Lo primero que debe verificar es si el autor está hablando de polinomios sin procesar frente a polinomios ortogonales .
Para polinomios ortogonales. el coeficiente no se está volviendo "más grande".
Aquí hay dos ejemplos de expansión polinómica de segundo y décimo quinto orden. Primero mostramos el coeficiente para la expansión de segundo orden.
summary(lm(mpg~poly(wt,2),mtcars))
Call:
lm(formula = mpg ~ poly(wt, 2), data = mtcars)
Residuals:
Min 1Q Median 3Q Max
-3.483 -1.998 -0.773 1.462 6.238
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 20.0906 0.4686 42.877 < 2e-16 ***
poly(wt, 2)1 -29.1157 2.6506 -10.985 7.52e-12 ***
poly(wt, 2)2 8.6358 2.6506 3.258 0.00286 **
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error: 2.651 on 29 degrees of freedom
Multiple R-squared: 0.8191, Adjusted R-squared: 0.8066
F-statistic: 65.64 on 2 and 29 DF, p-value: 1.715e-11
Luego mostramos el decimoquinto orden.
summary(lm(mpg~poly(wt,15),mtcars))
Call:
lm(formula = mpg ~ poly(wt, 15), data = mtcars)
Residuals:
Min 1Q Median 3Q Max
-5.3233 -0.4641 0.0072 0.6401 4.0394
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 20.0906 0.4551 44.147 < 2e-16 ***
poly(wt, 15)1 -29.1157 2.5743 -11.310 4.83e-09 ***
poly(wt, 15)2 8.6358 2.5743 3.355 0.00403 **
poly(wt, 15)3 0.2749 2.5743 0.107 0.91629
poly(wt, 15)4 -1.7891 2.5743 -0.695 0.49705
poly(wt, 15)5 1.8797 2.5743 0.730 0.47584
poly(wt, 15)6 -2.8354 2.5743 -1.101 0.28702
poly(wt, 15)7 2.5613 2.5743 0.995 0.33459
poly(wt, 15)8 1.5772 2.5743 0.613 0.54872
poly(wt, 15)9 -5.2412 2.5743 -2.036 0.05866 .
poly(wt, 15)10 -2.4959 2.5743 -0.970 0.34672
poly(wt, 15)11 2.5007 2.5743 0.971 0.34580
poly(wt, 15)12 2.4263 2.5743 0.942 0.35996
poly(wt, 15)13 -2.0134 2.5743 -0.782 0.44559
poly(wt, 15)14 3.3994 2.5743 1.320 0.20525
poly(wt, 15)15 -3.5161 2.5743 -1.366 0.19089
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error: 2.574 on 16 degrees of freedom
Multiple R-squared: 0.9058, Adjusted R-squared: 0.8176
F-statistic: 10.26 on 15 and 16 DF, p-value: 1.558e-05
Tenga en cuenta que estamos utilizando polinomios ortogonales , por lo que el coeficiente del orden inferior es exactamente el mismo que los términos correspondientes en los resultados del orden superior. Por ejemplo, la intersección y el coeficiente para primer orden es 20.09 y -29.11 para ambos modelos.
106 6
> summary(lm(mpg~poly(wt,15, raw=T),mtcars))
Call:
lm(formula = mpg ~ poly(wt, 15, raw = T), data = mtcars)
Residuals:
Min 1Q Median 3Q Max
-5.6217 -0.7544 0.0306 1.1678 5.4308
Coefficients: (3 not defined because of singularities)
Estimate Std. Error t value Pr(>|t|)
(Intercept) 6.287e+05 9.991e+05 0.629 0.537
poly(wt, 15, raw = T)1 -2.713e+06 4.195e+06 -0.647 0.526
poly(wt, 15, raw = T)2 5.246e+06 7.893e+06 0.665 0.514
poly(wt, 15, raw = T)3 -6.001e+06 8.784e+06 -0.683 0.503
poly(wt, 15, raw = T)4 4.512e+06 6.427e+06 0.702 0.491
poly(wt, 15, raw = T)5 -2.340e+06 3.246e+06 -0.721 0.480
poly(wt, 15, raw = T)6 8.537e+05 1.154e+06 0.740 0.468
poly(wt, 15, raw = T)7 -2.184e+05 2.880e+05 -0.758 0.458
poly(wt, 15, raw = T)8 3.809e+04 4.910e+04 0.776 0.447
poly(wt, 15, raw = T)9 -4.212e+03 5.314e+03 -0.793 0.438
poly(wt, 15, raw = T)10 2.382e+02 2.947e+02 0.809 0.429
poly(wt, 15, raw = T)11 NA NA NA NA
poly(wt, 15, raw = T)12 -5.642e-01 6.742e-01 -0.837 0.413
poly(wt, 15, raw = T)13 NA NA NA NA
poly(wt, 15, raw = T)14 NA NA NA NA
poly(wt, 15, raw = T)15 1.259e-04 1.447e-04 0.870 0.395
Residual standard error: 2.659 on 19 degrees of freedom
Multiple R-squared: 0.8807, Adjusted R-squared: 0.8053
F-statistic: 11.68 on 12 and 19 DF, p-value: 2.362e-06