AxisArrays.jl

Performant arrays where each dimension can have a named axis with values
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February 2015

AxisArrays.jl

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This package for the Julia language provides an array type (the AxisArray) that knows about its dimension names and axis values. This allows for indexing by name without incurring any runtime overhead. This permits one to implement algorithms that are oblivious to the storage order of the underlying arrays. AxisArrays can also be indexed by the values along their axes, allowing column names or interval selections.

In contrast to similar approaches in Images.jl and NamedArrays.jl, this allows for type-stable selection of dimensions and compile-time axis lookup. It is also better suited for regularly sampled axes, like samples over time.

Collaboration is welcome! This is still a work-in-progress. See the roadmap for the project's current direction.

Note about Axis{} and keywords

An AxisArray stores an object of type Axis{:name} for each dimension, containing both the name (a Symbol) and the "axis values" (an AbstractVector). These types are what made compile-time lookup possible. Instead of providing them explicitly, it is now possible to use keyword arguments for both construction and indexing:

V = AxisArray(rand(10); row='a':'j')  # AxisArray(rand(10), Axis{:row}('a':'j'))
V[row='c'] == V[Axis{:row}('c')] == V[row=3] == V[3] 

Note about axes() and indices()

The function AxisArrays.axes returns the tuple of such Axis objects. Since Julia version 0.7, Base.axes(V) == (1:10,) gives instead the range of possible ordinary integer indices. (This was called Base.indices.) Since both names are exported, this collision results in a warning if you try to use axes without qualification:

julia> axes([1,2])
WARNING: both AxisArrays and Base export "axes"; uses of it in module Main must be qualified
ERROR: UndefVarError: axes not defined

Packages that are upgrading to support Julia 0.7+ should:

  • Replace all uses of the axes function with the fully-qualified AxisArrays.axes
  • Replace all uses of the deprecated indices function with the un-qualified axes
  • Immediately after using AxisArrays, add const axes = Base.axes

In the future, AxisArrays will be looking for a new name for its functionality. This will allow you to use the idiomatic Base name and offers an easy upgrade path to whatever the new name will be.

Example of currently-implemented behavior:

julia> using Pkg; pkg"add AxisArrays Unitful"
julia> using AxisArrays, Unitful, Random

julia> fs = 40000; # Generate a 40kHz noisy signal, with spike-like stuff added for testing
julia> import Unitful: s, ms, µs
julia> rng = Random.MersenneTwister(123); # Seed a random number generator for repeatable examples
julia> y = randn(rng, 60*fs+1)*3;
julia> for spk = (sin.(0.8:0.2:8.6) .* [0:0.01:.1; .15:.1:.95; 1:-.05:.05] .* 50,
                  sin.(0.8:0.4:8.6) .* [0:0.02:.1; .15:.1:1; 1:-.2:.1] .* 50)
           i = rand(rng, round(Int,.001fs):1fs)
           while i+length(spk)-1 < length(y)
               y[i:i+length(spk)-1] += spk
               i += rand(rng, round(Int,.001fs):1fs)
           end
       end

julia> A = AxisArray(hcat(y, 2 .* y); time = (0s:1s/fs:60s), chan = ([:c1, :c2]))
2-dimensional AxisArray{Float64,2,...} with axes:
    :time, 0.0 s:2.5e-5 s:60.0 s
    :chan, Symbol[:c1, :c2]
And data, a 2400001×2 Array{Float64,2}:
  3.5708     7.14161
  6.14454   12.2891  
  3.42795    6.85591
  1.37825    2.75649
 -1.19004   -2.38007
 -1.99414   -3.98828
  2.9429     5.88581
 -0.226449  -0.452898
  0.821446   1.64289
 -0.582687  -1.16537-3.50593   -7.01187
  2.26783    4.53565
 -0.16902   -0.33804
 -3.84852   -7.69703
  0.226457   0.452914
  0.560809   1.12162
  4.67663    9.35326
 -2.41005   -4.8201  
 -3.71612   -7.43224

AxisArrays behave like regular arrays, but they additionally use the axis information to enable all sorts of fancy behaviors. For example, we can specify indices in any order, just so long as we annotate them with the axis name:

julia> A[time=4] # or A[Axis{:time}(4)]
1-dimensional AxisArray{Float64,1,...} with axes:
    :chan, Symbol[:c1, :c2]
And data, a 2-element Array{Float64,1}:
 1.37825
 2.75649

julia> A[chan = :c2, time = 1:5] # or A[Axis{:chan}(:c2), Axis{:time}(1:5)]
1-dimensional AxisArray{Float64,1,...} with axes:
    :time, 0.0 s:2.5e-5 s:0.0001 s
And data, a 5-element Array{Float64,1}:
  7.14161
 12.2891
  6.85591
  2.75649
 -2.38007

We can also index by the values of each axis using an Interval type that selects all values between two endpoints a .. b or the axis values directly. Notice that the returned AxisArray still has axis information itself... and it still has the correct time information for those datapoints!

julia> A[40µs .. 220µs, :c1]
1-dimensional AxisArray{Float64,1,...} with axes:
    :time, 5.0e-5 s:2.5e-5 s:0.0002 s
And data, a 7-element Array{Float64,1}:
  3.42795
  1.37825
 -1.19004
 -1.99414
  2.9429  
 -0.226449
  0.821446

julia> AxisArrays.axes(ans, 1)
AxisArrays.Axis{:time,StepRangeLen{Quantity{Float64, Dimensions:{𝐓}, Units:{s}},Base.TwicePrecision{Quantity{Float64, Dimensions:{𝐓}, Units:{s}}},Base.TwicePrecision{Quantity{Float64, Dimensions:{𝐓}, Units:{s}}}}}(5.0e-5 s:2.5e-5 s:0.0002 s)

You can also index by a single value using atvalue(t). This function is not needed for categorical axes like :chan here, as :c1 is a Symbol which can't be confused with an integer index.

Using atvalue() will drop a dimension (like using a single integer). Indexing with an Interval(lo, hi) type retains dimensions, even when the ends of the interval are equal (like using a range 1:1):

julia> A[atvalue(2.5e-5s), :c1]
6.14453912336772

julia> A[2.5e-5s..2.5e-5s, :c1]
1-dimensional AxisArray{Float64,1,...} with axes:
    :time, 2.5e-5 s:2.5e-5 s:2.5e-5 s
And data, a 1-element Array{Float64,1}:
 6.14454

You can even index by multiple values by broadcasting atvalue over an array:

julia> A[atvalue.([2.5e-5s, 75.0µs])]
2-dimensional AxisArray{Float64,2,...} with axes:
    :time, Quantity{Float64, Dimensions:{𝐓}, Units:{s}}[2.5e-5 s, 7.5e-5 s]
    :chan, Symbol[:c1, :c2]
And data, a 2×2 Array{Float64,2}:
 6.14454  12.2891
 1.37825   2.75649

Sometimes, though, what we're really interested in is a window of time about a specific index. One of the operations above (looking for values in the window from 40µs to 220µs) might be more clearly expressed as a symmetrical window about a specific index where we know something interesting happened. To represent this, we use the atindex function:

julia> A[atindex(-90µs .. 90µs, 5), :c2]
1-dimensional AxisArray{Float64,1,...} with axes:
    :time_sub, -7.5e-5 s:2.5e-5 s:7.500000000000002e-5 s
And data, a 7-element Array{Float64,1}:
  6.85591
  2.75649
 -2.38007
 -3.98828
  5.88581
 -0.452898
  1.64289

Note that the returned AxisArray has its time axis shifted to represent the interval about the given index! This simple concept can be extended to some very powerful behaviors. For example, let's threshold our data and find windows about those threshold crossings.

julia> idxs = findall(diff(A[:,:c1] .< -15) .> 0);

julia> spks = A[atindex(-200µs .. 800µs, idxs), :c1]
2-dimensional AxisArray{Float64,2,...} with axes:
    :time_sub, -0.0002 s:2.5e-5 s:0.0008 s
    :time_rep, Quantity{Float64, Dimensions:{𝐓}, Units:{s}}[0.162 s, 0.20045 s, 0.28495 s, 0.530325 s, 0.821725 s, 1.0453 s, 1.11967 s, 1.1523 s, 1.22085 s, 1.6253 s  …  57.0094 s, 57.5818 s, 57.8716 s, 57.8806 s, 58.4353 s, 58.7041 s, 59.1015 s, 59.1783 s, 59.425 s, 59.5657 s]
And data, a 41×247 Array{Float64,2}:
   0.672063    7.25649      0.6333751.54583     5.81194    -4.706
  -1.65182     2.57487      0.477408       3.09505     3.52478     4.13037
   4.46035     2.11313      4.78372        1.23385     7.2525      3.57485
   5.25651    -2.19785      3.05933        0.965021    6.78414     5.94854
   7.8537      0.345008     0.960533       0.812989    0.336715    0.303909
   0.466816    0.643649    -3.670873.92978    -3.1242      0.789722
  -6.0445    -13.2441      -4.60716        0.265144   -4.50987    -8.84897
  -9.21703   -13.2254     -14.4409        -8.6664    -13.3457    -11.6213
 -16.1809    -22.7037     -25.023        -15.9376    -28.0817    -16.996
 -23.2671    -31.2021     -25.3787       -24.4914    -32.2599    -26.1118
   ⋮                                  ⋱                ⋮
  -0.301629    0.0683982   -4.36574        1.92362    -5.12333    -3.4431
   4.7182      1.18615      4.40717       -4.51757    -8.64314     0.0800021
  -2.43775    -0.151882    -1.40817       -3.38555    -2.23418     0.728549
   3.2482     -0.60967      0.4712882.53395     0.468817   -3.65905
  -4.26967     2.24747     -3.13758        1.74967     4.5052     -0.145357
  -0.752487    1.69446     -1.20491        1.71429     1.81936     0.290158
   4.64348    -3.94187     -1.59213        7.15428    -0.539748    4.82309
   1.09652    -2.66999      0.521931      -3.80528     1.70421     3.40583
  -0.94341     2.60785     -3.342911.10584     4.31118     3.6404

By indexing with a repeated interval, we have added a dimension to the output! The returned AxisArray's columns specify each repetition of the interval, and each datapoint in the column represents a timepoint within that interval, adjusted by the time of the theshold crossing. The best part here is that the returned matrix knows precisely where its data came from, and has labeled its dimensions appropriately. Not only is there the proper time base for each waveform, but we also have recorded the event times as the axis across the columns.

Indexing

Two main types of Axes supported by default include:

  • Categorical axis -- These are vectors of labels, normally symbols or strings. Elements or slices can be selected by elements or vectors of elements.

  • Dimensional axis -- These are sorted vectors or iterators that can be selected by Intervals. These are commonly used for sequences of times or date-times. For regular sample rates, ranges can be used.

Here is an example with a Dimensional axis representing a time sequence along rows and a Categorical axis of symbols for column headers.

B = AxisArray(reshape(1:15, 5, 3), .1:.1:0.5, [:a, :b, :c])
B[row = (0.2..0.4)] # restrict the AxisArray along the time axis
B[0.0..0.3, [:a, :c]]   # select an interval and two of the columns

User-defined axis types can be added along with custom indexing behaviors.

Example: compute the intensity-weighted mean along the z axis

B = AxisArray(randn(100,100,100), :x, :y, :z)
Itotal = sumz = 0.0
for iter in CartesianIndices(Base.axes(B))  # traverses in storage order for cache efficiency
    global Itotal, sumz
    I = B[iter]  # intensity in a single voxel
    Itotal += I
    sumz += I * iter[axisdim(B, Axis{:z})]  # axisdim "looks up" the z dimension
end
meanz = sumz/Itotal

The intention is that all of these operations are just as efficient as they would be if you used traditional position-based indexing with all the inherent assumptions about the storage order of B.