CN107871180A - GM based on Newton-Cotes formulas tectonic setting value（1,1）Model prediction method - Google Patents
GM based on Newton-Cotes formulas tectonic setting value（1,1）Model prediction method Download PDFInfo
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Abstract
The invention discloses the GM based on Newton-Cotes formulas tectonic setting value (1,1) model prediction method, comprise the following steps：1, according to prediction Object selection forecast model used by original data sequence, this original data sequence is one group of nonnegative number data sequence, is designated as X(0)；2, to original data sequence X(0)One-accumulate processing is done, generates one-accumulate sequence X(1)；3, to one-accumulate sequence X(1)Using Newton-Cotes formulas tectonic setting value z(1)(k), and parameter a, u is obtained；4, based on solving parameter a, u, settling time response sequenceAnd reduce the predicted value for solving initial point
Description
Technical Field
The invention relates to the technical field of data prediction, in particular to a GM (1, 1) model prediction method for constructing a background value based on a Newton-Cootz formula.
Background
The goods turnover amount prediction methods are many, wherein the common methods include: time series method, BP neural network, regression analysis method, grey prediction, prediction of combination of methods and the like. The gray prediction is widely applied due to the advantages of less required samples, high prediction precision and the like.
The gray GM (1, 1) model is one of the core contents of the gray system theory, and the GM (1, 1) model is widely applied to various fields at present. In the prediction of a classic GM (1, 1) model proposed by professor Dungpo, the difference between the background value function structure and the actual value is large, the research finds that the structure of the background value plays a decisive role in the prediction result, the traditional background value is calculated by adopting a trapezoidal formula, and the calculation method is shown as the following formula.
In the formula, z (1) (k) Is the k-th background value, x (1) (k)、x (1) And (k-1) are respectively the k-th and k-1 terms of the primary accumulation sequence. FIG. 1 depicts the sources of error generated using the trapezoidal equation. The actual value of the background value is And withThe area enclosed is the area enclosed by the classical GM (1, 1) model, in order to simplify the calculation of the background value, the area enclosed by the curve is replaced by the area of the trapezoidal formula, and the background value is changed intoAnd withThe area enclosed by the cover is as follows. When the curve between k-1-k is relatively flat, the approximation substitution error by using the trapezoidal formula is relatively small, and the use error is gradually increased along with the increase of the gradient of the curve.
Based on many ideas of background value optimization, the conventional optimization equally divides a primary accumulation curve between k-1 and k, calculates the function value of each interpolation point between k-1 and k through interpolation, calculates a background value through formulas such as Simpson and Newton-Cookies, and effectively improves the prediction precision of the model.
However, after the construction of the background value reaches a certain precision along with the increase of the interpolation points, the precision of the model rather shows a descending trend, and the fitting effect of random data without obvious change rules is poor, generally, the calculation amount of the calculation method is large, and the fitting error is large for some cases, so that the construction of the GM (1, 1) background value still has a large improvement space.
Disclosure of Invention
The invention aims to overcome the defects in the prior art, and provides a GM (1, 1) model prediction method for constructing a background value based on a Newton-Cookies formula, so that the influence of uncertain factors on the stability and reliability of a system is reduced, and the synchronous control of a driving system and a response system is realized.
In order to solve the technical problem, the invention provides a GM (1, 1) model prediction method for constructing a background value based on a Newton-Cookies formula, which is characterized by comprising the following steps of:
s1, selecting an original data sequence adopted by a prediction model according to a prediction target, wherein the original data sequence is a group of non-negative data sequences marked as X (0) ；
Step S2, for the original data sequence X (0) Performing a summation process to generate a summation sequence X (1) ；
Step S3, for the primary accumulation sequence X (1) Construction of background value z using Newton-Cootz formula (1) (k) And calculating parameters a and u;
s4, establishing a time response sequence based on the solved parameters a and uAnd reducing to obtain the predicted value of the initial pointThe value is a predicted value sequence of the original data sequence;
and S5, after the predicted value of the original data sequence is solved according to the previous step, carrying out error check to judge the prediction precision of the GM (1, 1) model.
Further, in step S2, a primary accumulation sequence is generated by the following calculation:
in the formula, x (1) (k) As raw data x (0) (k) The accumulation sequence is recorded as:
X (1) ＝{x (1) (1),…,x (1) (n)}
further, according to a once-accumulated sequence X (1) The whitening differential equation and the gray differential equation are calculated:
generation of whitening differential equation:
generating a sequence { x for first order accumulation (1) (k) The differential equation of GM (1, 1) with respect to the time variable t is shown below:
in the formula, a and u are constants to be identified;
gray differential equation:
x (0) (k)+az (1) (k)＝u
further, in step S3, the specific steps of calculating the background value and determining the constants a and u are as follows:
(3.1) the whitening equation after deformation gives the following formula:
dx (1) (t)+ax (1) (t)dt＝udt
(3.2) the integration of the above formula over the interval [ k-2, k +2] gives the following formula:
(3.3) solving the integral term by using the Newton-Cookies formulaObtaining:
(3.4) Gray differential equation x for the above equation (0) (k)+az (1) (k) The format of = u is collated, resulting in the following formula:
(3.5) determination of the background value z by analogy with the classical grey prediction (1) (k)：
(3.6) determining matrix B and matrix Y by analogy with the above formula and classical grey prediction:
(3.7) the ash differential equation (3.4) is evaluated by the least squares method to determine the parameters a, u:
(a,u) T ＝(B T B) -1 B T Y。
further, in step S4, by solving the white differential equation, the time response function obtained by substituting the parameters a and u is:
discretizing the above formula to obtain:
in the formula, x (1) (0)＝x (0) (1)，Is a predicted value.
And (3) reduction of a predicted value:
further, in step S5, a calculation formula of the relative error during error checking is as follows:
compared with the prior art, the invention has the following beneficial effects: the invention constructs GM (1, 1) modeling of the background value based on the Newton-Karsts formula, and has simple calculation process and lower error.
Drawings
FIG. 1 is a graph of background error constructed using a trapezoidal equation;
FIG. 2 is a flow chart of selecting an original data sequence;
FIG. 3 is a flow chart of an accumulation process performed on an original data sequence;
FIG. 4 is a flow chart of background value calculation and parameter determination;
FIG. 5 is a flow chart for solving a sequence of predicted values for an original data sequence;
fig. 6 is a line graph of fit values versus actual values for the three methods.
Detailed Description
The invention is further described below with reference to the accompanying drawings. The following examples are only for illustrating the technical solutions of the present invention more clearly, and the protection scope of the present invention is not limited thereby.
The invention relates to a GM (1, 1) model prediction method for constructing a background value based on a Newton-Counters formula, which comprises the following steps of:
s1, selecting an original data sequence adopted by a prediction model according to a prediction target, wherein the original data sequence is a group of non-negative data sequences marked as X (0) 。
Let the original data sequence be:
X (0) ＝{x (0) (1),…,x (0) (n)}
in the formula, x (0) (i)>, 0, i =1, \8230;, n. Specific process for constructing original data sequence referring to FIG. 2, if there is prediction dataIn the negative case, the processing method comprises the following steps: 1) If the initial data are all negative numbers, all the data are used after absolute values are taken; 2) If part of the data is negative, all the data is added with the absolute value of the minimum negative number for use.
Step S2, for the original data sequence X (0) Performing an accumulation process to generate an accumulation sequence X (1) 。
Generating a first accumulation sequence X by the following calculation (1) With reference to fig. 3, the specific process is:
in the formula, x (1) (k) Is x (0) (1)……x (0) (k) The accumulation sequence is recorded as:
X (1) ＝{x (1) (1),…,x (1) (n)}
step S3, for the primary accumulation sequence X (1) As background value z (1) (k) Parameters a and u are generated and calculated by a least square method.
Generation of whitening differential equation:
generating a sequence { x for first order accumulation (1) (k) The differential equation of GM (1, 1) with respect to the time variable t is shown below:
in the formula, a and u are constants to be identified.
Gray differential equation:
x (0) (k)+az (1) (k)＝u
the specific steps of calculating the background value and determining the constants a and u are as follows, referring to fig. 4:
(3.1) deformation of the whitening equation yields the following formula:
dx (1) (t)+ax (1) (t)dt＝udt
(3.2) the integration of the above formula over the interval [ k-2, k +2] gives the following formula:
(3.3) solving the integral term by using Newton-Cowster's equationObtaining:
(3.4) Gray differential equation x for the above equation (0) (k)+az (1) (k) The format of = u is collated, giving the following formula:
(3.5) determining the background value z by analogy of the above equation with classical grey prediction (1) (k)：
(3.6) determining matrix B and matrix Y by analogy with the above formula and classical grey prediction:
(3.7) the parameters a, u can be found by the least squares method for the ash differential equation (3.4):
(a,u) T ＝(B T B) -1 B T Y
s4, establishing a time response sequence based on the solved parameters a and uReducing and solving to obtain the predicted value of the initial pointThis value is the sequence of predicted values for the original data sequence.
Referring to fig. 5, by solving the white differential equation, the parameters a and u are substituted to obtain the time response function:
discretizing the above formula to obtain:
in the formula, x (1) (0)＝x (0) (1)，Is a predicted value.
And (3) reduction of a predicted value:
and S5, after the predicted value of the original data sequence is solved according to the previous step, carrying out error check to judge the prediction precision of the GM (1, 1) model.
Calculation of relative error:
the lower the relative error, the higher the fitting accuracy of the representation model, and the greater the reliability of the result obtained by the prediction.
Examples
In order to verify the effectiveness of the invention, taking the railway transportation turnover number of 2000-2009 in Heilongjiang province as an example, a new background value construction method of a GM (1, 1) model based on interpolation and Newton Cotes formula (see document 1: li Peak, davinghai. Jun. GM (1, 1) model based on interpolation and Newton-Cotes formula) is compared with the method provided by the invention by applying [ J ]. Systematic engineering theory and practice, 2004, 24 (10): 122-126.), and a classical GM (1, 1) model (see document 2: dungpo. Gray system basic method [ M ]. Wuhan: university of Mandarin China, 1987), and the advantages and disadvantages of the method are analyzed according to the comparison of relative errors.
(1) The original data sequence of the railway transportation turnover number in 2000-2009 in Heilongjiang province is as follows (unit: hundred million tons kilometers):
X (0) ＝{736.7，761.6，766.7，808.9，874.5，919.7，937.3，978.9，1029.3，980.7}
(2) The primary accumulation sequence is calculated as:
X (1) ＝{736.7，1498.3，2265，3073.9，3948.4，4868.1，5805.4，6784.3，7813.6，8794.3}
the method comprises the following steps:
(3) For one-time accumulation sequence X (1) Generating background values, and calculating B and Y
Background value z (1) (k)：
(4) Calculation of parameters a, u:
(5) Determination of the response function:
(6) And (3) reduction of a predicted value:
the calculation is performed by using a GM (1, 1) model based on interpolation and Newton's Cortes equation in document 1:
(3) Calculation of newton interpolation between once-accumulated sequence points:
and (2) constructing a nine-order interpolation polynomial for the primary accumulation sequence, wherein the detailed process is visible in high mathematics of engineering, and quartering between any two points, and the function value of the equant points calculated by the nine-order polynomial is as follows:
(4) Calculating B and Y:
background value z (1) (k) Three equally divided points are arranged between any two accumulation sequence points, corresponding function values are shown in step (3), and the five points are used for calculating the background value z by using a Newton's Cookies formula (1) (k) Comprises the following steps:
(5) Calculation of parameters a, u:
(6) Determination of the response function:
(7) And (3) reduction of a predicted value:
the calculation was carried out using the classical GM (1, 1) model proposed by professor dunghong in document 2:
determination of the response function:
and (3) reduction of a predicted value:
comparison of the fitting results of the three methods:
the fitting results of the three methods are shown in table 1 below, calculated by relative error equations.
TABLE 1 fitting results of the three methods
As can be seen from comparison of the fitting results of the methods shown in table 1 and fig. 6, taking the turnover number of railway transportation in 2000-2009 from heilongjiang as an example, the fitting relative errors based on the methods of documents 1 and 2 are 2.317% and 2.32%, respectively, and the relative error of the optimized GM (1, 1) method proposed herein is at least 1.94%.
The invention provides a background value construction method with simple calculation, which uses a Newton's Cockets formula and proves the effectiveness of the method by taking the railway turnover number of 2000-2009 in Heilongjiang province as an example.
The above description is only a preferred embodiment of the present invention, and it should be noted that, for those skilled in the art, it is possible to make various improvements and modifications without departing from the technical principle of the present invention, and those improvements and modifications should be also considered as the protection scope of the present invention.
Claims (6)
1. The GM (1, 1) model prediction method for constructing the background value based on the Newton-Cowster formula is characterized by comprising the following steps of:
s1, selecting an original data sequence adopted by a prediction model according to a prediction target, wherein the original data sequence is a group of non-negative data sequences marked as X (0) ；
Step S2, for the original data sequence X (0) Performing an accumulation process to generate an accumulation sequence X (1) ；
Step S3, for the primary accumulation sequence X (1) Construction of background value z using Newton-Cootz formula (1) (k) And calculating parameters a and u;
s4, establishing a time response sequence based on the solved parameters a and uAnd reducing to obtain the predicted value of the initial pointThe value is a predicted value sequence of the original data sequence;
and S5, after the predicted value of the original data sequence is solved according to the previous step, carrying out error check to judge the prediction precision of the GM (1, 1) model.
2. The GM (1, 1) model prediction method for constructing background values based on newton-kowski equation as claimed in claim 1, wherein in step S2, the first cumulative sequence is calculated by the following formula:
in the formula, x (1) (k) As raw data x (0) (k) The accumulation sequence is recorded as:
X (1) ＝{x (1) (1),…,x (1) (n)}
3. the GM (1, 1) model prediction method for constructing background values based on Newton-Cowski equation as claimed in claim 1, wherein the background values are constructed according to a once-accumulated sequence X (1) Calculate whitening differential equation and gray differential equation:
generation of whitening differential equation:
generating a sequence { x for a first order accumulation (1) (k) The differential equation of GM (1, 1) with respect to the time variable t is shown below:
in the formula, a and u are constants to be identified;
gray differential equation:
x (0) (k)+az (1) (k)＝u
4. the GM (1, 1) model prediction method for constructing the background value based on Newton-Cookies' formula as claimed in claim 3, wherein in step S3, the specific steps of calculating the background value and determining the constants a and u are as follows:
(3.1) the whitening equation after deformation gives the following formula:
dx (1) (t)+ax (1) (t)dt＝udt
(3.2) the above formula is integrated over the interval [ k-2, k +2] to give the following formula:
(3.3) solving the integral term by using Newton-Cowster's equationObtaining:
(3.4) Gray differential equation x for the above equation (0) (k)+az (1) (k) The format of = u is collated, resulting in the following formula:
(3.5) determining the background value z by analogy of the above equation with classical grey prediction (1) (k)：
(3.6) determining matrix B and matrix Y by analogy of the above equation with classical grey prediction:
(3.7) the ash differential equation (3.4) is evaluated by the least squares method to determine the parameters a, u:
(a,u) T ＝(B T B) -1 B T Y。
5. the GM (1, 1) model prediction method for constructing background values based on Newton-Cowski equation as claimed in claim 4, wherein in step S4, the time response function is obtained by solving the white differential equation by substituting the parameters a, u as:
discretizing the above formula to obtain:
in the formula, x (1) (0)＝x (0) (1)，Is a predicted value.
And (3) reduction of a predicted value:
6. the GM (1, 1) model prediction method for constructing the background value based on the newton-kowski equation as claimed in claim 1, wherein in step S5, the calculation formula of the relative error during error checking is:
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| CN109902593A (en) * | 2019-01-30 | 2019-06-18 | 济南大学 | A kind of gesture occlusion detection method and system based on Kinect |
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| Publication number | Priority date | Publication date | Assignee | Title |
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| CN108665110A (en) * | 2018-05-16 | 2018-10-16 | 淮阴工学院 | A kind of GM based on improvement background value（1,1）The Port Throughput Capacity Forecast Method of model |
| CN109902593A (en) * | 2019-01-30 | 2019-06-18 | 济南大学 | A kind of gesture occlusion detection method and system based on Kinect |
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