Spline Kubik Dalam Regresi Nonparametrik Dan Aplikasinya

Takaria, Johannis (2006) Spline Kubik Dalam Regresi Nonparametrik Dan Aplikasinya. Masters thesis, Institut Teknologi Sepuluh November.

Text 1304201009-Master_thesis-1.pdf Restricted to Repository staff only Download (12MB)

Abstract

Diberikan model regresi nonparametrik: y_j = f(t_j) + ε_j, j = 1, 2, ..., n, dimana f(t_j) merupakan kurva regresi yang tidak diketahui bentuknya, dan error ε_j diasumsikan identik, independen dengan mean nol dan variansi σ². Estimator spline kubik dapat ditulis menjadi: f̃_α(t) = K(α)Ỹ, dimana K(α) = X(XᵀX + nαM)⁻¹Xᵀ, dan M = {6δ_jk}(j, k = 1, 2, ..., n) diperoleh dengan optimasi bersyarat, yaitu meminimumkan: n⁻¹ ∑(j=1)^n (y_j - f(t_j))² dengan syarat ∫_a^b (f''(t))² dt < ρ, ρ ≥ 0. Estimator yang dihasilkan adalah bias, tetapi tak bias asimtotik.

Spline kubik dalam penelitian ini diaplikasikan dalam mengkaji bentuk botol, salah satunya adalah botol kimia berbentuk buah. Model yang dihasilkan dengan pendekatan spline kubik adalah:

Ŝ(t) = 0,2034524 t³ - 0,334083(t - 5,5)₊³ - 2,386697(t - 9,01)₊³

= { 0,2034t³ ; t < 5,5
{ -0,131t³ + 5,512t² - 30,318t + 55,583 ; 5,5 ≤ t < 9,01
{ -2,313t³ + 70,025t² - 611,575t + 1801,29 ; t ≥ 9,01

dengan dua titik knot optimal 5,5 dan 9,01 yang memiliki skor GCV sebesar 2,77578.
====================================================================================================================================
Consider a nonparametric regression model: y_j = f(t_j) + ε_j, j = 1, 2, ..., n, where f(t_j) represents unidentified form of a regression curve, and error ε_j is assumed as
independently, identical with the zero mean and variance σ². Cubic spline estimator can be
written as: f̃_α(t) = K(α)Ỹ, where K(α) = X(XᵀX + nαM)⁻¹Xᵀ, and M = {6δ_jk}(j, k = 1, 2, ..., n) is obtained through a conditional optimization, by minimizing: n⁻¹ ∑(j=1)^n (y_j - f(t_j))² with conditions ∫_a^b (f''(t))² dt < ρ, ρ ≥ 0. The estimator produced is biased, but not asymptotic biased.
In this research the cubic spline is applied in observing the shape of a bottle, an example being the fruit shaped chemical bottle. The model yielded through the cubic spline
approximation is: Ŝ(t) = 0,2034524 t³ - 0,334083(t - 5,5)₊³ - 2,386697(t - 9,01)₊³
= { 0,2034t³ ; t < 5,5
{ -0,131t³ + 5,512t² - 30,318t + 55,583 ; 5,5 ≤ t < 9,01
{ -2,313t³ + 70,025t² - 611,575t + 1801,29 ; t ≥ 9,01
With two optimal knot points 5,5 and 9,01 reaching the GCV score of2,77578.

Item Type:	Thesis (Masters)
Additional Information:	RTSt 519.536 Tak s
Uncontrolled Keywords:	Regresi Nonparametrik, Spline· Kubik, Titik Knot, GCV, Nonparametric Regression, Cubic spline, Knot points, GCV
Subjects:	Q Science > QA Mathematics > QA278.2 Regression Analysis. Logistic regression
Divisions:	Faculty of Mathematics and Science > Statistics > 49101-(S2) Master Thesis
Depositing User:	magang .
Date Deposited:	01 Oct 2026 01:30
Last Modified:	01 Oct 2026 01:30
URI:	http://repository.its.ac.id/id/eprint/145109

Actions (login required)

View Item