision is improved to a considerable extent compared with the traditional intelligent algorithms.

The flood disaster exists in a complex time sequence characteristic all the time, and in the face of the natural disaster which is flood and has great destructiveness, the flood model is inevitably required to have the characteristics of high precision, strong effectiveness and the like. The flood model based on the traditional scheme has the problems of subjective factor intervention, one-sided characteristic response, low prediction precision, complex calculation and the like.

Aiming at the problems of the existing model, the invention is based on the computer intelligent algorithm, constructs an accurate, stable and efficient flood prediction and early warning method, points out the defects of the existing algorithm on the basis of deeply researching the working principle and the core idea of the existing algorithm, and provides a new flood prediction and early warning scheme based on the intelligent calculation method according to the distribution characteristics of flood monitoring parameters, thereby being successfully applied to flood prediction and early warning.

Disclosure of Invention

The invention aims to provide a flood disaster prediction and early warning method based on QR-BC-ELM, which is based on a computer intelligent algorithm, constructs an accurate, stable and efficient flood prediction and early warning method, provides a new flood prediction and early warning scheme based on an intelligent calculation method according to the distribution characteristics of flood monitoring parameters, and is successfully applied to flood prediction and early warning.

The invention provides a flood disaster prediction and early warning method based on QR-BC-ELM, which comprises the following steps:

s1: extracting multi-factor indexes based on main cause analysis;

s2: constructing a flood disaster risk evaluation system;

s3: improving the extreme learning model, constructing a QR decomposition extreme learning model and a BC decomposition extreme learning model, combining the BC decomposition extreme learning model with a geographic information system to establish a flood model, and simulating a flood easily-developing area;

s4: flood prediction results and analysis are obtained based on the improved extreme learning model;

s5: and carrying out flood risk classification early warning and analysis according to the risk grade.

Further, the multi-factor index in the step S1 includes 49 calculated indexes in two criterion layers and three index layers.

Further, the risk assessment system evaluation step of step S2 includes:

s201: collecting rainfall, terrain humidity index, river power index and normalized vegetation index of different hydrological stations within three days as input of a model to obtain the correlation degree among the different hydrological stations;

s202: predicting the water level change condition of the dependent variable observation station according to the index change condition of the independent variable observation station;

s203: the non-time sequence data takes a digital elevation model, a soil texture index, a gradient index and a land utilization mode of each variable station as the input of the model, takes the same index of the variable station as the output, and calculates the index characteristic of the height correlation point;

s204: taking the river distance index, the river power index and the socioeconomic index as input, and counting and early warning the flood grade and the emergency degree.

Further, after the characteristic engineering of the multi-factor indexes is extracted, the characteristic engineering is input into an improved extreme learning model in a time sequence data mode to calculate the contribution degree of each index to flood.

Further, the computation process of the QR decomposition extreme learning model is as follows:

calculating the singular value decomposition of the matrix H of M x N, wherein the singular value decomposition formula is H ═ U Σ V^T；

Wherein

H^HIs the conjugate transpose of H, sigma is a diagonal matrix with H singular value lambda as diagonal element, V^TIs H^HFeature vectors corresponding to the orthonormal basis;

the QR decomposition limit learning model divides H into Q R, namely H Q R, wherein Q is an orthogonal matrix, R is an upper triangular matrix, and the inverse matrix of Q can be obtained by solving the transposition of Q.

Further, the calculation process of the BC decomposition extreme learning model is as follows:

let A be C^m*nThe decomposition is carried out by taking the matrix A as (a)_ij)_m*nAnd rank a ≦ r min (m, n), then a may be full rank decomposed, a ═ BC, where B is an m × r order matrix, C is an r × n order matrix, and rank B ═ rank C ═ r;

the Moore-Penrose generalized inverse product made by the full rank decomposition of matrix a in BC decomposition is unique: c^H(CC^H)^-1(B^HB)B^H。

Compared with the prior art, the invention has the following remarkable advantages:

the invention provides a flood disaster prediction and early warning method based on QR-BC-ELM, which considers various flood causes, establishes a flood model based on an extreme learning machine and a Geographic Information System (GIS) so as to simulate a flood prone area of a yellow river basin and verify the efficiency and precision advantages of the extreme learning machine relative to an artificial neural network.

The learning speed of the improved extreme learning model provided by the invention is 32 times that of the artificial neural network and 1.2 times that of the traditional extreme learning model. Moreover, the noise processing capacity of the orthogonal triangular decomposition extreme learning model and the full-rank decomposition extreme learning model is greatly superior to that of an artificial neural network, and the BC-ELM and the QR-ELM have great advantages in the aspects of prediction precision and prediction efficiency, and are more suitable for selecting a flood forecasting model.

Drawings

FIG. 1 is a diagram of method steps provided by an embodiment of the present invention;

FIG. 2 is an index system relationship diagram provided by an embodiment of the present invention;

FIG. 3 is a diagram of model prediction results of an improved extreme learning model provided by an embodiment of the present invention;

FIG. 4 is a diagram of the results of model evaluation provided by the embodiments of the present invention;

FIG. 5 is a comparison chart of model evaluation index results provided by the embodiments of the present invention;

FIG. 6 is a graph comparing risk level changes provided by embodiments of the present invention;

FIG. 7 is a chart illustrating the proportion of annual risk classes according to an embodiment of the present invention;

fig. 8 is a flood level estimation diagram provided in the embodiment of the present invention;

fig. 9 is a monitoring data graph of three water level stations of luo (yang), long (door) and black (stone gate) according to an embodiment of the present invention;

FIG. 10 is a diagram illustrating predicted results of three water stations according to an embodiment of the present invention;

fig. 11 is a graph of performance-to-contrast parameters in four types of scenes according to an embodiment of the present invention.

Detailed Description

The technical solutions of the embodiments of the present invention are clearly and completely described below with reference to the drawings in the present invention, and it is obvious that the described embodiments are some embodiments of the present invention, but not all embodiments. All other embodiments, which can be obtained by a person skilled in the art without any inventive step based on the embodiments of the present invention, shall fall within the scope of protection of the present invention.

The current flood prediction methods are mainly classified into the following categories, and the general outline of each category is as follows:

(1) a flood prediction model based on a traditional hydrological physical mechanism comprises the following steps: the method comprises the following steps that artificial subjective factors influence model feature extraction, so that prediction accuracy is influenced, meanwhile, in the face of monitoring data which are large in quantity, high in latitude and complex in data heterogeneous conditions, a reasonable calculation scheme is usually difficult to provide by a traditional water conservancy method, and the problems are important factors for restricting the accuracy of a flood prediction model;

(2) the neural network combines a traditional hydrological physical mechanism prediction model: generally, the method has the advantages that the consideration factor is single, the model calculation is complex, and the accuracy and the algorithm efficiency are to be improved;

(3) the flood prediction model taking cluster analysis, dimension reduction analysis and collaborative filtering as the leading factors comprises the following steps: the user is required to have certain prior knowledge on the observed object, and the treatment process cannot be interfered by a parameterization method, so that the precision and the efficiency are not high; meanwhile, the principal factor (characteristic) derived from the PCA method is not optimal under non-gaussian distribution.

(4) Flood prediction model based on deep learning: the overall performance of the system mainly depends on the number of parameters, the values of the parameters, the network topology, the compatibility of the transfer function and other factors. Thus, improper structural design can lead to performance degradation of the deep neural network. Meanwhile, the deep neural network has a complex structure and low algorithm operation efficiency.

In the prior art, rainfall is used as input and runoff is used as model output based on time sequence data, influence of other factors on flood is not considered, high error is caused, characteristics of the flood cannot be well reflected, and prediction and submerged point simulation cannot be performed on a flood water body.

Referring to fig. 1, the invention provides a flood disaster prediction and early warning method based on QR-BC-ELM, which comprises the following steps:

s1: extracting multi-factor indexes based on main cause analysis;

s2: constructing a flood disaster risk evaluation system;

s3: improving the extreme learning model, constructing a QR decomposition extreme learning model and a BC decomposition extreme learning model, combining the BC decomposition extreme learning model with a Geographic Information System (GIS) to establish a flood model, and simulating a flood easily-developing area;

Example 1

Referring to fig. 2, the multi-factor index in the step S1 includes a total of 49 calculated indexes in two criterion layers and three index layers.

Example 2

The risk assessment system evaluation step of step S2 includes:

s201: collecting rainfall (M3PD) of three days at most, a terrain humidity index (TWI), a river power index (SPI) and a normalized vegetation index (NDVI) of different hydrological stations as input of a model, and obtaining the correlation degree among the different hydrological stations;

s203: the non-time sequence data takes a Digital Elevation Model (DEM), a soil texture index (ST), a slope index (SL) and a land utilization model (LUP) of each variable station as the input of the model, takes the same index of the variable station as the output, and calculates the index characteristic of the height correlation point;

s204: taking a river distance index (DR), a river power index (SPI) and a socioeconomic index as input, and counting and early warning the flood level and the emergency degree.

Taking the yellow river basin as an example, hydrological data meteorological data and remote sensing data of relevant hydrological stations in the flowing process are respectively extracted, hydrological changes are evaluated and calculated through index relations and water level relations among the hydrological stations, so that the characteristic relation between flood characteristics and an index system is calculated, and the relation among indexes is deduced from the flood water level changes according to a priori probability theory.

Example 3

In a traditional extreme learning machine, after the characteristic engineering of each index is extracted, the characteristic engineering is input into an extreme learning machine model in the form of time series data, the time series data is used as an input matrix of the model, the runoff is used as output, and the weight is corrected in a reverse error propagation mode, so that the contribution degree of each index to flood is obtained. In the prior least square solution solving, a singular value decomposition is generally used for solving a matrix H, then a plus sign generalized inverse is solved, an optimal least square solution is solved as a training weight, and based on the analysis, the most core part of extreme learning is to H⁺And (4) performing the operation of (1).

After the characteristic engineering of the multi-factor indexes is extracted, the characteristic engineering is input into an improved extreme learning model in a time sequence data mode to calculate the contribution degree of each index to flood.

The calculation process of the QR decomposition extreme learning model is as follows:

Wherein

the QR decomposition limit learning model divides H into Q R, namely H Q R, wherein Q is an orthogonal matrix, R is an upper triangular matrix, and the inverse matrix of Q can be obtained by solving the transposition of Q. According to the property of the orthogonal matrix, when the inverse matrix of Q is solved, the transpose of Q is directly solved, so that the inverse matrix can be solved, and the matrix after decomposition is not required to be specially subjected to inversion.

The QR (orthogonal triangle) decomposition extreme learning model shows excellent performance when solving a linear problem, the calculation process is simple and effective, the QR decomposition is adopted to solve the output weight, compared with the singular value decomposition, the orthogonal decomposition method has the advantages of high efficiency and stable performance, the calculation process is greatly simplified, and the QR decomposition shows high-efficiency performance advantages no matter the size of a Hussain matrix.

(II) the calculation process of the BC decomposition limit learning model is as follows:

BC decomposition is full rank decomposition, also called maximum rank decomposition, and A belongs to C^m*nThe decomposition is carried out by taking the matrix A as (a)_ij)_m*nAnd rank a ≦ r min (m, n), then a may be subjected to full rank decomposition, a ═ BC, where B is an m × r order matrix, C is an r × n order matrix, and rank B ═ rank C ═ r;

Example 4

Referring to fig. 3, the improved effect of the present application is embodied by comparing various schemes of artificial neural networks and conventional extreme learning, respectively. Extreme learning with only regularization was used as a comparison. In the experiment, the rainfall capacity of the maximum three days is greatly related to the occurrence of flood, on the other hand, the influence coefficient of the soil texture and the distance from the river to the flood is relatively small, in order to more fully explain the advantages of the model, the data are divided into four parts, three parts of data are trained, one part of data is verified and named as model one to model four, and the first part of data to the fourth part of data are sequentially added and used in a cross verification mode. The prediction of the runoff trend from the correlation index is shown in fig. 3.

It can be seen from fig. 3 that the extreme learning machine except the model one shows a satisfactory fitting degree in time series data prediction, the prediction accuracy of the improved extreme learning in the models one to four is higher than that of the artificial neural network, and the expressions of the models one, three and four are particularly obvious, the unmodified extreme learning shows a large deviation in the prediction effects of the models two and four, while the artificial neural network shows that the larger the processing data amount in prediction, the better the prediction effect, because the iterative algorithm continuously performs error correction, the artificial neural network also achieves a satisfactory prediction result. The first model shows that the artificial neural network and the extreme learning are not excellent in extreme learning under the condition of processing outlier data, the phenomenon of data outlier is called noise, and if the problem is to be avoided, data needs to be regularized. In the aspect of learning efficiency, the extreme learning machine has strong advantages, and in the research, the learning efficiency of the extreme learning machine is 588 times that of the artificial neural network under the condition that the learning rate of the artificial neural network is set to be lower than 0.02, and the learning efficiency of the QR extreme learning machine is improved by 1.12 times.

Referring to fig. 4, in order to better evaluate the accuracy of the model, a scatter point subsection is drawn, an x axis represents an observed value, a y axis represents a predicted value, and a y-x function is given for better observed data, so that it can be known that points of the subsection around the y-x are points with more accurate prediction, and from the comprehensive model one to the model four, no matter improved limit learning or non-improved limit learning is superior to a neural network in the small sample model one, the deviation condition of the neural network in processing the small sample needs to be corrected by continuously and repeatedly increasing sample adjustment iteration parameters, but the limit learning avoids such problems, and thus the generalization capability is improved. With the increase of sample data, the problem of the neural network algorithm is gradually alleviated, such as model two and model three. However, in the model four, the neural network algorithm shows a large deviation in the last data segment, which is that the same-period extreme learning such as SVD and QR show strong stability, the cause of the phenomenon is often caused by noise points in the data, and it is known through the above analysis that the noise points are caused by outlier data, but in an actual situation, the noise points may be caused by collection equipment failure, but more cases are that the water level changes have large fluctuation, and the situation is often a high risk period of flood outbreak.

The comprehensive table and fig. 5 can find a model three and a model four, wherein the pearson correlation coefficient r of the extreme learning machine is larger than that of the artificial neural network, the pearson correlation coefficient is widely used for measuring the correlation degree between two variables, and the coefficient is used for evaluating the correlation degree between a predicted value and an observed value so as to evaluate the accuracy of the model. The value of the Ens Nash efficiency coefficient, which is usually used for evaluating the model in a hydrological model, is negative infinity to 1, the Ens is close to 1, the quality of the representation mode is good, and the reliability of the model is high; ens is close to 0, and represents that the simulation result is close to the average value level of the observed value, namely the overall result is credible, but the simulation error is large; ens is far less than 0, the model is not credible, a Walter index WI corresponding to a Nash index is used for proving the credibility of the model, the Walter index is an absolute value of prediction and observation used unlike the Nash index, and the improved extreme learning machines in the four types of models are all higher than the artificial neural network in view of the Nash index Ens and the Walter index WI, which shows that the credibility of the model is also higher than that of the artificial neural network. In model evaluation, a complementary message between the RMSE and the MAE is a root mean square error, and the MAE is an average absolute error, from which the error level of extreme learning is found to be smaller than that of the artificial neural network, which further indicates that both the error level of the artificial neural network and the model prediction accuracy are smaller than that of extreme learning. The comprehensive four models can find the performances of MAE and RMSE, the BC-based extreme learning precision is higher than QR and SVD, and the comprehensive four models can see that QR extreme learning and SVD extreme learning basically have the lowest error but QR extreme learning is higher than SVD extreme learning in computational efficiency.

Example 5

Referring to fig. 6 and 7, the proportion of each risk proportion in the calendar year is given at first at the beginning of the experiment, and by combining fig. 7, the frequency of the three-grade risk and the four-grade risk is continuously increased in years from 2002.07 beginning to 2019, wherein the three-grade risk and the four-grade risk reach 0.0497 (73.7%) once in years from 2011 to 2019, and the probability of the one-grade risk and the two-grade risk is also continuously increased in years from 02.07 to 19. On one hand, the water level pressure is increased due to the fact that strong rainfall is increased along with the climate change, and the water level can be macroscopically regulated and controlled by combining the water level with the water level regulation method in the figures 6 and 7, so that flood can be scientifically and effectively managed.

Referring to fig. 8, which depicts the distribution and trend of the flood risk levels estimated since 2002 to 2019, it can be seen that substantially no flood risk above three occurs since 2002.01 to 2005.12, but the high risk (fourth level) occurs twice in the four years between 2006.01 and 2010.12. The risk of 2011.06-2019.12 appears for the first time, and the frequency of the risk is higher, the frequency of the risk of four grades is also increased relative to the frequency of 2006.01-2010.12, and the data are observed, the high risk period lasting this time in 19 years reaches hundreds of days, and it can be seen from the combination of fig. 8 that the low risk proportion is also rising substantially, the three-stage risk proportions 2002.07 to 2005.12,2006.01 to 2010.12,2011.01 to 2019.12, the four-stage risk proportions being 3.5%, 22.8%, 73.7%, respectively, and the proportions are seen to be rising, multiple high risk floods occurred during periods 2011.01 to 2019.12, the flood peak this time 19 years according to the trend shown in model 4 of fig. 8 although the water level was high, however, the risk level does not approach four levels as usual, and observing the water level change can find the process in which the short water level drop occurs, which may be related to the dispatching of reservoir flood and also to factors such as rainfall. Leading to an increase in the flood storage capacity of the reservoir, which is, in turn, extremely dangerous if not allocated, since the stations are greatly affected by slight water level changes upstream.

Example 6

Referring to fig. 9, taking the yellow river branch as an example, the yellow river branch under the three gorges, the inland river and the lohe generate centuries flood many times. The iloxowo river is one of the main areas of flood sources in the three gorges-garden mouth area, and the area for analyzing the flood comes from luo (yang), long (gate) and black (stone gate). Meanwhile, according to decision data provided by an intelligent algorithm, the black stone gateway hydrologic station in the Ilo river section, the white horse temple hydrologic station in the Luo river section and the Longmen hydrologic station in the Ilo river section have strong relevance, and by combining the conditions, the hydrologic station is supposed to be adopted as an experimental object, and the white horse temple and the Longmen town are used for predicting the black stone gateway water level. The monitoring data of the three water stations are shown in fig. 8.

The selection method comprises the following steps: traditional neural network models, traditional extreme learning machine models, SVD-ELM, QR-ELM and BC-ELM models. The prediction results of the black stone water level using the mosque and the town of longmen are shown in fig. 8. According to the experimental result, the three aspects of the prediction precision, the outlier data processing and the learning efficiency of the model are evaluated respectively:

in the aspect of precision: the prediction accuracy of the improved extreme learning machine in the scenes 1 to 4 is higher than that of the artificial neural network; in the

scenes

2 and 4, the prediction effect shows that the unmodified extreme learning has larger deviation, and the artificial neural network shows that the larger the processing data amount is, the better the prediction effect is in the prediction, because the iterative algorithm continuously performs error correction, and the artificial neural network also achieves a more satisfactory prediction result.

Processing outlier data: the performance of the artificial neural network in the scene 1 is not excellent in extreme learning, but the improved extreme learning machine has great progress, and the improved extreme learning machine, particularly BC, SVD and QR schemes, has strong stability and robustness in the process of time series data prediction.

Learning efficiency: the extreme learning machine has strong advantages, and researches show that if the learning rate of the artificial neural network is set to be lower 0.02, the learning efficiency of the extreme learning machine is 588 times that of the artificial neural network, and the improved learning efficiency of the QR extreme learning machine is improved by 1.12 times.

Referring to fig. 10, in the small sample scenario 1, both the improved extreme learning and the non-improved extreme learning are better than the neural network, and the deviation of the neural network in processing the small sample needs to be corrected by adjusting iterative parameters through increasing samples repeatedly, but the extreme learning avoids such problems, thereby improving the generalization ability.

As the sample data increases, the problem of neural network algorithms gradually alleviates, for example, scenario 2 and scenario 3. In the scenario 4, the neural network algorithm shows a large deviation in the last data segment, the contemporaneous extreme learning such as SVD, QR shows a strong stability, and a noise point may be caused by a failure of the collection equipment in an actual situation, but more situations are that a large fluctuation occurs in a water level change, and such situations are often a high risk period of flood outbreak. As shown in fig. 10.

Referring to fig. 11, in the comparison of the relevant parameters of the model, a plurality of model performance comparison parameters are selected:

the pearson correlation coefficient is widely used to measure the degree of correlation between two variables, and is used to evaluate the degree of correlation between a predicted value and an observed value to evaluate the accuracy of a model.

The value of the Ens Nash efficiency coefficient, which is usually used for evaluating the model in a hydrological model, is negative infinity to 1, the Ens is close to 1, the quality of the representation mode is good, and the reliability of the model is high; ens is close to 0, and represents that the simulation result is close to the average value level of the observed value, namely the overall result is credible, but the simulation error is large; ens is much less than 0 and the model is not trusted.

The witness index WI is also used to prove the reliability of the model, and is different from the nash index in that the witness index is the absolute value of the prediction and observation used.

In view of the combination of the Nash index Ens and the special index WI, the improved extreme learning machines in the four types of scenes are all higher than the artificial neural network, and the credibility is also higher than the artificial neural network.

This conclusion also leads to the same conclusions in the root mean square error RMSE, the mean absolute error MAE.

The above disclosure is only for a few specific embodiments of the present invention, however, the present invention is not limited to the above embodiments, and any variations that can be made by those skilled in the art are intended to fall within the scope of the present invention.

Claims

1. A flood disaster prediction and early warning method based on QR-BC-ELM is characterized by comprising the following steps:

s1: extracting multi-factor indexes based on main cause analysis;

s2: constructing a flood disaster risk evaluation system;

2. The QR-BC-ELM-based flood disaster prediction and early warning method according to claim 1, wherein the multi-factor indexes in the step S1 include 49 calculated indexes in two criterion layers and three index layers.

3. The QR-BC-ELM-based flood disaster prediction and early warning method according to claim 1, wherein the risk assessment system evaluation step of the step S2 comprises the following steps:

4. The flood disaster prediction and early warning method based on QR-BC-ELM as claimed in claim 1, wherein the characteristic engineering of the multi-factor index is extracted and then input into the improved extreme learning model in the form of time series data to calculate the contribution degree of each index to the flood.

5. The flood disaster prediction and early warning method based on QR-BC-ELM as claimed in claim 4, wherein the QR decomposition extreme learning model is calculated as follows:

Wherein

6. The QR-BC-ELM-based flood disaster prediction and early warning method according to claim 4, wherein the BC decomposition limit learning model is calculated as follows:

let A be C^m*nThe decomposition is carried out by taking the matrix A as (a)_ij)_m*nAnd rank a ≦ r min (m, n), then a may be subjected to full rank decomposition, a ═ BC, where B is an m × r order matrix, C is an r × n order matrix, and rank B ═ rank C ═ r;