Without calling polym(), how can I count the number of interactions it will return?

Question

How can I count number of interactions poly will return?

If I have two variables, then the number of interactions poly will return in function of degree is given by:

degree <- 2

dim(poly(rnorm(10), rnorm(10), degree = degree))[2]

That is the same as:

(degree^2+3*degree)/2

Is there anyway to count the number of interactions depending on the number of degree and variables (in case I use more than two)?

Answer 1

Math result from combinations

Suppose you have p variables, the number of interactions associated with degree d is computed by:

fd <- function (p, d) {
  k <- choose(p, d)
  if (d > 1) k <- k + p * sum(choose(p-1, 0:(d-2)))
  return(k)
  }

The function poly (actually polym in this case), with p input variables and a degree = D, will construct interactions from degree = 1 up to degree = D. So the following function counts it:

fD <- function (p, D) {
   if (D < 1) return(0)
   component <- sapply(1:D, fd, p = p)
   list(component = component, ncol = sum(component))
   }

The entry component gives the number of interaction for each degree from 1 to D, and ncol component gives total number of interactions.

---

A quick test:

a <- runif(50)
b <- runif(50)
c <- runif(50)
d <- runif(50)
X <- poly(a, b, c, d, degree = 3)
ncol(X)
# 34

fD(4, 3)
# component
# [1] 4  10  20
#
# ncol
# [1] 34

---

How R does this?

The first few lines of the source code for polym explains how R addresses this problem. An expand.grid is first called to get all possible interactions, then a rowSums is called to compute the degree of all available interactions. Finally, a filter is applied to retain only interactions terms with degree between 1 and D.