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> using DimensionalData

julia> A = rand(5, 10)

5×10 Matrix{Float64}:
 0.67178   0.760205  0.796812   0.545129  …  0.661715  0.321923   0.0697249
 0.677207  0.553558  0.280686   0.054613     0.557524  0.342437   0.392848
 0.768325  0.912065  0.917543   0.8234       0.874037  0.847981   0.625296
 0.514177  0.413756  0.0621546  0.700218     0.15388   0.912676   0.222681
 0.764613  0.7826    0.625257   0.366217     0.344408  0.0140644  0.870094

julia> da = DimArray(A, (X, Y))

┌ 5×10 DimArray{Float64, 2} ┐
├───────────────────── dims ┤
  ↓ X, → Y
└───────────────────────────┘
 0.67178   0.760205  0.796812   0.545129  …  0.661715  0.321923   0.0697249
 0.677207  0.553558  0.280686   0.054613     0.557524  0.342437   0.392848
 0.768325  0.912065  0.917543   0.8234       0.874037  0.847981   0.625296
 0.514177  0.413756  0.0621546  0.700218     0.15388   0.912676   0.222681
 0.764613  0.7826    0.625257   0.366217     0.344408  0.0140644  0.870094

We can access a value with the same dimension wrappers:

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

0.6772070772014024

There are shortcuts for creating DimArray:

julia> A = rand(5, 10)

5×10 Matrix{Float64}:
 0.677974   0.468066   0.347379  0.250913  …  0.385262  0.52496    0.468184
 0.0599255  0.247198   0.813229  0.391395     0.208542  0.750146   0.106224
 0.557369   0.0558987  0.44104   0.178902     0.561365  0.475747   0.605217
 0.104867   0.497181   0.570254  0.906883     0.986207  0.0287126  0.108968
 0.176197   0.675467   0.208384  0.438447     0.704858  0.299799   0.540566

julia> DimArray(A, (X, Y))

┌ 5×10 DimArray{Float64, 2} ┐
├───────────────────── dims ┤
  ↓ X, → Y
└───────────────────────────┘
 0.677974   0.468066   0.347379  0.250913  …  0.385262  0.52496    0.468184
 0.0599255  0.247198   0.813229  0.391395     0.208542  0.750146   0.106224
 0.557369   0.0558987  0.44104   0.178902     0.561365  0.475747   0.605217
 0.104867   0.497181   0.570254  0.906883     0.986207  0.0287126  0.108968
 0.176197   0.675467   0.208384  0.438447     0.704858  0.299799   0.540566

julia> DimArray(A, (X, Y); name=:DimArray, metadata=Dict())

┌ 5×10 DimArray{Float64, 2} DimArray ┐
├────────────────────────────── dims ┤
  ↓ X, → Y
└────────────────────────────────────┘
 0.677974   0.468066   0.347379  0.250913  …  0.385262  0.52496    0.468184
 0.0599255  0.247198   0.813229  0.391395     0.208542  0.750146   0.106224
 0.557369   0.0558987  0.44104   0.178902     0.561365  0.475747   0.605217
 0.104867   0.497181   0.570254  0.906883     0.986207  0.0287126  0.108968
 0.176197   0.675467   0.208384  0.438447     0.704858  0.299799   0.540566

Constructing DimArray with arbitrary dimension names

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

julia> da1 = DimArray(rand(5, 5), (:a, :b))

┌ 5×5 DimArray{Float64, 2} ┐
├──────────────────── dims ┤
  ↓ a, → b
└──────────────────────────┘
 0.137436   0.206406  0.811947   0.444708  0.358625
 0.252633   0.94412   0.481184   0.286785  0.828395
 0.573147   0.917457  0.758865   0.127356  0.680352
 0.0760848  0.774452  0.0641884  0.303772  0.680225
 0.0992684  0.784295  0.970459   0.498764  0.530271

and get a value, here another smaller DimArray:

julia> da1[a=3, b=1:3]

┌ 3-element DimArray{Float64, 1} ┐
├────────────────────────── dims ┤
  ↓ b
└────────────────────────────────┘
 0.573147
 0.917457
 0.758865

Dimensional Indexing

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

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> da2 = rand(X(10), Y(7), Z(5))

┌ 10×7×5 DimArray{Float64, 3} ┐
├─────────────────────── dims ┤
  ↓ X, → Y, ↗ Z
└─────────────────────────────┘
[:, :, 1]
 0.0987741  0.959907  0.0726015  0.0442881  0.877156  0.725503   0.86641
 0.848841   0.877295  0.55884    0.896093   0.976255  0.249525   0.609585
 0.0851053  0.893793  0.858332   0.556985   0.857562  0.0105671  0.991037
 0.732574   0.337006  0.870536   0.542646   0.924924  0.241047   0.0889246
 0.385228   0.251766  0.292204   0.0326231  0.299078  0.771093   0.804936
 0.798934   0.146259  0.689557   0.0129614  0.217848  0.665549   0.919702
 0.829194   0.395143  0.0342701  0.0470287  0.752915  0.109204   0.649382
 0.972201   0.537364  0.475142   0.868227   0.302211  0.793706   0.777313
 0.453674   0.819533  0.481182   0.538717   0.821808  0.216046   0.554297
 0.199504   0.733034  0.578734   0.818616   0.210304  0.660868   0.404788

julia> da2[(X(3), Z(5))]

┌ 7-element DimArray{Float64, 1} ┐
├────────────────────────── dims ┤
  ↓ Y
└────────────────────────────────┘
 0.639398
 0.144391
 0.370946
 0.974725
 0.664311
 0.805566
 0.359254

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

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> da2[inds]

┌ 7×3 DimArray{Float64, 2} ┐
├──────────────────────────┴─────────────────────────────────────── dims ┐
  ↓ Y,
  → XZ MergedLookup{Tuple{Int64, Int64}} [(3, 5), …, (8, 2)] ↓ X, → Z
└────────────────────────────────────────────────────────────────────────┘
  (3, 5)    (7, 4)    (8, 2)
 0.639398  0.559092  0.504242
 0.144391  0.241109  0.159094
 0.370946  0.190387  0.849392
 0.974725  0.668515  0.425368
 0.664311  0.236262  0.976043
 0.805566  0.965946  0.709747
 0.359254  0.299806  0.115367

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

julia> da2[DimIndices(dims(da2, (X, Z))), Y(3)]

┌ 10×5 DimArray{Float64, 2} ┐
├───────────────────── dims ┤
  ↓ X, → Z
└───────────────────────────┘
 0.0726015  0.189031   0.478776   0.636878  0.607268
 0.55884    0.242935   0.647999   0.823697  0.327547
 0.858332   0.507807   0.761782   0.750189  0.370946
 0.870536   0.65229    0.0977422  0.474751  0.852659
 0.292204   0.788363   0.129544   0.937845  0.373488
 0.689557   0.0563826  0.88885    0.58454   0.0073424
 0.0342701  0.11003    0.160568   0.190387  0.598658
 0.475142   0.849392   0.103129   0.834542  0.566304
 0.481182   0.126864   0.135832   0.898508  0.302643
 0.578734   0.669608   0.569461   0.698929  0.141916

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> da1[X(3), 4]

ERROR: ArgumentError: invalid index: X{Int64}(3) of type X{Int64}

Begin End indexing

WARNING

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.

julia> da[X=Begin+1, Y=End]

0.39284805878379914

It also works in ranges, even with basic math:

julia> da[X=Begin:Begin+1, Y=Begin+1:End-1]

┌ 2×8 DimArray{Float64, 2} ┐
├──────────────────── dims ┤
  ↓ X, → Y
└──────────────────────────┘
 0.760205  0.796812  0.545129  0.989571  …  0.3037   0.661715  0.321923
 0.553558  0.280686  0.054613  0.925873     0.36706  0.557524  0.342437

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> da5 = rand(X(3), Y(4), Ti(5))

┌ 3×4×5 DimArray{Float64, 3} ┐
├────────────────────── dims ┤
  ↓ X, → Y, ↗ Ti
└────────────────────────────┘
[:, :, 1]
 0.888027  0.537621  0.507171  0.519136
 0.906864  0.220951  0.207805  0.0766932
 0.91711   0.537011  0.95494   0.877763

julia> sum(da5; dims=Ti)

┌ 3×4×1 DimArray{Float64, 3} ┐
├────────────────────── dims ┤
  ↓ X, → Y, ↗ Ti
└────────────────────────────┘
[:, :, 1]
 3.00706  1.78108  1.45863  2.91389
 2.09261  2.65699  2.40165  1.2761
 2.09856  1.04     2.83859  3.14096

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> 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> @benchmark $da4[X(1), Y(2)]

BenchmarkTools.Trial: 10000 samples with 1000 evaluations per sample.
 Range (min … max):  3.396 ns … 20.849 ns  ┊ GC (min … max): 0.00% … 0.00%
 Time  (median):     3.406 ns              ┊ GC (median):    0.00%
 Time  (mean ± σ):   3.464 ns ±  0.601 ns  ┊ GC (mean ± σ):  0.00% ± 0.00%

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  3.4 ns         Histogram: frequency by time        5.43 ns <

 Memory estimate: 0 bytes, allocs estimate: 0.

the same as accessing the parent array directly:

julia> @benchmark parent($da4)[1, 2]

BenchmarkTools.Trial: 10000 samples with 1000 evaluations per sample.
 Range (min … max):  3.396 ns … 25.338 ns  ┊ GC (min … max): 0.00% … 0.00%
 Time  (median):     3.407 ns              ┊ GC (median):    0.00%
 Time  (mean ± σ):   3.440 ns ±  0.528 ns  ┊ GC (mean ± σ):  0.00% ± 0.00%

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  3.4 ns         Histogram: frequency by time        3.42 ns <

 Memory estimate: 0 bytes, allocs estimate: 0.