CN107918808A - Grey forecasting model method based on inverse cosine function and power function transformation and background value Combinatorial Optimization - Google Patents
Grey forecasting model method based on inverse cosine function and power function transformation and background value Combinatorial Optimization Download PDFInfo
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Abstract
The invention discloses a kind of grey forecasting model method based on inverse cosine function and power function transformation and background value Combinatorial Optimization, include the following steps：Step 1, generates original data sequence；Step 2, pre-processes initiation sequence；Step 3, inverse cosine function and power function combined transformation are carried out by pretreated sequence；Step 4, generates one-accumulate sequence；Step 5, based on method of exhaustion tectonic setting value, carries out gray prediction；Step 6, determines receptance function；Step 7, reduces predicted value；Step 8, calculates relative error；Step 9, selects the relative error magnitudes of minimum；The present invention can effectively raise the smoothness of initiation sequence, adapt to various data variation types；Calculating process is simple；Error is low, the effective precision of prediction for improving model.
Description
Technical field
The present invention relates to technical field of data prediction, and in particular to the grey of a kind of initiation sequence and background value Combinatorial Optimization
Prediction model method.
Background technology
Science accurately predicts Highway Passenger Transportation Volume, grasps trend, feature, rule and the number of the Highway Passenger Transportation Volume development in future
Amount, to formulate related roads decision-making form the basis.Traditional Forecasting Methodology has kind more than 300, and wherein passenger traffic volume forecast is more common
It is Regression Forecast, but the precision of prediction of this method is limited to the selection of passenger traffic volume influence factor.The present invention will use improved
GM (1,1) model is predicted；GM (1,1) model prediction accuracy is high, it is simple to calculate.
But classical GM (1,1) models are there are many defects, according to investigation, the smoothness and background of initiation sequence
The accuracy of value has a great influence precision of prediction.Traditional research is all parked in the optimization to one side, and the present invention is to existing
Method improved, power function is incorporated into inverse cosine function, and the construction of background value is improved, it is proposed that be based onFunctional transformation and GM (1,1) prediction model of background value Combinatorial Optimization.
The content of the invention
To solve the deficiencies in the prior art, it is an object of the invention to propose one kind to be based onFunctional transformation with
GM (1,1) prediction model method of background value Combinatorial Optimization, can effectively adapt to various data variation types, effectively improve mould
The precision of prediction of type.
In order to realize above-mentioned target, the present invention adopts the following technical scheme that：
Grey forecasting model method based on inverse cosine function and power function transformation and background value Combinatorial Optimization, including it is as follows
Step：
Step 1, according to original data sequence used by prediction Object selection prediction model, this original data sequence is
One group of non-negative incrementally several data sequence, is denoted as X(0)；
If original data sequence is：
X(0)={ x(0)(1),···,x(0)(n)}；
In formula, x(0)(i) ＞ 0i=1, n；
Step 2, to initiation sequence X(0)Pre-processed, by the scope control of all data in section [0,1], be denoted as
F(0), the formula of pretreatment is shown below：
In formula, x(0)(k) it is k-th of value in initiation sequence, M is more than X(0)All numbers in sequence, can empirically personnel
Preference is voluntarily chosen, f(0)(k) it is k-th of value in the sequence after pretreatment；
Step 3, to F(0)Sequence carries outFunctional transformation, obtains new data sequence R(0)Calculation formula such as following formula
It is shown：
In formula, r(0)(k) it is processFunctional transformation obtains k-th of value of sequence；
Step 4, to original data sequence R(0)One-accumulate processing is done, calculation formula is shown below：
In formula, r(1)(k) it is initial data r(0)(k) one-accumulate sequence, cumulative sequence are denoted as：
R(1)={ r(1)(1),···,r(1)(n)}；
Step 5, one-accumulate sequence R(1)Meet the white differential equationCalculated using background value public
Formula calculates z(1)(k) and matrix B, Y, parameter a, u is obtained using least square method,
z(1)(k) calculated by following formula：
z(1)(k+1)=β r(1)(k)+(1-β)·r(1)(k+1) (k=1, n-1；0≤β≤1)；
Matrix B, Y and parameter a, u are calculated by following formula：
(a,u)T=(BTB)-1BTY；
Step 6, based on parameter a, u solved, settling time response sequenceAnd reduce and solveSequence
The predicted value of row
By solving the white differential equationBring parameter a, u into and can obtain time response function and be：
Above formula discretization obtains：
In formula, r(1)(1)=r(0)(1),For predicted value；
Calculated by following formula：
Step 7, reduces predicted value；The reducing value of predicted value is calculated by reducing formula, is denoted asReduce formula such as
Shown in following formula：
Step 8, calculates relative error；After the predicted value that original data sequence is solved according to step 7, according to following public affairs
Formula carries out error-tested to judge the precision of prediction of GM (1,1) model；
The foregoing grey forecasting model method based on inverse cosine function and power function transformation and background value Combinatorial Optimization, step
In rapid three,0.1,0.5,0.9 is taken respectively, according toEach value is taken to useGM (1,1) mould is reused after functional transformation
Type is predicted to obtain the relative error of each value, the final minimum relative error calculating for choosing minimum relative error as the present invention
Value.
The foregoing grey forecasting model method based on inverse cosine function and power function transformation and background value Combinatorial Optimization, step
In rapid five, background value z(1)(k) value of constant β is not known in can determine that β is increased since 0 with precision 0.01 by the method for exhaustion
Long, when reaching 1 end loop, final selection makes the β value of relative error minimum.
The invention has the beneficial effects that：
Initiation sequence proposed by the present invention is based onThe prioritization scheme of functional transformation, power function is dissolved into initially
In series processing function, there is provided a kind of new initiation sequence transforming function transformation function, compared with traditional arccosx functional transformations, this
Invention is the further expansion of conventional method, can not only improve the smoothness of initiation sequence, compared to conventional method, prediction essence
Degree has also been greatly improved.
Background value construction proposed by the present invention based on the method for exhaustion, compared with traditional trapezoid formula tectonic setting value, this hair
Bright is the further expansion of conventional method, it is possible to increase the search range of background value, has compared to conventional method precision of prediction
Very big raising.
Brief description of the drawings
Fig. 1 is the operational flow diagram of prediction model of the present invention；
Fig. 2 is the present inventionThe ginseng of functional image makes figure；
Fig. 3 is the flow chart that the present invention does original data sequence one-accumulate processing；
Fig. 4 is the flow chart of β systems of selection of the present invention；
Fig. 5 is the present invention and two methods of the match value of the prior art and the line chart of actual value.
Embodiment
Make specific introduce to the present invention below in conjunction with the drawings and specific embodiments.
Grey forecasting model method based on inverse cosine function and power function transformation and background value Combinatorial Optimization, including it is as follows
Step：
Step 1, according to original data sequence used by prediction Object selection prediction model, this original data sequence is
One group of non-negative incrementally several data sequence, is denoted as X(0)；
If original data sequence is：
X(0)={ x(0)(1),···,x(0)(n)}；
In formula, x(0)(i) ＞ 0i=1, n；
Step 2, to initiation sequence X(0)Pre-processed, by the scope control of all data in section [0,1], be denoted as
F(0), the formula of pretreatment is shown below：
In formula, x(0)(k) it is k-th of value in initiation sequence, M is more than X(0)All numbers in sequence, can empirically personnel
Preference is voluntarily chosen, f(0)(k) it is k-th of value in the sequence after pretreatment；
Step 3, to F(0)Sequence carries outFunctional transformation, obtains new data sequence R(0)Calculation formula is as follows
Shown in formula：
In formula, r(0)(k) it is processFunctional transformation obtains k-th of value of sequence；
Step 4, to original data sequence R(0)One-accumulate processing is done, detailed process is with reference to Fig. 3 calculation formula such as following formula
It is shown：
In formula, r(1)(k) it is initial data r(0)(k) one-accumulate sequence, cumulative sequence are denoted as：
R(1)={ r(1)(1),···,r(1)(n)}；
Step 5, one-accumulate sequence R(1)Meet the white differential equationUsing background value calculation formula
Calculate z(1)(k) and matrix B, Y, parameter a, u is obtained using least square method,
z(1)(k) calculated by following formula：
z(1)(k+1)=β r(1)(k)+(1-β)·r(1)(k+1) (k=1, n-1；0≤β≤1)；
Matrix B, Y and parameter a, u are calculated by following formula：
(a,u)T=(BTB)-1BTY；
Step 6, based on parameter a, u solved, settling time response sequenceAnd reduce and solveSequence
The predicted value of row
By solving the white differential equationBring parameter a, u into and can obtain time response function and be：
Above formula discretization obtains：
In formula, r(1)(1)=r(0)(1),For predicted value；
Calculated by following formula：
Step 7, reduces predicted value；The reducing value of predicted value is calculated by reducing formula, is denoted asReduce formula such as
Shown in following formula：
Step 8, generates one-accumulate sequence, calculates relative error；The pre- of original data sequence is solved according to step 7
After measured value, error-tested is carried out according to the following formula to judge the precision of prediction of GM (1,1) model；
In the step of need to further illustrating three,Value is 0.1,0.5,0.9, according toEach value is taken to useGroup
GM (1,1) model prediction, which is reused, after conjunction conversion obtains the relative error of each value, it is final to choose minimum relative error conduct
The minimum relative error calculated value of the present invention.
In the step of need to further illustrating five, background value z(1)(k) value of constant β is not known in can pass through the method for exhaustion
Determine, β is increased since 0 with precision 0.01, when reaching 1 end loop.It is final to choose the β value for making relative error minimum, specifically
Exhaustion process is with reference to Fig. 4
The theoretical proof that smoothness improves
In order to prove that the present invention can improve the smoothness of initiation sequence, the present invention is demonstrate,proved in terms of the definition of smoothness
It is bright.
Define 1.1 and set X(0)={ x(0)(k), k=1,2, n } and it is non-negative data sequence,IfThen claim X(0)For smooth discrete series.
Theorem 1.1X(0)={ x(0)(k), k=1,2, n } for the necessary and sufficient condition of smooth discrete series beIt is the decreasing function of k.
It is respectively F (x) and G (x) that theorem 1.2, which is equipped with two kinds of data conversion, if having to arbitrary nonnegative sequence x (k)：
Then claim the effect of the smoothness of conversion F (x) raising sequences better than conversion G (x).
Prove 1：BySequence after functional transformation is smooth discrete series.
Initiation sequence X(0)For non-negative monotonically increasing function, it is set to meet the monotonic increasing function of section [0,1] by pretreatment
Sequence F(0), it is known that inverse cosine function is monotonic decreasing function on [0,1], the specific image of transforming function transformation function used in the present invention
As shown in Figure 2.
0 ＜ f(0)(k) ＜ 1
I.e.：
That is functionOn the monotonic decreasing function of k, that is, have：
Again because inverse cosine function is greater than zero on section [0,1], that is, have：
Because inequality both sides are all greater than zero number, both the above formula inequality both sides are multiplied, inequality
Symbol is constant, obtains following formula：
Following formula is obtained after being adjusted：
Passed through according to theorem 1.1Sequence after functional transformation is smooth discrete series.
Prove 2：BySequence smoothness after functional transformation is better than the smoothness of initiation sequence.
Order,
Derivation is carried out to above formula, obtains following formula：
The h'(x according to above formula)≤0, that is, prove that h (x) is subtraction function, that is, is had：
Inequality both sides obtain following formula after being arranged：
To 1, k-1 carries out above formula calculating respectively, and sign of inequality direction does not change, obtained after all formulas summations as
Under：
Following formula is obtained after above formula is arranged：
According to theorem 1.2,Functional transformation can lift initiation sequence smoothness.
In order to prove that the present invention has the average relative error of smaller, the effective precision of prediction for improving model；Passing through will
The passenger traffic volume sequence of China 1990-2002 is as research object (unit：Ten thousand people), by classical gray model (referring to document
1：Deng gathers imperial gray systems basic skills [M] Wuhan:Huazhong University of Science and Technology publishing house, 1987.) it is based on initiation sequenceGM (1,1) model of conversion is (referring to document 2：The grey that Li Ye, Lu Shan, Dong Fen justice are converted based on inverse cosine function
The practice of prediction model research [J] mathematics and understanding, 2016,46 (11)：252-254) compared with method proposed by the present invention
Compared with according to the comparison of relative error come the quality of analysis method.
The calculation procedure of proposition method of the present invention：
(1) original data sequence of the passenger traffic volume of China 1990-2002 is：
(2) to initiation sequence X(0)Pre-processed, by the scope control of all data in section [0,1], be denoted as F(0)；
(3) to F(0)Sequence carries outFunctional transformation, obtains new data sequence, is denoted as R(0)；
(4) to original data sequence R(0)It is one-accumulate processing, generation one-accumulate sequence R(1), calculation formula and gained
Sequence is as follows：
(5) to one-accumulate sequence R(1)Z is calculated using background value calculation formula(1)(k) and matrix B, Y, utilizes a most young waiter in a wineshop or an inn
Multiplication obtains parameter a, u；
z(1)(k) calculated by following formula：
z(1)(k+1)=β r(1)(k)+(1-β)·r(1)(k+1) (k=1, n-1；0≤β≤1)
Matrix B, Y and parameter a, u are calculated by following formula：
(a,u)T=(BTB)-1BTY
(6) based on parameter a, u solved, settling time response sequenceAnd reduce and solveSequence
Predicted value
By the solution of the white differential equation, bring parameter a, u into and can obtain time response function and be：
Above formula discretization obtains：
In formula, r(1)(1)=r(0)(1),For predicted value.
Calculated by following formula：
(7) reducing value of predicted value is calculated by reducing formula, is denoted asReduction formula is shown below：
(8) present invention takesFor 0.1,0.5,0.9, according toEach value is taken to useGM is reused after combined transformation
(1,1) model prediction obtains the relative error of each value, final to choose minimum relative error as the minimum opposite of the present invention
Error calculation value.
Prove that the present invention passes through in initiation sequence with table 1After conversion, background value Combinatorial Optimization energy is used in combination
Enough reduce error；
Table 1 does not use background value optimization and the fitting result using background value optimization
By table 1 the experiment proves that in parameterIn the case that value determines, prediction essence can be improved using background value optimization
Degree, it was demonstrated that the validity of background value prioritization scheme, andWhen obtained precision of prediction be up to
1.5606%, precision of prediction of the value as the present invention.
Classical GM (1,1) model that professor Deng Julong proposes in document 1 calculates：
Receptance function determines：
The reduction of predicted value：
GM (1,1) model that 2 initiation sequence of document is converted based on cosx
(1) original data sequence of the passenger traffic volume of China 1990-2002 is：
(2) to initiation sequence X(0)Pre-processed, by the scope control of all data in section [0,1], be denoted as F(0)；
(3) to F(0)Sequence carries out arccosx functional transformations, obtains new data sequence, is denoted as R(0)；
r(0)(k)=arccosf(0)(k)
(4) to original data sequence R(0)It is one-accumulate processing, generation one-accumulate sequence R(1)
To one-accumulate sequence R(1)Z is calculated using background value calculation formula(1)(k) and matrix B, Y, utilizes least square method
Obtain parameter a, u；
z(1)(k) calculated by following formula：
(6) based on parameter a, u solved, settling time response sequenceAnd reduce and solveSequence
Predicted value
r(1)(k+1)=- 13.3891e-0.0876k+14.4881
Calculated by following formula：
(7) it is by reducing the reducing value of formula calculating predicted value
Passenger traffic volume row the Fitting Calculation by above-mentioned three kinds of computational methods to China 1990-2002, result of calculation such as table 2
It is shown.
The fitting result of 2 three kinds of computational methods of table
According to upper table, improved GM (1,1) model average relative error proposed by the present invention is 1.56%, Classical Grey
The average relative error of color model is 3.83%, the average phase of GM (1, the 1) model of initiation sequence based on arccosx functional transformations
It is 2.07% to error, therefore improved GM proposed by the present invention (1,1) method error is relatively low, precision improvement is more obvious, intends
Result is closed with reference to shown in Fig. 5.According to it is demonstrated experimentally that this method is applicable not only to the prediction of China's passenger traffic volume, can also apply to
In the otherwise prediction of traffic, there is good prediction result.
Initiation sequence proposed by the present invention is based onThe prioritization scheme of functional transformation, power function is dissolved into initially
In series processing function, there is provided a kind of new initiation sequence transforming function transformation function, compared with traditional arccosx functional transformations, this
Invention is the further expansion of conventional method, can not only improve the smoothness of initiation sequence, compared to conventional method, prediction essence
Degree has also been greatly improved.
Background value construction proposed by the present invention based on the method for exhaustion, compared with traditional trapezoid formula tectonic setting value, this hair
Bright is the further expansion of conventional method, it is possible to increase the search range of background value, has compared to conventional method precision of prediction
Very big raising.
The basic principles, main features and advantages of the invention have been shown and described above.The technical staff of the industry should
Understand, the invention is not limited in any way for above-described embodiment, all to be obtained by the way of equivalent substitution or equivalent transformation
Technical solution, all falls within protection scope of the present invention.
Claims (3)
1. the grey forecasting model method based on inverse cosine function and power function transformation and background value Combinatorial Optimization, its feature exist
In including the following steps：
Step 1, according to prediction Object selection prediction model used by original data sequence, this original data sequence be one group
It is non-negative incrementally to count data sequence, it is denoted as X(0)；
If original data sequence is：
X(0)={ x(0)(1),···,x(0)(n)}；
In formula, x(0)(i) ＞ 0i=1, n；
Step 2, to initiation sequence X(0)Pre-processed, by the scope control of all data in section [0,1], be denoted as F(0),
The formula of pretreatment is shown below：
<mrow>
<msup>
<mi>f</mi>
<mrow>
<mo>(</mo>
<mn>0</mn>
<mo>)</mo>
</mrow>
</msup>
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<mo>(</mo>
<mi>k</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
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<mi>M</mi>
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<mo>(</mo>
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<mo>=</mo>
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<mo>,</mo>
<mo>...</mo>
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<mi>n</mi>
<mo>)</mo>
</mrow>
<mo>;</mo>
</mrow>
In formula, x(0)(k) it is k-th of value in initiation sequence, M is more than X(0)All numbers in sequence, can empirically personnel's preference
Voluntarily choose, f(0)(k) it is k-th of value in the sequence after pretreatment；
Step 3, to F(0)Sequence carries outFunctional transformation, obtains new data sequence R(0)Calculation formula such as following formula institute
Show：
<mrow>
<msup>
<mi>r</mi>
<mrow>
<mo>(</mo>
<mn>0</mn>
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<mo>(</mo>
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<mo>&part;</mo>
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<mo>;</mo>
</mrow>
In formula, r(0)(k) it is processFunctional transformation obtains k-th of value of sequence；
Step 4, to original data sequence R(0)One-accumulate processing is done, calculation formula is shown below：
<mrow>
<msup>
<mi>r</mi>
<mrow>
<mo>(</mo>
<mn>1</mn>
<mo>)</mo>
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<mo>(</mo>
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<mo>=</mo>
<munderover>
<mo>&Sigma;</mo>
<mrow>
<mi>i</mi>
<mo>=</mo>
<mn>1</mn>
</mrow>
<mi>k</mi>
</munderover>
<msup>
<mi>r</mi>
<mrow>
<mo>(</mo>
<mn>0</mn>
<mo>)</mo>
</mrow>
</msup>
<mrow>
<mo>(</mo>
<mi>i</mi>
<mo>)</mo>
</mrow>
<mo>,</mo>
<mi>k</mi>
<mo>=</mo>
<mn>1</mn>
<mo>,</mo>
<mo>...</mo>
<mo>,</mo>
<mi>n</mi>
<mo>;</mo>
</mrow>
In formula, r(1)(k) it is initial data r(0)(k) one-accumulate sequence, cumulative sequence are denoted as：
R(1)={ r(1)(1),···,r(1)(n)}；
Step 5, one-accumulate sequence R(1)Meet the white differential equationCalculated using background value calculation formula
z(1)(k) and matrix B, Y, parameter a, u is obtained using least square method,
z(1)(k) calculated by following formula：
z(1)(k+1)=β r(1)(k)+(1-β)·r(1)(k+1) (k=1, n-1；0≤β≤1)；
Matrix B, Y and parameter a, u are calculated by following formula：
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(a,u)T=(BTB)-1BTY；
Step 6, based on parameter a, u solved, settling time response sequenceAnd reduce and solveSequence
Predicted value
By solving the white differential equationBring parameter a, u into and can obtain time response function and be：
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Above formula discretization obtains：
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In formula, r(1)(1)=r(0)(1),For predicted value；
Calculated by following formula：
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Step 7, reduces predicted value；The reducing value of predicted value is calculated by reducing formula, is denoted asReduce formula such as following formula institute
Show：
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Step 8, calculates relative error；After step 7 solves the predicted value of original data sequence, missed according to the following formula
Difference is examined to judge the precision of prediction of GM (1,1) model；
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2. the gray prediction according to claim 1 based on inverse cosine function and power function transformation and background value Combinatorial Optimization
Model method, it is characterised in that in step 3,0.1,0.5,0.9 is taken respectively, according toEach value is taken to useFunction
GM (1,1) model prediction is reused after conversion and obtains the relative error of each value, it is final to choose minimum relative error as originally
The minimum relative error calculated value of invention.
3. the gray prediction according to claim 1 based on inverse cosine function and power function transformation and background value Combinatorial Optimization
Model method, it is characterised in that in step 5, background value z(1)(k) value that constant β is not known in can be true by the method for exhaustion
Fixed, β is increased since 0 with precision 0.01, and when reaching 1 end loop, final selection makes the β value of relative error minimum.
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