Engineering project management based on multiple regression equation and building information modelling technology

The selection of samples at the initial stage of predictive model establishment is more important. Therefore, we follow the requirements of the sampling conditions and consult relevant information. At the same time, it is reasonable to choose the average value of the tunnelling parameters in each round of BIM technology as the statistical value [5]. Following the above principles and combined with the actual situation of the data, we eliminate weak points, too large parameter values, and too long holes in the original data. The article selects 807 samples from the lithological section of the Yang Formation. The structure of the sample rock mass is mainly layered, of which the principal type III surrounding rock.

The best combination of sample variables is expressed as follows: (1) There is no correlation between variables, that is, non-collinearity. (2) The correlation between the variables entering the model and the dependent variable (geological score) is good. However, in practice, the natural geological conditions vary greatly, and the heterogeneity and anisotropy of rock and soil reduce the correlation between the collected parameters and the dependent variables. Therefore, we adopt two methods when establishing the prediction equation to improve the model's prediction accuracy: (1) consider the BIM technology tunnelling parameters. (2) Combination of tunnelling parameters and geological factors. Table 1 shows the correlation coefficients between the excavation parameter samples and the geological factors. It can be seen that the correlation coefficient between tunnelling parameters and geological factors is low. Therefore, the correlation is lacking, which is in line with the best combination of sample variables.

The construction speed of BIM technology is greatly affected by the surrounding rock geological conditions. The better the geological conditions of the surrounding rock, the faster the construction and excavation speed. On the contrary, the slower. Why is the correlation between the BIM technology excavation parameters and surrounding rock geological factors not apparent, as shown in Table 1? This is determined by the characteristics of the diversion tunnel of the hydropower station [6]. The water diversion tunnel has a considerable buried depth, and the effect of in-situ stress is pronounced, and the rockburst damage is severe. The better the geological conditions during the BIM technology excavation process, the easier it is for rock bursts. This resulted in the excavation speed not being too fast in the tunnel section with better geological conditions.

We use multiple linear regression and multiple stepwise regression to perform regression analysis on the tunnelling parameters and the combined model of tunnelling parameters and rock strength.

Let y_kl = y(x_tkl|x_1k,x_2k, ⋯ , x_qk) be the l observation on the k sample function of the multivariate linear process y. x_tk1 is the process independent variable on the k sample function. The i value (x_tk1 < x_tk2 < ⋯ < x_tknk) : x_1k,x_2k, ⋯ ,x_qk of x_t is the value of the state independent variable x₁,x₂, ⋯ ,x_q on the k-th sample function. k = 1,2, ⋯ ,m;l = 1,2,3, ⋯ ,n_k and m > q respectively [7]. When the first test time is the same, we can set x_t1 = x_t11 = x_t21 = ⋯ = x_tm1 in Eq. (6). Then the likelihood function of the multivariate linear process y is (8) L=∏k=1m{12πσ1exp[−12σ12(yk1−a−bxt1−∑i=1qbixik)]2×∏l=2nk12πΔxtklI(x¯tkl)exp[−(Δykl−bΔxtkl)22σ02ΔxtklI(x¯tkl)]} L = \prod\limits_{k = 1}^m \left\{{{1 \over {\sqrt {2\pi {\sigma _1}}}}\exp {{\left[ {- {1 \over {2\sigma _1^2}}\left({{y_{k1}} - a - b{x_{t1}} - \sum\limits_{i = 1}^q {b_i}{x_{ik}}} \right)} \right]}^2} \times \prod\limits_{l = 2}^{{n_k}} {1 \over {\sqrt {2\pi \Delta {x_{tkl}}I({{\overline x}_{tkl}})}}}\exp \left[ {- {{{{(\Delta {y_{kl}} - b\Delta {x_{tkl}})}^2}} \over {2\sigma _0^2\Delta {x_{tkl}}I({{\overline x}_{tkl}})}}} \right]} \right\}

According to the maximum likelihood principle and unbiasedness of the estimator, the estimators of a,b,b, σ02 \sigma _0^2 , σ12 \sigma _1^2 and θ can be obtained as (9) a^=y¯1−b^xt1−∑i=1qb^ix¯1=y¯1−b^xt1−b^xt1−b^Tx¯ \widehat a = {\overline y _1} - \widehat b{x_{t1}} - \sum\limits_{i = 1}^q {\widehat b_i}{\overline x _1} = {\overline y _1} - \widehat b{x_{t1}} - \widehat b{x_{t1}} - {\widehat b^T}\overline x (10) b^=ltxyltxx \widehat b = {{{l_{txy}}} \over {{l_{txx}}}} (11) σ^02=1υ∑k=1m∑l=2nk(Δykl−b^Δxtkl)2(ΔxtklI(x¯tkl)=1υ(ltyy−ltxy2ltxx) \widehat \sigma _0^2 = {1 \over \upsilon}\sum\limits_{k = 1}^m \sum\limits_{l = 2}^{{n_k}} {{{{(\Delta {y_{kl}} - \widehat b\Delta {x_{tkl}})}^2}} \over {(\Delta {x_{tkl}}I({{\overline x}_{tkl}})}} = {1 \over \upsilon}\left({{l_{tyy}} - {{l_{txy}^2} \over {{l_{txx}}}}} \right) (12) b^=(b^1,b^2,⋯,b^q)T=Lxx−1lxy \widehat b = {({\widehat b_1},{\widehat b_2}, \cdots,{\widehat b_q})^T} = L_{xx}^{- 1}{l_{xy}} (13) σ^12=1m−q−1∑k=1m(yk1−a^−b^xt1−∑i=1qb^ixik)2=1m−q−1(lyy−lxyTLxx−1lxy) \widehat \sigma _1^2 = {1 \over {m - q - 1}}\sum\limits_{k = 1}^m {\left({{y_{k1}} - \widehat a - \widehat b{x_{t1}} - \sum\limits_{i = 1}^q {{\widehat b}_i}{x_{ik}}} \right)^2} = {1 \over {m - q - 1}}\left({{l_{yy}} - l_{xy}^TL_{xx}^{- 1}{l_{xy}}} \right) (14) E(θ)=2ltxyltxxltxyθ−ltxy2ltxxltxxθ−ltyyθ+n*+υ+m−nυ(ltyy−ltxy2ltxx)=0 E(\theta) = 2{{{l_{txy}}} \over {{l_{txx}}}}{l_{txy\theta}} - {{l_{txy}^2} \over {{l_{txx}}}}{l_{txx\theta}} - {l_{tyy\theta}} + {{{n^*} + \upsilon + m - n} \over \upsilon}\left({{l_{tyy}} - {{l_{txy}^2} \over {{l_{txx}}}}} \right) = 0

Where v is the degree of freedom of σ^02 \widehat \sigma _0^2 . When θ ≠ 0 is v = n − m − 2 when θ = 0 is v = n − m − 1, n=∑k=1mnk n = \sum\limits_{k = 1}^m {n_k} ; n*=∑k=1m∑l=2nk1I(x¯tkl) {n^*} = \sum\limits_{k = 1}^m \sum\limits_{l = 2}^{{n_k}} {1 \over {I({{\overline x}_{tkl}})}} and (15) x¯t=1n∑k=1m∑l=1nkxtkl {\overline x _t} = {1 \over n}\sum\limits_{k = 1}^m \sum\limits_{l = 1}^{{n_k}} {x_{tkl}} (16) x¯i=1m∑k=1mxik {\overline x _i} = {1 \over m}\sum\limits_{k = 1}^m {x_{ik}} (17) y¯1=1m∑k=1myk1 {\overline y _1} = {1 \over m}\sum\limits_{k = 1}^m {y_{k1}} (18) x¯tkl=xtkl+xtk(l−1)2 {\overline x _{tkl}} = {{{x_{tkl}} + {x_{tk(l - 1)}}} \over 2}

4.2.1

Multiple linear regression

We consider two situations. (1) The article selects 5 tunnelling parameters to enter the model. (2) Six parameters of tunnelling parameters and rock strength are entered into the model. The paper uses SPSS software to perform regression analysis on a large amount of data to obtain the summary table (Table 2) and model equations. That is, 2 equations are obtained. Equation (1) is a linear equation composed of BIM technology tunnelling parameters. Equation (2) is a linear equation composed of the BIM technology tunnelling parameters and the saturated uniaxial compressive strength of the rock. The establishment of the two equations is to allow all selected parameters to enter the equation [8]. The number of variables is not significant. Although there is collinearity between the parameters, it does not affect the use of the equation. (19) Tscore=55.071+1.161r+0.321T+0.0928S−0.377F−0.180P {T_{score}} = 55.071 + 1.161r + 0.321T + 0.0928S - 0.377F - 0.180P (20) Tscore=−21.738+0.99r−0.123T+0.043S−0.128F−0.055P+1.015Rb {T_{score}} = - 21.738 + 0.99r - 0.123T + 0.043S - 0.128F - 0.055P + 1.015{R_b}

Table 2

Model overview table

Model		Model 1	Model 2
Correlation coefficient		0.243	0.82
Fitting coefficient		0.059	0.673
Adjusted fitting coefficient		0.053	0.67
Standard error of the estimate		6.9209	4.0857
Corrected statistics	Fitting coefficient	0.059	0.673
	F	10.078	273.827
	Degree of freedom 1	5	6
	Degrees of freedom 2	801	800
	Significant level	0	0

Equation (2) is much better than that of Eq. (1). The introduction of rock strength significantly improves the linear relationship between model dependent variables and independent variables. Use 2 regression equations to verify the selected 807 sets of data [9]. The score span of the two equations in the original 807 sets of data and the coincidence length of the scores are shown in Figure 1, Tables 3 and 4.

Fig. 1

Table 3

Equations (1), (2) score span table

Score span	Equation (1)	Equation (2)
Minimum	50.17	38.92
Max	61.46	77.46

Table 4

Equations (1), (2) points value coincidence rate table

Surrounding rock category	Field length	Equation (1)	Equation (2)
IIs	148.2	0	148.2
II	80.3	0	77.4
IIIs	30.7	30.7	13.6
III	1002.4	1002.4	787.7
IV	14.7	0	14.7
Coincidence rate	-	80.94%	81.61%

The following points can be obtained from Tables 3, 4, and Figure 1. (1) From the perspective of the prediction score range, the prediction score range of Eq. (2) is more extensive (38–78), and the range of Eq. (1) is smaller (50–61). Thus, there is a big difference between the surrounding rock score and the original geological score. (2) From the perspective of the coincidence rate of geological scores, the coincidence rates of Eqs. (1) and (2) are 80.94% and 81.61%, respectively. Still, equation one can only reflect type III surrounding rock, and Eq. (2) covers three types of surrounding rock: II, III, and IV. (3) The type of surrounding rock above the red area in Figure 2 is II or IIb, and the score of Eq. (1) does not reach this height [10]. The scores of Eq. (1) in the red area are all distributed between 50 and 60. Class IV surrounding rocks below the red zone cannot be effectively predicted. This makes the prediction of the equation meaningless. Although the predicted score of Eq. (2) does not partly match the geological field score, it does not exceed the category of the surrounding rock. From the graph, the trend of scattered points is the same. Therefore, it can be seen that the accuracy and practicability of Eq. (2) are better than that of Eq. (1).

Fig. 2