findInterval: Find Interval Numbers or Indices

Description

Given a vector of non-decreasing breakpoints in vec, find the interval containing each element of x; i.e., if i <- findInterval(x,v), for each index j in x \(v_{i_j} \le x_j < v_{i_j + 1}\) where \(v_0 := -\infty\), \(v_{N+1} := +\infty\), and N <- length(v). At the two boundaries, the returned index may differ by 1, depending on the optional arguments rightmost.closed and all.inside.

Usage

findInterval(x, vec, rightmost.closed = FALSE, all.inside = FALSE,
             left.open = FALSE)

Arguments

numeric.

vec

numeric, sorted (weakly) increasingly, of length N, say.

rightmost.closed

logical; if true, the rightmost interval, vec[N-1] .. vec[N] is treated as closed, see below.

all.inside

logical; if true, the returned indices are coerced into 1,…,N-1, i.e., 0 is mapped to 1 and N to N-1.

left.open

logical; if true all the intervals are open at left and closed at right; in the formulas below, \(\le\) should be swapped with \(<\) (and \(>\) with \(\ge\)), and rightmost.closed means ‘leftmost is closed’. This may be useful, e.g., in survival analysis computations.

Value

vector of length length(x) with values in 0:N (and NA) where N <- length(vec), or values coerced to 1:(N-1) if and only if all.inside = TRUE (equivalently coercing all x values inside the intervals). Note that NAs are propagated from x, and Inf values are allowed in both x and vec.

Details

The function findInterval finds the index of one vector x in another, vec, where the latter must be non-decreasing. Where this is trivial, equivalent to apply( outer(x, vec, ">="), 1, sum), as a matter of fact, the internal algorithm uses interval search ensuring \(O(n \log N)\) complexity where n <- length(x) (and N <- length(vec)). For (almost) sorted x, it will be even faster, basically \(O(n)\). This is the same computation as for the empirical distribution function, and indeed, findInterval(t, sort(X)) is identical to \(n F_n(t; X_1,\dots,X_n)\) where \(F_n\) is the empirical distribution function of \(X_1,\dots,X_n\). When rightmost.closed = TRUE, the result for x[j] = vec[N] (\( = \max vec\)), is N - 1 as for all other values in the last interval. left.open = TRUE is occasionally useful, e.g., for survival data. For (anti-)symmetry reasons, it is equivalent to using “mirrored” data, i.e., the following is always true:

    identical(
          findInterval( x,  v,      left.open= TRUE, ...) ,
      N - findInterval(-x, -v[N:1], left.open=FALSE, ...) )

where N <- length(vec) as above.

Examples

Run this code

x <- 2:18
v <- c(5, 10, 15) # create two bins [5,10) and [10,15)
cbind(x, findInterval(x, v))

N <- 100
X <- sort(round(stats::rt(N, df = 2), 2))
tt <- c(-100, seq(-2, 2, len = 201), +100)
it <- findInterval(tt, X)
tt[it < 1 | it >= N] # only first and last are outside range(X)

##  'left.open = TRUE' means  "mirroring" :
N <- length(v)
stopifnot(identical(
                  findInterval( x,  v,  left.open=TRUE) ,
              N - findInterval(-x, -v[N:1])))