Kvantilna regresija

master's thesis

• Author: Andrea Rumenjak
• Mentor: Miljenko Huzak (mentor)
• Committee member: Miljenko Huzak (predsjednik povjerenstva)
• Committee member: Bojan Basrak (član povjerenstva)
• Committee member: Eduard Marušić-Paloka (član povjerenstva)
• Committee member: Mladen Jurak (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2014-07-14, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics

Abstract

Linearna regresija je postupak u kojem pokušavamo opisati zavisnu varijablu kao linearnu kombinaciju jedne ili više nezavisnih varijabli. Ona procjenjuje uvjetno očekivanje zavisne varijable uz danu nezavisnu varijablu te ju prikazuje kao afinu funkciju od zavisne varijable. Za taj tip regresije metoda najmanjih kvadrata je dovoljno učinkovita, ali problemi nastaju kada se žele proučavati ekstremi nekih podataka ili njihovi kvantili. Tu do izražaja dolazi kvantilna regresija, gdje kvantilni regresijski parametar predstavlja promjenu zavisne varijable (varijable odaziva) u određenom kvantilu za jednu jedinicu promjene nezavisne varijable (varijable poticaja). U kvantilnoj regresiji procjenitelj za uvjetni medijan je procijenjen minimiziranjem simetrične težinske sume apsolutnih pogrešaka (gdje je težina jednaka 0.5), a procjenitelj za druge uvjetne kvantilne funkcije se procjenjuje minimiziranjem asimetrične težinske sume apsolutnih pogrešaka (gdje su težine jednake funkcijama od kvantila). Zbog toga je kvantilna regresija robustna na outliere što smo objasnili u radu. Još jedna prednost kvantilne regresije koju smo prikazali u radu je svojstvo ekvivarijance koje nam omogućuje da imamo da je kvantil transformirane slučajne varijable (transformirana pomoću strogo rastuće afine funkcije) jednak transformiranom kvantilu originalne slučajne varijable, što nije slučaj kod matematičkog očekivanja. Zbog toga je interpretacija modela kvantilne regresije jednostavnija nego ona kod metode najmanjih kvadrata.

Abstract (english)

Linear regression is a process in which we try to describe the dependent variable as a linear combination of one or more independent variables. It estimates the conditional expectation of the dependent variable for a given independent variable and it is shown as an affine function of the dependent variable. For this type of regression least squares method is efficient, but problems arise when we want to study the extreme values of some data or their quintiles. This is where quantile regression is useful. The quantile regression parameter represents the change in the dependent variable (response variable) in a specified quantile of a unit change in the independent variable (predictor variable). In quantile regression, the conditional median estimator is estimated by minimizing the symetrically weighted sum of absolute error (where the weight is equal to 0.5). The estimator for other conditional quantile function is estimated by minimizing the asymmetric weighted sum of absolute errors (where the weights are the functions of quintiles). Therefore, the quantile regression is robust to the presence of outliers as explained in this paper. Another advantage of quantile regression, shown in this paper, is equivariance that allows the quantile transformed random variables (transformed by strictly increasing affine functions) to be equal to the transformed quantiles of the original random variables, which is not the case with mathematical expectation. Therefore the interpretation of the model of quantile regression is simpler than in the method of least squares.

• Keywords: Linearna regresija; kvantilna regresija; ekvivarijanca
• Keywords (english): linear regression; quantile regression; equivariance
• Language: croatian
• URN:NBN: urn:nbn:hr:217:168519
• Study programme: Title: Mathematical Statistics; Study programme type: university; Study level: graduate; Academic / professional title: magistar/magistra matematike (magistar/magistra matematike)
• Type of resource: Text
• File origin: Born digital
• Access conditions: Open access
• Created on: 2019-02-05 15:44:17