Dimensional broadcasts with @d and broadcast_dims

Broadcasting over AbstractDimArray works as usual with Base Julia broadcasts, except that dimensions are checked for compatibility with each other, and that values match. Strict checks can be turned off globally with strict_broadcast!(false). To avoid even dimension name checks, broadcast over parent(dimarray).

The @d macro is a dimension-aware extension to regular dot broadcasting. broadcast_dims is analogous to Base Julia's broadcast.

Because we know the names of the dimensions, there is no ambiguity in which ones we mean to broadcast together. This means we can permute and reshape dims so that broadcasts that would fail with a regular Array just work with a DimArray.

As an added bonus, broadcast_dims even works on DimStacks. Currently, @d does not work on DimStack.

Example: scaling along the time dimension

Define some dimensions:

using DimensionalData
using Dates
using Statistics

julia> x, y, t = X(1:100), Y(1:25), Ti(DateTime(2000):Month(1):DateTime(2000, 12))

(↓ X 1:100,
→ Y 1:25,
↗ Ti DateTime("2000-01-01T00:00:00"):Month(1):DateTime("2000-12-01T00:00:00"))

A DimArray from 1:12 to scale with:

julia> month_scalars = DimArray(month, t)

┌ 12-element DimArray{Int64, 1} month(Ti) ┐
├─────────────────────────────────────────┴────────────────────────────── dims ┐
  ↓ Ti Sampled{DateTime} DateTime("2000-01-01T00:00:00"):Month(1):DateTime("2000-12-01T00:00:00") ForwardOrdered Regular Points
└──────────────────────────────────────────────────────────────────────────────┘
 2000-01-01T00:00:00   1
 2000-02-01T00:00:00   2
 2000-03-01T00:00:00   3
 2000-04-01T00:00:00   4
 2000-05-01T00:00:00   5
 ⋮
 2000-08-01T00:00:00   8
 2000-09-01T00:00:00   9
 2000-10-01T00:00:00  10
 2000-11-01T00:00:00  11
 2000-12-01T00:00:00  12

And a larger DimArray for example data:

julia> data = rand(x, y, t)

┌ 100×25×12 DimArray{Float64, 3} ┐
├────────────────────────────────┴─────────────────────────────────────── dims ┐
  ↓ X Sampled{Int64} 1:100 ForwardOrdered Regular Points,
  → Y Sampled{Int64} 1:25 ForwardOrdered Regular Points,
  ↗ Ti Sampled{DateTime} DateTime("2000-01-01T00:00:00"):Month(1):DateTime("2000-12-01T00:00:00") ForwardOrdered Regular Points
└──────────────────────────────────────────────────────────────────────────────┘
[:, :, 1]
   ↓ →  1         2          3          …  23         24         25
   1    0.333692  0.46747    0.618895       0.808742   0.576437   0.657325
   2    0.5207    0.95715    0.534996       0.25951    0.877483   0.287422
   3    0.757128  0.978177   0.692047       0.274087   0.803685   0.933502
   ⋮                                    ⋱                         ⋮
  97    0.335082  0.14166    0.290357       0.393876   0.177009   0.826134
  98    0.249064  0.0313839  0.0966582      0.857851   0.80082    0.547268
  99    0.171798  0.0649413  0.671101       0.826641   0.259839   0.728112
 100    0.380872  0.393821   0.314181   …   0.931262   0.616368   0.767844

A regular broadcast fails:

julia> scaled = data .* month_scalars

ERROR: DimensionMismatch: arrays could not be broadcast to a common size: a has axes X(Base.OneTo(100)) and b has axes Ti(Base.OneTo(12))

But @d knows to broadcast over the Ti dimension:

julia> scaled = @d data .* month_scalars

┌ 100×25×12 DimArray{Float64, 3} ┐
├────────────────────────────────┴─────────────────────────────────────── dims ┐
  ↓ X Sampled{Int64} 1:100 ForwardOrdered Regular Points,
  → Y Sampled{Int64} 1:25 ForwardOrdered Regular Points,
  ↗ Ti Sampled{DateTime} DateTime("2000-01-01T00:00:00"):Month(1):DateTime("2000-12-01T00:00:00") ForwardOrdered Regular Points
└──────────────────────────────────────────────────────────────────────────────┘
[:, :, 1]
   ↓ →  1         2          3          …  23         24         25
   1    0.333692  0.46747    0.618895       0.808742   0.576437   0.657325
   2    0.5207    0.95715    0.534996       0.25951    0.877483   0.287422
   3    0.757128  0.978177   0.692047       0.274087   0.803685   0.933502
   ⋮                                    ⋱                         ⋮
  97    0.335082  0.14166    0.290357       0.393876   0.177009   0.826134
  98    0.249064  0.0313839  0.0966582      0.857851   0.80082    0.547268
  99    0.171798  0.0649413  0.671101       0.826641   0.259839   0.728112
 100    0.380872  0.393821   0.314181   …   0.931262   0.616368   0.767844

We can see the means of each month are scaled by the broadcast :

julia> mean(eachslice(data; dims=(X, Y)))

┌ 12-element DimArray{Float64, 1} ┐
├─────────────────────────────────┴────────────────────────────────────── dims ┐
  ↓ Ti Sampled{DateTime} DateTime("2000-01-01T00:00:00"):Month(1):DateTime("2000-12-01T00:00:00") ForwardOrdered Regular Points
└──────────────────────────────────────────────────────────────────────────────┘
 2000-01-01T00:00:00  0.501086
 2000-02-01T00:00:00  0.50466
 2000-03-01T00:00:00  0.498604
 2000-04-01T00:00:00  0.503934
 2000-05-01T00:00:00  0.497502
 ⋮
 2000-08-01T00:00:00  0.491305
 2000-09-01T00:00:00  0.501987
 2000-10-01T00:00:00  0.512221
 2000-11-01T00:00:00  0.506841
 2000-12-01T00:00:00  0.503408

julia> mean(eachslice(scaled; dims=(X, Y)))

┌ 12-element DimArray{Float64, 1} ┐
├─────────────────────────────────┴────────────────────────────────────── dims ┐
  ↓ Ti Sampled{DateTime} DateTime("2000-01-01T00:00:00"):Month(1):DateTime("2000-12-01T00:00:00") ForwardOrdered Regular Points
└──────────────────────────────────────────────────────────────────────────────┘
 2000-01-01T00:00:00  0.501086
 2000-02-01T00:00:00  1.00932
 2000-03-01T00:00:00  1.49581
 2000-04-01T00:00:00  2.01574
 2000-05-01T00:00:00  2.48751
 ⋮
 2000-08-01T00:00:00  3.93044
 2000-09-01T00:00:00  4.51789
 2000-10-01T00:00:00  5.12221
 2000-11-01T00:00:00  5.57525
 2000-12-01T00:00:00  6.04089

You can also use broadcast_dims the same way:

julia> broadcast_dims(*, data, month_scalars)

┌ 100×25×12 DimArray{Float64, 3} ┐
├────────────────────────────────┴─────────────────────────────────────── dims ┐
  ↓ X Sampled{Int64} 1:100 ForwardOrdered Regular Points,
  → Y Sampled{Int64} 1:25 ForwardOrdered Regular Points,
  ↗ Ti Sampled{DateTime} DateTime("2000-01-01T00:00:00"):Month(1):DateTime("2000-12-01T00:00:00") ForwardOrdered Regular Points
└──────────────────────────────────────────────────────────────────────────────┘
[:, :, 1]
   ↓ →  1         2          3          …  23         24         25
   1    0.333692  0.46747    0.618895       0.808742   0.576437   0.657325
   2    0.5207    0.95715    0.534996       0.25951    0.877483   0.287422
   3    0.757128  0.978177   0.692047       0.274087   0.803685   0.933502
   ⋮                                    ⋱                         ⋮
  97    0.335082  0.14166    0.290357       0.393876   0.177009   0.826134
  98    0.249064  0.0313839  0.0966582      0.857851   0.80082    0.547268
  99    0.171798  0.0649413  0.671101       0.826641   0.259839   0.728112
 100    0.380872  0.393821   0.314181   …   0.931262   0.616368   0.767844

And with the @d macro you can set the dimension order and other properties of the output array, by passing a single assignment or a NamedTuple argument to @d after the broadcast:

julia> @d data .* month_scalars dims=(Ti, X, Y)

┌ 12×100×25 DimArray{Float64, 3} ┐
├────────────────────────────────┴─────────────────────────────────────── dims ┐
  ↓ Ti Sampled{DateTime} DateTime("2000-01-01T00:00:00"):Month(1):DateTime("2000-12-01T00:00:00") ForwardOrdered Regular Points,
  → X Sampled{Int64} 1:100 ForwardOrdered Regular Points,
  ↗ Y Sampled{Int64} 1:25 ForwardOrdered Regular Points
└──────────────────────────────────────────────────────────────────────────────┘
[:, :, 1]
 ↓ →                   1         2        …  98         99         100
  2000-01-01T00:00:00  0.333692  0.5207       0.249064   0.171798    0.380872
  2000-02-01T00:00:00  1.32867   1.30479      1.91317    1.77189     1.86003
  2000-03-01T00:00:00  2.96646   2.41472      0.411866   2.80454     0.726928
 ⋮                                        ⋱                          ⋮
  2000-09-01T00:00:00  3.85246   4.69114      7.54669    4.75139     2.80107
  2000-10-01T00:00:00  9.27949   9.94204  …   1.02435    9.31906     0.507183
  2000-11-01T00:00:00  8.06674   8.57906      9.12414    9.52623     7.20807
  2000-12-01T00:00:00  9.36757   5.16825      2.69071    4.7913      3.15659

Or

julia> @d data .* month_scalars (dims=(Ti, X, Y), name=:scaled)

┌ 12×100×25 DimArray{Float64, 3} scaled ┐
├───────────────────────────────────────┴──────────────────────────────── dims ┐
  ↓ Ti Sampled{DateTime} DateTime("2000-01-01T00:00:00"):Month(1):DateTime("2000-12-01T00:00:00") ForwardOrdered Regular Points,
  → X Sampled{Int64} 1:100 ForwardOrdered Regular Points,
  ↗ Y Sampled{Int64} 1:25 ForwardOrdered Regular Points
└──────────────────────────────────────────────────────────────────────────────┘
[:, :, 1]
 ↓ →                   1         2        …  98         99         100
  2000-01-01T00:00:00  0.333692  0.5207       0.249064   0.171798    0.380872
  2000-02-01T00:00:00  1.32867   1.30479      1.91317    1.77189     1.86003
  2000-03-01T00:00:00  2.96646   2.41472      0.411866   2.80454     0.726928
 ⋮                                        ⋱                          ⋮
  2000-09-01T00:00:00  3.85246   4.69114      7.54669    4.75139     2.80107
  2000-10-01T00:00:00  9.27949   9.94204  …   1.02435    9.31906     0.507183
  2000-11-01T00:00:00  8.06674   8.57906      9.12414    9.52623     7.20807
  2000-12-01T00:00:00  9.36757   5.16825      2.69071    4.7913      3.15659