# GenomicTuples: Classes and Methods

Peter Hickey

#### Modified: 12 February 2015. Compiled: 26 Mar 2020

# Contents

# 1 Introduction

The GenomicTuples R package defines general purpose containers for storing genomic tuples. It aims to provide functionality for tuples of genomic co-ordinates that are analogous to those available for genomic ranges in the GenomicRanges Bioconductor package.

As you will see, the functionality of the GenomicTuples package is based almost entirely on the wonderful GenomicRanges package. Therefore, I have tried to keep the user interface as similar as possible. This vignette is also heavily based on the vignette “An Introduction to Genomic Ranges Classes”, which is included with the GenomicRanges package1. While not essential, familiarity with the GenomicRanges will be of benefit in understanding the GenomicTuples package.

## 1.1 What is a genomic tuple?

A genomic tuple is defined by a sequence name (`seqnames`), a strand (`strand`) and a tuple (`tuples`). All positions in a genomic tuple must be on the same strand and sorted in ascending order. Each tuple has an associated `size`, which is a positive integer. For example, `chr1:+:{34, 39, 60}` is a 3-tuple (`size` = 3) of the positions `chr1:34`, `chr1:39` and `chr1:60` on the `+` strand.

When referring to genomic tuples of a general (fixed) `size`, I will abbreviate these to \(m\)-tuples, where \(m\) = `size`. I will refer to the first position as \(pos_{1}\) (`pos1`), the second as \(pos_{2}\) (`pos2`), \(\ldots{}\), and the final position as \(pos_{m}\) (`posm`).

The difference between a genomic tuple and a genomic range can be thought of as the difference between a set and an interval. For example, the genomic tuple `chr10:-:{800, 900}` only includes the positions `chr10:-:800` and `chr10:-:900` whereas the genomic range `chr10:-:[800, 900]` includes the positions `chr10:-:800`, `chr10:-:801`, `chr10:-:802`, \(\ldots{}\), `chr10:-:900`.

## 1.2 When might you need a genomic tuple?

In short, whenever the co-ordinates of your genomic data are better defined by a set than by an interval.

The original use case for the GTuples class was to store the genomic co-ordinates of “methylation patterns”. I am currently developing these ideas in a separate R package, MethylationTuples, which makes heavy use of the GTuples class. Other genomic data, such as long reads containing multiple variants, may also be better conceptualised as genomic tuples rather than as genomic ranges and therefore may benefit from the GenomicTuples infrastructure.

# 2 GTuples

The GTuples class represents a collection of genomic tuples, where each tuple has the same `size`. These objects can be created by using the `GTuples` constructor function. For example, the following code creates a GTuples object with 10 genomic tuples:

    library(GenomicTuples)
    #> Loading required package: GenomicRanges
    #> Loading required package: stats4
    #> Loading required package: BiocGenerics
    #> Loading required package: parallel
    #>
    #> Attaching package: 'BiocGenerics'
    #> The following objects are masked from 'package:parallel':
    #>
    #>     clusterApply, clusterApplyLB, clusterCall, clusterEvalQ,
    #>     clusterExport, clusterMap, parApply, parCapply, parLapply,
    #>     parLapplyLB, parRapply, parSapply, parSapplyLB
    #> The following objects are masked from 'package:stats':
    #>
    #>     IQR, mad, sd, var, xtabs
    #> The following objects are masked from 'package:base':
    #>
    #>     Filter, Find, Map, Position, Reduce, anyDuplicated, append,
    #>     as.data.frame, basename, cbind, colnames, dirname, do.call,
    #>     duplicated, eval, evalq, get, grep, grepl, intersect, is.unsorted,
    #>     lapply, mapply, match, mget, order, paste, pmax, pmax.int, pmin,
    #>     pmin.int, rank, rbind, rownames, sapply, setdiff, sort, table,
    #>     tapply, union, unique, unsplit, which, which.max, which.min
    #> Loading required package: S4Vectors
    #>
    #> Attaching package: 'S4Vectors'
    #> The following object is masked from 'package:base':
    #>
    #>     expand.grid
    #> Loading required package: IRanges
    #>
    #> Attaching package: 'IRanges'
    #> The following object is masked from 'package:grDevices':
    #>
    #>     windows
    #> Loading required package: GenomeInfoDb

    seqinfo <- Seqinfo(paste0("chr", 1:3), c(1000, 2000, 1500), NA, "mock1")
    gt3 <- GTuples(seqnames = Rle(c("chr1", "chr2", "chr1", "chr3"),
                                  c(1, 3, 2, 4)),
                   tuples = matrix(c(1:10, 2:11, 3:12), ncol = 3),
                   strand = Rle(strand(c("-", "+", "*", "+", "-")),
                                c(1, 2, 2, 3, 2)),
                   score = 1:10, GC = seq(1, 0, length = 10), seqinfo = seqinfo)
    names(gt3) <- letters[1:10]
    gt3
    #> GTuples object with 10 x 3-tuples and 2 metadata columns:
    #>     seqnames pos1 pos2 pos3 strand | score       GC
    #>   a     chr1    1    2    3      - |     1 1.000000
    #>   b     chr2    2    3    4      + |     2 0.888889
    #>   c     chr2    3    4    5      + |     3 0.777778
    #>   d     chr2    4    5    6      * |     4 0.666667
    #>   e     chr1    5    6    7      * |     5 0.555556
    #>   f     chr1    6    7    8      + |     6 0.444444
    #>   g     chr3    7    8    9      + |     7 0.333333
    #>   h     chr3    8    9   10      + |     8 0.222222
    #>   i     chr3    9   10   11      - |     9 0.111111
    #>   j     chr3   10   11   12      - |    10 0.000000
    #>   ---
    #>   seqinfo: 3 sequences from mock1 genome

The output of the GTuples `show` method is very similar to that of the `show` method for GenomicRanges::GRanges objects. Namely, it separates the information into a left and right hand region that are separated by `|` symbols. The genomic coordinates (`seqnames`, `tuples`, and `strand`) are located on the left-hand side and the metadata columns (annotation) are located on the right. For this example, the metadata is comprised of `score` and `GC` information, but almost anything can be stored in the metadata portion of a GTuples object.

The main difference between a GTuples object and GenomicRanges::GRanges object is that the former uses tuples while the latter uses ranges in the genomic coordinates.

For even more information on the GTuples class, be sure to consult the documentation:

    ?GTuples

## 2.1 GTuples methods

Most methods defined for GenomicRanges::GRanges are also defined for GTuples. Those that are not yet defined, which are those that make sense for ranges but generally not for tuples, return error messages.

If you require a method that is not defined for GTuples but is defined for GenomicRanges::GRanges, then this can be achieved by first coercing the GTuples object to a GenomicRanges::GRanges object; Warning: coercing a GTuples object to a GenomicRanges::GRanges is generally a destructive operation.

    as(gt3, "GRanges")
    #> GRanges object with 10 ranges and 2 metadata columns:
    #>     seqnames    ranges strand |     score        GC
    #>        <Rle> <IRanges>  <Rle> | <integer> <numeric>
    #>   a     chr1       1-3      - |         1  1.000000
    #>   b     chr2       2-4      + |         2  0.888889
    #>   c     chr2       3-5      + |         3  0.777778
    #>   d     chr2       4-6      * |         4  0.666667
    #>   e     chr1       5-7      * |         5  0.555556
    #>   f     chr1       6-8      + |         6  0.444444
    #>   g     chr3       7-9      + |         7  0.333333
    #>   h     chr3      8-10      + |         8  0.222222
    #>   i     chr3      9-11      - |         9  0.111111
    #>   j     chr3     10-12      - |        10  0.000000
    #>   -------
    #>   seqinfo: 3 sequences from mock1 genome

### 2.1.1 Basic GTuples accessors

The components of the genomic coordinates within a GTuples object can be extracted using the `seqnames`, `tuples`, and `strand` accessor functions. Warning: The `tuples` accessor should be used in place of the `ranges` accessor. While the `ranges` method is well-defined, namely it accesses \(pos_{1}\) and \(pos_{m}\) of the object, this is not generally what is desired or required.

    seqnames(gt3)
    #> factor-Rle of length 10 with 4 runs
    #>   Lengths:    1    3    2    4
    #>   Values : chr1 chr2 chr1 chr3
    #> Levels(3): chr1 chr2 chr3
    tuples(gt3)
    #>       pos1 pos2 pos3
    #>  [1,]    1    2    3
    #>  [2,]    2    3    4
    #>  [3,]    3    4    5
    #>  [4,]    4    5    6
    #>  [5,]    5    6    7
    #>  [6,]    6    7    8
    #>  [7,]    7    8    9
    #>  [8,]    8    9   10
    #>  [9,]    9   10   11
    #> [10,]   10   11   12
    strand(gt3)
    #> factor-Rle of length 10 with 5 runs
    #>   Lengths: 1 2 2 3 2
    #>   Values : - + * + -
    #> Levels(3): + - *

Stored annotations for these coordinates can be extracted as a DataFrame object using the `mcols` accessor:

    mcols(gt3)
    #> DataFrame with 10 rows and 2 columns
    #>       score        GC
    #>   <integer> <numeric>
    #> a         1  1.000000
    #> b         2  0.888889
    #> c         3  0.777778
    #> d         4  0.666667
    #> e         5  0.555556
    #> f         6  0.444444
    #> g         7  0.333333
    #> h         8  0.222222
    #> i         9  0.111111
    #> j        10  0.000000

Seqinfo can be extracted using the `seqinfo` accessor:

    seqinfo(gt3)
    #> Seqinfo object with 3 sequences from mock1 genome:
    #>   seqnames seqlengths isCircular genome
    #>   chr1           1000         NA  mock1
    #>   chr2           2000         NA  mock1
    #>   chr3           1500         NA  mock1

Methods for accessing the length and names are also defined:

    length(gt3)
    #> [1] 10
    names(gt3)
    #>  [1] "a" "b" "c" "d" "e" "f" "g" "h" "i" "j"

### 2.1.2 Splitting and combining GTuples objects}

GTuples objects can be divided into groups using the `split` method. This produces a GTuplesList object, a class that will be discussed in detail in the next section:

    sp <- split(gt3, rep(1:2, each=5))
    sp
    #> GTuplesList object of length 2:
    #> $`1`
    #> GTuples object with 5 x 3-tuples and 2 metadata columns:
    #>     seqnames pos1 pos2 pos3 strand | score       GC
    #>   a     chr1    1    2    3      - |     1 1.000000
    #>   b     chr2    2    3    4      + |     2 0.888889
    #>   c     chr2    3    4    5      + |     3 0.777778
    #>   d     chr2    4    5    6      * |     4 0.666667
    #>   e     chr1    5    6    7      * |     5 0.555556
    #>   ---
    #>   seqinfo: 3 sequences from mock1 genome
    #>
    #> $`2`
    #> GTuples object with 5 x 3-tuples and 2 metadata columns:
    #>     seqnames pos1 pos2 pos3 strand | score       GC
    #>   f     chr1    6    7    8      + |     6 0.444444
    #>   g     chr3    7    8    9      + |     7 0.333333
    #>   h     chr3    8    9   10      + |     8 0.222222
    #>   i     chr3    9   10   11      - |     9 0.111111
    #>   j     chr3   10   11   12      - |    10 0.000000
    #>   ---
    #>   seqinfo: 3 sequences from mock1 genome

If you then grab the components of this `GenomicTuplesList`, they can also be combined by using the `c` and `append` methods:

    c(sp[[1]], sp[[2]])
    #> GTuples object with 10 x 3-tuples and 2 metadata columns:
    #>     seqnames pos1 pos2 pos3 strand | score       GC
    #>   a     chr1    1    2    3      - |     1 1.000000
    #>   b     chr2    2    3    4      + |     2 0.888889
    #>   c     chr2    3    4    5      + |     3 0.777778
    #>   d     chr2    4    5    6      * |     4 0.666667
    #>   e     chr1    5    6    7      * |     5 0.555556
    #>   f     chr1    6    7    8      + |     6 0.444444
    #>   g     chr3    7    8    9      + |     7 0.333333
    #>   h     chr3    8    9   10      + |     8 0.222222
    #>   i     chr3    9   10   11      - |     9 0.111111
    #>   j     chr3   10   11   12      - |    10 0.000000
    #>   ---
    #>   seqinfo: 3 sequences from mock1 genome

### 2.1.3 Subsetting GTuples objects

The expected subsetting operations are also available for GTuples objects:

    gt3[2:3]
    #> GTuples object with 2 x 3-tuples and 2 metadata columns:
    #>     seqnames pos1 pos2 pos3 strand | score       GC
    #>   b     chr2    2    3    4      + |     2 0.888889
    #>   c     chr2    3    4    5      + |     3 0.777778
    #>   ---
    #>   seqinfo: 3 sequences from mock1 genome

A second argument to the `[` subset operator can be used to specify which metadata columns to extract from the GTuples object. For example:

    gt3[2:3, "GC"]
    #> GTuples object with 2 x 3-tuples and 1 metadata column:
    #>     seqnames pos1 pos2 pos3 strand |       GC
    #>   b     chr2    2    3    4      + | 0.888889
    #>   c     chr2    3    4    5      + | 0.777778
    #>   ---
    #>   seqinfo: 3 sequences from mock1 genome

You can also assign into elements of the GTuples object. Here is an example where the 2nd row of a GTuples object is replaced with the 1st row of `gt3`:

    gt3_mod <- gt3
    gt3_mod[2] <- gt3[1]
    head(gt3_mod, n = 3)
    #> GTuples object with 3 x 3-tuples and 2 metadata columns:
    #>     seqnames pos1 pos2 pos3 strand | score       GC
    #>   a     chr1    1    2    3      - |     1 1.000000
    #>   b     chr1    1    2    3      - |     1 1.000000
    #>   c     chr2    3    4    5      + |     3 0.777778
    #>   ---
    #>   seqinfo: 3 sequences from mock1 genome

There are also methods to repeat, reverse, or select specific portions of GTuples objects:

    rep(gt3[2], times = 3)
    #> GTuples object with 3 x 3-tuples and 2 metadata columns:
    #>     seqnames pos1 pos2 pos3 strand | score       GC
    #>   b     chr2    2    3    4      + |     2 0.888889
    #>   b     chr2    2    3    4      + |     2 0.888889
    #>   b     chr2    2    3    4      + |     2 0.888889
    #>   ---
    #>   seqinfo: 3 sequences from mock1 genome
    rev(gt3)
    #> GTuples object with 10 x 3-tuples and 2 metadata columns:
    #>     seqnames pos1 pos2 pos3 strand | score       GC
    #>   j     chr3   10   11   12      - |    10 0.000000
    #>   i     chr3    9   10   11      - |     9 0.111111
    #>   h     chr3    8    9   10      + |     8 0.222222
    #>   g     chr3    7    8    9      + |     7 0.333333
    #>   f     chr1    6    7    8      + |     6 0.444444
    #>   e     chr1    5    6    7      * |     5 0.555556
    #>   d     chr2    4    5    6      * |     4 0.666667
    #>   c     chr2    3    4    5      + |     3 0.777778
    #>   b     chr2    2    3    4      + |     2 0.888889
    #>   a     chr1    1    2    3      - |     1 1.000000
    #>   ---
    #>   seqinfo: 3 sequences from mock1 genome
    head(gt3, n = 2)
    #> GTuples object with 2 x 3-tuples and 2 metadata columns:
    #>     seqnames pos1 pos2 pos3 strand | score       GC
    #>   a     chr1    1    2    3      - |     1 1.000000
    #>   b     chr2    2    3    4      + |     2 0.888889
    #>   ---
    #>   seqinfo: 3 sequences from mock1 genome
    tail(gt3, n = 2)
    #> GTuples object with 2 x 3-tuples and 2 metadata columns:
    #>     seqnames pos1 pos2 pos3 strand | score       GC
    #>   i     chr3    9   10   11      - |     9 0.111111
    #>   j     chr3   10   11   12      - |    10 0.000000
    #>   ---
    #>   seqinfo: 3 sequences from mock1 genome
    window(gt3, start = 2, end = 4)
    #> GTuples object with 3 x 3-tuples and 2 metadata columns:
    #>     seqnames pos1 pos2 pos3 strand | score       GC
    #>   b     chr2    2    3    4      + |     2 0.888889
    #>   c     chr2    3    4    5      + |     3 0.777778
    #>   d     chr2    4    5    6      * |     4 0.666667
    #>   ---
    #>   seqinfo: 3 sequences from mock1 genome

### 2.1.4 Basic tuple operations for GTuples objects

Basic tuple characteristics of GTuples objects can be extracted using the `start`, `end`, and `tuples` methods. Warning: While the `width` method is well-defined, namely as \(pos_{m} - pos_{1} + 1\), this may not be what is required. Instead, please see the `IPD` method that will be discussed in the next section.

    start(gt3)
    #>  [1]  1  2  3  4  5  6  7  8  9 10
    end(gt3)
    #>  [1]  3  4  5  6  7  8  9 10 11 12
    tuples(gt3)
    #>       pos1 pos2 pos3
    #>  [1,]    1    2    3
    #>  [2,]    2    3    4
    #>  [3,]    3    4    5
    #>  [4,]    4    5    6
    #>  [5,]    5    6    7
    #>  [6,]    6    7    8
    #>  [7,]    7    8    9
    #>  [8,]    8    9   10
    #>  [9,]    9   10   11
    #> [10,]   10   11   12

#### 2.1.4.1 Intra-tuple operations

Most of the intra-range methods defined for GenomicRanges::GRanges objects are not currently defined via extension for GTuples objects due to the differences between ranges and tuples. Those not currently defined, and which return an error message, are:

- `narrow`
- `flank`
- `promoters`
- `resize`
- `Ops`

I am happy to add these methods if appropriate, so please contact me if you have suggestions for good definitions.

Both the `trim` and `shift` methods are well-defined, although the former is somewhat limited since it will return an error if the internal positions exceed the `seqlengths`:

    shift(gt3, 500)
    #> GTuples object with 10 x 3-tuples and 2 metadata columns:
    #>     seqnames pos1 pos2 pos3 strand | score       GC
    #>   a     chr1  501  502  503      - |     1 1.000000
    #>   b     chr2  502  503  504      + |     2 0.888889
    #>   c     chr2  503  504  505      + |     3 0.777778
    #>   d     chr2  504  505  506      * |     4 0.666667
    #>   e     chr1  505  506  507      * |     5 0.555556
    #>   f     chr1  506  507  508      + |     6 0.444444
    #>   g     chr3  507  508  509      + |     7 0.333333
    #>   h     chr3  508  509  510      + |     8 0.222222
    #>   i     chr3  509  510  511      - |     9 0.111111
    #>   j     chr3  510  511  512      - |    10 0.000000
    #>   ---
    #>   seqinfo: 3 sequences from mock1 genome

    # Raises warning due to tuple being outside of seqlength
    x <- shift(gt3[1], 999)
    #> Warning in valid.GenomicRanges.seqinfo(x, suggest.trim = TRUE): GRanges object contains 1 out-of-bound range located on sequence chr1.
    #>   Note that ranges located on a sequence whose length is unknown (NA) or
    #>   on a circular sequence are not considered out-of-bound (use
    #>   seqlengths() and isCircular() to get the lengths and circularity flags
    #>   of the underlying sequences). You can use trim() to trim these ranges.
    #>   See ?`trim,GenomicRanges-method` for more information.
    x
    #> GTuples object with 1 x 3-tuple and 2 metadata columns:
    #>     seqnames pos1 pos2 pos3 strand | score GC
    #>   a     chr1 1000 1001 1002      - |     1  1
    #>   ---
    #>   seqinfo: 3 sequences from mock1 genome

    # Returns an error because internal position exceeds sequence length, resulting
    # in a malformed tuple when trimmed.
    trim(x)
    #> Error in validObject(object): invalid class "GTuples" object:
    #>     positions in each tuple must be sorted in strictly increasing order,
    #>     i.e. 'pos1' < ... < 'pos3'

#### 2.1.4.2 Inter-tuple operations

None of the inter-range methods defined for GenomicRanges::GRanges objects are currently defined via extension for GTuples objects due to the differences between ranges and tuples. Those not currently defined, and which return an error message, are:

- `range`
- `reduce`
- `gaps`
- `disjoin`
- `isDisjoint`
- `disjointBins`

I am happy to add these methods if appropriate, so please contact me if you have suggestions for good definitions.

#### 2.1.4.3 Interval set operations for GTuples objects

None of the interval set operations defined for GenomicRanges::GRanges objects are currently defined via extension for GTuples objects due to the differences between ranges and tuples. Those not currently defined, and which return an error message, are:

- `union`
- `intersect`
- `setdiff`
- `punion`
- `pintersect`
- `psetdiff`

I am happy to add these methods if appropriate, so please contact me if you have suggestions for good definitions.

### 2.1.5 Additional methods unique to GTuples

GTuples have a few specifically defined methods that do not exist for GenomicRanges::GRanges. These are `tuples`, `size` and `IPD`.

The `tuples` method we have already seen and is somewhat analogous to the `ranges` method for GenomicRanges::GRanges, although returning an integer matrix rather than an IRanges::IRanges object:

    tuples(gt3)
    #>       pos1 pos2 pos3
    #>  [1,]    1    2    3
    #>  [2,]    2    3    4
    #>  [3,]    3    4    5
    #>  [4,]    4    5    6
    #>  [5,]    5    6    7
    #>  [6,]    6    7    8
    #>  [7,]    7    8    9
    #>  [8,]    8    9   10
    #>  [9,]    9   10   11
    #> [10,]   10   11   12

The `size` method returns the size of the tuples stored in the object:

    size(gt3)
    #> [1] 3

Every m-tuple with \(m \geq 2\) has an associated vector of intra-pair distances (\(IPD\)). This is defined as \(IPD = (pos_{2} - pos_{1}, \ldots, pos_{m} - pos_{m - 1})\). The `IPD` method returns this as an integer matrix, where the \(i^{th}\) row contains the \(IPD\) for the \(i^{th}\) tuple:

    IPD(gt3)
    #>       [,1] [,2]
    #>  [1,]    1    1
    #>  [2,]    1    1
    #>  [3,]    1    1
    #>  [4,]    1    1
    #>  [5,]    1    1
    #>  [6,]    1    1
    #>  [7,]    1    1
    #>  [8,]    1    1
    #>  [9,]    1    1
    #> [10,]    1    1

## 2.2 Implementation details

While the GTuples class can be thought of as a matrix-link object, with the number of columns equal to the `size` of the tuples plus two (one for the `seqname` and one for the `strand`), internally, it extends the GenomicRanges::GRanges class. Specifically, the `ranges` slot stores an IRanges::IRanges object containing \(pos_{1}\) and \(pos_{m}\) and, if `size` \(> 2\), a matrix is used to store the co-ordinates of the “internal positions”, \(pos_{2}, \ldots, pos_{m - 1}\) in the `internalPos` slot. If `size` \(\leq 2\) then the `internalPos` slot is set to `NULL`. The `size` is stored as an integer in the `size` slot.

While there are arguments for creating stand-alone GTuples and GTuplesList classes, by extending the GenomicRanges::GRanges and GenomicRanges::GRangesList classes we get a lot of very useful functionality “for free” via appropriately defined inheritance.

# 3 GTuplesList

The GTuplesList class is a container to store a S4Vectors::List of GTuples objects. It extends the GenomicRanges::GRangesList class.

Currently, all GTuples in a GTuplesList must have the same `size` 2. I expect that users will mostly use GTuples objects and have little need to directly use GTuplesList objects.

    seqinfo <- Seqinfo(paste0("chr", 1:3), c(1000, 2000, 1500), NA, "mock1")
    gt3 <- GTuples(seqnames = Rle(c("chr1", "chr2", "chr1", "chr3"),
                                  c(1, 3, 2, 4)),
                   tuples = matrix(c(1:10, 2:11, 3:12), ncol = 3),
                   strand = Rle(strand(c("-", "+", "*", "+", "-")),
                                c(1, 2, 2, 3, 2)),
                   score = 1:10, GC = seq(1, 0, length = 10), seqinfo = seqinfo)
    gtl3 <- GTuplesList(A = gt3[1:5], B = gt3[6:10])
    gtl3
    #> GTuplesList object of length 2:
    #> $A
    #> GTuples object with 5 x 3-tuples and 2 metadata columns:
    #>       seqnames pos1 pos2 pos3 strand | score       GC
    #>   [1]     chr1    1    2    3      - |     1 1.000000
    #>   [2]     chr2    2    3    4      + |     2 0.888889
    #>   [3]     chr2    3    4    5      + |     3 0.777778
    #>   [4]     chr2    4    5    6      * |     4 0.666667
    #>   [5]     chr1    5    6    7      * |     5 0.555556
    #>   ---
    #>   seqinfo: 3 sequences from mock1 genome
    #>
    #> $B
    #> GTuples object with 5 x 3-tuples and 2 metadata columns:
    #>       seqnames pos1 pos2 pos3 strand | score       GC
    #>   [1]     chr1    6    7    8      + |     6 0.444444
    #>   [2]     chr3    7    8    9      + |     7 0.333333
    #>   [3]     chr3    8    9   10      + |     8 0.222222
    #>   [4]     chr3    9   10   11      - |     9 0.111111
    #>   [5]     chr3   10   11   12      - |    10 0.000000
    #>   ---
    #>   seqinfo: 3 sequences from mock1 genome

For even more information on the GTuplesList class, be sure to consult the documentation:

    ?GTuplesList

## 3.1 GTuplesList methods

Most methods defined for GenomicRanges::GRangesList are also applicable to GTuplesList. Those that are not yet defined, which are those that make sense for ranges but generally not for tuples, return error messages.

If a method that is not defined for GTuplesList but is defined for GenomicRanges::GRangesList is truly required, then this can be achieved by first coercing the GTuplesList object to a GenomicRanges::GRangesList object, noting that this is generally a destructive operation:

    as(gtl3, "GRangesList")
    #> GRangesList object of length 2:
    #> $A
    #> GRanges object with 5 ranges and 2 metadata columns:
    #>       seqnames    ranges strand |     score        GC
    #>          <Rle> <IRanges>  <Rle> | <integer> <numeric>
    #>   [1]     chr1       1-3      - |         1  1.000000
    #>   [2]     chr2       2-4      + |         2  0.888889
    #>   [3]     chr2       3-5      + |         3  0.777778
    #>   [4]     chr2       4-6      * |         4  0.666667
    #>   [5]     chr1       5-7      * |         5  0.555556
    #>   -------
    #>   seqinfo: 3 sequences from mock1 genome
    #>
    #> $B
    #> GRanges object with 5 ranges and 2 metadata columns:
    #>       seqnames    ranges strand |     score        GC
    #>          <Rle> <IRanges>  <Rle> | <integer> <numeric>
    #>   [1]     chr1       6-8      + |         6  0.444444
    #>   [2]     chr3       7-9      + |         7  0.333333
    #>   [3]     chr3      8-10      + |         8  0.222222
    #>   [4]     chr3      9-11      - |         9  0.111111
    #>   [5]     chr3     10-12      - |        10  0.000000
    #>   -------
    #>   seqinfo: 3 sequences from mock1 genome

### 3.1.1 Basic GTuplesList accessors}

These are very similar to those available for GTuples objects, except that they typically return a list since the input is now essentially a list of GTuples objects:

    seqnames(gtl3)
    #> RleList of length 2
    #> $A
    #> factor-Rle of length 5 with 3 runs
    #>   Lengths:    1    3    1
    #>   Values : chr1 chr2 chr1
    #> Levels(3): chr1 chr2 chr3
    #>
    #> $B
    #> factor-Rle of length 5 with 2 runs
    #>   Lengths:    1    4
    #>   Values : chr1 chr3
    #> Levels(3): chr1 chr2 chr3
    # Returns a list of integer matrices
    tuples(gtl3)
    #> List of length 2
    #> names(2): A B
    tuples(gtl3)[[1]]
    #>      pos1 pos2 pos3
    #> [1,]    1    2    3
    #> [2,]    2    3    4
    #> [3,]    3    4    5
    #> [4,]    4    5    6
    #> [5,]    5    6    7
    strand(gtl3)
    #> RleList of length 2
    #> $A
    #> factor-Rle of length 5 with 3 runs
    #>   Lengths: 1 2 2
    #>   Values : - + *
    #> Levels(3): + - *
    #>
    #> $B
    #> factor-Rle of length 5 with 2 runs
    #>   Lengths: 3 2
    #>   Values : + -
    #> Levels(3): + - *

The `length` and `names` methods will return the length and names of the list, respectively:

    length(gtl3)
    #> [1] 2
    names(gtl3)
    #> [1] "A" "B"

Seqinfo can be extracted using the `seqinfo` accessor:

    seqinfo(gtl3)
    #> Seqinfo object with 3 sequences from mock1 genome:
    #>   seqnames seqlengths isCircular genome
    #>   chr1           1000         NA  mock1
    #>   chr2           2000         NA  mock1
    #>   chr3           1500         NA  mock1

The `elementNROWS` method returns a list of integers corresponding to the result of calling `length` on each individual GTuples object contained by the GTuplesList. This is a faster alternative to calling `lapply` on the GTuplesList:

    elementNROWS(gtl3)
    #> A B
    #> 5 5

You can also use `isEmpty` to test if a GTuplesList object contains anything:

    isEmpty(gtl3)
    #> [1] FALSE
    isEmpty(GTuplesList())
    #> [1] TRUE

Finally, in the context of a GTuplesList object, the `mcols` method performs a similar operation to what it does on a GTuples object. However, this metadata now refers to information at the list level instead of the level of the individual GTuples objects:

    mcols(gtl3) <- c("Feature A", "Feature B")
    mcols(gtl3)
    #> DataFrame with 2 rows and 1 column
    #>         value
    #>   <character>
    #> A   Feature A
    #> B   Feature B

### 3.1.2 Combining GTuplesList objects

GTuplesList objects can be unlisted to combine the separate GTuples objects that they contain as an expanded GTuples:

    ul <- unlist(gtl3)
    ul
    #> GTuples object with 10 x 3-tuples and 2 metadata columns:
    #>     seqnames pos1 pos2 pos3 strand | score       GC
    #>   A     chr1    1    2    3      - |     1 1.000000
    #>   A     chr2    2    3    4      + |     2 0.888889
    #>   A     chr2    3    4    5      + |     3 0.777778
    #>   A     chr2    4    5    6      * |     4 0.666667
    #>   A     chr1    5    6    7      * |     5 0.555556
    #>   B     chr1    6    7    8      + |     6 0.444444
    #>   B     chr3    7    8    9      + |     7 0.333333
    #>   B     chr3    8    9   10      + |     8 0.222222
    #>   B     chr3    9   10   11      - |     9 0.111111
    #>   B     chr3   10   11   12      - |    10 0.000000
    #>   ---
    #>   seqinfo: 3 sequences from mock1 genome

You can also combine GTuplesList objects together using `append` or `c`.

### 3.1.3 Subsetting GTuplesList objects

Subsetting of GTuplesList objects is identical to subsetting of GenomicRanges::GRangesList objects:

    gtl3[1]
    #> GTuplesList object of length 1:
    #> $A
    #> GTuples object with 5 x 3-tuples and 2 metadata columns:
    #>       seqnames pos1 pos2 pos3 strand | score       GC
    #>   [1]     chr1    1    2    3      - |     1 1.000000
    #>   [2]     chr2    2    3    4      + |     2 0.888889
    #>   [3]     chr2    3    4    5      + |     3 0.777778
    #>   [4]     chr2    4    5    6      * |     4 0.666667
    #>   [5]     chr1    5    6    7      * |     5 0.555556
    #>   ---
    #>   seqinfo: 3 sequences from mock1 genome
    gtl3[[1]]
    #> GTuples object with 5 x 3-tuples and 2 metadata columns:
    #>       seqnames pos1 pos2 pos3 strand | score       GC
    #>   [1]     chr1    1    2    3      - |     1 1.000000
    #>   [2]     chr2    2    3    4      + |     2 0.888889
    #>   [3]     chr2    3    4    5      + |     3 0.777778
    #>   [4]     chr2    4    5    6      * |     4 0.666667
    #>   [5]     chr1    5    6    7      * |     5 0.555556
    #>   ---
    #>   seqinfo: 3 sequences from mock1 genome
    gtl3["A"]
    #> GTuplesList object of length 1:
    #> $A
    #> GTuples object with 5 x 3-tuples and 2 metadata columns:
    #>       seqnames pos1 pos2 pos3 strand | score       GC
    #>   [1]     chr1    1    2    3      - |     1 1.000000
    #>   [2]     chr2    2    3    4      + |     2 0.888889
    #>   [3]     chr2    3    4    5      + |     3 0.777778
    #>   [4]     chr2    4    5    6      * |     4 0.666667
    #>   [5]     chr1    5    6    7      * |     5 0.555556
    #>   ---
    #>   seqinfo: 3 sequences from mock1 genome
    gtl3$B
    #> GTuples object with 5 x 3-tuples and 2 metadata columns:
    #>       seqnames pos1 pos2 pos3 strand | score       GC
    #>   [1]     chr1    6    7    8      + |     6 0.444444
    #>   [2]     chr3    7    8    9      + |     7 0.333333
    #>   [3]     chr3    8    9   10      + |     8 0.222222
    #>   [4]     chr3    9   10   11      - |     9 0.111111
    #>   [5]     chr3   10   11   12      - |    10 0.000000
    #>   ---
    #>   seqinfo: 3 sequences from mock1 genome

When subsetting a GTuplesList, you can also pass in a second parameter (as with a GTuples object) to again specify which of the metadata columns you wish to select:

    gtl3[1, "score"]
    #> GTuplesList object of length 1:
    #> $A
    #> GTuples object with 5 x 3-tuples and 1 metadata column:
    #>       seqnames pos1 pos2 pos3 strand | score
    #>   [1]     chr1    1    2    3      - |     1
    #>   [2]     chr2    2    3    4      + |     2
    #>   [3]     chr2    3    4    5      + |     3
    #>   [4]     chr2    4    5    6      * |     4
    #>   [5]     chr1    5    6    7      * |     5
    #>   ---
    #>   seqinfo: 3 sequences from mock1 genome
    gtl3["B", "GC"]
    #> GTuplesList object of length 1:
    #> $B
    #> GTuples object with 5 x 3-tuples and 1 metadata column:
    #>       seqnames pos1 pos2 pos3 strand |       GC
    #>   [1]     chr1    6    7    8      + | 0.444444
    #>   [2]     chr3    7    8    9      + | 0.333333
    #>   [3]     chr3    8    9   10      + | 0.222222
    #>   [4]     chr3    9   10   11      - | 0.111111
    #>   [5]     chr3   10   11   12      - | 0.000000
    #>   ---
    #>   seqinfo: 3 sequences from mock1 genome

The `head`, `tail`, `rep`, `rev`, and `window` methods all behave as you would expect them to for a List object. For example, the elements referred to by `window` are now list elements instead of GTuples elements:

    rep(gtl3[[1]], times = 3)
    #> GTuples object with 15 x 3-tuples and 2 metadata columns:
    #>        seqnames pos1 pos2 pos3 strand | score       GC
    #>    [1]     chr1    1    2    3      - |     1 1.000000
    #>    [2]     chr2    2    3    4      + |     2 0.888889
    #>    [3]     chr2    3    4    5      + |     3 0.777778
    #>    [4]     chr2    4    5    6      * |     4 0.666667
    #>    [5]     chr1    5    6    7      * |     5 0.555556
    #>    ...      ...  ...  ...  ...    ... .   ...      ...
    #>   [11]     chr1    1    2    3      - |     1 1.000000
    #>   [12]     chr2    2    3    4      + |     2 0.888889
    #>   [13]     chr2    3    4    5      + |     3 0.777778
    #>   [14]     chr2    4    5    6      * |     4 0.666667
    #>   [15]     chr1    5    6    7      * |     5 0.555556
    #>   ---
    #>   seqinfo: 3 sequences from mock1 genome
    rev(gtl3)
    #> GTuplesList object of length 2:
    #> $B
    #> GTuples object with 5 x 3-tuples and 2 metadata columns:
    #>       seqnames pos1 pos2 pos3 strand | score       GC
    #>   [1]     chr1    6    7    8      + |     6 0.444444
    #>   [2]     chr3    7    8    9      + |     7 0.333333
    #>   [3]     chr3    8    9   10      + |     8 0.222222
    #>   [4]     chr3    9   10   11      - |     9 0.111111
    #>   [5]     chr3   10   11   12      - |    10 0.000000
    #>   ---
    #>   seqinfo: 3 sequences from mock1 genome
    #>
    #> $A
    #> GTuples object with 5 x 3-tuples and 2 metadata columns:
    #>       seqnames pos1 pos2 pos3 strand | score       GC
    #>   [1]     chr1    1    2    3      - |     1 1.000000
    #>   [2]     chr2    2    3    4      + |     2 0.888889
    #>   [3]     chr2    3    4    5      + |     3 0.777778
    #>   [4]     chr2    4    5    6      * |     4 0.666667
    #>   [5]     chr1    5    6    7      * |     5 0.555556
    #>   ---
    #>   seqinfo: 3 sequences from mock1 genome
    head(gtl3, n = 1)
    #> GTuplesList object of length 1:
    #> $A
    #> GTuples object with 5 x 3-tuples and 2 metadata columns:
    #>       seqnames pos1 pos2 pos3 strand | score       GC
    #>   [1]     chr1    1    2    3      - |     1 1.000000
    #>   [2]     chr2    2    3    4      + |     2 0.888889
    #>   [3]     chr2    3    4    5      + |     3 0.777778
    #>   [4]     chr2    4    5    6      * |     4 0.666667
    #>   [5]     chr1    5    6    7      * |     5 0.555556
    #>   ---
    #>   seqinfo: 3 sequences from mock1 genome
    tail(gtl3, n = 1)
    #> GTuplesList object of length 1:
    #> $B
    #> GTuples object with 5 x 3-tuples and 2 metadata columns:
    #>       seqnames pos1 pos2 pos3 strand | score       GC
    #>   [1]     chr1    6    7    8      + |     6 0.444444
    #>   [2]     chr3    7    8    9      + |     7 0.333333
    #>   [3]     chr3    8    9   10      + |     8 0.222222
    #>   [4]     chr3    9   10   11      - |     9 0.111111
    #>   [5]     chr3   10   11   12      - |    10 0.000000
    #>   ---
    #>   seqinfo: 3 sequences from mock1 genome
    window(gtl3, start = 1, end = 1)
    #> GTuplesList object of length 1:
    #> $A
    #> GTuples object with 5 x 3-tuples and 2 metadata columns:
    #>       seqnames pos1 pos2 pos3 strand | score       GC
    #>   [1]     chr1    1    2    3      - |     1 1.000000
    #>   [2]     chr2    2    3    4      + |     2 0.888889
    #>   [3]     chr2    3    4    5      + |     3 0.777778
    #>   [4]     chr2    4    5    6      * |     4 0.666667
    #>   [5]     chr1    5    6    7      * |     5 0.555556
    #>   ---
    #>   seqinfo: 3 sequences from mock1 genome

### 3.1.4 Basic tuple operations for GTuplesList objects

Basic tuple characteristics of GTuplesList objects can be extracted using the `start`, `end`, and `tuples` methods. These are very similar to those available for GTuples objects, except that they typically return a list since the input is now essentially a list of GTuples objects.

WARNING: While the `width` method is well-defined, namely it returns an IntegerList of \(pos_{m} - pos_{1} + 1\), this is not generally what is desired or required. Instead, please see the `IPD` method that is discussed later.

    start(gtl3)
    #> IntegerList of length 2
    #> [["A"]] 1 2 3 4 5
    #> [["B"]] 6 7 8 9 10
    end(gtl3)
    #> IntegerList of length 2
    #> [["A"]] 3 4 5 6 7
    #> [["B"]] 8 9 10 11 12
    tuples(gtl3)
    #> List of length 2
    #> names(2): A B

#### 3.1.4.1 Intra-tuple operations

Most of the intra-range methods defined for GenomicRanges::GRangesList objects are not currently defined via extension for GTuples objects due to the differences between ranges and tuples. Those not currently defined, and which return an error message, are:

- `flank`
- `promoters`
- `resize`
- `restrict`

I am happy to add these methods if appropriate, so please contact me if you have suggestions for good definitions.

The `shift` method is well-defined:

    shift(gtl3, 500)
    #> GTuplesList object of length 2:
    #> $A
    #> GTuples object with 5 x 3-tuples and 2 metadata columns:
    #>       seqnames pos1 pos2 pos3 strand | score       GC
    #>   [1]     chr1  501  502  503      - |     1 1.000000
    #>   [2]     chr2  502  503  504      + |     2 0.888889
    #>   [3]     chr2  503  504  505      + |     3 0.777778
    #>   [4]     chr2  504  505  506      * |     4 0.666667
    #>   [5]     chr1  505  506  507      * |     5 0.555556
    #>   ---
    #>   seqinfo: 3 sequences from mock1 genome
    #>
    #> $B
    #> GTuples object with 5 x 3-tuples and 2 metadata columns:
    #>       seqnames pos1 pos2 pos3 strand | score       GC
    #>   [1]     chr1  506  507  508      + |     6 0.444444
    #>   [2]     chr3  507  508  509      + |     7 0.333333
    #>   [3]     chr3  508  509  510      + |     8 0.222222
    #>   [4]     chr3  509  510  511      - |     9 0.111111
    #>   [5]     chr3  510  511  512      - |    10 0.000000
    #>   ---
    #>   seqinfo: 3 sequences from mock1 genome
    shift(gtl3, IntegerList(A = 300L, B = 500L))
    #> GTuplesList object of length 2:
    #> $A
    #> GTuples object with 5 x 3-tuples and 2 metadata columns:
    #>       seqnames pos1 pos2 pos3 strand | score       GC
    #>   [1]     chr1  301  302  303      - |     1 1.000000
    #>   [2]     chr2  302  303  304      + |     2 0.888889
    #>   [3]     chr2  303  304  305      + |     3 0.777778
    #>   [4]     chr2  304  305  306      * |     4 0.666667
    #>   [5]     chr1  305  306  307      * |     5 0.555556
    #>   ---
    #>   seqinfo: 3 sequences from mock1 genome
    #>
    #> $B
    #> GTuples object with 5 x 3-tuples and 2 metadata columns:
    #>       seqnames pos1 pos2 pos3 strand | score       GC
    #>   [1]     chr1  506  507  508      + |     6 0.444444
    #>   [2]     chr3  507  508  509      + |     7 0.333333
    #>   [3]     chr3  508  509  510      + |     8 0.222222
    #>   [4]     chr3  509  510  511      - |     9 0.111111
    #>   [5]     chr3  510  511  512      - |    10 0.000000
    #>   ---
    #>   seqinfo: 3 sequences from mock1 genome

#### 3.1.4.2 Inter-tuple operations

None of the inter-range methods defined for GenomicRanges::GRangesList objects are currently defined via extension for GTuplesList objects due to the differences between ranges and tuples. Those not currently defined, and which return an error message, are:

- `range`
- `reduce`
- `disjoin`
- `isDisjoint`

I am happy to add these methods if appropriate, so please contact me if you have suggestions for good definitions.

#### 3.1.4.3 Interval set operations for GTuplesList objects

None of the interval set operations defined for GenomicRanges::GRangesList objects are currently defined via extension for GTuplesList objects due to the differences between ranges and tuples. Those not currently defined, and which return an error message, are:

- `punion`
- `pintersect`
- `psetdiff`

I am happy to add these methods if appropriate, so please contact me if you have suggestions for good definitions.

### 3.1.5 Looping over GTuplesList objects

Like for GenomicRanges::GRangesList objects, for GTuplesList objects there is a family of apply methods. These include `lapply`, `sapply`, `mapply`, `endoapply`, `mendoapply`, `Map`, and `Reduce`. The different looping methods defined for GTuplesList objects are useful for returning different kinds of results. The standard `lapply` and `sapply` behave according to convention, with the `lapply` method returning a list and `sapply` returning a more simplified output:

    lapply(gtl3, length)
    #> $A
    #> [1] 5
    #>
    #> $B
    #> [1] 5
    sapply(gtl3, length)
    #> A B
    #> 5 5

As with GenomicRanges::GRangesList objects, there is also a multivariate version of `sapply`, called `mapply`, defined for GTuplesList objects. And, if you don’t want the results simplified, you can call the `Map` method, which does the same things as `mapply` but without simplifying the output:

    gtl3_shift <- shift(gtl3, 10)
    names(gtl3) <- c("shiftA", "shiftB")
    mapply(c, gtl3, gtl3_shift)
    #> $shiftA
    #> GTuples object with 10 x 3-tuples and 2 metadata columns:
    #>        seqnames pos1 pos2 pos3 strand | score       GC
    #>    [1]     chr1    1    2    3      - |     1 1.000000
    #>    [2]     chr2    2    3    4      + |     2 0.888889
    #>    [3]     chr2    3    4    5      + |     3 0.777778
    #>    [4]     chr2    4    5    6      * |     4 0.666667
    #>    [5]     chr1    5    6    7      * |     5 0.555556
    #>    [6]     chr1   11   12   13      - |     1 1.000000
    #>    [7]     chr2   12   13   14      + |     2 0.888889
    #>    [8]     chr2   13   14   15      + |     3 0.777778
    #>    [9]     chr2   14   15   16      * |     4 0.666667
    #>   [10]     chr1   15   16   17      * |     5 0.555556
    #>   ---
    #>   seqinfo: 3 sequences from mock1 genome
    #>
    #> $shiftB
    #> GTuples object with 10 x 3-tuples and 2 metadata columns:
    #>        seqnames pos1 pos2 pos3 strand | score       GC
    #>    [1]     chr1    6    7    8      + |     6 0.444444
    #>    [2]     chr3    7    8    9      + |     7 0.333333
    #>    [3]     chr3    8    9   10      + |     8 0.222222
    #>    [4]     chr3    9   10   11      - |     9 0.111111
    #>    [5]     chr3   10   11   12      - |    10 0.000000
    #>    [6]     chr1   16   17   18      + |     6 0.444444
    #>    [7]     chr3   17   18   19      + |     7 0.333333
    #>    [8]     chr3   18   19   20      + |     8 0.222222
    #>    [9]     chr3   19   20   21      - |     9 0.111111
    #>   [10]     chr3   20   21   22      - |    10 0.000000
    #>   ---
    #>   seqinfo: 3 sequences from mock1 genome
    Map(c, gtl3, gtl3_shift)
    #> $shiftA
    #> GTuples object with 10 x 3-tuples and 2 metadata columns:
    #>        seqnames pos1 pos2 pos3 strand | score       GC
    #>    [1]     chr1    1    2    3      - |     1 1.000000
    #>    [2]     chr2    2    3    4      + |     2 0.888889
    #>    [3]     chr2    3    4    5      + |     3 0.777778
    #>    [4]     chr2    4    5    6      * |     4 0.666667
    #>    [5]     chr1    5    6    7      * |     5 0.555556
    #>    [6]     chr1   11   12   13      - |     1 1.000000
    #>    [7]     chr2   12   13   14      + |     2 0.888889
    #>    [8]     chr2   13   14   15      + |     3 0.777778
    #>    [9]     chr2   14   15   16      * |     4 0.666667
    #>   [10]     chr1   15   16   17      * |     5 0.555556
    #>   ---
    #>   seqinfo: 3 sequences from mock1 genome
    #>
    #> $shiftB
    #> GTuples object with 10 x 3-tuples and 2 metadata columns:
    #>        seqnames pos1 pos2 pos3 strand | score       GC
    #>    [1]     chr1    6    7    8      + |     6 0.444444
    #>    [2]     chr3    7    8    9      + |     7 0.333333
    #>    [3]     chr3    8    9   10      + |     8 0.222222
    #>    [4]     chr3    9   10   11      - |     9 0.111111
    #>    [5]     chr3   10   11   12      - |    10 0.000000
    #>    [6]     chr1   16   17   18      + |     6 0.444444
    #>    [7]     chr3   17   18   19      + |     7 0.333333
    #>    [8]     chr3   18   19   20      + |     8 0.222222
    #>    [9]     chr3   19   20   21      - |     9 0.111111
    #>   [10]     chr3   20   21   22      - |    10 0.000000
    #>   ---
    #>   seqinfo: 3 sequences from mock1 genome

The `endoapply` method will return the results as a GTuplesList object rather than as a list:

    endoapply(gtl3, rev)
    #> GTuplesList object of length 2:
    #> $shiftA
    #> GTuples object with 5 x 3-tuples and 2 metadata columns:
    #>       seqnames pos1 pos2 pos3 strand | score       GC
    #>   [1]     chr1    5    6    7      * |     5 0.555556
    #>   [2]     chr2    4    5    6      * |     4 0.666667
    #>   [3]     chr2    3    4    5      + |     3 0.777778
    #>   [4]     chr2    2    3    4      + |     2 0.888889
    #>   [5]     chr1    1    2    3      - |     1 1.000000
    #>   ---
    #>   seqinfo: 3 sequences from mock1 genome
    #>
    #> $shiftB
    #> GTuples object with 5 x 3-tuples and 2 metadata columns:
    #>       seqnames pos1 pos2 pos3 strand | score       GC
    #>   [1]     chr3   10   11   12      - |    10 0.000000
    #>   [2]     chr3    9   10   11      - |     9 0.111111
    #>   [3]     chr3    8    9   10      + |     8 0.222222
    #>   [4]     chr3    7    8    9      + |     7 0.333333
    #>   [5]     chr1    6    7    8      + |     6 0.444444
    #>   ---
    #>   seqinfo: 3 sequences from mock1 genome

There is also a multivariate version of the `endoapply` method in the form of the `mendoapply` method:

    mendoapply(c, gtl3, gtl3_shift)
    #> GTuplesList object of length 2:
    #> $shiftA
    #> GTuples object with 10 x 3-tuples and 2 metadata columns:
    #>        seqnames pos1 pos2 pos3 strand | score       GC
    #>    [1]     chr1    1    2    3      - |     1 1.000000
    #>    [2]     chr2    2    3    4      + |     2 0.888889
    #>    [3]     chr2    3    4    5      + |     3 0.777778
    #>    [4]     chr2    4    5    6      * |     4 0.666667
    #>    [5]     chr1    5    6    7      * |     5 0.555556
    #>    [6]     chr1   11   12   13      - |     1 1.000000
    #>    [7]     chr2   12   13   14      + |     2 0.888889
    #>    [8]     chr2   13   14   15      + |     3 0.777778
    #>    [9]     chr2   14   15   16      * |     4 0.666667
    #>   [10]     chr1   15   16   17      * |     5 0.555556
    #>   ---
    #>   seqinfo: 3 sequences from mock1 genome
    #>
    #> $shiftB
    #> GTuples object with 10 x 3-tuples and 2 metadata columns:
    #>        seqnames pos1 pos2 pos3 strand | score       GC
    #>    [1]     chr1    6    7    8      + |     6 0.444444
    #>    [2]     chr3    7    8    9      + |     7 0.333333
    #>    [3]     chr3    8    9   10      + |     8 0.222222
    #>    [4]     chr3    9   10   11      - |     9 0.111111
    #>    [5]     chr3   10   11   12      - |    10 0.000000
    #>    [6]     chr1   16   17   18      + |     6 0.444444
    #>    [7]     chr3   17   18   19      + |     7 0.333333
    #>    [8]     chr3   18   19   20      + |     8 0.222222
    #>    [9]     chr3   19   20   21      - |     9 0.111111
    #>   [10]     chr3   20   21   22      - |    10 0.000000
    #>   ---
    #>   seqinfo: 3 sequences from mock1 genome

Finally, the `Reduce` method will allow the GTuples objects to be collapsed across the whole of the GTuplesList object:

    Reduce(c, gtl3)
    #> GTuples object with 10 x 3-tuples and 2 metadata columns:
    #>        seqnames pos1 pos2 pos3 strand | score       GC
    #>    [1]     chr1    1    2    3      - |     1 1.000000
    #>    [2]     chr2    2    3    4      + |     2 0.888889
    #>    [3]     chr2    3    4    5      + |     3 0.777778
    #>    [4]     chr2    4    5    6      * |     4 0.666667
    #>    [5]     chr1    5    6    7      * |     5 0.555556
    #>    [6]     chr1    6    7    8      + |     6 0.444444
    #>    [7]     chr3    7    8    9      + |     7 0.333333
    #>    [8]     chr3    8    9   10      + |     8 0.222222
    #>    [9]     chr3    9   10   11      - |     9 0.111111
    #>   [10]     chr3   10   11   12      - |    10 0.000000
    #>   ---
    #>   seqinfo: 3 sequences from mock1 genome

### 3.1.6 Additional methods unique to GTuplesList

Like GTuples, GTuplesList have a few specifically defined methods that do not exist for GenomicRanges::GRangesList. These are `tuples`, `size` and `IPD`. These are identical to the methods for GTuples, except that they typically return a list since the input is now essentially a List of GTuples objects.

    tuples(gtl3)
    #> List of length 2
    #> names(2): shiftA shiftB
    tuples(gtl3)[[1]]
    #>      pos1 pos2 pos3
    #> [1,]    1    2    3
    #> [2,]    2    3    4
    #> [3,]    3    4    5
    #> [4,]    4    5    6
    #> [5,]    5    6    7
    size(gtl3)
    #> [1] 3
    IPD(gtl3)
    #> List of length 2
    #> names(2): shiftA shiftB
    IPD(gtl3)[[1]]
    #>      [,1] [,2]
    #> [1,]    1    1
    #> [2,]    1    1
    #> [3,]    1    1
    #> [4,]    1    1
    #> [5,]    1    1

## 3.2 Implementation details

The GTuplesList class extends the GenomicRanges::GRangesList class.

# 4 `findOverlaps`-based methods

The definition of what constitutes an “overlap” between genomic tuples, or between genomic tuples and genomic ranges, lies at the heart of all `findOverlaps`-based methods3 for GTuples and GTuplesList objects.

I have chosen a definition that matches my intuition of what constitutes an “overlap” between genomic tuples or between genomic tuples and genomic ranges. However, I am open to suggestions on amending or extending this behaviour in future versions of GenomicTuples.

## 4.1 Definition of overlapping genomic tuples

I consider two genomic tuples to be equal (`type = "equal"`) if they have identical sequence names (`seqnames`), strands (`strand`) and tuples (`tuples`). For 1-tuples and 2-tuples, this means we can simply defer to the `findOverlaps`-based methods for GenomicRanges::GRanges and GenomicRanges::GRangesList objects via inheritance. However, we cannot do the same for m-tuples with \(m > 2\) since this would ignore the “internal positions”. Therefore, I have implemented a special case of the `findOverlaps` method for when `size` \(> 2\) and `type = "equal"`, which ensures that the “internal positions” are also checked for equality.

In all other cases genomic tuples are treated as genomic ranges. This means that when `type = "any"`, `type = "start"`, `type = "end"` or `type = "within"` then the genomic tuples are treated as if they were genomic ranges. Specifically, GTuples (resp. GTuplesList) are treated as though they were GenomicRanges::GRanges (resp. GenomicRanges::GRangesList) with `pos1` = `start` and `posm` = `end`.

## 4.2 Definition of overlapping genomic tuples and ranges

Genomic tuples are always treated as genomic ranges when searching for overlaps between genomic tuples and genomic ranges.

## 4.3 Examples

It is easiest to understand the above definitions by studying a few examples.

Firstly, for 1-tuples where the GTuples methods use the GenomicRanges::GRanges methods:

    # Construct example 1-tuples
    gt1 <- GTuples(seqnames = c('chr1', 'chr1', 'chr1', 'chr2'),
                   tuples = matrix(c(10L, 10L, 10L, 10L), ncol = 1),
                   strand = c('+', '-', '*', '+'))
    # GRanges version of gt1
    gr1 <- as(gt1, "GRanges")
    findOverlaps(gt1, gt1, type = 'any')
    #> Hits object with 8 hits and 0 metadata columns:
    #>       queryHits subjectHits
    #>       <integer>   <integer>
    #>   [1]         1           1
    #>   [2]         1           3
    #>   [3]         2           2
    #>   [4]         2           3
    #>   [5]         3           1
    #>   [6]         3           2
    #>   [7]         3           3
    #>   [8]         4           4
    #>   -------
    #>   queryLength: 4 / subjectLength: 4
    # GTuples and GRanges methods identical
    identical(findOverlaps(gt1, gt1, type = 'any'),
              findOverlaps(gr1, gr1, type = 'any'))
    #> [1] TRUE
    findOverlaps(gt1, gt1, type = 'start')
    #> Hits object with 8 hits and 0 metadata columns:
    #>       queryHits subjectHits
    #>       <integer>   <integer>
    #>   [1]         1           1
    #>   [2]         1           3
    #>   [3]         2           2
    #>   [4]         2           3
    #>   [5]         3           1
    #>   [6]         3           2
    #>   [7]         3           3
    #>   [8]         4           4
    #>   -------
    #>   queryLength: 4 / subjectLength: 4
    # GTuples and GRanges methods identical
    identical(findOverlaps(gt1, gt1, type = 'start'),
              findOverlaps(gr1, gr1, type = 'start'))
    #> [1] TRUE
    findOverlaps(gt1, gt1, type = 'end')
    #> Hits object with 8 hits and 0 metadata columns:
    #>       queryHits subjectHits
    #>       <integer>   <integer>
    #>   [1]         1           1
    #>   [2]         1           3
    #>   [3]         2           2
    #>   [4]         2           3
    #>   [5]         3           1
    #>   [6]         3           2
    #>   [7]         3           3
    #>   [8]         4           4
    #>   -------
    #>   queryLength: 4 / subjectLength: 4
    # GTuples and GRanges methods identical
    identical(findOverlaps(gt1, gt1, type = 'end'),
              findOverlaps(gr1, gr1, type = 'end'))
    #> [1] TRUE
    findOverlaps(gt1, gt1, type = 'within')
    #> Hits object with 8 hits and 0 metadata columns:
    #>       queryHits subjectHits
    #>       <integer>   <integer>
    #>   [1]         1           1
    #>   [2]         1           3
    #>   [3]         2           2
    #>   [4]         2           3
    #>   [5]         3           1
    #>   [6]         3           2
    #>   [7]         3           3
    #>   [8]         4           4
    #>   -------
    #>   queryLength: 4 / subjectLength: 4
    # GTuples and GRanges methods identical
    identical(findOverlaps(gt1, gt1, type = 'within'),
              findOverlaps(gr1, gr1, type = 'within'))
    #> [1] TRUE
    findOverlaps(gt1, gt1, type = 'equal')
    #> Hits object with 8 hits and 0 metadata columns:
    #>       queryHits subjectHits
    #>       <integer>   <integer>
    #>   [1]         1           1
    #>   [2]         1           3
    #>   [3]         2           2
    #>   [4]         2           3
    #>   [5]         3           1
    #>   [6]         3           2
    #>   [7]         3           3
    #>   [8]         4           4
    #>   -------
    #>   queryLength: 4 / subjectLength: 4
    # GTuples and GRanges methods identical
    identical(findOverlaps(gt1, gt1, type = 'equal'),
              findOverlaps(gr1, gr1, type = 'equal'))
    #> [1] TRUE
    # Can pass other arguments, such as select and ignore.strand
    findOverlaps(gt1, gt1, type = 'equal', ignore.strand = TRUE, select = 'last')
    #> [1] 3 3 3 4

Next, for 2-tuples where the GTuples methods use the GenomicRanges::GRanges methods:

    # Construct example 2-tuples
    gt2 <- GTuples(seqnames = c('chr1', 'chr1', 'chr1', 'chr1', 'chr2'),
                   tuples = matrix(c(10L, 10L, 10L, 10L, 10L, 20L, 20L, 20L, 25L,
                                     20L), ncol = 2),
                   strand = c('+', '-', '*', '+', '+'))
    # GRanges version of gt2
    gr2 <- as(gt2, "GRanges")
    findOverlaps(gt2, gt2, type = 'any')
    #> Hits object with 13 hits and 0 metadata columns:
    #>        queryHits subjectHits
    #>        <integer>   <integer>
    #>    [1]         1           4
    #>    [2]         1           1
    #>    [3]         1           3
    #>    [4]         2           2
    #>    [5]         2           3
    #>    ...       ...         ...
    #>    [9]         3           3
    #>   [10]         4           4
    #>   [11]         4           1
    #>   [12]         4           3
    #>   [13]         5           5
    #>   -------
    #>   queryLength: 5 / subjectLength: 5
    # GTuples and GRanges methods identical
    identical(findOverlaps(gt2, gt2, type = 'any'),
              findOverlaps(gr2, gr2, type = 'any'))
    #> [1] TRUE
    findOverlaps(gt2, gt2, type = 'start')
    #> Hits object with 13 hits and 0 metadata columns:
    #>        queryHits subjectHits
    #>        <integer>   <integer>
    #>    [1]         1           4
    #>    [2]         1           1
    #>    [3]         1           3
    #>    [4]         2           2
    #>    [5]         2           3
    #>    ...       ...         ...
    #>    [9]         3           3
    #>   [10]         4           4
    #>   [11]         4           1
    #>   [12]         4           3
    #>   [13]         5           5
    #>   -------
    #>   queryLength: 5 / subjectLength: 5
    # GTuples and GRanges methods identical
    identical(findOverlaps(gt2, gt2, type = 'start'),
              findOverlaps(gr2, gr2, type = 'start'))
    #> [1] TRUE
    findOverlaps(gt2, gt2, type = 'end')
    #> Hits object with 9 hits and 0 metadata columns:
    #>       queryHits subjectHits
    #>       <integer>   <integer>
    #>   [1]         1           1
    #>   [2]         1           3
    #>   [3]         2           2
    #>   [4]         2           3
    #>   [5]         3           1
    #>   [6]         3           2
    #>   [7]         3           3
    #>   [8]         4           4
    #>   [9]         5           5
    #>   -------
    #>   queryLength: 5 / subjectLength: 5
    # GTuples and GRanges methods identical
    identical(findOverlaps(gt2, gt2, type = 'end'),
              findOverlaps(gr2, gr2, type = 'end'))
    #> [1] TRUE
    findOverlaps(gt2, gt2, type = 'within')
    #> Hits object with 11 hits and 0 metadata columns:
    #>        queryHits subjectHits
    #>        <integer>   <integer>
    #>    [1]         1           4
    #>    [2]         1           1
    #>    [3]         1           3
    #>    [4]         2           2
    #>    [5]         2           3
    #>    [6]         3           4
    #>    [7]         3           1
    #>    [8]         3           2
    #>    [9]         3           3
    #>   [10]         4           4
    #>   [11]         5           5
    #>   -------
    #>   queryLength: 5 / subjectLength: 5
    # GTuples and GRanges methods identical
    identical(findOverlaps(gt2, gt2, type = 'within'),
              findOverlaps(gr2, gr2, type = 'within'))
    #> [1] TRUE
    findOverlaps(gt2, gt2, type = 'equal')
    #> Hits object with 9 hits and 0 metadata columns:
    #>       queryHits subjectHits
    #>       <integer>   <integer>
    #>   [1]         1           1
    #>   [2]         1           3
    #>   [3]         2           2
    #>   [4]         2           3
    #>   [5]         3           1
    #>   [6]         3           2
    #>   [7]         3           3
    #>   [8]         4           4
    #>   [9]         5           5
    #>   -------
    #>   queryLength: 5 / subjectLength: 5
    # GTuples and GRanges methods identical
    identical(findOverlaps(gt2, gt2, type = 'equal'),
              findOverlaps(gr2, gr2, type = 'equal'))
    #> [1] TRUE
    # Can pass other arguments, such as select and ignore.strand
    findOverlaps(gt2, gt2, type = 'equal', ignore.strand = TRUE, select = 'last')
    #> [1] 3 3 3 4 5

Finally, for m-tuples with \(m > 2\) where GTuples methods use the GenomicRanges::GRanges methods unless `type = "equal"`:

    # Construct example 3-tuples
    gt3 <- GTuples(seqnames = c('chr1', 'chr1', 'chr1', 'chr1', 'chr2'),
                   tuples = matrix(c(10L, 10L, 10L, 10L, 10L, 20L, 20L, 20L, 25L,
                                     20L, 30L, 30L, 35L, 30L, 30L), ncol = 3),
                   strand = c('+', '-', '*', '+', '+'))
    # GRanges version of gt3
    gr3 <- as(gt3, "GRanges")
    findOverlaps(gt3, gt3, type = 'any')
    #> Warning in .local(query, subject, maxgap, minoverlap, type, select, ...): 'type'
    #> is not 'equal' so coercing 'query' and 'subject' to 'GRanges' objects (see docs
    #> for details)
    #> Hits object with 13 hits and 0 metadata columns:
    #>        queryHits subjectHits
    #>        <integer>   <integer>
    #>    [1]         1           3
    #>    [2]         1           1
    #>    [3]         1           4
    #>    [4]         2           3
    #>    [5]         2           2
    #>    ...       ...         ...
    #>    [9]         3           4
    #>   [10]         4           3
    #>   [11]         4           1
    #>   [12]         4           4
    #>   [13]         5           5
    #>   -------
    #>   queryLength: 5 / subjectLength: 5
    # GTuples and GRanges methods identical
    identical(findOverlaps(gt3, gt3, type = 'any'),
              findOverlaps(gr3, gr3, type = 'any')) # TRUE
    #> Warning in .local(query, subject, maxgap, minoverlap, type, select, ...): 'type'
    #> is not 'equal' so coercing 'query' and 'subject' to 'GRanges' objects (see docs
    #> for details)
    #> [1] TRUE

    findOverlaps(gt3, gt3, type = 'start')
    #> Warning in .local(query, subject, maxgap, minoverlap, type, select, ...): 'type'
    #> is not 'equal' so coercing 'query' and 'subject' to 'GRanges' objects (see docs
    #> for details)
    #> Hits object with 13 hits and 0 metadata columns:
    #>        queryHits subjectHits
    #>        <integer>   <integer>
    #>    [1]         1           3
    #>    [2]         1           1
    #>    [3]         1           4
    #>    [4]         2           3
    #>    [5]         2           2
    #>    ...       ...         ...
    #>    [9]         3           4
    #>   [10]         4           3
    #>   [11]         4           1
    #>   [12]         4           4
    #>   [13]         5           5
    #>   -------
    #>   queryLength: 5 / subjectLength: 5
    # GTuples and GRanges methods identical
    identical(findOverlaps(gt3, gt3, type = 'start'),
              findOverlaps(gr3, gr3, type = 'start')) # TRUE
    #> Warning in .local(query, subject, maxgap, minoverlap, type, select, ...): 'type'
    #> is not 'equal' so coercing 'query' and 'subject' to 'GRanges' objects (see docs
    #> for details)
    #> [1] TRUE

    findOverlaps(gt3, gt3, type = 'end')
    #> Warning in .local(query, subject, maxgap, minoverlap, type, select, ...): 'type'
    #> is not 'equal' so coercing 'query' and 'subject' to 'GRanges' objects (see docs
    #> for details)
    #> Hits object with 7 hits and 0 metadata columns:
    #>       queryHits subjectHits
    #>       <integer>   <integer>
    #>   [1]         1           1
    #>   [2]         1           4
    #>   [3]         2           2
    #>   [4]         3           3
    #>   [5]         4           1
    #>   [6]         4           4
    #>   [7]         5           5
    #>   -------
    #>   queryLength: 5 / subjectLength: 5
    # GTuples and GRanges methods identical
    identical(findOverlaps(gt3, gt3, type = 'end'),
              findOverlaps(gr3, gr3, type = 'end')) # TRUE
    #> Warning in .local(query, subject, maxgap, minoverlap, type, select, ...): 'type'
    #> is not 'equal' so coercing 'query' and 'subject' to 'GRanges' objects (see docs
    #> for details)
    #> [1] TRUE

    findOverlaps(gt3, gt3, type = 'within')
    #> Warning in .local(query, subject, maxgap, minoverlap, type, select, ...): 'type'
    #> is not 'equal' so coercing 'query' and 'subject' to 'GRanges' objects (see docs
    #> for details)
    #> Hits object with 10 hits and 0 metadata columns:
    #>        queryHits subjectHits
    #>        <integer>   <integer>
    #>    [1]         1           3
    #>    [2]         1           1
    #>    [3]         1           4
    #>    [4]         2           3
    #>    [5]         2           2
    #>    [6]         3           3
    #>    [7]         4           3
    #>    [8]         4           1
    #>    [9]         4           4
    #>   [10]         5           5
    #>   -------
    #>   queryLength: 5 / subjectLength: 5
    # GTuples and GRanges methods identical
    identical(findOverlaps(gt3, gt3, type = 'within'),
              findOverlaps(gr3, gr3, type = 'within')) # TRUE
    #> Warning in .local(query, subject, maxgap, minoverlap, type, select, ...): 'type'
    #> is not 'equal' so coercing 'query' and 'subject' to 'GRanges' objects (see docs
    #> for details)
    #> [1] TRUE

    findOverlaps(gt3, gt3, type = 'equal')
    #> Hits object with 5 hits and 0 metadata columns:
    #>       queryHits subjectHits
    #>       <integer>   <integer>
    #>   [1]         1           1
    #>   [2]         2           2
    #>   [3]         3           3
    #>   [4]         4           4
    #>   [5]         5           5
    #>   -------
    #>   queryLength: 5 / subjectLength: 5
    # GTuples and GRanges methods **not** identical because  GRanges method ignores
    # "internal positions".
    identical(findOverlaps(gt3, gt3, type = 'equal'),
              findOverlaps(gr3, gr3, type = 'equal')) # FALSE
    #> [1] FALSE
    # Can pass other arguments, such as select and ignore.strand
    findOverlaps(gt3, gt3, type = 'equal', ignore.strand = TRUE, select = 'last')
    #> [1] 2 2 3 4 5

# 5 Comparison of genomic tuples

I have chosen a definition that matches my intuition of what constitutes a
comparison between genomic tuples. However, I am open to suggestions on amending or extending this behaviour in future versions of GenomicTuples.

## 5.1 Definition of comparison methods for genomic tuples

The comparison of two genomic tuples, `x` and `y`, is done by first comparing the `seqnames(x)` to `seqnames(y)`, then `strand(x)` to `strand(y)` and finally `tuples(x)` to `tuples(y)`.

Ordering of `seqnames` and `strand` is as implemented GenomicRanges::GRanges. Ordering of `tuples` is element-wise, i.e. \(pos_{1}, \ldots, pos_{m}\) are compared in turn. For example, `chr1:+:10, 20, 30` is considered less than `chr1:+:10, 20, 40`. This defines what I will refer to as the “natural order” of genomic tuples.

The above is implemented in the `pcompare` method for GTuples, which performs “generalized range-wise comparison” of two GTuples objects, `x` and `y`. That is, `pcompare(x, y)` returns an integer vector where the \(i^{th}\) element is a code describing how the \(i^{th}\) element in `x` is qualitatively positioned relatively to the \(i^{th}\) element in `y`. A code that is `< 0`, `= 0`, or `> 0`, corresponds to `x[i] < y[i]`, `x[i] == y[i]`, or `x[i] > y[i]`, respectively.

The 6 traditional binary comparison operators (`==`, `!=`, `<=`, `>=`, `<`, and `>`), other comparison operators (`match`, `order`, `sort`, and `rank`) and duplicate-based methods (`duplicated` and `unique`) all use this “natural order”.

## 5.2 Examples

It is easiest to understand the above definitions by studying a few examples, here using 3-tuples:

    # Construct example 3-tuples
    gt3 <- GTuples(seqnames = c('chr1', 'chr1', 'chr1', 'chr1', 'chr2', 'chr1',
                                'chr1'),
                   tuples = matrix(c(10L, 10L, 10L, 10L, 10L, 5L, 10L, 20L, 20L,
                                     20L, 25L, 20L, 20L, 20L, 30L, 30L, 35L, 30L,
                                     30L, 30L, 35L),
                                   ncol = 3),
                   strand = c('+', '-', '*', '+', '+', '+', '+'))
    gt3
    #> GTuples object with 7 x 3-tuples and 0 metadata columns:
    #>       seqnames pos1 pos2 pos3 strand
    #>   [1]     chr1   10   20   30      +
    #>   [2]     chr1   10   20   30      -
    #>   [3]     chr1   10   20   35      *
    #>   [4]     chr1   10   25   30      +
    #>   [5]     chr2   10   20   30      +
    #>   [6]     chr1    5   20   30      +
    #>   [7]     chr1   10   20   35      +
    #>   ---
    #>   seqinfo: 2 sequences from an unspecified genome; no seqlengths

    # pcompare each tuple to itself
    pcompare(gt3, gt3)
    #> [1] 0 0 0 0 0 0 0
    gt3 < gt3
    #> [1] FALSE FALSE FALSE FALSE FALSE FALSE FALSE
    gt3 > gt3
    #> [1] FALSE FALSE FALSE FALSE FALSE FALSE FALSE
    gt3 == gt3
    #> [1] TRUE TRUE TRUE TRUE TRUE TRUE TRUE

    # pcompare the third tuple to all tuples
    pcompare(gt3[3], gt3)
    #> [1]  2  1  0  2 -1  2  2
    gt3[3] < gt3
    #> [1] FALSE FALSE FALSE FALSE  TRUE FALSE FALSE
    gt3[3] > gt3
    #> [1]  TRUE  TRUE FALSE  TRUE FALSE  TRUE  TRUE
    gt3[3] == gt3
    #> [1] FALSE FALSE  TRUE FALSE FALSE FALSE FALSE

    ## Some comparisons where tuples differ only in one coordinate

    # Ordering of seqnames
    # 'chr1' < 'chr2' for tuples with otherwise identical coordinates
    gt3[1] < gt3[5] # TRUE
    #> [1] TRUE

    # Ordering of strands
    # '+' < '-' < '*' for tuples with otherwise identical coordiantes
    gt3[1] < gt3[2] # TRUE
    #> [1] TRUE
    gt3[1] < gt3[2] # TRUE
    #> [1] TRUE
    gt3[1] < unstrand(gt3[2]) # TRUE
    #> [1] TRUE
    gt3[2] < unstrand(gt3[2]) # TRUE
    #> [1] TRUE

    # Ordering of tuples
    # Tuples checked sequentially from pos1, ..., posm for tuples with otherwise
    # identical coordinates
    gt3[6] < gt3[1] # TRUE due to pos1
    #> [1] TRUE
    gt3[2] < gt3[4] # TRUE due to pos2
    #> [1] FALSE
    gt3[1] < gt3[7] # TRUE due to pos3
    #> [1] TRUE

    # Sorting of tuples
    # Sorted first by seqnames, then by strand, then by tuples
    sort(gt3)
    #> GTuples object with 7 x 3-tuples and 0 metadata columns:
    #>       seqnames pos1 pos2 pos3 strand
    #>   [1]     chr1    5   20   30      +
    #>   [2]     chr1   10   20   30      +
    #>   [3]     chr1   10   20   35      +
    #>   [4]     chr1   10   25   30      +
    #>   [5]     chr1   10   20   30      -
    #>   [6]     chr1   10   20   35      *
    #>   [7]     chr2   10   20   30      +
    #>   ---
    #>   seqinfo: 2 sequences from an unspecified genome; no seqlengths

    # Duplicate tuples
    # Duplicate tuples must have identical seqnames, strand and positions (tuples)
    duplicated(c(gt3, gt3[1:3]))
    #>  [1] FALSE FALSE FALSE FALSE FALSE FALSE FALSE  TRUE  TRUE  TRUE
    unique(c(gt3, gt3[1:3]))
    #> GTuples object with 7 x 3-tuples and 0 metadata columns:
    #>       seqnames pos1 pos2 pos3 strand
    #>   [1]     chr1   10   20   30      +
    #>   [2]     chr1   10   20   30      -
    #>   [3]     chr1   10   20   35      *
    #>   [4]     chr1   10   25   30      +
    #>   [5]     chr2   10   20   30      +
    #>   [6]     chr1    5   20   30      +
    #>   [7]     chr1   10   20   35      +
    #>   ---
    #>   seqinfo: 2 sequences from an unspecified genome; no seqlengths

# 6 Acknowledgements

I am very grateful to all the Bioconductor developers but particularly wish to thank the developers of GenomicRanges (Lawrence, M. et al. Software for computing and annotating genomic ranges. PLoS Comput. Biol. 9, e1003118 (2013).), which GenomicTuples uses heavily and is based upon. A special thanks to Hervé Pagès for his assistance and fixes when making upstream changes to GenomicRanges.

# 7 Session info

Here is the output of `sessionInfo` on the system on which this document was compiled:

    sessionInfo()
    #> R Under development (unstable) (2020-03-10 r77920)
    #> Platform: x86_64-w64-mingw32/x64 (64-bit)
    #> Running under: Windows Server 2012 R2 x64 (build 9600)
    #>
    #> Matrix products: default
    #>
    #> locale:
    #> [1] LC_COLLATE=C
    #> [2] LC_CTYPE=English_United States.1252
    #> [3] LC_MONETARY=English_United States.1252
    #> [4] LC_NUMERIC=C
    #> [5] LC_TIME=English_United States.1252
    #>
    #> attached base packages:
    #> [1] parallel  stats4    stats     graphics  grDevices utils     datasets
    #> [8] methods   base
    #>
    #> other attached packages:
    #> [1] GenomicTuples_1.21.5 GenomicRanges_1.39.3 GenomeInfoDb_1.23.14
    #> [4] IRanges_2.21.8       S4Vectors_0.25.14    BiocGenerics_0.33.3
    #> [7] BiocStyle_2.15.6
    #>
    #> loaded via a namespace (and not attached):
    #>  [1] Rcpp_1.0.4             knitr_1.28             XVector_0.27.2
    #>  [4] magrittr_1.5           zlibbioc_1.33.1        rlang_0.4.5
    #>  [7] stringr_1.4.0          tools_4.0.0            data.table_1.12.8
    #> [10] xfun_0.12              htmltools_0.4.0        yaml_2.2.1
    #> [13] digest_0.6.25          bookdown_0.18          GenomeInfoDbData_1.2.2
    #> [16] BiocManager_1.30.10    bitops_1.0-6           RCurl_1.98-1.1
    #> [19] evaluate_0.14          rmarkdown_2.1.1        stringi_1.4.6
    #> [22] compiler_4.0.0

# 8 References

- Lawrence, M. et al. Software for computing and annotating genomic ranges. PLoS Comput. Biol. 9, e1003118 (2013).