CN103106256A - Gray model (GM) (1,1) prediction method of orthogonal interpolation based on Markov chain - Google Patents

Abstract

The invention discloses a gray model (GM) (1,1) prediction method of orthogonal interpolation based on Markov chain. According to the grey orthogonal method and the principle of the Markov chain, the Gauss-Chebyshev orthogonal thought is used to predict overall trend of time-series data. Accuracy of the prediction is time-varying, the principle of the Markov chain has good advantages during the process of processing a time-varying system, and choosing the GM (1,1) prediction method of the orthogonal interpolation based on the Markov chain can solve the instability of the prediction result better, and therefore the grey Markov orthogonal model is put forward for the prediction of data of electricity consumption, and is suitable for a dynamic prediction process which is of a short-middle term, low in demanded quantity of data and large in data amplitude. The GM (1,1) prediction method of the orthogonal interpolation based on the Markov chain is scientific in inventive concept, simple to calculate, small in work load, and high in prediction accuracy, and good in usage value and wide in application range in the technical field of prediction.

Description

A kind of GM (1,1) model prediction method of the orthogonalization interpolation based on the Markov chain

Technical field

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

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.

Summary of the invention

The purpose of this invention is to provide a kind of orthogonalization model prediction method based on Grey Markov Chain, 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 orthogonalization model prediction method based on Grey Markov Chain, its special permission is: 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

,

Estimated value for a; The process of trying to achieve background value is as follows:

(a) make f (t)=x ⁽¹⁾(t)

{。##.##1},

$\begin{array}{c}{\∫}_k^{k+1}{x}^{\left(1\right)}\left(t\right)\mathrm{dt}={\∫}_k^{k+1}f\left(t\right)\mathrm{dt}=\frac{1}{2}{\∫}_{-1}^{1}f(\frac{1}{2}u+k+\frac{1}{2})\mathrm{du}\\ ={\∫}_{-1}^{1}f\left(v\right)\mathrm{dv}={\∫}_{-1}^1\frac{1}{\sqrt{1-v^2}}\sqrt{1-v^2}f\left(v\right)\mathrm{dv}={\∫}_{-1}^1\frac{1}{\sqrt{1-v^2}}F\left(v\right)\mathrm{dv}\end{array}$

$\≈A_{0}F\left(v_0\right)+A_{1}F\left(v_1\right)+A_{2}F\left(v_2\right)$

(b) Gauss point is the zero point of Chebyshev polynomials, therefore T ³(v)=4v ³-3v=0 has

$v_0=-\frac{\sqrt{3}}{2},v_1=0,v_2=\frac{\sqrt{3}}{2}$

(c) for the Gauss-Chebyshev polynomial expression quadrature F (v)=1 that has twice algebraic accuracy, v, v ²All accurately set up.

Simultaneous Equations

$\left\{\begin{array}{c}{A}_0+A_1+A_2={\∫}_{-1}^1\frac{1}{\sqrt{1-v^2}}\mathrm{dv}=\mathrm{\π}\\ -\frac{\sqrt{3}}{2}{A}_0+0A_1+\frac{\sqrt{3}}{2}{A}_2={\∫}_{-1}^1\frac{v}{\sqrt{1-v^2}}\mathrm{dv}=0\\ \frac{3}{4}{A}_0+0A_1+\frac{3}{4}{A}_2={\∫}_{-1}^1\frac{v^2}{\sqrt{1-v^2}}\mathrm{dv}=\frac{\mathrm{\π}}{2}\end{array}\right.$

Solve, $A_0=A_1=A_2=\frac{\mathrm{\π}}{3}$

(d) background value that is optimized is as follows

$\begin{array}{c}{z}^{\left(1\right)}(k+1)={\∫}_k^{k+1}{x}^{\left(1\right)}\left(t\right)\mathrm{dt}\≈{\∫}_k^{k+1}{S}_k\mathrm{dt}=\frac{\mathrm{\π}}{3}\sqrt{1-{(-\frac{\sqrt{3}}{2})}^2}{x}^{\left(1\right)}(k+\frac{2-\sqrt{3}}{4})\\ +\frac{\mathrm{\π}}{3}\sqrt{1-{\left(0\right)}^2}{x}^{\left(1\right)}(k+\frac{1}{2})+\frac{\mathrm{\π}}{3}\sqrt{1-{\left(\frac{\sqrt{3}}{2}\right)}^2}{x}^{\left(1\right)}(k+\frac{2+\sqrt{3}}{4})\end{array}$

Algorithm computer with the decimal node in following formula can't realize, so be the decimal Node integer node by suitable interpolation method, makes chebyshev algorithm be able to predict by computer realization.Method is as follows:

$z^{\left(1\right)}(k+1)={\∫}_k^{k+1}{x}^{\left(1\right)}\left(t\right)\mathrm{dt}\≈{\∫}_k^{k+1}{S}_k\left(t\right)\mathrm{dt}$

$=\frac{\mathrm{\π}}{3}\sqrt{1-{(-\frac{\sqrt{3}}{2})}^2}{x}^{\left(1\right)}(k+\frac{2-\sqrt{3}}{4})+\frac{\mathrm{\π}}{3}\sqrt{1-{\left(0\right)}^2}{x}^{\left(1\right)}(k+\frac{1}{2})+\frac{\mathrm{\π}}{3}\sqrt{1-{\left(\frac{\sqrt{3}}{2}\right)}^2}{x}^{\left(1\right)}(k+\frac{2+\sqrt{3}}{4})$

$\begin{array}{c}=\frac{\mathrm{\π}}{6}[\frac{15+8\sqrt{3}}{32}{x}^{\left(1\right)}\left(k\right)+\frac{9-4\sqrt{3}}{16}{x}^{\left(1\right)}(k+1)-\frac{1}{32}{x}^{\left(1\right)}(k+2)]\\ +\frac{\mathrm{\π}}{3}[\frac{3}{8}{x}^{\left(1\right)}\left(k\right)+\frac{3}{4}{x}^{\left(1\right)}(k+1)-\frac{1}{8}{x}^{\left(1\right)}(k+2)]+\frac{\mathrm{\π}}{6}[\frac{15-8\sqrt{3}}{32}{x}^{\left(1\right)}\left(k\right)+\frac{9+4\sqrt{3}}{16}{x}^{\left(1\right)}(k+1)-\frac{1}{32}{x}^{\left(1\right)}(k+2)]\end{array}$

$=\frac{9\mathrm{\π}}{32}{x}^{\left(1\right)}\left(k\right)+\frac{7\mathrm{\π}}{16}{x}^{\left(1\right)}(k+1)-\frac{5\mathrm{\π}}{96}{x}^{\left(1\right)}(k+2)$

Following formula is GM(1,1) the new background value of model.

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

With

Replace, and 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 according to step (4), recycle residual test method or the degree of association method of inspection or the poor method of inspection of posteriority and 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.

GM (1,1) the model prediction method of described a kind of orthogonalization interpolation based on the Markov chain is characterized in that: original 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 the orthogonalization interpolation, the background value of major effect precision of prediction to be reconstructed rule, and provided on this basis a kind of data message prediction method for digging of the orthogonalization interpolation based on the Markov chain, 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 adopts the method for orthogonalization interpolation, is satisfying under constringent condition, and structure Gauss interpolation function 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.

This paper of the present invention proposes a kind of new orthogonalization Forecasting Methodology based on Grey Markov Chain, take full advantage of the Gauss-Chebyshev orthogonalization method and process the characteristic that the advantage of non-Equidistant Nodes aspect and Markov chain are processed the large and markov property data of undulatory property aspect, building method to traditional gray model carries out the orthogonalization improvement, and the data point of big rise and fall is converted into interval prediction, thereby improve precision and the accuracy of model prediction.

Description of drawings

Fig. 1 is the inventive method process flow diagram.

Embodiment

As shown in Figure 1.A kind of orthogonalization model prediction method based on Grey Markov Chain comprises 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

,

For

Estimated value; The process of trying to achieve background value is as follows:

(a) make f (t)=x ⁽¹⁾(t)

{。##.##1},

$\begin{array}{c}{\∫}_k^{k+1}{x}^{\left(1\right)}\left(t\right)\mathrm{dt}={\∫}_k^{k+1}f\left(t\right)\mathrm{dt}=\frac{1}{2}{\∫}_{-1}^{1}f(\frac{1}{2}u+k+\frac{1}{2})\mathrm{du}\\ ={\∫}_{-1}^{1}f\left(v\right)\mathrm{dv}={\∫}_{-1}^1\frac{1}{\sqrt{1-v^2}}\sqrt{1-v^2}f\left(v\right)\mathrm{dv}={\∫}_{-1}^1\frac{1}{\sqrt{1-v^2}}F\left(v\right)\mathrm{dv}\end{array}$

$\≈A_{0}F\left(v_0\right)+A_{1}F\left(v_1\right)+A_{2}F\left(v_2\right)$

(b) Gauss point is the zero point of Chebyshev polynomials, therefore T ₃(v)=4v ³-3v=0 has

$v_0=-\frac{\sqrt{3}}{2},v_1=0,v_2=\frac{\sqrt{3}}{2}$

(c) for the Gauss-Chebyshev polynomial expression quadrature F (v)=1 that has twice algebraic accuracy, v, v ²All accurately set up.

Simultaneous Equations

$\left\{\begin{array}{c}{A}_0+A_1+A_2={\∫}_{-1}^1\frac{1}{\sqrt{1-v^2}}\mathrm{dv}=\mathrm{\π}\\ -\frac{\sqrt{3}}{2}{A}_0+0A_1+\frac{\sqrt{3}}{2}{A}_2={\∫}_{-1}^1\frac{v}{\sqrt{1-v^2}}\mathrm{dv}=0\\ \frac{3}{4}{A}_0+0A_1+\frac{3}{4}{A}_2={\∫}_{-1}^1\frac{v^2}{\sqrt{1-v^2}}\mathrm{dv}=\frac{\mathrm{\π}}{2}\end{array}\right.$

Solve, $A_0=A_1=A_2=\frac{\mathrm{\π}}{3}$

(d) background value that is optimized is as follows

$\begin{array}{c}{z}^{\left(1\right)}(k+1)={\∫}_k^{k+1}{x}^{\left(1\right)}\left(t\right)\mathrm{dt}\≈{\∫}_k^{k+1}{S}_k\mathrm{dt}=\frac{\mathrm{\π}}{3}\sqrt{1-{(-\frac{\sqrt{3}}{2})}^2}{x}^{\left(1\right)}(k+\frac{2-\sqrt{3}}{4})\\ +\frac{\mathrm{\π}}{3}\sqrt{1-{\left(0\right)}^2}{x}^{\left(1\right)}(k+\frac{1}{2})+\frac{\mathrm{\π}}{3}\sqrt{1-{\left(\frac{\sqrt{3}}{2}\right)}^2}{x}^{\left(1\right)}(k+\frac{2+\sqrt{3}}{4})\end{array}$

Algorithm computer with the decimal node in following formula can't realize, so be the decimal Node integer node by suitable interpolation method, makes chebyshev algorithm be able to predict by computer realization.Method is as follows:

$z^{\left(1\right)}(k+1)={\∫}_k^{k+1}{x}^{\left(1\right)}\left(t\right)\mathrm{dt}\≈{\∫}_k^{k+1}{S}_k\left(t\right)\mathrm{dt}$

$=\frac{\mathrm{\π}}{3}\sqrt{1-{(-\frac{\sqrt{3}}{2})}^2}{x}^{\left(1\right)}(k+\frac{2-\sqrt{3}}{4})+\frac{\mathrm{\π}}{3}\sqrt{1-{\left(0\right)}^2}{x}^{\left(1\right)}(k+\frac{1}{2})+\frac{\mathrm{\π}}{3}\sqrt{1-{\left(\frac{\sqrt{3}}{2}\right)}^2}{x}^{\left(1\right)}(k+\frac{2+\sqrt{3}}{4})$

$\begin{array}{c}=\frac{\mathrm{\π}}{6}[\frac{15+8\sqrt{3}}{32}{x}^{\left(1\right)}\left(k\right)+\frac{9-4\sqrt{3}}{16}{x}^{\left(1\right)}(k+1)-\frac{1}{32}{x}^{\left(1\right)}(k+2)]\\ +\frac{\mathrm{\π}}{3}[\frac{3}{8}{x}^{\left(1\right)}\left(k\right)+\frac{3}{4}{x}^{\left(1\right)}(k+1)-\frac{1}{8}{x}^{\left(1\right)}(k+2)]+\frac{\mathrm{\π}}{6}[\frac{15-8\sqrt{3}}{32}{x}^{\left(1\right)}\left(k\right)+\frac{9+4\sqrt{3}}{16}{x}^{\left(1\right)}(k+1)-\frac{1}{32}{x}^{\left(1\right)}(k+2)]\end{array}$

$=\frac{9\mathrm{\π}}{32}{x}^{\left(1\right)}\left(k\right)+\frac{7\mathrm{\π}}{16}{x}^{\left(1\right)}(k+1)-\frac{5\mathrm{\π}}{96}{x}^{\left(1\right)}(k+2)$

Following formula is GM(1,1) the new background value of model.

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

With

Replace, and 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 according to step (4), recycle residual test method or the degree of association method of inspection or the poor method of inspection of posteriority and 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.

Original data sequence commonly used in step (1) has scientific experimental data, empirical data, production data, decision data.

According to utilizing GM(1,1) model carries out the target of data prediction,

(1) make f (t)=x ⁽¹⁾(t)

{。##.##1},

$\begin{array}{c}{\∫}_k^{k+1}{x}^{\left(1\right)}\left(t\right)\mathrm{dt}={\∫}_k^{k+1}f\left(t\right)\mathrm{dt}=\frac{1}{2}{\∫}_{-1}^{1}f(\frac{1}{2}u+k+\frac{1}{2})\mathrm{du}\\ ={\∫}_{-1}^{1}f\left(v\right)\mathrm{dv}={\∫}_{-1}^1\frac{1}{\sqrt{1-v^2}}\sqrt{1-v^2}f\left(v\right)\mathrm{dv}={\∫}_{-1}^1\frac{1}{\sqrt{1-v^2}}F\left(v\right)\mathrm{dv}\end{array}$

$\≈A_{0}F\left(v_0\right)+A_{1}F\left(v_1\right)+A_{2}F\left(v_2\right)$

(2) Gauss point is the zero point of Chebyshev polynomials, therefore T ₃(v)=4v ³-3v=0 has

$v_0=-\frac{\sqrt{3}}{2},v_1=0,v_2=\frac{\sqrt{3}}{2}$

(3) for the Gauss-Chebyshev polynomial expression quadrature F (v)=1 that has twice algebraic accuracy, v, v ²All accurately set up.

(4) Simultaneous Equations

$\left\{\begin{array}{c}{A}_0+A_1+A_2={\∫}_{-1}^1\frac{1}{\sqrt{1-v^2}}\mathrm{dv}=\mathrm{\π}\\ -\frac{\sqrt{3}}{2}{A}_0+0A_1+\frac{\sqrt{3}}{2}{A}_2={\∫}_{-1}^1\frac{v}{\sqrt{1-v^2}}\mathrm{dv}=0\\ \frac{3}{4}{A}_0+0A_1+\frac{3}{4}{A}_2={\∫}_{-1}^1\frac{v^2}{\sqrt{1-v^2}}\mathrm{dv}=\frac{\mathrm{\π}}{2}\end{array}\right.$

Solve, $A_0=A_1=A_2=\frac{\mathrm{\π}}{3}$

The background value that is optimized is as follows

$\begin{array}{c}{z}^{\left(1\right)}(k+1)={\∫}_k^{k+1}{x}^{\left(1\right)}\left(t\right)\mathrm{dt}\≈{\∫}_k^{k+1}{S}_k\mathrm{dt}=\frac{\mathrm{\π}}{3}\sqrt{1-{(-\frac{\sqrt{3}}{2})}^2}{x}^{\left(1\right)}(k+\frac{2-\sqrt{3}}{4})\\ +\frac{\mathrm{\π}}{3}\sqrt{1-{\left(0\right)}^2}{x}^{\left(1\right)}(k+\frac{1}{2})+\frac{\mathrm{\π}}{3}\sqrt{1-{\left(\frac{\sqrt{3}}{2}\right)}^2}{x}^{\left(1\right)}(k+\frac{2+\sqrt{3}}{4})\end{array}$

Algorithm computer with the decimal node in following formula can't realize, so be the decimal Node integer node by suitable interpolation method, makes chebyshev algorithm be able to predict by computer realization.

Method is as follows:

$z^{\left(1\right)}(k+1)={\∫}_k^{k+1}{x}^{\left(1\right)}\left(t\right)\mathrm{dt}\≈{\∫}_k^{k+1}{S}_k\left(t\right)\mathrm{dt}$

$=\frac{\mathrm{\π}}{3}\sqrt{1-{(-\frac{\sqrt{3}}{2})}^2}{x}^{\left(1\right)}(k+\frac{2-\sqrt{3}}{4})+\frac{\mathrm{\π}}{3}\sqrt{1-{\left(0\right)}^2}{x}^{\left(1\right)}(k+\frac{1}{2})+\frac{\mathrm{\π}}{3}\sqrt{1-{\left(\frac{\sqrt{3}}{2}\right)}^2}{x}^{\left(1\right)}(k+\frac{2+\sqrt{3}}{4})$

$\begin{array}{c}=\frac{\mathrm{\π}}{6}[\frac{15+8\sqrt{3}}{32}{x}^{\left(1\right)}\left(k\right)+\frac{9-4\sqrt{3}}{16}{x}^{\left(1\right)}(k+1)-\frac{1}{32}{x}^{\left(1\right)}(k+2)]\\ +\frac{\mathrm{\π}}{3}[\frac{3}{8}{x}^{\left(1\right)}\left(k\right)+\frac{3}{4}{x}^{\left(1\right)}(k+1)-\frac{1}{8}{x}^{\left(1\right)}(k+2)]+\frac{\mathrm{\π}}{6}[\frac{15-8\sqrt{3}}{32}{x}^{\left(1\right)}\left(k\right)+\frac{9+4\sqrt{3}}{16}{x}^{\left(1\right)}(k+1)-\frac{1}{32}{x}^{\left(1\right)}(k+2)]\end{array}$

$=\frac{9\mathrm{\π}}{32}{x}^{\left(1\right)}\left(k\right)+\frac{7\mathrm{\π}}{16}{x}^{\left(1\right)}(k+1)-\frac{5\mathrm{\π}}{96}{x}^{\left(1\right)}(k+2)$

Be background value z ⁽¹⁾(k+1) generation and the x in original series ⁽¹⁾(k) point, x ⁽¹⁾(k+1) and x ⁽¹⁾(k+2) the point Linear table goes out, and these three points are respectively the previous point of background value, this site and the point that lags behind.

Following formula is exactly to adopt the Gauss-Chebyshev method of quadrature to improve the GM(1 that obtains, 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 ⁽¹⁾(1), x ⁽⁰⁾(2), 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) ^-1+B ^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

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

(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: ${\hat{x}}^{\left(0\right)}\left(k\right)={\hat{x}}^{\left(1\right)}\left(k\right)-{\hat{x}}^{\left(1\right)}(k-1)$

(14) Markov process refers to for stochastic process X (t), when at moment t ₀The time residing state known, state after this is only and t ₀The time state relevant, and and t ₀State in the past is irrelevant, and such X (t) is exactly Markov process.The basic ideas of Markov Chain Forecast method are to try to achieve the state-transition matrix of sequence by original data sequence, according to state-transition matrix, the variation tendency in future are made estimation.

The form of one step state transition matrix is

It has described n the probability distribution that state shifts mutually.

(15) in above-mentioned matrix, P _ijFor by Markov chain state S _iTransfer to state S _jProbability.A step transition probability of obtaining state is the key of Markov Chain Forecast method.The theoretical distribution of one step transition probability is unknown, therefore can utilize the frequency of the transfer between state as the estimated value of probability.

For Gauss-ChebyshevGM (1, what 1) model obtained predicts the outcome, can obtain Gauss-ChebyshevGM (1 according to the method for Markov chain, 1) model is in the deviation rule in known month, and according to this rule, Gauss-ChebyshevGM (1,1) model result is revised, by Gauss-ChebyshevGM (1,1) model result predicted numerical value, revising becomes estimation range interval and that probability forms.The basic ideas that Markov chain that Here it is is improved one's methods.

Claims (2)

1. orthogonalization model prediction method based on Grey Markov Chain, its special permission is: 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

,

Estimated value for a; The process of trying to achieve background value is as follows:

(a) make f (t)=x ⁽¹⁾(t)

{。##.##1},

$\begin{array}{c}{\∫}_k^{k+1}{x}^{\left(1\right)}\left(t\right)\mathrm{dt}={\∫}_k^{k+1}f\left(t\right)\mathrm{dt}=\frac{1}{2}{\∫}_{-1}^{1}f(\frac{1}{2}u+k+\frac{1}{2})\mathrm{du}\\ ={\∫}_{-1}^{1}f\left(v\right)\mathrm{dv}={\∫}_{-1}^1\frac{1}{\sqrt{1-v^2}}\sqrt{1-v^2}f\left(v\right)\mathrm{dv}={\∫}_{-1}^1\frac{1}{\sqrt{1-v^2}}F\left(v\right)\mathrm{dv}\end{array}$

$\≈A_{0}F\left(v_0\right)+A_{1}F\left(v_1\right)+A_{2}F\left(v_2\right)$

(b) Gauss point is the zero point of Chebyshev polynomials, therefore T ₃=4v ³-3v=0 has

$v_0=-\frac{\sqrt{3}}{2},v_1=0,v_2=\frac{\sqrt{3}}{2}$

(c) for the Gauss-Chebyshev polynomial expression quadrature F (v)=1 that has twice algebraic accuracy, v, v ²All accurately set up.

Simultaneous Equations

$\left\{\begin{array}{c}{A}_0+A_1+A_2={\∫}_{-1}^1\frac{1}{\sqrt{1-v^2}}\mathrm{dv}=\mathrm{\π}\\ -\frac{\sqrt{3}}{2}{A}_0+0A_1+\frac{\sqrt{3}}{2}{A}_2={\∫}_{-1}^1\frac{v}{\sqrt{1-v^2}}\mathrm{dv}=0\\ \frac{3}{4}{A}_0+0A_1+\frac{3}{4}{A}_2={\∫}_{-1}^1\frac{v^2}{\sqrt{1-v^2}}\mathrm{dv}=\frac{\mathrm{\π}}{2}\end{array}\right.$

Solve, $A_0=A_1=A_2=\frac{\mathrm{\π}}{3}$

(d) background value that is optimized is as follows

$\begin{array}{c}{z}^{\left(1\right)}(k+1)={\∫}_k^{k+1}{x}^{\left(1\right)}\left(t\right)\mathrm{dt}\≈{\∫}_k^{k+1}{S}_k\mathrm{dt}=\frac{\mathrm{\π}}{3}\sqrt{1-{(-\frac{\sqrt{3}}{2})}^2}{x}^{\left(1\right)}(k+\frac{2-\sqrt{3}}{4})\\ +\frac{\mathrm{\π}}{3}\sqrt{1-{\left(0\right)}^2}{x}^{\left(1\right)}(k+\frac{1}{2})+\frac{\mathrm{\π}}{3}\sqrt{1-{\left(\frac{\sqrt{3}}{2}\right)}^2}{x}^{\left(1\right)}(k+\frac{2+\sqrt{3}}{4})\end{array}$

Algorithm computer with the decimal node in following formula can't realize, so be the decimal Node integer node by suitable interpolation method, makes chebyshev algorithm be able to predict by computer realization.Method is as follows:

$z^{\left(1\right)}(k+1)={\∫}_k^{k+1}{x}^{\left(1\right)}\left(t\right)\mathrm{dt}\≈{\∫}_k^{k+1}{S}_k\left(t\right)\mathrm{dt}$

$=\frac{\mathrm{\π}}{3}\sqrt{1-{(-\frac{\sqrt{3}}{2})}^2}{x}^{\left(1\right)}(k+\frac{2-\sqrt{3}}{4})+\frac{\mathrm{\π}}{3}\sqrt{1-{\left(0\right)}^2}{x}^{\left(1\right)}(k+\frac{1}{2})+\frac{\mathrm{\π}}{3}\sqrt{1-{\left(\frac{\sqrt{3}}{2}\right)}^2}{x}^{\left(1\right)}(k+\frac{2+\sqrt{3}}{4})$

$\begin{array}{c}=\frac{\mathrm{\π}}{6}[\frac{15+8\sqrt{3}}{32}{x}^{\left(1\right)}\left(k\right)+\frac{9-4\sqrt{3}}{16}{x}^{\left(1\right)}(k+1)-\frac{1}{32}{x}^{\left(1\right)}(k+2)]\\ +\frac{\mathrm{\π}}{3}[\frac{3}{8}{x}^{\left(1\right)}\left(k\right)+\frac{3}{4}{x}^{\left(1\right)}(k+1)-\frac{1}{8}{x}^{\left(1\right)}(k+2)]+\frac{\mathrm{\π}}{6}[\frac{15-8\sqrt{3}}{32}{x}^{\left(1\right)}\left(k\right)+\frac{9+4\sqrt{3}}{16}{x}^{\left(1\right)}(k+1)-\frac{1}{32}{x}^{\left(1\right)}(k+2)]\end{array}$

$=\frac{9\mathrm{\π}}{32}{x}^{\left(1\right)}\left(k\right)+\frac{7\mathrm{\π}}{16}{x}^{\left(1\right)}(k+1)-\frac{5\mathrm{\π}}{96}{x}^{\left(1\right)}(k+2)$

Following formula is GM(1,1) the new background value of model.

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

With

Replace, and 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 according to step (4), recycle residual test method or the degree of association method of inspection or the poor method of inspection of posteriority and 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. the GM (1 of a kind of orthogonalization interpolation based on the Markov chain according to claim 1,1) model prediction method is characterized in that: original data sequence commonly used in step (1) has scientific experimental data, empirical data, production data, decision data.