========================= Quantile Digest Functions

Presto implements two algorithms for estimating rank-based metrics, quantile digest <http://dx.doi.org/10.1145/347090.347195>_ and T-digest <https://doi.org/10.1016/j.simpa.2020.100049>. T-digest has better performance <https://arxiv.org/abs/1902.04023> in general while the Presto implementation of quantile digests supports more numeric types. T-digest has better accuracy at the tails, often dramatically better, but may have worse accuracy at the median, depending on the compression factor used. In comparison, quantile digests supports a maximum rank error, which guarantees relative uniformity of precision along the quantiles. Quantile digests are also formally proven to support lossless merges, while T-digest is not (but does empirically demonstrate lossless merges).

Presto implements the approx_percentile function with the quantile digest data structure. The underlying data structure, :ref:qdigest <qdigest_type>, is exposed as a data type in Presto, and can be created, queried and stored separately from approx_percentile.

Data Structures

A quantile digest is a data sketch which stores approximate percentile information. The presto type for this data structure is called qdigest, and it takes a parameter which must be one of bigint, double or real which represent the set of numbers that may be ingested by the qdigest. They may be merged without losing precision, and for storage and retrieval they may be cast to/from VARBINARY.

Functions

.. function:: merge(qdigest) -> qdigest :noindex:

Merges all input ``qdigest``\ s into a single ``qdigest``.

.. function:: value_at_quantile(qdigest(T), quantile) -> T

Returns the approximate percentile values from the quantile digest given
the number ``quantile`` between 0 and 1.

.. function:: quantile_at_value(qdigest(T), T) -> quantile

Returns the approximate ``quantile`` number between 0 and 1 from the
quantile digest given an input value. Null is returned if the quantile digest
is empty or the input value is outside of the range of the quantile digest.

.. function:: scale_qdigest(qdigest(T), scale_factor) -> qdigest(T)

Returns a ``qdigest`` whose distribution has been scaled by a factor
specified by ``scale_factor``.

.. function:: values_at_quantiles(qdigest(T), quantiles) -> T

Returns the approximate percentile values as an array given the input
quantile digest and array of values between 0 and 1 which
represent the quantiles to return.

.. function:: qdigest_agg(x) -> qdigest<[same as x]>

Returns the ``qdigest`` which is composed of  all input values of ``x``.

.. function:: qdigest_agg(x, w) -> qdigest<[same as x]>

Returns the ``qdigest`` which is composed of  all input values of ``x`` using
the per-item weight ``w``.

.. function:: qdigest_agg(x, w, accuracy) -> qdigest<[same as x]>

Returns the ``qdigest`` which is composed of  all input values of ``x`` using
the per-item weight ``w`` and maximum error of ``accuracy``. ``accuracy``
must be a value greater than zero and less than one, and it must be constant
for all input rows.