On Constructing Super-Models to Enhance Failure. Forecasting: A Comparative Study of Four Real. Data Sets

Transcription

1 Applied Mathematical Scieces, Vol. 12, 2018, o. 31, HIKARI Ltd, O Costructig Super-Models to Ehace Failure Forecastig: A Comparative Study of Four Real Data Sets Lutfiah Ismail Al turk Statistics Departmet Faculty of Scieces Kig Abdulaziz Uiversity Jeddah, Kigdom of Saudi Arabia Wejda Saleem Al ahmadi Mathematics Departmet Faculty of Scieces Taibah Uiversity Media, Kigdom of Saudi Arabia Copyright 2018 Lutfiah Ismail Al turk ad Wejda Saleem Al ahmadi. This article is distributed uder the Creative Commos Attributio Licese, which permits urestricted use, distributio, ad reproductio i ay medium, provided the origial work is properly cited. Abstract To ehace the predictive capability of Software Reliability Growth Models (SRGMs), researchers have proposed techique to establish Super-Models (SMs) by combiig two or more SRGMs, which have better fittig for the same software project. I this paper, No-Homogeeous Poisso Process (NHPP) model is created based o Type-II Geeralized Half-Logistic Distributio (GHLD-II). Three methods for estimatig the parameters of the NHPP GHLD-II model are cosidered i the case of failure-occurrece time data. Fudametally, our mai attetio i this work is the SM techique based o combiig predictios geerated by the NHPP GHLD-II model. A applicatio usig four published real data sets are coducted to verify the effectiveess of our proposed models based o three evaluatio criteria ad useful results are obtaied.

2 1572 Lutfiah Ismail Al turk ad Wejda Saleem Al ahmadi Keywords: Software reliability growth models, Geeralized half-logistic distributio, Maximum likelihood estimatio, No-liear least square estimatio, Weighted o-liear least square estimatio, Super-model 1. Itroductio Software reliability growth models (SRGMs) have bee used extesively i the software reliability aalysis [1-4]. Researchers have oticed that combiig more tha oe SRGM may ehace predictio precisio tha selectig a sigle model. Combig multiple SRGMs or the Super-Model (SM) techique is more geeral tha others ehacemet techiques. I this techique istead of lookig solely at previous predictios give by oe model, we lear from all or from the most recet predictios obtaied from all available models. These predictios are combied i order to predict the future failure behavior of the software. Literature witesses that combiig techique have bee effectively prove i various real software applicatios [5-8]. Hece i this paper, a NHPP software reliability model is created usig Type-II Geeralized Half-Logistic Distributio (GHLD-II) that proposed by Katam et al. [9], which hopefully will give a good descriptio of the software failure pheomea. By chagig its shape parameter, several sub-models are offered to be examied for selectio of appropriate model for a give project. The parameters of the NHPP GHLD-II model are estimated usig the Maximum Likelihood (ML), No-Liear Least Squares (NLS), ad Weighted No-Liear Least Squares (WNLS) estimatio methods. For the WNLS estimatio method three weightig fuctios are used i our study. Furthermore, several liear combiatios are formed based o either: cosiderig several values of the shape parameter the choosig the best fit resulted models as compoets or usig the five differet estimatio methods to build up the liear combiatio based o the three resulted best performace models. The reliability fuctio of the GHLD-II distributio with scale parameter ad shape parameter is give by: R(t) = ( 2e t 1+e t ), 0 < t <, > 0, > 0. (1) Accordig to Eq. (1), the correspodig Cumulative Distributio Fuctio (CDF) ca be writte as: F(t) = 1 R(t) = 1 ( 2e t 1+e t ). (2)

3 O costructig super-models to ehace failure forecastig 1573 While, by differetiatig Eq. (2) with respect to the time t, we get the Probability Desity Fuctios (PDF) of the GHLD-II distributio as follows: f(t) = df(t) dt t = (2e ). (3) (1+e ) t +1 Also, the associated hazard fuctios ca be obtaied usig Eqs. (1) ad (3) as follows: H(t) = f(t) R(t) = (1+e t ). (4) This paper is orgaized as follows: Sectio 2 displays the formulatio ad some measures of reliability of the NHPP GHLD-II model. Sectio 3 describes the ML, NLS, ad WNLS estimatio approaches of the NHPP GHLD-II model. Sectio 4 itroduces the SMs costructio algorithm ad possible methods for selectig weight for each model that with i the SM compositio. Sectio 5 illustrates the evaluatio criteria that will be used i our evaluatio study. Sectio 6 shows real data applicatio. Sectio 7 displays the coclusio of this paper. 2. The NHPP GHLM-II Model I this sectio, the formulatio ad some essetial characteristics of the NHPP GHLD-II model is preseted. The NHPP GHLD-II model will be costructed by Lyu [10] as follows: The mea value fuctio of the NHPP GHLD-II model ca be calculated as: m(t) = af(t) t = a (1 ( 2e 1+e t Ad, the failure itesity fuctio is obtaied by: λ(t) = af(t) = a ) ). (5) (2e t ) +1, (6) (1+e t )

4 1574 Lutfiah Ismail Al turk ad Wejda Saleem Al ahmadi where a is the umber of iitial faults i the software, is the scale parameter ad is the shape parameter of the NHPP GHLD-II model. By usig Eq. (5), the umber of remaiig errors takes the followig formula: (t) = a m(t) t = a ( 2e 1+e t ). (7) By usig Eqs. (6) ad (7), we ca get the error detectio rate as follow: d(t) = λ (t) (t) = (1+e t ). (8) The MTBF of the NHPP GHLD-II model ca be foud from Eq. (6) as: MTBF = 1 λ(t) = (1+e t +1 ) a(2e t. (9) ) Accordig to Eq. (5), the coditioal reliability fuctio is: R(x t) = exp[ (m (t + x) m(t))] t+x = exp [ 2a + a(( 2e 1+e t+x ) t + ( 2e 1+e t ) )]. (10) 3. Parameter Estimatio of the NHPP GHLM-II Model I this sectio, differet methods to estimate the parameters of the NHPP GHLD- II model are give. 3.1 The Maximum Likelihood Estimatio (MLE) method Let T i is a radom variable represetig the time betwee (i-1) st ad i th failure, i The, S i = i=1 T i is a radom variable idicatig the i th failure occurrece time where T i = S i S i 1 (i = 1, 2,, ; S 0 = 0) the joit desity of the ukow parameters, Θ, of a NHPP model with m(s ; Θ) is give by:

5 O costructig super-models to ehace failure forecastig 1575 L MLE (S Θ) = e m(s;θ) i=1 λ(s i ; Θ). (11) Now, for estimatig the ukow three parameters a, ad of the NHPP GHLD- II model usig the MLE method ad based o the data of failure occurrece time s i (i = 1, 2,, ; 0 s 1 s 2 s ), we substitute Eqs. (5) ad (6) i Eq. (11), the by takig the atural logarithm of both sides we obtai: s l L MLE (S a,, ) = a + a [ 2e 1+e s ] + l a + l l + l 2 i=1 s i ( + 1) l(1 + e s i ) i=1. (12) Takig the partial derivative of Eq. (12) with respect to a, ad we get: { l L MLE (S a,,) l L MLE (S a,,) a l L MLE (S a,,) = a s (2e s ) 2(1+e s ) +1 s = 1 + [ 2e 1+e s ] +. a = l 2 + i=1 s i + l (1 + e s i ) + i=1 s i 2 i=1. ( + 1) s i e s i i=1. 2(1+e s i ) (13) By settig the derivatives equal to 0, the followig o-liear equatios are obtaied: { l L MLE (S a,,) l L MLE (S a,,) = a s (2e s ) 2(1+e s ) +1 a = 1 [ 2e s 1+e s ] = l 2 + i=1 s i +. + l (1 + e s i i=1 ) = 0. i=1 s i 2 s i e s i ( + 1) i=1 = 0. 2(1+e s i ) (14)

6 1576 Lutfiah Ismail Al turk ad Wejda Saleem Al ahmadi Solvig the secod ad third expressios of Eq. (14) by applyig umerical methods, ad will be obtaied, the by substitutig them i the first expressio of Eq. (14), a is foud. 3.2 The No-Liear Least Squares Estimatio (NLSE) method The No-liear Least Square Estimatio (NLSE) method aim to miimize the followig fuctio: S NLSE (Θ) = i=1 [i m( ; Θ))] 2, (15) where Θ is the parameters of a NHPP model, ad m( ; Θ)) is its mea value fuctio. By substitutig Eq. (5) i Eq. (15) we obtai: S NLSE (a,, ) = (i a + a ( 2e i=1 ) ), (16) 1+e 2 Takig the partial derivative of Eq. (16) with respect to a, ad we obtai: S NLSE (S a,,) a S NLSE (S a,,) ) (i 2a + a ( 2e i=1 ) )]. 1+e 1+e = 2 [ i + a + ( 2e ) ) a ( 2e ) l ( 2e i=1 )]. 1+e 1+e 1+e = 2 [(i a + a ( 2e { S NLSE (S a,,) = 2a ti (2e [ 2 t i ) +1 (i a + a ( 2e i=1 ) )]. (1+e ) 1+e By settig the derivatives equal to 0, the followig o-liear equatios are obtaied: (17)

7 O costructig super-models to ehace failure forecastig 1577 S NLSE (Θ) a = t i [i i( 2e i=1 ) ] 1+e t i t 2 i [1 2( 2e ) +( 2e i=1 ) ] 1+e 1+e 1+e = 2 [(i a + a ( 2e ) ) a ( 2e ) l ( 2e i=1 )] = 0. 1+e. 1+e { S NLSE (Θ) = 2a t i ) t [ i (2e 2 +1 (i a + a ( 2e i=1 ) )] = 0. (1+e ) 1+e Solvig the secod ad third expressios of Eq. (18) by applyig umerical methods, ad will be obtaied, the by substitutig them i the first expressio of Eq. (18), a is foud. (18) 3.3 The Weighted No-Liear Least Squares Estimatio (WNLSE) method The Weighted No-liear Least Square Estimatio (WNLSE) method aim to miimize the followig fuctio: S WNLSE (Θ) = i=1 w i [i m( ; Θ))] 2, (19) where Θ is the parameters of a NHPP model, m( ; Θ)) is its mea value fuctio, ad w i (i = 1,2,, ) are positive weights. By substitutig Eq. (5) i Eq. (19) we obtai: S WNLSE (a,, ) = w i (i a + a ( 2e i=1 ) ), (20) 1+e Takig the partial derivative of Eq. (20) with respect to a, ad we obtai: 2

8 1578 Lutfiah Ismail Al turk ad Wejda Saleem Al ahmadi S WNLSE (a,,) a S WNLSE (a,,) ) (i 2a + a ( 2e i=1 ) )]. 1+e 1+e = 2 w i [ i + a + ( 2e ) ) a ( 2e ) l ( 2e i=1 )]. 1+e 1+e 1+e = 2 w i [(i a + a ( 2e { S WNLSE (a,,) = 2a (2e t ) i +1 (i a + a ( 2e i=1 ) )]. (1+e ) 1+e 2 w i [ By settig the derivatives equal to 0, the followig o-liear equatios are obtaied: (21) S WNLSE (a,,) a = i=1 i=1 w i [i i( 2e w i [1 2( 2e 1+e 1+e 1+e ) ] t 2 i ) +( 2e ) ] 1+e = 2 w i [(i a + a ( 2e ) ) a ( 2e ) l ( 2e i=1 )] = 0. (22) 1+e. 1+e { S WNLSE (a,,) = 2a (2e ) w 2 i [ +1 (i a + a ( 2e i=1 ) )] = 0. (1+e ) 1+e Solvig the secod ad third expressios of Eq. (22) by applyig umerical methods, ad will be obtaied, the by substitutig them i the first expressio of Eq. (22), a is foud. Accordig to the work of Ishii et al. [11], the weightig fuctio w i ca be defied as: w i = 1 m( ;Θ)) i=1m( ;Θ)). (23)

9 O costructig super-models to ehace failure forecastig 1579 For our applicatio, three weightig fuctios for estimatig the parameters of the NHPP GHLM-II model usig the WNLSE method are cosidered: the first weight is obtaied by substitutig Eq. (5) i Eq. (23) as follows: [( 1 e w i(1) = ) ). (24) 1+e ( 1+e i=1 ] 1 e The secod ad third weightig fuctio is empirically suggested as follows: [( 1 e w i(2) = ) ). (25) 1+e ( 1+e i=1 ] 1 e w i(3) = [( 1 e 1+e ) ( 1+e i=1 ) ] 1 e. (26) 4. Buildig Super-Models (SMs) This sectio is iteded as a guide to buildig Super-Models (SMs). Buildig these models is based o predictios attaied from all cosidered models alog with assigig suitable weights to these predictios. We first preset efficiet algorithm for steps of costructig the SMs. The, some weighig methods are reviewed SM costructio algorithm For givig better reliability predictios to the failure data, the SM techique is proposed to establish ew reliability models by combiig two or more SRGMs, which have better fittig effects with assigig proper weight to each icluded SRGM. The followig algorithm illustrates the steps of costructig SM i detail. Is formulized as liear combiatio of several SRGMs predictio as follow [12]: (i) (ii) (iii) (iv) Choose a set of SRGMs with good predictive ability so their biased predictios ca be egated, Keep track of the failure data with all the SRGMs, Assig appropriate weight to the selected compoet SRGMs, Form oe or several liear combiatio models for fial predictio.

10 1580 Lutfiah Ismail Al turk ad Wejda Saleem Al ahmadi 4.2. Weight criteria As metioed above, the SM techique gives each SRGM a specific weight, either a fixed umber or a fuctio i the previous predictio performace of these SRGMs. There are several methods suggested i literature to select the weight, oe of them deped o the values of the prequetial likelihood fuctios as calculated for each SRGM. The disadvatage of this approach is that whe the umber of parameters is more tha three parameters is difficult to be implemeted ad it cosumes a log time. Aother approach has bee suggested based o a Bayesia iferece weighted decisio [8]. Also, Lyu ad Nikora [12] formulated four SMs as follows: [1] Equally-weighted Liear Combiatio (ELC) model that use a arithmetic mea for the predictios of SRGMs as a predictio of SMs for each time. [2] Media-orieted Liear Combiatio (MLC) model that use the media value for the predictios of SRGMs as a predictio of SMs for each time. [3] Uequally-weighted Liear Combiatio (ULC) model that determied the weighig similar of Program Evaluatio ad Review Techique (PERT). [4] Fourth formula is Dyamically-weighted Liear Combiatio (DLC) model that use meta-predictor to form a SM from several SRGMs with the weightig chose i some optimal way e.g. posterior probability. 5. Goodess-of-Fit Measures The Mea of Square Errors (MSE), Sum of Square Errors (SSE), ad variace criteria are used for the evaluatio purpose i our applicatio. These criteria illustrate the variatio betwee the actual ad predicted values. The lower the criteria value, the better model performace. These criteria are described as follows. MSE= (y i m ( )) 2 i=1, (27) k SSE= i=1 (y i m ( )) 2, (28) Variace= ( 1 1 ) (y i m ( ) Bias) 2 i=1, where (29) m ( ) is the estimated cumulative umber of faults at time. y i is the total umber of faults observed at time. Bias is the average of predictio error, ad give by:

11 O costructig super-models to ehace failure forecastig 1581 Bias= i=1 [m ( ) y i ]. : The umber of faults. k: The umber of model parameters. 6. Numerical Applicatio I this sectio, four sets of failure data are used as iputs to our suggested reliability predictio models. The MLE, NLSE, ad WNLSE methods are cosidered for parameter estimatio. For the WNLSE method three differet weightig fuctio are studied. Eight super-models are costructed ad compared with its compoets based o three evaluatio criteria Real data sets I our applicatio, four real data sets (DS1, DS2, DS3, ad DS4) are used to examie the performace of the costructed SMs. DS1: represets 34 software failures of Navel Tactical Data System (NTDS), preseted i [13]. DS2: is from the procedure of a failures cotrol chart for failure software process, used by [14], the data cosists of 30 software failures. DS3: gives the time betwee 136 failures of a software product, used i [15]. DS4: cosists of time uit ad 41 failures, itroduced by [16]. The four data sets are summarized i Tables [1-4]. Failure Number Table 1: Failure time data (DS1). Failure Time (Days) Failure Number Failure Time (Days)

12 1582 Lutfiah Ismail Al turk ad Wejda Saleem Al ahmadi Table 2: Failure time data (DS2). Failure Time (Hours) Failure Number Failure Time (Hours) Failure Number Table 3: Failure time data (DS3). Failure Time (Hours) Failure Number Failure Time (Hours) Failure Number Failure Time (Hours) Failure Number Failure Time (Hours) Failure Number

13 O costructig super-models to ehace failure forecastig Table 4: Failure time data (DS4). Failure Number Failure Time (Hours) Failure Number Failure Time (Hours)

14 1584 Lutfiah Ismail Al turk ad Wejda Saleem Al ahmadi 6.2. Buildig SMs based o the NHPP GHLD-II model I this sectio, we apply eight liear combiatios of some geerated sub-models of the NHPP GHLD-II model usig four real data sets. Our sub-models are formed by cosiderig two cases. I the two cases, we costruct our SMs based o the work of [5]. Specifically, we select the ULC model to formulate our SMs. The geeral mathematical formula of the ULC model is give by: SM = 1 O + 4 M + 1 P, (30) where O: represets a optimistic predictio, M: represets a media predictio ad P: represets a pessimistic predictio. The two cases we cosider are illustrated as follows: Case I: All the model s parameters are ukow I this case four SMs are formed through the followig steps: Step1: Assume that all the three parameters (a, ad ) of the NHPP GHLD-II model are ukow. Step2: Apply the MLE, NLSE, WNLSE methods to estimate the model s parameters, (accordig to this step five model predictios will be obtaied). Step3: Select the optimistic, media, ad pessimistic predictio models from Step 2. Step4: Form the SM by substitutig the three selected predictio models i Eq. (30). Step5: Repeat step1 to step 4 for each data sets, as a result we obtai four SMs (SM1, SM2, SM3, ad SM4). Case II: Two of the model s parameters are ukow, ad oe kow I this case four SMs are formed through the followig steps: Step1: Assume assumig two ukow parameters(a ad ) ad a kow shape parameter with the values (0.8, 1, 1.2, ad 1.5). Step2: Apply the MLE, NLSE, WNLSE methods to estimate the model s parameters (accordig to this step twety predictio models will be obtaied). Step3: Select the optimistic, media, ad pessimistic predictio models from Step 2. Step4: Form the SM by substitutig the three selected predictio models i Eq. (30).

15 O costructig super-models to ehace failure forecastig 1585 Step5: Repeat Step1 to Step 4 for each data sets, as a result we obtai four SMs (SM5, SM6, SM7, ad SM8) Results Our umerical results will be illustrated i this sectio as follows: Case I results: The ML, NLS, ad WNLS estimates of the parameters of the five predictio models with its evaluatio criteria for each of the studied data sets: DS1, DS2, DS3, ad DS4, are summarized i Tables [5-8]. Table 5: The NHPP GHLD-II model parameter estimatio ad evaluatio criteria usig the MLE method. Data sets Ds1 Ds2 Ds3 Ds4 Parameters estimates Model selectio criteria a MLE MLE MLE SSE MSE Variace Table 6: The NHPP GHLD-II model parameter estimatio ad evaluatio criteria usig the NLSE method. Data sets Ds1 Ds2 Ds3 Ds4 Parameters estimates Model selectio criteria a NLSE NLSE NLSE SSE MSE Variace

16 1586 Lutfiah Ismail Al turk ad Wejda Saleem Al ahmadi Table 7: The NHPP GHLD-II model parameter estimatio usig the WNLSE method. Parameters estimates a WLSE (w i(1) ) WLSE (w i(1) ) WLSE (w i(1) ) Data sets Ds1 Ds2 Ds3 Ds a WLSE (w i(2) ) WLSE (w i(2) ) WLSE (w i(2) ) a WLSE (w i(3) ) WLSE (w i(3) ) WLSE (w i(3) ) Table 8: The evaluatio criteria of the NHPP GHLD-II model based o the WNLSE method. Parameters estimates Data sets SSE MSE Variace a WLSE (w i(1) ) Ds Ds Ds Ds a WLSE (w i(2) ) Ds Ds Ds Ds a WLSE (w i(3) ) Ds Ds Ds Ds Accordig to Tables [5-8] the three best predictios models amog the five predictio models for each of the four selected data sets are chose as compoets to costruct the first four SMs (SM1, SM2, SM3, ad SM4) usig Eq. (30) as follows: SM1 = 1 6 O (WNLSE (wi(3) ) ) M (NLSE) P (WNLSE (wi(2) ) ). (31)

17 O costructig super-models to ehace failure forecastig 1587 SM2 = 1 6 O (NLSE) M (WNLSE (wi(1) ) ) P (WNLSE (wi(3) ) ). (32) SM3 = 1 6 O (WNLSE (wi(1) ) ) M (WNLSE (wi(2) ) ) p (WNLSE (wi(3) ) ). (33) SM4 = 1 6 O (NLSE) M (WNLSE (wi(1) ) ) P (WNLSE (wi(2) ) ). (34) The predictio results of the four SMs alog with the predictio results of their three compoets for the last 16 failures based o DS1, DS2, DS3, ad DS4, respectively, are summarized i Table 9, Table 11, Table 13 ad Table 15 while the evaluatio criteria results of the four predictio models ad their compoets are summarized i Table 10, Table 12, Table 14, ad Table 16. Table 9: The predictio results of the SM1 ad its compoets. Actual O (WNLSE(wi(3) ) ) M (NLSE) P (WNLSE(wi(2) ) ) SM

18 1588 Lutfiah Ismail Al turk ad Wejda Saleem Al ahmadi Table 10: The evaluatio criteria results of the SM1 ad its compoets. Evaluatio Criteria NHPP GHLD-II O (WNLSE(wi(3) ) ) M (NLSE) P (WNLSE(wi(2) ) ) SM1 SSE MSE Variace For DS1: Accordig to Table 10, the SSE ad MSE values show that the SM1 is more efficiet tha P (WNLSE(wi(2) ) ) ad less efficiet tha O (WNLSE(wi(3) ) ) ad M (NLSE). While the values of the variace show that the SM1 is more efficiet tha the O (WNLSE(wi(3) ) ) ad M (NLSE) ad less efficiet tha P (WNLSE(wi(2) ) ). Table 11: The predictio results of the SM2 ad its compoets. Actual O (NLSE) M (WNLSE(wi(1) ) ) P (WNLSE(wi(3) ) ) SM Table 12: The evaluatio criteria results of the SM2 ad its compoets. Evaluatio Criteria NHPP GHLD-II O (NLSE) M (WNLSE(wi(1) ) ) P (WNLSE(wi(3) ) ) SM2 SSE MSE Variace

19 O costructig super-models to ehace failure forecastig 1589 For DS2: Table 12 shows that accordig to the SSE ad MSE values, the SM2 is more efficiet tha P (WNLSE(wi(3) ) ) ad less efficiet tha O (NLSE) ad M (WNLSE(wi(1) ) ). The values of the variace show that the SM2 is more efficiet tha (O (NLSE) ad M (WNLSE(wi(1) ) ) ), ad less efficiet tha P (WNLSE(wi(3) ) ). Table 13: The predictio results of the SM3 ad its compoets. Actual O (WNLSE(wi(1) ) ) M (WNLSE(wi(2) ) ) P (WNLSE(wi(3) ) ) SM Table 14: The evaluatio criteria results of the SM3 ad its compoets. NHPP GHLD-II Evaluatio O (WNLSE(wi(1) ) Criteria M (WNLSE(wi(2) ) ) P (WNLSE(wi(3) ) ) SM3 SSE MSE Variace For DS3: Table 14 illustrates that accordig to the SSE ad MSE results, the SM3 has the highest values amog its compoets; cosequetly, the SM3 is the least efficiet model for this data set while it has the same predictive capability as its compoets accordig to the variace criteria.

20 1590 Lutfiah Ismail Al turk ad Wejda Saleem Al ahmadi Table 15: The predictio results of the SM4 ad its compoets. Actual O (NLSE) M (WNLSE(wi(1) ) ) M (WNLSE(wi(2) ) ) SM Table 16: The evaluatio criteria results of the SM4 ad its compoets. Evaluatio Criteria NHPP GHLD-II O (NLSE) M (WNLSE(wi(1) ) ) P (WNLSE(wi(2) ) ) SM4 SSE MSE Variace For DS4: Table 14 illustrates that accordig to the values of the SSE, MSE, ad variace, the SM4 is more efficiet tha P (WNLSE(wi(2) ) ) ad less efficiet tha (O (NLSE) ad M (WNLSE(wi(1) ) )). Case II results: The ML, NLS, ad WNLS estimates of the parameters of the twety predictio models with their evaluatio criteria for each of the studied data sets: DS1, DS2, DS3, ad DS4, are summarized i Tables [17-22].

21 O costructig super-models to ehace failure forecastig 1591 Table 17: Estimates of the parameters of some sub-models of the NHPP GHLD- II model usig the MLE method. Model M 1 = 0.8 M 2 = 1 M 3 = 1.2 M 4 = 1.5 Parameters estimates a MLE MLE a MLE MLE a MLE MLE a MLE MLE Data set Ds1 Ds2 Ds3 Ds Table 18: The evaluatio criteria of some sub-models of the NHPP GHLD-II model usig the MLE method. Model Data sets SSE MSE Variace Ds M 1 Ds = 0.8 Ds Ds M 2 = 1 M 3 = 1.2 M 4 = 1.5 Ds Ds Ds Ds Ds Ds Ds Ds Ds Ds Ds Ds Table 19: Estimates of the parameters of some sub-models of the NHPP GHLD-II model usig the NLSE method. Model M 1 = 0.8 M 2 = 1 Parameters estimates a MLE MLE a MLE MLE Data set Ds1 Ds2 Ds3 Ds

22 1592 Lutfiah Ismail Al turk ad Wejda Saleem Al ahmadi Table 19: (Cotiued): Estimates of the parameters of some sub-models of the NHPP GHLD-II model usig the NLSE method. M 3 = 1.2 M 4 = 1.5 a MLE MLE a MLE MLE Table 20: The evaluatio criteria of some sub-models of the NHPP GHLD-II model usig the NLSE method. Model Data set SSE MSE Variace Ds M 1 Ds = 0.8 Ds Ds M 2 = 1 M 3 = 1.2 M 4 = 1.5 Ds Ds Ds Ds Ds Ds Ds Ds Ds Ds Ds Ds Table 21: Estimates of the parameters of some sub-models of the NHPP GHLD-II model usig the WNLSE method. Model Parameters Data sets estimates Ds1 Ds2 Ds3 Ds4 a WNLSE (w i(1) ) WNLSE (w i(1) ) M 1 a WNLSE (w i(2) ) = 0.8 WNLSE (w i(2) ) a WNLSE (w i(3) ) WNLSE (w i(3) ) a WNLSE (w i(1) ) WNLSE (w i(1) ) M 2 = 1

23 O costructig super-models to ehace failure forecastig 1593 M 3 = 1.2 M 4 = 1.5 a WNLSE (w i(2) ) WNLSE (w i(2) ) a WNLSE (w i(3) ) WNLSE (w i(3) ) a WNLSE (w i(1) ) WNLSE (w i(1) ) a WNLSE (w i(2) ) WNLSE (w i(2) ) a WNLSE (w i(3) ) WNLSE (w i(3) ) a WNLSE (w i(1) ) WNLSE (w i(1) ) a WNLSE (w i(2) ) WNLSE (w i(2) ) a WNLSE (w i(3) ) WNLSE (w i(3) ) Table 22: The evaluatio criteria of some sub-models of the NHPP GHLD-II model usig the WNLSE method. Model M 1 = 0.8 M 2 = 1 Parameters estimates a WNLSE (w i(1) ) WNLSE (w i(1) ) a WNLSE (w i(2) ) WNLSE (w i(2) ) a WNLSE (w i(3) ) WNLSE (w i(3) ) a WNLSE (w i(1) ) WNLSE (w i(1) ) a WNLSE (w i(2) ) WNLSE (w i(2) ) Data set SSE MSE Variace Ds Ds Ds Ds Ds Ds Ds Ds Ds Ds Ds Ds Ds Ds Ds Ds Ds Ds Ds Ds

24 1594 Lutfiah Ismail Al turk ad Wejda Saleem Al ahmadi Table 22: (Cotiued): The evaluatio criteria of some sub-models of the NHPP GHLD-II model usig the WNLSE method. M 3 = 1.2 M 4 = 1.5 a WNLSE (w i(3) ) WNLSE (w i(3) ) a WNLSE (w i(1) ) WNLSE (w i(1) ) a WNLSE (w i(2) ) WNLSE (w i(2) ) a WNLSE (w i(3) ) WNLSE (w i(3) ) a WNLSE (w i(1) ) WNLSE (w i(1) ) a WNLSE (w i(2) ) WNLSE (w i(2) ) a WNLSE (w i(3) ) WNLSE (w i(3) ) Ds Ds Ds Ds Ds Ds Ds Ds Ds Ds Ds Ds Ds Ds Ds Ds Ds Ds Ds Ds Ds Ds Ds Ds Ds Ds Ds Ds Accordig to Tables [17-22] the three best predictios models amog the twety predictio models for each of the four data sets are chose as compoets to costruct aother four SMs (SM5, SM6, SM7, ad SM8) usig Eq. (30) as follows: SM5 = 1 6 O M1 (NLSE) M M1 (WNLSE (wi(3) ) ) P M1 (WNLSE (wi(2) ) ). (35) SM6 = 1 6 O M1 (NLSE) M M1 (WNLSE (wi(3) ) ) P M1 (WNLSE (wi(2) ) ). (36) SM7 = 1 6 O M1 (NLSE) M M1 (WNLSE (wi(3) ) ) P M1 (WNLSE (wi(2) ) ). (37) SM8 = 1 6 O M4 (NLSE) M M4 (WNLSE (wi(3) ) ) P M3 (NLSE). (38)

25 O costructig super-models to ehace failure forecastig 1595 Table 23: The predictio results of the SM5 ad its compoets. Actual O M1 (NLSE) M M1 (WNLSE(wi(3) ) ) P M1 (WNLSE(wi(2) ) ) SM Table 24: The evaluatio criteria results of the SM5 ad its compoets. Evaluatio Criteria The NHPP GHLD-II model O M1 (NLSE) M M1 (WNLSE(wi(3) ) ) P M1 (WNLSE(wi(2) ) ) SM5 SSE MSE Variace For DS1: Table 24 shows that, accordig to the values of the SSE ad MSE, the SM5 is more efficiet tha P M1 (WNLSE(wi(2) ) ) ad less efficiet tha (O M1 (NLSE) ad M M1 (WNLSE(wi(3) ) ) ). While the values of the variace show that the SM5 is less efficiet tha P M1 (WNLSE(wi(2) ) ) ad more efficiet tha (O M1 (NLSE) ad M M1 (WNLSE(wi(3) ) ) ).

26 1596 Lutfiah Ismail Al turk ad Wejda Saleem Al ahmadi Table 25: The predictio results of the SM6 ad its compoets. Actual O M1 (NLSE) M M1 (WNLSE(wi(3) ) ) P M1 (WNLSE(wi(2) ) ) SM Table 26: The evaluatio criteria results of the SM6 ad its compoets. Evaluatio The NHPP GHLD-II model Criteria O M1 (NLSE) M M1 (WNLSE(wi(3) ) ) P M1 (WNLSE(wi(2) ) ) SM6 SSE MSE Variace For DS2: Table 26 shows that, accordig to the values of the SSE, MSE, the SM6 is more efficiet tha P M1 (WNLSE(wi(2) ) ) ad less efficiet tha (O M1 (NLSE) ad M M1 (WNLSE(wi(3) ) ) ). While the values of the variace show that the SM6 is less efficiet tha P M1 (WNLSE(wi(2) ) ) ad more efficiet tha (O M1 (NLSE) ad M M1 (WNLSE(wi(3) ) ) ). Table 27: The predictio results of the SM7 ad its compoets. Actual O M1 (NLSE) M M1 (WNLSE(wi(3) ) ) P M1 (WNLSE(wi(2) ) ) SM

27 O costructig super-models to ehace failure forecastig 1597 Table 27: (Cotiued): The predictio results of the SM7 ad its compoets Table 28: The evaluatio criteria results of the SM7ad its compoets. Evaluatio The NHPP GHLD-II model Criteria O M1 (NLSE) M M1 (WNLSE(wi(3) ) ) P M1 (WNLSE(wi(2) ) ) SM7 SSE MSE Variace For DS3: Table 28 shows that, accordig to the SSE ad MSE values, the SM7 has the lowest value amog its compoets predictios; cosequetly, the SM7 is the most efficiet model for this data set while the variace show that the SM7 is less efficiet tha P M1 (WNLSE(wi(2) ) ) ad more efficiet tha (O M1 (NLSE) ad M M1 (WNLSE(wi(3) ) ) ). Table 29: The predictio results of the SM8 ad its compoets. Actual O M4 (NLSE) M M4 (WNLSE(wi(3) ) ) P M3 (NLSE) SM

28 1598 Lutfiah Ismail Al turk ad Wejda Saleem Al ahmadi Table 30: The evaluatio criteria results of the SM8 ad its compoets. Evaluatio Criteria The NHPP GHLD-II model O M4 (NLSE) M M4 (WNLSE(wi(3) ) ) P M3 (NLSE) SM8 SSE MSE Variace For DS4: Accordig to Table 30, the values of the SSE, MSE ad variace show that the SM8 is more efficiet tha P M3 (NLSE) ad less efficiet tha (O M4 (NLSE) ad M M4 (WNLSE(wi(3) ) ) ). 7. Cocludig Remarks I this paper, the SM approach for the NHPP GHLD-II model have bee cosidered. Eight liear combiatio models have bee formed based o the work of [5]. Our proposed SMs have bee validated through real data applicatio. Due to various choices of method of estimatio, goodess-of-fit tests, values of the model s shape parameter, ad real failure data sets 72 compariso scearios have bee geerated. Accordig to these scearios, the followig poits are cocluded: Differet evaluatio criteria give differet selected proper predictio models, this idicates choosig the correct model evaluatio process ca be difficult ad eed more cosideratio. Case II which icludes the geeratio of four SMs by assumig the value of the shape parameter to be kow gives higher values of the evaluatio criteria ad hece worse performace of its costructed SMs tha the costructed SMs i Case I. Amog all the geerated 72 compariso scearios, 30 scearios reveal that the SM techique ehace the predictio model performace: 11 i Case I ad 19 i Case II ad 3 cases give similar predictio accuracy of the SM as some of its compoets. The rest of the cosidered cases show that the SM is less efficiet tha some of its compoets. Also, i most of the cases, the results show that the NLSE ad WNLSE methods has better evaluatio criteria tha the MLE method, while the NLSE ad WNLSE show approximately similar behavior especially for the case whe the weightig fuctio is w i(3). The weightig fuctio w i(1), w i(2) ad w i(3) show very close evaluatio criteria values with little prefereces for the sake of w i(3) i most of the cases, the w i(2) takes the secod-best choice. Thus, the weightig fuctio provides a foudatio for more ivestigatio Overall, the SM techique gives good results i our applicatio, but still more work eed to be doe i this area.

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