A class of Liu-type estimators based on ridge regression under multicollinearity with an application to mixture experiments

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1 A class of Lu-te estmators based o rdge regresso uder multcolleart wth a alcato to mture eermets Preseter: A-Chu Che 陳愛群 Advsor: aesh Emura Jue 6, 5 Graduate Isttute of Statstcs, NCU

2 Outle Itroducto Methodolog hoer Numercal aalss Cocluso

3 Itroducto Model Lear model wth tercet N ] [ wth, ~, I ε ε Stadardzato of the desg matr j for j j j j...,, ad

4 Itroducto Bacgroud Ordar least square OLS estmator Mmze the resdual sum of squares RSS RSS Pros: Ubased ad wth mmum varace

5 Itroducto Multcolleart roblem Multcolleart [ ] [ ] Nearl lear deedet Motgomer et al., Problems here est a least oe small egevalue of j var OLS j j oo large Jmch, 5

6 Itroducto Mture eermets Resose deeds ol o the roortos of the gredets the mture Corel, Eamle: Mae a cae 3 : Flour Satsf : Water : Egg 3 j j Costrats L U, j, j j j, 3

7 Itroducto Motvatg eamle Portlad cemet data Woods et al., 93 Al O 3 4CaO Al O3 FeO3 3 3CaO Al O3 SO CaO SO 4 3CaO SO

8 Itroducto Rdge regresso Hoerl ad Keard 97 Rdge I, Pros: Solve the multcolleart roblem o OLS What ca mrove: Itercet term cosderato? Mea squared error erformace?

9 Itroducto Rdge regresso I the vew of RSS RSS Rdge RSS ealt, rue Rdge OLS Rdge Rdge

10 Itroducto Other rdge-te estmators Lu 993 Lu estmator Lu d I d OLS, d Lu 3 Lu-te estmator Lu *, d I d,, d * ca be a estmator of. Saallıoğlu ad Kaçıralar 6 SK, d I - { d, Rdge }, d

11 Methodolog Proosed method ealt RSS RSS ealt * * *, where * ca be a estmator of. estmator I * Note: Proosed s a secal class of Lu-te estmator Lu, 3 * How to estmate?

12 Methodolog Proosed method Defto Comoud Uvarate Estmator Use uvarate model,,..., to estmate * Use uvarate model * estmate, j,..., j * Comoud uvarate estmator s the comoud of all the uvarate estmators. * j j,,..., * j j j to Emura et al.

13 Methodolog Proosed method Comoud uvarate estmator * } {dag I I I I * ] } {dag [ ] } {dag [ Emura et al.

14 Methodolog Proosed method If forget or do ot wat to stadardze * Comoud uvarate estmator does t have smle form Stll use to estmate } { dag *

15 Methodolog Proosed method Shrage scheme for the roosed method rue OLS { dag }

16 heor Mea squared error calculato otal mea squared error MSE calculato Cosder a lear estmator ~ C ~ ~ ~ MSE E ~ ~ ~ bas bas v ~ bas C I ad ~ v tracecc

17 heor Mea squared error calculato We cosder estmators Lu Lu Rdge Rdge OLS d C d C C Rdge SK Rdge Lu, Lu Rdge OLS ] } { dag [ } {, } { } {, I I I I I I I C dc d C dc C d d C C C d,, SK SK Lu Lu, C d C d C,d d

18 heor Model caocal form Let be the egevalues of ad γ,, where γ be the corresod egevectors Model caocal form A Γ, Γ α [ γ,, γ Γ ε Γ ] ad Aα ε wth Γ ad A A dag,,, Γ Γ Λ Λ dag,,

19 heor Mea squared error calculato Lemma bas ad total varace of ew estmator Bas square otal varace } bas{ } bas{ } v{

20 heor Mea squared error calculato heorem MSE of the ew estmator } MSE{ Lemma Dervatves for the bas square of ew estmator d d 3 } bas{ } bas{ d d 4 } bas{ } bas{

21 heor Mea squared error calculato Lemma 3 Dervatves for total varace of ew estmator d d 3 } v{ d d } ] v{ tr[ Lemma 4 Dervatves for MSE of ew estmator d d 3 3 } MSE{ d d } MSE{

22 heor Estece theorem We have that Note: lm d d bas{ lm Rdge OLS d d v{ } { } } Bas square s flat at otal varace s decreasg at

23 heor Otmal value of shrage arameter Several algorthm estmate Wth-tercet-te model seldom cosder Ofte searate to: tercet term ad other term Numercal mmzato Rdge arg m m arg I real data, we use OLS estmator to relace true value

24 Numercal aalss Smulato desg Four cases: 3, 4 Case Case Case 3 Case 4 5,,,, 5,,,,,,,,,,,, ad ad ad ad

25 Numercal aalss Smulato desg Geerate,, b 3 5 ~ N, 5I, ~ N, 5I 4 5 for,, Samle correlato matr of Samle Corr

26 Numercal aalss Smulato result MSE ad bas-varace trade off Case 5,,,, ad

27 Numercal aalss Smulato result MSE ad bas-varace trade off Case 5,,,, ad

28 Numercal aalss Smulato result MSE ad bas-varace trade off Case 3,,,, ad

29 Numercal aalss Smulato result MSE ad bas-varace trade off Case 4,,,, ad

30 Numercal aalss Smulato result Effects of tercet term ad o rdge & ew method he ew method s more robust agast the chagg of tercet

31 Numercal aalss Smulato result MSE & Shrage arameter estmato

32 Numercal aalss Data aalss Portlad cemet data Woods et al., Heat evolved durg cemet hardeg 3CaO Al O CaO SO CaO Al O Fe 3 O CaO SO

33 Numercal aalss Data aalss Portlad cemet data Woods et al., Heat evolved durg cemet hardeg 3CaO Al O CaO SO CaO Al O Fe 3 O CaO SO

34 Numercal aalss Data aalss Samle correlato matr Samle Corr Chec egevalues matr Λ Λ

35 Numercal aalss Data aalss MSE o Portlad cemet data Mmu MSE Rdge OLS 3.766

Numercal aalss Data aalss Case that do ot stadardze Portlad cemet data 3 Bas Var MSE OLS 6.4.55.5..4 49.9 49.

8 4887.3 ot SK, d, HK, d dot 7.6.9.87.47..46 959.5 7.96 HK, HK 8.7.36.3.9.33 335. 96.78 96., 88.99.8.4.8.4 77.

36 Numercal aalss Data aalss Case that do ot stadardze Portlad cemet data 3 Bas Var MSE OLS Rdge, HK Rdge Rdge , Lu d, d d ot SK, d, HK, d dot HK, HK , /{ Rdge.53 3 NEW OLS OLS }.535 d ot 4 HK Hoerl & Keard, 97 HK Saallıoğlu ad Kaçıralar, 6

37 Numercal aalss Data aalss Case that do ot stadardze Portlad cemet data

38 Numercal aalss Data aalss Flare data McLea ad Aderso, : 3 4 : Amout of llumato, cadles : : : Magesum Sodum trate Strotum trate Bder Costrats

39 Numercal aalss Data aalss Uder stadardzato flare data OLS Does ot est * , Lu d, d d Does ot est Rdge * ot SK *, d,, d dot * , U stadardzato flare data OLS Does ot est * , Lu d, d d Does ot est Rdge * ot SK *, d,, d dot *, * * * * /{ dot * dot }

40 Cocluso We wors o model wth tercet. Acheves the smallest MSE amog OLS ad rdge regresso some case, esecall large tercet cases. Proosed method wors o ustadardzed model.

41 Future wor Scheffé te model mture eermet From mture eermets to other eermet desg

42 HE END HANK YOU

Training Sample Model: Given n observations, [[( Yi, x i the sample model can be expressed as (1) where, zero and variance σ

Training Sample Model: Given n observations, [[( Yi, x i the sample model can be expressed as (1) where, zero and variance σ Stat 74 Estmato for Geeral Lear Model Prof. Goel Broad Outle Geeral Lear Model (GLM): Trag Samle Model: Gve observatos, [[( Y, x ), x = ( x,, xr )], =,,, the samle model ca be exressed as Y = µ ( x, x,,