#statistics #machine-learning #multidimensional #vector-algebra #geometric-median


Statistical Measures, Vector Algebra, Geometric Median, Data Analysis and Machine Learning

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Rstats - Rust Stats

Crates.io GitHub last commit (branch)


Insert into your Cargo.toml file [dependencies] section:

rstats = "^0.7" 

and import into your source file(s) any of these functions and/or traits that you want:

use rstats::{GI,GV,here,functions,Stats,Vecf64,Vecu8,VecVecf64,VecVecu8,Mutvectors};


Rstats is primarily about characterising multidimensional sets of points, with applications to Machine Learning and Data Analysis. It begins with statistical measures and vector algebra, which provide some basic self-contained tools for the more interesting algorithms but can also be used in their own right.

Our treatment of multidimensional sets of points is constructed from the first principles. Some original concepts, not found elsewhere, are introduced and implemented here. Specifically, the new multidimensional (geometric) median algorithm. Also, the comediance matrix; a replacement for the covariance matrix. It is obtained simply by supplying covar with the geometric median instead of the centroid.

Zero median vectors are generally preferable to the commonly used zero mean vectors.

Most 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 not even any easier to compute.

Specifically, all such 1-d measures are sensitive to the choice of axis.

Our methods based on the True Geometric Median, computed here by gmedian, are axis (rotation) independent from the first step.


The constituent parts of Rstats are Rust traits grouping together functions applicable to vectors of data of relevant end types.End type f64 is most commonly used. Facilities for other end types are limited. For lots of data of other end types, it is always possible to clone to f64, see for example the included utility function vecu8asvecf64.


Follow the documentation link. Then select a trait of interest to see the skeletal comments on the prototype function declarations in lib.rs. To see more detailed comments, plus some examples from the implementation files, scroll to the bottom of the trait and unclick [+] to the left of the implementations of the trait. To see the tests, consult tests.rs.

To run the tests, use single thread. It will be slower but will produce the results in the right order:

cargo test --release -- --test-threads=1 --nocapture --color always

Macro, structs and functions

  • macro here!() for easy diagnostics

  • pub struct GI for printing in green any singular type that has display implemented

  • pub struct GV for printing in green any vector whose end type has display implemented

  • pub struct Med to hold median and quartiles

  • pub struct MStats to hold a mean and standard deviation

  • functions wsum, genvec, genvecu8 (see documentation for the module functions).



One dimensional statistical measures implemented for &[i64] and &[f64].

All these methods operate on one vector of data and take no arguments. For example, s.amean() returns the arithmetic mean of slice s of either type. This is the only attempt at genericity.
This trait is carefully checked and will report all kinds of errors, such as empty input. This means you have to call .unwrap() or something similar on its results.

Included in this trait are:

  • means (arithmetic, geometric and harmonic),
  • standard deviations,
  • linearly weighted means (useful for time dependent data analysis),
  • median and quartiles.


Vector algebra implemented on one or two &[f64] slices of any length (dimensionality):

  • Autocorrelation, Pearson's, Spearman's and Kendall's correlations.
  • Finding minimum and maximum, linear transformation to [0,1].
  • Indirect merge sort, binary search.

This trait is sometimes unchecked (for speed), so some caution with data is advisable.


  • Some vector algebra as above for vectors of u8 (bytes).
  • Frequency count of bytes by their values (Histogram or Probability Density Function).
  • Entropy measures in units of e (using natural logarithms).


Some of the above functions are for memory efficiency reasons reimplemented in this trait so that they mutate self in place, instead of creating a new Vec. Clearly, they can only be applied to a mutable variable. They are useful in vector iterative methods. Beware that they work by side-effect and do not return anything, so they can not be chained.


Relationships of one vector to a set of vectors (of &[f64] end types):

  • sums of distances, eccentricity,
  • centroid, medoid, true geometric median,
  • transformation to zero (geometric) median data,
  • relationship between sets of multidimensional vectors: trend.

Trait VecVec is entirely unchecked, so check your data upfront. This is the more sophisticated part of the library. The true geometric median is found iteratively.


Some of the above for vectors of vectors of bytes.

Appendix I: Terminology (and some new definitions) for sets of nD points

  • Centroid\Centre\Mean is the (generally non member) point that minimises the sum of squares of distances to all member points. Thus it is susceptible to outliers. Specifically, it is the n-dimensional arithmetic mean. By drawing physical analogy with gravity, 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 (if somewhat confusing) 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 use here.

  • 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 not yet implemented here, as its advantages over GM are not clear.

  • Medoid is the member of the set with the least sum of distances to all other members.

  • Outlier is the member of the set with the greatest sum of distances to all other members.

  • 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 and is rotation independent.

  • Zero median vector is obtained by subtracting the geometric median. This is a proposed alternative to the commonly used zero mean vector, obtained by subtracting the centroid.

  • Comediance is similar to covariance, except zero median vectors are used to compute it instead of zero mean vectors.

Appendix II: Recent Releases

  • Version 0.7.14 Added weighted centroid wacentroid. Updated weighted geometric median wgmedian. Updated gmedian and wgmedian in vecvecu8 to include the optimisations of v. 0.7.11. Ensured compatibility with the latest release of crate indxvec.

  • Version 0.7.12 Split off Index trait and associated functions into a new crate indxvec.

  • Version 0.7.11 Removed Kazutsugi (too specialised). Added gcentroid (geometric centroid). Further optimisations to gmedian.

  • Version 0.7.10 Added symmatrix to reconstruct full symmetric matrix from its lower triangular part (for compatibility with crates which duplicate data). Renamed mergerank to plain rank and added boolean argument to facilitate ranking in ascending or descending order. Expanded vecf64() tests (see it for instructive example usage).

  • Version 0.7.9 Added wcovar of weighted points. Improved struct GV and tests. Replaced emsg with macro here!() for easier diagnostics. Moved all structs into lib.rs.

  • Version 0.7.8 Added covar = covariance or comediance matrix computations. Some changes to this text (Readme.md).

  • Version 0.7.7 Fixed merge_immutable and added a test. Added cityblockd and vaddu8.

  • Version 0.7.6 Added merge_immutable and merge_indices. Simplified mergesort.

  • Version 0.7.5 Renamed VecVec trait to VecVecf64 to make the naming consistent. Added unindexu8. Removed wsortedeccs and wsortedcos for being too application specific.

  • Version 0.7.4 Added merge of two sorted &[f64]. Added ascending boolean flag to unindex, sortm and functions that call them, to facilitate easy sorting in ascending or descending order. Added genvecu8 to functions to generate sets of random u8 vectors. Normalised cummulative probability density functions to [0,1].

  • Version 0.7.3 Replaced varc with vector similarity and dissimilarity in [0,1] in terms of their cosines. Similar to unstandardised Pearson's correlation.

  • Version 0.7.2 Added weighted wgmedian and wsortedeccs to VecVecu8. Created new source file for VecVecu8 trait.

  • Version 0.7.1 Ported the improved gmedian also to VecVecu8.

  • Version 0.7.0 Made gmedian slightly more accurate. Added Weighted Geometric Median and supporing functions. Added vecu8asvecf64 utility conversion.