Machine Learning: Lorentzian Classification

/ ====================
// ==== Background ====
// ====================

// When using Machine Learning algorithms like K-Nearest Neighbors, choosing an
// appropriate distance metric is essential. Euclidean Distance is often used as
// the default distance metric, but it may not always be the best choice. This is
// because market data is often significantly impacted by proximity to significant
// world events such as FOMC Meetings and Black Swan events. These major economic
// events can contribute to a warping effect analogous a massive object's
// gravitational warping of Space-Time. In financial markets, this warping effect
// operates on a continuum, which can analogously be referred to as "Price-Time".

// To help to better account for this warping effect, Lorentzian Distance can be
// used as an alternative distance metric to Euclidean Distance. The geometry of
// Lorentzian Space can be difficult to visualize at first, and one of the best
// ways to intuitively understand it is through an example involving 2 feature
// dimensions (z=2). For purposes of this example, let's assume these two features
// are Relative Strength Index (RSI) and the Average Directional Index (ADX). In
// reality, the optimal number of features is in the range of 3-8, but for the sake
// of simplicity, we will use only 2 features in this example.

// Fundamental Assumptions:
// (1) We can calculate RSI and ADX for a given chart.
// (2) For simplicity, values for RSI and ADX are assumed to adhere to a Gaussian
// distribution in the range of 0 to 100.
// (3) The most recent RSI and ADX value can be considered the origin of a coordinate
// system with ADX on the x-axis and RSI on the y-axis.

// Distances in Euclidean Space:
// Measuring the Euclidean Distances of historical values with the most recent point
// at the origin will yield a distribution that resembles Figure 1 (below).

// [RSI]
// |
// |
// |
// ...:::....
// .:.:::••••••:::•::..
// .:•:.:•••::::••::••....::.
// ....:••••:••••••••::••:...:•.
// ...:.::::::•••:::•••:•••::.:•..
// ::•:.:•:•••••••:.:•::::::...:..
// |--------.:•••..•••••••:••:...:::•:•:..:..----------[ADX]
// 0 :•:....:•••••::.:::•••::••:.....
// ::....:.:••••••••:•••::••::..:.
// .:...:••:::••••••••::•••....:
// ::....:.....:•::•••:::::..
// ..:..::••..::::..:•:..
// .::..:::.....:
// |
// |
// |
// |
// _|_ 0
//
// Figure 1: Neighborhood in Euclidean Space

// Distances in The Space:
// However, the same set of historical values measured using The Distance will
// yield a different distribution that resembles Figure 2 (below).

//
// [RSI]
// ::.. | ..:::
// ..... | ......
// .••••::. | :••••••.
// .:•••••:. | :::••••••.
// .•••••:... | .::.••••••.
// .::•••••::.. | :..••••••..
// .:•••••••::.........::••••••:..
// ..::::••••.•••••••.•••••••:.
// ...:•••••••.•••••••••::.
// .:..••.••••••.••••..
// |---------------.:•••••••••••••••••.---------------[ADX]
// 0 .:•:•••.••••••.•••••••.
// .••••••••••••••••••••••••:.
// .:••••••••••::..::.::••••••••:.
// .::••••••::. | .::•••:::.
// .:••••••.. | :••••••••.
// .:••••:... | ..•••••••:.
// ..:••::.. | :.•••••••.
// .:•.... | ...::.:••.
// ...:.. | :...:••.
// :::. | ..::
// _|_ 0
//
// Figure 2: Neighborhood in the Space


// Observations:
// (1) In the Space, the shortest distance between two points is not
// necessarily a straight line, but rather, a geodesic curve.
// (2) The warping effect of Lorentzian distance reduces the overall influence
// of outliers and noise.
// (3) The Distance becomes increasingly different from Euclidean Distance
// as the number of nearest neighbors used for comparison increases.