CN103116698A - GM (1, 1) model prediction method based on cubic spline - Google Patents

Abstract

The invention discloses a dynamic GM (1, 1) model based on a cubic spline interpolation. The dynamic GM (1, 1) model is applied for a thought of a piecewise polynomial interpolation and an overall trend of dynamic prediction time series data. The dynamic GM (1, 1) model based on the cubic spline interpolation theoretically analyzes a background value of a GM (1, 1) model. Firstly, a through of a cubic spline interpolation is put forward. Then, a method of combination of a piecewise linear interpolation function and a cubic spline interpolation formula to construct a new class grey prediction model. A constructing process of the background value can be improved so as to overcome the shortcomings of an existing grey improving model and provide a new means to improve prediction accuracy. Lastly, the model can be utilized to implement prediction. The dynamic GM (1, 1) model based on the cubic spline interpolation is scientific in concept, simple in calculation, small in workload and high in prediction accuracy. The dynamic GM (1, 1) model based on the cubic spline interpolation has a better use value and a wide application prospect in a prediction technique field.

Description

A kind of GM based on cubic spline (1,1) model prediction method

Technical field

The present invention relates to the data predication method field, be specially a kind of GM based on cubic spline (1,1) model prediction method.

Background technology

Gray theory is the mathematical method that a kind of use solves the incomplete system of information.This method is regarded each stochastic variable as a grey variable that changes in given range.And the method that need not add up is processed the grey variable, directly processes raw data, seeks inherent Changing Pattern.Owing to existing in a large number gray system at numerous areas such as economy, social science and engineerings, therefore this Forecasting Methodology is widely used.The basic thought of Grey Prediction Algorithm is: at first, original time series is carried out the one-accumulate operation, generate new time series; Then, according to gray theory, suppose that new time series has exponential relationship, set up the corresponding differential equation and carry out match, and then utilize the differential pair equation to carry out discretize to obtain a system of linear equations; At last, utilize least square method that unknown parameter is estimated, thereby finally obtain forecast model.

GM(1,1) grey forecasting model is tool exponential model devious.Since gray prediction theory was set up, the many aspects such as in order to adapt to the characteristics of each application, GM (1,1) grey forecasting model is chosen in starting condition, background value reconstruct, method for parameter estimation improvement had all obtained significant improvement.

Utilize grey GM(1,1) although model is predicted many successful stories,, the same with other Forecasting Methodologies, also there is certain limitation in it.Therefore, in recent years, GM(1,1) improvement of model and the concern that optimization research has been subject to many scholars.The more existing representational research methods of following brief description:

Article " GM(1,1) structure method of background value of model and application " (system engineering theory and practice, 2000) pointed out to cause GM(1,1) model error reason bigger than normal be in conventional model structure method of background value improper due to, and provided a kind of new building method, improved precision and the adaptability of model prediction;

The article scope of application of model " GM(1,1) " (system engineering theory and practice, 2000) is take simulation, experiment as the basis, to GM(1,1) scope of application of model is studied, and the relation of development coefficient and precision of prediction is quantized;

Article " gray model GM(1,1) optimize " (Chinese engineering science, 2003) utilize the exponential form solution of linear ordinary differential equation of first order to come the structural setting value, substitute the method take next-door neighbour's average as background value in conventional model, have certain superiority, reduced to a certain extent model error;

Article " based on the GM(1 of interpolation and Newton-Cores formula, 1) background value of model structure new method " (system engineering theory and practice, 2004) utilizes the Newton-Cores formula that background value is reconstructed, structure x ⁽¹⁾(t) n-1 Newton interpolation polynomial N (t), the N (t) that utilizes the Cores formula to calculate on interval [k, k+1] is worth, and is worth as improved background value with this.

Because it is usually larger that the approximate value of utilizing trapezoid formula to obtain definite integral is worth time error as a setting, thereby cause the deviation of model prediction also larger, precision of prediction does not reach requirement naturally.But study discovery by the present invention, even adopt more advanced interpolation algorithm reconstructed background value, also there is certain limitation, because research previously is all to adopt a certain individual event interpolation method, although improved to a certain extent the precision of prediction of model, also exist defective, be and covet high precision and increase nodal point number and cause oscillatory occurences to occur, distortion appears in prediction, causes the applicability of forecast model to reduce even not available.

The present invention proposes the thought of combination interpolation for this reason.The method that adopts piecewise linear interpolation to be combined with cubic spline interpolation is satisfying under constringent condition, and structure interpolating function N (t) makes it approach background value z on interval [k, k+1] ⁽¹⁾(k+1), and as the background value under new state.This invention individual event interpolation method as compared with the past, the unreliability problems such as node vibration have clearly been solved, avoided distortion, improved the theoretical degree of depth of model construction, having increased stability that model uses, also to possess algebraic accuracy simultaneously high, characteristics that relative error is little, and set up thus GM(1,1) forecast model is realized the accuracy prediction to data message.

Summary of the invention

The purpose of this invention is to provide a kind of GM based on cubic spline (1,1) model prediction method, the problem that exists to solve prior art.

In order to achieve the above object, the technical solution adopted in the present invention is:

A kind of GM based on cubic spline (1,1) model prediction method is characterized in that: comprise the following steps:

(1) original data sequence is chosen: choose according to target of prediction the original data sequence that forecast model adopts, and data sequence is necessary for one group of nonnegative number according to sequence, i.e. X ⁽⁰⁾

(2) the 1-AGO sequence is set up: with the original data sequence X that chooses ⁽⁰⁾As GM(1,1) basic data of forecast model, and to X ⁽⁰⁾Make 1-AGO, obtain result 1-AGO sequence X ⁽¹⁾, then respectively to X ⁽⁰⁾And X ⁽¹⁾Valid slickness check and the judgement of accurate index law, judgement original data sequence X ⁽⁰⁾With the 1-AGO sequence X ⁽¹⁾Whether satisfy GM(1,1) the applicable requirement of forecast model;

(3) background value generates: to the 1-AGO sequence X ⁽¹⁾Make background value Z ⁽¹⁾Generate, can calculate B and Y, wherein, $B=\left[\begin{array}{cc}-z^{\left(1\right)}\left(2\right)& 1\\ -z^{\left(1\right)}\left(3\right)& 1\\ .& .\\ .& .\\ .& .\\ -z^{\left(1\right)}\left(n\right)& 1\end{array}\right]$ ，Y _n=[x ⁽⁰⁾(2),x ⁽⁰⁾(3),…,x ⁽⁰⁾(n)] ^T，

z ⁽¹⁾(k) be the background value sequence, x ⁽⁰⁾(i) be original data sequence; Because least-squares estimation can be so that the indifference quadratic sum reaches minimum, therefore utilize least-squares estimation can obtain Argument List ,

Be the estimated value of a, the process of trying to achieve background value is as follows:

(a) for GM(1,1) model carries out the target of data prediction, take piecewise linear interpolation and cubic spline interpolation and calculation process as theoretical foundation, realizes the segmentation of interval [k, k+1] is comprised the following steps:

If second derivative value the S " (x of S (x) _k)=M _k(k=1,2 ..., n), and the second derivative at two ends is known, S " (1)=S " (n)=0.Due to S (x) at interval [x _k, x _k+1] on be cubic polynomial, therefore S " (x) at [x _k, x _k+1] on be linear function, can be expressed as:

$S^{\′\′}\left(x\right)=M_k\frac{x_{k+1}-x}{h_k}+M_{k+1}\frac{x-x_k}{h_k},k=\mathrm{1,2},...,n-1$

H wherein _k=x _k+1-x _k, k=1,2 ..., n-1

To following formula integration twice, and utilize S (x _k)=x ⁽¹⁾(k) and S (x _k+1)=x ⁽¹⁾(k+1), can obtain integration constant, so, can get the cubic spline functions expression formula:

$\begin{array}{c}S\left(x\right)=M_k\frac{{(x_{k+1}-x)}^3}{6h_k}+M_{k+1}\frac{{(x-x_k)}^3}{6h_k}\\ +(x^{\left(1\right)}\left(k\right)-\frac{M_k{h}_k^2}{6})\frac{x_{k+1}-x}{h_k}+(x^{\left(1\right)}(k+1)-\frac{M_k{h}_k^2}{6})\frac{x-x_k}{h_k}\\ k=\mathrm{1,2},...,n-1\end{array},$

Due to x ⁽¹⁾(k) ≈ S (t), therefore:

$\begin{array}{c}{z}^{\left(1\right)}(k+1)={\∫}_{x_k}^{x_{k+1}}{x}^{\left(1\right)}\left(t\right)\mathrm{dt}={\∫}_{x_k}^{x_{k+1}}S\left(x\right)\mathrm{dx}\\ =\frac{M_{k+1}+M_k}{24h_k}{(x_{k+1}-x_k)}^4+\frac{x^{\left(1\right)}(k+1)+x^{\left(1\right)}\left(k\right)-\frac{M_{k+1}+M_k}{6}{h}_k^2}{2h_k}{(x_{k+1}-x_k)}^2\\ k=\mathrm{1,2},...,n-1\end{array}$

Here, M _k(k=1,2 ..., be n) unknown, so background value z ⁽¹⁾Finding the solution (k+1) can be converted into an x _kPlace's second order is led M _kFind the solution;

(b) differentiate gets to S (x):

$S{\left(x\right)}^{\′}=-M_k\frac{{(x_{k+1}-x)}^2}{2h_k}+M_{k+1}\frac{{(x-x_k)}^2}{2h_k}+\frac{x^{\left(1\right)}(k+1)-x^{\left(1\right)}\left(k\right)}{h_k}-\frac{M_{k+1}-M_k}{6}{h}_k$

By S ' (x) at interval [x _k-1, x _k] and [x _k, x _k+1] on expression formula, can try to achieve S ' (x) at an x _kLocating the left and right limit is respectively:

$S^{\′}(x_k-0)=\frac{h_{k-1}}{6}{M}_{k-1}+\frac{h_{k-1}}{3}{M}_k+\frac{x^{\left(1\right)}\left(k\right)-x^{\left(1\right)}(k-1)}{h_{k-1}}$

$S^{\′}(x_k+0)=-\frac{h_k}{3}{M}_k+\frac{h_k}{6}{M}_{k+1}+\frac{x^{\left(1\right)}(k+1)-x^{\left(1\right)}\left(k\right)}{h_k}$

Lead continuity S ' (x by cubic spline tie point place single order _k-0)=S ' (x _k+ 0) can get:

μ _kM _k-1+2M _k+λ _kM _k+1=d _k,j=2,3,…,n-1

That is:

$\left\{\begin{array}{c}{\mathrm{\&mu;}}_2{\mathrm{<figure-callout\; id="1"\; label="M"\; filenames="HDA00002767261200011.png"\; state="\{\{state\}\}">M</figure-callout>}}_1+2M_2+{\mathrm{\&lambda;}}_2{M}_3=d_2\\ {\mathrm{\&mu;}}_3{M}_2+2M_3+{\mathrm{\&lambda;}}_3{M}_4={d_3}_{}\\ ...\\ {\mathrm{\&mu;}}_{n-1}{M}_{n-2}+2M_{n-1}+{\mathrm{\&lambda;}}_{n-1}{M}_n=d_{n-1}\end{array}\right.$

Wherein:

$\begin{array}{c}{\mathrm{\&mu;}}_k=\frac{h_{k-1}}{h_{k-1}+h_k},{\mathrm{\&lambda;}}_i=\frac{h_k}{h_{k-1}+h_k}\\ d_k=6\frac{f[x_k,x_{k+1}]-f[x_{k-1},x_k]}{h_{k-1}+h_k}\end{array}$ j=2,3,…,n-1

(c) due to M ₁=M _n=0, following formula can be written as:

Change into matrix form:

The matrix of coefficients of following formula is full rank, thereby M=(M ₂, M ₃, M _n-1) ^TSolution exist and unique, so can try to achieve new background value: $\begin{array}{c}{z}^{\left(1\right)}(k+1)={\&Integral;}_{x_k}^{x_{k+1}}{x}^{\left(1\right)}\left(t\right)\mathrm{dt}={\&Integral;}_{x_k}^{x_{k+1}}S\left(x\right)\mathrm{dx}\\ =\frac{M_{k+1}+M_k}{24h_k}{(x_{k+1}-x_k)}^4+\frac{x^{\left(1\right)}(k+1)+x^{\left(1\right)}\left(k\right)-\frac{M_{k+1}+M_k}{6}{h}_k^2}{2h_k}{(x_{k+1}-x_k)}^2\\ k=\mathrm{1,2},...,n-1\end{array}$

(4) model is determined and is found the solution: a in step (3) and b are used estimated value

With

Replace, therefore can set up GM(1,1) model and time response sequence , then solve the predicted value of first point

The analogue value, reduction at last solves the predicted value of initial point

The analogue value namely

,

Value be the predicted value sequence of original data sequence;

(5) error-tested): after solving the predicted value of original data sequence, can utilize residual test method or the degree of association method of inspection or the poor method of inspection of posteriority to judge GM(1, the 1) precision of forecast model; GM(1,1) precision of forecast model can be passed through different background value generating modes, the choice of raw data, and the Residual GM of the conversion of data sequence, correction and different stage (1,1) model is improved.

Described a kind of GM based on cubic spline (1,1) model prediction method is characterized in that: data data sequence commonly used in step (1) has scientific experimental data, empirical data, production data, decision data.

the present invention is existing based on grey GM(1 in analysis, 1) on the basis of the data message of model prediction, Main Problems and limitation in its forecasting process have been considered, proposed by cubic spline interpolation, the background value of major effect precision of prediction to be reconstructed and the rule of combination of piecewise linear interpolation, and provided on this basis a kind of data message prediction method for digging based on the cubic spline piecewise linear interpolation, of the present inventionly realize that logic is the method that adopts piecewise linear interpolation to be combined with cubic spline interpolation, satisfying under its constringent condition, the structure interpolating function, make the interpolating function of constructing approach background value, and with it as the background value under new state.Build GM(1 on the basis of background value under new state, 1) model, by building the final accurate prediction that realizes data message of model.

The present invention utilizes cubic spline interpolation to use the low order polynomial spline to realize the way of less interpolation error, has avoided the Runge phenomenon of using higher order polynomial to occur, and whole Approximation effect is better.Adopt to roll simultaneously and the strategy of feedback compensation, predict according to up-to-date information at each time point, controlled preferably fluctuating error, improved model the time precision of prediction under changing environment.Dynamic GM (1, the 1) Forecasting Methodology based on cubic spline that the present invention proposes, for the electricity demand forecasting of intelligent grid provides a kind of effective approach, but as a kind of forecast model, predicted data can not fit like a glove with True Data.Therefore, in real work, rationally utilize forecast model, for the aid decision making of intelligent grid provides realistic, exercisable science reference.

Description of drawings

Fig. 1 is the inventive method process flow diagram.

Embodiment

As shown in Figure 1.A kind of GM based on cubic spline (1,1) model prediction method comprises the following steps:

(1) according to utilizing GM(1,1) model carries out the target of data prediction, take piecewise linear interpolation and cubic spline interpolation and calculation process as theoretical foundation, realizes the segmentation of interval [k, k+1] is comprised the following steps:

If second derivative value the S " (x of S (x) _k)=M _k(k=1,2 ..., n), and the second derivative at two ends is known, and S " (1)=S " is (n).Due to S (x) at interval [x _k, x _k+1] on be cubic polynomial, therefore S " (x) at [x _k, x _k+1] on be linear function, can be expressed as:

$S^{\&prime;\&prime;}\left(x\right)=M_k\frac{x_{k+1}-x}{h_k}+M_{k+1}\frac{x-x_k}{h_k},k=\mathrm{1,2},...,n-1$

H wherein _k=x _k+1-x _k, k=1,2 ..., n-1

(2) to following formula integration twice, and utilize S (x _k)=x ⁽¹⁾(k) and S (x _k+1)=x ⁽¹⁾(k+1), can obtain integration constant, so, the cubic spline functions expression formula can be got

$\begin{array}{c}S\left(x\right)=M_k\frac{{(x_{k+1}-x)}^3}{6h_k}+M_{k+1}\frac{{(x-x_k)}^3}{6h_k}\\ +(x^{\left(1\right)}\left(k\right)-\frac{M_k{h}_k^2}{6})\frac{x_{k+1}-x}{h_k}+(x^{\left(1\right)}(k+1)-\frac{M_k{h}_k^2}{6})\frac{x-x_k}{h_k}\\ k=\mathrm{1,2},...,n-1\end{array},$

Due to x ⁽¹⁾(k) ≈ S (t), therefore

$\begin{array}{c}{z}^{\left(1\right)}(k+1)={\&Integral;}_{x_k}^{x_{k+1}}{x}^{\left(1\right)}\left(t\right)\mathrm{dt}={\&Integral;}_{x_k}^{x_{k+1}}S\left(x\right)\mathrm{dx}\\ =\frac{M_{k+1}+M_k}{24h_k}{(x_{k+1}-x_k)}^4+\frac{x^{\left(1\right)}(k+1)+x^{\left(1\right)}\left(k\right)-\frac{M_{k+1}+M_k}{6}{h}_k^2}{2h_k}{(x_{k+1}-x_k)}^2\\ k=\mathrm{1,2},...,n-1\end{array}$

Here, M _k(k=1,2 ..., be n) unknown, so background value z ⁽¹⁾Finding the solution (k+1) can be converted into an x _kPlace's second order is led M _kFind the solution.

(3) differentiate gets to S (x):

$S{\left(x\right)}^{\&prime;}=-M_k\frac{{(x_{k+1}-x)}^2}{2h_k}+M_{k+1}\frac{{(x-x_k)}^2}{2h_k}+\frac{x^{\left(1\right)}(k+1)-x^{\left(1\right)}\left(k\right)}{h_k}-\frac{M_{k+1}-M_k}{6}{h}_k$

By at interval [x _k-1, x _k] and [, x _k+1] on expression formula, can try to achieve S ' (x) at an x _kLocating the left and right limit is respectively:

$S^{\&prime;}(x_k-0)=\frac{h_{k-1}}{6}{M}_{k-1}+\frac{h_{k-1}}{3}{M}_k+\frac{x^{\left(1\right)}\left(k\right)-x^{\left(1\right)}(k-1)}{h_{k-1}}$

$S^{\&prime;}(x_k+0)=-\frac{h_k}{3}{M}_k+\frac{h_k}{6}{M}_{k+1}+\frac{x^{\left(1\right)}(k+1)-x^{\left(1\right)}\left(k\right)}{h_k}$

Lead continuity S ' (x by cubic spline tie point place single order _k-0)=S ' (x _k+ 0) can get

μ _kM _k-1+2M _k+λ _kM _k-1=d _k,j=2,3,…,n-1

Namely

$\left\{\begin{array}{c}{\mathrm{\&mu;}}_2{\mathrm{<figure-callout\; id="1"\; label="M"\; filenames="HDA00002767261200011.png"\; state="\{\{state\}\}">M</figure-callout>}}_1+2M_2+{\mathrm{\&lambda;}}_2{M}_3=d_2\\ {\mathrm{\&mu;}}_3{M}_2+2M_3+{\mathrm{\&lambda;}}_3{M}_4={d_3}_{}\\ ...\\ {\mathrm{\&mu;}}_{n-1}{M}_{n-2}+2M_{n-1}+{\mathrm{\&lambda;}}_{n-1}{M}_n=d_{n-1}\end{array}\right.$

Wherein

$\begin{array}{c}{\mathrm{\&mu;}}_k=\frac{h_{k-1}}{h_{k-1}+h_k},{\mathrm{\&lambda;}}_i=\frac{h_k}{h_{k-1}+h_k}\\ d_k=6\frac{f[x_k,x_{k+1}]-f[x_{k-1},x_k]}{h_{k-1}+h_k}\end{array}$ j=2,3,…,n-1

(4) due to M ₁=M _n=0, following formula can be written as:

Change into matrix form:

The matrix of coefficients of following formula is full rank, thereby M=(M ₂, M ₃, M _n-1) ^TSolution exist and unique, so can try to achieve background value

$\begin{array}{c}{z}^{\left(1\right)}(k+1)={\&Integral;}_{x_k}^{x_{k+1}}{x}^{\left(1\right)}\left(t\right)\mathrm{dt}={\&Integral;}_{x_k}^{x_{k+1}}S\left(x\right)\mathrm{dx}\\ =\frac{M_{k+1}+M_k}{24h_k}{(x_{k+1}-x_k)}^4+\frac{x^{\left(1\right)}(k+1)+x^{\left(1\right)}\left(k\right)-\frac{M_{k+1}+M_k}{6}{h}_k^2}{2h_k}{(x_{k+1}-x_k)}^2\\ k=\mathrm{1,2},...,n-1\end{array}$

Here it is our GM(1 of adopting the combination interpolation optimization to improve to obtain, 1) the new background value of model.

(5) obtaining carrying out GM(1,1 on the basis of new background value) foundation of forecast model, comprising following steps;

Be provided with original data sequence:, x ⁽⁰⁾(3) ..., x ⁽⁰⁾(n), they satisfy x ⁽⁰⁾(k) 〉=0, k=1,2 ..., n. utilizes this data sequence to set up GM(1,1) and the step of model is as follows:

(6) establish X ⁽⁰⁾={ x ⁽⁰⁾(1), x ⁽⁰⁾(2) ..., x ⁽⁰⁾(n) } be original series, it carried out one-accumulate obtain:

X ⁽¹⁾={x ⁽¹⁾(1),x ⁽¹⁾(2),…,x ⁽¹⁾(n)}

Wherein

(k=1,2 ..., n), claim X ⁽¹⁾(k) be X ⁽⁰⁾(k) one-accumulate sequence is designated as 1-AGO;

(7) set up the albinism differential equation of GM (1,1) model

$\frac{d{x}^{\left(1\right)}}{\mathrm{dt}}+a{x}^{\left(1\right)}=u$

Its difference form is x ⁽⁰⁾(k)+az ⁽¹⁾(k)=u

A wherein, u is parameter to be identified, and claims that a is development coefficient, u is the grey action;

(8) found the solution computation model development coefficient and parameters u to be identified by least square method.[a,u] ^T=(B ^TB) ^-1B ^TY _n

Here $B=\left[\begin{array}{cc}-z^{\left(1\right)}\left(2\right)& 1\\ -z^{\left(1\right)}\left(3\right)& 1\\ .& .\\ .& .\\ .& .\\ -z^{\left(1\right)}\left(n\right)& 1\end{array}\right]$ ，Y _n=[x ⁽⁰⁾(2),x ⁽⁰⁾(3),…,x ⁽⁰⁾(n)] ^T

And z ⁽¹⁾(k+1) be the background value of GM (1,1) forecast model;

(9) above background value z is tried to achieve in step (1)-(4) ⁽¹⁾(k+1) be used in matrix B

$z^{\left(1\right)}(k+1)={\&Integral;}_k^{k+1}{x}^{\left(1\right)}\left(t\right)\mathrm{dt}\&ap;{\&Integral;}_k^{k+1}{S}_k\left(t\right)\mathrm{dt}=\frac{21}{50}{x}^{\left(1\right)}\left(k\right)+\frac{23}{50}{x}^{\left(1\right)}(k+1)-\frac{2}{25}{x}^{\left(1\right)}(k+2)$

(10) Time Created response model

: ${\hat{x}}^{\left(1\right)}(k+1)=(x^{\left(0\right)}\left(1\right)-u/a)e^{-\mathrm{ak}}+u/a$ 。

(11) with discretize time response: ${\hat{x}}^{\left(1\right)}(k+1)=(x^{\left(0\right)}\left(1\right)-u/a)e^{-\mathrm{ak}}+u/a$ 。

(12) k value substitution walk-off-mode pattern is calculated the prediction accumulated value

(13) will predict that accumulated value is reduced to predicted value: ${\hat{x}}^{\left(0\right)}\left(k\right)={\hat{x}}^{\left(1\right)}\left(k\right)-{\hat{x}}^{\left(1\right)}(k-1)$ 。

Claims (2)

1. the GM based on cubic spline (1,1) model prediction method is characterized in that: comprise the following steps:

(1) original data sequence is chosen: choose according to target of prediction the original data sequence that forecast model adopts, and data sequence is necessary for one group of nonnegative number according to sequence, i.e. X ⁽⁰⁾

(2) the 1-AGO sequence is set up: with the original data sequence X that chooses ⁽⁰⁾As GM(1,1) basic data of forecast model, and to X ⁽⁰⁾Make 1-AGO, obtain result 1-AGO sequence X ⁽¹⁾, then respectively to X ⁽⁰⁾And X ⁽¹⁾Valid slickness check and the judgement of accurate index law, judgement original data sequence X ⁽⁰⁾With the 1-AGO sequence X ⁽¹⁾Whether satisfy GM(1,1) the applicable requirement of forecast model;

(3) background value generates: to the 1-AGO sequence X ⁽¹⁾Make background value Z ⁽¹⁾Generate, can calculate B and Y, wherein, $B=\left[\begin{array}{cc}-z^{\left(1\right)}\left(2\right)& 1\\ -z^{\left(1\right)}\left(3\right)& 1\\ .& .\\ .& .\\ .& .\\ -z^{\left(1\right)}\left(n\right)& 1\end{array}\right]$ ，Y _n=[x ⁽⁰⁾(2),x ⁽⁰⁾(3),…,x ⁽⁰⁾(n)] ^T，

z ⁽¹⁾(k) be the background value sequence, x ⁽⁰⁾(i) be original data sequence; Because least-squares estimation can be so that the indifference quadratic sum reaches minimum, therefore utilize least-squares estimation can obtain Argument List

,

Be the estimated value of a, the process of trying to achieve background value is as follows:

(a) for GM(1,1) model carries out the target of data prediction, take piecewise linear interpolation and cubic spline interpolation and calculation process as theoretical foundation, realizes the segmentation of interval [k, k+1] is comprised the following steps:

If second derivative value the S " (x of S (x) _k)=M _k(k=1,2 ..., n), and the second derivative at two ends is known, S " (1)=S " (n)=0.Due to S (x) at interval [x _k, x _k+1] on be cubic polynomial, therefore S " (x) at [x _k, x _k+1] on be linear function, can be expressed as:

$S^{\&prime;\&prime;}\left(x\right)=M_k\frac{x_{k+1}-x}{h_k}+M_{k+1}\frac{x-x_k}{h_k},k=\mathrm{1,2},...,n-1$

H wherein _k=x _k+1-x _k, k=1,2 ..., n-1

To following formula integration twice, and utilize S (x _k)=x ⁽¹⁾(k) and S (x _k+1)=x ⁽¹⁾(k+1), can obtain integration constant, so, can get the cubic spline functions expression formula:

$\begin{array}{c}S\left(x\right)=M_k\frac{{(x_{k+1}-x)}^3}{6h_k}+M_{k+1}\frac{{(x-x_k)}^3}{6h_k}\\ +(x^{\left(1\right)}\left(k\right)-\frac{M_k{h}_k^2}{6})\frac{x_{k+1}-x}{h_k}+(x^{\left(1\right)}(k+1)-\frac{M_k{h}_k^2}{6})\frac{x-x_k}{h_k}\\ k=\mathrm{1,2},...,n-1\end{array},$

Due to x ⁽¹⁾(k) ≈ S (t), therefore:

$\begin{array}{c}{z}^{\left(1\right)}(k+1)={\&Integral;}_{x_k}^{x_{k+1}}{x}^{\left(1\right)}\left(t\right)\mathrm{dt}={\&Integral;}_{x_k}^{x_{k+1}}S\left(x\right)\mathrm{dx}\\ =\frac{M_{k+1}+M_k}{24h_k}{(x_{k+1}-x_k)}^4+\frac{x^{\left(1\right)}(k+1)+x^{\left(1\right)}\left(k\right)-\frac{M_{k+1}+M_k}{6}{h}_k^2}{2h_k}{(x_{k+1}-x_k)}^2\\ k=\mathrm{1,2},...,n-1\end{array}$

Here, M _k(k=1,2 ..., be n) unknown, so background value z ⁽¹⁾Finding the solution (k+1) can be converted into an x _kPlace's second order is led M _kFind the solution;

(b) differentiate gets to S (x):

$S{\left(x\right)}^{\&prime;}=-M_k\frac{{(x_{k+1}-x)}^2}{2h_k}+M_{k+1}\frac{{(x-x_k)}^2}{2h_k}+\frac{x^{\left(1\right)}(k+1)-x^{\left(1\right)}\left(k\right)}{h_k}-\frac{M_{k+1}-M_k}{6}{h}_k$

By S ' (x) at interval [x _k-1, x _k] and [x _k, x _k+1] on expression formula, can try to achieve S ' (x) at an x _kLocating the left and right limit is respectively:

$S^{\&prime;}(x_k-0)=\frac{h_{k-1}}{6}{M}_{k-1}+\frac{h_{k-1}}{3}{M}_k+\frac{x^{\left(1\right)}\left(k\right)-x^{\left(1\right)}(k-1)}{h_{k-1}}$

$S^{\&prime;}(x_k+0)=-\frac{h_k}{3}{M}_k+\frac{h_k}{6}{M}_{k+1}+\frac{x^{\left(1\right)}(k+1)-x^{\left(1\right)}\left(k\right)}{h_k}$

Lead continuity S ' (x by cubic spline tie point place single order _k-0)=S ' (x _k+ 0) can get:

μ _kM _k-1+2M _k+λ _kM _k+1=d _k,j=2,3,…,n-1

That is:

$\left\{\begin{array}{c}{\mathrm{\&mu;}}_2{M}_1+2M_2+{\mathrm{\&lambda;}}_2{M}_3=d_2\\ {\mathrm{\&mu;}}_3{M}_2+2M_3+{\mathrm{\&lambda;}}_3{M}_4={d_3}_{}\\ ...\\ {\mathrm{\&mu;}}_{n-1}{M}_{n-2}+2M_{n-1}+{\mathrm{\&lambda;}}_{n-1}{M}_n=d_{n-1}\end{array}\right.$

Wherein:

$\begin{array}{c}{\mathrm{\&mu;}}_k=\frac{h_{k-1}}{h_{k-1}+h_k},{\mathrm{\&lambda;}}_i=\frac{h_k}{h_{k-1}+h_k}\\ d_k=6\frac{f[x_k,x_{k+1}]-f[x_{k-1},x_k]}{h_{k-1}+h_k}\end{array}$ j=2,3,…,n-1

(c) due to M ₁=M _n=0, following formula can be written as:

Change into matrix form:

The matrix of coefficients of following formula is full rank, thereby M=(M ₂, M ₃, M _n-1) ^TSolution exist and unique, so can try to achieve new background value: $\begin{array}{c}{z}^{\left(1\right)}(k+1)={\&Integral;}_{x_k}^{x_{k+1}}{x}^{\left(1\right)}\left(t\right)\mathrm{dt}={\&Integral;}_{x_k}^{x_{k+1}}S\left(x\right)\mathrm{dx}\\ =\frac{M_{k+1}+M_k}{24h_k}{(x_{k+1}-x_k)}^4+\frac{x^{\left(1\right)}(k+1)+x^{\left(1\right)}\left(k\right)-\frac{M_{k+1}+M_k}{6}{h}_k^2}{2h_k}{(x_{k+1}-x_k)}^2\\ k=\mathrm{1,2},...,n-1\end{array}$

(4) model is determined and is found the solution: a in step (3) and b are used estimated value

With

Replace, therefore can set up GM(1,1) model and time response sequence

, then solve the predicted value of first point

The analogue value, reduction at last solves the predicted value of initial point

The analogue value namely

, Value be the predicted value sequence of original data sequence;

(5) error-tested): after solving the predicted value of original data sequence, can utilize residual test method or the degree of association method of inspection or the poor method of inspection of posteriority to judge GM(1, the 1) precision of forecast model; GM(1,1) precision of forecast model can be passed through different background value generating modes, the choice of raw data, and the Residual GM of the conversion of data sequence, correction and different stage (1,1) model is improved.

2. a kind of GM based on cubic spline according to claim 1 (1,1) model prediction method is characterized in that: data data sequence commonly used in step (1) has scientific experimental data, empirical data, production data, decision data.

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