CN106295869A - A kind of based on the building settlement Forecasting Methodology improving unbiased function - Google Patents

Abstract

The invention discloses a kind of based on the building settlement Forecasting Methodology improving unbiased function, set up unbiased grey-forecasting model, utilize cubic spline interpolation that initial data is carried out pretreatment, set up and improve unbiased function, increase certain data volume, make data sequence more smooth, inherit the characteristic of unbiased function, and improve the fitting and prediction precision improving unbiased function.

Description

A kind of based on the building settlement Forecasting Methodology improving unbiased function

Technical field

The present invention relates to the deformation monitoring of building, sink based on the building improving unbiased function particularly to a kind of Fall Forecasting Methodology.

Background technology

Society rapid economic development, the rapid expansion of urbanization so that civil high-rise buildings becomes universalness and close Collectionization, the construction of skyscraper is increasing on the impact of perimeter security so that the settlement monitoring work to these groups of building becomes Obtaining particularly significant, the most effectively being analyzed predicting to settlement monitoring data also becomes important process.

Unbiased function has the characteristic eliminating tradition gray model constant error, calculates desired data amount few, calculates Convenient, it is suitable for medium-term and long-term prediction.Owing to unbiased function remains a kind of exponential function, during solution differential functional eqution just May require that the fluctuation of original data sequence can not be excessive, the error that predicts the outcome of forecast model otherwise can be made to increase, it is impossible to correct The effect of reflection prediction.

Summary of the invention

The present invention solves the weak point existing for above-mentioned prior art, it is provided that a kind of based on improving unbiased function Building settlement Forecasting Methodology, utilize cubic spline interpolation that initial data is carried out pretreatment, increase certain data volume, Make data sequence more smooth, inherit the characteristic of unbiased function, and improve the matching prediction improving unbiased function Precision.

The present invention solves technical problem and adopts the following technical scheme that

A kind of based on the building settlement Forecasting Methodology improving unbiased function, it is characterised in that to comprise the steps:

Step one: the equally spaced original data sequence being provided with the quasi-optical cunning of non-negative is X⁽⁰⁾={ x⁽⁰⁾(1),x⁽⁰⁾(2),x⁽⁰⁾ (3),…,x⁽⁰⁾(k-1),x⁽⁰⁾(k) }, k=1,2 ... n, the establishment step of unbiased function is as follows:

(1), initial data is cumulative

Initial data is carried out one-accumulate, obtains cumulative sequence X⁽¹⁾={ x⁽¹⁾(1),x⁽¹⁾(2),x⁽¹⁾(3),…x⁽¹⁾ (k) }, in formulaSolve the differential equation (1) and draw X⁽¹⁾。

$\frac{{\mathrm{dx}}_k^{\left(1\right)}}{dt}+{\mathrm{ax}}_k^{\left(1\right)}=b---\left(1\right)$

(2) data matrix B, Y, are determined_N

$B=\left[\begin{array}{cc}-\frac{1}{2}\left(x^{\left(1\right)}\right(2)+x^{\left(1\right)}(1\left)\right)& 1\\ -\frac{1}{2}\left(x^{\left(1\right)}\right(3)+x^{\left(1\right)}(2\left)\right)& 1\\ .& .\\ .& .\\ .& .\\ -\frac{1}{2}\left(x^{\left(1\right)}\right(n)+x^{\left(1\right)}(n-1\left)\right)& 1\end{array}\right],Y_N=\left[\begin{array}{c}{x}^{\left(0\right)}\left(2\right)\\ x^{\left(0\right)}\left(3\right)\\ .\\ .\\ .\\ x^{\left(0\right)}\left(n\right)\end{array}\right]---\left(2\right)$

(3), method of least square is utilized to seek parameterWith

$\left[\begin{array}{c}\hat{a}\\ \hat{b}\end{array}\right]={\left(B^{T}B\right)}^{-1}{B}^T{Y}_N---\left(3\right)$

(4) unbiased function parameter, is solvedWith

${\hat{a}}^{\′}=ln\frac{2-\hat{a}}{2+\hat{a}},{\hat{b}}^{\′}=\frac{2\hat{b}}{2+\hat{a}}---\left(4\right)$

(5), unbiased grey-forecasting model is set up

${\hat{x}}^{\left(0\right)}\left(1\right)=x^{\left(0\right)}\left(1\right),x^{\left(0\right)}\left(k\right)={\hat{b}}^{\′}{e}^{{\hat{a}}^{\′}(k-1)},k=1,2,...,n---\left(5\right)$

As k ＜ n,Referred to as pattern die analog values；As k=n,Referred to as model filtering value；When During k ＞ n,Referred to as model predication value.

Step 2: carry out cubic spline interpolation: set and have division Δ a: a=x on [a, b]₀＜ x₁＜ ... ＜ x_n=b is given X at node_kThe functional value at place is y_k=f (x_k), (k=0,1 ..., n), if existence function S (x), meet

(1) interpolation condition: S (x_k)=y_k, k=0,1 ..., n；

(2) segmentation condition: at minizone [x_k,x_k+1], (k=0,1 ..., n-1) on, S (x) is cubic algebra multinomial, I.e. S (x)=a₀+a₁x+a₂x²+a₃x³；

(3) smoothness condition: S (x) ∈ C²[a,b]。

Then S (x) is called Spline Node x_kOn cubic spline functions (Cubic SplineInterpolation), The method seeking S (x) is called cubic spline interpolation method.

If original data sequence is X⁽⁰⁾={ x⁽⁰⁾(1),x⁽⁰⁾(2),x⁽⁰⁾(3),…,x⁽⁰⁾(k-1),x⁽⁰⁾(k) }, k=1, 2 ... n, it is S that initial data carries out the sequence after cubic spline interpolation process⁽⁰⁾={ s⁽⁰⁾(1),s⁽⁰⁾(2),s⁽⁰⁾(3),…, s⁽⁰⁾(k-1),s⁽⁰⁾(k) }, calculation expression is as follows

$S\left(x\right)=\left\{\begin{array}{c}{s}_0\left(x\right)=a_0+b_{0}x+c_0{x}^2+d_0{x}^3,x\∈\[x_0,x_1\]\\ s_1\left(x\right)=a_1+b_{1}x+c_1{x}^2+d_1{x}^3,x\∈\[x_1,x_2\]\\ .\\ .\\ .\\ s_{k-1}\left(x\right)=a_k+b_{k}x+c_k{x}^2+d_k{x}^3,x\∈\[x_{k-1},x_k\]\end{array}\right.,k=1,2,...n---\left(6\right)$

If when this function does not has first derivative and second derivative values at two-end-point, then replaced by corresponding difference coefficient value.

Step 3: set up and improve unbiased function.

Step 4: utilize improvement unbiased function that Building's Subsidence Survey Data is simulated, and predict in the future Monitoring Data value.

Beneficial effects of the present invention: utilize cubic spline interpolation that initial data is carried out pretreatment, increase certain number According to amount, make data sequence more smooth, inherit the characteristic of unbiased function, and improve the matching improving unbiased function Precision of prediction.

Accompanying drawing explanation

Fig. 1 improves the flow chart of unbiased function.

Fig. 2 Building's Subsidence Survey Data.

Data after the process of Fig. 3 cubic spline interpolation.

Two kinds of forecast models of Fig. 4 predict the outcome.

The match value of two kinds of forecast models of Fig. 5 compares.

Detailed description of the invention

Below with reference to drawings and Examples, technical scheme is described in further detail.

Certain high-rise settlement monitoring data totally 16 phase, as shown in Figure 2.With the 1st phase in Fig. 2 to the 12nd phase initial data Setting up unbiased function and improve unbiased function, drawn by two kinds of forecast models predicted the outcome with the 13rd phase to 16 phases Measured data carry out precision analysis and compare, carried out the effectiveness of judgment models by relative error.

1, unbiased function is set up

Carried out pretreatment by the 1st phase in Fig. 2 to the 12nd phase initial data and set up unbiased function, it can be deduced that ginseng NumberWithUnbiased function is

x⁽⁰⁾(k)=2.98745e^{-0.00068149(k-1)}, k=1,2 ..., n

2, improvement unbiased function is set up

Original data sequence is carried out cubic spline interpolation process, and interpolation knot is set to the midpoint of each minizone, It is [x_k,x_k+1] midpoint x_k+0.5, owing to data do not have first derivative and second dervative, then derivative value is replaced by difference coefficient value.With In setting up initial data totally 12 phase of model, the data cycle after being spaced apart the cubic spline interpolation of 0.5 and processing becomes 23 phases, after pretreatment, data value is as shown in Figure 3.The 1st phase in Fig. 3 is carried out unbiased function to the 23rd phase initial data Foundation can obtain parameterWithImprovement unbiased function is

x⁽⁰⁾(k)=2.98735e^{-0.00033788(k-1)}, k=1,2 ..., n

By unbiased function and improvement unbiased function, the 1st phase of Building's Subsidence Survey Data was entered to 12 phases Row simulation, and predict the Monitoring Data value of the 13rd phase to the 16th phase, the match value of two kinds of methods and predictive value are as shown in Figure 4.

Contrasting two kinds of forecast models, improve unbiased function higher than the fitting and prediction precision of unbiased function, it is put down All relative error and mean absolute errors are respectively 0.00348%, 0.00980% and 0.00786%, 0.01139%, such as Fig. 5 Shown in.

Claims (1)

1. a building settlement Forecasting Methodology based on improvement unbiased function, it is characterised in that comprise the steps:

Step one: the equally spaced original data sequence being provided with the quasi-optical cunning of non-negative is X⁽⁰⁾={ x⁽⁰⁾(1),x⁽⁰⁾(2),x⁽⁰⁾ (3),…,x⁽⁰⁾(k-1),x⁽⁰⁾(k) }, k=1,2 ... n, the establishment step of unbiased function is as follows: (1), initial data It is cumulative,

Initial data is carried out one-accumulate, obtains cumulative sequence X⁽¹⁾={ x⁽¹⁾(1),x⁽¹⁾(2),x⁽¹⁾(3),…x⁽¹⁾(k) }, In formulaSolve the differential equation (1) and draw X⁽¹⁾,

$\frac{{\mathrm{dx}}_k^{\left(1\right)}}{dt}+{\mathrm{ax}}_k^{\left(1\right)}=b---\left(1\right)$

(2) data matrix B, Y, are determined_N

$B=\left[\begin{array}{cc}-\frac{1}{2}\left(x^{\left(1\right)}\right(2)+x^{\left(1\right)}\left(1\right))& 1\\ -\frac{1}{2}\left(x^{\left(1\right)}\right(3)+x^{\left(1\right)}\left(2\right))& 1\\ .& .\\ .& .\\ .& .\\ -\frac{1}{2}\left(x^{\left(1\right)}\right(n)+x^{\left(1\right)}(n-1))& 1\end{array}\right],Y_N=\left[\begin{array}{c}{x}^{\left(0\right)}\left(2\right)\\ x^{\left(0\right)}\left(3\right)\\ .\\ .\\ .\\ x^{\left(0\right)}\left(n\right)\end{array}\right]---\left(2\right)$

(3), method of least square is utilized to seek parameterWith

$\left[\begin{array}{c}\hat{a}\\ \hat{b}\end{array}\right]={\left(B^{T}B\right)}^{-1}{B}^T{Y}_N---\left(3\right)$

(4) unbiased function parameter, is solvedWith

${\hat{a}}^{\′}=\ln \frac{2-\hat{a}}{2+\hat{a}},{\hat{b}}^{\′}=\frac{2\hat{b}}{2+\hat{a}}---\left(4\right)$

(5), unbiased grey-forecasting model is set up

${\hat{x}}^{\left(0\right)}\left(1\right)=x^{\left(0\right)}\left(1\right),x^{\left(0\right)}\left(k\right)={\hat{b}}^{\′}{e}^{{\hat{a}}^{\′}(k-1)},k=1,2,\mathrm{...},n---\left(5\right)$

As k ＜ n,Referred to as pattern die analog values；As k=n,Referred to as model filtering value；As k ＞ n Time,Referred to as model predication value；

Step 2: carry out cubic spline interpolation: set and have division Δ a: a=x on [a, b]₀＜ x₁＜ ... ＜ x_n=b, given node Place x_kThe functional value at place is y_k=f (x_k), (k=0,1 ..., n), if existence function S (x), meet

(1) interpolation condition: S (x_k)=y_k, k=0,1 ..., n,

(2) segmentation condition: at minizone [x_k,x_k+1], (k=0,1 ..., n-1) on, S (x) is cubic algebra multinomial, i.e. S (x) =a₀+a₁x+a₂x²+a₃x³,

(3) smoothness condition: S (x) ∈ C²[a, b], then S (x) is called Spline Node x_kOn cubic spline functions (Cubic Spline Interpolation), the method seeking S (x) is called cubic spline interpolation method, if original data sequence is X⁽⁰⁾= {x⁽⁰⁾(1),x⁽⁰⁾(2),x⁽⁰⁾(3),…,x⁽⁰⁾(k-1),x⁽⁰⁾(k) }, k=1,2 ... n, initial data is carried out cubic spline Sequence after interpolation processing is S⁽⁰⁾={ s⁽⁰⁾(1),s⁽⁰⁾(2),s⁽⁰⁾(3),…,s⁽⁰⁾(k-1),s⁽⁰⁾(k) }, calculation expression is such as Under

$S\left(x\right)=\left\{\begin{array}{c}{s}_0\left(x\right)=a_0+b_{0}x+c_0{x}^2+d_0{x}^3,x\∈\[x_0,x_1\]\\ s_1\left(x\right)=a_1+b_{1}x+c_1{x}^2+d_1{x}^3,x\∈\[x_1,x_2\]\\ \mathrm{...}\\ s_{k-1}\left(x\right)=a_k+b_{k}x+c_k{x}^2+d_k{x}^3,x\∈\[x_{k-1},x_k\]\end{array}\right.,k=1,2,...n---\left(6\right)$

If when this function does not has first derivative and second derivative values at two-end-point, then replaced by corresponding difference coefficient value；

Step 3: set up and improve unbiased function；

Step 4: utilize improvement unbiased function that Building's Subsidence Survey Data is simulated, and predict monitoring in the future Data value.