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Linear Algebra and Statistics library for Scala.js, JVM, and Native.

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# S.L.A.S.H
Scala Linear Algebra & Statistics Hacks

[![javadoc](https://javadoc.io/badge2/ai.dragonfly/slash_3/javadoc.svg)](https://javadoc.io/doc/ai.dragonfly/slash_3)

<h3>Design goals:</h3>

<ol>
<li>Cross compile to JVM, Native, and JavaScript platforms</li>
<li>Maximize performance</li>
<li>Minimize memory footprint</li>
<li>Provide convenient syntax</li>
<li>Seamlessly interoperate with other math libraries</li>
<li>Seamlessly interoperate with native languages like JavaScript, C/C++, and Java</li>
<li>Serialize efficiently and compactly by default</li>
</ol>

```scala
libraryDependencies += "ai.dragonfly" %%% "slash" % "<LATEST_VERSION>"
```

<h3>Features:</h3>

- High performance Vector data types with convenient vector math syntax.
- Probability Distributions, Parametric and Estimated (Online/Streaming): Gaussian/Normal, Poisson, LogNormal, Binomial (parametric only), Beta, and PERT; each with support for sampling and probability density functions, PDFs.
- Sampleable trait for making types into generative models.
- Math functions: Beta, Factorial, and Gamma functions: B(α, β), x! and Γ(x).  Convenience macros, methods, and case classes for computing logarithms of arbitrary base. 
- Geometry: Sample points uniformly from volumes defined by 3D tetrahedrons.
- Bresenham Line Drawing Algorithm that invokes a lambda for each discrete point on a line.
- Kernels: Gaussian, Epanechnikov, Uniform, and Discrete.
- Flexible Histogram data structures with Console friendly Text Based Visualizations inspired by <a href="https://github.com/JuliaPlots/UnicodePlots.jl">Julia Plots</a>.
- Bijection[A, B]: an abstraction for bijective implicit conversions.
- BigRandom: scala.util.Random extension methods to generate random BigInt and BigDecimal values.
- Interval and Domain types and objects with support for random sampling.
- Unicode text formatting utility for writing numeric value types in superscript or subscript positions.
- Matrix math:
  + multiplication for Matrix * Matrix, Matrix * Vector, and Scalar * Matrix
  + element wise operations: add, subtract, multiply, and divide
  + sub-matrix, column, row, and element operations: get, set
  + determinant
  + transpose
  + inverse
  + norm operations: one, two, infinity, and Frobenius
  + decompositions: Cholesky, Eigen, LU, QR, and Singular Value
- In memory data sets: unsupervised and supervised
- Linear Regression based on both QR Decomposition and Singular Value Decomposition.
- Principal Components Analysis

&nbsp;&nbsp;&nbsp;Instead of case classes, traits, or wrappers, this library represents all runtime vector data as native arrays of double precision floating point values.  However, it also uses Scala 3 features like opaque types, dependent types, and extension methods to decorate the array primitives with convenient syntax, e.g. overloaded operators like `+ - * / += -= *= /=`, and also, by expressing vector dimensionality as a type parameter, can prevent runtime errors resulting from trying to perform vector operations on vectors of mismatched dimensions at compile time.  For example:

```scala
opaque type Vec[N <: Int] = NArray[Double]

val v2:Vec[2] = Vec[2](1.0, 2.0)
val v3:Vec[3] = Vec[3](1.0, 2.0, 3.0)

println((v2 + v3).show) // compiler error!

// Even though, at runtime, v2 and v3 have the same type,
// the compiler treats them as though they have different, distinct types.
// This eliminates runtime exceptions related to mismatched vector dimensions.
```

&nbsp;&nbsp;&nbsp;Why `NArray[Double]` and not `Array[Double]`?  Because Vec relies on <a href="https://github.com/dragonfly-ai/narr">NArr</a>, JavaScript environments store vector data as: `Float64Array` while JVM and Native environments rely on: `Array[Double]`.  This ensures that whichever compilation target you choose, Vector will always reduce to the native array type with the highest available performance.

&nbsp;&nbsp;&nbsp;For a more detailed explanation of the design decisions that have shaped this library, see the <a href="https://dragonfly-ai.github.io/vector/">design notes.</a>

<h3 id="vectormath">More Vector Math Syntax:</h3>

```scala
import slash.vector.*

// create a 3 dimensional vector
val v1:Vec[3] = Vec[3](1.0, 0.5, 0.0)

// print it to the console with expressive unicode text
println(v1.show)  // -> 《³↗〉1.0ᵢ 0.0ⱼ 0.0ₖ〉

// perform various vector math operations
val v2:Vec[3] = Vec[3](0.75, 1.0, 0.5)

val v3:Vec[3] = v1 + v2

println( v3.norm )

v3 -= v1

println( v3.euclideanDistanceTo(v2) )

// perform Vector3 specific operations
import slash.vector.Vector3.*

println(v1.x) // .x .y and .z provided by an extension method in Vector3

// compute 3D Vector Cross product
println( (v1 cross v2 ).show )
println( (v1 ⨯ v2 ).show ) // or with a unicode ⨯ operator

import slash.Random.*
val r:Random = defaultRandom

// higher dimensional vectors
val v42a:Vec[42] = r.nextVec[42]()
val v42b:Vec[42] = r.nextVec[42]()

println( v42a dot v42b )
println( (v42a - v42b).render() ) // fully customisable render method.
println( (v42a + v42b).csv() ) // output vector sum as comma separated values
println( (v42a + v42b).tsv() ) // output vector sum tab separated values
```

<h3>Matrix math</h3>
&nbsp;&nbsp;&nbsp;This matrix library differs most significantly from others like <a href="https://math.nist.gov/javanumerics/jama/">JAMA</a> and <a href="https://commons.apache.org/proper/commons-math/javadocs/api-3.6.1/org/apache/commons/math3/linear/">Apache Commons Math</a>, by providing compile time dimensionality checks.  Instead of encoding matrix row and column dimensions with method parameters or `Array[Double].length` values, this library relies on dependent types.  For example:

```scala
// create an 3 x 2 matrix of zeros.
val m:Mat[3, 2] = Mat.zeros[3, 2]

// create a 3 x 3 matrix with convenient matrix literal syntax:
val m0:Mat[3, 3] = Mat[3,3](
  1, 2, 3,
  4, 5, 6,
  7, 8, 9
)
```

&nbsp;&nbsp;&nbsp;By encoding the matrix's row and column dimensions into its type, the compiler can prevent a whole category of runtime errors that arise from mismatched matrix dimensions:

```scala
// create an 3 x 2 matrix of zeros:
val m0:Mat[3, 2] = Mat.zeros[3, 2]
// create an 2 x 3 matrix of zeros:
val m1:Mat[2, 3] = Mat.zeros[2, 3]

val m:Mat[3, 3] = m0 * m1
val m = m * m1 // compiler error!
```

&nbsp;&nbsp;&nbsp;Relatedly, many matrix operations like `determinant`, Cholesky decomposition, etc, only pertain to square matrices.  This library relies on type conditioned extension methods so that users simply cannot attempt to invoke these operations on rectangular matrices.  More specifically:

```scala
extension [MN <: Int] (m: Mat[MN, MN])(using ValueOf[MN]) {
  def determinant: Double = LU[MN, MN](m).determinant
}
```

&nbsp;&nbsp;&nbsp;Instead of including a `determinant` method directly in the `Matrix` class, this extension method makes a `determinant` method available only for square matrices.  Trying to invoke the `determinant` method on a rectangular matrix, for which M != N, will yield a compiler error.

<h3>JavaScript Optimization</h3>

&nbsp;&nbsp;&nbsp;Because matrix relies on <a href="https://github.com/dragonfly-ai/narr">NArr</a>, JavaScript environments store matrix data as:
```scala
var matrixArray:NArray[Double]
```
&nbsp;&nbsp;&nbsp;which is equivalent to:
```scala
var matrixArray:Float64Array
```
&nbsp;&nbsp;&nbsp;In JVM and Native environments, matrix data occupies normal scala `Array[Double]`.

<h3>History</h3>
&nbsp;&nbsp;&nbsp;Although it began as a 1:1 port of <a href="https://math.nist.gov/javanumerics/jama/">JAMA</a> for Scala 3 and Scala.js projects, it has expanded to include features that make matrix operations more comfortable in idiomatic Scala. Past versions of this library JAMA from the <a href="https://mvnrepository.com/artifact/gov.nist.math/jama/1.0.3">maven repository</a> on the JVM side, and provided facades for a JavaScript version of JAMA ported through <a href="http://www.jsweet.org">Jsweet</a> and included through the <a href="https://scalacenter.github.io/scalajs-bundler/">scalajs-bundler</a> sbt plugin.  As scalajs-bundler has slipped into an unmaintained status, this library evolved into a pure scala port of JAMA and has begun to take its own shape.

<h3>Exclusions</h3>
&nbsp;&nbsp;&nbsp;This implementation of JAMA excludes test and I/O functionality as well as some constructors that comments in the original JAMA library describe as dangerous and unnecessary.

<h3>Verification</h3>
&nbsp;&nbsp;&nbsp;See the verification subproject of this repository to evaluate the fidelity of this port from Java to Scala.  Given the original JaMa implementation of hypot, these two matrix libraries produce identical output, however, modern Java includes a more advanced implementation of the hypot function and using it produces tiny discrepancies between Jama and this scala implementation.

<h3>Parametric Probability Distributions</h3>

```scala
import slash.stats.probability.distributions.*

// create a gaussian distribution parametrically
val g:Gaussian = Gaussian(10.0, 42.0)
g.p(7.0)   // evaluate the Probability Density Function at 7.0, in other words: PDF(7.0)
g.random() // randomly sample a value from this gaussian model
```

<h3>Estimated, also called Online or Streaming, Probability Distributions</h3>

```scala
import slash.stats.probability.distributions.*

// create a gaussian distribution parametrically
val eg:stream.Gaussian = stream.Gaussian()
// observe some trials
eg.observe(7.0)
eg.observe(7.5)
eg.observe(9.1)
eg.observe(6.3)
eg.observe(8.4)

// estimate a parametric Gaussian distribution from the observations
val pg:Gaussian = eg.estimate
pg.p(8.0)   // estimate the Probability Density Function at 8.0, in other words: PDF'(8.0)
pg.random() // randomly sample a value from the estimated gaussian model
```

<h3>Compile time Logarithms</h3>

&nbsp;&nbsp;&nbsp;To compute `logₓ(y)`, where `x` is any value of type: `Double`, we compute: `log₁₀(y) / log₁₀(x)`.  Which can introduce a lot of computational overhead, especially when done in loops which repeatedly compute the same value for `log₁₀(x)`.  To improve performance and legibility, this library provides a `log[BASE <: Double | Int]` macro that computes the `log₁₀(x)` denominator at compile time; it also clarifies the operation by allowing users to write the base of the log into the type parameter and the operand as a method parameter.  As such, instead of: `log(2 /*base*/, 42 /*operand*/)` we can write: `log[2](42)`.

```scala
import slash.*

// Compile time optimized Logarithms of known base:
log[2](42.0)  // Computes log₂(42) at compile time
log[0.5](11.0)  // Computes log₀.₅(11) at compile time

var i: Int = 1; while (i > 0) {
  println( log[2](i) )  // computes half of this operation at compile time
  i = i << 1
}
```

<h3>Runtime Logarithms</h3>
&nbsp;&nbsp;&nbsp;The runtime `Log` class can yield comparable performance in cases when the base of the logarithm can't be known at compile time, or can't be expressed as a constant.

```scala
import slash.*
// Use the runtime Log class for:
// a base determined by a value:
import slash.Constant.π
val logBasePi: Log = Log(π)
logBasePi(13)
// or any base unknown at compile time,
val logBaseRandom:Log = Log(Math.random())
logBaseRandom(42)
```

Unicode Histogram Plot:

```
Histogram: { 
	[-17.00,-11.00 ) 🌑 ︙    ∝ 3.0E-4
	[-11.00, -5.00 ) 🌑 ▕█   ∝ 0.0094
	[ -5.00,  1.00 ) 🌑 ▕████████    ∝ 0.0732
	[  1.00,  7.00 ) 🌒 ▕███████████████████████████    ∝ 0.2447
	[  7.00, 13.00 ) 🌓 ▕███████████████████████████████████████   ∝ 0.3516
	[ 13.00, 19.00 ) 🌔 ▕██████████████████████████▋   ∝ 0.2411`
	[ 19.00, 25.00 ) 🌔 ▕███████▌   ∝ 0.069
	[ 25.00, 31.00 ) 🌔 ▕█   ∝ 0.0101
	[ 31.00, 37.00 ) 🌔 ︙    ∝ 5.0E-4
	[ 37.00, 43.00 ] 🌕 ︰    ∝ 1.0E-4
}
```
These plots label the bins with standardised math notation for open and closed intervals, but also use the unicode glyphs for the phases of the moon to represent the cumulative distribution.

<h3>Add the SBT dependency:</h3>

```scala
libraryDependencies += "ai.dragonfly" %%% "vector" % "<LATEST_VERSION>"
```

`NArray[Double]` is useful, but why exclude `NArray[Float]`?

In Scala.js, the runtime type of `Float` doesn't fully exist, while Scala.js, Scala JVM, and Scala Native all share the same implementation of `Double`.
</li>
<li>What if a vector's dimension depends on a variable?  Can this library support vectors with dimensionality defined at runtime?

Yes, but with some limitations.  Please consider the following examples:

```scala
// Vectors with lengths defined at runtime.

// method 1:
val l0:Int = r.nextInt (100)
type N0 = l0.type
val rtv0: Vec[N0] = r.nextVec[N0]()
println(rtv0.render())

// method 2:
val l1: Int = r.nextInt(100)
val rtv1: Vec[l1.type] = r.nextVec[l1.type]()
println(rtv1.render())

// For better or worse, this throws a compiler error even though l1 == l2 is true:
val l1: Int = r.nextInt(100)
val l2: Int = 0 + l1
val rtv1: Vec[l1.type] = r.nextVec[l1.type]()
val rtv2: Vec[l2.type] = r.nextVec[l2.type]()
println((rtv1 + rtv2).render())
  
//    [error] 57 |    println((rtv1 + rtv2).render())
//    [error]    |                    ^^^^
//    [error]    |             Found:    (rtv2 : slash.vector.Vec[(l2 : Int)])
//    [error]    |             Required: slash.vector.Vec[(l1 : Int)]

// However, you can do this:
val l1: Int = r.nextInt(100)
val l2: Int = 0 + l1
val rtv1: Vec[l1.type] = r.nextVec[l1.type]()
val rtv2: Vec[l2.type] = r.nextVec[l2.type]()
println((rtv1 + rtv2.asInstanceOf[Vec[l1.type]]).render())
```

To coordinate runtime vector dimensions across a system, SLASH provides the `VectorSpace` and `MatrixSpace` classes:

```scala
val l1: Int = r.nextInt(100)
val vs = VectorSpace(l1)
val rtv1: Vec[vs.N] = r.nextVec[vs.N]()
val rtv2: Vec[vs.N] = r.nextVec[vs.N]()
println((rtv1 + rtv2.asInstanceOf[Vec[vs.N]]).render())
```
Unfortunately, this means having to think about a concept that no other math library has had, but it can also head off all kinds of run time errors caused by mismatched dimension exceptions.
</li>
</ul>
<br />

Projects that rely on this Library:

https://github.com/dragonfly-ai/uriel

https://github.com/dragonfly-ai/spatial

https://github.com/dragonfly-ai/mesh

https://github.com/dragonfly-ai/img

Acknowledgements: <br />
&nbsp;&nbsp;&nbsp;This library has evolved over years, but owes a lot of its virtues to feedback from the Scala Discord community.  In no particular order:

https://github.com/ekrich - Performance tips and encouragement.

https://github.com/armanbilge - Continuous Integration tools and support.

https://github.com/s5bug - Design insights.

https://github.com/BalmungSan - Design insights.

https://github.com/Quafadas - Contributor!

https://github.com/philwalk - Design insights, and contributor!

https://github.com/JD557 - Design insights, named this library, and contributor!

https://github.com/J-mie6 - Git advice.

Thanks, scala fam!

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