Rstats
Statistics, Linear Algebra, Information Measures, Machine Learning, Data Analysis, Cholesky Matrix Decomposition, Mahalanobis Distance, Convex Hull and more ...
Usage
Insert rstats = "^1"
in the Cargo.toml
file, under [dependencies]
.
Use in your source files any of the following structs, when needed:
use ;
and any of the following rstats defined traits:
use ;
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.rs from the github repository as examples of usage. To run all the tests, use single thread in order to produce the results in the right order:
Introduction
Rstats
is primarily about characterising sets of n points, each represented as a Vec
, in space of d
dimensions (where d is vec.len()
). It has applications mostly in Machine Learning and Data Analysis.
Several branches of mathematics: statistics, linear algebra, information theory and set theory are combined in this one consistent crate, based on the fact that they all operate on these same objects. The only difference being that an ordering of their components is sometimes assumed (linear algebra, set theory) and sometimes it is not (statistics, information theory, set theory).
RStats
begins with basic statistical measures and vector algebra, which provide self-contained tools for the machine learning (ML) multidimensional algorithms but can also be used in their own right.
Non analytical statistics
is preferred, 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 is constructed from the first principles. Some original concepts, not found elsewhere, are introduced and implemented here:
-
median correlation
- in one dimension, ourmediancorr
method is to replacePearson's correlation
. We definemedian correlation
as cosine of an angle between two zero median samples (instead of Pearson's zero mean samples). This conceptual clarity is one of the benefits of our thinking of a sample as a vector in d dimensional space. -
gmedian and pmedian
- fast multidimensionalgeometric median (gm)
algorithms. -
madgm
- generalisation tond
of a robust data spread estimator known asMAD
(median of absolute deviations from median). -
contribution
- of a point to an nd set. Defined as a magnitude of gm adjustment, when the point is added to the set. It is related to the point's radius (distance from the gm) but not the same, as it depends on the radii of all the other points as well. -
comediance
- instead of covariance (matrix). It is obtained by supplyingcovar
with the geometric median instead of the usual centroid. Thuszero median vectors
are replacingzero mean vectors
in covariance calculations.
Zero median vectors are generally preferable to the commonly used zero mean vectors.
In n dimensions, 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 also 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.
Terminology
Including some new definitions for sets of nd points, i.e. n points in d dimensional space
-
Median correlation
between two samples. 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
of an nd set. It is the (generally non member) point that minimises its sum of squares of distances to all member points. Thus it is susceptible to outliers. Specifically, it is the d-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 medians along each axis). It is a mistaken concept which we do not recommend using. -
Tukey Median
is the point maximisingTukey'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 bytukeyvec
, 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 member points. This is the one we want. It is much less susceptible to outliers than the 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 thegm
. -
Outlier
is the member of the set with the greatest sum of distances to all other members. Equivalently, it is the point furthest from thegm
. -
Convex Hull
is the subset consisting of selected points p, such that no other member points lie outside the plane through p and normal to its radius vector. The remaining points are theinternal
points. -
Zero median vectors
are obtained by subtracting thegm
(placing the origin of the coordinate system at thegm
). This is a proposed alternative to the commonly usedzero mean vectors
, obtained by subtracting the centroid. -
MADGM
(median of distances from gm). This is our generalisation ofMAD
(median of absolute differences) measure from one dimension to any number of dimensions (d>1). It is a robust measure of nd data spread. -
Comediance
is similar tocovariance
, 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. Thus distances in the directions in which there are few points are penalised (increased) and vice versa. 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 = LL'. See more details in the code comments. -
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 thecontribution
of a point as the magnitude of adjustment to gm caused by adding 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 point in question but also on the radii of all other (existing) examples.
Implementation
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.
Errors
RStats crate produces custom errors RError
:
Each of its enum variants also carries a generic payload T
. Most commonly this will be simply a &'static str
message giving more helpful explanation, e.g.:
return Err;
There is a type alias shortening return declarations to, e.g.: Result<Vec<f64>,RE>
, where
pub type RE = ;
More error checking will be added in later versions, where it makes sense.
Documentation
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
struct MStats
holds the central tendency, e.g. some kind of mean or median, and dispersion, e.g. standard deviation or MAD.
Auxiliary Functions
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.
Trait Stats
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, depending on the variant.
Included in this trait are:
- means (arithmetic, geometric and harmonic),
- standard deviations,
- linearly weighted means (useful for time dependent data analysis),
- probability density function (pdf)
- autocorrelation, entropy
- linear transformation to [0,1],
- cholesky matrix decomposition: M = LL' (where ' denotes a transpose).
- other measures and vector algebra operators
Note that fast implementation of 1d medians is as of version 1.1.0 in crate medians
.
Trait Vecg
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.methodname::<f64>(arg)
This is because Rust is currently incapable of inferring the type ('the inference bug').
- Vector additions, subtractions and products (scalar, kronecker, outer),
- Other relationships and measures of difference,
- Pearson's, Spearman's and Kendall's correlations,
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).- Joint pdf, joint entropy, statistical independence (based on mutual information).
Contribution
measure of a point w.r.t gmMahalanobis distance
The simpler methods of this trait are sometimes unchecked (for speed), so some caution with data is advisable.
Trait MutVecg
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.
Trait Vecu8
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.
- Frequency count of bytes by their values (histogram, pdf, jointpdf)
- Entropy, jointentropy, independence.
Trait VecVec
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. Its variant pmedian
iterates point-by-point, which gives even better convergence.
- centroid, medoid, outliers, gm
- sums of distances, radius of a point (as its distance from gm)
- characterisation of a set of multidimensional points by the mean, standard deviation, median and MAD of its points' radii. These are useful recognition measures for the set.
- transformation to zero geometric median data,
- multivariate trend (regression) between two sets of nd points,
- covariance and comediance matrices.
- convex hull points
Trait VecVecg
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.
Appendix: Recent Releases
-
Version 1.2.13 - Updated dependency to
indxvec v1.4.2
. Addednormalize
andwvmean
. Addedradius
toVecVec
. Addedwtukeyvec
adottukey
. Removed bulky test/tests.rs from the crate, get them from the github repository. Householder decomposition/orthogonalization is 'to do'. -
Version 1.2.12 - Updated dependency `indxvec v1.3.4'.
-
Version 1.2.11 - Added
convex_hull
to trait VecVec. Added more error checking: VecVecg trait is now fully checked, be prepared to append?
after most method calls. -
Version 1.2.10 - Minor: corrected some examples, removed all unnecessary
.as_slice()
conversions. -
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 aliasRStats::RE
to shorten method headings returningRErrors
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 traitStatsg
for efficient matrix decomposition. -
Version 1.2.3 - Fixed
hwmeanstd
. Some more tidying up using RError.Autocorr
andlintrans
now also check their data and returnResult
. -
Version 1.2.2 - Introduced custom error RError, potentially returned by some methods of trait
Statsg
. Removed the dependency on crateanyhow
. -
Version 1.2.1 - Code pruning - removed
wsortedcos
of questionable utility from traitVecVecg
.