I'm trying to do some regression on variables with the interval [0,1]. I'd like to include orthogonal quadratic and cubic components. The polynomials I am after are Shifted Legendre polynomials(Wikipedia).
I am baffled by the behaviour of the poly() function in R. Not only does it not return a vector in [0,1], the vector it does return varies with the length of the input vector.
This code generates an illustrative example. The code generates some first-order polynomials (lines) from x. The intervals of the resultant polynomials range from [-0.36,0.36] to [-0.017,0.017].
x <- seq(0,1,by = 0.1) # base variable
plot(x,x,col=2)
M<-matrix(nrow=6,ncol=4)
dfpoly<-data.frame(M)
v<- c(0.05,0.01,0.005,0.001,0.0005,0.0001) # This vector alters the length of x
for (i in 1:length(v)){
x <- seq(0,1,by = v[i])
y <- poly(x,degree = 3)[,1] #first order polynomial, should in my mind be the same as x
points(x,y)
dfpoly[i,1] <- length(x)
dfpoly[i,2] <- min(y)
dfpoly[i,3] <- max(y)
dfpoly[i,4] <- mean(diff(y)/diff(x))
}
names(dfpoly) <- c("x length","y min","y max","y slope")
dfpoly
graph of x vs generated first order polynomials
Summary of outputs:
x length y min y max y slope
1 21 -0.36037498508 0.36037498508 0.72074997016
2 101 -0.17064747029 0.17064747029 0.34129494057
3 201 -0.12156314064 0.12156314064 0.24312628128
4 1001 -0.05469022724 0.05469022724 0.10938045447
5 2001 -0.03870080906 0.03870080906 0.07740161813
6 10001 -0.01731791041 0.01731791041 0.03463582082
Now, I'd expect all the lines to inhabit the same interval [0,1] as x, and perfectly overlap with x (the red series of points) on the graph. But they don't. Nor do they have any pattern to them that I can identify by eye.
1. What is the cause of the weird interval behaviour with poly()?
2. Are there other techniques or functions I can use to coerce these polynomials to [0,1]?
The poly() function returns a matrix, whose columns are the values of polynomials evaluated at the values of x. From the help page ?poly, the columns are mutually orthogonal, and are also orthogonal to the constant polynomial p(x) = 1. Orthogonality is in the vector sense (i.e. $\sum x_i y_i = 0$).
I don't think the help page guarantees this, but in practice it appears that the columns are also unit length, i.e. $\sum x_i^2 = 1$.
The unit length condition explains your "weird interval behaviour". More terms means they have to be smaller to still have a sum of squares equal to 1.
To coerce the columns to the range [0,1], just subtract the minimum and divide by the range. This will lose both the orthogonality and unit length properties, but
will keep the degree and linear independence.
For example,
x <- seq(0,1,by = 0.1) # base variable
plot(x,x,col=2)
M<-matrix(nrow=6,ncol=4)
dfpoly<-data.frame(M)
v<- c(0.05,0.01,0.005,0.001,0.0005,0.0001) # This vector alters the length of x
for (i in 1:length(v)){
x <- seq(0,1,by = v[i])
y <- poly(x,degree = 3)[,1] #first order polynomial, should in my mind be the same as x
y <- (y - min(y))/diff(range(y))
points(x,y)
dfpoly[i,1] <- length(x)
dfpoly[i,2] <- min(y)
dfpoly[i,3] <- max(y)
dfpoly[i,4] <- mean(diff(y)/diff(x))
}
names(dfpoly) <- c("x length","y min","y max","y slope")
dfpoly
This prints
x length y min y max y slope
1 21 0 1 1
2 101 0 1 1
3 201 0 1 1
4 1001 0 1 1
5 2001 0 1 1
6 10001 0 1 1