DimensionalData.jl

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DimArrays ​

DimArrays are wrappers for other kinds of AbstractArray that add named dimension lookups.

Here we define a Matrix of Float64, and give it X and Y dimensions

julia
julia> using DimensionalData

julia> A = rand(5, 10)
5×10 Matrix{Float64}:
 0.254966   0.0542736  0.174367   0.52276   …  0.257671  0.433249   0.37079
 0.0196069  0.492355   0.0284599  0.681254     0.266973  0.841434   0.723008
 0.585697   0.945226   0.82122    0.834784     0.622533  0.860144   0.430653
 0.556947   0.342535   0.0125467  0.75126      0.636407  0.0268885  0.0670457
 0.268822   0.0363362  0.53834    0.319352     0.153277  0.543613   0.783523
julia
julia> da = DimArray(A, (X, Y))
╭──────────────────────────╮
│ 5×10 DimArray{Float64,2} │
├──────────────────── dims ┤
  ↓ X, → Y
└──────────────────────────┘
 0.254966   0.0542736  0.174367   0.52276   …  0.257671  0.433249   0.37079
 0.0196069  0.492355   0.0284599  0.681254     0.266973  0.841434   0.723008
 0.585697   0.945226   0.82122    0.834784     0.622533  0.860144   0.430653
 0.556947   0.342535   0.0125467  0.75126      0.636407  0.0268885  0.0670457
 0.268822   0.0363362  0.53834    0.319352     0.153277  0.543613   0.783523

We can access a value with the same dimension wrappers:

julia
julia> da[Y(1), X(2)]
0.01960686848243043

There are shortcuts for creating DimArray:

julia
julia> A = rand(5, 10)
5×10 Matrix{Float64}:
 0.473731  0.478363  0.490415  0.973651  …  0.906265    0.209399  0.234222
 0.84213   0.197936  0.50529   0.802567     0.74795     0.877415  0.274847
 0.12631   0.431317  0.191017  0.487295     0.510128    0.478535  0.285446
 0.314178  0.328764  0.615541  0.862096     0.00462179  0.873724  0.860179
 0.474796  0.906484  0.536741  0.232471     0.525516    0.273485  0.277565
julia
julia> DimArray(A, (X, Y))
╭──────────────────────────╮
│ 5×10 DimArray{Float64,2} │
├──────────────────── dims ┤
  ↓ X, → Y
└──────────────────────────┘
 0.473731  0.478363  0.490415  0.973651  …  0.906265    0.209399  0.234222
 0.84213   0.197936  0.50529   0.802567     0.74795     0.877415  0.274847
 0.12631   0.431317  0.191017  0.487295     0.510128    0.478535  0.285446
 0.314178  0.328764  0.615541  0.862096     0.00462179  0.873724  0.860179
 0.474796  0.906484  0.536741  0.232471     0.525516    0.273485  0.277565
julia
julia> DimArray(A, (X, Y); name=:DimArray, metadata=Dict())
╭───────────────────────────────────╮
│ 5×10 DimArray{Float64,2} DimArray │
├───────────────────────────── dims ┤
  ↓ X, → Y
├───────────────────────── metadata ┤
  Dict{Any, Any}()
└───────────────────────────────────┘
 0.473731  0.478363  0.490415  0.973651  …  0.906265    0.209399  0.234222
 0.84213   0.197936  0.50529   0.802567     0.74795     0.877415  0.274847
 0.12631   0.431317  0.191017  0.487295     0.510128    0.478535  0.285446
 0.314178  0.328764  0.615541  0.862096     0.00462179  0.873724  0.860179
 0.474796  0.906484  0.536741  0.232471     0.525516    0.273485  0.277565

Constructing DimArray with arbitrary dimension names ​

For arbitrary names, we can use the Dim{:name} dims by using Symbols, and indexing with keywords:

julia
julia> da1 = DimArray(rand(5, 5), (:a, :b))
╭─────────────────────────╮
│ 5×5 DimArray{Float64,2} │
├─────────────────── dims ┤
  ↓ a, → b
└─────────────────────────┘
 0.630672  0.411295  0.390882  0.93114    0.25481
 0.459446  0.223727  0.672225  0.960334   0.366746
 0.926632  0.602688  0.420311  0.0614758  0.0500562
 0.912899  0.407069  0.692156  0.435304   0.455434
 0.799753  0.698676  0.408161  0.0036255  0.892026

and get a value, here another smaller DimArray:

julia
julia> da1[a=3, b=1:3]
╭───────────────────────────────╮
│ 3-element DimArray{Float64,1} │
├───────────────────────── dims ┤
  ↓ b
└───────────────────────────────┘
 0.926632
 0.602688
 0.420311

Dimensional Indexing ​

When used for indexing, dimension wrappers free us from knowing the order of our objects axes. These are the same:

julia
julia> da[X(2), Y(1)] == da[Y(1), X(2)]
true

We also can use Tuples of dimensions like CartesianIndex, but they don't have to be in order of consecutive axes.

julia
julia> da2 = rand(X(10), Y(7), Z(5))
╭────────────────────────────╮
│ 10×7×5 DimArray{Float64,3} │
├────────────────────── dims ┤
  ↓ X, → Y, ↗ Z
└────────────────────────────┘
[:, :, 1]
 0.494198  0.891012  0.35591   0.879507   0.306018  0.171146  0.900921
 0.90456   0.581675  0.662664  0.475175   0.34708   0.318687  0.125854
 0.49259   0.51576   0.435284  0.829837   0.491404  0.808217  0.488783
 0.674695  0.776362  0.740192  0.301411   0.869942  0.585707  0.197516
 0.179302  0.72292   0.982988  0.846096   0.267424  0.423173  0.121698
 0.101133  0.311115  0.106988  0.401742   0.906479  0.607331  0.194732
 0.949688  0.10438   0.839559  0.875198   0.850127  0.617745  0.347996
 0.66229   0.679423  0.752713  0.360941   0.358949  0.180831  0.732175
 0.688789  0.879564  0.350734  0.851987   0.8019    0.673176  0.436439
 0.96233   0.462039  0.903333  0.0222559  0.521157  0.173682  0.11301
julia
julia> da2[(X(3), Z(5))]
╭───────────────────────────────╮
│ 7-element DimArray{Float64,1} │
├───────────────────────── dims ┤
  ↓ Y
└───────────────────────────────┘
 0.399328
 0.0768945
 0.725217
 0.0782474
 0.480972
 0.739601
 0.796302

We can index with Vector of Tuple{Vararg(Dimension}} like vectors of CartesianIndex. This will merge the dimensions in the tuples:

julia
julia> inds = [(X(3), Z(5)), (X(7), Z(4)), (X(8), Z(2))]
3-element Vector{Tuple{X{Int64}, Z{Int64}}}:
 ↓ X 3, → Z 5
 ↓ X 7, → Z 4
 ↓ X 8, → Z 2
julia
julia> da2[inds]
╭─────────────────────────╮
│ 7×3 DimArray{Float64,2} │
├─────────────────────────┴────────────────────────────────────────── dims ┐
  ↓ Y ,
  → XZ MergedLookup{Tuple{Int64, Int64}} [(3, 5), (7, 4), (8, 2)] ↓ X, → Z
└──────────────────────────────────────────────────────────────────────────┘
  (3, 5)     (7, 4)      (8, 2)
 0.399328   0.222351    0.845011
 0.0768945  0.622988    0.996835
 0.725217   0.428327    0.433753
 0.0782474  0.056375    0.172586
 0.480972   0.00889783  0.533393
 0.739601   0.0311501   0.768995
 0.796302   0.660471    0.409441

DimIndices can be used like CartesianIndices but again, without the constraint of consecutive dimensions or known order.

julia
julia> da2[DimIndices(dims(da2, (X, Z))), Y(3)]
╭──────────────────────────╮
│ 10×5 DimArray{Float64,2} │
├──────────────────── dims ┤
  ↓ X, → Z
└──────────────────────────┘
 0.35591   0.425549  0.897045  0.390221   0.343369
 0.662664  0.538327  0.57397   0.771498   0.512143
 0.435284  0.107479  0.25401   0.103998   0.725217
 0.740192  0.603077  0.536009  0.332878   0.878539
 0.982988  0.396291  0.686698  0.92885    0.310618
 0.106988  0.696538  0.758446  0.0499511  0.615719
 0.839559  0.822314  0.607239  0.428327   0.24181
 0.752713  0.433753  0.528599  0.169286   0.445842
 0.350734  0.653801  0.264563  0.536341   0.657623
 0.903333  0.842037  0.669504  0.206849   0.90977

The Dimension indexing layer sits on top of regular indexing and can not be combined with it! Regular indexing specifies order, so doesn't mix well with our dimensions.

Mixing them will throw an error:

julia
julia> da1[X(3), 4]
ERROR: ArgumentError: invalid index: X{Int64}(3) of type X{Int64}

Begin End indexing ​

julia
julia> da1[X=Begin+1, Y=End]
┌ Warning: (X, Y) dims were not found in object.
└ @ DimensionalData.Dimensions ~/work/DimensionalData.jl/DimensionalData.jl/src/Dimensions/primitives.jl:775
╭─────────────────────────╮
│ 5×5 DimArray{Float64,2} │
├─────────────────── dims ┤
  ↓ a, → b
└─────────────────────────┘
 0.630672  0.411295  0.390882  0.93114    0.25481
 0.459446  0.223727  0.672225  0.960334   0.366746
 0.926632  0.602688  0.420311  0.0614758  0.0500562
 0.912899  0.407069  0.692156  0.435304   0.455434
 0.799753  0.698676  0.408161  0.0036255  0.892026

It also works in ranges, even with basic math:

julia
julia> da1[X=Begin:Begin+1, Y=Begin+1:End-1]
┌ Warning: (X, Y) dims were not found in object.
└ @ DimensionalData.Dimensions ~/work/DimensionalData.jl/DimensionalData.jl/src/Dimensions/primitives.jl:775
╭─────────────────────────╮
│ 5×5 DimArray{Float64,2} │
├─────────────────── dims ┤
  ↓ a, → b
└─────────────────────────┘
 0.630672  0.411295  0.390882  0.93114    0.25481
 0.459446  0.223727  0.672225  0.960334   0.366746
 0.926632  0.602688  0.420311  0.0614758  0.0500562
 0.912899  0.407069  0.692156  0.435304   0.455434
 0.799753  0.698676  0.408161  0.0036255  0.892026

In base julia the keywords begin and end can be used to index the first or last element of an array. But this doesn't work when named indexing is used. Instead you can use the types Begin and End.

Indexing

Indexing AbstractDimArrays works with getindex, setindex! and view. The result is still an AbstracDimArray, unless using all single Int or Selectors that resolve to Int inside Dimension.

dims keywords ​

In many Julia functions like, size or sum, you can specify the dimension along which to perform the operation as an Int. It is also possible to do this using Dimension types with AbstractDimArray:

julia
julia> da5 = rand(X(3), Y(4), Ti(5))
╭───────────────────────────╮
│ 3×4×5 DimArray{Float64,3} │
├───────────────────── dims ┤
  ↓ X, → Y, ↗ Ti
└───────────────────────────┘
[:, :, 1]
 0.0119886  0.752378  0.209112  0.473908
 0.354549   0.598904  0.190014  0.187816
 0.0929424  0.899065  0.366571  0.024557
julia
julia> sum(da5; dims=Ti)
╭───────────────────────────╮
│ 3×4×1 DimArray{Float64,3} │
├───────────────────── dims ┤
  ↓ X, → Y, ↗ Ti
└───────────────────────────┘
[:, :, 1]
 1.0199   3.55923  2.08786  3.0408
 2.09372  2.19568  2.25177  1.6164
 1.5156   3.16999  2.77272  2.77511

Dims keywords

Methods where dims, dim types, or Symbols can be used to indicate the array dimension:

  • size, axes, firstindex, lastindex

  • cat, reverse, dropdims

  • reduce, mapreduce

  • sum, prod, maximum, minimum

  • mean, median, extrema, std, var, cor, cov

  • permutedims, adjoint, transpose, Transpose

  • mapslices, eachslice

Performance ​

Indexing with Dimensions has no runtime cost. Let's benchmark it:

julia
julia> using BenchmarkTools

julia> da4 = ones(X(3), Y(3))
╭─────────────────────────╮
│ 3×3 DimArray{Float64,2} │
├─────────────────── dims ┤
  ↓ X, → Y
└─────────────────────────┘
 1.0  1.0  1.0
 1.0  1.0  1.0
 1.0  1.0  1.0
julia
julia> @benchmark $da4[X(1), Y(2)]
BenchmarkTools.Trial: 10000 samples with 1000 evaluations.
 Range (min … max):  2.785 ns … 25.538 ns  ┊ GC (min … max): 0.00% … 0.00%
 Time  (median):     2.795 ns              ┊ GC (median):    0.00%
 Time  (mean ± σ):   2.805 ns ±  0.372 ns  ┊ GC (mean ± σ):  0.00% ± 0.00%

  ▆▄                █ ▆                ▃ ▁                ▂  ▁
  ██▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁█▁█▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁█▁█▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁██ █
  2.78 ns      Histogram: log(frequency) by time     2.82 ns <

 Memory estimate: 0 bytes, allocs estimate: 0.

the same as accessing the parent array directly:

julia
julia> @benchmark parent($da4)[1, 2]
BenchmarkTools.Trial: 10000 samples with 1000 evaluations.
 Range (min … max):  2.785 ns … 32.962 ns  ┊ GC (min … max): 0.00% … 0.00%
 Time  (median):     2.795 ns              ┊ GC (median):    0.00%
 Time  (mean ± σ):   2.806 ns ±  0.411 ns  ┊ GC (mean ± σ):  0.00% ± 0.00%

  ▆▄            █▆            ▃ ▁            ▃▁              ▁
  ██▁▁▁▁▁▁▁▁▁▁▁▁██▁▁▁▁▁▁▁▁▁▁▁▁█▁█▁▁▁▁▁▁▁▁▁▁▁▁██▁▁▁▁▁▁▁▁▁▁▁▁▆ █
  2.78 ns      Histogram: log(frequency) by time     2.82 ns <

 Memory estimate: 0 bytes, allocs estimate: 0.