Mathematical Principles of Gray Modeling

Gray modeling prediction is the core of gray system theory and methodology. It is to establish a model that describes the dynamic change characteristics of the research system, referred to as GM (h, N) model, there are a variety of models and methods, each model has its own characteristics and scope of application, of which the GM (0, N) model, which does not contain the derivative, is the zero-order N variables based on the generating function of the non-differential model of X (1), is a kind of static model, is suitable for the analysis of the state of the indicators between the model, similar to multiple linear regression models However, there is a fundamental difference with the general multiple linear regression model, multiple linear regression modeling based on a large amount of raw data, GM (0, N) modeling based on a one-time cumulative sequence of raw data (1-AG0).

8.1.1 Mathematical model

Set: X(0)1 = (X(0)1(1), X(0)1(2), ..., X(0)1(N))

For the system characteristic data sequence that is the known borehole coal seam methane content data sequence.

Coalbed methane geological conditions and storage pattern in the Hansheng mining area

is the sequence of relevant factors, i.e., the sequence of main factors affecting the methane content of coal beds.

Coalbed methane geological conditions and storage pattern in the Hancheng mining area

The sequence X(1)i(t) can not only provide intermediate information for modeling, but also make the randomness of the original data sequence weaker. As shown in Fig. 8.1, (a) is the original data sequence curve with obvious oscillation; (b) is the one-time cumulative generation curve, obviously the regularity is enhanced and the randomness is weakened. Generally speaking, for non-negative data series, the more times of accumulation, the more significant the randomness is weakened, the stronger the regularity, showing an exponential law.GM (0, N) model is exactly the sequence (1)

Figure 8.1 data series curve

(a) the original data series; (b) one-time accumulation generation

as the basis of modeling, and its mathematical expression is:

Coalbed methane geological conditions and storage pattern in the Hancheng mining area

8.1.2 Calculation method

(1) Data processing

The sequence X(1)i(t) (i = 1, 2, ..., N) is generated from the original data sequence X(0)i (i = 1, 2, ..., N) by one-time corresponding cumulative generation of the sequence X(1)i(t) (i = 1, 2, ..., N). ..., N, t = 1, 2, ..., n).

(2) Construct matrix x (B) and vector Y

Coalbed methane geological conditions and the pattern of coalbed methane in the Hancheng mining area

8.1.3 Model test

Gray system theory adopts three methods of testing to judge the accuracy of the model:

1) Residual magnitude test: it is a point-by-point test for the error between the model values and the actual values;

2) correlation test: the test is carried out by examining the degree of similarity between the model value curve and the modeling sequence curve;

①Calculation of the residual value ε(0) and the relative error q

Based on the original data sequence x(0)1 = [x(0)1(1), x(0)1(2), ..., x(0)1(n)] and the corresponding The difference between the model simulation sequence and the corresponding residual sequence ε(0) can be found:

Coalbed methane geological conditions and storage patterns in the Hansheng mine area

Relative error sequence

Coalbed methane geological conditions and storage patterns in the Hansheng mine area

The average simulation relative error is 1--q.

②Model Accuracy level assessment

If -q is within the permissible range, the predicted value can be calculated, otherwise residual correction is required, according to experience, the model is generally tested according to the following accuracy level (Table 8.1).

Table 8.1 Accuracy test level reference table

3) Posterior difference test: it is to test the statistical characteristics of the residual analysis.

In general, the most commonly used is the relative error test index.