## Distances.jl

A Julia package for evaluating distances (metrics) between vectors.
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August 2014

# Distances.jl

A Julia package for evaluating distances (metrics) between vectors.

This package also provides optimized functions to compute column-wise and pairwise distances, which are often substantially faster than a straightforward loop implementation. (See the benchmark section below for details).

## Supported distances

• Euclidean distance
• Squared Euclidean distance
• Periodic Euclidean distance
• Cityblock distance
• Total variation distance
• Jaccard distance
• Rogers-Tanimoto distance
• Chebyshev distance
• Minkowski distance
• Hamming distance
• Cosine distance
• Correlation distance
• Chi-square distance
• Kullback-Leibler divergence
• Generalized Kullback-Leibler divergence
• Rényi divergence
• Jensen-Shannon divergence
• Mahalanobis distance
• Squared Mahalanobis distance
• Bhattacharyya distance
• Hellinger distance
• Haversine distance
• Spherical angle distance
• Mean absolute deviation
• Mean squared deviation
• Root mean squared deviation
• Normalized root mean squared deviation
• Bray-Curtis dissimilarity
• Bregman divergence

For `Euclidean distance`, `Squared Euclidean distance`, `Cityblock distance`, `Minkowski distance`, and `Hamming distance`, a weighted version is also provided.

## Basic use

The library supports three ways of computation: computing the distance between two iterators/vectors, "zip"-wise computation, and pairwise computation. Each of these computation modes works with arbitrary iterable objects of known size.

### Computing the distance between two iterators or vectors

Each distance corresponds to a distance type. You can always compute a certain distance between two iterators or vectors of equal length using the following syntax

```r = evaluate(dist, x, y)
r = dist(x, y)```

Here, `dist` is an instance of a distance type: for example, the type for Euclidean distance is `Euclidean` (more distance types will be introduced in the next section). You can compute the Euclidean distance between `x` and `y` as

```r = evaluate(Euclidean(), x, y)
r = Euclidean()(x, y)```

Common distances also come with convenient functions for distance evaluation. For example, you may also compute Euclidean distance between two vectors as below

`r = euclidean(x, y)`

### Computing distances between corresponding objects ("column-wise")

Suppose you have two `m-by-n` matrix `X` and `Y`, then you can compute all distances between corresponding columns of `X` and `Y` in one batch, using the `colwise` function, as

`r = colwise(dist, X, Y)`

The output `r` is a vector of length `n`. In particular, `r[i]` is the distance between `X[:,i]` and `Y[:,i]`. The batch computation typically runs considerably faster than calling `evaluate` column-by-column.

Note that either of `X` and `Y` can be just a single vector -- then the `colwise` function computes the distance between this vector and each column of the other argument.

### Computing pairwise distances

Let `X` and `Y` have `m` and `n` columns, respectively, and the same number of rows. Then the `pairwise` function with the `dims=2` argument computes distances between each pair of columns in `X` and `Y`:

`R = pairwise(dist, X, Y, dims=2)`

In the output, `R` is a matrix of size `(m, n)`, such that `R[i,j]` is the distance between `X[:,i]` and `Y[:,j]`. Computing distances for all pairs using `pairwise` function is often remarkably faster than evaluting for each pair individually.

If you just want to just compute distances between all columns of a matrix `X`, you can write

`R = pairwise(dist, X, dims=2)`

This statement will result in an `m-by-m` matrix, where `R[i,j]` is the distance between `X[:,i]` and `X[:,j]`. `pairwise(dist, X)` is typically more efficient than `pairwise(dist, X, X)`, as the former will take advantage of the symmetry when `dist` is a semi-metric (including metric).

To compute pairwise distances for matrices with observations stored in rows use the argument `dims=1`.

### Computing column-wise and pairwise distances inplace

If the vector/matrix to store the results are pre-allocated, you may use the storage (without creating a new array) using the following syntax (`i` being either `1` or `2`):

```colwise!(r, dist, X, Y)
pairwise!(R, dist, X, Y, dims=i)
pairwise!(R, dist, X, dims=i)```

Please pay attention to the difference, the functions for inplace computation are `colwise!` and `pairwise!` (instead of `colwise` and `pairwise`).

## Distance type hierarchy

The distances are organized into a type hierarchy.

At the top of this hierarchy is an abstract class PreMetric, which is defined to be a function `d` that satisfies

``````d(x, x) == 0  for all x
d(x, y) >= 0  for all x, y
``````

SemiMetric is a abstract type that refines PreMetric. Formally, a semi-metric is a pre-metric that is also symmetric, as

``````d(x, y) == d(y, x)  for all x, y
``````

Metric is a abstract type that further refines SemiMetric. Formally, a metric is a semi-metric that also satisfies triangle inequality, as

``````d(x, z) <= d(x, y) + d(y, z)  for all x, y, z
``````

This type system has practical significance. For example, when computing pairwise distances between a set of vectors, you may only perform computation for half of the pairs, derive the values immediately for the remaining half by leveraging the symmetry of semi-metrics. Note that the types of `SemiMetric` and `Metric` do not completely follow the definition in mathematics as they do not require the "distance" to be able to distinguish between points: for these types `x != y` does not imply that `d(x, y) != 0` in general compared to the mathematical definition of semi-metric and metric, as this property does not change computations in practice.

Each distance corresponds to a distance type. The type name and the corresponding mathematical definitions of the distances are listed in the following table.

type name convenient syntax math definition
Euclidean `euclidean(x, y)` `sqrt(sum((x - y) .^ 2))`
SqEuclidean `sqeuclidean(x, y)` `sum((x - y).^2)`
PeriodicEuclidean `peuclidean(x, y, w)` `sqrt(sum(min(mod(abs(x - y), w), w - mod(abs(x - y), w)).^2))`
Cityblock `cityblock(x, y)` `sum(abs(x - y))`
TotalVariation `totalvariation(x, y)` `sum(abs(x - y)) / 2`
Chebyshev `chebyshev(x, y)` `max(abs(x - y))`
Minkowski `minkowski(x, y, p)` `sum(abs(x - y).^p) ^ (1/p)`
Hamming `hamming(k, l)` `sum(k .!= l)`
RogersTanimoto `rogerstanimoto(a, b)` `2(sum(a&!b) + sum(!a&b)) / (2(sum(a&!b) + sum(!a&b)) + sum(a&b) + sum(!a&!b))`
Jaccard `jaccard(x, y)` `1 - sum(min(x, y)) / sum(max(x, y))`
BrayCurtis `braycurtis(x, y)` `sum(abs(x - y)) / sum(abs(x + y))`
CosineDist `cosine_dist(x, y)` `1 - dot(x, y) / (norm(x) * norm(y))`
CorrDist `corr_dist(x, y)` `cosine_dist(x - mean(x), y - mean(y))`
ChiSqDist `chisq_dist(x, y)` `sum((x - y).^2 / (x + y))`
KLDivergence `kl_divergence(p, q)` `sum(p .* log(p ./ q))`
GenKLDivergence `gkl_divergence(x, y)` `sum(p .* log(p ./ q) - p + q)`
RenyiDivergence `renyi_divergence(p, q, k)` `log(sum( p .* (p ./ q) .^ (k - 1))) / (k - 1)`
JSDivergence `js_divergence(p, q)` `KL(p, m) / 2 + KL(q, m) / 2 with m = (p + q) / 2`
SpanNormDist `spannorm_dist(x, y)` `max(x - y) - min(x - y)`
BhattacharyyaDist `bhattacharyya(x, y)` `-log(sum(sqrt(x .* y) / sqrt(sum(x) * sum(y)))`
HellingerDist `hellinger(x, y)` `sqrt(1 - sum(sqrt(x .* y) / sqrt(sum(x) * sum(y))))`
Haversine `haversine(x, y, r = 6_371_000)` Haversine formula
SphericalAngle `spherical_angle(x, y)` Haversine formula
Mahalanobis `mahalanobis(x, y, Q)` `sqrt((x - y)' * Q * (x - y))`
SqMahalanobis `sqmahalanobis(x, y, Q)` `(x - y)' * Q * (x - y)`
MeanAbsDeviation `meanad(x, y)` `mean(abs.(x - y))`
MeanSqDeviation `msd(x, y)` `mean(abs2.(x - y))`
RMSDeviation `rmsd(x, y)` `sqrt(msd(x, y))`
NormRMSDeviation `nrmsd(x, y)` `rmsd(x, y) / (maximum(x) - minimum(x))`
WeightedEuclidean `weuclidean(x, y, w)` `sqrt(sum((x - y).^2 .* w))`
WeightedSqEuclidean `wsqeuclidean(x, y, w)` `sum((x - y).^2 .* w)`
WeightedCityblock `wcityblock(x, y, w)` `sum(abs(x - y) .* w)`
WeightedMinkowski `wminkowski(x, y, w, p)` `sum(abs(x - y).^p .* w) ^ (1/p)`
WeightedHamming `whamming(x, y, w)` `sum((x .!= y) .* w)`
Bregman `bregman(F, ∇, x, y; inner=dot)` `F(x) - F(y) - inner(∇(y), x - y)`

Note: The formulas above are using Julia's functions. These formulas are mainly for conveying the math concepts in a concise way. The actual implementation may use a faster way. The arguments `x` and `y` are iterable objects, typically arrays of real numbers; `w` is an iterator/array of parameters (like weights or periods); `k` and `l` are iterators/arrays of distinct elements of any kind; `a` and `b` are iterators/arrays of Bools; and finally, `p` and `q` are iterators/arrays forming a discrete probability distribution and are therefore both expected to sum to one.

### Precision for Euclidean and SqEuclidean

For efficiency (see the benchmarks below), `Euclidean` and `SqEuclidean` make use of BLAS3 matrix-matrix multiplication to calculate distances. This corresponds to the following expansion:

`(x-y)^2 == x^2 - 2xy + y^2`

However, equality is not precise in the presence of roundoff error, and particularly when `x` and `y` are nearby points this may not be accurate. Consequently, `Euclidean` and `SqEuclidean` allow you to supply a relative tolerance to force recalculation:

```julia> x = reshape([0.1, 0.3, -0.1], 3, 1);

julia> pairwise(Euclidean(), x, x)
1×1 Array{Float64,2}:
7.45058e-9

julia> pairwise(Euclidean(1e-12), x, x)
1×1 Array{Float64,2}:
0.0```

## Benchmarks

The implementation has been carefully optimized based on benchmarks. The script in `benchmark/benchmarks.jl` defines a benchmark suite for a variety of distances, under column-wise and pairwise settings.

Here are benchmarks obtained running Julia 1.5 on a computer with a quad-core Intel Core i5-2300K processor @ 3.2 GHz. Extended versions of the tables below can be replicated using the script in `benchmark/print_table.jl`.

### Column-wise benchmark

Generically, column-wise distances are computed using a straightforward loop implementation. For `[Sq]Mahalanobis`, however, specialized methods are provided in Distances.jl, and the table below compares the performance (measured in terms of average elapsed time of each iteration) of the generic to the specialized implementation. The task in each iteration is to compute a specific distance between corresponding columns in two `200-by-10000` matrices.

distance loop colwise gain
SqMahalanobis 0.089470s 0.014424s 6.2027
Mahalanobis 0.090882s 0.014096s 6.4475

### Pairwise benchmark

Generically, pairwise distances are computed using a straightforward loop implementation. For distances of which a major part of the computation is a quadratic form, however, the performance can be drastically improved by restructuring the computation and delegating the core part to `GEMM` in BLAS. The table below compares the performance (measured in terms of average elapsed time of each iteration) of generic to the specialized implementations provided in Distances.jl. The task in each iteration is to compute a specific distance in a pairwise manner between columns in a `100-by-200` and `100-by-250` matrices, which will result in a `200-by-250` distance matrix.

distance loop pairwise gain
SqEuclidean 0.001273s 0.000124s 10.2290
Euclidean 0.001445s 0.000194s 7.4529
CosineDist 0.001928s 0.000149s 12.9543
CorrDist 0.016837s 0.000187s 90.1854
WeightedSqEuclidean 0.001603s 0.000143s 11.2119
WeightedEuclidean 0.001811s 0.000238s 7.6032
SqMahalanobis 0.308990s 0.000248s 1248.1892
Mahalanobis 0.313415s 0.000346s 906.1836

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