20.500.12556/RUL-173509 Chatterjeejev ksi delo diplomskega seminarja Chatterjee's xi V diplomski nalogi preučujemo Chatterjeejev koeficient odvisnosti $\xi$ in njegovo posplošitev na pogojno odvisnost $T(Y,Z| X)$, ki kvantificira dodatno informacijo v $Z$ glede na $X$ pri pojasnjevanju $Y$. Za Chatterjeejev $\xi$ obstajata vzorčni koeficient $\xi_n$, ki skoraj gotovo konvergira k $\xi(X,Y)$, ter oblika $\xi$, ki jo dobimo prek parcialnih odvodov kopule. Z obsežnimi simulacijami preverimo (i) nepristranskost ocene $\xi_n$, (ii) odziv na prisotnost šuma, (iii) primerjavo v različnih funkcijskih razmerjih s klasičnimi koeficienti korelacije (Pearsonov $r$, Spearmanov $\rho$ in Kendallov $\tau$) in (iv) približevanje empirične porazdelitve $\sqrt{n}\,\xi_n$ k normalni asimptoti tako pri majhnih kot pri velikih vzorcih. Na podatkovni bazi Galtonovih grahov prikažemo izrazito asimetrijo $\xi$, ki je ena ključnih značilnosti te mere odvisnosti: vrednosti $\xi(X,Y)$ in $\xi(Y,X)$ se lahko bistveno razlikujeta ter s tem razkrijeta smerno (funkcijsko) strukturo povezave. Posplošitev $T(Y,Z| X)$ je prikazana pri oceni prispevka dodatnih pojasnjevalnih spremenljivk: na podatkih o portugalskih vinih empirično ovrednotimo, v kolikšni meri kemijske značilnosti vina prispevajo dodatno informacijo o kakovosti prek že upoštevanih spremenljivk. In this thesis we study Chatterjee's coefficient of dependence $\xi$ and its generalization to conditional dependence $T(Y,Z| X)$, which quantifies the additional information in $Z$ relative to $X$ for explaining $Y$. For Chatterjee's $\xi$ there is a sample version $\xi_n$ that converges almost surely to $\xi(X,Y)$, as well as a representation of $\xi$ via partial derivatives of the copula. Through extensive simulations we examine (i) the equitability of the estimator $\xi_n$, (ii) its response to noise, (iii) comparisons across a range of functional relationships with classical correlation coefficients (Pearson's $r$, Spearman's $\rho$, and Kendall's $\tau$) and (iv) the approach of the empirical distribution of $\sqrt{n}\,\xi_n$ to its normal asymptotic limit for both small and large sample sizes. Using Galton's peas data set, we demonstrate the pronounced asymmetry of $\xi$ - a key feature of this measure of dependence - where $\xi(X,Y)$ and $\xi(Y,X)$ can differ substantially, thereby revealing the directional (functional) structure of the association. The generalization $T(Y,Z| X)$ is illustrated in assessing the contribution of additional explanatory variables: on Portuguese wine data we empirically evaluate the extent to which chemical characteristics provide information about quality beyond variables already taken into account. mera odvisnosti Chatterjeejev ksi koeficient $T$ nepristranskost kopula measure of dependence Chatterjee's xi $T$ coefficient equitability copula true false false Slovenski jezik Angleški jezik Delo diplomskega seminarja/zaključno seminarsko delo/naloga 2025-09-18 08:15:23 2025-09-18 08:15:26 2025-11-10 03:34:57 0000-00-00 00:00:00 2025 0 0 0000-00-00 NiDoloceno NiDoloceno NiDoloceno 0000-00-00 0000-00-00 0000-00-00 519.2 153904 250109699 19529.pdf 19529.pdf 1 D603376CB50A1AC4DCC3AEC6EFFFECE0 b5d7d036b8702dce5b7b79633ff8ea3cb37dd4191e773b3f6c52b7cd2ecf9ed3 b1e07712-9456-11f0-9328-0050569b8976 https://repozitorij.uni-lj.si/Dokument.php?lang=slv&id=218024 Fakulteta za matematiko in fiziko 0 0 0