Estimation on Monotone Partial Functional Linear Regression
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1 A^VÇÚO 1 33 ò 1 4 Ï 217 c 8 Chinese Journl of Applied Probbility nd Sttistics Aug., 217, Vol. 33, No. 4, pp doi: /j.issn Estimtion on Monotone Prtil Functionl Liner Regression DING JinHu (Deprtment of Sttistics, Shnxi Dtong University, Dtong, 379, Chin) Abstrct: We propose new estimtion method for the prmeters of prtil functionl liner model when the prmeter curve is subject to monotone constrint. The proposed estimtors re implemented under the nonliner mixed effects model frmework. The smll smple properties re illustrted through simultion experiment. Keywords: functionl dt; monotone estimtor; nonliner mixed effects model 21 Mthemtics Subject Clssifiction: 62G8 Cittion: Ding J H. Estimtion on monotone prtil functionl liner regression J. Chinese J. Appl. Probb. Sttist., 217, 33(4): Introduction Dt tht tke the form of rndom curves rther thn sclrs or vectors re now common feture of sttisticl problems. It is often the sitution tht response is relted to both vector of finite length nd function-vlued rndom vrible s predictor vribles. However, the direct use of clssicl regression methods in the cse where predictor vribles re digitized recording from rndom curve often fils. Severl techniques hve been proposed for deling with dt of such nture. Crmbes et l. 1 proposed estimtion methods for functionl liner model using functionl principl components nlysis or smoothing techniques nd investigted the symptotic behviors of the estimtors. Aneiros-Pérez nd Vieu 2 introduced semi-functionl prtil liner regression model by modeling the reltionship of functionl predictor with the response nonprmetriclly in order to cpture the dvntges of prtil liner modeling nd nonprmetric functionl dt nlysis. Recently, Zhng et l. 3 introduced liner model for mixed dt. They clled their model prtil functionl liner model since the predictor vribles include The project ws supported by the Doctor Reserch Fund of Shnxi Dtong University (Grnt No. 215-B-8). Received November 24, 216. Revised December 2, 216.
2 434 Chinese Journl of Applied Probbility nd Sttistics Vol. 33 those hving functionl nture. The prtil functionl liner model (PFLM) is defined s: y = z T β + x(t)γ(t)dt + ε, (1) where y is rel-vlued response vrible, z = (z 1, z 2,..., z d ) is d-dimensionl covrite vrible, β is vector of unknown regression prmeters, {x(t) : t, b} is covrite function, γ is unknown smooth prmeter curve nd ε is rndom error with zero men nd vrince σε 2 tht is independent of z nd x. When investigting the ssocition between the covrite functions nd the sclr response it could be the cse tht we hve prior informtion, for exmple, tht the prmeter curve γ(t) is subject to monotonicity constrint. Enforcing monotonicity on the prmeter curve ws recently pproched by 4 nd 5 in bio-medicl field pplictions. In this pper, we develop new estimtors for the prmeters of prtil functionl liner model when the prmeter curve is smooth monotone. The rest of the pper is orgnized s follows. In Section 2, we introduce the monotone functionl nonliner model nd propose new estimtion method bsed on nonliner mixed effects model frmework. Numericl simultions re conducted to illustrte the ppliction of the proposed method in Section A Monotone Functionl Nonliner Model Extending the PFLM to incorporte monotone function, we ssume tht the prmeter curve γ(t) lies in the spce of continuous twice differentible strictly monotone functions. Note tht ll twice differentil strictly monotone functions γ(t) stisfy the following differentil eqution by 6, 2 γ(t) t 2 = ω(t) γ(t), t where ω(t) is smooth unconstrined function, the solution to this differentil eqution is γ(t) = c + c 1 t ( v ) exp ω(u)du dv. (2) By ssuming this form of γ(t) the prtil functionl liner model (1) is trnsformed into functionl nonliner model due to the nonliner dependence of γ(t) on the prmeter curve ω(t).
3 No. 4 DING J. H.: Estimtion on Monotone Prtil Functionl Liner Regression 435 The prmeter vectors β = (β 1, β 2,..., β d ) T, c, c 1, nd ω(u) cn be estimted by minimizing the penlized sum of squres n ( y i z T i β ) 2 b x i (t)γ(t)dt + λ (ω (t)) 2 dt, (3) where λ is clled the smoothing prmeter tht my depend on the dt, nd determines the weight of the penlty (ω (t)) 2 dt. A lrge vlue of λ gives much influence to the penlty term nd hence leds to smooth estimte of ω, nd conversely. Intermedite vlues re best. The choice of λ is given in the following section. To incorporte regulrized bsis function pproch into our fitting criterion, let b = (b 1, b 2,..., b m ) T, δ(t) = (δ 1 (t), δ 2 (t),..., δ m (t)) T, Q = δ (t)δ (t) T dt, where δ i (t) is the i th bsis function evluted t t, for i = 1, 2,..., m. Define the following (v) = v δ(u)du, ω(t) b T δ(t), f(t, b) = For fixed λ, fitting criterion (3) my now be pproximted by n b (y t i zi T β c k i c 1 x i (t) n n t exp(b T (v))dv, k i = ( v ) exp ω(u)du dv dt (y t 2 i zi T β c k i c 1 x i (t) exp(b T (v))dv dt) + λbt Qb x i (t)dt. ) 2 b + λ (ω (t)) 2 dt 2 (y i zi T β c k i c 1 x i (t)f(t, b)dt) + λbt Qb. (5) Nextly, we reprmetrize our functionl nonliner model s nonliner mixed effects model to estimte ω(u), with the lest squres estimtion of β, c nd c 1, nd the smoothing prmeter λ chosen by restricted mximum likelihood (REML). The functionl nonliner model with prmeter curve (2) my be reprmeterized s nonliner mixed effects model. Let (v) = ( 1 (v), 2 (v),..., m (v)) T, where (v) is defined in (4). We express the functionl nonliner model with prmeter curve (2) s y = zβ + c k + c 1 η(x(t), b) + ε, (6) where y = (y 1, y 2,..., y n ) T, z = (z 1, z 2,..., z n ) T, ε = (ε 1, ε 2,..., ε n ), x(t) = (x 1 (t), x 2 (t),..., x n (t)) T, nd η(x(t), b) = x(t) t (4) exp(b T (v))dvdt. (7)
4 436 Chinese Journl of Applied Probbility nd Sttistics Vol. 33 The function (7) is nonliner function of the prmeter b. The fitting criterion (5) in terms of η(x(t), b) is (y zβ c k c 1 η(x(t), b)) T (y zβ c k c 1 η(x(t), b)) + λb T Qb. (8) We ssume tht penlty mtrix Q m m is non-negtive definite nd symmetric, so tht reordered eigenvlue decomposition of Q cn be obtined Q = V DV T, (9) where the columns of V contin the eigenvectors v 1, v 2,..., v m nd D is digonl mtrix with eigenvlues e 1 e 2 e m. If Q hs rnk r, then the first l = m r digonl elements in D will be zero, = e 1 = e 2 = = e l, with the remining r eigenvlues being non-zero. Let d be digonl mtrix contining these non-zero eigenvlues so tht l l l r Q m m = V V T. r l d r r Prtition the mtrix V = V V 1, where V is m l with corresponding columns v 1, v 2,..., v l, V 1 is m r with corresponding columns v l+1, v l+2,..., v m. Since the v i s re orthogonl we hve V V T = V T V = I m m nd V T V 1 = l r. (1) Let d 1/2 be digonl mtrix with elements d 1/2 i,i = e 1/2 l+i for i = 1, 2,..., r. Define s the projection of b on the spce spnned by v 1, v 2,..., v l, i.e. = V T b. Define u s the projection of b on the spce spnned by v l+1, v l+2,..., v m nd the digonl elements of d 1/2, i.e. u = d 1/2 V T 1 b. Thus, u = V Tb d 1/2 V1 Tb. Since V V T = I, we hve Then we my express (v) T b = (v) T V b = V. (11) = (v) T V (v) T V 1 = (v) T V + (v) T V 1.
5 No. 4 DING J. H.: Estimtion on Monotone Prtil Functionl Liner Regression 437 Using (9) (11) the penlty term becomes λb T Qb = λ = λ T V T QV T V T V V T V d = λu T u. The prmeter b cn now be expressed s nonliner mixed model of the form y = η(ν)+ε with prmeter vector ν = (v) T V + (v) T V 1, nd design mtrices M = (v) T V nd N = (v) T V 1 d 1/2. Assuming u N(, σ 2 ui) nd ε N(, σ 2 εi), for fixed σ u, σ ε, c, c 1 nd β, estimtes for nd u cn be obtined by minimizing the function (y zβ + c k + c 1 η(x(t), b)) T (y zβ + c k + c 1 η(x(t), b)) + σ2 ε σu 2 u T u, (12) where η(ν) is reprmetriztion of η(x(t), b) in (7). By setting λ = σ 2 ε/σ 2 u, minimizing the penlized lest squres criterion (12) over (, u) is equivlent to minimizing the penlized lest squres criterion (5) over the prmeter b. Computtionl detils for minimizing fitting criterion (12) re given in 7. The following itertive two-stge minimiztion procedure with performnce itertion cn be used for estimting β, c, c 1, ω(u) nd λ: 1. Select initil vlues of β, c, c Conditionl on β, c, c 1, obtin estimtes of η(ν) nd λ in nonliner mixed model frmework. 3. Conditionl on η(ν) obtined in step 2, estimte β, c, c 1 using lest squres. 4. Repet steps 2 3 until convergence. The initil vlues cn be chosen by ny nonprmetric pproch nd R function nlme cn be employed in step Simultion Study In this section, we provide numericl exmple to exmine the performnce of our proposed estimtors. We generted dt from model (1) in the cse where z = (z 1, z 2,..., z 5 )
6 438 Chinese Journl of Applied Probbility nd Sttistics Vol. 33 hs multivrite norml distribution with zero men vector nd vrince-covrince mtrix I, {x(t), t, 1} is stndrd Brownin motion, ε is norml with zero men nd unit vrince nd using the following monotone-restricted prtil functionl liner regression model: y = z T β + x(t)γ(t)dt + ε, (13) where β = (2, 1, 1.5, 5, 1.7) T nd γ(t) = 9/(1 + exp( 2t + 1)). In this instnce, x(t) cn be represented s x(t) = u j φ j (t), t, 1, j=1 where the u j re distributed s independent norml with men nd vrince λ j = ((j.5)π) 2 nd φ j (t) = 2 sin((j.5)πt). Rndom smples of n = 1 i.i.d. (z i, x i, y i ) were generted with ech x i ws observed t 1 eqully spced points on, 1. This experiment ws then replicted 5 times. Rndom smples of size n = 1 were lso generted to ssess the behvior of the estimtors with lrge smple. We employ B-spline to pproximte ω(t). Precise knot plcement of B-spline, usully hs less impct compred to selection of the number of bsis functions. Typiclly the knots re chosen to be evenly spced throughout the rnge or t the quntiles of {z 1, z 2,..., z n }. The informtion criterion AIC = n ln(sse/n) + 2 edf, BIC = n ln(sse/n) + edf ln(n) or GCV = SSE/(1 edf/n) 2 my be used for the number of bsis functions, where SSE = n ( y i z T i β + x i (t) γ(t)dt) 2, nd edf is effective degrees of freedom. For the constrined model, BIC usully hs the smller RMSE thn AIC nd GCV from our experiment of simultion. Here the number of knots is selected by BIC informtion criteri nd locted on eqully spced quntiles. The performnce of monotone estimtor γ of γ is mesured by the root-men-squre error: ( 1 RMSE = n n ( γ(t i ) γ(t i )) 2) 1/2. Figure 1 displys the true regression coefficient function γ(t), unconstrined estimte nd monotone-constrined estimte γ over 5 replicte smples when n = 1 nd n = 1. As seen in Figure 1, the unconstrined estimte does not stisfy monotonicity in the both ends of the domin due to the rtificil error. The shded grey region round the curve describes pointwise 1 nd 9 percentiles of the γ s for the monotone prmetric curve. Root-men-squre errors of γ re.3873 nd.1265, respectively, when n = 1 nd
7 No. 4 DING J. H.: Estimtion on Monotone Prtil Functionl Liner Regression 439 n = 1. The essentil shpe for γ is recovered for both smple sizes. The bis, vrince nd men squred error (MSE) of the resulting β k, k = 1, 2,..., 5 re given in Tble 1. Tble 1 The bis, vrince nd MSE of the β s from 5 repetitions n = 1 n = 1 Bis Vrince MSE Bis Vrince MSE β β β β β gmm(t) true function Monotonic Unconstrined gmm(t) true function Monotonic Unconstrined Figure 1 t t The doted curve denotes the true function, the dot-dshed curve denotes the unconstrined estimte, the solid curve denotes monotone estimte for the function γ(t). The shded grey region describes pointwise 1 nd 9 percentiles of the γ s from 5 repetitions with n = 1 (left) nd n = 1 (right). References 1 Crmbes C, Kneip A, Srd P. Smoothing splines estimtors for functionl liner regression J. Ann. Sttist., 29, 37(1): Aneiros-Pérez G, Vieu P. Semi-functionl prtil liner regression J. Sttist. Probb. Lett., 26, 76(11): Zhng D W, Lin X H, Sowers M F. Two-stge functionl mixed models for evluting the effect of longitudinl covrite profiles on sclr outcome J. Biometrics, 27, 63(2):
8 44 Chinese Journl of Applied Probbility nd Sttistics Vol Schipper M, Tylor J M G, Lin X H. Byesin generlized monotonic functionl mixed models for the effects of rdition dose histogrms on norml tissue complictions J. Stt. Med., 27, 26(25): Schipper M, Tylor J M G, Lin X H. Generlized monotonic functionl mixed models with ppliction to modelling norml tissue complictions J. J. Roy. Sttist. Soc. Ser. C, 28, 57(2): Rmsy J O. Estimting smooth monotone functions J. J. R. Stt. Soc. Ser. B Stt. Methodol., 1998, 6(2): Lindstrom M J, Btes D M. Nonliner mixed effects models for repeted mesures dt J. Biometrics, 199, 46(3): ün¼ê.êâœëê.o ï u (ìüœóœæúox, ŒÓ, 379) Á : ïäünå^ e¼ê.êâœëê.o K, ^š 5 ÜA. {, Ñ. ëê þúünëê O, ÏLéJÑ {?1êŠ Û. ' c: ¼ê.êâ; üno; š 5 ÜA. ã Ò: O212.7
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