A method for predicting building settlement based on improved non-uniformly spaced grey model

Abstract

The present invention provides a building settlement prediction method based on an improved non-equidistant grey model, comprising the following steps: Step 1, transforming the cumulative settlement data sequence to obtain the cotx function to transform the settlement data sequence x1 0 , for x1 0 The first-order cumulative settlement data sequence is obtained by weighting and first-order accumulation; step 2, a non-uniformly spaced GM (1, 1) model of the cotx function transformation is constructed using the first-order cumulative settlement data sequence, and the initial fitness of the model is calculated; step 3, an optimal weight parameter of the background value of the model is obtained by using a particle swarm algorithm; step 4, an improved non-uniformly spaced GM (1, 1) model of the cotx function transformation is established using the weight parameter optimized in step 3; step 5, a fitting prediction is performed on the trend of the building cumulative settlement data, and the fitting prediction result is subjected to an accuracy test. The method of the present invention is simple and easy to implement, has a good settlement prediction effect, does not require a large amount of preliminary settlement data, and saves settlement monitoring data costs.

Description

Technical Field

[0001] The invention belongs to the technical field of settlement prediction, and in particular relates to a building settlement prediction method based on an improved non-equidistant grey model. Background Art

[0002] With the development of China's infrastructure model, various tall and majestic engineering buildings have sprung up. While promoting economic development, they have also changed the ground morphology, exerting certain pressure on the foundation, causing changes in the foundation and surrounding areas. Predicting this settlement through limited observations and understanding its changing trends is of great significance for relevant safety prevention.

[0003] Building settlement prediction methods primarily rely on multiple linear regression, gray theory, time series methods, and neural networks. Multiple linear regression requires a linear relationship between the independent and dependent variables, which makes it somewhat restrictive. Time series methods, on the other hand, require a certain amount of time series data to summarize patterns, which is time-consuming and expensive in actual settlement monitoring. Finally, neural network methods also require a large amount of building settlement monitoring data samples.

[0004] The gray GM(1,1) model has been widely used in various industries due to its simple principle, low computational effort, and verifiability, achieving rapid development in both theory and practice. However, the gray GM(1,1) model also has certain limitations, and its prediction performance for some building settlement data often falls short of expectations. For example, data smoothness has a certain impact on the accuracy of building settlement prediction using the gray GM(1,1) model. Poor data smoothness indicates significant fluctuations in settlement. Functional transformation of the original sequence can improve smoothness and thus the accuracy of model predictions. It is important to note that different functional transformations, or transformations of the same trigonometric function within different intervals, can significantly affect the prediction results. Therefore, it is necessary to select an appropriate function interval based on the data characteristics. For unequal-time settlement monitoring, due to the uneven temporal intervals of the data, a non-equally spaced GM(1,1) model is required to predict the settlement data. However, using equal weights of the settlement data to construct the model background value increases the error in the model background value construction. Therefore, it is necessary to select simpler and more effective means to improve the model based on its shortcomings and problems, so as to better predict the future development trend of engineering construction settlement. Summary of the Invention

[0005] The purpose of the embodiment of the present invention is to provide a building settlement prediction method based on an improved non-uniformly spaced gray model to solve the problems of uneven accumulation of settlement data, large fluctuations in settlement amount and construction errors of model background values in the gray GM (1,1) model in the prior art in building settlement prediction.

[0006] To solve the above technical problems, the technical solution adopted by the present invention is a building settlement prediction method based on an improved non-uniformly spaced grey model, comprising the following steps:

[0007] Step 1: For the cumulative settlement data sequence x 0 Transform the cotx function to transform the sedimentation data sequence x1 0 , for x1 0 Weighted and first-order cumulative sedimentation data sequence x1 1 ;

[0008] Step 2: Use the first-order cumulative settlement data sequence x1 1 Construct a non-uniformly spaced GM(1,1) model transformed by the cotx function and calculate the initial fitness of the model; the initial fitness is the cotx function transformed sedimentation data sequence x1 0 and the model fitting sequence The average relative error of

[0009] Step 3: Use the particle swarm algorithm to obtain the optimal weight parameter λ′ of the background value of the non-uniformly spaced GM(1,1) model transformed by the cotx function; each particle in the particle swarm algorithm represents a first-order cumulative sedimentation transformation value x1 at the i-th observation (1) (t i ) and the first-order cumulative sedimentation transformation value x1 at the i-1th observation (1) (t i-1 ) between the fitting coefficients;

[0010] Step 4: Use the weight parameter λ′ optimized in step 3 to establish the non-uniformly spaced GM(1,1) model of the improved cotx function transformation;

[0011] Step 5: Use the improved model from step 4 to fit and predict the trend of building cumulative settlement data. The fitting prediction results are tested for accuracy using relative error and grey correlation index.

[0012] Furthermore, the step 1 is specifically as follows:

[0013] Cumulative settlement data series x 0 ={x (0) (t1),x (0) (t2),…,x (0) (t n)}, where t1, t2…t n is the observation time corresponding to the cumulative settlement data, x (0) Represents the cumulative settlement data of the building; for the cumulative settlement data sequence x 0 Perform compression transformation to satisfy the cotx function transformation interval:

[0014]

[0015] where t i ={t1,t2,…,t n}, N and q are constants, intermediate variables Y 0 (t i ) represents t i Compression transformation of time; cotx function transforms the sedimentation data sequence x1 0 ={x1 (0) (t1),x1 (0) (t2),…,x1 (0) (t n )}={cot(Y 0 (t1)),cot(Y 0 (t2)),…,cot(Y 0 (t n ))} When performing first-order accumulation generation, add the time weight Δt i =t i -t i-1 , i=2,3,…,n, and obtain the first-order cumulative settlement data sequence x1 1 ={x1 (1) (t1),x1 (1) (t2),…,x1 (1) (t i ),…,x1 (1) (t n )},in

[0016]

[0017] Where Δt i It represents the time difference between the cumulative settlement data at the time of observation i and observation i-1, when i=1, Δt1=1; x1 (0) Indicates the cumulative settlement data x (0) Data after cotx function transformation, x1 (1) Represents the first-order cumulative settlement data, x1 (1) (t i ) represents t i First-order cumulative settlement data at time t.

[0018] Furthermore, the step 2 is specifically as follows:

[0019] Take the first-order cumulative settlement data sequence x1 1 Establish the grey system theory about the observation time t of settlement data i The whitened differential equation of :

[0020]

[0021] Where a is the development coefficient, which indicates the development law and trend of the cumulative settlement sequence; u is the gray action, which reflects the changing relationship between the cumulative settlement sequences. Both a and u are parameters to be solved; t i ={t1,t2,…,t n} is the data observation time corresponding to the cumulative settlement data; x1 (1) (t i ) represents t i First-order cumulative settlement data at time;

[0022] Using formula (3) to replace differential, we get

[0023] x1 (0) (t i )+az (1) (t i )=u (4)

[0024] Formula (4) is the basic form of the GM (1,1) model, where x1 (0) Represents the building's cumulative settlement data x (0) Data after cotx function transformation; z (1) (t i ) is the cotx function transforming the sedimentation data sequence x1 0 The sequence of adjacent mean values is generated, which represents the average value between the two cumulative settlement values. Shifting formula (4) yields:

[0025] Y=BU

[0026] Among them, Y represents the array of settlement data values transformed by the cotx function, B is the coefficient matrix of a and u, and U is the parameter column to be calculated, that is:

[0027]

[0028] And the sequence z is generated next to the mean (1) (t i ) From the GM(1,1) model, we know that:

[0029] z (1) (t i )=0.5[x1 (1) (i)+x1 (1)(i-1)],i=1,2…,n. (5)

[0030] The estimated values of the development coefficient a and the gray action u in the parameter list U are obtained using the least squares principle. and The estimated parameter list is:

[0031]

[0032] where x1 (1) (t1) = x1 (0) (t1), the solution of equation (3) is:

[0033]

[0034] Formula (7) is the time response function of the non-uniformly spaced GM (1, 1) model. The prediction equation of the non-uniformly spaced GM (1, 1) model transformed by the cotx function is obtained by cumulative reduction of formula (7), namely:

[0035]

[0036] Among them, e is a natural constant, x1 (1) The fitting sequence of x1 (0) The fitting sequence of

[0037] Use formula (8) to find the cotx function to transform the settlement data sequence x1 0 The fitting sequence of Transform the sedimentation data series x1 by the cotx function 0 and fitting series Use formula (9) to find the fitting sequence The average relative error provides the initial fitness for the particle swarm algorithm to find the optimal background value of the model:

[0038]

[0039] Where n is the number of cotx function transformed settlement data participating in the non-uniformly spaced GM(1,1) model calculation.

[0040] Furthermore, the step of using the particle swarm algorithm to find the optimal weight parameter λ′ of the background value in step 3 is:

[0041] S31, particle initialization, set the number of particles, randomize particle position λ = [0, 1] and particle speed vlimit = [-1, 1], set the maximum number of iterations, set the individual initial optimal position λ best = 0.5 and the initial optimal fitness and the initial optimal position λ of the group best = 0.5 and the initial optimal fitness

[0042] S32, calculating the fitness of each particle;

[0043] S33, updating the individual optimal position of each particle and the group optimal position of the particle swarm based on the calculated fitness of each particle;

[0044] S34, updating the speed and position of each particle in the particle swarm according to the new individual optimal position and the group optimal position, and then determining whether the speed and position of each particle exceeds the particle position limit and the speed limit. If so, setting the speed and position of each particle to be equal to the corresponding limit boundary value;

[0045] S35, determine whether the maximum number of iterations has been reached, if not, go to S32; if the maximum number of iterations has been reached, output the group's best position gbest k That is the optimal weight parameter λ′.

[0046] Furthermore, the initial optimal fitness in S31

[0047]

[0048] Where n is the number of cotx function transformed settlement data participating in the non-uniformly spaced GM(1,1) model calculation, x1 (0) (t i ) represents t i Cumulative settlement data at time x (0) (t i ) The data after cotx function transformation, x1 (0) The fitting sequence, t i ={t1,t2,…,t n} is the data observation time corresponding to the cumulative settlement data.

[0049] Furthermore, the specific method of S33 is:

[0050] If the calculated fitness of particle k is better than the fitness of the individual best position, the individual best position pbest of particle k is updated. k If the fitness of particle k is better than the fitness of the group's best position, the individual position of particle k is the new group's best position gbest k .

[0051] Furthermore, step four is specifically as follows:

[0052] Based on the process of constructing the non-uniformly spaced GM(1,1) model of the cotx function transformation in step 2, the background value calculation formula is changed to z (1) (t i )=λ′x1 (1) (t i-1 )+(1-λ′)x1 (1) (t i ),λ′∈[0,1], and use the least squares method to obtain a new parameter sequence According to the parameter sequence pass The time response function of the non-uniformly spaced GM(1,1) model based on the cotangent function and the improved background value is obtained, and the time response function is cumulatively reduced to obtain the non-uniformly spaced GM(1,1) model of the improved Cotx function transformation;

[0053] In the above formula, λ′ is the optimal weight parameter, z (1) (t i ) is the cotx function transforming the sedimentation data sequence x1 0 The adjacent mean value of the sequence is generated, which represents the average of the two cumulative settlement values before and after, x1 (0) (t i ) represents t i Cumulative settlement data at time x (0) (t i ) The data after cotx function transformation, x1 (0) The fitting sequence, t i ={t1,t2,…,t n} is the data observation time corresponding to the cumulative settlement data, and are the estimated values of the development coefficient a and the gray action u in the parameter column U.

[0054] Furthermore, the fitting prediction method in step five is specifically as follows:

[0055] Transform the sedimentation data sequence x1 using the cotx function 0 Substitute the non-uniformly spaced GM(1,1) model transformed by the improved cotx function to obtain the fitted prediction sequence Finally, perform the inverse cotangent and then use the compression formula The cumulative sedimentation data sequence x is obtained by reduction 0 The fitted predicted value of In the formula, t i ={t1,t2,…,t n} is the data observation time corresponding to the cumulative settlement data, N and q are constants, n is the number of settlement data.

[0056] Furthermore, the precision test method of step five is specifically as follows:

[0057] is the absolute value of the ratio of the residual to the cumulative settlement of the building, where Δ k Represents the relative error, x (0) Represents the cumulative settlement data of the building, t n is the data observation time corresponding to the cumulative settlement data; ε represents the residual value;

[0058] is the average relative error; the smaller the average relative error, the lower the deviation between the cumulative settlement value and the fitting prediction value, and the more accurate the fitting and prediction results;

[0059] Grey relational degree n is the cumulative settlement data sequence x 0 The number of cumulative settlement data, x0(k) and x i (k) are the actual settlement value and the fitted predicted value, ξ is the resolution coefficient, which is taken as 0.5; the larger the grey correlation degree, the higher the consistency of the trend between the fitted sequence and the actual settlement value over time.

[0060] The beneficial effects of the present invention are:

[0061] Compared with other function transformation methods, this method takes into account the characteristics of the cotx function to select a suitable transformation interval when reducing the fluctuation of the settlement amount of the settlement data series. Without changing the model structure, the particle swarm algorithm is used to automatically search for the optimal parameters of its background value, avoiding other complex formulas for improving the background value structure, and improving the accuracy of the grey GM (1,1) model for settlement prediction.

[0062] The method of the present invention is simple and easy to implement. Its settlement prediction effect is better than that of the single improved GM (1,1) model, and it does not require a large amount of early settlement data, thus saving the cost of settlement monitoring data. BRIEF DESCRIPTION OF THE DRAWINGS

[0063] In order to more clearly illustrate the embodiments of the present invention or the technical solutions in the prior art, the following briefly introduces the drawings required for use in the embodiments or the description of the prior art. Obviously, the drawings described below are only some embodiments of the present invention. For ordinary technicians in this field, other drawings can be obtained based on these drawings without paying any creative work.

[0064] Figure 1 4 is a flow chart of a prediction method according to an embodiment of the present invention. DETAILED DESCRIPTION

[0065] The following will clearly and completely describe the technical solutions in the embodiments of the present invention in conjunction with the accompanying drawings. Obviously, the described embodiments are only part of the embodiments of the present invention, not all of the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by ordinary technicians in this field without making creative efforts are within the scope of protection of the present invention.

[0066] The present invention provides a building settlement prediction method based on an improved non-equidistant grey model. Figure 1 As shown, the following steps are included:

[0067] Step 1: For the cumulative settlement data sequence x 0 Transform the cotx function to transform the sedimentation data sequence x1 0 , for x1 0 Weighted first-order accumulation to obtain the first-order accumulation settlement data sequence x1 1 .

[0068] Assume that the cumulative settlement data sequence x 0 ={x (0) (t1),x (0) (t2),…,x (0) (t n )}, where t1, t2…t n is the observation time corresponding to the cumulative settlement data, x (0) Indicates the cumulative settlement data of buildings (structures), such as x (0) (t1) represents the cumulative settlement data of the building at time t1, x (0) (t i ) represents t i The cumulative settlement data of the building at the moment; for the cumulative settlement data sequence x 0 Perform compression transformation of formula (1) to satisfy the cotx function transformation interval, and then perform cotx function transformation to improve the smoothness of the cumulative settlement data:

[0069]

[0070] where t i ={t1,t2,…,t n}, N and q are constants, Y 0 (t i ) represents t i Compression transformation of time.

[0071] Transform the sedimentation data sequence x1 using the cotx function 0 ={x1 (0) (t1),x1 (0) (t2),…,x1(0) (t n )}={cot(Y 0 (t1)),cot(Y 0 (t2)),…,cot(Y 0 (t n ))} When performing first-order accumulation generation, add the time weight Δt i =t i -t i-1 ,i=2,3,…,n, where Δt1=1, Δt i represents the time difference between the cumulative settlement data at the time of observation i and observation i-1, x1 (0) Indicates the cumulative settlement data x (0) Data after cotx function transformation, such as x1 (0) (t1) represents the cumulative settlement data x at time t1 (0) (t1) Data after cotx function transformation, x1 (0) (t i ) represents t i Cumulative settlement data at time x (0) (t i ) After the data is transformed by the cotx function, the first-order cumulative settlement data sequence x1 is obtained by the above method. 1 ={x1 (1) (t1),x1 (1) (t2),…,x1 (1) (t i ),…,x1 (1) (t n )},in

[0072]

[0073] Where x1 (1) Represents the first-order cumulative settlement data, x1 (1) (t i ) represents t i First-order cumulative settlement data at time t.

[0074] Step 2: Construct a non-uniformly spaced GM(1,1) model transformed by the cotx function and calculate the initial fitness of the model (i.e., the cotx function transforms the sedimentation data sequence x1 0 and the model fitting sequence The average relative error of

[0075] Take the first-order cumulative settlement data sequence x1 1 Establish the grey system theory about the observation time t of settlement data i The whitened differential equation of :

[0076]

[0077] Where a is the development coefficient, which represents the development law and trend of the first-order cumulative settlement data series; u is the grey action, which reflects the changing relationship between the first-order cumulative settlement data series. Both a and u are parameters to be solved.

[0078] Using formula (3) to replace differential, we get

[0079] x1 (0) (t i )+az (1) (t i )=u (4)

[0080] Formula (4) is the basic form of the GM (1,1) model, where z (1) (t i ) is the cotx function transforming the sedimentation data sequence x1 0 The adjacent mean value of the sequence is generated, which represents the average of the two cumulative settlement values before and after. Shifting formula (4) yields

[0081] Y=BU

[0082] Among them, Y represents the array of settlement data values transformed by the cotx function, B is the coefficient matrix of a and u, and U is the parameter column to be calculated, that is,

[0083]

[0084] And the sequence z is generated next to the mean (1) (t i ) From the basic knowledge of GM(1,1) model, we know that:

[0085] z (1) (t i )=0.5[x1 (1) (t i )+x1 (1) (t i-1 )],i=1,2,…,n (5)

[0086] The estimated values of the development coefficient a and the gray action u in the parameter list U are obtained using the least squares principle. and The estimated parameter list

[0087]

[0088] Where Δt1=1, then x1 (1) (t1) = x1 (0) (t1), the solution of equation (3) is

[0089]

[0090] Formula (7) is the time response function of the non-uniformly spaced GM (1, 1) model. The prediction equation of the non-uniformly spaced GM (1, 1) model transformed by the cotx function is obtained by cumulative reduction of formula (7), that is,

[0091]

[0092] e is a natural constant, and the cotx function is used to transform the sedimentation data sequence x1 using formula (8): 0 The fitting sequence of Transform the sedimentation data series x1 by the cotx function 0 and fitting series Use formula (9) to find the fitting sequence The average relative error provides the initial fitness for the particle swarm algorithm to find the optimal background value of the model:

[0093]

[0094] Where n is the number of cotx function transformed settlement data participating in the non-uniformly spaced GM(1,1) model calculation.

[0095] Step 3: Use the particle swarm algorithm to obtain the optimal weight parameter λ′ of the background value of the non-uniformly spaced GM(1,1) model of the cotx function transformation, so that the cumulative settlement data sequence x with different observation time intervals obtained during deformation monitoring of buildings (structures) is 0 More closely aligned with the actual background value of the model.

[0096] The traditional GM(1,1) model generates background values in a manner close to the mean, but in the non-uniformly spaced GM(1,1) model, the data intervals are not equal, and the traditional construction method may not be ideal. Transform the sedimentation data sequence x1 with the cotx function 0 and cotx function transform non-uniformly spaced GM(1,1) model prediction series The average relative error is the fitness function of the particle swarm algorithm to find the optimal weight parameter λ′, that is

[0097] z (1) (t i )=λ′x1 (1) (t i-1 )+(1-λ′)x1 (1) (t i ),λ′∈[0,1] (10)

[0098] Using the particle swarm algorithm (each particle represents a first-order cumulative sedimentation transformation value x1 at the i-th observation (1) (t i) and the first-order cumulative sedimentation transformation value x1 at the i-1th observation (1) (t i-1 ) between the fitting coefficient λ) to find the optimal weight parameter λ′ of the background value:

[0099] Step (1): Particle initialization, set the number of particles to 50, randomize the particle position λ = [0, 1] and the particle speed vlimit = [-1, 1], the maximum number of iterations is 3000 times, and the initial optimal position and initial optimal fitness of the individual and group are set to λ best = 0.5 and the fitness obtained in step 2

[0100] Step (2): Substitute the particle positions λ of all particles into formula (10) and calculate the fitness of each particle according to step 2, that is, the average relative error:

[0101]

[0102] Step (3): Based on the fitness of each particle obtained in the previous step Update the individual best position of each particle and the group best position of the particle swarm.

[0103] Specifically: If the fitness of particle k Fitness that is better than its individual best position Then update the individual best position pbest of particle k k , if the fitness of particle k A fitness that is better than the best position of its group Then the individual position of particle i is the new group optimal position gbest k The initial fitness of the individual's best position and the fitness of the optimal position of the group That is, the fitness in step (1) The best position is then compared with the best position in the mth and m+1th iterations by the algorithm iteration. and as well as and And change.

[0104] Step (4) According to the new pbest k and gbest k Use the following formula to update the speed and position of each particle in the particle swarm, and then determine whether it exceeds the particle position limit and speed limit. If so, set it equal to its limit boundary value.

[0105] v k m+1 =c0v k m+c1r k (pbest k -x k m )+c2r′ k (gbest k -x k m )

[0106] x k m+1 =x k m +v k m+1

[0107] where v k m 、x k m 、v k m+1 and x k m+1 are the velocity and position of the kth particle in the mth and m+1th iterations, respectively. c0∈[0.5,1] is the inertia weight factor. c1,c2∈(0,4) are the self- and group learning factors. r k and r′ k is a random number between [0,1]. The best position pbest of each particle in the mth iteration k and the group best position gbest k Update its own speed and position as the initial speed and position of the m+1th particle until the iteration is completed, and record the mth best position of its group gbest k and the optimal fitness of the group

[0108] Step (5) determines whether the maximum number of iterations has been reached. If not, go to step (2). If the maximum number of iterations has been reached, output the group's best position gbest. k That is the optimal weight parameter λ′.

[0109] Step 4: Use the weight parameter λ′ optimized in step 3 to establish the improved model. When the development coefficient and gray action are calculated according to formula (6) in step 2, the background value calculation formula is changed from formula (5) to formula (10), and the new parameter sequence is obtained by the least squares method. Based on the parameter sequence, the time response function of the non-uniformly spaced GM(1,1) model based on the cotangent function and the improved background value is obtained through equation (7). The time response function is then cumulatively reduced to obtain the non-uniformly spaced GM(1,1) model transformed with the improved Cotx function. This can, to a certain extent, reduce the impact of large fluctuations in overall settlement and unequal observations on the settlement prediction performance of the GM(1,1) model.

[0110] Step 5: Use the improved new model in step 4 to fit and predict the trend of engineering construction cumulative settlement data. The fitting prediction results are tested for accuracy using relative error and grey correlation index.

[0111] Transform the sedimentation data sequence x1 using the cotx function 0 Substitute the non-uniformly spaced GM(1,1) model transformed by the improved cotx function to obtain the fitted prediction sequence Finally, the inverse cotangent and the reduction of formula (1) are performed to obtain the cumulative settlement data sequence x 0 The fitted predicted value of

[0112] The relative error is the absolute value of the ratio of the residual to the cumulative settlement of the engineering building, is the average relative error. The smaller the average relative error, the lower the deviation between the cumulative settlement value and the fitting prediction value, and the more accurate the fitting and prediction results of the grey GM (1,1) model.

[0113] Grey relational degree n is the cumulative settlement data sequence x 0 The number of cumulative settlement data, x0(k) and x i (k) represents the actual settlement value and the fitted predicted value, and ξ represents the resolution coefficient, which is set to 0.5. The larger the grey correlation degree, the higher the consistency of the trend between the fitted sequence and the actual settlement value over time.

[0114] Example

[0115] A non-uniformly spaced GM(1,1) model, a non-uniformly spaced GM(1,1) model using a cotangent function transformation, and a non-uniformly spaced GM(1,1) model based on a cotangent function and an improved background value were established for comparative analysis. For ease of presentation, these are referred to as Model 1, Model 2, and Model 3, respectively.

[0116] Example 1

[0117] The ten-period cumulative settlement monitoring data of a settlement monitoring point at a construction site are shown in Table 1. Model 3 was constructed using the cumulative settlement monitoring data of the first eight periods. The relative error test was used to compare and analyze the results with Model B and Model C in the existing literature (Zhang Zhenchao, Yuan Debao, Zhang Jun, Wu Ziruo. Optimization of non-uniformly spaced GM (1,1) model and its application in deformation monitoring [J]. Journal of Surveying and Mapping Science and Technology, 2020, 37(02): 124-132.). The cumulative settlement monitoring data of the first eight periods were used to predict the cumulative settlement monitoring results of the last two periods.

[0118] Table 1 Cumulative settlement monitoring values at monitoring points at construction sites

[0119]

[0120]

[0121] The cumulative settlement monitoring sequence of the construction site monitoring points x 0 The cotangent function transformation interval is fixed in the range close to π / 2. The sequence after cotangent function transformation is obtained by using formula (1) and then performing cotangent function transformation:

[0122] x1 0 =[0.1266,0.0913,0.0777,0.0708,0.0628,0.0580,0.0415,0.0394]

[0123] The optimized background value weight parameter λ′ of model 3 in Table 2 is 0.34948, and the time response functions of the four GM(1,1) models established are:

[0124] Non-uniformly spaced GM(1,1) model (Model 1):

[0125] Model B:

[0126] Model C:

[0127] Model 3 (λ′=0.34948):

[0128] Model 3 uses the time response function to calculate the corresponding value and then performs cumulative reduction reduction, and the inverse cotangent function and compression function are used to restore the final cumulative settlement monitoring fitting and prediction value.

[0129] Table 2 Comparison of modeling accuracy of settlement monitoring data

[0130]

[0131]

[0132] From Table 2, we can see that the average relative errors of Model B and Model C are 0.75% and 0.73% respectively, which is about 70% less than the 2.09% of the non-equally spaced GM (1,1) model, and its fitting and prediction accuracy are greatly improved. The average relative error of Model 3 is 0.72%, and the relative errors of its predicted values are 0.37% and 0.00% respectively. Compared with Model C, it can be seen that Model 3 has better fitting and prediction effects and higher accuracy. From the perspective of gray correlation, the gray correlation between Model 3 and the cumulative settlement observation value is 0.883, which is higher than the gray correlation of other models. This shows that compared with several other models, Model 3 has better similarity and correlation with the development trend of the cumulative settlement observation value over time, and has a higher degree of consistency with the development trend of the cumulative settlement prediction value. In addition, Models 1, 2, and 3 are established to convert the cumulative settlement monitoring sequence x of the monitoring point into the gray correlation. 0 Substitute the time response function of model 1 and perform cumulative reduction to restore the cumulative settlement monitoring point x1 after the cotangent function transformation 0 The sequence is substituted into the time response function of model 2 and 3, and then the cumulative subtraction and function restoration are performed to obtain the cumulative settlement monitoring sequence x of the monitoring points of model 1, 2, and 3 respectively. 0 The fitted predicted value of The results are shown in Table 3. The time response function of model 2 is As in Model 3, the time response function is used to obtain the corresponding value and then the cumulative reduction is performed. The inverse cotangent function and compression function are used to restore the final cumulative settlement monitoring fitting and prediction value.

[0133] Table 3 Modeling results of settlement monitoring data

[0134]

[0135] Table 3 shows that compared with Model 1 (non-uniformly spaced GM(1,1)), the accuracy of Models 2 and 3 is significantly improved, with average relative errors of 0.74% and 0.72%, respectively, and gray correlation degrees of 0.875 and 0.883, respectively. Model 3 achieves the best prediction results and the highest overall accuracy. This shows that it is feasible to improve the overall prediction accuracy of the GM(1,1) model through combined optimization of both function transformation and background value.

[0136] Example 2

[0137] Select a bridge construction project 6 # Table 4 shows the cumulative settlement monitoring data from the pier pile foundation, monitored by settlement meters. The cumulative settlement monitoring results for the last two periods were predicted using the data from the first nine periods. Models 1, 2, and 3 were developed, and a comparative analysis using relative error tests was performed. The results are shown in Table 5.

[0138] Table 4 6 #Cumulative settlement monitoring results of pier pile foundation settlement meter

[0139]

[0140] Fix the cotangent function transformation interval to the range close to π / 2, use formula (1) to calculate and then perform the cotangent function transformation to obtain the sequence after the cotangent function transformation:

[0141] x1 0 =[0.1494,0.1046,0.0866,0.0765,0.0604,0.0532,0.0470,0.0427,0.0189]

[0142] The time response functions of the three GM(1,1) models established are:

[0143] Model 1:

[0144] Model 2:

[0145] Model 3 (λ′=0.64066):

[0146] Table 5 Comparison of modeling data for pier pile foundation cumulative settlement monitoring

[0147]

[0148]

[0149] Comparing Model 2 and Model 3, the average relative errors for Model 2 and Model 3 were 3.88% and 3.68%, respectively, and the grey correlation degrees were 0.910 and 0.915, respectively. Overall, Model 3 performed better than Model 2 in terms of fitting and prediction accuracy. Furthermore, a comparison of the two examples revealed that the GM(1,1) model, with a cotangent function transformation and a fixed transformation interval close to π / 2, performed better in fitting and predicting when the cumulative settlement monitoring data grew more slowly.

[0150] To solve the problems of unequal time and non-smoothness of monitoring data and construction error of model background value, the fitting and prediction accuracy of non-uniformly spaced GM(1,1) model are improved by performing cotangent function transformation on the original sequence of settlement monitoring and optimizing the background value through PSO algorithm.

[0151] Analysis of the examples shows that compared with the non-equally spaced GM (1, 1) model, the non-equally spaced GM (1, 1) model based on the cotangent function and the two optimized GM (1, 1) models in Example 1, the non-equally spaced GM (1, 1) model based on the cotangent function and the improved background value has better effects in fitting and predicting the settlement amount. The average relative error of the fitting and prediction of the cumulative settlement observation value is smaller than that of several other models, and the similarity and correlation with the development trend of the cumulative settlement observation value over time are better. Through comparison of the two examples, it is found that when the settlement data grows relatively slowly, the cotangent function transformation interval is fixed at about the function flat interval π / 2, and the fitting and prediction effects of the data are better.

[0152] Each embodiment in this specification is described in a related manner. Similar parts between the various embodiments can be referred to in conjunction with each other. Each embodiment focuses on the differences between the other embodiments. In particular, the system embodiment is generally similar to the method embodiment, so the description is relatively simple. For related parts, refer to the description of the method embodiment.

[0153] The above description is only a preferred embodiment of the present invention and is not intended to limit the scope of protection of the present invention. Any modifications, equivalent replacements, improvements, etc. made within the spirit and principles of the present invention are included in the scope of protection of the present invention.

Claims

1. A building settlement prediction method based on an improved non-uniformly spaced grey model, characterized in that: The following steps are involved: Step 1: For the cumulative settlement data sequence x 0 Transform the cotx function to transform the sedimentation data sequence x1 0 , for x1 0 Weighted and first-order cumulative sedimentation data sequence x1 1 ; Cumulative settlement data series x 0 ={x (0) (t1),x (0) (t2),…,x (0) (t n )}, where t1, t2…t n is the observation time corresponding to the cumulative settlement data, x (0) Indicates the cumulative settlement data of the building; For the cumulative settlement data series x 0 Perform compression transformation to satisfy the cotx function transformation interval: where t i ={t1,t2,…,t n }, N and q are constants, intermediate variables Y 0 (t i ) represents t i Compression transformation of moments; The cotx function transforms the sedimentation data sequence x1 0 ={x1 (0) (t1),x1 (0) (t2),…,x1 (0) (t n )}={cot(Y 0 (t1)),cot(Y 0 (t2)),…,cot(Y 0 (t n ))} When performing first-order accumulation generation, add the time weight Δt i =t i -t i-1 , i=2,3,…,n, and obtain the first-order cumulative settlement data sequence x1 1 ={x1 (1) (t1),x1 (1) (t2),…,x1 (1) (t i ),…,x1 (1) (t n )},in Where, Δt i It represents the time difference between the cumulative settlement data at the time of observation i and observation i-1, when i=1, Δt1=1; x1 (0) Indicates the cumulative settlement data x (0) Data after cotx function transformation, x1 (1) Represents the first-order cumulative settlement data, x1 (1) (t i ) represents t i First-order cumulative settlement data at time; Step 2: Use the first-order cumulative settlement data sequence x1 1 Construct a non-uniformly spaced GM(1,1) model transformed by the cotx function and calculate the initial fitness of the model; the initial fitness is the cotx function transformed sedimentation data sequence x1 0 and the model fitting sequence The average relative error of Step 3: Use the particle swarm algorithm to obtain the optimal weight parameter λ′ of the background value of the non-uniformly spaced GM(1,1) model transformed by the cotx function; each particle in the particle swarm algorithm represents a first-order cumulative sedimentation transformation value x1 at the i-th observation (1) (t i ) and the first-order cumulative sedimentation transformation value x1 at the i-1th observation (1) (t i-1 ), including: S31, particle initialization, set the number of particles, randomize particle position λ = [0, 1] and particle speed vlimit = [-1, 1], set the maximum number of iterations, set the individual initial optimal position λ best = 0.5 and the initial optimal fitness and the initial optimal position λ of the group best = 0.5 and the initial optimal fitness S32, calculating the fitness of each particle; S33, updating the individual optimal position of each particle and the group optimal position of the particle swarm based on the calculated fitness of each particle; S34. Update the speed and position of each particle in the particle swarm according to the new individual optimal position and group optimal position, and then determine whether the speed and position of each particle exceeds the particle position limit and speed limit. If so, set the speed and position of each particle to be equal to the corresponding limit boundary value. The speed and position of the particle are specifically: v k m+1 =c0v k m +c1r k (pbest k -x k m )+c2r k ′(gbest k -x k m ) x k m+1 =x k m +v k m+1 where v k m 、x k m 、v k m+1 and x k m+1 are the velocity and position of the kth particle in the mth and m+1th iterations, respectively. c0∈[0.5,1] is the inertia weight factor. c1,c2∈(0,4) are the self- and group learning factors. r k and r k ′ is a random number between [0,1]; each particle has its best individual position pbest after the mth iteration k and the group's best position gbest k Update its own speed and position as the initial speed and position of the m+1th particle until the iteration is completed, and record the mth best position of its group gbest k and the optimal fitness of the group S35, determine whether the maximum number of iterations has been reached, if not, go to S32; if the maximum number of iterations has been reached, output the group's best position gbest k That is, the optimal weight parameter λ′; Step 4: Use the weight parameter λ′ optimized in step 3 to establish the non-uniformly spaced GM(1,1) model of the improved function cotx transformation; Step 5: Use the improved model from step 4 to perform fitting prediction on the trend of building cumulative settlement data. The fitting prediction results are tested for accuracy using relative error and grey correlation index. The fitting prediction is: Transform the sedimentation data sequence x1 using the cotx function 0 Substitute the non-uniformly spaced GM(1,1) model transformed by the improved cotx function to obtain the fitted prediction sequence Finally, perform the inverse cotangent and then use the compression formula The cumulative sedimentation data sequence x is obtained by reduction 0 The fitted predicted value of In the formula, t i ={t1,t2,…,t n } is the data observation time corresponding to the cumulative settlement data, N and q are constants, n is the number of settlement data; When using the particle swarm algorithm to find the optimal weight parameter λ′ of the background value, the number of particles is set, the position and speed of the particles are randomized, and the maximum number of iterations is set.

2. A building settlement prediction method based on an improved non-equidistant grey model according to claim 1, characterized in that: The step 2 is specifically as follows: Take the first-order cumulative settlement data sequence x1 1 Establish the grey system theory about the observation time t of settlement data i The whitened differential equation of : Where a is the development coefficient, which indicates the development law and trend of the cumulative settlement sequence; u is the gray action, which reflects the changing relationship between the cumulative settlement sequences. Both a and u are parameters to be solved; t i ={t1,t2,…,t n } is the data observation time corresponding to the cumulative settlement data; x1 (1) (t i ) represents t i First-order cumulative settlement data at time; Using formula (3) to replace differential, we get x1 (0) (t i )+az (1) (t i )=u (4) Formula (4) is the basic form of the GM (1,1) model, where x1 (0) Represents the building's cumulative settlement data x (0) Data after cotx function transformation; z (1) (t i ) is the cotx function transforming the sedimentation data sequence x1 0 The sequence of adjacent mean values is generated, which represents the average value between the two cumulative settlement values. Shifting formula (4) yields: Y=BU Among them, Y represents the array of settlement data values transformed by the cotx function, B is the coefficient matrix of a and u, and U is the parameter column to be calculated, that is: And the sequence z is generated next to the mean (1) (t i ) From the GM(1,1) model, we know that: z (1) (t i )=0.5[x1 (1) (i)+x1 (1) (i-1)],i=1,2…,n. (5) The estimated values of the development coefficient a and the gray action u in the parameter list U are obtained using the least squares principle. and The estimated parameter list is: where x1 (1) (t1) = x1 (0) (t1), the solution of equation (3) is: Formula (7) is the time response function of the non-uniformly spaced GM (1, 1) model. The prediction equation of the non-uniformly spaced GM (1, 1) model transformed by the cotx function is obtained by cumulative reduction of formula (7), namely: Among them, e is a natural constant, x1 (1) The fitting sequence of x1 (0) The fitting sequence of Using formula (8) to obtain the cotx function to transform the settlement data sequence x1 0 The fitting sequence of Transform the sedimentation data series x1 by the cotx function 0 and fitting series Use formula (9) to find the fitting sequence The average relative error provides the initial fitness for the particle swarm algorithm to find the optimal background value of the model: Where n is the number of cotx function transformed settlement data participating in the non-uniformly spaced GM(1,1) model calculation.

3. A building settlement prediction method based on an improved non-equidistant grey model according to claim 1, characterized in that: The initial optimal fitness in S31 Where n is the number of cotx function transformed settlement data participating in the non-uniformly spaced GM(1,1) model calculation, x1 (0) (t i ) represents t i Cumulative settlement data at time x (0) (t i ) The data after cotx function transformation, x1 (0) The fitting sequence, t i ={t1,t2,…,t n } is the data observation time corresponding to the cumulative settlement data.

4. The building settlement prediction method based on the improved non-equidistant grey model according to claim 1, characterized in that: The specific method of S33 is: If the calculated fitness of particle k is better than the fitness of the individual best position, the individual best position pbest of particle k is updated. k If the fitness of particle k is better than the fitness of the group's best position, the individual position of particle k is the new group's best position gbest k .

5. A building settlement prediction method based on an improved non-equidistant grey model according to claim 1 or 2, characterized in that: Step 4 is as follows: Based on the process of constructing the non-uniformly spaced GM(1,1) model of the cotx function transformation in step 2, the background value calculation formula is changed to z (1) (t i )=λ′x1 (1) (t i-1 )+(1-λ′)x1 (1) (t i ),λ′∈[0,1], and use the least squares method to obtain a new parameter sequence According to the parameter sequence pass The time response function of the non-uniformly spaced GM(1,1) model based on the cotangent function and the improved background value is obtained, and the time response function is cumulatively reduced to obtain the non-uniformly spaced GM(1,1) model of the improved Cotx function transformation; In the above formula, λ′ is the optimal weight parameter, z (1) (t i ) is the cotx function transforming the sedimentation data sequence x1 0 The adjacent mean value of the sequence is generated, which represents the average of the two cumulative settlement values before and after, x1 (0) (t i ) represents t i Cumulative settlement data at time x (0) (t i ) The data after cotx function transformation, x1 (0) The fitting sequence, t i ={t1,t2,…,t n } is the data observation time corresponding to the cumulative settlement data, and are the estimated values of the development coefficient a and the gray action u in the parameter column U.

6. The building settlement prediction method based on the improved non-equidistant grey model according to claim 1, characterized in that: The specific method for step 5 precision inspection is as follows: is the absolute value of the ratio of the residual to the cumulative settlement of the building, where Δ k Represents the relative error, x (0) Represents the cumulative settlement data of the building, t n The data observation time corresponding to the cumulative settlement data; ε represents the residual value; is the average relative error; the smaller the average relative error, the lower the deviation between the cumulative settlement value and the fitting prediction value, and the more accurate the fitting and prediction results; Grey relational degree n is the cumulative settlement data sequence x 0 The number of cumulative settlement data, x0(k) and x i (k) are the actual settlement value and the fitted predicted value, ξ is the resolution coefficient, which is taken as 0.5; the larger the grey correlation degree, the higher the consistency of the trend between the fitted sequence and the actual settlement value over time.

Images