In R, how to estimate a polynomial function by Least Square method? - r

Question

n = 50
set.seed(100)
x = matrix(runif(n, -2, 2), nrow=n)
y = 2 + 0.75*sin(x) - 0.75*cos(x) + rnorm(n, 0, 0.2)
.
In R,
I want to estimate the above polynomial function by Least Square method.
Which means I want to know the estimate of γ0, γ1, γ2 and γ3.
And I also want to know the MSE under this estimator function.
I used this
summary(lm(y ~ x+ x^2+ x^3))
But just get this output:
Call:
lm(formula = y ~ x + (x^2) + (x^3))
Residuals:
Min 1Q Median 3Q Max
-0.66448 -0.22251 -0.07694 0.20647 0.79429
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 1.54972 0.04761 32.55 <2e-16 ***
x 0.65279 0.04633 14.09 <2e-16 ***
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error: 0.3319 on 48 degrees of freedom
Multiple R-squared: 0.8053, Adjusted R-squared: 0.8013
F-statistic: 198.6 on 1 and 48 DF, p-value: < 2.2e-16
Please tell me in R, what function or package can I use to do it.
Thank you.

Answer 1

You need to wrap I(.) around your polynomials to indicate R that the ^ operator should be used rather as an arithmetical operator than a formula operator.
Let's create some example data to illustrate.
set.seed(42)
x <- runif(100)
y <- x + x^2 + x^3 + rnorm(100)
What you want is this:
lm(y ~ x + I(x^2) + I(x^3))$coe
# (Intercept) x I(x^2) I(x^3)
# -0.4069207 3.4580770 -4.0516060 4.3227674
or:
lm(y ~ poly(x, 3, raw=TRUE))$coe
# (Intercept) poly(x, 3, raw = TRUE)1 poly(x, 3, raw = TRUE)2 poly(x, 3, raw = TRUE)3
# -0.4069207 3.4580770 -4.0516060 4.3227674
Without the I(.) the result is different,
lm(y ~ x + x^2 + x^3)$coe
# (Intercept) x
# -0.6149136 3.3590395
because for some reason R interprets the forumla just as
lm(y ~ x)$coe
# (Intercept) x
# -0.6149136 3.3590395
all(lm(y ~ x)$coe - lm(y ~ x + x^2 + x^3)$coe == 0)
# [1] TRUE
However, consider:
lm(y ~ I(x + x^2 + x^3))$coe
# (Intercept) I(x + x^2 + x^3)
# -0.191611 1.141912
Creating a variable z
z <- x + x^2 + x^3
gives the same result:
lm(y ~ z)$coe
# (Intercept) z
# -0.191611 1.141912
Or, more explicit, say we want to calculate the interaction x with a, we want to use * as formula operator:
a <- runif(100)
lm(y ~ x*a)$coe
# (Intercept) x a x:a
# -0.71920356 3.42008631 0.19180499 -0.07049342
When we use * as an arithmetical operator,
lm(y ~ I(x*a))$coe
# (Intercept) I(x * a)
# 0.5317547 2.7361742
the result is the same as:
xa <- x*a
lm(y ~ xa)$coe
# (Intercept) xa
# 0.5317547 2.7361742