Aero-Material Consumption Prediction Based on Linear Regression Model

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1 Avalable ole at SceceDrect Proceda Computer Scece 3 () 5 3 th Iteratoal Cogress of Iformato ad Commucato Techology (ICICT-) Aero-Materal Cosumpto Predcto Based o Lear Regresso Model Yamg Yag 一, Lulu Su, Chaora Guo Naval Avato Uversty Qgdao Campus, Qgdao,, PR Cha Abstract It s dspesable to scetfcally predct the cosumpto of aero-materal spare parts ad to make scetfc decsos o avato equpmet mateace resources ad make full use of exstg resources to mprove mateace capablty. I ths paper, the mathematc model ad calculato method of lear regresso model are troduced. Ad the parameter estmato ad model test method of lear regresso model s dscussed. A learzato method s proposed for olear problems. The applcato of lear regresso model forecastg the cosumpto of aero-materal spare parts s aalyzed by examples. Fally, the regresso equato s aalyzed for sgfcace aalyss, varace aalyss ad resdual aalyss. Accordg to the aalyss results, the regresso equato s modfed as ecessary to further mprove the predcto accuracy. The results show that the lear regresso model s feasble ad effectve for the predcto of aero-materal spare parts cosumpto. The Authors. Publshed by Elsever B.V. The Authors. Publshed by Elsever Ltd. Ths s a ope access artcle uder the CC BY-NC-ND lcese ( Selecto ad peer-revew uder resposblty of the scetfc commttee of the th Iteratoal Cogress of Iformato ad Commucato Techology. Keywords: aero-materal cosumpto, lear regresso model, parameter estmato, model test, resdual aalyss, predcto;. Itroducto The aero-materal spare parts are essetal materal bass for avato equpmet mateace egeerg. Wth the developmet of avato equpmet, more ad more complex equpmet, mateace of spare parts requred for the varety ad quatty are more ad more spare parts facg, supply ad storage process s also more complex. I ths paper, we use lear regresso method to forecast the cosumpto of spare parts []. The regresso aalyss ad forecastg method s based o the aalyss of the correlato betwee depedet varables ad depedet varables, the regresso model betwee varables s bult, ad the regresso model s used as the forecastg method. There are a lot of ways of regresso aalyss. Depedg o the umber of depedet varables the relatoshp, * Correspodg author. Tel.: E-mal address:yymqd@.com The Authors. Publshed by Elsever B.V. Ths s a ope access artcle uder the CC BY-NC-ND lcese The Authors. Publshed by Elsever Ltd. Ths s a ope access artcle uder the CC BY-NC-ND lcese ( Selecto ad peer-revew uder resposblty of the scetfc commttee of the th Iteratoal Cogress of Iformato ad Commucato Techology./j.procs...7

2 Yamg Yag et al. / Proceda Computer Scece 3 () 5 3 the regresso models ca be dvded to smple regresso aalyss ad multvarate regresso aalyss. Depedg o the correlato betwee depedet varables ad depedet varables, the regresso models ca be dvded to lear regresso forecastg ad olear regresso forecastg. Ths paper focuses o smple lear regresso predcto.. The prcple of smple lear regresso model The lear regresso s a lear method used to smulate the relatoshp betwee oe depedet varable ad may explaatory varables. The case of oe explaatory varable s called smple lear regresso model, whle the case of multple explaatory varables s called multvarate lear regresso model [-]. As a commoly used statstcal methods ad for ts prcples s clear, model s smple ad easy to use, classcal lear regresso model has bee a very wde rage of applcatos the avato equpmet mateace ad support... Regresso model The smple lear regresso model s based o the approxmate lear relatoshp betwee a depedet varable ad a depedet varable, ad s ftted wth a lear equato to predct the lear equato. A smple lear regresso model s [5]: y a bx () where y s the forecast object, kow as depedet varables or explaatory varables; x s the fluecg factor, kow as depedet varables or explaatory varables; a, b for the pedg regresso coeffcet... Parameter estmato Estmatg the parameters a, b the model, from the pot of vew of curve fttg, least square method ca be adopted. Suppose you have collected pars of data that predct the target y ad the fluecg factor x: ( x, y) (,..., ). After aalyzg the hstorcal data, assumg a lear relatoshp betwee y ad x, you ca use the regresso model of equato (). Applyg the least squares method: ( x x)( y y) b ( x x) a y bx () where x x ; y y..3. Model test After the regresso model s establshed, whether the model ca be used for predcto or ot, the model test s also eeded. Commo statstcal tests are stadard devato test ad correlato coeffcet test. () Stadard devato test. Stadard devato s, used to test the accuracy of regresso predcto model, the formula s:

3 Yamg Yag et al. / Proceda Computer Scece 3 () s y y ˆ (3) where y s the predcted object actual value of the estmated value (or aalog value). ˆ The stadard devato s reflects the average error betwee the estmated ad actual values obtaed by the regresso predcto model, so the smaller the value of s, the better. I geeral, the followg requremets should be met: s % ~ 5% y () () Correlato coeffcet test. The correlato coeffcet s used to test the sgfcace of the lear correlato betwee two varables, whch s calculated as: r y yˆ y y (5) It s easy to see that whe r=l, the actual value completely falls o the regresso le, ad y s completely related to x; whe r=, y ad x are completely ucorrelated; whe <r<l, y ad x has a certa correlato. Geerally oly whe r s close to, we ca descrbe the relatoshp betwee y ad x usg a lear regresso model. To what extet s r, regresso predcto model has practcal sgfcace? The actual test s through the crtcal correlato coeffcet r (usually take a sgfcat level.5 ) to determe, ths process s called correlato test... Learzato of olear problems I practcal problems, sometmes the relatoshp betwee y ad x s ot ecessarly a lear relatoshp, but a certa curve relatoshp. For such problems, we should geerally use curvlear regresso to descrbe them, but drectly fdg regresso curves s rather dffcult. For some specal cases, t ca be treated by varable replacemet for lear regresso problems. To a olear regresso problem to lear regresso, we must frst determe (or approxmate) type of olear fuctos, ad the see f you ca use the lear varable replacemet, geeral; determe the type of olear fucto s ot easy, but some problems ca be determed wth professoal kowledge ad experece, or by mathematcal method estmated. If the scatter plot of measured data s roughly aroud the followg curve, that s, roughly showg a certa curve, t ca take the correspodg replacemet ad tur t to a lear regresso problem. The learzato methods of several commo curves are as follows. The power fucto: y a x b () Take the logarthm o both sdes of the equato, the l y bl x l a ; let y l y, a l a, x l x, the lear equatos s y a bx. The expoetal curve fucto: y ae bx (7)

4 Yamg Yag et al. / Proceda Computer Scece 3 () 5 3 Take the logarthm o both sdes of the equato, the l equatos s y a bx. The logarthmc curve fucto: y bx l a ; let y l y, a l a, the lear y a b lg x () Let x lg x, the lear equatos s y a bx. The hyperbolc fucto: b a y x (9) Let y, x, the lear equatos s y a bx. y x The s curve fucto: y a bc x () Let y, y x c x, the lear equatos s y a bx. 3. Applcato example aalyss 3.. Problem descrpto The umber of arcraft ladg ad ladg ad the ma tre cosumpto - are show Table. Try to establsh a smple lear regresso model to predct the ma tre cosumpto. Table. Arcraft ladg frequecy ad ma tre cosumpto statstcs -. Arcraft ladg frequecy Ma tre cosumpto Year Frst Secod Thrd Fourth Frst Secod Thrd Fourth Costruct regresso predcto model Assumg that y s the ma tre cosumpto, x s the umber of arcraft ladg, accordg to the equato () to calculate the regresso coeffcet: b=.93, a=7.77. The get a lear regresso model: y x () Usg Mtab software regresso aalyss fucto to get the followg ftted le plot, as show Fg..

5 Yamg Yag et al. / Proceda Computer Scece 3 () Tre Cosumpto = Ladg Frequecy Tre Cosumpto Regresso 95% CI 95% PI S 7. R-Sq.% R-Sq(adj) 79.3% 3 5 Ladg Frequecy 7 Fg.. Ftted le plot of arcraft ladg frequecy ad ma tre cosumpto Regresso model test ad aalyss () Sgfcace test of regresso equato. Accordg to Table, aalyze the results the ANOVA table frst. The P-value correspodg to 73.9 of the F-Value s. <.5, so judgg the regresso equato as a whole s remarkably effectve. () A measure of the total effect of the regresso equato. Accordg to Table 3, R-Sq s very close to R-Sq (adjustmet), ad R-Sq (adjustmet) s 79.3%, dcatg that the ftted model ca expla 79.3% of the varato tre cosumpto y, so the model fttg has a better overall effect. (3) Sgfcace test of regresso coeffcet. Accordg to Table, the depedet varable x coeffcet p = <.5, dcatg that the depedet varable x s a sgfcat factor. But the costat p=.7>.5 shows that the costat s ot sgfcat, ad the regresso equato should ot cota the costat. Therefore, the equato () ecessary amedmets to obta a ew regresso predcto model: y.x () The model test ad aalyss are carred out for the equato (). The test results of the lear regresso model are show Table 5. The total effect measure of the regresso model s show Table, ad the sgfcace test of the regresso coeffcet s show Table 7. Accordg to the above model test ad aalyss results, all the tests of equato () passed. Therefore, regresso equato () s more approprate as a predcto model. Table : Varace aalyss of regresso model () Source DF Adj SS Adj MS F-Value P-Value Regresso Ladg frequecy Error Total Table 3: Model summary of regresso model () S R-sq R-sq(adj) R-sq(pred) 7..% 79.33% 75.77% Table : Coeffcets of regresso model () Term Coef SE Coef T-Value P-Value VIF Costat Ladg frequecy

6 3 Yamg Yag et al. / Proceda Computer Scece 3 () 5 3 Table 5: Varace aalyss of regresso model () Source DF Adj SS Adj MS F-Value P-Value Regresso Ladg frequecy Error Total 597 Table : Model summary of regresso model () S R-sq R-sq(adj) R-sq(pred) % 9.% 9.7% Table 7: Coeffcets of regresso model () Term Coef SE Coef T-Value P-Value VIF Ladg frequecy plots aalyss aalyss was used to test the goodess of ft of the model regresso ad aalyss of varace. Checkg the resdual plot helps determe f the ordary least-squares assumpto s fulflled. If these assumptos are satsfed, the ordary least squares regresso wll yeld a ubased estmate of the smallest varace. []. Mtab software provdes the followg resdual plots. The resdual plot of the regresso equato () s show Fg. (a). Ad the resdual plot of the regresso equato () s show Fg. (b). () s versus order of data. To observe the scatter plot of the resduals the order of the observatoal order as the trasverse axs, we focus o the radom fluctuato of the resdual value the map the horzotal axs. If radom fluctuatos, the resdual values are mutually depedet. I ths example, the resduals fluctuate radomly ad are depedet of each other. () s versus ftted values. The emphass s o whether the resdual dfferece the scatter plot s mataed or ot, that s, whether there s a "fuel-shaped" or a "hor shape". If there s a obvous fuel shape or a hor shape, t s ecessary to do some kd of trasformato for the respose varable y, ad the ft the model after the trasformato, ad the fttg effect wll be better. I ths example, the graph s ormal ad the resdual s equal varace. (3) Normal probablty plot of resduals. The ormal probablty graph of the resdual s observed to see f the resdual value s the ormal dstrbuto. If the resduals are a ormal dstrbuto, the pot the graph wll geerally form a straght le. I ths example, the pots are bascally a straght le, ad the resduals ca be regarded as ormal dstrbutos. The resdual hstogram the lower left corer ca be used to check the dstrbuto of resduals. If you have oe or two bars that are farther away from other bars, these may be outlers. Normal Probablty Plot Versus Fts Normal Probablty Plot Versus Fts Percet Percet Ftted Value Ftted Value Hstogram Versus Order Hstogram Versus Order Frequecy - Frequecy Observato Order Observato Order (a) (b) Fg.. (a) resdual plot of the regresso equato (); (b) resdual plot of the regresso equato ().

7 Yamg Yag et al. / Proceda Computer Scece 3 () Comparatve aalyss of predcto results The regresso equatos () ad () are respectvely used to predct the ma tre cosumpto of the arcraft. The predcted value ad the true value are show Fg. 3 (a). The resduals of the two models are compared as show Fg. 3 (b). As ca be see from the fgure, the regresso model better ftted the actual value. At the same tme, the resdual values radomly fluctuate rregularly above ad below the horzotal axs ad are depedet of each other wth o abormaltes. Therefore, t s feasble ad effectve to use the lear regresso model to forecast the aero-materal cosumpto. 9 Actual value Forecasts(y= x) Forecasts(y=.x) (y=.x) (y= x) 7 Data 5-3 Idex (a) - (b) Idex Fg. 3. (a) comparso of predcto results ad actual values; (b) comparso of resduals betwee two regresso equatos.. Coclusos The regresso aalyss ad predcto method s a classcal ad practcal forecastg scheme. It s precsely because of ts classc, so t s also mature, ad t s easer to uderstad, ad more wdely used. I the actual use process, f we choose specfc methods ad models, we ca aalyze the data detal, ad observe ad aalyze the scatters, whch ca also be more detaled ad the predcto results wll be satsfactory. Whe the regresso predcto method s appled, t s ecessary to determe whether there s a correlato betwee the varables. If there s o correlato betwee the varables, the regresso predcto method of these varables wll result the wrog result. Whe the regresso aalyss s appled correctly, we should pay atteto to the qualtatve aalyss of the relatoshp betwee the pheomea, avod the extrapolato of the regresso predcto ad apply the approprate data. At the same tme, the lear regresso predcto method proposed ths paper ot oly apples to the predcto of aero-materal spare parts cosumpto, but also to other equpmet dexes or parameters, whch provdes a scetfc method ad meas for equpmet support forecast. Refereces. Yag, Yamg, W. Wag, ad C. Guo. "Forecastg for Ar Materal Cosumpto Based o Wters Expoetal Smoothg Model." Iteratoal Coferece o Automato, Mechacal Cotrol ad Computatoal Egeerg. Davd A. Freedma. Statstcal Models: Theory ad Practce. Cambrdge Uversty Press. p.. (9). 3. Recher, Alv C.; Chrstese, Wllam F., "Chapter, Multvarate regresso Secto., Itroducto", Methods of Multvarate Aalyss, Wley Seres Probablty ad Statstcs, 79 (3rd ed.), Joh Wley & Sos, p. 9, ().. Iformato o 5. Yamg YANG, Yag YANG, Rul ZHANG. Statstcal Aalyss ad Applcato of Qualty Maagemet. Bejg: Tsghua Uversty Press, 5.. Yag, Ya Mg, H. Yu, ad Z. Su. "Arcraft falure rate forecastg method based o Holt-Wters seasoal model." IEEE, Iteratoal Coferece o Cloud Computg ad Bg Data Aalyss IEEE, 7.