research article coupled model of artificial neural

Research ArticleCoupled Model of Artificial Neural Network and Grey Model forTendency Prediction of Labor Turnover

Yueru Ma and Lijun Peng

Business School Central South University Changsha Hunan 410083 China

Correspondence should be addressed to Lijun Peng penlijincaozhisinacom

Received 20 December 2013 Revised 9 March 2014 Accepted 11 March 2014 Published 15 April 2014

Academic Editor Gongnan Xie

Copyright copy 2014 Y Ma and L Peng 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

The tendency of labor turnover in theChinese enterprise shows the characteristics of seasonal fluctuations and irregular distributionof various factors especially the Chinese traditional social and cultural characteristics In this paper we present a coupled modelfor the tendency prediction of labor turnover In the model a time series of tendency prediction of labor turnover was expressedas trend item and its random item Trend item of tendency prediction of labor turnover is predicted using Grey theory Randomitem of trend item is calculated by artificial neural network model (ANN) A case study is presented by the data of 24 months ina Chinese matured enterprise The model uses the advantages of ldquoaccumulative generationrdquo of a Grey prediction method whichweakens the original sequence of random disturbance factors and increases the regularity of data It also takes full advantage ofthe ANN model approximation performance which has a capacity to solve economic problems rapidly describes the nonlinearrelationship easily and avoids the defects of Grey theory

1 Introduction

New classical economists are sticking to a fundamentalassumption the social and economic phenomenon can bepredictedThemathematicalmodel can predict the long-termtrend of the economic problem development and help ratio-nal decision-makers to achieve rapid development [1ndash3] Dueto the effect of social and economic cyclical demographiccharacteristics and the causes of work-life balance the humanresource is one of the most unstable productive factors inenterprises [4 5] The mathematical model is needed toestablish a set of index systems and analysis method Bythe model we can find the inner logic and relatively stablestructure of employee turnover analyze the trend of humanresource flow identify potential crisis shorten the differencetime between searching and using the labor reasonable planand control the strategy combination of human resourceas well as deposit the completion of sustainable productivetasks in enterprises [6ndash8] Many models and technologies forforecasting the trends of economic problem such as Greymodel (GM) artificial neural network (ANN) time seriesanalysis and the combination model were developed in thepast years However the predictive models do not meet the

practical conditions with the characteristics of random andcomplex nonlinear and volatility as well as localization andcontextualized [9 10] In this paper the combinative modelof grey model and artificial neural network (CM) is usedto handle the Chinese enterprise employee flowing data andpredict the current trend of Chinese enterprise employee

In this work the following aspects will be explainedand analyzed (1) designing concrete model to provide keyindicators for economic data with characteristic of timeseries which can be used to track the trend change andmutant of economic phenomenon The model can explainthe relationship between the current and previous data ineconomic problems and focus on the change in a specificperiod (2) analyzing the annual turnover rate and the detailsand trends of annual turnover flow and identifying orexpecting the shortage or excess of employee number bythe application of the model Thus it can provide the basisof hiring allocating or dismissing employee for rationaldecision makers The rational decision makers can bettercontrol cost and comply with budget constraints

The core of these methods is to establish the appropriateforecasting model Because of the complexity of time seriesand the nonlinear characteristics of the tendency prediction

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2014 Article ID 918307 6 pageshttpdxdoiorg1011552014918307

2 Mathematical Problems in Engineering

system it is difficult to establish the ideal forecast modelIn addition the given uncertain tendency prediction is achaotic system so the pursuit of the prediction of long-term development and evolution of tendency prediction isunrealistic but the prediction of short-term behavior of theirdegree is possible

Therefore a short-term ideal forecast model should meettwo conditions (i) a theory that can predict time series (ii)the advantages of a theory that are modeling of nonlinearfunctions and good performance in noisy environmentsHowever it is difficult to find a theory to meet the twoconditions It is possible to establish amodel based on the twotheories Grey theory has been successfully used to predicttime series ANN Random Forest Support Vector MachineBP Neural Network Uncertainty Measurement and Bayesmethods are also used to solve nonlinear problems [11ndash16] Inthe work the ANN was selected to fit the nonlinear perfor-mance of the labor turnover since the major advantages ofan artificial neural network (ANN) are [12 17] fast trainingmodeling of nonlinear functions and good performance innoisy environments that gives enough data

We present a coupled model of Grey model and artificialneural network for predicting tendency prediction of laborturnover [18] In this model a time series of tendencyprediction of labor turnover was expressed as trend item andits random item The trend item of tendency prediction oflabor turnover is predicted with Grey theory Random itemof trend item is calculated by ANN The model uses theadvantages of ldquoaccumulative generationrdquo of a Grey predictionmethod which weakens the original sequence of randomdisturbance factors and increases the regularity of data It alsotakes full advantage of the ANN model approximation per-formance because it has fast solving speed and can describethe nonlinear relationship easily and avoid the defects of Greytheory [19ndash22]

2 Methodology

21 Brief Mathematical Explanation of an Artificial NeuralNetwork Neural networkmodels in artificial intelligence areusually referred to as artificial neural networks (ANNs) [1217] these are essentially simplemathematicalmodels defininga function or a distribution but sometimes models are alsointimately associated with a particular learning algorithmor learning rule A common use of the phrase ANN modelreally means the definition of a class of such functions (wheremembers of the class are obtained by varying parametersconnection weights or specifics of the architecture such asthe number of neurons or their connectivity)

Models The models presented in this section appear fairlydifficult mathematically However they eventually boil downto just multiplication and addition The use of matrices andvectors simplifies the notation but is not absolutely requiredfor this application

Neuron Model A model of a neuron has three basic partsinput weights a summer and an output function The inputweights scale values used as inputs to the neuron the summer

B

x a

W1

W2

W3

I1

I2

I3

f(x)sum

Figure 1 Neuron model

adds all the scaled values together and the output functionproduces the final output of the neuronOne additional inputknown as the bias is often added to the system If a bias isused it can be represented by a weight with a constant inputof one This description is laid out visually in Figure 11198681 1198682 and 119868

3are the inputs 119882

11198822 and 119882

3are the

weights 119861 is the bias 119909 is an intermediate output and a isthe final output The equation for 119886 is given by

119886 = 119891 (11988211198681+11988221198682+11988231198683+ 119861) (1)

where 119891 could be of any function Most often 119891 is the signof the argument (ie 1 if the argument is positive and minus1 ifthe argument is negative) linear (ie the output is simplythe input times some constant factor) or some complexcurve used in function matching (not needed here) Forthis model we will use the first case where 119891 is the sign ofthe argument for two reasons it closely matches the ldquoall ornothingrdquo property seen in biological neurons and it is fairlyeasy to be implemented

When artificial neurons are implemented vectors arecommonly used to represent the inputs and the weights sothe first of two brief reviews of linear algebra is appropriatehere The dot product of two vectors = (119909

1 1199092 119909

119899) and

119910 = (1199101 1199102 119910

119899) is given by

∙ 119910 = 11990911199101+ 11990921199102+ sdot sdot sdot + 119909

119899119910119899 (2)

Using this notation the output is simplified to

119886 = 119891 ( ∙ 119868 + 119861) (3)

where all the inputs are contained in 119868 and all the weights arecontained in

Neuron Layer In a neuron layer each input is tied to everyneuron and each neuron produces its own output This canbe represented mathematically by the following series ofequations

1198861= 1198911(1∙ 119868 + 119861

1) (4a)

1198862= 1198912(2∙ 119868 + 119861

2) (4b)

1198863= 1198913(3∙ 119868 + 119861

3)

sdot sdot sdot

(4c)

Mathematical Problems in Engineering 3

Representing the input vector and the biases as onecolumn matrices we can simplify the above notation to

119886 = 119891 (119882 sdot 119868 + 119861) (5)

which is the final form of the mathematical representation ofone layer of artificial neurons

Neural Network A neural network is simply a collectionof neuron layers where the output of each previous layerbecomes the input to the next layer So for example theinputs to layer two are the outputs of layer one In this exercisewe are keeping it relatively simple by not having feedback (ieoutput from layer 119899 being input for some previous layer) Tomathematically represent the neural network we only have tochain together the equations The finished equation for thethree-layer network in this equation is given by

119886 = 119891 (1198823sdot 119891 (119882

2sdot 119891 (119882

1sdot 119868 +

1) + 2) + 3) (6)

22 GM-ANNModel In thismodel a time series of tendencyprediction of labor turnover was expressed as trend item andits random item Trend item of deformation which is relatedby time is predicted with Grey theory [22] Random itemof trend item which is a complex nonlinear sequence iscalculated by ANN [18] We resolve a time series of tendencyprediction of labor turnover119882(119905) as

119882(119905) = 119884 (119883 (119905) 119905) (7)

where series of trend item 119883(119905) can be expressed as the greymodel and series of random item of trend item 119884(119883(119905) 119905) canbe expressed as the ANN model For time series 119883(119905) afterone accumulative generation we obtain

119883(1)(119905) =

119905

sum

119896=1

119883(119896) (119905 = 1 2 119872) (8)

and series119883(1)(119905) satisfies GM (11) model

119889119909(1)(119905)

119889119905+ 119886119909(1)(119905) = 119887 (9)

We can know easily that

119909 (119905) = 119909(1)(119905) minus 119909

(1)(119905 minus 1) 119905 = 1 2 119872

119883(1)(119905) = (119883

(1)(0) minus119887

119886) 119890minus119886119905+119887

119886

(10)

Suppose series of random item 119884(119883(119905) 119905) satisfied ANNmodel The architecture of a basic ANN model is shown inFigure 2 The model has three layers input summation andoutput with theweighted connections between the inputs andthe summation layers

Themodel used for the present study has two input nodes(spatial coordinates) 119895 pattern two (1 + 1) summation nodesand one output nodes The function of the input layer is topass forward the activity patterns presented to the network toall nodes in the pattern layer The nodes in the pattern layer

X(t)

t

fW(t)

sum

Figure 2 Architecture of ANN

Table 1 Data of turnover of employees from the Chinese Aenterprises in 2011

Month (119905)1 2 3 4 5 6 7 8 9 10 11 12

(1) Entry 119909(119905) 3 5 8 8 4 1 2 3 3 1 2 0(2) Exit 119910(119905) 0 3 21 9 17 9 6 21 11 6 3 8

perform a nonlinear transformation of the input patternsWhen a new vector 119883 is entered into the network it issubtracted from the stored weight vector representing eachcluster centre [23] Based on the above theories in this paperwe produce a program of ANNmodel using MATLAB 2011aApplied steps of the program are as follows (i) according tothe measured data of tendency prediction of labor turnoverthe predicted sequence is obtained using the established GM(11) model part of the program (ii) In the ANN modelthe predicted sequence and time can be selected as inputvariables the measured data of tendency prediction of laborturnover can be selected as expected variables The ANNmodel is trained to predict the tendency prediction of laborturnover according to the time and Grey prediction As thesmooth factor affects network performance it is needed tokeep trying to determine the best value (iii) if we want toknow the the measured data of tendency prediction of laborturnover in a futuremonth we will get the accurate predictedvalue after only entering the required time into the well-established ANN model Then it will serve scientific andreasonable decision makers

3 Establishing Models and Discussions

This model is based on data from the Chinese enterpriseA Enterprise A is relatively matured and stable economiesIts business develops steadily in the process of the rapidexpansion of the Chinese economyThis study used two yearsof data of enterprise A (specific as shown in Tables 1 and2) The data is a mapping of staff flow trend in the Chineseenterprises in recent years The data follow the two aspectsof the law a law of time series data is followed and thenonlinear relationship between the data On the basis of GM(11)models (ie model 1model 2model 3 andmodel 4) theoptimized models based on ANN of GM (11) (ie couplingmodel 1 coupling model 2 coupling model 3 and couplingmodel 4) are obtained

The GM model of entry number for employees in 2011the GMmodel of exit number for employees in 2011 the GM


0123456789

0 2 4 6 8 10 12

Entr

y nu

mbe

r

Month

ObservationsGMCM

(a) Comparisons of entry number in 2011

ObservationsGMCM

0

5

10

15

20

25

0 2 4 6 8 10 12

Qui

t num

ber

Month

(b) Comparisons of quit number in 2011

ObservationsGMCM

0

2

4

6

8

10

12

14

0 2 4 6 8 10 12

Entr

y nu

mbe

r

Month

(c) Comparisons of entry number in 2012

ObservationsGMCM

0

5

10

15

20

25

0 2 4 6 8 10 12

Qui

t num

ber

Month

(d) Comparisons of quit number in 2012

Figure 3 Results and comparisons of entry and quit number

model of entry number for employees in 2012 and the GMmodel of exit number for employees in 2012 were obtained as

119909 (119905 + 1) = minus 158563366119890minus0015188119905

+ 160710719

119909 (119905 + 1) = minus 362298701119890minus0035088119905

+ 362298701

119909 (119905 + 1) = 214134731198900134079119905minus 14761375

119909 (119905 + 1) = minus 279593023119890minus0054556119905

+ 279593023

(11)

GM often only considers time series which is used toincrease exponential time betrayal Apparently the reality ofthe economic data is more complex although the time seriesand nonlinear data can be considered together Consideringthe complexity of the Chinese employee turnover trend itis necessary for time series and nonlinear to fit the methodin model building Therefore according to the GM (11) timeseries model is established and the ANNwas used to modifythe GM (11) models

The calculated results of 2ndash11 months with GM CM andobservations are shown in Figure 3 The predicted result of


Month (119905)1 2 3 4 5 6 7 8 9 10 11 12

(1) Entry 119909(119905) 6 12 9 5 4 0 2 0 1 2 8 7(2) Exit 119910(119905) 0 22 16 9 5 5 13 16 11 8 13 5

Table 3 Predicted results of December

Entry in2011

Quit in2011

Entry in2012

Quit in2012

Observation 0 8 7 5GM 117815 879513 135858 860282Error minus117815 minus079513 564142 minus360282CM 1 75691932 78233 583Error minus1 04308068 minus08233 minus083

December is listed in Table 3 and the errors were discussedFigure 3(a) shows the results of CM for months 2 4 5 7


9 10 and 11 that are consistent with observations and theerrors are smaller than the results of GMThe errors of resultsfor months 3 and 6 by GM are smaller than that of CMFigure 3(b) shows that results of CM for months 1 2 and9 are not well consistent with observations but the averageerrors are smaller than the results of GM Figure 3(c) showsthat results of CM for months 3 4 5 6 7 9 10 and 11 arewell consistent with observations and the errors are smallerthan the results of GM The errors of results for months 2and 8 by GM are smaller than that of CM Figure 3(d) showsthat results of CM for months 2 3 4 5 6 9 and 11 are wellconsistent with observations and the errors are smaller thanthe results of GMThe errors of results for months 2 7 8 and10 by GM are smaller than that of CM Table 3 shows thaterrors of predicted results for entry in 2011 quit in 2011 andquit in 2012 by CM are smaller than that of GM The CM isfailed to predict the entry in 2012 From above analysis it canbe clearly seen that the average accuracy ofCM is smaller thanthat of GM which has demonstrated that the proposed CMcan analyze the tendency prediction of labor turnover thoughthere still exist some errors in CM

4 Conclusions

In the present work a coupling model with use of the GMand ANN is presented The aim of the present study wasto develop a reliable model for the problem of tendencyprediction of labor turnover According to the existing data ofthe enterprise the predicted sequence is obtained using theestablished GM (11) model in that part of the program Inpart of general regression neural network model predictedsequence and time are input variables the existing data isexpected variables The ANN model is trained to predicttendency prediction of labor turnover The model is appliedto the data of 24 months in a Chinese matured enterprise Itshows that the approximation error of the proposed model issmall Results show that the proposed model has a capacityto solve economic problems rapidly which can describe thenonlinear relationship easily and can avoid the defects ofGreytheory

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] A Denvir and F McMahon ldquoLabour turnover in Londonhotels and the cost effectiveness of preventative measuresrdquoInternational Journal of Hospitality Management vol 11 no 2pp 143ndash154 1992

[2] L A Peletier and J Gabrielsson ldquoNonlinear turnover modelsfor systems with physiological limitsrdquo European Journal ofPharmaceutical Sciences vol 37 no 1 pp 11ndash26 2009

[3] Y Zhang T Gong and Y Zhang ldquoA forecasting model foremployees1015840 exceptional turnover based on calamities GreyPredictionTheoryrdquoMathematics in Practice andTheory vol 20article 9 2008

[4] P C Pendharkar J A Rodger and K Wibowo ldquoArtificialneural network based data mining application for learningdiscovering and forecasting human resources requirement inPennsylvania Health Care Industryrdquo Pennsylvania Journal ofBusiness and Economics vol 7 no 1 pp 91ndash111 2000

[5] A-L Balduck A Prinzie andM Buelens ldquoThe effectiveness ofcoach turnover and the effect on home team advantage teamquality and team rankingrdquo Journal of Applied Statistics vol 37no 4 pp 679ndash689 2010

[6] R S Sexton S McMurtrey J O Michalopoulos and AM Smith ldquoEmployee turnover a neural network solutionrdquoComputers and Operations Research vol 32 no 10 pp 2635ndash2651 2005

[7] A R AlBattat A P M Som and A S Helalat ldquoOvercomestaff turnover in the Hospitality Industry using Mobley ModelrdquoInternational Journal of Learning and Development vol 3 no 6pp 64ndash71 2013

[8] C-T Lin Y-HWang andW-R Lin ldquoInformation effect of topexecutive turnover testing hybrid grey-market modelrdquo Journalof Grey System vol 20 no 1 pp 53ndash64 2008

[9] N M Abbas ldquoNeural network solution to economic forecast-ingrdquo in Proceedings of the International Conference on BusinessIntelligence and Knowledge Economy Amman Jordan April2012

[10] A Subasi and E ilgun ldquoEconomic variable forecasting usingartificial neural network A Case Study in Turkeyrdquo in Pro-ceedings of the 1st International Symposium on SustainableDevelopment Sarajevo Bosnia and Herzegovina June 2009

[11] J L Deng A Course on Grey SystemTheory Huazhong Univer-sity of Science and Technology Press Wuhan China 1990

[12] R J Schalkoff Artificial Neural Networks McGraw-Hill HigherEducation 1997

[13] G J Burghouts ldquoSoft-assignment random-forest with an appli-cation to discriminative representation of human actions invideosrdquo International Journal of Pattern Recognition and Arti-ficial Intelligence vol 27 no 4 2013

[14] J A K Suykens and J Vandewalle ldquoLeast squares supportvector machine classifiersrdquo Neural Processing Letters vol 9 no3 pp 293ndash300 1999

[15] D Zhou ldquoOptimization modeling for GM (11) Model Based onBPNeuralNetworkrdquo International Journal of ComputerNetworkand Information Security vol 4 no 1 article 24 2012

[16] I Dialsingh ldquoLarge-scale inference empirical Bayes methodsfor estimation testing and predictionrdquo Journal of AppliedStatistics vol 39 no 10 pp 2305ndash2305 2012

[17] V Cherkassky J H Friedman and H Wechsler From Statisticsto Neural Networks Theory and Pattern Recognition Applica-tions Springer 2012

[18] J Li A Yang and W Dai ldquoModeling mechanism of greyneural network and its applicationrdquo in Proceedings of the IEEEInternational Conference onGrey Systems and Intelligent Services(GSIS rsquo07) pp 404ndash408 November 2007

[19] L Dong and X Li ldquoAn application of grey-general regressionneural network for predicting landslide deformation of DahuMine in Chinardquo Advanced Science Letters vol 6 no 1 pp 577ndash581 2012

[20] H Duan B Shen and S Mo ldquoStudy on combined predictablemodel of Gray GM (1 N) self-memory and neural networkrdquoJournal of Water Resources andWater Engineering vol 5 article23 2010


[21] P Xiong Y Dang X Wu and X Li ldquoCombined model basedon optimized multi-variable grey model and multiple linearregressionrdquo Journal of Systems Engineering and Electronics vol22 no 4 pp 615ndash620 2011

[22] L Dong and X Li ldquoComprehensive models for evaluatingrockmass stability based on statistical comparisons of multipleclassifiersrdquo Mathematical Problems in Engineering vol 2013Article ID 395096 9 pages 2013

[23] Yueya L Mingjie and Z Y Shi ldquoTurnover of passenger trafficprediction for civil aviation based on GM (11) modelrdquo Journalof Civil Aviation Flight University of China vol 5 article 9 2011

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

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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


system it is difficult to establish the ideal forecast modelIn addition the given uncertain tendency prediction is achaotic system so the pursuit of the prediction of long-term development and evolution of tendency prediction isunrealistic but the prediction of short-term behavior of theirdegree is possible

Therefore a short-term ideal forecast model should meettwo conditions (i) a theory that can predict time series (ii)the advantages of a theory that are modeling of nonlinearfunctions and good performance in noisy environmentsHowever it is difficult to find a theory to meet the twoconditions It is possible to establish amodel based on the twotheories Grey theory has been successfully used to predicttime series ANN Random Forest Support Vector MachineBP Neural Network Uncertainty Measurement and Bayesmethods are also used to solve nonlinear problems [11ndash16] Inthe work the ANN was selected to fit the nonlinear perfor-mance of the labor turnover since the major advantages ofan artificial neural network (ANN) are [12 17] fast trainingmodeling of nonlinear functions and good performance innoisy environments that gives enough data

We present a coupled model of Grey model and artificialneural network for predicting tendency prediction of laborturnover [18] In this model a time series of tendencyprediction of labor turnover was expressed as trend item andits random item The trend item of tendency prediction oflabor turnover is predicted with Grey theory Random itemof trend item is calculated by ANN The model uses theadvantages of ldquoaccumulative generationrdquo of a Grey predictionmethod which weakens the original sequence of randomdisturbance factors and increases the regularity of data It alsotakes full advantage of the ANN model approximation per-formance because it has fast solving speed and can describethe nonlinear relationship easily and avoid the defects of Greytheory [19ndash22]

2 Methodology

21 Brief Mathematical Explanation of an Artificial NeuralNetwork Neural networkmodels in artificial intelligence areusually referred to as artificial neural networks (ANNs) [1217] these are essentially simplemathematicalmodels defininga function or a distribution but sometimes models are alsointimately associated with a particular learning algorithmor learning rule A common use of the phrase ANN modelreally means the definition of a class of such functions (wheremembers of the class are obtained by varying parametersconnection weights or specifics of the architecture such asthe number of neurons or their connectivity)

Models The models presented in this section appear fairlydifficult mathematically However they eventually boil downto just multiplication and addition The use of matrices andvectors simplifies the notation but is not absolutely requiredfor this application

Neuron Model A model of a neuron has three basic partsinput weights a summer and an output function The inputweights scale values used as inputs to the neuron the summer

B

x a

W1

W2

W3

I1

I2

I3

f(x)sum

Figure 1 Neuron model

adds all the scaled values together and the output functionproduces the final output of the neuronOne additional inputknown as the bias is often added to the system If a bias isused it can be represented by a weight with a constant inputof one This description is laid out visually in Figure 11198681 1198682 and 119868

3are the inputs 119882

11198822 and 119882

3are the

weights 119861 is the bias 119909 is an intermediate output and a isthe final output The equation for 119886 is given by

119886 = 119891 (11988211198681+11988221198682+11988231198683+ 119861) (1)

where 119891 could be of any function Most often 119891 is the signof the argument (ie 1 if the argument is positive and minus1 ifthe argument is negative) linear (ie the output is simplythe input times some constant factor) or some complexcurve used in function matching (not needed here) Forthis model we will use the first case where 119891 is the sign ofthe argument for two reasons it closely matches the ldquoall ornothingrdquo property seen in biological neurons and it is fairlyeasy to be implemented

When artificial neurons are implemented vectors arecommonly used to represent the inputs and the weights sothe first of two brief reviews of linear algebra is appropriatehere The dot product of two vectors = (119909

1 1199092 119909

119899) and

119910 = (1199101 1199102 119910

119899) is given by

∙ 119910 = 11990911199101+ 11990921199102+ sdot sdot sdot + 119909

119899119910119899 (2)

Using this notation the output is simplified to

119886 = 119891 ( ∙ 119868 + 119861) (3)

where all the inputs are contained in 119868 and all the weights arecontained in

Neuron Layer In a neuron layer each input is tied to everyneuron and each neuron produces its own output This canbe represented mathematically by the following series ofequations

1198861= 1198911(1∙ 119868 + 119861

1) (4a)

1198862= 1198912(2∙ 119868 + 119861

2) (4b)

1198863= 1198913(3∙ 119868 + 119861

3)

sdot sdot sdot

(4c)



119886 = 119891 (119882 sdot 119868 + 119861) (5)



119886 = 119891 (1198823sdot 119891 (119882

2sdot 119891 (119882

1sdot 119868 +

1) + 2) + 3) (6)


119882(119905) = 119884 (119883 (119905) 119905) (7)


119883(1)(119905) =

119905

sum

119896=1

119883(119896) (119905 = 1 2 119872) (8)


119889119909(1)(119905)

119889119905+ 119886119909(1)(119905) = 119887 (9)


119909 (119905) = 119909(1)(119905) minus 119909

(1)(119905 minus 1) 119905 = 1 2 119872

119883(1)(119905) = (119883

(1)(0) minus119887

119886) 119890minus119886119905+119887

119886

(10)



X(t)

t

fW(t)

sum



Month (119905)1 2 3 4 5 6 7 8 9 10 11 12

(1) Entry 119909(119905) 3 5 8 8 4 1 2 3 3 1 2 0(2) Exit 119910(119905) 0 3 21 9 17 9 6 21 11 6 3 8






0123456789

0 2 4 6 8 10 12

Entr

y nu

mbe

r

Month

ObservationsGMCM


ObservationsGMCM

0

5

10

15

20

25

0 2 4 6 8 10 12

Qui

t num

ber

Month


ObservationsGMCM

0

2

4

6

8

10

12

14

0 2 4 6 8 10 12

Entr

y nu

mbe

r

Month


ObservationsGMCM

0

5

10

15

20

25

0 2 4 6 8 10 12

Qui

t num

ber

Month




119909 (119905 + 1) = minus 158563366119890minus0015188119905

+ 160710719

119909 (119905 + 1) = minus 362298701119890minus0035088119905

+ 362298701

119909 (119905 + 1) = 214134731198900134079119905minus 14761375

119909 (119905 + 1) = minus 279593023119890minus0054556119905

+ 279593023

(11)




Month (119905)1 2 3 4 5 6 7 8 9 10 11 12

(1) Entry 119909(119905) 6 12 9 5 4 0 2 0 1 2 8 7(2) Exit 119910(119905) 0 22 16 9 5 5 13 16 11 8 13 5


Entry in2011

Quit in2011

Entry in2012

Quit in2012





4 Conclusions




References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119886 = 119891 (119882 sdot 119868 + 119861) (5)



119886 = 119891 (1198823sdot 119891 (119882

2sdot 119891 (119882

1sdot 119868 +

1) + 2) + 3) (6)


119882(119905) = 119884 (119883 (119905) 119905) (7)


119883(1)(119905) =

119905

sum

119896=1

119883(119896) (119905 = 1 2 119872) (8)


119889119909(1)(119905)

119889119905+ 119886119909(1)(119905) = 119887 (9)


119909 (119905) = 119909(1)(119905) minus 119909

(1)(119905 minus 1) 119905 = 1 2 119872

119883(1)(119905) = (119883

(1)(0) minus119887

119886) 119890minus119886119905+119887

119886

(10)



X(t)

t

fW(t)

sum



Month (119905)1 2 3 4 5 6 7 8 9 10 11 12

(1) Entry 119909(119905) 3 5 8 8 4 1 2 3 3 1 2 0(2) Exit 119910(119905) 0 3 21 9 17 9 6 21 11 6 3 8






0123456789

0 2 4 6 8 10 12

Entr

y nu

mbe

r

Month

ObservationsGMCM


ObservationsGMCM

0

5

10

15

20

25

0 2 4 6 8 10 12

Qui

t num

ber

Month


ObservationsGMCM

0

2

4

6

8

10

12

14

0 2 4 6 8 10 12

Entr

y nu

mbe

r

Month


ObservationsGMCM

0

5

10

15

20

25

0 2 4 6 8 10 12

Qui

t num

ber

Month




119909 (119905 + 1) = minus 158563366119890minus0015188119905

+ 160710719

119909 (119905 + 1) = minus 362298701119890minus0035088119905

+ 362298701

119909 (119905 + 1) = 214134731198900134079119905minus 14761375

119909 (119905 + 1) = minus 279593023119890minus0054556119905

+ 279593023

(11)




Month (119905)1 2 3 4 5 6 7 8 9 10 11 12

(1) Entry 119909(119905) 6 12 9 5 4 0 2 0 1 2 8 7(2) Exit 119910(119905) 0 22 16 9 5 5 13 16 11 8 13 5


Entry in2011

Quit in2011

Entry in2012

Quit in2012





4 Conclusions




References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0123456789

0 2 4 6 8 10 12

Entr

y nu

mbe

r

Month

ObservationsGMCM


ObservationsGMCM

0

5

10

15

20

25

0 2 4 6 8 10 12

Qui

t num

ber

Month


ObservationsGMCM

0

2

4

6

8

10

12

14

0 2 4 6 8 10 12

Entr

y nu

mbe

r

Month


ObservationsGMCM

0

5

10

15

20

25

0 2 4 6 8 10 12

Qui

t num

ber

Month




119909 (119905 + 1) = minus 158563366119890minus0015188119905

+ 160710719

119909 (119905 + 1) = minus 362298701119890minus0035088119905

+ 362298701

119909 (119905 + 1) = 214134731198900134079119905minus 14761375

119909 (119905 + 1) = minus 279593023119890minus0054556119905

+ 279593023

(11)




Month (119905)1 2 3 4 5 6 7 8 9 10 11 12

(1) Entry 119909(119905) 6 12 9 5 4 0 2 0 1 2 8 7(2) Exit 119910(119905) 0 22 16 9 5 5 13 16 11 8 13 5


Entry in2011

Quit in2011

Entry in2012

Quit in2012





4 Conclusions




References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











4 Conclusions




References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra




















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









research article coupled model of artificial neural

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