t-digest

A new data structure for accurate on-line accumulation of rank-based statistics such as quantiles
and trimmed means. The t-digest algorithm is also very friendly to parallel programs making it
useful in map-reduce and parallel streaming applications implemented using, say, Apache Spark.

The t-digest construction algorithm uses a variant of 1-dimensional k-means clustering to produce a
very compact data structure that allows accurate estimation of quantiles. This t-digest data
structure can be used to estimate quantiles, compute other rank statistics or even to estimate
related measures like trimmed means. The advantage of the t-digest over previous digests for
this purpose is that the t-digest handles data with full floating point resolution. With small
changes, the t-digest can handle values from any ordered set for which we can compute something akin to a mean.
The accuracy of quantile estimates produced by t-digests can be orders of magnitude more accurate than
those produced by previous digest algorithms in spite of the fact that t-digests are much more
compact when stored on disk.

In summary, the particularly interesting characteristics of the t-digest are that it

• has smaller summaries when serialized
• works on double precision floating point as well as integers.
• provides part per million accuracy for extreme quantiles and typically <1000 ppm accuracy for middle quantiles
• is very fast (~ 140 ns per add)
• is very simple (~ 5000 lines of code total, <1000 for the most advanced implementation alone)
• has a reference implementation that has > 90% test coverage
• can be used with map-reduce very easily because digests can be merged
• requires no dynamic allocation after initial creation (MergingDigest only)
• has no runtime dependencies

Recent News

Lots has happened in t-digest lately. The basic gist of all the
changes are that the core algorithms have been made much more rigorous
and the associated papers have been updated to match the reality of
the most advanced implementations. The general areas of improvement
include substantial speedups, a new framework for dealing with scale
functions, real proofs of size bounds and invariants for all current
scale functions, much improved interpolation algorithms, better
accuracy testing and splitting the entire distribution into parts for
the core algorithms, quality testing, benchmarking and documentation.

I am working on a 4.0 release that incorporates all of these
improvements. The remaining punch list for the release is roughly:

• verify all tests are clean and not disabled (done!)
• integrate all scale functions into AVLTreeDigest (done!)
• describe accuracy using the quality suite
• extend benchmarks to include AVLTreeDigest as first-class alternative
• measure merging performance
• consider issue #87
• review all outstanding issues (add unit tests if necessary or close if not)

I have submitted the main paper describing t-digest to RSS Series B
and will be preparing a more implementation-oriented paper intended
for submission to the Journal of Statistical Software. Potential
co-authors who
could accelerate these submissions are encouraged to speak up! In
the mean time, an
archived pre-print of the paper is available.

Scale Functions

The idea of scale functions is the heart of the t-digest. But things
don't quite work the way that we originally thought. Originally, it
was presumed that accuracy should be proportional to the square of the
size of a cluster. That isn't true in practice. That means that scale
functions need to be much more aggressive about controlling cluster
sizes near the tails. We now have 4 scale functions supported for both
major digest forms (MergingDigest and AVLTreeDigest) to allow
different trade-offs in terms of accuracy.

These scale functions now have associated proofs that they all
preserve the key invariants
necessary to build an accurate digest and that they all give
tight bounds on the size of a digest.
Having new scale functions means that we can get much better tail
accuracy than before without losing much in terms of median accuracy.
It also means that insertion into a MergingDigest is faster than
before since we have been able to eliminate all fancy functions like
sqrt, log or sin from the critical path (although sqrt is faster
than you might think).

Better Interpolation

The better accuracy achieved by the new scale functions partly comes
from the fact that the most extreme clusters near q=0 or q=1 are
limited to only a single sample. Handling these singletons well makes
a huge difference in the accuracy of tail estimates. Handling the
transition to non-singletons is also very important.

Both cases are handled much better than before.

The better interpolation has been fully integrated and tested in both
the MergingDigest and AVLTreeDigest with very good improvements in
accuracy. The bug detected in the AVLTreeDigest that affected data
with many repeated values has also been fixed.

Two-level Merging

We now have a trick for the MergingDigest that uses a higher value
of the compression parameter (delta) while we are accumulating a
t-digest and a lower value when we are about to store or display a
t-digest. This two-level merging has a small (negative) effect on
speed, but a substantial (positive) effect on accuracy because
clusters are ordered more strongly. This better ordering of clusters
means that the effects of the improved interpolation are much easier
to observe.

Extending this to AVLTreeDigest is theoretically possible, but it
isn't clear the effect it will have.

Repo Reorg

The t-digest repository is now split into different functional
areas. This is important because it simplifies the code used in
production by extracting the (slow) code that generates data for
accuracy testing, but also because it lets us avoid any dependencies
on GPL code (notable the jmh benchmarking tools) in the released
artifacts.

The major areas are

• core - this is where the t-digest and unit tests live
• docs - the main paper and auxillary proofs live here
• benchmarks - this is the code that tests the speed of the digest algos
• quality - this is the code that generates and analyzes accuracy information

Within the docs sub-directory, proofs of invariant preservation and size
bounds are moved to docs/proofs and all figures in docs/t-digest-paper
are collected into a single directory to avoid cluster.

LogHistogram and FloatHistogram

This package also has an implementation of FloatHistogram which is
another way to look at distributions where all measurements are
positive and where you want relative accuracy in the measurement space
instead of accuracy defined in quantiles. This FloatHistogram makes
use of the floating point hardware to implement variable width bins so
that adding data is very fast (5ns/data point in benchmarks) and the
resulting sketch is small for reasonable accuracy levels. For
instance, if you require dynamic range of a million and are OK with
about bins being about ±10%, then you only need 80 counters.

Since the bins for FloatHistogram's are static rather than adaptive,
they can be combined very easily. Thus you can store a histogram for
short periods of time and combined them at query time if you are
looking at metrics for your system. You can also reweight histograms
to avoid errors due to structured omission.

Another class called LogHistogram is also available in
t-digest. The LogHistogram is very much like the FloatHistogram,
but it incorporates a clever quadratic update step (thanks to Otmar
Ertl) so that the bucket widths vary more precisely and thus the
number of buckets can be decreased by about 40% while getting the same
accuracy. This is particularly important when you are maintaining only
modest accuracy and want small histograms.

In the future, I will incorporate some of the interpolation tricks
from the main t-digest into the LogHistogram implementation.

Compile and Test

You have to have Java 1.8 to compile and run this code. You will also
need maven (3+ preferred) to compile and test this software. In order
to build the images that go into the theory paper, you will need R.
In order to format the paper, you will need latex. Pre-built pdf
versions of all figures and papers are provided so you won't need latex
if you don't need to make changes to these documents.

On Ubuntu, you can get the necessary pre-requisites for compiling the
code with the following:

sudo apt-get install  openjdk-8-jdk git maven

Once you have these installed, use this to build and test the software:

mvn test

Most of the very slow tests are in the quality module so if you just run
the tests in core module, you can save considerable time.

Testing Accuracy and Comparing to Q-digest

The normal test suite produces a number of diagnostics that describe
the scaling and accuracy characteristics of t-digests. In order to
produce nice visualizations of these properties, you need to have more
samples. To get this enhanced view, run the tests in the quality module

cd quality; mvn test

The data from these tests are stored in a variety of data files in the
quality directory. Some of these files are quite large.

I have prepared detailed instructions on producing all of the figures
used in the main paper.

Most of these scripts will complete almost instantaneously; one or two
will take a few tens of seconds.

The output of these scripts are a collection of PDF files that can be
viewed with any suitable viewer such as Preview on a Mac. Many of
these images are used as figures in the
main t-digest paper.

Implementations in Other Languages

The t-digest algorithm has been ported to other languages:

• Python: tdigest
• Go: github.com/spenczar/tdigest https://github.com/influxdata/tdigest
• JavaScript: tdigest
• C++: CPP TDigest, FB's Folly Implementation (high performance)
• Scala: need link!
• C: tdigestc (w/ bindings to Go, Java, Python, JS via wasm)

Continuous Integration

The t-digest project makes use of Travis integration with Github for testing whenever a change is made.

You can see the reports at:

https://travis-ci.org/tdunning/t-digest

travis update

Installation

The t-Digest library Jars are released via Maven Central Repository.
The current version is 3.2.

     <dependency>
         <groupId>com.tdunning</groupId>
         <artifactId>t-digest</artifactId>
         <version>3.2</version>
     </dependency>