1. quantification theory and its application in groundwater management
"Quantification" is the quantitative expression of specified things. Qualitative data is only a description of a state. It does not have the nature of arithmetic operation. In fact, quantitative theoretical method is the analysis method of qualitative data, which is a branch of multivariate analysis.
In the problems we study, variables can often be classified according to their nature: some variables can be regarded as the causes of change, called explanatory variables or independent variables; Another variable is regarded as the result of change, which is called standard variable or dependent variable. On the other hand, according to its changes, it can be divided into two situations: one is what we usually call variables, such as length, weight and volume, which are called quantitative variables; Another variable does not really change in quantity, but only in nature, such as weather (cloudy or sunny), lithology (clay, sand), hydrogeological conditions (good or bad), etc., which is called qualitative variable.
In the practical work of hydrogeology, the role of qualitative variables can not be ignored. As we know, hydrogeology is a highly practical science, and in many cases, practical experience is very important. For example, in hydrogeological generalization and parameter selection, we need to accumulate rich experience on the basis of various materials. This is why it is not enough to have a good mathematical foundation when establishing a hydrogeological prediction and management model. In fact, this kind of "experience" is a qualitative thing, which is difficult to express with numbers.
Using the quantitative theory, experts' experience can be "calculated" through the description of some qualitative variables, and used for forecasting and management decisions. This is of great significance to the establishment of hydrogeology expert system.
In fact, quantitative variables and qualitative variables can be transformed into each other. If we divide the number axis into several disjoint intervals, when some quantitative variables take values in the same interval, they are considered to be at the same level, so these quantitative variables are transformed into qualitative variables and the corresponding data are transformed into qualitative data. On the contrary, it is feasible to transform qualitative variables and their data into quantitative variables according to reasonable principles, and to classify and predict according to the obtained quantitative data, which is also the content and purpose of quantitative theory. Quantification theory enables us to use not only quantitative variables, but also qualitative variables to study problems, so that we can make full use of information and comprehensively study and discover the relationships and laws between things, so it is widely used. However, the quantification theory belongs to the developing theory, and there are still many problems worth studying. For example, how to choose projects, how to classify them, and how quantitative data will affect the results when converted into qualitative data. Especially, the application of this theory in groundwater prediction and management has just begun.
Quantification theory can be divided into four types according to different research purposes, which are called quantification theory ⅰ, ⅱ, ⅲ and ⅳ respectively. Among them, the main purpose of the study of theories ⅱ, ⅲ and ⅳ is to classify quantitative and qualitative problems into variables or samples, so I won't repeat them here. We mainly introduce the model I use to predict and discover relationships.
Second, the quantitative theory Ⅰ
In quantitative theory, qualitative variables are often called terms, and various "values" of qualitative variables are called categories. For a forecasting problem (called benchmark variable Y), its influencing factor variables (qualitative) can be called terms: such as x 1, x2, …, xn; Each qualitative variable has several possibilities, so the range of all possibilities of qualitative variable is the corresponding category of the variable. For example, in order to predict the degree of groundwater pollution (benchmark variable), the shape, lithology and distribution of vadose zone of pollution sources can be regarded as terms. Each project includes several categories, such as the form of groundwater pollution sources, which can be divided into three categories: point pollution sources, line pollution sources and non-point pollution sources; The lithologic items of vadose zone can be divided into clay, silt, coarse sand and gravel (Zhao Yongsheng, 1992). Suppose we observe n samples. If the J-th project has rj category, you can list the reaction table of the project and category (table 15-3).
In the table: y is the reference variable, δi(j, k) (I = 1, 2, …, n; j= 1，2，…，m； K = 1, 2, ..., rj) The reaction of class K called item J in sample I:
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If x 1 is the pollution source shape, there are three types, C 1 1, C 12 and c13; ; They are point pollution source, line pollution source and non-point pollution source respectively. If in a given example (a sample), x 1 can only be one of three situations: C 1, C 12 and C 13, the corresponding category is1,and the other two situations are zero. If it is a linear pollution source, the item x 1 belongs to the category C 12, so that C1,C1,C 13 are all zero, which is the formula (/kloc
Table 15-3 Project Category Table
According to the properties of δi(j, k):
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The reaction matrix x consists of all Δ I (j, k), and x is a matrix of order n× p. ..
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(A) the mathematical model of quantitative theory I and its solution
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or
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Here bjk is the constant coefficient of K category that only depends on J item, εi is the random error of the ith sampling, the reference variable vector, X is the response matrix, coefficient vector, error vector, and other symbolic meanings are the same as before.
According to the known samples and benchmark variables, the unknown coefficient bjk in the model can be obtained. Use the least square method, that is, find bjk, like this:
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Reach the minimum value. To this end, find the partial derivative of Q with respect to buv and make it equal to zero, and get:
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Because this is a necessary condition for the minimum point, when bjk reaches the minimum value, it satisfies:
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The matrix form of the formula (15-20) is:
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Where: y=(y 1, y2, …, yn) t;
We call equation (15-20) or equation (15-2 1) a normal system of equations. It can be proved that: ① the coefficient matrix of normal equations is symmetrical; ② Because the response matrix X is the assignment matrix of qualitative variables, the coefficient matrix X ′ x of the normal equation (15-2 1) does not satisfy the rank, and its rank R (x ′ x) is at most rj-m+ 1. Therefore, the solution of the equation is infinite.
Assuming that the rank of x ′ x is rj-m+ 1 (in practical problems, it can be guaranteed when n is large enough), then we can delete the first equation of the j-th term (j=2, …, m) and take =0. The deleted matrix is a full rank matrix, so the rest can be solved uniquely. Mathematically, it can be proved that the solution obtained in this way does not lose generality, and q in the formula (15- 19) can be minimized. The expression is:
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(2) prediction accuracy analysis
(1) complex correlation coefficient r:
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Where: the predicted value and the measured value respectively; Is the average of the measurements. The closer the r value is to 1, the higher the prediction accuracy is.
(2) residual mean square:
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Where: n is the number of samples; M is the number of influencing variables. The smaller the residual mean square value, the higher the interpretation accuracy.
(3) A model with both qualitative and quantitative variables
When the problem under consideration has both qualitative variables and quantitative variables, of course, the quantitative variables can be transformed into qualitative variables by dividing them into several grades, and then treated by the method of quantitative theory I. However, this transformation is sometimes inappropriate, because the transformation from quantitative variables to qualitative variables actually loses the information in the data.
Let a problem have m terms and h quantitative variables, and their data in the ith sample are Xi (u) (u = 1, 2, …, h; I= 1, 2, …, n). Using the above method to derive the quantitative theoretical model I, we can get similar results with quantitative variables and qualitative variables.
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