Power Weighted Quantile Regression and Its Application

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1 Joural of Data Sciece 12(2014), Power Weighted Quatile Regressio ad Its Applicatio Xue J. Ma 1 ad Feg X. He 2 1 Remi Uiversity of Chia 2 North Chia Electric Power Uiversity Abstract: I the paper, we propose power weighted quatile regressio(pwqr), which ca reduce the effect of heterogeeous of the coditioal desities of the respose effectively ad improve efficiecy of quatile regressio). I additio to PWQR, this article also proves that all the weightig of those that the actual value is less tha the estimated value of PWQR ad the proportio of all the weightig is very close to the correspodig quatile. At last, this article establishes the relatioship betwee Geomagetic Idices ad GIC. Accordig to the problems of power system security operatio, we make GIC risk value table. This table ca have stroger practical operatio ability, ca provide power system security operatio with importat ifereces. Key words: Quatile regressio, Power Weighted Quatile Regressio, GIC, Geomagetic Idices 1. Itroductio Weighted quatile regressio is proposed by Koeker(2005). Whe the coditioal desities of the respose are heterogeeous, he maitaied weighted quatile regressio might lead to efficiecy improvemets. It is cosidered, but reweightig based o estimated desities is somewhat difficult, that is the estimatio of the weights is hard to do well, so it is t usually doe. Ofte researchers prefer to try to correct sparsity fuctio or bootstrap (Hedricks ad Koeker 1991,Parze et al 1994) for o iid error behavior. If the covariace structure were kow, weighted quatile regressio would be more efficiet. With regard to research weighted quatile regressio, Taylor(2008) proposed expoetially weighted quatile regressio ad applied i Value at Risk ad expected shortfall. But if data exists ode, expoetially weighted effect would be meaigless. For example, whe oe idepedet variable value correspodig to more tha oe depedet variable values, the method of expoetially weight is clearly ureasoable. Because the time sequece of the observed values are differet, weights are differet. So that this paper proposes a ew weightig method, that is Power Weighted Quatile Regressio. Correspodig author.

2 536 Power Weighted Quatile Regressio ad Its Applicatio 2. Power Weighted Quatile Regressio I the light of the developmet of Quatile regressio, we propose power weighted quatile regressio: mi x i α ρ βεr p τ (y i x T i β ) (1) where x is orm. Quatile regressio is the special case of power weighted quatile regressio. Because whe α = 0, power weighted quatile regressio is quatle regressio. Theorem 1 If X cotais i quatile regressio model Q τ (y x)= x T β, the for ay β, solvig (1), we have x i α I(y i < x T i β ) x i α τ x i α I(y i < x T i β ) + x i α I(y i = x T i β ) x i α x i α I(y i > x T i β ) x i α 1 τ x i α I(y i > x T i β ) + x i α I(y i = x T i β ) x i α (2) (3) Proof. See Appedix. The theorem idicates that the estimator of PWQR essetially partitios divided the y i observatios ito three parts y i < x i T β, yi = x i T β, yi < x i T β. The sum of the weights o y i < x i T β, as a proportio of the sum of all the weights, is close to τ. The proportio is ot exactly equal to τ, because y i = x i T β may exist. That is to say, give τ, the weighted proportio of those observatios, which uderlie quatile curve, is close to τ. Similarly, the sum of the weights o y i > x i T β, as a proportio of the sum of all the weights, is close to 1 τ. I (1), we must deal with the importat α selectio problem, as the quality of the liear estimates depeds sesitively o the choice of it. A coveiet ad effective method is that we ca use goodess of fit, proposed by Koeker ad Machado(1999). Cosider the quatile regressio model Q T (y i, x i ) = x i T β which we ca partitio as Q T (y i, x i ) = x i1 T β 1 + x i2 T β 2 ad let V (τ, α) = mi βεr p (y i x T 1i β 1 ), so the goodess of fit is x α i ρτ (y i x T i β ), V (τ, α) = mi βεr p q x i R 1 (τ, α) = 1 V (τ, α) V (τ, α) Covetioally, x i1 icludig oly a itercept parameter (x i1 =1) yields ther 1 usually reported. R 1 (τ, α) measures the relative success of the correspodig quatile regressio α ρτ

3 Xue J. Ma ad Feg X. He 537 models at a specific quatile with referece to a appropriately weighted sum of absolute residuals, which ca assess goodess of fit for quatile regressio models. R 1 (τ, α) lies betwee 0 ad 1.Geerally speakig, it s better to have a larger R 1 (τ, α). The optimal α ca be selected by maximizig R 1 (τ, α), that is α opt (τ) = arg max a R 1 (τ, α) 3. Data Geomagetic Idices depicts the overall itesity of geomageitic disturbace i a time slot or the idex of certai geomageitic disturbace, such as frequetly-used idexes of K, Kp, ap ad so o, of which the idex K is employed to describe geomageitic disturbace itesity of a sigle geomagetic observatory i a time slot of three hours, which meas that idex K could oly describe the geomagetic activity i certai district, ot the globe. I order to preset the geomagetic activity itesity of the globe, it eeds a ew idex Kp by stadardizig ad the averagig 12 geomagetic observatories from global Geomagetic statio etwork. Because of the oliear relatioship betwee Kp ad geomageitic disturbace, it is trasformed ito liear relatioship to get the idex Kp. The size of a geomagetic storm is classified as moderate, itese ad super-storm accordig to the size of Geomagetic Idices. Geerally speakig, the higher of the geomagetic storm itesity, the larger of GIC caused by geomagetic storm, which demostrates the close relatio betwee GIC ad Geomagetic Idices. As for the evaluatio of GIC, there are basically two models: physical model ad statistical model, of which the later oe is the focus of this article. Trichtcheko ad Boteler(2004) from Caada with others has established class liear regressio of GIC usig ap. He ad Ma(2011)used quatile regressio. This article tries to improve the quatile regressio ad make advatage of power weighted quatile regressio. Data sources: the data of GIC is based o the moitorig of Guagzhou ligao uclear power plat whe geomagetic storm took place i 2004 to 2005, ad the data of ap come from Ceter for Space Eviromet Research ad Forecast. 3.1 Comparig quatile regressio ad power weighted quatile regressio As for GIC ad ap, we eed to establish uitary quatile regressio model ad uitary power weighted quatile regressio model respectively. What s more, we also eed to give the regressio coefficiet whe the quatile is0.05,0.1,0.2,0.3,0.4,0.5,0.6,0.7,0.8,0.9,0.95 respectively. Accordig to goodess of fit, the correspodig selectio of α is 0.0,-0.4,-0.1,-2.6,-0.4,- 0.4,-0.9,-0.4,-0.9,-1.4, 0.0. From the table 1, the goodess of fit of PWQR is larger tha QR except α = 0. The all the coefficiet give by PWQR is remarkable at the level of 0.01, with oly part of the coefficiet give by QR is remarkable at the level of Whe tau equals to 0.3, 0.4, 0.5, 0.6 respectively, the sigificace of QR fails the test, which meas that it is ot otable, while PWQR is all very otable. Except for the value of PWQR ad QR are the same, the fact that the

4 538 Power Weighted Quatile Regressio ad Its Applicatio coefficiet before ap of PWQR are all smaller tha those of QR ad costat term of PWQR are all bigger tha those of QR. From the aspect of the sigificace of regressio coefficiet ad goodess of fit, it is easy to judge that the PWQR model is superior to the quatile regressio model. 3.2 Assessig GIC by PWQR From the above sectio, we kow the coefficiet whe tau=0.05, 0.1, 0.3, 0.5, 0.7, 0.9, 0.95 respectively. As follows: Q 0.05 (GIC ap ) = ap Q 0.1 (GIC ap ) = ap Q 0.3 (GIC ap ) = ap Q 0.5 (GIC ap ) = ap Q 0.7 (GIC ap ) = ap Q 0.9 (GIC ap ) = ap Q 0.95 (GIC ap ) = ap

5 Xue J. Ma ad Feg X. He 539 Table 1: Rregressio coefficiet of PWQR ad QR. By the expressios, we could see that regressio costat give by power weighted quatile ad the coefficiet of ap are iclied to icrease with the icrease of tau, of which tau=0.05 represets the low tail distributio of GIC, tau=0.3 represets mid low tail distributio, tau=0.5 represets middle iformatio of GIC, tau=0.7 represets mid high tail iformatio ad tau=0.9 or 0.95 represets the high tail iformatio. So by the aalysis, we could fid that PWQR is able to depict the overall iformatio of GIC. I Figure 1, from the bottom to the top, these straight lies represets regressio lies whe quatiles are 0.05, 0.1, 0.3, 0.5, 0.7, 0.9, 0.95 respectively. Furthermore, with the growig of ap, GIC teds to go up, ad whe ap < 110, GIC is desely distributed i the low positio, with a few i higher positio. At this poit, we could use differet quatile to fit the value of GIC, of which for certai ap, we could get differet GIC fitted value of differet tau, rederig the value give by quatile regressio beig able to reflect the overall iformatio of GIC. For example,

6 540 Power Weighted Quatile Regressio ad Its Applicatio Figure 1: Fitted lies of PWQR whe ap=179, the actual value are 3.172, 3.755, 8.966, or The fitted value of differet tau i the models is as followig: For the actual value, we could fid the fitted value of tau by quatile regressio, such as GIC=3.75, tau=0.1 is used to fit. I practice, our cocer is those GIC that could pose damage to electric grid, i other words, the high tail iformatio of GIC distributio; or the middle level of GIC, ad that is to say the PWQR whe tau=0.5 Table 2: Fitted values of differet quatile Accordig to PWQR, we get the fitted value of high tail ad middle GIC whe ap > 111 (table 3). What we should highlight is that we could ot oly get the fitted value of GIC ad also its probability by PWQR. For example, whe ap=179, we are coviced that the probability of GIC that is o less tha 9.57 is 50% ad of GIC that is o less tha15.762, ad is 80%,90% ad 95% respectively. This is very importat for the predictio of geomagetic storm. I order to predict a geomagetic storm ad ap, we could fid the probability of GIC that is o less tha certai value from the table 3, thus helpig us to avoid ecoomic losses caused by uderestimatio or overestimatio of GIC.

7 Xue J. Ma ad Feg X. He 541 Table 3: Fitted values of differet quatile 4. Coclusios This article is tryig to put forward a ew method of power weighted quatile regressio (PWRQ), especially whe there are a lot of kots i statistics, PWQR is proved to be superior to EWQR. This article proves that all the weightig of those whose actual value is less tha its estimated value by PWQR ad all the weightig proportio are very close to the correspodig τ; that all the weightig of those whose actual value is more tha its estimated value by PWQR ad all the weightig proportio are very close to the correspodig 1 τ. I this article, it also gives examples to testify the superiority of power weighted quatile regressio to quatile regressio. I the ed, by power weighted quatile regressio, we could get GIC value (which has little differece with actual value), ad predict the probability of GIC i a certai rage, reducig uecessary losses. Appedix: Proof of Theorem 1 Proof of Theorem 1: The EWQR objective fuctio R(β) is R(β) = mi x i α ρ βεr p τ (y i x T i β) = mi x i α ρ βεr p τ (y i x T i β) (τ I(y i < x T i β)) The fuctio is piecewise liear cotiuous fuctio, which is ot differetiable at the poit y i = x i T β. But there exists directioal derivatives at all the poits. I order to compute the miimum of R(β), we cosider its directioal derivatives. The directioal derivative of the objective fuctio i directio w is give by

8 542 Power Weighted Quatile Regressio ad Its Applicatio where R(β, w) = d ds R(β + sw) s=0 = d ds x i α (y i x i T β x i T sw)[τ I(y i < x i T β + x i T sw)] s=0 = x i α ψ τ (y i x T i β, x T i w)x T i w τ I(u < 0) if u 0 ψ τ (u, v) = { τ I(u < 0) if u = 0 The parameter vector β miimizes of R(β) if ad oly if R(β,w) 0 for w. R(β, w) = x i α ψ τ (y i x i T β, x i T w)x i T w Because X icludes a itercept term, there exists a vector γ R p such that x i T γ = 1, i = 1, 2,,. If we let = γ i expressio (4), we get That is x i α ψ τ (y i x i T β, 1) 0 τ x i α I(y i > x T i β ) + (τ 1) x i α I(y i < x T i β ) + τ x i α I(y i = x T i β ) 0 (5) Form (6) Thereby The iequalities of expressios (5) ca be rewritte as the iequalities of expressio: τ[ x i α I(y i > x T i β ) + x i α I(y i < x T i β ) + x i α I(y i = x i T β )] x i α I(y i < x i T β ) τ x i α x i α I(y i < x T i β ) x i α I(y i < x T i β ) x i α τ (6)

9 Xue J. Ma ad Feg X. He 543 So the left of (2) has proved. Both sides subtractig x i α I(y i > x T i β ) x i α I(y i = x i T β ) ito (5),we obtai 1 τ x i α I(y i > x T i β ) + x i α I(y i = x T i β ) x i α So the right of (3) has proved. I a similar way, we ca let w = γ (so tha x T i w = 1), the right of (2) ad the left of (3) are proved Ackowledgmets The research was supported by the Fudametal Research Fuds for the Cetral Uiversities, ad the Research Fuds of Remi Uiversity of Chia (14XNH107). We thak the editor ad aoymous referees whose commets led to improve the paper. Refereces [1] He, F. X. ad Ma, X. J.(2011). A ew method for assessmet of GIC usig geomagetic idices. Multimedia Techology (ICMT), 2011 Iteratioal Coferece o [2] Hedricks,W. ad Koeker,R.(1991).Hierarchical splie models for coditioal quatiles ad the demad for electricity. Joural of the America Statistical Associatio 87, [3] Taylor,J. W.(2008)Usig expoetially weighted quatile regressio to estimate value at risk ad expected shortfall. Joural of Fiacial Ecoometrics 6, [4] Koeker R.(2005). Quatile regressio.cambridge Uiversity Press, Lodo. [5] Trichtcheko, L. ad Boteler,D.H.(2004). Modelig geomagetically iduced currets usig geomagetic idices ad data. IEEE Trasactios o Plasma Sciece 32, [6] Parze, M. I., Wei,L.,Yig, Z..(1994). A resamplig method based o pivotal estimatig fuctios. Biometrika 81, [7] Koeker. R. ad Machado,J. A. F. (1999). Goodess of fit ad related ifer ece processes for quatile regressio. Joural of the America Statistical Associatio 94,

10 544 Power Weighted Quatile Regressio ad Its Applicatio Received December 12, 2013; accepted July 26, Xue J. Ma Ceter for Applied Statistics Remi Uiversity of Chia 59 Zhogguacu Ave., Beijig, , Chia Feg X. He Departmet of Mathematics ad Physics North Chia Electric Power Uiversity 2 Beiog Road, Huiloggua Tow, Chagpig District, Beijig,102206, Chia he fx@126.com

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