CN101644595A - Fitting method of complex water level process - Google Patents

Abstract

The invention discloses a fitting method of complex water level process-layered transformation screening fitting method, which organically integrates polynomial regression, stepwise regression, parameter ridge estimation, etc., and introduces progressive transformation to form a new method systematically. The core difference between this method and similar methods is that it considers the strong coupling between the common weak influencing factors in the complex water level process; it uses a variety of methods to minimize the error of the fitting model; and it introduces the necessary progressive transformation. This method organically integrates the strengths of multiple theories and methods, is easy to use, and has universal applicability to similar complex fitting problems.

Description

A Fitting Method for Complicated Water Level Process

technical field

The invention relates to the fields of hydrology and water resources, in particular to a fitting method for complex water level processes.

Background technique

In water regime forecasting, it is necessary to establish an effective relationship model based on historical data, especially for rivers with complex water regimes, it is very difficult to establish a relationship model.

Taking the Yellow River as an example, the internal relationship in the observation data of water and sediment in the flood season in the lower reaches of the Yellow River is very complicated. First, relatively less water and more sand. The Sanmenxia hydrological station in the middle reaches of the Yellow River has an average annual sediment concentration of 35kg/m ³ and a sediment load of about 1.6 billion tons. At the same time, the sediment particles of the Yellow River are very fine, and sometimes the river water is even in a muddy state; second, the temporal and spatial distribution of water and sand is uneven. . 60% of the water and 80% of the sediment in the whole year come from the flood season, which mainly comes from several storms and floods.

These particularities make the water level in the flood season show strong different characteristics. First, the water levels of the same flow rate (at different times) at the same section in the same period can differ by more than 0.6m; second, when two flood peaks with the same water level in the upstream section evolve to the downstream, the water levels shown can differ by more than 0.2m; third, The water level of the section rises and falls steeply. Due to the complexity of the problem itself, there are few researches on the effective fitting of the Yellow River's water-sediment process in the world.

In terms of fitting the complex water level process in the lower reaches of the Yellow River, some literatures have adopted models of hydrology and hydraulics, and the applicant has also used models and methods such as half-parameter and nonlinear high-dimensional regression in the research, but the fitting results are not satisfactory. After improving the analysis of variance in multivariate statistics, the fitting effect is more obvious, but the calculation is too complicated, and it is necessary to obtain the (different) values of the corresponding response variables under the same conditions (at different times) of the influencing factor values.

It is often the case in engineering problems that the coupling of some influencing factors produces a significantly stronger effect on the response variable (such as coupled resonances).

The theories and methods of statistics are all about analyzing certain types of laws in data in a targeted manner. Polynomial regression provides one of the options for the model structure, but the universality of the model in the application is often poor; stepwise regression can eliminate the items that are not significant in the regression to obtain the optimal regression model, but does not consider the strong coupling of the influencing factors; nonlinear regression The processing method of the existing nonlinear relationship items in the model is given, and the form of the nonlinear items cannot be given; when there are multiple correlations among the model components, the ridge estimation can provide more stable model parameters than the least squares estimation The variance of is also a smaller estimate, but also cannot give the form of non-linear terms etc.

The inherent laws of many engineering problems are very complex. When analyzing these laws, it is often difficult to achieve good results by using only one or two theories or methods. At this time, it is necessary to organically integrate the strengths of several similar theories and methods according to the characteristics of the specific problem, introduce necessary new processing, and theoretically improve the processing process to form a new method that can effectively analyze the internal laws of this type of problem.

Contents of the invention

The technical problem to be solved by the present invention is to provide a fitting method of complex water level process - layered transformation screening fitting method, which can effectively separate significant nonlinear coupling disturbances and improve model accuracy.

The fitting method of complex water level process of the present invention comprises the following steps:

1) For the fitting of the water level _process y, determine all possible influencing factors x ₁ , ..., x _n of y, and organize the corresponding original data according to the principle that the influencing factors correspond to the corresponding water level; , ..., x _n pairs of scatter diagrams or linear correlation coefficients, remove the collinear relationship between x ₁ , ..., x _n , suppose that after eliminating the collinear relationship, the remaining influencing factors are z ₁ ,..., z _m ( m≤n);

2) According to the relationship between y and z ₁ , ..., z _m shown in pairwise scatter diagrams, linearize the factors in z ₁ , ..., z _m that have a nonlinear relationship with y, and use the transformed The form replaces the original influencing factors, and together with the original influencing factors that do not need to be transformed are used as basic parameters to form a multiple regression polynomial;

3) According to the scatter diagram or linear correlation coefficient between each basic parameter and each compound nonlinear term, the collinear relationship between each order item in the regression polynomial is eliminated;

4) According to the relationship between the remaining composite nonlinear items and the scatter diagram of y, perform linear transformation on the composite nonlinear items that have a nonlinear relationship with y, and completely replace the corresponding regression polynomial in the transformed form. Composite nonlinear terms;

5) According to the pairwise scatter diagram or linear correlation coefficient between the remaining items of each order and y, remove all items that have no significant impact on y, and then fit the model;

6) Calculate the fitting model parameters by ridge estimation method, and check the fitting effect.

The invention considers the strong effect of coupling between common weak influencing factors on the response variable in engineering problems, and comprehensively uses collinearity elimination, linear transformation, weak influence item elimination, etc. to the greatest extent, effectively reducing model errors. This method organically integrates the strengths of multiple theories and methods, and is easy to use. Each step of this method has sufficient theory to guarantee its rationality and necessity, and has universal applicability under similar fitting problems.

Description of drawings

Figure 1 is a scatter diagram of the relationship between y and _x1 ;

Figure 2 is a scatter diagram of the relationship between y and x ₂ ;

Figure 3 is a scatter diagram of the relationship between y and 1/x ₃ ;

Figure 4 is a scatter diagram of the relationship between y and x ₄ ;

Figure 5 is a scatter diagram of the relationship between y and x1x4;

Figure 6 is a scatter diagram of the relationship between y and x2/x3.

Detailed ways

1, the concrete steps of the present invention are as follows:

Step 1 Aiming at the fitting of the water level process y, determine all possible influencing factors x ₁ ,..., x _n of y, and organize the corresponding original data according to the principle that the influencing factors correspond to the corresponding water level. According to the collated data, according to the scatter diagram or linear correlation coefficient between x ₁ , ..., x _n , eliminate the collinear relationship among x ₁ , ..., x _n .

Do not exclude any possible influencing factors with corresponding observation data here, because considering the different collocations of these factors, they may have a joint strong effect on y.

The collinear relationship between the influencing factors indicates that these factors have almost the same physical meaning, and only those factors with sample observation values and relatively simple forms are kept.

It is assumed that after the collinear relationship is eliminated, the remaining influencing factors are z ₁ , ..., z _m (m≤n).

Step 2 According to the relationship between y and z ₁ , ..., z _m shown in pairwise scatter diagrams, linearize the factors in z ₁ , ..., z _m that have a nonlinear relationship with y ^{[18, 19]} . And replace the original influencing factors with the transformed form, and use the original influencing factors that do not need to be transformed as the basic parameters to form a multiple regression polynomial.

The solvability of engineering problems makes regression polynomials generally valid. It is generally difficult to find corresponding physical explanations for the third-order and higher-order terms in engineering problems, and the third-order and higher-order terms are generally omitted. For the convenience of description, the second-order and above-order terms are called compound nonlinear terms.

Step 3: According to the scatter diagram or linear correlation coefficient between each basic parameter and each compound nonlinear term, the collinear relationship between each order item in the regression polynomial is eliminated.

At this time, the compound nonlinear term is removed from the collinear relationship.

Step 4: According to the relationship shown in the pairwise scatter diagram between each remaining composite nonlinear item and y, linearize the composite nonlinear item that has a nonlinear relationship with y. and completely replace the corresponding composite non-linear term in the regression polynomial in the transformed form.

Since some of the basic parameters that make up the compound nonlinear term here have been linearized, the transformation made on the compound nonlinear term here is called progressive transform.

For the composite non-linear item which has a linear relationship with y, it can also be properly transformed to make the linear relationship with y stronger, which can further improve the accuracy of the final model.

Step 5: According to the pairwise scatter diagram or linear correlation coefficient between the remaining items of each order and y, remove all items that have no significant impact on y, and then fit the model. At this time, all items with insignificant influence in the model constituent items should be eliminated.

Step 6 Calculate the fitting model parameters by ridge estimation method, and check the fitting effect.

There is likely to be a strong correlation between the model components here. Therefore, the choice of ridge estimation will give a parameter estimate that makes the model more accurate, and the ridge estimate is often more stable than the least squares estimate, although there is a slight deviation between the expectation of the ridge estimate and the real parameter value.

The sample value of the compound nonlinear item in the fitting model is determined by the sample value of the corresponding original influencing factor according to the mathematical relationship.

Retain weak influencing factors first, eliminate collinear items repeatedly, progressive linear transformation, replace corresponding items with transformed forms to form models, etc. The organic comprehensive adoption of these uncommon methods makes the method proposed in this paper significantly different from existing similar methods.

Considering the coupling effects that exist in a large number of engineering problems, note that step 5 cannot be performed before. Steps 1 to 5 in the method can properly eliminate the random error of the final model, especially steps 2 and 4.

For the convenience of description, the complete method embodied in the above six steps is called hierarchical transformation screening and fitting method. In a nutshell, variables are introduced and only collinearity between influencing factors is eliminated, linearization and y are non-linear factors and multiple regression polynomials are introduced, regression polynomial collinearity is eliminated, linearization and y are non-linear composite non-linear items, Eliminate all items with insignificant linear trends in the regression polynomial, and calculate the model parameters by ridge estimation. The order of these six steps cannot be reversed.

2. The following is the fitting of the present invention to the complex water level process in the lower reaches of the Yellow River, in order to illustrate the effectiveness of the method.

The river bed erosion and deposition in the middle and lower reaches of the Yellow River are very severe, and the hydrological laws implied in the hydrological process are very complicated.

2.1 Determine the water level process to be fitted and the corresponding influencing factors, and organize the corresponding data according to the corresponding principles

According to the relevant theories of hydrology and sediment science, when a certain water body appears in the upper monitoring section of the lower reaches of the Yellow River, the factors affecting the corresponding downstream water level y of the water body are: the water level x ₁ , the sediment content x ₂ , the water level when the water body appears in the upper section Sand factor x ₃ and downstream simultaneous water level x ₄ . Here x ₃ has a strong correlation with x ₁ and x ₃ has a strong correlation with x ₂ , and x ₂ has a certain correlation with x ₁ . x ₃ is called the water-sediment coefficient in sediment science, which reflects the amount of sediment carried by water flow per unit flow.

Since the fitting model needs to be further used for forecasting, the fitting of the corresponding downstream water level y is considered here. With the help of the corresponding water level process lines in the upper and lower reaches, according to the correspondence between each influencing factor and the corresponding water level y, the corresponding values of y and x ₁ , x ₂ , x ₃ , and x ₄ in a certain year between Huayuankou and Jiahe Beach of the Yellow River are accurately extracted, as shown in Table 1 . The maximum sediment concentration in the flood season of this year was above 150kg/m ³ , which was a typical and complicated year.

According to the corresponding scatter diagram analysis, there is no collinear relationship among x ₁ , x ₂ , x ₃ , and x ₄ .

Table 1 The measured data and fitting results of y and x ₁ , x ₂ , x ₃ , x ₄ in a certain year in the lower reaches of the Yellow River

Note: The date 711 in the table is July 11, the times 1800 and 1720 are 18:00 and 17:20 respectively, and so on.

2.2 Linearization and y are the influencing factors of the nonlinear relationship, and introduce multivariate polynomials

The scatter relationship between y and x ₁ , x ₂ , 1/x ₃ , and x ₄ is shown in Figures 1 to 4, and the ordinates in Figures 1 to 4 are all y values. Figure 1 has a relatively clear linear trend, Figure 2 has a certain linear trend in the main part, Figure 3 is a linear trend with a large bandwidth (because there are some weak hyperbolic trends between y and _x3 ), and Figure 4 also has a large bandwidth linear trend. According to the requirements of hierarchical transformation screening and fitting method, 1/x ₃ was used instead of x ₃ for further analysis. From Figures 1 to 4, it can be seen that there is no collinear relationship between y and x ₁ , x ₂ , x ₃ , and x ₄ , and they all take more than two different values. According to step 2 of hierarchical transformation screening and fitting method, x ₁ , x ₂ , 1/x ₃ , and x ₄ are taken as basic parameters to form the quaternary regression polynomial (1) of y.

y＝a ₀ +a ₁ x ₁ +a ₂ x ₂ +a ₃ (1/x ₃ )+a ₄ x ₄ +a ₅ x ₁ ² +a ₆ x ₂ ² +a ₇ (1/x ₃ ) ² +a ₈ x ₄ ²

+a ₉ x ₁ x ₂ +a ₁₀ x ₁ /x ₃ +a ₁₁ x ₁ x ₄ +a ₁₂ x ₂ /x ₃ +a ₁₃ x ₂ x ₄ +a ₁₄ x ₄ /x ₃ +ε (1)

In the formula, a _i , i=0, 1, ..., 14 are undetermined parameters, and ε is a random error.

2.3 Eliminate all collinear relationships between basic parameters and composite nonlinear terms

The relationship between x ₁ and x ₁ ² , and x ₄ and x ₄ ² is a parabolic relationship, but the values of x ₁ and x ₄ in the flood season each year are only far from zero and change within a relatively small range (see Table 1 ), this parabola is almost a straight line segment over this small defined interval. The changing characteristics of x ₁ and x ₄ also make x ₁ x ₂ and x ₂ x ₄ equivalent to multiplying x ₂ by two different constants. In fact, x ₁ and x ₁ ² , x ₄ and x ₄ ² , x ₂ and x ₁ x ₂ (or x ₂ x ₄ ), 1/x ₃ and x ₁ /x ₃ (or x ₄ /x ₃ ) The linear correlation coefficients between them are all above 0.9999, that is to say, they are each collinear, so six relatively complicated composite non-linear items are eliminated. After testing, on the right side of the equal sign in formula (1), there is no collinearity between the four basic parameters and the four residual composite non-linear items x ₁ x ₄ , x ₂ /x ₃ , x ₂ ² , 1/x ₃ ² relation.

2.4 Progressive transformation of residual compound nonlinear items with nonlinear relationship between pair and y

See Figure 5 for the scatter distribution of y and x ₁ x ₄ , which has a strong linear trend. See Figure 6 for the scatter distribution of y and x ₂ /x ₃ , and there is an obvious nonlinear logarithmic relationship on the whole. Therefore, transform x ₂ /x ₃ into ln(x ₂ /x ₃ ), and replace x ₂ /x ₃ with ln(x ₂ /x ₃ ).

The overall characteristics of the scatter-point relationship between y and x ₂ ² are similar to Figure 2, and the overall characteristics of the scatter-point relationship between y and 1/x ₃ ² are similar to Figure 3, both showing that the relationship is weak.

2.5 Select the items with significant linear trend and give the fitting model

Based on the above and the linear correlation coefficient between y and each item in Table 2, three items x ₁ , x ₁ x ₄ , and ln(x ₂ /x ₃ ) are used to form the fitting model of y

y＝b ₀ +b ₁ x ₁ +b ₂ x ₁ x ₄ +b ₃ ln(x ₂ /x ₃ )+e (2)

Among them, b ₀ ,..., b ₃ are undetermined parameters with corresponding dimensions. e is the model error.

Table 2 Linear correlation coefficients between y and four basic parameters and four residual composite non-linear items

2.6 Determine the fitting model parameters

The composite nonlinear term is regarded as a new variable, and the fitting model parameters are calculated according to the ridge estimation, and the variance inflation factor method is used to determine the ridge parameters in the calculation. The fitting model of y is shown in formula (3), and the fitting effect is shown in the fitting value and absolute error in Table 1. y＝66.3997-0.091198x ₁ +0.00183257x ₁ x ₄ +0.46400697ln(x ₂ /x ₃ ) (3)

Taking Huayuankou-Jiahetan and Jiahetan-Gaocun cross-sections in the lower reaches of the Yellow River, a long series (more than 20 consecutive years) of water and sediment observation data in each flood season were used to predict the water level of the lower reaches of the Yellow River and its application (Rui Xiaofang, Chen Jieyun, Chang Xingyuan, et al. Water level forecasting model and its application in the lower reaches of the Yellow River. Advances in Water Science, 1998, 9(3): 245-250); water level calculation model and its application in water level forecasting (Huang Guoru, Zhu Qingping, Ma Jun, et al. Water level calculus model and its application in water level forecasting. Hydrology, 2(1999): 1-6.); I model equations and numerical methods (Zhang Hongwu, Huang Yuandong, Zhao Lianjun, etc. Mathematics of unsteady sediment transport in the lower reaches of the Yellow River Model—I Model Equation and Numerical Method. Advances in Water Science, 2002, (3): 265-271.) etc. Model and method fitting, the accuracy of the model obtained is obviously lower than the present invention. The hydrological law implied in the data is very complex, and the simulation effect is remarkable, which also shows the scientific nature of the method proposed in this paper.

2.7 Interpretation of the physical meaning of the compound nonlinear term in the fitting model

x ₁ x ₄ is the coupling term of the upstream water level and the downstream simultaneous water level. x ₂ /x ₃ is essentially the upstream flow, and the logarithmic relationship between the upstream flow and the corresponding downstream water level conforms to the physical background.

There is such a class of research indicators in engineering problems, which is characterized in that some of the multiple influencing factors are significant, and the individual effects of others are not necessarily significant, but when they reach a certain coupling, the coupling effect on the research indicators will be significant. Obviously, at the same time, there are more than two measured differences between the research indicators and their influencing factors. When fitting such research indicators, the hierarchical transformation screening fitting method given in this paper is more suitable. The core difference between this method and similar methods is that it considers the coupling between the common weak influencing factors in engineering problems and the strong effect on the response variable; it maximizes the comprehensive use of collinearity elimination, linearization transformation, and elimination of weak influence items, etc. Effectively reduce the model error; and carry out the necessary progressive transformation. This method organically integrates the strengths of multiple theories and methods, and is easy to use. Each step of the method has sufficient theory to guarantee its rationality and necessity, and has universal applicability under similar fitting problems.

Claims (1)

1, a kind of fitting method of complex water level process is characterized in that comprising the following steps: 1) For the fitting of the water level _process y, determine all possible influencing factors x ₁ , ..., x _n of y, and organize the corresponding original data according to the principle that the influencing factors correspond to the corresponding water level; , ..., x _n pairs of scatter diagrams or linear correlation coefficients, remove the collinear relationship between x ₁ , ..., x _n , suppose that after eliminating the collinear relationship, the remaining influencing factors are z ₁ ,..., z _m ( m≤n); 2) According to the relationship between y and z ₁ , ..., z _m shown in pairwise scatter diagrams, linearize the factors in z ₁ , ..., z _m that have a nonlinear relationship with y, and use the transformed The form replaces the original influencing factors, and together with the original influencing factors that do not need to be transformed are used as basic parameters to form a multiple regression polynomial; 3) According to the scatter diagram or linear correlation coefficient between each basic parameter and each compound nonlinear term, the collinear relationship between each order item in the regression polynomial is eliminated; 4) According to the relationship between the remaining composite nonlinear items and the scatter diagram of y, perform linear transformation on the composite nonlinear items that have a nonlinear relationship with y, and completely replace the corresponding regression polynomial in the transformed form. Composite nonlinear terms; 5) According to the pairwise scatter diagram or linear correlation coefficient between the remaining items of each order and y, remove all items that have no significant impact on y, and then fit the model; 6) Calculate the fitting model parameters by ridge estimation method, and check the fitting effect.

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