Add Wasserstein distance

src/Distances.jl

@@ -58,6 +58,7 @@ export
     RMSDeviation,
     NormRMSDeviation,
     Bregman,
+    Wasserstein,
 
     # convenient functions
     euclidean,
@@ -91,6 +92,7 @@ export
     bhattacharyya,
     hellinger,
     bregman,
+    wasserstein,
 
     haversine,
 
@@ -107,5 +109,6 @@ include("haversine.jl")
 include("mahalanobis.jl")
 include("bhattacharyya.jl")
 include("bregman.jl")
+include("wasserstein.jl")
 
 end # module end

src/wasserstein.jl

@@ -0,0 +1,115 @@
+#####
+##### Wasserstein distance
+#####
+
+export Wasserstein
+
+# TODO: Make concrete
+struct Wasserstein{T<:AbstractFloat} <: PreMetric
+    u_weights::Union{AbstractArray{T}, Nothing}
+    v_weights::Union{AbstractArray{T}, Nothing}
+end
+
+Wasserstein(u_weights, v_weights) = Wasserstein{eltype(u_weights)}(u_weights, v_weights)
+
+(w::Wasserstein)(u, v) = wasserstein(u, v, w.u_weights, w.v_weights)
+
+evaluate(dist::Wasserstein, u, v) = dist(u,v)
+
+abstract type Side end
+struct Left <: Side end
+struct Right <: Side end
+
+"""
+    pysearchsorted(a,b;side="left")
+
+Based on accepted answer in:
+    https://stackoverflow.com/questions/55339848/julia-vectorized-version-of-searchsorted
+"""
+pysearchsorted(a,b,::Left) = searchsortedfirst.(Ref(a),b) .- 1
+pysearchsorted(a,b,::Right) = searchsortedlast.(Ref(a),b)
+
+function compute_integral(u_cdf, v_cdf, deltas, p)
+    if p == 1
+        return sum(abs.(u_cdf - v_cdf) .* deltas)
+    end
+    if p == 2
+        return sqrt(sum((u_cdf - v_cdf).^2 .* deltas))
+    end
+    return sum(abs.(u_cdf - v_cdf).^p .* deltas)^(1/p)
+end
+
+function _cdf_distance(p, u_values, v_values, u_weights=nothing, v_weights=nothing)
+    _validate_distribution(u_values, u_weights)
+    _validate_distribution(v_values, v_weights)
+
+    u_sorter = sortperm(u_values)
+    v_sorter = sortperm(v_values)
+
+    all_values = vcat(u_values, v_values)
+    sort!(all_values)
+
+    # Compute the differences between pairs of successive values of u and v.
+    deltas = diff(all_values)
+
+    # Get the respective positions of the values of u and v among the values of
+    # both distributions.
+    u_cdf_indices = pysearchsorted(u_values[u_sorter],all_values[1:end-1], Right())
+    v_cdf_indices = pysearchsorted(v_values[v_sorter],all_values[1:end-1], Right())
+
+    # Calculate the CDFs of u and v using their weights, if specified.
+    if u_weights == nothing
+        u_cdf = (u_cdf_indices) / length(u_values)
+    else
+        u_sorted_cumweights = vcat([0], cumsum(u_weights[u_sorter]))
+        u_cdf = u_sorted_cumweights[u_cdf_indices.+1] / u_sorted_cumweights[end]
+    end
+
+    if v_weights == nothing
+        v_cdf = (v_cdf_indices) / length(v_values)
+    else
+        v_sorted_cumweights = vcat([0], cumsum(v_weights[v_sorter]))
+        v_cdf = v_sorted_cumweights[v_cdf_indices.+1] / v_sorted_cumweights[end]
+    end
+
+    # Compute the value of the integral based on the CDFs.
+    return compute_integral(u_cdf, v_cdf, deltas, p)
+end
+
+function _validate_distribution(vals, weights)
+    # Validate the value array.
+    length(vals) == 0 && throw(ArgumentError("Distribution can't be empty."))
+    # Validate the weight array, if specified.
+    if weights ≠ nothing
+        if length(weights) != length(vals)
+            throw(DimensionMismatch("Value and weight array-likes for the same empirical distribution must be of the same size."))
+        end
+        any(weights .< 0) && throw(ArgumentError("All weights must be non-negative."))
+        if !(0 < sum(weights) < Inf)
+            throw(ArgumentError("Weight array-like sum must be positive and finite. Set as None for an equal distribution of weight."))
+        end
+    end
+    return nothing
+end
+
+"""
+    wasserstein(u_values, v_values, u_weights=nothing, v_weights=nothing)
+
+Compute the first Wasserstein distance between two 1D distributions.
+This distance is also known as the earth mover's distance, since it can be
+seen as the minimum amount of "work" required to transform ``u`` into
+``v``, where "work" is measured as the amount of distribution weight
+that must be moved, multiplied by the distance it has to be moved.
+
+ - `u_values` Values observed in the (empirical) distribution.
+ - `v_values` Values observed in the (empirical) distribution.
+
+ - `u_weights` Weight for each value.
+ - `v_weights` Weight for each value.
+
+If the weight sum differs from 1, it must still be positive
+and finite so that the weights can be normalized to sum to 1.
+"""
+function wasserstein(u_values, v_values, u_weights=nothing, v_weights=nothing)
+    return _cdf_distance(1, u_values, v_values, u_weights, v_weights)
+end