Fast algorithm for finding 1d medians, implemented in Rust.
rust
use medians::{Med,MStats,Median};
Finding the medians is a common task in statistics and general data analysis. At least it should be, if only it did not take so long. We argue in rstats, that using the Geometric Median is the most stable way to characterise multidimensional data (nd). That leaves the one dimensional (1d) medians, addressed here. Medians are more stable measure of central tendency than means but they are not used nearly enough. One suspects that this is mostly due to being slower to compute and the fast algorithm developed here being non-trivial.
See tests.rs as examples of usage. Their automatically generated output can be found by clicking the 'test' icon at the top of this document and then examining the latest log.
The naive median is found by sorting the list of data and then picking the midpoint. In this case, the fastest hashsort from crate indxvec was used, which is a lot faster than the standard Rust Quicksort.
Nevertheless, the problem with this approach is that, even when using a good quality sort, its complexity is at best O(nlog(n)). The quest for faster median algorithms, with complexity O(n) is based on the observation that not all items need to be fully sorted. Only partitioned and counted off.
Therefore the naive median could not compete and it has now been deleted (as of version 2.0.0).
median
Iteratively partitions data around a pivot estimate (the arithmetic mean of the data). This is reasonably well centred and it is very fast to compute (summation being faster than comparisons and memory manipulations of 'median of medians'). This algorithm has linear complexity.
odd_strict_median
Returns the midpoint of type T, which could be any complex unquantifiable struct. Traits Ord and Clone have to be implemented for T.
The algorithm uses BinaryHeap<T> to find the unsorted minimum n/2+1 items and then picks their maximum (which is at the root of the max heap already). Thus all comparisons and swaps are kept to the minimum. Furthermore, only pointers to T items are being manipulated, minimising also the moving of the potentially bulky original data items.
even_strict_median
As the data items T are unquantifiable, we can not simply average the two midpoints of even length data, as we did in median. So we return them both as a tuple, the smaller one first. Otherwise very similar to odd_strict_median.
median
is the main public entry point, implemented as a method of trait Median.
rust
/// Fast 1D medians and associated information and tasks
pub trait Median<T> {
/// Finds the median of `&[T]`, fast
fn median(self, quantify: &mut impl FnMut(&T) -> f64) -> Result<f64, ME>;
/// Finds the median of odd sized nonquantifiable Ord data
fn odd_strict_median(self) -> T
where
T: Ord + Clone;
/// Finds the two mid values of even sized nonquantifiable Ord data
fn even_strict_median(self) -> (T,T)
where
T: Ord + Clone;
/// Zero median f64 data produced by finding and subtracting the median.
fn zeromedian(self, quantify: &mut impl FnMut(&T) -> f64) -> Result<Vec<f64>, ME>;
/// Median correlation = cosine of an angle between two zero median vecs
fn mediancorr(
self,
v: &[T],
quantify: &'static mut impl FnMut(&T) -> f64,
) -> Result<f64, MedError<String>>;
/// Median of absolute differences (MAD).
fn mad(self, med: f64, quantify: &mut impl FnMut(&T) -> f64) -> Result<f64, ME>;
/// Median and MAD.
fn medstats(self, quantify: &mut impl FnMut(&T) -> f64) -> Result<MStats, ME>;
/// Median, quartiles, MAD, Stderr
fn medinfo(self, quantify: &mut impl FnMut(&T) -> f64) -> Result<Med, ME>;
}
Version 2.0.4 - Updated indxvec dependency to 1.4.12. Improved the text test.
Version 2.0.3 - Added methods odd_strict_median and even_strict_median to trait Median<T>.
These methods apply in classical situations where T is unquantifiable, only Ord(erable). They are about 1.75 times slower.
However, this is only a constant factor which does not grow with the length of data.
Version 2.0.2 - Removed trait parameter Q to ease external usage.
Version 2.0.1 - Moved all methods directly associated with 1d medians from rstats to here. Removed all remaining trait bounds from end data type T. This is one of the benefits of passing explicit quantify closures.
Version 2.0.0 - Better, leaner, faster! Drastically reduced stack usage. Significant speed up using iterative implementation. More concise code. Deleted all old algorithms with inferior performance, such as naive_median. Pivot value estimates are now simple arithmetic means. This is not as sophisticated as secant but is fast to evaluate, giving better overall performance. Introduced closure argument quantify, allowing dynamic application to any (quantifiable) data types. Yanked versions 1.0.9 & 1.0.10 as returning Result was a breaking change which according to semver requires major new version, i.e. this one.
Version 1.0.9 - Added custom MedError and wrapped outputs in Result. Updated times dependency.
Version 1.0.8 - Added fully automated tests by github actions.
Version 1.0.7 - Updated to ran 1.0.4
Version 1.0.6 - Updated to times 1.0.4. Changed the comparison test accordingly.
Version 1.0.5 - Simplification. Deleted unnecessary w_median. Simplified error test. Updated dev-dependencies ran 1.0.3 and times 1.0.3.
Version 1.0.4 - Updated dependency indxvec v.1.4.2.
Version 1.0.3 - Added ratio mad/median (standard error) to struct Med and improved its Display.
Version 1.0.2 - Removed unnecessary extra reference from method median.
Version 1.0.1 - Added for convenience struct MStats and method medstats returning it. It holds here the median and MAD. More generally, any centre and dispersion. Moved the low level and private functions to module algos.rs. Updated times dev-dependency.
Version 1.0.0 - Updated to the latest indxvec dependency, v. 1.2.11. Added times crate for timing comparison test.
Version 0.1.2 - The public methods are now in trait Median.