# Relational operators for intervals with the intrval R package

December 02, 2016

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).

## Motivation

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)
# [1] TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE


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

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

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

### 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 %[<o]% and %[o>]%.

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

### 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

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

library(devtools)
install_github("psolymos/intrval")


## Examples

library(intrval)

## bounding box
set.seed(1)
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)$hour %[]% c(0, 11) plot(sin(x) ~ dt, type="l", col="grey", main = "Filtering date/time objects") points(sin(x) ~ dt, pch = 19, col = f + 1) ## QCC library(qcc) data(pistonrings) mu <- mean(pistonrings$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 %
#  TRUE

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


For more examples, see the unit-testing script.

## Feedback

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

#### Update (2016-12-04)

Sergey Kashin pointed out that some operators are redundant. It is now explained in the manual:

Note that some operators return identical results but are syntactically different: %[<]% and %[<)% both evaluate x < a; %[>]% and %(>]% both evaluate x > b; %(<]% and %(<)% evaluate x <= a; %[>)% and %(>)% both evaluate x >= b. This is so because we evaluate only one end of the interval but still conceptually referring to the relationship defined by the right-hand-side interval object. This implies 2 conditional logical evaluations instead of treating it as a single 3-level ordered factor.

#### Update (2016-12-06)

intrval R package v0.1 is on CRAN: https://CRAN.R-project.org/package=intrval

#### Fitting removal models with the detect R package

In a paper recently published in the Condor, titled Evaluating time-removal models for estimating availability of boreal birds during point-count surveys: sample size requirements and model complexity, we assessed different ways of controlling for point-count duration in bird counts using data from the Boreal Avian Modelling Project. As the title indicates, the paper describes a cost-benefit analysis to make recommendations about when to use different types of the removal model. The paper is open access, so feel free to read the whole paper here.