Relational operators for intervals with the intrval R package

December 2, 2016

(This article was first published on Peter Solymos - R related posts, and kindly contributed to R-bloggers)

I recently posted a piece about how to write and document special functions in R. I meant that as a prelude for the topic I am writing about in this post. Let me start at the beginning. The other day Dirk Eddelbuettel tweeted about the new release of the data.table package (v1.9.8).
There were new features announced for joins based on %inrange% and %between%. That got me thinking: it would be really cool to generalize this idea for different intervals, for example as x %[]% c(a, b).


We want to evaluate if values of x satisfy the condition x >= a & x <= b given that a <= b. Typing x %[]% c(a, b) instead of the previous expression is not much shorter (14 vs. 15 characters with counting spaces). But considering the a <= b condition as well, it becomes a saving (x >= min(a, b) & x <= mmax(a, b) is 31 characters long). And sorting is really important, because by flipping a and b, we get quite different answers:

x <- 5
x >= 1 & x <= 10
# [1] TRUE
x >= 10 & x <= 1
# [1] FALSE

Also, min and max will not be very useful when we want to vectorize the expression. We need to use pmin and pmax for obvious reasons:

x >= min(1:10, 10:1) & x <= max(10:1, 1:10)
# [1] TRUE
x >= pmin(1:10, 10:1) & x <= pmax(10:1, 1:10)

If interval endpoints can also be open or closed, and allowing them to flip around makes the semantics of left/right closed/open interval definitions hard. We can thus all agree that there is a need for an expression, like x %[]% c(a, b), that is compact, flexible, and invariant to endpoint sorting. This is exactly what the intrval package is for!

What’s in the package

Functions for evaluating if values of vectors are within
different open/closed intervals
(x %[]% c(a, b)), or if two closed
intervals overlap (c(a1, b1) %[o]% c(a2, b2)).
Operators for negation and directional relations also implemented.

Value-to-interval relations

Values of x are compared to interval endpoints a and b (a <= b).
Endpoints can be defined as a vector with two values (c(a, b)): these values will be compared as a single interval with each value in x.
If endpoints are stored in a matrix-like object or a list,
comparisons are made element-wise.

x <- rep(4, 5)
a <- 1:5
b <- 3:7
cbind(x=x, a=a, b=b)
x %[]% cbind(a, b) # matrix
x %[]% data.frame(a=a, b=b) # data.frame
x %[]% list(a, b) # list

If lengths do not match, shorter objects are recycled. Return values are logicals.
Note: interval endpoints are sorted internally thus ensuring the condition
a <= b is not necessary.

These value-to-interval operators work for numeric (integer, real) and ordered vectors, and object types which are measured at least on ordinal scale (e.g. dates).

Closed and open intervals

The following special operators are used to indicate closed ([, ]) or open ((, )) interval endpoints:

Operator Expression Condition
%[]% x %[]% c(a, b) x >= a & x <= b
%[)% x %[)% c(a, b) x >= a & x < b
%(]% x %(]% c(a, b) x > a & x <= b
%()% x %()% c(a, b) x > a & x < b

Negation and directional relations

Eqal Not equal Less than Greater than
%[]% %)(% %[<]% %[>]%
%[)% %)[% %[<)% %[>)%
%(]% %](% %(<]% %(>]%
%()% %][% %(<)% %(>)%

The helper function intrval_types can be used to
print/plot the following summary:

Interval types

Interval-to-interval relations

The overlap of two closed intervals, [a1, b1] and [a2, b2],
is evaluated by the %[o]% operator (a1 <= b1, a2 <= b2).
Endpoints can be defined as a vector with two values
(c(a1, b1))or can be stored in matrix-like objects or a lists
in which case comparisons are made element-wise.
Note: interval endpoints
are sorted internally thus ensuring the conditions
a1 <= b1 and a2 <= b2 is not necessary.

c(2:3) %[o]% c(0:1)
list(0:4, 1:5) %[o]% c(2:3)
cbind(0:4, 1:5) %[o]% c(2:3)
data.frame(a=0:4, b=1:5) %[o]% c(2:3)

If lengths do not match, shorter objects are recycled.
These value-to-interval operators work for numeric (integer, real)
and ordered vectors, and object types which are measured at
least on ordinal scale (e.g. dates).

%)o(% is used for the negation,
directional evaluation is done via the operators %[ and %[o>]%.

Eqal Not equal Less than Greater than
%[0]% %)0(% %[<0]% %[0>]%

Operators for discrete variables

The previous operators will return NA for unordered factors.
Set overlap can be evaluated by the base %in% operator and its negation
%nin%. (This feature is really redundant, I know, but decided to include regardless…)


Install development version from GitHub (not yet on CRAN):


The package is licensed under GPL-2.


Interval examples


## bounding box
n <- 10^4
x <- runif(n, -2, 2)
y <- runif(n, -2, 2)
d <- sqrt(x^2 + y^2)
iv1 <- x %[]% c(-0.25, 0.25) & y %[]% c(-1.5, 1.5)
iv2 <- x %[]% c(-1.5, 1.5) & y %[]% c(-0.25, 0.25)
iv3 <- d %()% c(1, 1.5)
plot(x, y, pch = 19, cex = 0.25, col = iv1 + iv2 + 1,
    main = "Intersecting bounding boxes")
plot(x, y, pch = 19, cex = 0.25, col = iv3 + 1,
     main = "Deck the halls:\ndistance range from center")

## time series filtering
x <- seq(0, 4*24*60*60, 60*60)
dt <- as.POSIXct(x, origin="2000-01-01 00:00:00")
f <- as.POSIXlt(dt)$hour %[]% c(0, 11)
plot(sin(x) ~ dt, type="l", col="grey",
    main = "Filtering date/time objects")
points(sin(x) ~ dt, pch = 19, col = f + 1)

## QCC
mu <- mean(pistonrings$diameter[pistonrings$trial])
SD <- sd(pistonrings$diameter[pistonrings$trial])
x <- pistonrings$diameter[!pistonrings$trial]
iv <- mu + 3 * c(-SD, SD)
plot(x, pch = 19, col = x %)(% iv +1, type = "b", ylim = mu + 5 * c(-SD, SD),
    main = "Shewhart quality control chart\ndiameter of piston rings")
abline(h = mu)
abline(h = iv, lty = 2)

## Annette Dobson (1990) "An Introduction to Generalized Linear Models".
## Page 9: Plant Weight Data.
ctl <- c(4.17,5.58,5.18,6.11,4.50,4.61,5.17,4.53,5.33,5.14)
trt <- c(4.81,4.17,4.41,3.59,5.87,3.83,6.03,4.89,4.32,4.69)
group <- gl(2, 10, 20, labels = c("Ctl","Trt"))
weight <- c(ctl, trt)

lm.D9 <- lm(weight ~ group)
## compare 95% confidence intervals with 0
(CI.D9 <- confint(lm.D9))
#                2.5 %    97.5 %
# (Intercept)  4.56934 5.4946602
# groupTrt    -1.02530 0.2833003
0 %[]% CI.D9
# (Intercept)    groupTrt
#       FALSE        TRUE

lm.D90 <- lm(weight ~ group - 1) # omitting intercept
## compare 95% confidence of the 2 groups to each other
(CI.D90 <- confint(lm.D90))
#            2.5 %  97.5 %
# groupCtl 4.56934 5.49466
# groupTrt 4.19834 5.12366
CI.D90[1,] %[o]% CI.D90[2,]
# 2.5 %

DATE <- as.Date(c("2000-01-01","2000-02-01", "2000-03-31"))
DATE %[<]% as.Date(c("2000-01-151", "2000-03-15"))
DATE %[]% as.Date(c("2000-01-151", "2000-03-15"))
DATE %[>]% as.Date(c("2000-01-151", "2000-03-15"))

For more examples, see the unit-testing script.


Please check out the package and use the issue tracker
to suggest a new feature or report a problem.

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