CN101644595A - Fitting method of complex water level process - Google Patents
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Abstract
本发明公开了一种复杂水位过程的拟合方法——分层变换筛选拟合法,其将多项式回归、逐步回归、参数的岭估计等有机集成,并引进累进变换,系统形成了一个新方法。本方法与同类方法的核心区别,在于考虑了复杂水位过程中常见的弱影响因素间的强耦合作用;综合采用了多种方法最大限度地降低拟合模型误差;并引进了必要的累进变换。本方法有机集成了多个理论和方法的长处,且使用方便,有着同类复杂拟合问题下的普遍适用性。
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
技术领域 technical field
本发明涉及水文学及水资源领域,具体是一种复杂水位过程的拟合方法。The invention relates to the fields of hydrology and water resources, in particular to a fitting method for complex water level processes.
背景技术 Background technique
水情预报中需要根据历史数据,建立有效的关系模型,特别是对于水情复杂的河流,建立关系模型的难度是很大的。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.
以黄河为例,黄河下游汛期水沙观测数据中的内在关系,有很强的复杂性。其一,相对水少沙多。黄河中游的三门峡水文站多年年平均含沙量35kg/m3、输沙量约16亿吨,同时黄河泥沙颗粒很细,有时河水甚至呈泥浆状态;其二,水、沙时空分布不均。全年60%的水量和80%的泥沙集中来自汛期,汛期又主要来自几场暴雨洪水。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.
这些特殊性使其汛期水位表现出很强的不同特征。第一,同期同断面相同流量(不同时刻)的水位能相差0.6m以上;第二,在上游断面相同水位的两个洪峰演进到下游时,表现出来的水位能相差0.2m以上;第三,断面水位陡升陡降。由于问题本身的复杂性,世界上在黄河水沙过程有效拟合方面的研究较少。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.
在黄河下游复杂水位过程的拟合方面,一些文献采用了水文学、水力学的模型,申请人在研究中也使用过半参数、非线性高维回归等模型和方法,拟合效果均不理想。改进了多元统计中的方差分析后,拟合效果较为明显,但计算过于复杂,且需取得影响因素值分别相同条件下(不同时刻),相应响应变量的(不同)值。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.
工程问题中常常出现这样的情况,一些影响因素的耦合,对响应变量产生了显著更强的影响(如耦合共振)。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).
统计学的理论和方法,都是有针对性地分析数据中的某类规律。多项式回归提供了模型结构的选择之一,但应用中模型普适性往往较差;逐步回归能剔除回归不显著的项得到最优回归模型,但未考虑影响因素的强耦合作用;非线性回归给出了模型中已存在的非线性关系项的处理方式,不能给出非线性项的形式;当模型构成项间存在多重相关性时,岭估计可以比最小二乘估计提供模型参数的更稳定的方差也更小的估计,但也不能给出非线性项的形式等。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.
许多工程问题的内在规律很复杂,在分析这些规律时,仅使用一两种理论或方法往往难以取得好的效果。这时需要针对具体问题特点,将几个同类理论和方法的长处有机集成,引进必要的新的处理,并从理论上使处理过程完善,形成能有效分析该类问题内在规律的新方法。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.
发明内容 Contents of the invention
本发明所要解决的技术问题是提供一种复杂水位过程的拟合方法-分层变换筛选拟合法,该方法能有效分离出显著非线性耦合扰动,提高模型精度。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.
本发明所述的复杂水位过程的拟合方法,包括以下步骤:The fitting method of complex water level process of the present invention comprises the following steps:
1)针对水位过程y的拟合,确定y的所有可能影响因素x1,…,xn,并按影响因素与相应水位对应的原则整理相应的原始数据;由整理出的数据,依x1,…,xn两两间散点图或线性相关系数,剔除x1,…,xn间的共线性关系,设剔除共线性关系后,剩余的影响因素为z1,…,zm(m≤n);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)依y分别与z1,…,zm两两间散点图体现的关系,对z1,…,zm中与y间是非线性关系的因素作线性化变换,并以变换后的形式取代原影响因素,与不需变换的原影响因素一起作为基本参量,组成多元回归多项式;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)依各基本参量与各复合非线性项两两之间的散点图或线性相关系数,剔除回归多项式中各阶项之间的共线性关系;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)依剩余各复合非线性项与y两两间散点图体现的关系,对与y间是非线性关系的复合非线性项作线性化变换,并以变换后形式完全取代回归多项式中的相应复合非线性项;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)依剩余各阶项与y两两间散点图或线性相关系数,剔除所有对y影响不显著的项,得拟合模型;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)以岭估计法计算拟合模型参数,并检验拟合效果。6) Calculate the fitting model parameters by ridge estimation method, and check the fitting effect.
本发明考虑了工程问题中常见的弱影响因素间的耦合对响应变量的强作用,其最大限度地综合使用剔除共线性、线性化变换、剔除弱影响项等,有效降低了模型误差。该方法有机集成了多个理论和方法的长处,且使用方便。本方法的每一步都有充分的理论保证其合理性、必要性,有着同类拟合问题下的普遍适用性。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.
附图说明 Description of drawings
图1是y与x1关系散点图;Figure 1 is a scatter diagram of the relationship between y and x1 ;
图2是y与x2关系散点图;Figure 2 is a scatter diagram of the relationship between y and x 2 ;
图3是y与1/x3关系散点图;Figure 3 is a scatter diagram of the relationship between y and 1/x 3 ;
图4是y与x4关系散点图;Figure 4 is a scatter diagram of the relationship between y and x 4 ;
图5是y与x1x4关系散点图;Figure 5 is a scatter diagram of the relationship between y and x1x4;
图6是y与x2/x3关系散点图。Figure 6 is a scatter diagram of the relationship between y and x2/x3.
具体实施方式 Detailed ways
1、本发明的具体步骤如下:1, the concrete steps of the present invention are as follows:
步骤1针对水位过程y的拟合,确定y的所有可能影响因素x1,…,xn,并按影响因素与相应水位对应的原则整理相应的原始数据。由整理出的数据,依x1,…,xn两两间散点图或线性相关系数,剔除x1,…,xn间的共线性关系。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 .
这里不要剔除任何有相应观测数据的可能影响因素,因为考虑到这些因素状态的不同搭配,可能产生对y的联合强作用。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.
影响因素间共线性关系,表明这些因素间有几乎完全相同的物理意义,只保留其中有样本观察值,且形式相对简单的因素。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.
设剔除共线性关系后,剩余的影响因素为z1,…,zm(m≤n)。It is assumed that after the collinear relationship is eliminated, the remaining influencing factors are z 1 , ..., z m (m≤n).
步骤2依y分别与z1,…,zm两两间散点图体现的关系,对z1,…,zm中与y间是非线性关系的因素作线性化变换[18,19]。并以变换后的形式取代原影响因素,与不需变换的原影响因素一起作为基本参量,组成多元回归多项式。
工程问题的有解性,使回归多项式一般能成立。三阶及以上高阶项在工程问题中一般难以找到对应的物理解释,一般略去三阶及以上项。为叙述方便,二阶及二阶以上项称为复合非线性项。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.
步骤3依各基本参量与各复合非线性项两两之间的散点图或线性相关系数,剔除回归多项式中各阶项之间的共线性关系。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.
这时的共线性关系中剔除复合非线性项。At this time, the compound nonlinear term is removed from the collinear relationship.
步骤4依剩余各复合非线性项与y两两间散点图体现的关系,对与y间是非线性关系的复合非线性项作线性化变换。并以变换后形式完全取代回归多项式中的相应复合非线性项。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.
由于组成这里复合非线性项的基本参量,有的是已作过线性化变换的,所以对这里复合非线性项所作的变换称为累进变换。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.
对于与y间是线性关系的复合非线性项,也可作适当变换,使之与y间线性关系更强,这样可以更进一步提高最终模型精度。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.
步骤5依剩余各阶项与y两两间散点图或线性相关系数,剔除所有对y影响不显著的项,得拟合模型。这时应剔除模型构成项中所有影响不显著的项。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.
步骤6以岭估计法计算拟合模型参数,并检验拟合效果。Step 6 Calculate the fitting model parameters by ridge estimation method, and check the fitting effect.
这里模型构成项间很可能有较强的相关性。因此选用岭估计将能给出使模型精度更高的参数估计,且岭估计往往比最小二乘估计更稳定,尽管岭估计的期望与真实参数值间有微小偏差。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.
拟合模型中复合非线性项的样本值,由相应原始影响因素的样本值按数学关系确定。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.
先保留弱影响因素,累次剔除共线性项,累进线性化变换,变换后形式取代相应项构成模型等等,这些非常用方法的有机综合采用,使得本文所提方法显著区别于现有同类方法。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.
考虑到工程问题中大量存在的耦合作用,注意步骤5不可在前面执行。方法中步骤1~5都能适当消除最终模型的随机误差,特别是2、4步。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.
为叙述方便,上述六步体现的完整方法称为分层变换筛选拟合法。概括起来说即,引进变量并仅剔除影响因素间共线性,线性化与y是非线性关系的因素并引进多元回归多项式,剔除回归多项式中共线性,线性化与y是非线性关系的复合非线性项,剔除回归多项式中所有线性趋势不显著的项,以岭估计计算模型参数。这六步的次序不能颠倒。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、以下是采用本发明对黄河下游复杂水位过程的拟合,据以说明本方法的有效性。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.
黄河中下游河床冲刷和淤积都很剧烈,其水文过程中隐含的水文规律很复杂。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确定待拟合水位过程和相应影响因素,按相应原则整理对应数据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
依水文和泥沙学科相关理论,黄河下游上监测断面出现某水体时,该水体的相应下游水位y的影响因素有:该水体在上断面出现时的水位x1、含沙量x2、水沙系数x3和下游同时水位x4。这里x3与x1以及x3与x2关联较强,x2与x1有一定关联。x3在泥沙学科中称为水沙系数,体现单位流量水流的挟沙量。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.
由于拟合模型需进一步用于预报,这里考虑相应下游水位y的拟合。借助上下游相应水位过程线,按各影响因素与相应水位y的对应,精确摘录到黄河花园口-夹河滩间某年y与x1、x2、x3、x4的对应值见表1。该年汛期最大含沙量在150kg/m3以上,属于典型复杂的年份。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.
经相应散点图分析,x1、x2、x3、x4两两间均没有共线性关系。According to the corresponding scatter diagram analysis, there is no collinear relationship among x 1 , x 2 , x 3 , and x 4 .
表1黄河下游某年y与x1、x2、x3、x4的实测数据及拟合结果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
注:表中日期711即7月11日,时刻1800、1720分别为18时0分和17时20分,余类推。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线性化与y间是非线性关系的影响因素,并引进多元多项式2.2 Linearization and y are the influencing factors of the nonlinear relationship, and introduce multivariate polynomials
y分别与x1、x2、1/x3、x4两两间散点关系见图1~4,图1~4中纵坐标均为y值。图1有比较明确的线性趋势,图2主体部分有一定的线性趋势,图3是带宽较大的线性趋势(因为y与x3间有一些弱的双曲线趋势),图4也是带宽偏大的线性趋势。根据分层变换筛选拟合法要求,以1/x3取代x3作进一步分析。图1~4可见y与x1、x2、x3、x4间均无共线性关系,且均取两个以上的不同值。依分层变换筛选拟合法步骤2,取x1、x2、1/x3、x4作为基本参量构成y的四元回归多项式(1)。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
y=a0+a1x1+a2x2+a3(1/x3)+a4x4+a5x1 2+a6x2 2+a7(1/x3)2+a8x4 2 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
+a9x1x2+a10x1/x3+a11x1x4+a12x2/x3+a13x2x4+a14x4/x3+ε (1)+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)
式中ai,i=0,1,…,14为待定参数,ε为随机误差。In the formula, a i , i=0, 1, ..., 14 are undetermined parameters, and ε is a random error.
2.3剔除基本参量与复合非线性项间的所有共线性关系2.3 Eliminate all collinear relationships between basic parameters and composite nonlinear terms
x1与x1 2、x4与x4 2间各是抛物线关系,但每年汛期x1、x4的值均分别只在离零点较远,且相对较小的范围内变动(参见表1),这一抛物线在这一小定义区间上几乎是直线段。x1、x4变化的特点,也使x1x2、x2x4相当于在x2上分别乘上两个不同的常数。事实上,x1与x1 2、x4与x4 2、x2与x1x2(或x2x4)、1/x3与x1/x3(或x4/x3)之间的线性相关系数均在0.9999以上,也就是说,他们之间各是共线性关系,故剔除相对复杂的六个复合非线性项。经检验,式(1)中等号右侧,四个基本参量和四个剩余复合非线性项x1x4、x2/x3、x2 2、1/x3 2两两间无共线性关系。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对与y间是非线性关系的剩余复合非线性项累进变换2.4 Progressive transformation of residual compound nonlinear items with nonlinear relationship between pair and y
y与x1x4散点分布参见图5,有较强线性趋势。y与x2/x3散点分布参见图6,总体上有明显非线性对数关系。故变换x2/x3为ln(x2/x3),并以ln(x2/x3)取代x2/x3。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 ).
y与x2 2散点关系总体特征类似图2,y与1/x3 2散点关系总体特征类似图3,均显示关系较弱。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选择线性趋势显著的各项,并给出拟合模型2.5 Select the items with significant linear trend and give the fitting model
综上及表2中y与各项间线性相关系数,取x1、x1x4、ln(x2/x3)三项构成y的拟合模型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=b0+b1x1+b2x1x4+b3ln(x2/x3)+e (2)y=b 0 +b 1 x 1 +b 2 x 1 x 4 +b 3 ln(x 2 /x 3 )+e (2)
其中b0,…,b3为待定参数,都有相应量纲。e为模型误差。Among them, b 0 ,..., b 3 are undetermined parameters with corresponding dimensions. e is the model error.
表2y与四个基本参量及四个剩余复合非线性项两两间线性相关系数Table 2 Linear correlation coefficients between y and four basic parameters and four residual composite non-linear items
2.6确定拟合模型参数2.6 Determine the fitting model parameters
将复合非线性项看成新变量,依岭估计计算拟合模型参数,计算中岭参数的确定采用方差膨胀因子法。得y的拟合模型见式(3),拟合效果参见表1中拟合值和绝对误差。y=66.3997-0.091198x1+0.00183257x1x4+0.46400697ln(x2/x3) (3)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)
取黄河下游花园口-夹河滩、夹河滩-高村两对断面,较长系列(连续20余年)各年汛期的水沙观测数据,分别用黄河下游水位预报模型及其应用(芮孝芳,陈洁云,常星源,等.黄河下游水位预报模型及其应用.水科学进展,1998,9(3):245-250);水位演算模型及其在水位预报中的应用(黄国如,朱庆平,马俊,等.水位演算模型及其在水位预报中的应用.水文,2(1999):1-6.);I模型方程与数值方法(张红武,黄远东,赵连军,等.黄河下游非恒定输沙数学模型——I模型方程与数值方法.水科学进展,2002,(3):265-271.)等中的模型和方法拟合,所得模型精度都比本发明明显要低。数据中隐含的水文规律很复杂,模拟效果显著也说明了本文所提方法的科学性。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拟合模型中复合非线性项物理意义解释2.7 Interpretation of the physical meaning of the compound nonlinear term in the fitting model
x1x4是上游水位与下游同时水位的耦合项。x2/x3实质上是上游流量,上游流量与相应下游水位是对数关系符合物理背景。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.
工程问题中有这样一类研究指标,其特点是,多个影响因素中有一些是显著的,另一些的单独作用不一定显著,但当它们达到某种耦合时,对研究指标的耦合作用会很显著,同时研究指标与其影响因素的实测不同值都在两个以上。在拟合这类研究指标时,本文给出的分层变换筛选拟合法更适宜。本方法与同类方法的核心区别,在于考虑了工程问题中常见的弱影响因素间的耦合,对响应变量的强作用;最大限度地综合使用剔除共线性、线性化变换、剔除弱影响项等,有效降低模型误差;并进行必要的累进变换。该方法有机集成了多个理论和方法的长处,且使用方便。方法的每一步都有充分的理论保证其合理性、必要性,有着同类拟合问题下的普遍适用性。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.
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