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Jackknife

Jackknife. Motiveeriv näide. Soovime hinnata suurust f (E X ), näiteks log(EX) = ? (EX) 4 = ? sin(EX) = ? Idee: kui kasutaks valimikeskmist? log(EX) ≈ log(x) (EX) 4 ≈ x 4. Paraku saame nihkega hinnangu.

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Jackknife

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  1. Jackknife

  2. Motiveeriv näide • Soovime hinnata suurust f(EX), näiteks log(EX) = ? (EX)4 = ? sin(EX) = ? Idee: kui kasutaks valimikeskmist? log(EX) ≈ log(x) (EX)4≈ x4 .....

  3. Paraku saame nihkega hinnangu... Statistikas on hinnangu nihe üllatavalt sageli suurusjärgus k/n ( +o(n-1) ) Kuidas sellisest nihkest lahti saada (või vähemalt teda pisendada)?

  4. Algidee • Olgu antud statistik S • Vaatame statistikut S*(i) = nS – (n-1) S(i) kus S(i) on statistiku S väärtus arvutatud ilma i. vaatlust kasutamata. Antud statistiku nihe on: nihe( S*(i)) = n nihe(S) - (n - 1) nihe(S(i)) nihe( S*(i)) = n (c/n+...) - (n - 1) (c/(n-1)+...) nihe( S*(i)) = c + n∙... - c – (n-1)∙...

  5. Miks algidee ei kõlba? • Millise vaatluse me paljude seast välja valime? Tulemus sõltub ju sellest? • Kui statistikuks on valimikeskmine, siis mis tuleb suuruse S*(i) väärtuseks?

  6. Lahendus • Leiame n erinevat nihketa (vähendatud nihkega) statistikut ja lõpphinnanguks kasutame nende hinnangute keskmist:

  7. Usaldusintervallid Tukey (1958) ligikaudsed usaldusintervallid: S*+ta/2; df=n-1 s* ... S*+t1-a/2; df=n-1 s*

  8. Näide: (EX)4 • n=10 • Valimikeskmine4 : Nihe: (2.3...2.6) Jackknife: Nihe: (-0.1...0.2)

  9. Determinatsioonikordaja ja uhke mudel > x=1:5 > y=0*x+rnorm(5) > m=lm(y~x+I(x^2)) > summary(m)$r.square [1] 0.3085117 R2 = 0,31

  10. > S=summary(m)$r.square > > Sn_1=rep(NA, 5) > for(i in 1:5){ + ym=y[-i] + xm=x[-i] + m1=lm(ym~xm+I(xm^2)) + Sn_1=summary(m1)$r.square + } > > Si_tarn=5*S-4*Sn_1 > mean(Si_tarn) [1] -0.9277159 R2 = -0,93 ???

  11. R2 = ( D(Y) – D(prognoosivead) ) / D(Y) > m=lm(y~x+I(x^2)) > xx=rep(x, 10000) > yy=0*x+rnorm(5*10000) > prognoosiviga=yy-predict(m, data.frame(x=xx)) > (var(yy)-var(prognoosiviga))/var(yy) [1] -0.225787

  12. Mudeli valik

  13. Mudeli valik

  14. Ristvalideerimine(Cross-Validation)

  15. Ristvalideerimine(Cross-Validation)

  16. Ristvalideerimine(Cross-Validation)

  17. Ristvalideerimine(Cross-Validation)

  18. Ristvalideerimine(Cross-Validation)

  19. Ristvalideerimine(Cross-Validation)

  20. Ristvalideerimine(Cross-Validation)

  21. Ristvalideerimine(Cross-Validation) RV= 2,55 Ristvalideerimisel RV=17,71

  22. RV=3,69 Ristvalideerimisel RV=5,77

  23. Leave-One-Out Cross-Validation > m1=lm(y~x+I(x^2)) > sum((y-predict(m1))**2) [1] 2.554625 > > m2=lm(y~1) > sum((y-predict(m2))**2) [1] 3.694387 prog1=rep(NA, 5) prog2=rep(NA, 5) for (i in 1:5){ ym=y[-i] xm=x[-i] m1=lm(ym~xm+I(xm^2)) prog1[i]=predict(m1, data.frame(xm=x[i])) m2=lm(ym~1) prog2[i]=predict(m2, data.frame(xm=x[i])) } > sum((prog1-y)**2) [1] 17.71242 > sum((prog2-y)**2) [1] 5.77248

  24. > summary(m1) Call: lm(formula = y ~ x + I(x^2)) Residuals: 1 2 3 4 5 0.009211 0.336618 -1.065121 1.083543 -0.364252 Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) -1.21858 2.42397 -0.503 0.665 x 0.63302 1.84723 0.343 0.764 I(x^2) -0.05011 0.30205 -0.166 0.883 Residual standard error: 1.13 on 2 degrees of freedom Multiple R-squared: 0.3085, Adjusted R-squared: -0.383 F-statistic: 0.4462 on 2 and 2 DF, p-value: 0.6915

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