intrval: Relational Operators for Intervals

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.

Install

Install from CRAN:

install.packages("intrval")

Install development version from GitHub:

if (!requireNamespace("remotes")) install.packages("remotes")
remotes::install_github("psolymos/intrval")

User visible changes are listed in the NEWS file.

Use the issue tracker to report a problem.

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
%[]%	%)(%	%[<]%	%[>]%
%[)%	%)[%	%[<)%	%[>)%
%(]%	%](%	%(<]%	%(>]%
%()%	%][%	%(<)%	%(>)%

Dividing a range into 3 intervals

The functions %[c]%, %[c)%, %(c]%, and %(c)% return an integer vector taking values (the c within the brackets refer to 'cut'):

• -1L when the value is less than or equal to a (a <= b), depending on the interval type,
• 0L when the value is inside the interval, or
• 1L when the value is greater than or equal to b (a <= b), depending on the interval type.

Expression	Evaluates to -1	Evaluates to 0	Evaluates to 1
x %[c]% c(a, b)	x < a	x >= a & x <= b	x > b
x %[c)% c(a, b)	x < a	x >= a & x < b	x >= b
x %(c]% c(a, b)	x <= a	x > a & x <= b	x > b
x %(c)% c(a, b)	x <= a	x > a & x < b	x >= b

Interval-to-interval relations

The operators define the open/closed nature of the lower/upper limits of the intervals on the left and right hand side of the o in the middle.

Intervals	Int. 2: []	Int. 2: [)	Int. 2: (]	Int. 2: ()
Int. 1: []	%[]o[]%	%[]o[)%	%[]o(]%	%[]o()%
Int. 1: [)	%[)o[]%	%[)o[)%	%[)o(]%	%[)o()%
Int. 1: (]	%(]o[]%	%(]o[)%	%(]o(]%	%(]o()%
Int. 1: ()	%()o[]%	%()o[)%	%()o(]%	%()o()%

The overlap of two closed intervals, [a1, b1] and [a2, b2], is evaluated by the %[o]% (alias for %[]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. 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), see Examples. 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 of two closed interval overlap, directional evaluation is done via the operators %[<o]% and %[o>]%. The overlap of two open intervals is evaluated by the %(o)% (alias for %()o()%). %]o[% is used for the negation of two open interval overlap, directional evaluation is done via the operators %(<o)% and %(o>)%.

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

Overlap operators with mixed endpoint do not have negation and directional counterparts.

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 %ni% (as in not in, the opposite of in). %nin% and %notin% are aliases for better code readability (%in% can look very much like %ni%).

Examples

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)

Quality control chart (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)

Confidence intervals and hypothesis testing

## 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

Dates

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

dt1 <- as.Date(c("2000-01-01", "2000-03-15"))
dt2 <- as.Date(c("2000-03-15", "2000-06-07"))
dt1 %[]o[]% dt2
# [1] TRUE
dt1 %[]o[)% dt2
# [1] TRUE
dt1 %[]o(]% dt2
# [1] FALSE
dt1 %[]o()% dt2
# [1] FALSE

Watch precedence!

(2 * 1:5) %[]% (c(2, 3) * 2)
# [1] FALSE  TRUE  TRUE FALSE FALSE
2 * 1:5 %[]% (c(2, 3) * 2)
# [1] 0 0 0 2 2
(2 * 1:5) %[]% c(2, 3) * 2
# [1] 2 0 0 0 0
2 * 1:5 %[]% c(2, 3) * 2
# [1] 0 4 4 0 0

Truncated distributions

Find the math here, as implemented in the package truncdist.

dtrunc <- function(x, ..., distr, lwr=-Inf, upr=Inf) {
    f <- get(paste0("d", distr), mode = "function")
    F <- get(paste0("p", distr), mode = "function")
    Fx_lwr <- F(lwr, ..., log=FALSE)
    Fx_upr <- F(upr, ..., log=FALSE)
    fx     <- f(x,   ..., log=FALSE)
    fx / (Fx_upr - Fx_lwr) * (x %[]% c(lwr, upr))
}
n <- 10^4
curve(dtrunc(x, distr="norm"), -2.5, 2.5, ylim=c(0, 2), ylab="f(x)")
curve(dtrunc(x, distr="norm", lwr=-0.5, upr=0.1), add=TRUE, col=4, n=n)
curve(dtrunc(x, distr="norm", lwr=-0.75, upr=0.25), add=TRUE, col=3, n=n)
curve(dtrunc(x, distr="norm", lwr=-1, upr=1), add=TRUE, col=2, n=n)

Shiny example 1: regular slider

library(shiny)
library(intrval)
library(qcc)

data(pistonrings)
mu <- mean(pistonrings$diameter[pistonrings$trial])
SD <- sd(pistonrings$diameter[pistonrings$trial])
x <- pistonrings$diameter[!pistonrings$trial]

## UI function
ui <- fluidPage(
  plotOutput("plot"),
  sliderInput("x", "x SD:",
    min=0, max=5, value=0, step=0.1,
    animate=animationOptions(100)
  )
)

# Server logic
server <- function(input, output) {
  output$plot <- renderPlot({
    Main <- paste("Shewhart quality control chart", 
        "diameter of piston rings", sprintf("+/- %.1f SD", input$x),
        sep="\n")
    iv <- mu + input$x * c(-SD, SD)
    plot(x, pch = 19, col = x %)(% iv +1, type = "b", 
        ylim = mu + 5 * c(-SD, SD), main = Main)
    abline(h = mu)
    abline(h = iv, lty = 2)
  })
}

## Run shiny app
if (interactive()) shinyApp(ui, server)

Shiny example 2: range slider

library(shiny)
library(intrval)

set.seed(1)
n <- 10^4
x <- round(runif(n, -2, 2), 2)
y <- round(runif(n, -2, 2), 2)
d <- round(sqrt(x^2 + y^2), 2)

## UI function
ui <- fluidPage(
  titlePanel("intrval example with shiny"),
  sidebarLayout(
    sidebarPanel(
      sliderInput("bb_x", "x value:",
        min=min(x), max=max(x), value=range(x), 
        step=round(diff(range(x))/20, 1), animate=TRUE
      ),
      sliderInput("bb_y", "y value:",
        min = min(y), max = max(y), value = range(y),
        step=round(diff(range(y))/20, 1), animate=TRUE
      ),
      sliderInput("bb_d", "radial distance:",
        min = 0, max = max(d), value = c(0, max(d)/2),
        step=round(max(d)/20, 1), animate=TRUE
      )
    ),
    mainPanel(
      plotOutput("plot")
    )
  )
)

# Server logic
server <- function(input, output) {
  output$plot <- renderPlot({
    iv1 <- x %[]% input$bb_x & y %[]% input$bb_y
    iv2 <- x %[]% input$bb_y & y %[]% input$bb_x
    iv3 <- d %()% input$bb_d
    op <- par(mfrow=c(1,2))
    plot(x, y, pch = 19, cex = 0.25, col = iv1 + iv2 + 3,
        main = "Intersecting bounding boxes")
    plot(x, y, pch = 19, cex = 0.25, col = iv3 + 1,
         main = "Deck the halls:\ndistance range from center")
    par(op)
  })
}

## Run shiny app
if (interactive()) shinyApp(ui, server)