R: difficulties generating orthogonal polynomials between 0 and 1

Question

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]?

Answer 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 (ie $\\sum x_i y_i = 0$).

I don't think the help page guarantees this, but in practice it appears that the columns are also unit length, ie $\\sum x_i^2 = 1$.

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