research article research on the concentration prediction...

Research ArticleResearch on the Concentration Prediction of Nitrogen inRed Tide Based on an Optimal Grey Verhulst Model

Xiaomei Hu1 Yubin Wang1 Yue Yu1 Dong Wang1 and Yuan Tian2

1The Key Laboratory of Intelligent Manufacturing and Robotics School of Mechatronic Engineering and AutomationShanghai University Mailbox 232 No 149 Yanchang Road Shanghai 200072 China2Department of Mechanical Engineering College of Engineering University of Michigan Ann Arbor MI 48105 USA

Correspondence should be addressed to Xiaomei Hu sufeimasohxm163com

Received 21 March 2016 Revised 2 August 2016 Accepted 15 August 2016

Academic Editor Rosana Rodriguez-Lopez

Copyright copy 2016 Xiaomei Hu et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

In order to reduce the harm of red tide to marine ecological balance marine fisheries aquatic resources and human health anoptimal Grey Verhulst model is proposed to predict the concentration of nitrogen in seawater which is the key factor in red tideThe Grey Verhulst model is established according to the existing concentration data series of nitrogen in seawater which is thenoptimized based on background value and time response formula to predict the future changes in the nitrogen concentration inseawater Finally the accuracy of the model is tested by the posterior test The results show that the prediction value based onthe optimal Grey Verhulst model is in good agreement with the measured nitrogen concentration in seawater which proves theeffectiveness of the optimal Grey Verhulst model in the forecast of red tide

1 Introduction

With the population expansion land resources are becomingmore and more precious which leads to the shortage ofmaterial resources and the crisis of energy The developmentofmarine resources has become an effective way to relieve thepressure of resources and environment in the 21st centuryWith rapid development of marine resources a varietyof marine disasters follow as a result In particular theoccurrence of red tide as well as the harm caused by it isfrequently increasing [1] Many researches have shown thatthe eutrophication of the seawater is the primary conditionof the occurrence of red tide The increase of nitrogenphosphorus and other nutrient salts in seawater greatlypromotes the eutrophication of seawater [2] Moreover thenitrogen concentration in seawater is regarded as a key factorto predict the occurrence of red tide Measured results haveshown that the change of the nitrogen concentration inseawater is not monotonous

Through the analysis of Grey system model and tradi-tional Verhulst model it is found that Grey system model issuitable to describe the monotonous change process but it

can be used in small sample data as well [3 4] In contrasttraditional Verhulst model is suitable for nonmonotonousdata but large samples are required [5] In light of thecharacteristics of the change of the nitrogen concentrationin seawater Grey Verhulst model is applied to predict thenitrogen concentration in seawater In Grey Verhulst modelan accumulation result of the original data is used to expandthe scope of the application of the traditional Verhulst model[6 7] Therefore Grey Verhulst model has been widely usedin recent years [8ndash11]

In order to improve the accuracy of the prediction anoptimal Grey Verhulst model is proposed to predict thenitrogen concentration in seawater The experimental resultsshow its high precision and small error compared with othermodels [12] a testament to the effectiveness of the optimalGreyVerhulstmodel So the optimalGreyVerhulstmodel canbe applied to forecast red tide

2 Related Work

21 Research on Red Tide Disaster Red tide is an abnormalecological phenomenon which is caused by fulminating

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2016 Article ID 9786107 9 pageshttpdxdoiorg10115520169786107

2 Mathematical Problems in Engineering

proliferation or accumulation of plankton in seawater [13]According to statistics the frequency and the cumulativeoccurrence area of red tide are both increasing year by year

Although themechanismof the occurrence of red tide hasnot been determined yet the main reasons that increase thefrequency of red tide are widely recognized as follows [14]

(1) More and more eutrophic seawater(2) The increase of the utilization and the development of

coastal water such as the development of aquaculturewhich leads to marine pollution

(3) The increasing marine traffic which is considered toexpand the distribution of some harmful algae

(4) Abnormal climate events such as Nino and SouthernOscillation phenomenon

(5) Decreasing efforts in the marine environmental pro-tection and careless attitude towards the red tide

A large number of studies have shown that the occurrenceof red tide is most strongly associated with seawater eutroph-ication [2] Therefore the research on the forecast of theconcentration of nitrogen in seawater has great significancein the prediction of red tide disaster

22 Grey VerhulstModel Grey system theorywas establishedand developed by Professor Julong Deng at the beginningof 1980s which has been successfully applied in indus-trial agricultural economic and other fields solving manypractical problems in production and scientific researchSpecifically Grey system theory is mainly used in smallsample monotonous data Grey system theory can effectivelydeal with incomplete and uncertain information The Greymodel (GM) is the core of Grey system theory which collectsavailable data to obtain the internal regularity without usingany assumptions The forecasting accuracy is related to thesample number 119899 in GM However Gray model alwaysneeds to be combined with other methods to optimize themodel which can increase the accuracy of the predictionFor example the combination of Grey model GM(1 1) withthree-point moving average proposed by Professor Mao andChirwa has been proven to be a more powerful forecastingtool and yields far much better predictions for vehicle fatalityrisk rates [15] Its application to the UK and US data setsyields exact predictions that are of high repeatability withcharacteristics depicting high reliability and efficiency [16]The paper is based on the Grey theory combined with theVerhulst model to predict nitrogen concentration which isthe key factor of red tide Traditional Verhulst model was putforward by Verhulst in the study of biological reproductionrulesThemodel is mainly used in large amount of data GreyVerhulst model extends traditional Verhulst model so that itcan be used in the unimodal type data

In order to improve the accuracy of prediction someresearchers have optimized Grey Verhulst model Evansproposed a Generalized Grey Verhulst model in which anew parameter estimation method was proposed on thebasis of the relationship of background value and simulativefunction The amount of British steel input was predicted by

Generalized Grey Verhulst model to prove its effectiveness[17] Chunguang et al established an unbiased Grey Verhulstmodel according to the objective function which is theminimum value of the square of subtraction between recip-rocal accumulating generating sequence and its inverselysimulative value [16] Wang et al established a new GreyVerhulst model and its application is put forward [18] Julongimproved the simulative accuracy by using Fourier transformto correct simulation residual and the trend of the euroagainst the dollar was predicted by this model to prove thegood forecasting effect [19] According to the analysis of theexisting Grey Verhulst models there are few researches onthe model from the perspective of the initial value and thesimulative value

In order to predict the nitrogen concentration in seawaterand avoid the error accumulation problem a new method tooptimize the time response function of Grey Verhulst modelis proposed according to the criterion of minimum sum-square of difference between the raw data vector and thesimulated data vector The Logistic curve is used to fit theraw data which optimizes background value and improvesthe prediction accuracy

3 The Optimization ofthe Grey Verhulst Model

31 The Optimization of the Background Value The GreyVerhulst model GM(1 1) is constituted by a first-orderdifferential equation containing only one variable [20ndash22]Assuming that 119883(0) is a nonnegative raw data sequence andthat 119883(1) is an accumulative sequence of 119883(0) 119883(1) can bedefined as follows [23ndash26]

119883(1)= 119909(1)(1) 119909

(1)(2) 119909

(1)(119899)

119909(1)(119896) =

119896

sum

119895=1

119909(0)(119895) 119896 = 1 2 119899

(1)

In (1) 119899 is the number of data in the sequenceThe generated mean sequence 119885(1) of119883(1) is defined as

119885(1)= 119911(1)(2) 119911(1)(3) 119911

(1)(119899)

119911(1)(119896) = 05119909

(1)(119905119896) + 05119909

(1)(119905119896minus1) 119896 = 2 3 119899

(2)

At this time the power model of GM(1 1) is defined asfollows

119909(0)(119896) + 119886119911

(1)(119896) = 119887 (119911

(1)(119896))

120572 (3)

The whitening equation of GM(1 1) is defined as follows[27ndash30]

119889119909(1)

119889119905

+ 119886119909(1)= 119887 (119909

(1))

120572 (4)

When 120572 = 2 according to (4)119883(1) is calculated as [31ndash33]

(1)(119896 + 1) =

119886119909(1)(1)

119887119909(1)(1) + (119886 minus 119887119909

(1)(1)) 119890119886119896 (5)

Mathematical Problems in Engineering 3

According to (5) 119909(1)(119896) has S-type growth which is shownin Figure 1 (the sequence of 119909(1)(119896) is shown in Figure 1)

The integration results of (4) in (119896 minus 1 119896) are shown as

119909(0)(119896) + 119886int

119896

119896minus1

119909(1)(119896) 119889119905 = 119887int

119896

119896minus1

119909(1)(119896)2119889119905 (6)

By comparison of (3) and (6) it can be seen thatthe definition equation of Grey Verhulst model uses thetrapezoidal area to replace the curve graphics areaThereforethe definition equation of Grey Verhulst model has the loweraccuracy Curve fitting method is proposed in this paper tofit raw data of the curve and the background value and theaccuracy of the model are improved by using (6) to solve thevalues of the parameters 119886 and 119887

From (5) and Figure 1 Logistic curve is used to fit the rawdata and it can be expressed as

119909(1)(119896) =

1

119901 + 119902119890119898(119896minus1)

(7)

The weight function 119890(119905) equiv 1 so

1

119909(1)(119896)

= 119901 + 119902119890119898(119896minus1)

(8)

Assume that

119910(0)(119896) =

1

119909(1)(119896)

minus

1

119909(1)(119896 + 1)

119910(0)(119896)

119910(0)(119896 minus 1)

=

119902119890119898(119896minus1)

(1 minus 119890119898)

119902119890119898(119896minus2)

(1 minus 119890119898)

= 119890119898

(9)

The parameter119898 is calculated as follows

119898 = ln119910(0)(119896)

119910(0)(119896 minus 1)

(10)

According to (8) the parameters of 119901 119902 are obtained byusing the least squares approximation which is shown asfollows

119899119901 + 119902

119899

sum

119896=2

(

119910(0)(119896)

119910(0)(119896 minus 1)

)

(119896minus1)

minus

119899

sum

119896=1

1

119909(1)(119896)

= 0

119901

119899

sum

119896=2

(

119910(0)(119896)

119910(0)(119896 minus 1)

)

(119896minus1)

minus 119902

119899

sum

119896=2

(

119910(0)(119896)

119910(0)(119896 minus 1)

)

(2119896minus2)

minus

119899

sum

119896=2

1

119909(1)(119896)

(

119910(0)(119896)

119910(0)(119896 minus 1)

)

(119896minus1)

= 0

(11)

From (11) the solution of 119901 119902 is shown as

119901 =

1

119899

119899

sum

119896=1

1

119909(1)(119896)

minus

1

119899

sdot

sum119899

119896=1 (1119909(1)(119896)) (sum

119899

119896=2 (119910(0)(119896) 119910

(0)(119896 minus 1))

(119896minus1))

2

minus 119899sum119899

119896=2 (1119909(1)(119896)) (119910

(0)(119896) 119910

(0)(119896 minus 1))

(119896minus1)sum119899

119896=2 (119910(0)(119896) 119910

(0)(119896 minus 1))

(119896minus1)

(sum119899

119896=2 (119910(0)(119896) 119910

(0)(119896 minus 1))

(119896minus1))

2

+ 119899sum119899

119896=2 (119910(0)(119896) 119910

(0)(119896 minus 1))

(2119896minus2)

119902 =

sum119899

119896=1 (1119909(1)(119896))sum

119899

119896=2 (119910(0)(119896) 119910

(0)(119896 minus 1))

(119896minus1)minus 119899sum119899

119896=2 (1119909(1)(119896)) (119910

(0)(119896) 119910

(0)(119896 minus 1))

(119896minus1)

(sum119899

119896=2 (119910(0)(119896) 119910

(0)(119896 minus 1))

(119896minus1))

2

+ 119899sum119899

119896=2 (119910(0)(119896) 119910

(0)(119896 minus 1))

(2119896minus2)

(12)

The background value is calculated as follows [12]

1199111(119896) = int

119896

119896minus1

119909(1)(119896) 119889119905 = minus

1

119898119901

ln100381610038161003816100381610038161003816100381610038161003816

119901119890minus119898(119896minus1)

+ 119902

119901119890minus119898(119896minus2)

+ 119902

100381610038161003816100381610038161003816100381610038161003816

1199112(119896) = int

119896

119896minus1

119909(1)(119896)2119889119905

=

1

119898

[

1

119901

119902 [119890119898(119896minus2)

minus 119890119898(119896minus1)

]

(119901 + 119902119890119898(119896minus1)

) (119901 + 119902119890119898(119896minus2)

)

+

1

1199012ln10038161003816100381610038161003816100381610038161003816100381610038161003816

119890119898(119901 + 119902119890

119898(119896minus2))

119901 + 119902119890119898(119896minus1)

10038161003816100381610038161003816100381610038161003816100381610038161003816

]

(13)

(119886 119887)119879 is a sequence of parameters that can be expressed

as

= (119886 119887)119879= (119861119879119861)

minus1119861119879119884 (14)

In (14) 119884 can 119861 be expressed as follows

119884 =

[

[

[

[

[

[

119909(0)(2)

119909(0)(3)

119909(0)(119899)

]

]

]

]

]

]

119861 =

[

[

[

[

[

[

minus1199111(2) 119911

2(2)

minus1199111(3) 119911

2(3)

minus1199111(119899) 119911

2(119899)

]

]

]

]

]

]

(15)


kk minus 1

x(1)

(k)

x(1)

(t)

x(1)

(k minus 1)

Figure 1 The trend graph of the accumulative sequence

According to (14) and (15) the values of 119886 119887 can beobtained with the results of optimized background value

32 The Optimization of the Time Response The generalsolution of (4) for the time response function is shown as

(1)(119896) =

1

119888119890119886119896+ 119887119886

(16)

By comparison of (16) and (5) the simulated curve passesby the first point of the raw data in the traditional solutionwhich does not necessarily fit the facts The least squaresmethod does not need the simulated curve to pass by thefirst point and the parameter 119888 can be solved according tothe known information According to the criterion of theminimum sum of square between the reciprocal of the rawdata sequence and the reciprocal of the predictive value thefunction 119865(119888) is defined as

119865 (119888) =

119899

sum

119896=1

(119888119890119886(119896+1)

minus 119888119890119886119896

minus (

minus119909(0)(119896 + 1)

sum119896+1

119894=1 119909(0)(119894) sum119896

119894=1 119909(0)(119894)

))

2

(17)

According to the extreme conditions 1198651015840(119888) = 0 theparameter 119888 can be calculated as119888

=

minussum119899

119894=1 (119909(0)(119896 + 1) sum

119896+1

119894=1 119909(0)(119894) sum119896

119894=1 119909(0)(119894)) (119890

119886119896(119890119886minus 1))

sum119899

119894=1 (119890119886119896(119890119886minus 1))2

(18)

According to the above equation the optimal generalsolution of time response function is obtained

33 Grey Verhulst Model Accuracy Test The accuracy of GreyVerhulst model can be tested by three methods pretestintermediate test and post hoc test [34 35] The posteriorvariance test method which is a kind of intermediate test isapplied to test the accuracy of Grey Verhulst model (0)(119899) isthe predictive value and the predictive sequence is shown as

(0)= [(0)(1)

(0)(2)

(0)(119899)] (19)

The residual is expressed as

119864 = [119890 (1) 119890 (2) 119890 (119899)] = 119883(0)minus

(0)

119890 (119896) = 119909(0)(119896) minus

(0)(119896) 119896 = 1 2 119899

(20)

The variance of the raw sequence and residual sequenceis shown as follows

1198782

1 =1

119899

119899

sum

119896=1

[119909(0)(119896) minus 119909]

2

1198782

2 =1

119899

119899

sum

119896=1

[119890 (119896) minus 119890]2

(21)

In (21) 119909 and 119890 are defined as

119909 =

1

119899

119899

sum

119896=1

119909(0)(119896)

119890 =

1

119899

119899

sum

119896=1

119890 (119896)

(22)

The posterior variance ratio 119862 is defined as

119862 =

1198782

1198781

(23)

The small error probability 119901 is defined as

119901 = 119875 |119890 (119896) minus 119890| lt 067451198781 (24)

119862 and 119901 are the two important indicators to validate theprecision of themodel According to (24)119862 is determined by1198782 and 1198781The bigger the value of 1198781 the bigger the dispersiondegree of the original data A low value of 1198782 indicates alow degree of residual dispersion Therefore 11987821198781 namelythe value of 119862 being small shows that although the originaldata is very discrete the relationship between the calculatedvalues and the actual value of themodel is not very discrete 119901indicates the number of dots of which the difference betweenthe residual and the residual mean value is less than thegiven value 006451198781 The bigger the value of 119901 is the moreuniformly distributed is the fitted value According to 119862 and119901 the accuracy of the model can be divided into four levelsas shown in Table 1 [36 37]

4 Application Analysis of Optimal GreyVerhulst Model

41 Example 1 As the Bohai Bay is a semiclosed harbor it isnot conducive for the pollutants to spread The pollution ofthe sea water is very serious which promotes the microbialgrowth As a result red tides often occur The optimalGrey Verhulst model is applied to predict the nitrogenconcentration in the Bohai Bay The measured sample dataof the nitrogen concentration in the Bohai Bay collected insummer is shown in Table 2 119909(119896) refers to the nitrogenconcentration in seawater on 119896 day


Table 1 Model accuracy grade table

The precision grade The posteriorvariance ratio 119862

Small errorprobability 119901

Level 1 (good) 119862 le 035 095 le 119901

Level 2 (qualified) 035 lt 119862 le 05 080 le 119901 lt 095

Level 3 (reluctant) 05 lt 119862 le 065 070 le 119901 lt 080

Level 4 (unqualified) 065 lt 119862 119901 lt 070

Table 2The sample table of the nitrogen concentration with 9 sets

Nitrogen samples Concentration (120583molL)119909(1) 30119909(2) 33119909(3) 37119909(4) 45119909(5) 55119909(6) 65119909(7) 72119909(8) 76119909(9) 80

Through the analysis of the measured raw data in Table 2the sequence has been saturated So the raw data are directlytaken as the first-order accumulative data sequence 119883(1)which approximately matches the following Logistic func-tion

119909(1)(119896) asymp

1

00317119890minus02661119896

+ 0009

119896 = 1 2 119899 (25)

The first eight sets of data in the sequence are taken as themodeling data which are used to establish the traditionalGrey Verhulst model the Grey Verhulst model based onoptimal time response and the Grey Verhulst model basedon background value optimization respectively The last setof data in the sequence is used to make a comparison withprediction data in order to prove the extrapolation of themodel

In order to test the accuracy of different Grey modelsvarious models are formed in this paper

GVM GVM(1 1) modelTPGVM Modified Grey Verhulst model at timeresponse using the processed data [12]BPGVM Modified Grey Verhulst model at back-ground value using the processed data [12]TRGVM Modified Grey Verhulst model at timeresponse using the raw dataBRGVM Modified Grey Verhulst model at back-ground value using the raw data

GVM is shown as

119909(1)(119896 + 1) =

7983

0072 + 01914119890minus02661119896

119896 = 1 2 119899

(26)

TPGVM is shown as

119909(1)(119896) =

02661

00086119890minus02661119896

+ 00024

119896 = 1 2 119899 (27)

TRGVM is shown as

119909(1)(119896) =

02661

00093119890minus02661119896

+ 00024

119896 = 1 2 119899 (28)

Table 3 gives a comparison between the different Mod-ified Grey Verhulst models at time response and the tradi-tional Grey Verhulst model The average relative error is thesumof absolute values of relative errorThe extrapolated valueis the modelrsquos predictive value

The posterior variance ratio is calculated as

119862 =

1198782

1198781

= 0130 le 035

067451198781 = 1137

|119890 (119896) minus 119890| lt 067451198781

(29)

So the small error probability 119901 = 1Although the average relative error results of threemodels

are almost the same Grey Verhulst model based on timeresponse value optimization excludes different predictivemodels caused by different selection of raw data

In the aspects of the extrapolation shown as the lastrecord in Table 3 TRGVM model is the best among threemodels since the actual value is 80

Therefore (28) can be used to make better predictions ofthe nitrogen concentration

BPGVM is shown as

119909(1)(119896 + 1) =

7953

0074 + 01911119890minus02651119896

119896 = 1 2 119899

(30)

BRGVM is shown as

119909(1)(119896 + 1) =

8296

0078 + 02068119890minus02673119896

119896 = 1 2 119899

(31)

Table 4 gives a comparison between the different Mod-ified Grey Verhulst models at background value and thetraditional Grey Verhulst model

The posterior variance ratio is shown as

119862 =

1198782

1198781

= 0003 le 035

067451198781 = 1137

|119890 (119896) minus 119890| lt 067451198781

(32)

So the small error probability 119901 = 1In the aspects of the extrapolation shown as the last

record in Table 4 BRGVMmodel is also the best among threemodels since the actual value is 80



Table 3 Accuracy comparison of GVM TPGVM and TRGVMmodels

Measuredraw data

GVM TPGVM TRGVMPrediction value The relative error () Prediction value The relative error () Prediction value The relative error ()

30 30 0 299 minus05 279 minus6933 365 106 359 91 338 2537 432 169 427 155 404 9245 505 121 499 109 475 5455 578 51 572 41 547 minus0465 651 02 646 minus07 621 minus4472 721 009 715 minus06 692 minus3976 784 33 780 26 759 minus01Averagerelative error()

69 55 41

Extrapolationvalues 842 836 819

Table 4 Accuracy comparison of GVM BPGVM and BRGVMmodels

Measuredraw data

GVM BPGVM BRGVMPrediction value The relative error () Prediction value The relative error () Prediction value The relative error ()

30 30 0 30 0 291 minus2933 365 106 361 92 351 6337 432 169 427 153 417 12545 505 121 496 102 485 7955 578 51 567 31 556 1265 651 02 637 minus19 627 minus3572 721 009 704 minus22 694 minus3676 784 33 765 07 755 00Averagerelative error()

69 53 48


42 Example 2 In order to further illustrate the advantagesof the proposed optimization model the sample data isincreased in this exampleThe 18 sets of the nitrogen concen-tration in Zhuhai estuary collected in summer are shown inTable 5 The last two sets of data are extrapolated data Thecomparison between the different Modified Grey Verhulstmodels and the traditional Grey Verhulst model is shown inTables 6 and 7

According to Tables 6 and 7 the Modified Grey Verhulstmodel using the raw data is the best model in contrast withthe Modified Grey Verhulst model using the processed dataand the traditional Grey Verhulst model because it has thebest prediction and extrapolation effect

5 Conclusion

After analyzing the trends of the nitrogen concentrationwhich is the key factor in red tide occurrence an optimalGrey Verhulst model is proposed to predict the nitrogenconcentration in seawater In order to improve the predictiveaccuracy two optimal methods are put forward the opti-mization of the background value and the time responseThe application results show that the optimal Grey Verhulst


Nitrogen samples Concentration (120583molL)119909(1) 28119909(2) 30119909(3) 32119909(4) 35119909(5) 36119909(6) 42119909(7) 46119909(8) 51119909(9) 57119909(10) 65119909(11) 74119909(12) 83119909(13) 90119909(14) 95119909(15) 99119909(16) 102119909(17) 100119909(18) 101

model can better forecast the trends of the nitrogen concen-tration than the other two methods Since the optimal Grey



Measuredraw data


28 28 0 273 minus25 261 minus6830 334 113 335 116 338 12732 392 225 382 19 348 8835 413 18 413 18 407 1636 435 175 422 141 392 6042 445 59 439 45 422 0546 488 61 453 minus15 449 minus2451 536 51 528 35 532 4357 603 58 602 56 589 3365 676 4 681 47 667 2674 778 51 726 minus19 756 2283 859 35 853 28 835 0690 936 4 925 28 921 2395 985 37 972 23 963 1399 1007 17 1011 21 1005 15102 1032 11 1027 07 1014 06Averagerelative error()

72 53 33

Extrapolationvalues

1042 1022 10151047 1038 1021


Measuredraw data


28 28 0 272 minus36 263 minus7130 334 113 331 103 336 1232 392 225 387 209 354 10635 413 18 407 162 405 15736 435 175 427 154 409 9442 445 59 435 36 426 1446 488 61 447 minus28 455 minus1151 536 51 533 45 522 2357 603 58 605 61 594 4265 676 4 672 33 665 2374 778 51 731 12 753 1883 859 35 861 37 848 2190 936 4 932 34 923 2695 985 37 976 27 967 1899 1007 17 1013 23 1003 13102 1032 11 1024 04 1025 05Averagerelative error()

72 54 38

Extrapolationvalues

1047 1033 10241036 1031 1022


Verhulst model is only suitable for S-type data combiningthe optimal Grey Verhulst model with other algorithms toovercome the defects in the optimal Grey Verhulst model willbe the focus of study in the future

Notations

GM Grey dynamic modelGVM Grey Verhulst modelTPGVM Modified Grey Verhulst model at time

response using the processed dataBPGVM Modified Grey Verhulst model at

background value using the processed dataTRGVM Modified Grey Verhulst model at time

response using the raw dataBRGVM Modified Grey Verhulst model at

background value using the raw data119883(0) Nonnegative raw data sequence

119883(1) Accumulative sequence of119883(0)

119883(119888) Accumulative sequence

119899 Number of data in the sequence119885(1) Generated mean sequence119890(119905) Weight function

(0) Predictive value(119886 119887)119879 Sequence of parameters

119864 Residual1198782 Variance of the raw sequence and residual

sequence119862 Posterior variance ratio119875 Small error probability

Competing Interests

The authors declare that there are no competing interestsregarding the publication of this article

Acknowledgments

This work was supported by the Grand Science amp TechnologyProgram Shanghai China (no 16111105900)

References

[1] Y Luo X Liu R S Wu and Y J Wu ldquoSystem constructionimprovement of the ability of marine disaster preventionrdquoApplied Mechanics and Materials vol 522 pp 501ndash504 2014

[2] R E Sipler D A Bronk S P Seitzinger et al ldquoTrichodesmium-derived dissolved organic matter is a source of nitrogen capableof supporting the growth of toxic red tide Karenia brevisrdquoMarine Ecology Progress Series vol 483 pp 31ndash45 2013

[3] L Sifeng G Tianbang D Yaoguo et alTheGrey SystemTheoryand Its Application Science Press Beijing China 1999

[4] D JulongGrey SystemMethod HuazhongUniversity of Scienceand Technology Press Wuhan China 1987

[5] X Xinping and Q Lifen ldquoA new type solution and bifurcationof grey Verhulst modelrdquo Journal of Grey System vol 24 no 2pp 165ndash174 2012

[6] D Julong Grey SystemTheory Huazhong University of Scienceand Technology Press Wuhan China 2002

[7] C Cuiwen and G Xingsheng ldquoThe application of themetabolism of grey dynamic model in the product priceforecasting and demand forecastingrdquo Information and Controlvol 34 no 8 pp 398ndash402 2005

[8] N Xie C Zhu S Liu and Y Yang ldquoOn discrete grey systemforecasting model corresponding with polynomial time-varysequencerdquo Journal of Grey System vol 25 no 4 pp 1ndash18 2013

[9] S Yanhui andNDexin ldquoGreyVerhulstmodel of the foundationsettlement predictionrdquo Rock and Soil Mechanics vol 24 no 1pp 123ndash126 2003

[10] J Ming Z Fan Z Xie Y Jiang and B Zuo ldquoA modifiedgrey verhulst model method to predict ultraviolet protectionperformance of aging Bmori silk fabricrdquo Fibers and Polymersvol 14 no 7 pp 1179ndash1183 2013

[11] L Yucheng ldquoThe improved Verhulst model of the buildingsettlementrdquo Chinese Journal of Geological Hazard and Controlvol 17 no 4 pp 61ndash63 2006

[12] X PingpingThe optimization method the grey MGM (1 m) andVerhulst model [MS thesis] Nanjing University of Aeronauticsamp Astronautics Nanjing China 2012

[13] W Hongli and F Jianfeng Ecological Dynamics and Predictionof the Red Tides Tianjin University Press 2006

[14] F Shizhai L Fengqi and L ShaozhuAn Introduction toMarineScience Higher Education Press Beijing China 1999

[15] M Mao and E C Chirwa ldquoApplication of grey model GM(11)to vehicle fatality risk estimationrdquo Journal of TechnologicalForecasting amp Social Change vol 73 no 5 pp 588ndash605 2006

[16] S Chunguang CWanming and P Lingling ldquoThe optimizationof the initial conditions of unbiased Grey Verhulst modelrdquoStatistics and Information BBS vol 26 no 5 pp 3ndash6 2011

[17] M Evans ldquoAn alternative approach to estimating the parame-ters of a generalised grey verhulst model an application to steelintensity of use in the UKrdquo Expert Systems with Applicationsvol 41 no 4 pp 1236ndash1244 2014

[18] Z Wang Y Dang and Y Wang ldquoA new grey Verhulst modeland its applicationrdquo in Proceedings of the IEEE InternationalConference on Grey Systems and Intelligent Services (GSIS rsquo07)pp 571ndash574 Nanjing China November 2007

[19] D Julong ldquoOn judging the admissibility of grey modeling viaclass ratiordquoThe Journal of Grey System no 4 p 249 1993

[20] F Zhang F Liu W Zhao et al ldquoApplication of grey verhulstmodel in middle and long term load forcastingrdquo Power SystemTechnology vol 5 article 8 2003

[21] Z Gou X Song and J Ye ldquoA Verhulst model on time serieserror corrected for port throughput forecastingrdquo Journal of theEastern Asia Society for Transportation Studies vol 6 pp 881ndash891 2005

[22] K-L Wen and Y-F Huang ldquoThe development of Grey Ver-hulst toolbox and the analysis of population saturation statein Taiwan-Fukienrdquo in Proceedings of the IEEE InternationalConference on Systems Man and Cybernetics (SMC rsquo04) vol 6pp 5007ndash5012 IEEE October 2004

[23] D JulongGrey Prediction andGreyDecisionMaking HuazhongUniversity of Science and Technology Press Wuhan China2000

[24] D Julong and G Hong Method and Application of GreyForecasting Model High Book Company 1999

[25] L Chen and L Zhang ldquoCombination grey verhulst modelbased on initial value modificationrdquo Mathematics in Practiceand Theory vol 11 article 26 2010


[26] W Liu and W A Xu ldquoA new algorithm for estimating param-eters of grey verhulst modelrdquo Computer Simulation vol 11 pp119ndash123 2008

[27] J Xu T Tan M Tu and L Qi ldquoImprovement of grey models byleast squaresrdquo Expert Systems with Applications vol 38 no 11pp 13961ndash13966 2011

[28] W Zhixin D Yaoguo and S Chunguang ldquoThe research of theimprovement of the Grey Verhulst derivative modelrdquo Statisticsand Information BBS no 6 pp 19ndash22 2010

[29] X Kaigui H Bin Z Jiming et al ldquoThe discussion of the mod-eling method of Grey forecasting modelrdquo Journal of ChongqingInstitute of Post and Telecommunications no 3 pp 56ndash60 1998

[30] C Fangqiang T Fan and J Yonggang ldquoThe application ofVerhulst model in the prediction of the soft ground settlementof embankmentrdquo Journal of Rock Mechanics and Engineeringvol 26 no 7 pp 3122ndash3126 2007

[31] Q Li ldquoApplication of grey Verhulst model to commercial flightsat the Macau International Airportrdquo in Proceedings of the 24thIEEE International Conference on Grey Systems and IntelligentServices (GSIS rsquo13) pp 161ndash163 November 2013

[32] E Kayacan B Ulutas and O Kaynak ldquoGrey system theory-based models in time series predictionrdquo Expert Systems withApplications vol 37 no 2 pp 1784ndash1789 2010

[33] T C Lin F P Hsu and B Y Chen ldquoComparing accuracyof GM(11) and grey Verhulst model in Taiwan dental clinicsforecastingrdquoThe Journal of Grey System vol 19 no 1 pp 31ndash382007

[34] Z-X Wang Y-G Dang and S-F Liu ldquoUnbiased grey Verhulstmodel and its applicationrdquo Systems EngineeringmdashTheory ampPractice vol 29 no 10 pp 138ndash144 2009

[35] H Wenzhang and W Aidi ldquoThe method and applicationof the estimated Verhulst model parameters in the linearprogrammingrdquo Systems EngineeringTheory andPractice vol 26no 8 pp 141ndash144 2006

[36] G-D Li D Yamaguchi and M Nagai ldquoThe development ofstock exchange simulation prediction modeling by a hybridgrey dynamic modelrdquo The International Journal of AdvancedManufacturing Technology vol 36 no 1 pp 195ndash204 2008

[37] L-C Hsu ldquoApplying the Grey prediction model to the globalintegrated circuit industryrdquo Technological Forecasting and SocialChange vol 70 no 6 pp 563ndash574 2003

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


proliferation or accumulation of plankton in seawater [13]According to statistics the frequency and the cumulativeoccurrence area of red tide are both increasing year by year

Although themechanismof the occurrence of red tide hasnot been determined yet the main reasons that increase thefrequency of red tide are widely recognized as follows [14]

(1) More and more eutrophic seawater(2) The increase of the utilization and the development of

coastal water such as the development of aquaculturewhich leads to marine pollution

(3) The increasing marine traffic which is considered toexpand the distribution of some harmful algae

(4) Abnormal climate events such as Nino and SouthernOscillation phenomenon

(5) Decreasing efforts in the marine environmental pro-tection and careless attitude towards the red tide

A large number of studies have shown that the occurrenceof red tide is most strongly associated with seawater eutroph-ication [2] Therefore the research on the forecast of theconcentration of nitrogen in seawater has great significancein the prediction of red tide disaster

22 Grey VerhulstModel Grey system theorywas establishedand developed by Professor Julong Deng at the beginningof 1980s which has been successfully applied in indus-trial agricultural economic and other fields solving manypractical problems in production and scientific researchSpecifically Grey system theory is mainly used in smallsample monotonous data Grey system theory can effectivelydeal with incomplete and uncertain information The Greymodel (GM) is the core of Grey system theory which collectsavailable data to obtain the internal regularity without usingany assumptions The forecasting accuracy is related to thesample number 119899 in GM However Gray model alwaysneeds to be combined with other methods to optimize themodel which can increase the accuracy of the predictionFor example the combination of Grey model GM(1 1) withthree-point moving average proposed by Professor Mao andChirwa has been proven to be a more powerful forecastingtool and yields far much better predictions for vehicle fatalityrisk rates [15] Its application to the UK and US data setsyields exact predictions that are of high repeatability withcharacteristics depicting high reliability and efficiency [16]The paper is based on the Grey theory combined with theVerhulst model to predict nitrogen concentration which isthe key factor of red tide Traditional Verhulst model was putforward by Verhulst in the study of biological reproductionrulesThemodel is mainly used in large amount of data GreyVerhulst model extends traditional Verhulst model so that itcan be used in the unimodal type data

In order to improve the accuracy of prediction someresearchers have optimized Grey Verhulst model Evansproposed a Generalized Grey Verhulst model in which anew parameter estimation method was proposed on thebasis of the relationship of background value and simulativefunction The amount of British steel input was predicted by

Generalized Grey Verhulst model to prove its effectiveness[17] Chunguang et al established an unbiased Grey Verhulstmodel according to the objective function which is theminimum value of the square of subtraction between recip-rocal accumulating generating sequence and its inverselysimulative value [16] Wang et al established a new GreyVerhulst model and its application is put forward [18] Julongimproved the simulative accuracy by using Fourier transformto correct simulation residual and the trend of the euroagainst the dollar was predicted by this model to prove thegood forecasting effect [19] According to the analysis of theexisting Grey Verhulst models there are few researches onthe model from the perspective of the initial value and thesimulative value

In order to predict the nitrogen concentration in seawaterand avoid the error accumulation problem a new method tooptimize the time response function of Grey Verhulst modelis proposed according to the criterion of minimum sum-square of difference between the raw data vector and thesimulated data vector The Logistic curve is used to fit theraw data which optimizes background value and improvesthe prediction accuracy

3 The Optimization ofthe Grey Verhulst Model

31 The Optimization of the Background Value The GreyVerhulst model GM(1 1) is constituted by a first-orderdifferential equation containing only one variable [20ndash22]Assuming that 119883(0) is a nonnegative raw data sequence andthat 119883(1) is an accumulative sequence of 119883(0) 119883(1) can bedefined as follows [23ndash26]

119883(1)= 119909(1)(1) 119909

(1)(2) 119909

(1)(119899)

119909(1)(119896) =

119896

sum

119895=1

119909(0)(119895) 119896 = 1 2 119899

(1)

In (1) 119899 is the number of data in the sequenceThe generated mean sequence 119885(1) of119883(1) is defined as

119885(1)= 119911(1)(2) 119911(1)(3) 119911

(1)(119899)

119911(1)(119896) = 05119909

(1)(119905119896) + 05119909

(1)(119905119896minus1) 119896 = 2 3 119899

(2)

At this time the power model of GM(1 1) is defined asfollows

119909(0)(119896) + 119886119911

(1)(119896) = 119887 (119911

(1)(119896))

120572 (3)

The whitening equation of GM(1 1) is defined as follows[27ndash30]

119889119909(1)

119889119905

+ 119886119909(1)= 119887 (119909

(1))

120572 (4)

When 120572 = 2 according to (4)119883(1) is calculated as [31ndash33]

(1)(119896 + 1) =

119886119909(1)(1)

119887119909(1)(1) + (119886 minus 119887119909

(1)(1)) 119890119886119896 (5)




119909(0)(119896) + 119886int

119896

119896minus1

119909(1)(119896) 119889119905 = 119887int

119896

119896minus1

119909(1)(119896)2119889119905 (6)



119909(1)(119896) =

1

119901 + 119902119890119898(119896minus1)

(7)


1

119909(1)(119896)

= 119901 + 119902119890119898(119896minus1)

(8)

Assume that

119910(0)(119896) =

1

119909(1)(119896)

minus

1

119909(1)(119896 + 1)

119910(0)(119896)

119910(0)(119896 minus 1)

=

119902119890119898(119896minus1)

(1 minus 119890119898)

119902119890119898(119896minus2)

(1 minus 119890119898)

= 119890119898

(9)


119898 = ln119910(0)(119896)

119910(0)(119896 minus 1)

(10)


119899119901 + 119902

119899

sum

119896=2

(

119910(0)(119896)

119910(0)(119896 minus 1)

)

(119896minus1)

minus

119899

sum

119896=1

1

119909(1)(119896)

= 0

119901

119899

sum

119896=2

(

119910(0)(119896)

119910(0)(119896 minus 1)

)

(119896minus1)

minus 119902

119899

sum

119896=2

(

119910(0)(119896)

119910(0)(119896 minus 1)

)

(2119896minus2)

minus

119899

sum

119896=2

1

119909(1)(119896)

(

119910(0)(119896)

119910(0)(119896 minus 1)

)

(119896minus1)

= 0

(11)


119901 =

1

119899

119899

sum

119896=1

1

119909(1)(119896)

minus

1

119899

sdot

sum119899

119896=1 (1119909(1)(119896)) (sum

119899

119896=2 (119910(0)(119896) 119910

(0)(119896 minus 1))

(119896minus1))

2

minus 119899sum119899

119896=2 (1119909(1)(119896)) (119910

(0)(119896) 119910

(0)(119896 minus 1))

(119896minus1)sum119899

119896=2 (119910(0)(119896) 119910

(0)(119896 minus 1))

(119896minus1)

(sum119899

119896=2 (119910(0)(119896) 119910

(0)(119896 minus 1))

(119896minus1))

2

+ 119899sum119899

119896=2 (119910(0)(119896) 119910

(0)(119896 minus 1))

(2119896minus2)

119902 =

sum119899

119896=1 (1119909(1)(119896))sum

119899

119896=2 (119910(0)(119896) 119910

(0)(119896 minus 1))

(119896minus1)minus 119899sum119899

119896=2 (1119909(1)(119896)) (119910

(0)(119896) 119910

(0)(119896 minus 1))

(119896minus1)

(sum119899

119896=2 (119910(0)(119896) 119910

(0)(119896 minus 1))

(119896minus1))

2

+ 119899sum119899

119896=2 (119910(0)(119896) 119910

(0)(119896 minus 1))

(2119896minus2)

(12)


1199111(119896) = int

119896

119896minus1

119909(1)(119896) 119889119905 = minus

1

119898119901

ln100381610038161003816100381610038161003816100381610038161003816

119901119890minus119898(119896minus1)

+ 119902

119901119890minus119898(119896minus2)

+ 119902

100381610038161003816100381610038161003816100381610038161003816

1199112(119896) = int

119896

119896minus1

119909(1)(119896)2119889119905

=

1

119898

[

1

119901

119902 [119890119898(119896minus2)

minus 119890119898(119896minus1)

]

(119901 + 119902119890119898(119896minus1)

) (119901 + 119902119890119898(119896minus2)

)

+

1

1199012ln10038161003816100381610038161003816100381610038161003816100381610038161003816

119890119898(119901 + 119902119890

119898(119896minus2))

119901 + 119902119890119898(119896minus1)

10038161003816100381610038161003816100381610038161003816100381610038161003816

]

(13)


as

= (119886 119887)119879= (119861119879119861)

minus1119861119879119884 (14)


119884 =

[

[

[

[

[

[

119909(0)(2)

119909(0)(3)

119909(0)(119899)

]

]

]

]

]

]

119861 =

[

[

[

[

[

[

minus1199111(2) 119911

2(2)

minus1199111(3) 119911

2(3)

minus1199111(119899) 119911

2(119899)

]

]

]

]

]

]

(15)


kk minus 1

x(1)

(k)

x(1)

(t)

x(1)

(k minus 1)




(1)(119896) =

1

119888119890119886119896+ 119887119886

(16)


119865 (119888) =

119899

sum

119896=1

(119888119890119886(119896+1)

minus 119888119890119886119896

minus (

minus119909(0)(119896 + 1)

sum119896+1

119894=1 119909(0)(119894) sum119896

119894=1 119909(0)(119894)

))

2

(17)


=

minussum119899

119894=1 (119909(0)(119896 + 1) sum

119896+1

119894=1 119909(0)(119894) sum119896

119894=1 119909(0)(119894)) (119890

119886119896(119890119886minus 1))

sum119899

119894=1 (119890119886119896(119890119886minus 1))2

(18)



(0)= [(0)(1)

(0)(2)

(0)(119899)] (19)


119864 = [119890 (1) 119890 (2) 119890 (119899)] = 119883(0)minus

(0)

119890 (119896) = 119909(0)(119896) minus

(0)(119896) 119896 = 1 2 119899

(20)


1198782

1 =1

119899

119899

sum

119896=1

[119909(0)(119896) minus 119909]

2

1198782

2 =1

119899

119899

sum

119896=1

[119890 (119896) minus 119890]2

(21)


119909 =

1

119899

119899

sum

119896=1

119909(0)(119896)

119890 =

1

119899

119899

sum

119896=1

119890 (119896)

(22)


119862 =

1198782

1198781

(23)


119901 = 119875 |119890 (119896) minus 119890| lt 067451198781 (24)








Level 1 (good) 119862 le 035 095 le 119901







119909(1)(119896) asymp

1

00317119890minus02661119896

+ 0009

119896 = 1 2 119899 (25)




GVM is shown as

119909(1)(119896 + 1) =

7983

0072 + 01914119890minus02661119896

119896 = 1 2 119899

(26)

TPGVM is shown as

119909(1)(119896) =

02661

00086119890minus02661119896

+ 00024

119896 = 1 2 119899 (27)

TRGVM is shown as

119909(1)(119896) =

02661

00093119890minus02661119896

+ 00024

119896 = 1 2 119899 (28)



119862 =

1198782

1198781

= 0130 le 035

067451198781 = 1137

|119890 (119896) minus 119890| lt 067451198781

(29)





BPGVM is shown as

119909(1)(119896 + 1) =

7953

0074 + 01911119890minus02651119896

119896 = 1 2 119899

(30)

BRGVM is shown as

119909(1)(119896 + 1) =

8296

0078 + 02068119890minus02673119896

119896 = 1 2 119899

(31)



119862 =

1198782

1198781

= 0003 le 035

067451198781 = 1137

|119890 (119896) minus 119890| lt 067451198781

(32)






Measuredraw data



69 55 41



Measuredraw data



69 53 48




5 Conclusion







Measuredraw data



72 53 33

Extrapolationvalues

1042 1022 10151047 1038 1021


Measuredraw data



72 54 38

Extrapolationvalues

1047 1033 10241036 1031 1022



Notations












Competing Interests


Acknowledgments


References














































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












119909(0)(119896) + 119886int

119896

119896minus1

119909(1)(119896) 119889119905 = 119887int

119896

119896minus1

119909(1)(119896)2119889119905 (6)



119909(1)(119896) =

1

119901 + 119902119890119898(119896minus1)

(7)


1

119909(1)(119896)

= 119901 + 119902119890119898(119896minus1)

(8)

Assume that

119910(0)(119896) =

1

119909(1)(119896)

minus

1

119909(1)(119896 + 1)

119910(0)(119896)

119910(0)(119896 minus 1)

=

119902119890119898(119896minus1)

(1 minus 119890119898)

119902119890119898(119896minus2)

(1 minus 119890119898)

= 119890119898

(9)


119898 = ln119910(0)(119896)

119910(0)(119896 minus 1)

(10)


119899119901 + 119902

119899

sum

119896=2

(

119910(0)(119896)

119910(0)(119896 minus 1)

)

(119896minus1)

minus

119899

sum

119896=1

1

119909(1)(119896)

= 0

119901

119899

sum

119896=2

(

119910(0)(119896)

119910(0)(119896 minus 1)

)

(119896minus1)

minus 119902

119899

sum

119896=2

(

119910(0)(119896)

119910(0)(119896 minus 1)

)

(2119896minus2)

minus

119899

sum

119896=2

1

119909(1)(119896)

(

119910(0)(119896)

119910(0)(119896 minus 1)

)

(119896minus1)

= 0

(11)


119901 =

1

119899

119899

sum

119896=1

1

119909(1)(119896)

minus

1

119899

sdot

sum119899

119896=1 (1119909(1)(119896)) (sum

119899

119896=2 (119910(0)(119896) 119910

(0)(119896 minus 1))

(119896minus1))

2

minus 119899sum119899

119896=2 (1119909(1)(119896)) (119910

(0)(119896) 119910

(0)(119896 minus 1))

(119896minus1)sum119899

119896=2 (119910(0)(119896) 119910

(0)(119896 minus 1))

(119896minus1)

(sum119899

119896=2 (119910(0)(119896) 119910

(0)(119896 minus 1))

(119896minus1))

2

+ 119899sum119899

119896=2 (119910(0)(119896) 119910

(0)(119896 minus 1))

(2119896minus2)

119902 =

sum119899

119896=1 (1119909(1)(119896))sum

119899

119896=2 (119910(0)(119896) 119910

(0)(119896 minus 1))

(119896minus1)minus 119899sum119899

119896=2 (1119909(1)(119896)) (119910

(0)(119896) 119910

(0)(119896 minus 1))

(119896minus1)

(sum119899

119896=2 (119910(0)(119896) 119910

(0)(119896 minus 1))

(119896minus1))

2

+ 119899sum119899

119896=2 (119910(0)(119896) 119910

(0)(119896 minus 1))

(2119896minus2)

(12)


1199111(119896) = int

119896

119896minus1

119909(1)(119896) 119889119905 = minus

1

119898119901

ln100381610038161003816100381610038161003816100381610038161003816

119901119890minus119898(119896minus1)

+ 119902

119901119890minus119898(119896minus2)

+ 119902

100381610038161003816100381610038161003816100381610038161003816

1199112(119896) = int

119896

119896minus1

119909(1)(119896)2119889119905

=

1

119898

[

1

119901

119902 [119890119898(119896minus2)

minus 119890119898(119896minus1)

]

(119901 + 119902119890119898(119896minus1)

) (119901 + 119902119890119898(119896minus2)

)

+

1

1199012ln10038161003816100381610038161003816100381610038161003816100381610038161003816

119890119898(119901 + 119902119890

119898(119896minus2))

119901 + 119902119890119898(119896minus1)

10038161003816100381610038161003816100381610038161003816100381610038161003816

]

(13)


as

= (119886 119887)119879= (119861119879119861)

minus1119861119879119884 (14)


119884 =

[

[

[

[

[

[

119909(0)(2)

119909(0)(3)

119909(0)(119899)

]

]

]

]

]

]

119861 =

[

[

[

[

[

[

minus1199111(2) 119911

2(2)

minus1199111(3) 119911

2(3)

minus1199111(119899) 119911

2(119899)

]

]

]

]

]

]

(15)


kk minus 1

x(1)

(k)

x(1)

(t)

x(1)

(k minus 1)




(1)(119896) =

1

119888119890119886119896+ 119887119886

(16)


119865 (119888) =

119899

sum

119896=1

(119888119890119886(119896+1)

minus 119888119890119886119896

minus (

minus119909(0)(119896 + 1)

sum119896+1

119894=1 119909(0)(119894) sum119896

119894=1 119909(0)(119894)

))

2

(17)


=

minussum119899

119894=1 (119909(0)(119896 + 1) sum

119896+1

119894=1 119909(0)(119894) sum119896

119894=1 119909(0)(119894)) (119890

119886119896(119890119886minus 1))

sum119899

119894=1 (119890119886119896(119890119886minus 1))2

(18)



(0)= [(0)(1)

(0)(2)

(0)(119899)] (19)


119864 = [119890 (1) 119890 (2) 119890 (119899)] = 119883(0)minus

(0)

119890 (119896) = 119909(0)(119896) minus

(0)(119896) 119896 = 1 2 119899

(20)


1198782

1 =1

119899

119899

sum

119896=1

[119909(0)(119896) minus 119909]

2

1198782

2 =1

119899

119899

sum

119896=1

[119890 (119896) minus 119890]2

(21)


119909 =

1

119899

119899

sum

119896=1

119909(0)(119896)

119890 =

1

119899

119899

sum

119896=1

119890 (119896)

(22)


119862 =

1198782

1198781

(23)


119901 = 119875 |119890 (119896) minus 119890| lt 067451198781 (24)








Level 1 (good) 119862 le 035 095 le 119901







119909(1)(119896) asymp

1

00317119890minus02661119896

+ 0009

119896 = 1 2 119899 (25)




GVM is shown as

119909(1)(119896 + 1) =

7983

0072 + 01914119890minus02661119896

119896 = 1 2 119899

(26)

TPGVM is shown as

119909(1)(119896) =

02661

00086119890minus02661119896

+ 00024

119896 = 1 2 119899 (27)

TRGVM is shown as

119909(1)(119896) =

02661

00093119890minus02661119896

+ 00024

119896 = 1 2 119899 (28)



119862 =

1198782

1198781

= 0130 le 035

067451198781 = 1137

|119890 (119896) minus 119890| lt 067451198781

(29)





BPGVM is shown as

119909(1)(119896 + 1) =

7953

0074 + 01911119890minus02651119896

119896 = 1 2 119899

(30)

BRGVM is shown as

119909(1)(119896 + 1) =

8296

0078 + 02068119890minus02673119896

119896 = 1 2 119899

(31)



119862 =

1198782

1198781

= 0003 le 035

067451198781 = 1137

|119890 (119896) minus 119890| lt 067451198781

(32)






Measuredraw data



69 55 41



Measuredraw data



69 53 48




5 Conclusion







Measuredraw data



72 53 33

Extrapolationvalues

1042 1022 10151047 1038 1021


Measuredraw data



72 54 38

Extrapolationvalues

1047 1033 10241036 1031 1022



Notations












Competing Interests


Acknowledgments


References














































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










kk minus 1

x(1)

(k)

x(1)

(t)

x(1)

(k minus 1)




(1)(119896) =

1

119888119890119886119896+ 119887119886

(16)


119865 (119888) =

119899

sum

119896=1

(119888119890119886(119896+1)

minus 119888119890119886119896

minus (

minus119909(0)(119896 + 1)

sum119896+1

119894=1 119909(0)(119894) sum119896

119894=1 119909(0)(119894)

))

2

(17)


=

minussum119899

119894=1 (119909(0)(119896 + 1) sum

119896+1

119894=1 119909(0)(119894) sum119896

119894=1 119909(0)(119894)) (119890

119886119896(119890119886minus 1))

sum119899

119894=1 (119890119886119896(119890119886minus 1))2

(18)



(0)= [(0)(1)

(0)(2)

(0)(119899)] (19)


119864 = [119890 (1) 119890 (2) 119890 (119899)] = 119883(0)minus

(0)

119890 (119896) = 119909(0)(119896) minus

(0)(119896) 119896 = 1 2 119899

(20)


1198782

1 =1

119899

119899

sum

119896=1

[119909(0)(119896) minus 119909]

2

1198782

2 =1

119899

119899

sum

119896=1

[119890 (119896) minus 119890]2

(21)


119909 =

1

119899

119899

sum

119896=1

119909(0)(119896)

119890 =

1

119899

119899

sum

119896=1

119890 (119896)

(22)


119862 =

1198782

1198781

(23)


119901 = 119875 |119890 (119896) minus 119890| lt 067451198781 (24)








Level 1 (good) 119862 le 035 095 le 119901







119909(1)(119896) asymp

1

00317119890minus02661119896

+ 0009

119896 = 1 2 119899 (25)




GVM is shown as

119909(1)(119896 + 1) =

7983

0072 + 01914119890minus02661119896

119896 = 1 2 119899

(26)

TPGVM is shown as

119909(1)(119896) =

02661

00086119890minus02661119896

+ 00024

119896 = 1 2 119899 (27)

TRGVM is shown as

119909(1)(119896) =

02661

00093119890minus02661119896

+ 00024

119896 = 1 2 119899 (28)



119862 =

1198782

1198781

= 0130 le 035

067451198781 = 1137

|119890 (119896) minus 119890| lt 067451198781

(29)





BPGVM is shown as

119909(1)(119896 + 1) =

7953

0074 + 01911119890minus02651119896

119896 = 1 2 119899

(30)

BRGVM is shown as

119909(1)(119896 + 1) =

8296

0078 + 02068119890minus02673119896

119896 = 1 2 119899

(31)



119862 =

1198782

1198781

= 0003 le 035

067451198781 = 1137

|119890 (119896) minus 119890| lt 067451198781

(32)






Measuredraw data



69 55 41



Measuredraw data



69 53 48




5 Conclusion







Measuredraw data



72 53 33

Extrapolationvalues

1042 1022 10151047 1038 1021


Measuredraw data



72 54 38

Extrapolationvalues

1047 1033 10241036 1031 1022



Notations












Competing Interests


Acknowledgments


References














































Volume 2014




Journal of











Journal of


Function Spaces






Algebra













Level 1 (good) 119862 le 035 095 le 119901







119909(1)(119896) asymp

1

00317119890minus02661119896

+ 0009

119896 = 1 2 119899 (25)




GVM is shown as

119909(1)(119896 + 1) =

7983

0072 + 01914119890minus02661119896

119896 = 1 2 119899

(26)

TPGVM is shown as

119909(1)(119896) =

02661

00086119890minus02661119896

+ 00024

119896 = 1 2 119899 (27)

TRGVM is shown as

119909(1)(119896) =

02661

00093119890minus02661119896

+ 00024

119896 = 1 2 119899 (28)



119862 =

1198782

1198781

= 0130 le 035

067451198781 = 1137

|119890 (119896) minus 119890| lt 067451198781

(29)





BPGVM is shown as

119909(1)(119896 + 1) =

7953

0074 + 01911119890minus02651119896

119896 = 1 2 119899

(30)

BRGVM is shown as

119909(1)(119896 + 1) =

8296

0078 + 02068119890minus02673119896

119896 = 1 2 119899

(31)



119862 =

1198782

1198781

= 0003 le 035

067451198781 = 1137

|119890 (119896) minus 119890| lt 067451198781

(32)






Measuredraw data



69 55 41



Measuredraw data



69 53 48




5 Conclusion







Measuredraw data



72 53 33

Extrapolationvalues

1042 1022 10151047 1038 1021


Measuredraw data



72 54 38

Extrapolationvalues

1047 1033 10241036 1031 1022



Notations












Competing Interests


Acknowledgments


References














































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











Measuredraw data



69 55 41



Measuredraw data



69 53 48




5 Conclusion







Measuredraw data



72 53 33

Extrapolationvalues

1042 1022 10151047 1038 1021


Measuredraw data



72 54 38

Extrapolationvalues

1047 1033 10241036 1031 1022



Notations












Competing Interests


Acknowledgments


References














































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











Measuredraw data



72 53 33

Extrapolationvalues

1042 1022 10151047 1038 1021


Measuredraw data



72 54 38

Extrapolationvalues

1047 1033 10241036 1031 1022



Notations












Competing Interests


Acknowledgments


References














































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











Notations












Competing Interests


Acknowledgments


References














































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





























Volume 2014




Journal of











Journal of


Function Spaces






Algebra









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