Statistics, Linear Algebra, Information Measures, Cholesky Matrix Decomposition, Mahalanobis Distance, Multidimensional Data Analysis, Machine Learning and more ...
Insert rstats = "^1" in the Cargo.toml file, under [dependencies].
Use in your source files any of the following structs, when needed:
rust
use rstats::{RE,Mstats,MinMax,F64,Med};
and any of the following rstats defined traits:
rust
use rstats::{Stats,Vecg,Vecu8,MutVecg,VecVec,VecVecg};
The latest (nightly) version is always available in the github repository rstats. Sometimes it may be a little ahead of the crates.io release versions.
It is highly recommended to read and run tests/tests.rs, which shows examples of usage.
To run all the tests, use single thread in order to produce the results in the right order:
bash
cargo test --release -- --test-threads=1 --nocapture --color always
Rstats is primarily about characterising multidimensional sets of points, with applications to Machine Learning and Big Data Analysis. Several branches of mathematics: statistics, linear algebra, information theory and set theory are combined in this one consistent package, based on the fact that they all operate on the same objects: Vecs of data items.
RStats begins with basic statistical measures and vector algebra, which provide self-contained basic tools for the machine learning (ML) multidimensional (nd) algorithms but can also be used in their own right. Non analytical statistics is used, whereby the random variables are replaced by vectors of real data. Probabilities densities and other parameters are always obtained from the data, not from some assumed distributions.
Our treatment of multidimensional sets of points (nd vectors) is constructed from the first principles. Some original concepts, not found elsewhere, are introduced and implemented here:
median correlation- in one dimension (1d), our mediancorr method is to replace Pearson's correlation. We define median correlation as cosine of an angle between two zero median vectors (instead of Pearson's zero mean vectors).
gmedian and pmedian - fast multidimensional geometric median (gm) algorithms.
madgm - generalisation to nd of robust data spread estimator known as MAD (median of absolute deviations from median).
contribution - of a point to an nd set. Defined as gm displacement if the point was added/removed. Related to the point's radius (distance from gm) but not the same, as it depends on all the other points as well.
comediance - instead of covariance (matrix). It is obtained by supplying covar with the geometric median instead of the usual centroid. Thus zero median vectors are replacing zero mean vectors in covariance calculations.
Zero median vectors are generally preferable to the commonly used zero mean vectors.
In n dimensions (nd), many authors 'cheat' by using quasi medians (1-d medians along each axis). Quasi medians are a poor start to stable characterisation of multidimensional data. In a highly dimensional space, they are even much slower to compute than our gm.
Specifically, all such 1d measures are sensitive to the choice of axis and thus are affected by their rotation.
In contrast, analyses based on the true geometric median (gm) are axis (rotation) independent. Also, they are more stable, as medians have a 50% breakdown point (the maximum possible). They are computed here by methods gmedian and its weighted version wgmedian, in traits vecvec and vecvecg respectively.
Median correlation between
two 1d vectors. We define it analogously to Pearson, as cosine of an angle between two 'normalised' vectors. Pearson 'normalises' by subtracting the mean from all components, we subtract the median.
Centroid/Centre/Mean is the (generally non member) nd point that minimises the sum of squares of distances to all other member points. Thus it is susceptible to outliers. Specifically, it is the n-dimensional arithmetic mean. It is sometimes called 'the centre of mass'. Centroid can also sometimes mean the member of the set which is the nearest to the Centre. Here we follow the common usage: Centroid = Centre = Arithmetic Mean.
Quasi/Marginal Median is the point minimising sums of distances separately in each dimension (its coordinates are 1-d medians along each axis). It is a mistaken concept which we do not recommend using.
Tukey Median is the point maximising Tukey's Depth, which is the minimum number of (outlying) points found in a hemisphere in any direction. Potentially useful concept but only partially implemented here by tukeyvec, as its advantages over the geometric median are not clear.
Median or the true geometric median (gm), is the point (generally non member), which minimises the sum of distances to all members. This is the one we want. It is much less susceptible to outliers than centroid. In addition, unlike quasi median, gm is rotation independent.
Medoid is the member of the set with the least sum of distances to all other members. Equivalently, the member which is the nearest to the gm.
Outlier is the member of the set with the greatest sum of distances to all other members. Equivalently, it is the point furthest from the gm.
Zero median vectors are obtained by subtracting the gm (placing the origin of the coordinate system at the gm). This is a proposed alternative to the commonly used zero mean vectors, obtained by subtracting the centroid.
MADGM (median of distances from gm). This is our generalisation of MAD (median of absolute differences) measure from 1d to nd. It is a robust measure of nd data spread.
Comediance is similar to covariance, except that zero median vectors are used to compute it, instead of zero mean vectors.
Mahalanobis Distance is a weighted distace, where the weights are derived from the axis of variation of the nd data points cloud. Efficient Cholesky singular (eigen) value decomposition is used. Cholesky method decomposes the covariance/comediance positive definite matrix S into a product of two triangular matrices: S = LU. See more explanations in the code.
Contribution: one of the key questions of Machine Learning (ML) is how to quantify the contribution that each example point (typically a member of some large nd set) makes to the recognition concept, or class, represented by that set. In answer to this, we define the contribution of a point as the change to gm caused by adding/deleting that point. Generally, outlying points make greater contributions to the gm but not as much as they would to the centroid. The contribution depends not only on the radius of the example in question but also on the radii of all other (existing) examples.
The main constituent parts of Rstats are its traits. The selection of traits (to import) is primarily determined by the types of objects to be handled. These are mostly vectors of arbitrary length (dimensionality). The main traits are implementing methods applicable to:
Stats: a single vector (of numbers),Vecg: methods (of vector algebra) operating on two vectors, e.g. scalar productVecu8: some special methods for end-type u8MutVecg: some of the above methods, mutating selfVecVec: methods operating on n vectors, VecVecg: methods for n vectors, plus another generic argument, e.g. vector of weights.In other words, the traits and their methods operate on arguments of their required categories. In classical statistical terminology, the main categories correspond to the number of 'random variables'.
The vectors' end types (for the actual data) are mostly generic: usually some numeric type. There are also some traits specialised for input end type u8 and some that take mutable self. End type f64 is most commonly used for the results.
RStats crate produces custom errors RError:
rust
pub enum RError<T> where T:Sized+Debug {
/// Insufficient data
NoDataError(T),
/// Wrong kind/size of data
DataError(T),
/// Invalid result, such as prevented division by zero
ArithError(T),
/// Other error converted to RError
OtherError(T)
}
Each of its enum variants also carries a generic payload T. Most commonly this will be simply a &'static str message giving more specific information, e.g.:
rust
return Err(RError::ArithError("cholesky needs a positive definite matrix"));
There is a type alias shortening return declarations to, e.g.: Result<Vec<f64>,RE>, where
rust
pub type RE = RError<&'static str>;
More error checking will be added in later versions, where it makes sense.
For more detailed comments, plus some examples, see the source. You may have to unclick the 'implementations on foreign types' somewhere near the bottom of the page in the rust docs to get to it. (Since these traits are implemented over the pre-existing Rust Vec type).
struct MStats holds the central tendency, e.g. some kind of mean or median, and dispersion, e.g. standard deviation or MAD.i64tof64: converts an i64 vector to f64, sumn: sum of a sequence 1..n, also the size of a lower/upper triangular matrix below/above the diagonal (n*(n+1)/2.),seqtosubs returns row,column subsripts to 2d matrix from 1d flat index into a triangular matrix in scanning order.identity_lmatrix generates lower triangular identity matrix in flat scanning order.One dimensional statistical measures implemented for all numeric end types.
Its methods operate on one slice of generic data and take no arguments.
For example, s.amean() returns the arithmetic mean of the data in slice s.
These methods are checked and will report RError(s), such as an empty input. This means you have to apply ? to their results to pass the errors up, or explicitly match them to take recovery actions.
Included in this trait are:
Note that fast implementation of 1d medians is as of version 1.1.0 in crate medians.
Generic vector algebra operations between two slices &[T], &[U] of any length (dimensionality). It may be necessary to invoke some using the 'turbofish' ::<type> syntax to indicate the type U of the supplied argument, e.g.:
datavec.as_slice().methodname::<f64>(arg)
This is because Rust is currently incapable of inferring the type ('the inference bug').
Median correlation, which we define analogously to Pearson's, as cosine of an angle between two zero median vectors (instead of his zero mean vectors).Contribution measure of a point w.r.t gmMahalanobis distanceThe simpler methods of this trait are sometimes unchecked (for speed), so some caution with data is advisable.
A select few of the Stats and Vecg methods (e.g. mutable vector addition, subtraction and multiplication) are reimplemented under this trait, so that they can mutate self in-place. This is more efficient and convenient in some circumstances, such as in vector iterative methods.
Some vector algebra as above that can be more efficient when the end type happens to be u8 (bytes). These methods have u8 appended to their names to avoid confusion with Vecg methods. These specific algorithms are different to their generic equivalents in Vecg.
Relationships between n vectors (in d dimensions).
This general data domain is denoted here as (nd). It is in nd where the main original contribution of this library lies. True geometric median (gm) is found by fast and stable iteration, using improved Weiszfeld's algorithm gmedian. This algorithm solves Weiszfeld's convergence and stability problems in the neighbourhoods of existing set points.
Warning: trait VecVec is entirely unchecked, so check your data upfront.
Methods which take an additional generic vector argument, such as a vector of weights for computing weighted geometric medians (where each point has its own weight). Matrices multiplications.
Version 1.2.9 - More RError forwarding. Removed all deliberate panics.
Version 1.2.8 - Fixed a silly bug in symmatrix and made it return Result.
Version 1.2.7 - Added efficient mahalanobis distance and its test.
Version 1.2.6 - Added test matrices specifically for matrix operations. Added type alias RStats::RE to shorten method headings returning RErrors that carry &str payloads (see subsection Errors above).
Version 1.2.5 - Added some more matrix algebra. Added generic payload T to RError: RError<T> to allow it to carry more information.
Version 1.2.4 - Added Cholesky–Banachiewicz algorithm cholesky to trait Statsg for efficient matrix decomposition.
Version 1.2.3 - Fixed hwmeanstd. Some more tidying up using RError. Autocorr and lintrans now also check their data and return Result.
Version 1.2.2 - Introduced custom error RError, potentially returned by some methods of trait Statsg. Removed the dependency on crate anyhow.
Version 1.2.1 - Code pruning - removed wsortedcos of questionable utility from trait VecVecg.