Jádrové odhady regresní funkce pro korelovaná data
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1 Jádrové odhady regresní funkce pro korelovaná data Ústav matematiky a statistiky MÚ Brno Finanční matematika v praxi III., Podlesí
2 Obsah Motivace Motivace
3 Motivace Co se snažíme získat?
4 Motivace Co se snažíme získat?
5 Regresní model Motivace Standardní regresní model Y i =m(x i )+ε i, i = 1,..., n, kde m - neznámá regresní funkce Y i - měřená data x i - body, ve kterých se provádí měření ε i - chyby
6 Jádro Motivace Co je to jádro?
7 Jádro Motivace Co je to jádro? Jádro K je reálná funkce splňující podmínky: K Lip[ 1, 1], K( 1) = K(1) = 0 support(k) = [ 1, 1] 1 j = x j K(x)dx = 0 j = 1 β 2 0 j = 2
8 Epanečnikovo jádro
9 Jak se používá?
10 Jak se používá? Nadaraya-Watsonův odhad regresní funkce m(x, h) = n i=1 j=1 K h (x i x) Y n i, K h (x j x) kde K h (.) = 1 h K ( ḣ ) h - vyhlazovací parametr
11 Vyhlazovací parametr Co se stane, když ho zvoĺıme špatně?
12 Vyhlazovací parametr Co se stane, když ho zvoĺıme špatně?
13 Jaká je optimální hodnota?
14 Jaká je optimální hodnota?
15 Jaká je optimální hodnota? ( σ h opt = K 2 ) (x)dx 1/5 1 nβ2 2A, A 2 = m (x) 2 dx 2 0
16 Krosvalidační metoda Jak zvolit správné h?
17 Krosvalidační metoda Jak zvolit správné h? CV (h) = 1 n n ( ) 2 m j (x j, h) Y j min, j=1 kde m j (x j, h) je odhad m(x j, h), kde je x j smazáno
18 Regresní model s korelovanými chybami Jak se náš model změní?
19 Regresní model s korelovanými chybami Jak se náš model změní? Y i =m(x i )+ε i, i = 1,..., n, kde ε i - neznámý kauzální ARMA proces, neboli E(ε i ) = 0, var(ε i ) = σ 2 cov(ε i, ε j ) = γ i j = σ 2 ρ i j, corr(ε i, ε j ) = ρ i j
20 Proč prostě nepoužijeme předchozí metodu?
21 Proč prostě nepoužijeme předchozí metodu?
22
23 Plug-in přístup Motivace Můžeme nějak využít optimální hodnotu h?
24 Plug-in přístup Motivace Můžeme nějak využít optimální hodnotu h? S = k= ( 1 S 1 h opt = K 2 ) (x)dx 1/5 nβ2 2A, 2 γ k, A 2 = 1 0 m (x) 2 dx
25 Plug-in přístup Motivace Můžeme nějak využít optimální hodnotu h? S = k= ( 1 S 1 h opt = K 2 ) (x)dx 1/5 nβ2 2A, 2 γ k, A 2 = 1 0 m (x) 2 dx ĥ PI = (Ŝ 1 1 K 2 (x)dx nβ 2 2Â2 ) 1/5
26 Jak odhadnout S a A 2?
27 Jak odhadnout S a A 2? odhad S
28 Jak odhadnout S a A 2? odhad S odhadnutím chyb pomocí nějakého počátečního vyhlazení (průměr, přehlazení,...)
29 Jak odhadnout S a A 2? odhad S odhadnutím chyb pomocí nějakého počátečního vyhlazení (průměr, přehlazení,...) - nefunguje
30 Jak odhadnout S a A 2? odhad S odhadnutím chyb pomocí nějakého počátečního vyhlazení (průměr, přehlazení,...) - nefunguje odhadnutím spektrální hustoty f (λ) = 1 2π t= e itλ γ t
31 Jak odhadnout S a A 2? odhad S odhadnutím chyb pomocí nějakého počátečního vyhlazení (průměr, přehlazení,...) - nefunguje odhadnutím spektrální hustoty odhad A 2 f (λ) = 1 2π t= e itλ γ t
32 Jak odhadnout S a A 2? odhad S odhadnutím chyb pomocí nějakého počátečního vyhlazení (průměr, přehlazení,...) - nefunguje odhadnutím spektrální hustoty f (λ) = 1 2π t= odhad A 2 pracuje se na tom - viz Koláček (2008) e itλ γ t
33 Částečná krosvalidace (PCV) Dá se nějak obejít korelace?
34 Částečná krosvalidace (PCV) Dá se nějak obejít korelace? rozdělení pozorování do g skupin tak, že vezmeme každé g-té pozorování minimalizace průměru obyčejných CV každé skupiny CV (h) = g 1 g CV k (h) min k=1
35 Částečná krosvalidace (PCV) Dá se nějak obejít korelace? rozdělení pozorování do g skupin tak, že vezmeme každé g-té pozorování minimalizace průměru obyčejných CV každé skupiny CV (h) = g 1 g CV k (h) min k=1 h minimalizující CV (h) ĥ CV = arg min CV (h) = ĥpcv (g) = g 1/5 h CV
36 Částečná krosvalidace (PCV) Dá se nějak obejít korelace? rozdělení pozorování do g skupin tak, že vezmeme každé g-té pozorování minimalizace průměru obyčejných CV každé skupiny CV (h) = g 1 g CV k (h) min k=1 h minimalizující CV (h) ĥ CV = arg min CV (h) = ĥpcv (g) = g 1/5 h CV Ale volba g závisí γ k, K, A 2
37 Úroková míra CB Kanda Interest rates in Canada Interest rate (%) Time
38 Prvních 200 měsíců Motivace 22 Plug in method PCV method
39 Prvních 150 měsíců Motivace Interes rates in Canada Plug in method PCV method
40 References Motivace Altman, N.S. Kernel Smoothing of Data With Correlated Errors Journal of the American Statistical Association, Vol.85, No.411 (Sep., 1990), pp Bowman, A.W. - Azzalini, A. Applied Smoothing Techniques for Data Analysis Clarendon Press, Oxford, 1997 Chu, C.-K. - Marron, J.S. Comparison of two bandwidth selectors with dependent errors The Annals of statistics, Vol.19, No.4 (1991), pp Härdle, W. - Vieu, P. Kernel Regression Smoothing of Time Series Journal of Time Series Analysis, Vol.13, No.3, (May, 1992), pp
41 References Motivace Herrmann, E. - Gasser, T. - Kneip, A. Choice of Bandwidth for Kernel Regression when Residuals are Correlated Biometrika, Vol.79, No.4 (Dec., 1992), pp Koláček, J. Plug-in method for nonparametric regression Computational Statistics, Vol.23, No.1 (Jan., 2008), pp Opsomer, J. - Wang, Y. - Yang, Y. Nonparametric Regression with Correlated Errors Statistical Science, Vol.16, No.2 (May, 2001), pp
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