Abstract
Early warning of whether an enterprise will fall into decline stage in a near future is a new problemaroused by the enterprise life cycle theory and financial risk management. This paper presents anapproach by use of back propagation neural networks and rough set theory in order to give an early warn-ing whether enterprises will fall into a decline stage. Through attribute reduction by rough set, the influ-ence of noise data and redundant data are eliminated when training the networks. Our models obtainedfavorable accuracy, especially in predicting whether enterprises will fall into decline or not.
Keywords:Enterprise life cycleBP neural networkRough setEarly warningDecline
Introduction
  The enterprise life cycle is an important part of enterprise the-ory which views an enterprise from the longitudinal perspective,where enterprises may move through a fairly predictable sequenceof developmental stages. Over the past three decades, a consider-able study has been conducted on enterprise life cycles, which isusually linked to the study of organizational growth and develop-ment (Snyder, 2003). Since the majority of enterprise life cycle re-search is on investigation of the number of stages an enterprise canexperience and what characteristics an enterprise exhibits in dif-ferent stages, the approach of classifying different life cycle stages,is a crucial issue for enterprise life cycle researches.
  The number of enterprise life cycle stages proposed in existingworks has varied: some researchers have identified up to ten dif-ferent stages of an enterprise life cycle, while others may only pre-fer three stages. A widely used enterprise life cycle usually includesfour or five stages that can be encapsulated as start-up, growth,maturity, decline, and death (or revival). During enterprises’ differ-ent life cycle stages, the decline stage is much more crucial thanother stages. After experiencing original birth, rapid growth anda relative steady maturity stage, many enterprises fall into declineand confront crises in many aspects. Even though Miller andFriesen (1984) point out that there are wide variety of transitionalpaths open to enterprises and there is no definite, singular and irre-versible sequence of phases, progressing from birth, to growth, tomaturity, and then onto revival or perhaps decline, a number ofdeclining enterprises cannot walk out of the woods and perished,implying that the decline stage of an enterprise is much more cru-cial through the total enterprise life cycle stages.
  Existing literatures pay much attention to develop various earlywarning models to predicting the enterprises’ financial crises,however, little research have done on early warning the enter-prises falling into decline stage. Since this topic is important forthe reasons that a firm falls into decline compose a common sourceof financial crises, and enterprise’s managers and shareholders canadopt measures ahead to prevent the enterprise falling into de-cline, this paper we focus on construct an early warning modelto predicting and prewarning the enterprise falls into decline.
  One clue of this topic is from the research of enterprise life cycletheory. Previous studies have approached various methods to solvethe problem of classifying enterprise life cycle stages (Hwang,2004). Generally speaking, the existing life cycle classificationmethods can be divided into two categories in accordance withthe employed techniques. Most of life cycle studies were per-formed by applying simple self-assessment questionnaires to topmanagers of an enterprise, in an effort to ascertain where theirfirms were positioned within the enterprise life cycle framework.Another category of method of classification uses various financialand non-financial measures. The most common measures are age,sales, and growth rate with growth rate defined as income growth,asset growth, or sales growth (Hanks, Watson, Jansen, & Chandler,1993; Miller & Friesen, 1984; Olson & Terpstra, 1992 ). For example,Miller and Friesen (1984) have classified a firm with a decrease insales growth is in the decline stage.
  The mathematic methods used in previous life cycle researchesare commonly cluster analysis and factor analysis, etc., while an-other similar field, financial crises early warning models commonly0957-4174/$ - see front matterContents lists available atScienceDirectExpert Systems with Applicationsjournal homepage: www.else vier.com/locate/eswaadopt many much more complicated methods, such as discrimi-nate analysis, case-based reasoning and expert systems, et al. tosolve the problem. At present years, a new method which com-bined the rough set theory and neural network to prediction enter-prises’ financial crises arose and shows fine performance. Thispaper, referring to the measures used in financial crises, a multi-layer back propagation network (BP), combined with one tool ofsoft theory - rough set are used to construct an enterprise-declineearly warning model. This measure can integrate each advantage ofneural network and rough set, for BP network is often widely ap-plied in many areas such as predicting stock price, early detectingfinancial risks and many others, because of its ability to study andremember the relation between inputs and out-puts as well as toapproach any types of function, and for rough set can reduce theinfluence due to the drawbacks of BP such as low training speedand easily affected by noise and weak interdependency datathrough attribution reduction process (Xiao, Ye, Zhong, & Sun,2009). In other words, we take advantages of rough set – it coulddeal with incomplete, uncertain and ambiguous data and complexdata containing mass of variables; moreover, it can abstract knowl-edge or patterns from data. After attribution reduction, the noisedata and weak interdependency data are eliminated, so the influ-ences they have to BP during the initialization, study and trainingprocess are avoided, and then the accuracy of predictions is devel-oped and proved.
Rough sets and neural network
2.1. Rough sets
  Rough sets theory (RST) is a machine-learning method, which isintroduced by Pawlak (1991) in the early 1980s, has proved to be apowerful tool for uncertainty and has been applied to data reduc-tion, rule extraction, data mining and granularity computation(Yeh, Chi, & Hsu, 2009).
  The basic concept in rough set theory is an information systemwhich can be expressed by a 4-tuple S =(U, A, V, f ), whereU ={x1, x2, ..., xn} is a finite set of objects, called the universe;A = C [ D is a finite set of attributes, which is a union of the condi-tion attributes set C and decision attributes set D withC \ D = £;V = [a2AVais a domain of attribute a , and f : U  A ?V is an infor-mation function to determine each objectxi’s attribute value inset U that is: f (xi, a) 2 Va, for "xi2 U, a 2 A.
  In rough set theory, the objects in universe U can be describedby various attributes in attributes set A. When two different objectsare described by the same attributes, then these two objects areclassified as one kind in the information system S, thus we calltheir relationship is indiscernibility relation. In mathematicalword, an indiscernibility relation IND (B) generated by attributesubset B # A onU, is defined as follows:IND ðBÞ¼ ð xi; xj Þ2U  U j f ðxi; a Þ; ¼ f ð xj; aÞ ; 8 a 2 B
  The partition of U generated byIND (B ) is denoted by U/IND (B)={ C1, C2, ..., Ck} for every Ciis an equivalence class. For"x 2 U the equivalence class ofx in relation to U/IND (B ) is definedas follows:½ xU = IND ð B Þ¼ y 2 U jf ð y; aÞ¼ f ð x; aÞ ; 8 a 2 B fg Let X # U be a target set and P # A be a attribute subset, thatwe wish to represent X using attribute subsetP. In general, X can-not be expressed exactly, because the set may include and excludeobjects which are indistinguishable on the basis of attributes P.However, Pawlak (1991) present a method to approximating thetarget set P only by the information contained within P byconstructing the P-lower and P-upper approximations of X, whichis respectively defined as:P -lower approximations of X : P X ¼ xj½ xU =IND ð P Þ# XnoP -upper approximations of X : PX ¼ xj½ xU = IND ð P Þ\ X – £no
  The P-lower approximation, also called the positive region, isthe union of all equivalence classes in [x]U/IND ( P)which are con-tained by (i.e., are subsets of) the target set X. In another word,the lower approximation is the complete set of objects in U/IND (P)that can be positively (i.e. unambiguously) classified as belongingto target set X.
  The P-upper approximation is the union of all equivalence clas-ses in [x]U/IND (P)which have non-empty intersection with the targetset, that is the complete set of objects that in U/IND (P) that cannotbe unambiguously classified as belonging to the complement (X Þ ofthe target set X. In other words, the upper approximation is thecomplete set of objects that are possibly members of the targetset X.
  One of the most important aspects in rough set theory is thediscovery of attribute dependencies, that is, we wish to discoverwhich variables are strongly related to which other variables. Forthis purpose, given two attribute subset P, Q # A, Then, the depen-dency of attribute setQ on attribute setP, cP(Q), is given bycPð Q Þ¼card ð[X 2 U =IND ðQ ÞP X Þcard ðU Þwhere [X 2U/IND (Q )P⁄X can be denoted as POSP( Q), which means thatthe objects in it can be classified to one class of the classificationU/ IND (Q ) by attribute P.
  An attribute a is said to be dispensable in P with respect to Q ,ifcP(Q )=cP{ a}(Q ); otherwise a is an indispensable attribute in P withrespect to Q .Let (S = U, A, V, f ) be a decision table, the set of attributesP(P # C ) is a reduce of attribute, C if it satisfied the followingconditions:cPð D Þ¼ cCð DÞ ; cP0 ðD Þ – cCð DÞ 8 P0 P
  A reduction of condition attributes C is a subset that can discerndecision classes with the same accuracy as C, and none of the attri-butes in the reduced can be eliminated without decreasing itsdistrainable capability (Pawlak, 2002).
  Though it is a kernel concept in rough set, it is difficult to calcu-late the reduction if the size of information system is large. Manyscholars proposed a variety of attribute reduction algorithm, suchas: consistency of data ( Mi, Wei-zhi, & Wen-Xiu, 2004; Pawlak,1991), dependency of attributes (Wang, Hu, & Yang, 2002), mutualinformation (Skowron & Rauszer, 1992 ), discernibility matrix (JueFig. 1. A neuron of BP.Y. Cao et al. / Expert Systems with Applications 38 (2011) 6424–6429 6425& Duo-Qian, 1998) and genetic algorithm which are employed tofind reduction of an information system (Moradi, Gruzmala-Busse,& Roberts, 1998).
2.2. Neural network
  The BP neural network, which was first described by Paul Wer-bos in 1974, and gained recognition until 1986 through the work ofDavid E. Rumelhart, Geoffrey E. Hinton and Ronald J. Williams, ledto a ‘‘renaissance’’ in the field of artificial neural network research.The BP neural networks are the most widely used networks and areconsidered the workhorse of ANNs ( Basheer & Hajmeer, 2002).Thanks to its simplicity and excellent performance in extract usefulinformation from samples, the BP neural network is widely appliedrecently. Commonly the BP neural network is used to solve theproblems of classification and function approximation, which arisefrequently in loan risk warning ( Yang, Li, Ji, & Xu, 2001), stock mar-ket returns and price index prediction ( Enke & Thawornwong,2005; Quah & Srinivasan, 1999; Roh, 2007 ), the power system’short term load forecasting ( Xiao et al., 2009), box office revenueof movies forecasting (Zhang, Luo, & Yang, 2009 ), bank’s efficiencyevaluation (Wu, Yang, & Liang, 2006) and areas of decision supportsystems and management science ( Delen, Sharda, & Kumar, 2007;Wu, 2009).
  An elementary neuron withR inputs of BP is shown in Fig. 1 .Each input is weighted with an appropriatewi. The sum of theweighted inputs and the bias forms the input to the transfer func-tion f (), and f() transforms the sum of input values into outputvalues of the node. Typical choices of the transform function con-sist of the logistic, the tangent, the sign, and the linear.
  In this paper we apply a BP neural network with two hiddenlayer in which the neural neurons take tan-sigmoid function fortransform, and purelin, a linear function, is used in output layerfor transform to get a broad range of output values. The wholestructure of our network is shown in Fig. 2 , where a1 = tan-si-g( IW11⁄p 1+b1), a2 = tan-sig(LW21⁄a1+b2), and a 3 = pure-lin( LW32⁄a2+b3), besides the number of neural cells in hiddenlayers is determined by the training process.
Research design and methodology
3.1. Data collection
  The data used in this research is drawn from listed companies inChina’s Shanghai and Shenzhen stock markets covering the period2005–2008. We excluded the financial and insurance sector com-panies and missing data companies. After that, complete data foronly 107 such firms appear on the data base.
  We use the method proposed byAnthony and Ramesh (1992)todetermine which firms is under the decline. By this method, threevariables dividend payout ratio (DP), sales growth (SG), capitalexpenditure (CEV) are introduced. Each variable is defined asfollows:(1) Dividend payout ratio ( DPt): DPt=(DIVt/EPS )  100(2) Sales growth ( SGt): SGt=((SALESt  SALESt1)/ SALESt 1) 100(3) Capital expenditure ( CEVt): CEVt=(CEt/ VALUEt)  100whereDIVt= common dividends per share in yeartEPSt= the earnings per share in yeartSALES = net sales in yeartCEt= capital expenditure in yeartandVALUEt = market value of equity plus book value of long-termdebt at the end of yeart.
  On average, the firms in decline characterized a lower salesgrowth rate, and a lower rate of investment in production equip-ment. Therefore, their dividend payment rate will be relativelyhigh ( Chin, Tsao, & Chi, 2005 ). The firms in the other two periods(growth and mature) characterized other features of the abovethree variables. An overall description of the features of abovethree variables in different enterprise stages are shown in Table 1.
  Consistent with Anthony and Ramesh (1992), we rank on eachof the variables: from smallest to largest value of the dividend pay-out ratio, highest to lowest of the sales growth, highest to lowest ofthe capital expenditure. Each of the three rankings was then di-vided into 3 equal-sized groups and values of 1, 2, or 3 assignedto each group based on the criteria listed in Table 1. For example,observations falling into the group of smallest dividend payout ra-tios were assigned a value of 1, observations falling into the midstgroup were assigned a value of 2, and observations falling into thelargest group were assigned a value of 3. After that, we add thethree scores into a total score for each enterprise, and then rank to-tal score from the lowest to highest and divide it into 3 equal-sizedgroups again. After that, we assign the highest one third observa-tions as decline enterprises, and the other two third observationsas non-decline enterprises in the end. In order to get a more robustresult, we do the above process respectively for each year 2007 and2008, and only those firms that both classified as decline in 2 yearsare retained. By these measures we get a sample of 107 enterpriseswith 42 decline enterprises and 65 non-decline enterprises.
  Commonly, the neural network training process requires eachdecline enterprise is matched with a non-decline enterprises. Forthis purpose, we select 42 non-decline firms from the 62 non-de-cline firm samples to match the decline firms in terms of (1) retainas much manufacturing firms as possible, and (2) select the firmswith similar sectors and capital scale to the matching declineenterprises. By this way, we get a matched sample of 84 firms inthis research, 42 non-decline firms and 42 decline firms, 67 manu-facturing sector firms and 17 non-manufacturing sector firms inFig. 2. The structure of BP used in this paper.6426 Y. Cao et al. / Expert Systems with Applications 38 (2011) 6424–6429the end. The explicit description of the sample distribution isshown in Table 2.In Table 3 we give a descriptive analysis of thevariable DP, SG, CEV and asset size of the samples.
  The last step of data preparing, we randomly divide the sampleinto training samples and testing samples by the proportion 3 : 1.Thus we get 62 training sample firms and 22 testing sample firmsrespectively.
3.2. The firm decline early-warning index systems
  Since the firm decline early warning is less researched and it ishighly related to the areas of financial distress prediction, theselection of variables to be used as candidates for participation inthe input vector was based upon prior research work mainly linkedto the topic of financial distress prediction. Following Chen, Ren,and Cao (2008), we choose 29 variables and categorized then assix major types: profitability and its quality, period expense man-agement ability, asset management ability, growth ability, sol-vency and asset constitution. The details of each types and theseindicators belong to each type are listed in Table 4.
3.3. Neural network configuration
  In this paper, we proposed a model which combined the roughset theory and neural networks to predict firm decline, which is anew problem put forwarded by the author to differ from the com-mon problem of predicting firm financial crises.
  Firstly, in order to pick out the significant independent variablesthat are informative and closely related to firm’s decline condition,the RST-based application RSES (a collection of algorithms and datastructures for rough set computations, developed at the Group ofLogic, Inst. of Mathematical, University of Warsaw, Poland) andalso the genetic reduction algorithm ( Komorowski,/hrn, & Skow-ron, 2002 ) were used.
  In the training process, we apply the Levenberg–Marquardtalgorithm to enhance the speed since it appears to be the fastestmethod for training moderate-sized feed forward neural networks.The input vector is the variables depicted above after reduced byrough set and the element of target vector is coded by 1 if the firmfalls into decline, otherwise the element of target vector is codedby 0. We adopt the performance function as the form of a sum ofsquares and start the training of network by letting the error be0.01, learning rate be 0.7 and training times be 60000. We usedMATLAB 7.0 to construct the neural network which is a powerfulsimulation platform developed by Mathworks, and is very adaptiveto simulate intelligent algorithms and to solve complex problems(Zhang et al., 2009).
3.4. Results
  We construct three BP neural network combining rough setmodels which respectively based on the data of theT-0, T-1, andT-2 years prior the firm falls into decline.
  Our BP neural networks are equipped with two hidden layers asprevious section described. To determining the number of nodes inhidden layers we adopt the following measures: First, we calculatethe scope of the number of the nodes in hidden layers by use of aRule-of-Thumb equation (Yang & Zheng, 1992):ð2 p1þ p3Þ12< p2< 2 p1þ 1Table 1Expectations for firm-specific descriptors of life cycle stages.aLife cycle stages Life cycle descriptors Assigned valueDP SG CEVGrowth Low High High 1Mature Medium Medium Medium 2Decline High Low Low 3aDP, SG, CEV refer to dividend payout, sales growth, capital expenditure dividedby value of the firm (market value of equity plus book value of long-term debt), andfirm age, respectively.Table 2The sectoral distribution of the firms.Training sample Testing sampleManufacturingsectorNon-manufacturingsectorManufacturingsectorNon-manufacturingsectorNon-decline28 3 10 1Decline 18 13 9 2Total 46 26 19 3Table 3The descriptive analysis of the samples.Training sample Testing sampleDP (%) Non-decline 27.34 28.69Decline 44.59 49.27Average 35.96 38.98SG (%) Non-decline 45.82 55.52Decline 19.70 13.91Average 32.76 33.22CEV (%) Non-decline 4.39 5.37Decline 1.94 1.43Average 3.16 3.40Asset (million Yuan) Non-decline 1889.91 1277.44Decline 619.82 318.47Average 1254.87 797.96Table 4Indicators for decline-early-warning.Category Code IndicatorProfitability and its quality X1 Return on equity (ROE)X2 Return on asset (ROA)X3 Return on assets of core businessX4 Net sales marginX5 Gross sales marginX6 Main business profit marginsX7 EBITDA/main business incomeX8 Cost rate of main businessX9 The profit of main business/total profitX10 Operating profit/main business incomePeriod expensemanagement abilityX11 Operating expense ratioX12 Management expense ratioX13 Financial expense ratioAsset management ability X14 Receivables turnoverX15 Current assets turnoverX16 Fixed assets turnoverX17 Net assets turnoverX18 Total assets turnoverGrowth ability X19 Year-on-year growth rate of mainbusiness incomeX20 Year-on-year growth rate of net profitX21 Year-on-year growth rate of total assetsX22 Year-on-year growth rate of net assetsSolvency X23 Current ratioX24 Quick ratioX25 Conservative quick ratioX26 Asset-liability ratioX27 Equity ratioX28 Net debt/net assetsAsset constitution X29 Fixed assets/total assetsY. Cao et al. / Expert Systems with Applications 38 (2011) 6424–6429 6427where p1is the number of input values, p3is the number of outputvalues and p2is the number of nodes in hidden layers. After deter-mine the scope of the number of the nodes in hidden layers, we fur-thering do the trial and error method to determine the specificnumbers of nodes in hidden layers, that is by continuously tryingto increase the number of neurons to improve the networks’ con-vergence speed and fit ability until the network achieve a conditionwith highest convergence speed and the output error meet therequirement.Through this method, we determine the structure of each neu-ral network of T-0, T-1 and T-2 period.
  The structure of the T-0 per-iod network is 4-4-1 for the first hidden layer, the second hiddenlayer and the output layer respectively. The period T-1and T-2 net-works both have 5-6-1 structure.
3.4.1. The results of T-0 years prior the firms fall into decline
  For the period T-0 models, through RST, seven variables are se-lected as potential predictor variables: return on equity (ROE)(X1), management expense ratio (X12), receivables turnover(X14), fixed assets turnover (X16), year-on-year growth rate ofmain business income (X19), asset-liability ratio (X26) and netdebt/net assets (X28), which are respectively belongs to five as-pects of profitability and its quality, period expense managementability, asset management ability, growth ability and solvency.The selected variables are taken as input variables for the neuralnetwork of period T-0.
  After training, the variables of T-0 period of the training samplesand testing samples respectively used as input vectors in the sim-ulation of the BP neural network. The results are listed in Table 5.The model have a good prediction effect in T-0 period, for trainingsample, it has a average correct prediction rate 98.4%, and for test-ing samples, it has an average correct prediction rate of 90.9%.
  In order to measure the effects of the fitness of the actual outputto expected output, we did a regression analysis to the expectedoutput on actual output, and the regression equation is:A = 1.0001 T  0.055 with R = 0.828. Generally, as long as R > 0.7,we can believe the actual output get an ideal effect of prediction.So the regression result implies that the neural network modelfor firm decline early warning of T-0 period is effective.
3.4.2. The results of T-1 years of the firm fall into decline
  For the period T-0 models, through RST, eight variables are se-lected as potential predictor variables: return on equity (ROE)(X1), EBITDA/main business income (X7), management expense ra-tio (X12), current assets turnover (X15), year-on-year growth rateof main business income (X19), year-on-year growth rate of netprofit (X20), asset-liability ratio (X26) and fixed assets/total assets(X29), which are respectively belongs to six aspects of profitabilityand its quality, period expense management ability, asset manage-ment ability, growth ability, solvency and asset constitution. Theselected variables are taken as input variables for the neural net-work of period T-1.
  After training, the variables of T-1 period of the training samplesand testing samples respectively used as input vectors in the sim-ulation of the BP neural network. The results are listed in Table 6.The model still has a good prediction accuracy to training sample,however, the prediction accuracy of testing sample confront a ra-pid decline compared to the previous T-0 model. Only one firmswhich the model predict will fall into decline the next year but stillbe non-decline, whereas 54.5% firms which the model predict willnot fall into decline in the next year actually fall into decline. Theaverage prediction accurate is 68.1%, which is a great reductioncompared to the model of T-0 period. Even though, we believeour results is still meaningful, for the type I error of the model isjust 9.1%, which means that the enterprises which the model pre-dict will fall into decline will have great probability to fall into de-cline in the next year, thus gives a sound signal to thoseenterprises’ managers and shareholders.
  The regression equation of the expected output on predictionoutput isA = 0.537T + 0.364 with R = 0.461, implying the one-yearprediction results for the firms to fall into decline stage is not verysatisfactory. Three reasons can be used to explain such a result.Firstly, since classification of the life cycle stages is a very compli-cated problem, so our method to determine if a firm be in the de-cline stage which proposed by Anthony and Ramesh (1992) maynot be accurate to some degree, especially when we exclude allST enterprises due to data missing for some variables. Secondly,the size of training and testing samples in our research may betoo limited, which could induce a relative inaccurate predictionoutput for testing samples. Thirdly, the boundary between matu-rity and decline is sometimes ambiguous and it is difficult to differ-entiate between them (Miller & Friesen, 1984), which makes theone-year prediction problem very challengeable.
3.4.3. The results of T-2 years of the firm fall into decline
  For the period T-0 models, through RST, seven variables are se-lected as potential predictor variables: net sales margin (X4), mainbusiness profit margins (X6), management expense ratio (X12), to-tal assets turnover (X18), year-on-year growth rate of main busi-ness income (X19), year-on-year growth rate of total assets(X21), and fixed assets/total assets (X29), which are respectivelybelongs to five aspects of profitability and its quality, period ex-pense management ability, asset management ability, growth abil-ity and asset constitution. The selected variables are taken as inputvariables for the neural network of period T-2.After training, the variables of T-2 period of the training samplesand testing samples respectively used as input vectors in the sim-ulation of the BP neural network. The results are listed in Table 7.Though the model still has a good prediction accurate for trainingsample, the prediction accurate of testing sample is worse. 27.3%firms the model predict to fall into decline is actually not fall intodecline two years later, and 54.5% firms that the model predict willnot fall into decline is actually fall into decline after two year. Theregression equation of the expected output on prediction output isA = 0.255T + 0.402 with R = 0.275, implying it is more difficult topredict a firm if it will fall into decline two years later. EvenTable 5The results of training sample and testing sample for the period ofT-0.Prediction classTraining sample Testing sampleNon-decline Decline Non-decline DeclineActual class Non-decline 30 1 10 1Decline 0 31 1 10Percent Non-decline 96.8% 3.2% 90.9% 9.1%Decline 0% 100% 9.1% 90.9%Average correct predictionrate98.4% 90.9%Table 6The results of training sample and testing sample for the period ofT-1.Prediction classTraining sample Testing sampleNon-decline Decline Non-decline DeclineActual class Non-decline 30 1 10 1Decline 0 31 6 5Percent Non-decline 96.8% 3.2% 90.9% 9.1%Decline 0% 100% 54.5 % 45.5%Average correct predictionrate98.4% 68.1%6428 Y. Cao et al. / Expert Systems with Applications 38 (2011) 6424–6429Table 7The results of training sample and testing sample for the period ofT-0.Prediction classTraining sample Testing sampleNon-decline Decline Non-decline DeclineActual class Non-decline 30 1 8 3Decline 0 31 6 5Percent Non-decline 100.0% 0% 72.7% 27.3%Decline 0% 100% 54.5 % 45.5%Average correct predictionrate98.4% 59.1%Y. Cao et al. / Expert Systems with Applications 38 (2011) 6424–6429 6429though, the model is still somewhat usefulness for the first type er-ror is 27.3%, which means the firms which the model predict willfall into decline two years later will have a great probability toactually fall into decline, which gives a warning signal to the man-agers, shareholders, etc. to adopt measures to prevent the firm fallinto decline.
Conclusions
  In this paper we have combined the rough set and BP neuralnetworks to construct a model for early warning of the enterprisedecline in a life cycle. This is a new problem generated from theenterprises life cycle theory and enterprises’ financial crisis earlywarning models. Before the training of BP neural networks, weadd a preprocessing layer for attribute reduction by use of roughset theory, such that the problem caused by noise data and redun-dant data can be avoided when training the networks.
  We have taken a two hidden layer BP neural network model andapply the Levenberg–Marquardt algorithm to enhance the speed ofconvergence. To determine the number of nodes in hidden layers,we first get the scope of the amount by an experience equation,and then determine the definite amount of the notes in hidden lay-ers by trial and error method. We have built 4-4-1, 5-6-1 and 5-6-1models for the period T-0, T -1 and T-2 years prior to the firm fallinginto a decline stage respectively. The prediction based on trainingsamples all yield good accuracy rates in these three different mod-els. However, the prediction based on testing sample only have agood accuracy rate inT-0 models, which means our proposed ap-proach works well in short-term prediction of whether a firm willfall into decline stage. Results also indicate that prediction ofwhether the firm will fall into decline stage within one or twoyears yields high accuracy, because the first type error of our mod-els is relatively low.
  Further research should be focusing on comparison of otherdata mining approaches and evaluation of computational perfor-mance based on large empirical data (Wu, 2010; Wu & Lee, 2010).
Acknowledgment
  This paper was supported by the National Natural Science Foun-dation of China (No. 70921001), the National Natural Science Foun-dation of China (No. 71001108), the Youth Project of theHumanities and Social Sciences Fund of the Chinese EducationMinistry (No. 09YJC790262) and the Social Science Fund of the Hu-nan Province (No. 09YBA162).
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