hxd1011 - 4 months ago 29

R Question

Why I have the exact same model, but run predictions on **different grid size (by 0.001 vs by 0.01)** getting different predictions?

`set.seed(0)`

n_data=2000

x=runif(n_data)-0.5

y=0.1*sin(x*30)/x+runif(n_data)

plot(x,y)

poly_df=5

x_exp=as.data.frame(cbind(y,poly(x, poly_df)))

fit=lm(y~.,data=x_exp)

x_plt1=seq(-1,1,0.001)

x_plt_exp1=as.data.frame(poly(x_plt1,poly_df))

lines(x_plt1,predict(fit,x_plt_exp1),lwd=3,col=2)

x_plt2=seq(-1,1,0.01)

x_plt_exp2=as.data.frame(poly(x_plt2,poly_df))

lines(x_plt2,predict(fit,x_plt_exp2),lwd=3,col=3)

Answer

This is a coding / programming problem as on my quick run I can't reproduce this **with appropriate set-up by putting poly() inside model formula**. So I think this question better suited for Stack Overflow.

```
## quick test ##
set.seed(0)
x <- runif(2000) - 0.5
y <- 0.1 * sin(x * 30) / x + runif(2000)
plot(x,y)
x_exp <- data.frame(x, y)
fit <- lm(y ~ poly(x, 5), data = x_exp)
x1 <- seq(-1, 1, 0.001)
y1 <- predict(fit, newdata = list(x = x1))
lines(x1, y1, lwd = 5, col = 2)
x2 <- seq(-1, 1, 0.01)
y2 <- predict(fit, newdata = list(x = x2))
lines(x2, y2, lwd = 2, col = 3)
```

*cuttlefish44* has pointed out the fault in your implementation. **When making prediction matrix, we want to use the construction information in model matrix, rather than constructing a new set of basis.** If you wonder what such "construction information" is, perhaps you can go through this very long answer: How poly() generates orthogonal polynomials? How to understand the “coefs” returned?

Perhaps I can try making a brief summary and getting around that long, detailed answer.

- The construction of orthogonal polynomial always starts from centring the input covariate values
`x`

. If this centre is different, then all the rest will be different. Now, this is the difference between`poly(x, coef = NULL)`

and`poly(x, coef = some_coefficients)`

. The former will always construct a new set of basis using a new centre, while the latter, will use the existing centring information in`some_coefficients`

to predict basis value on given set-up. Surely this is what we want when making prediction. `poly(x, coef = some_coefficients)`

will actually call`predict.poly`

(which I explained in that long answer). It is relatively rare when we need to set`coef`

argument ourselves, unless we are doing testing. If we set up the linear model using the way I present in my quick run above,`predict.lm`

is smart enough to realize the correct way to predict`poly`

model terms, i.e., internally it will do the`poly(new_x, coef = some_coefficients)`

for us.- As an interesting contrast, ordinary polynomial don't have problem with this. For example, if you specify
`raw = TRUE`

in all`poly()`

calls in your code, you will have no trouble. This is because raw polynomial has no construction information; it is just taking powers`1, 2, ... degree`

of`x`

.