array_agg(x ORDER BY y DESC)
array_agg(x ORDER BY x, y, z)

This is an alias for :func:`!arbitrary`.
::

   SELECT any_value(t1.age),t1.gender
   FROM
   (
    SELECT *
    FROM
    (
        VALUES
            ('Alice', 30,'male'),
            ('Bob', 25,'male'),
            ('Charlie', 22,'female'),
            ('Lucy', 20,'female')
    ) AS t(name, age, gender)) t1
   group by t1.gender;
   --(22,female)
   --(30,male)

Returns an arbitrary non-null value of ``x``, if one exists.

Returns an array created from the input ``x`` elements.
::

    SELECT array_agg(name)
    FROM
    (
        VALUES
            ('Alice', 30,'male'),
            ('Bob', 25,'male'),
            ('Charlie', 22,'female'),
            ('Lucy', 20,'female')
    ) AS t(name, age, gender);
    --['Alice','Bob','Charlie','Lucy']

Returns the average (arithmetic mean) of all input values.
::

   SELECT avg(age)
   FROM
   (
   	VALUES
       ('Alice', 30,'male'),
       ('Bob', 25,'male'),
       ('Charlie', 22,'female'),
       ('Lucy', 20,'female')
    ) AS t(name, age, gender);
    --(24.25)

Returns the average interval length of all input values.
::

   SELECT avg(timediff)
   FROM
   (
   	VALUES
       (INTERVAL '10' DAY),
       (INTERVAL '20' DAY),
       (INTERVAL '30' DAY)
    ) AS t(timediff);
    --(20 00:00:00.000)//INTERVAL '20' DAY

Returns ``TRUE`` if every input value is ``TRUE``, otherwise ``FALSE``.
::

   SELECT bool_and(true_or_false)
   FROM
   (
   	VALUES
       (true),
       (true),
       (false)
    ) AS t(true_or_false);
    --(false)

Returns ``TRUE`` if any input value is ``TRUE``, otherwise ``FALSE``.
::

   SELECT bool_or(true_or_false)
   FROM
   (
   	VALUES
       (true),
       (true),
       (false)
    ) AS t(true_or_false);
    --(true)

Returns an order-insensitive checksum of the given values.
::

   SELECT checksum(name)
   FROM
   (
   	VALUES
       ('Alice', 30,'male'),
       ('Bob', 25,'male'),
       ('Charlie', 22,'female'),
       ('Lucy', 20,'female')
    ) AS t(name, age, gender);
    --(C0ACD56CF866E759)//hex format

   SELECT checksum(name)
   FROM
   (
   	VALUES
       ('Alice', 30,'male'),
       ('Bob', 25,'male'),
       ('Lucy', 20,'female'),
       ('Charlie', 22,'female')
    ) AS t(name, age, gender);
    --(C0ACD56CF866E759)//hex format

Returns the number of input rows.
::

   SELECT count(*)
   FROM
   (
   	VALUES
       ('Alice', 30,'male'),
       ('Bob', 25,'male'),
       ('Charlie', 22,'female'),
       ('Lucy', 20,'female')
    ) AS t(name, age, gender);
    --(4)

   SELECT count(*)
   FROM
   (
   	VALUES
       ('Alice', 30,'male'),
       ('Bob', 25,'male'),
       ('Charlie', 22,'female'),
       ('Lucy', null,'female')
    ) AS t(name, age, gender);
    --(4)

Returns the number of non-null input values.
::

   SELECT count(age)
   FROM
   (
   	VALUES
       ('Alice', 30,'male'),
       ('Bob', 25,'male'),
       ('Charlie', 22,'female'),
       ('Lucy', 20,'female')
    ) AS t(name, age, gender);
    --(4)

   SELECT count(age)
   FROM
   (
   	VALUES
       ('Alice', 30,'male'),
       ('Bob', 25,'male'),
       ('Charlie', 22,'female'),
       ('Lucy', null,'female')
    ) AS t(name, age, gender);
    --(3)

Returns the number of ``TRUE`` input values.
This function is equivalent to ``count(CASE WHEN x THEN 1 END)``.
::

    SELECT count_if(gender = 'female') AS female_count
    FROM (
    VALUES
        ('Alice', 30, 'female'),
        ('Bob', 25, 'male'),
        ('Lucy', 22, 'female')
    ) AS t(name, age, gender);
    --(2)

This is an alias for :func:`!bool_and`.

Returns the `geometric mean <https://en.wikipedia.org/wiki/Geometric_mean>`_ of all input values.
::

    SELECT geometric_mean(age) AS geo_mean_age
    FROM (
        VALUES
            ('Alice', 30, 'female'),
            ('Bob', 25, 'male'),
            ('Lucy', 22, 'female'),
            ('Tom', 28, 'male')
    ) AS t(name, age, gender);
    --(26.07116834203365)

Returns the value of ``x`` associated with the maximum value of ``y`` over all input values.
::

    SELECT max_by(name, age) AS oldest_person
    FROM (
        VALUES
        ('Alice', 30),
        ('Bob', 25),
        ('Lucy', 22),
        ('Tom', 35)
    ) AS t(name, age);
    --(Tom)

Returns ``n`` values of ``x`` associated with the ``n`` largest of all input values of ``y``
in descending order of ``y``.
::

    SELECT max_by(name, age, 2) AS top_2_oldest
    FROM (
        VALUES
        ('Alice', 30),
        ('Bob', 25),
        ('Lucy', 22),
        ('Tom', 35),
        ('Jerry', 33)
    ) AS t(name, age);
    --[Tom,Jerry]

Returns the value of ``x`` associated with the minimum value of ``y`` over all input values.
::

    SELECT min_by(name, age) AS youngest_person
    FROM (
    VALUES
        ('Alice', 30),
        ('Bob', 25),
        ('Lucy', 22),
        ('Tom', 35)
    ) AS t(name, age);
    --(Lucy)

Returns ``n`` values of ``x`` associated with the ``n`` smallest of all input values of ``y``
in ascending order of ``y``.
::

    SELECT min_by(name, age,2) AS youngest_person
    FROM (
    VALUES
        ('Alice', 30),
        ('Bob', 25),
        ('Lucy', 22),
        ('Tom', 35)
    ) AS t(name, age);
    --[Lucy,Bob]

Returns the maximum value of all input values.
::

    SELECT max(age) AS max_age
    FROM (
    VALUES
        ('Alice', 30),
        ('Bob', 25),
        ('Lucy', 22),
        ('Tom', 35)
    ) AS t(name, age);
    --(35)

Returns ``n`` largest values of all input values of ``x``.
::

    SELECT max(age, 3) AS top_3_ages
    FROM (
    VALUES
        ('Alice', 30),
        ('Bob', 25),
        ('Lucy', 22),
        ('Tom', 35),
        ('Jerry', 33)
    ) AS t(name, age);
    --[35,33,30]

Returns the minimum value of all input values.
::

    SELECT min(age) AS min_age
    FROM (
    VALUES
        ('Alice', 30),
        ('Bob', 25),
        ('Lucy', 22),
        ('Tom', 35)
    ) AS t(name, age);
    --(22)

Returns ``n`` smallest values of all input values of ``x``.
::

    SELECT min(age, 3) AS bottom_3_ages
    FROM (
    VALUES
        ('Alice', 30),
        ('Bob', 25),
        ('Lucy', 22),
        ('Tom', 35),
        ('Jerry', 33)
    ) AS t(name, age);
    --[22,25,30]

Reduces all input values into a single value. ``inputFunction`` will be invoked
for each input value. In addition to taking the input value, ``inputFunction``
takes the current state, initially ``initialState``, and returns the new state.
``combineFunction`` will be invoked to combine two states into a new state.
The final state is returned. Throws an error if ``initialState`` is NULL.
The behavior is undefined if ``inputFunction`` or ``combineFunction`` return a NULL.

Take care when designing ``initialState``, ``inputFunction`` and ``combineFunction``.
These must support evaluating aggregation in a distributed manner using partial
aggregation on many nodes, followed by shuffle over group-by keys, followed by
final aggregation. Consider all possible values of state to ensure that
``combineFunction`` is `commutative <https://en.wikipedia.org/wiki/Commutative_property>`_
and `associative <https://en.wikipedia.org/wiki/Associative_property>`_
operation with ``initialState`` as the
`identity <https://en.wikipedia.org/wiki/Identity_element>`_ value.::

    combineFunction(s, initialState) = s for any s

    combineFunction(s1, s2) = combineFunction(s2, s1) for any s1 and s2

    combineFunction(s1, combineFunction(s2, s3)) = combineFunction(combineFunction(s1, s2), s3) for any s1, s2, s3

In addition, make sure that the following holds for the inputFunction::

    inputFunction(inputFunction(initialState, x), y) = combineFunction(inputFunction(initialState, x), inputFunction(initialState, y)) for any x and y

::

    SELECT id, reduce_agg(value, 0, (a, b) -> a + b, (a, b) -> a + b)
    FROM (
        VALUES
            (1, 2),
            (1, 3),
            (1, 4),
            (2, 20),
            (2, 30),
            (2, 40)
    ) AS t(id, value)
    GROUP BY id;
    -- (1, 9)
    -- (2, 90)

    SELECT id, reduce_agg(value, 1, (a, b) -> a * b, (a, b) -> a * b)
    FROM (
        VALUES
            (1, 2),
            (1, 3),
            (1, 4),
            (2, 20),
            (2, 30),
            (2, 40)
    ) AS t(id, value)
    GROUP BY id;
    -- (1, 24)
    -- (2, 24000)

The state type must be a boolean, integer, floating-point, or date/time/interval.

Returns an array created from the distinct input ``x`` elements.

If the input includes ``NULL``, ``NULL`` will be included in the returned array.
If the input includes arrays with ``NULL`` elements or rows with ``NULL`` fields, they will
be included in the returned array.  This function uses ``IS DISTINCT FROM`` to determine
distinctness. ::

    SELECT set_agg(x) FROM (VALUES(1), (2), (null), (2), (null)) t(x) -- ARRAY[1, 2, null]
    SELECT set_agg(x) FROM (VALUES(ROW(ROW(1, null))), ROW((ROW(2, 'a'))), ROW((ROW(1, null))), (null)) t(x) -- ARRAY[ROW(1, null), ROW(2, 'a'), null]

Returns an array of all the distinct values contained in each array of the input.

When all inputs are ``NULL``, this function returns an empty array. If ``NULL`` is
an element of one of the input arrays, ``NULL`` will be included in the returned array.
If the input includes arrays with ``NULL`` elements or rows with ``NULL`` fields, they will
be included in the returned array.  This function uses ``IS DISTINCT FROM`` to determine
distinctness.

Example::

    SELECT set_union(elements)
    FROM (
        VALUES
            ARRAY[1, 2, 3],
            ARRAY[2, 3, 4]
    ) AS t(elements);

Returns ARRAY[1, 2, 3, 4]

Returns the sum of all input values.

Returns the bitwise AND of all input values in 2's complement representation.
::

    SELECT bitwise_and_agg(flags) AS result
    FROM (
    VALUES
        (7),   -- 0b0111
        (3),   -- 0b0011
        (1)    -- 0b0001
    ) AS t(flags);
    --(1) //0b0001

Returns the bitwise OR of all input values in 2's complement representation.
::

    SELECT bitwise_or_agg(flags) AS result
    FROM (
    VALUES
        (7),   -- 0b0111
        (3),   -- 0b0011
        (1)    -- 0b0001
    ) AS t(flags);
    --(7) //0b0111

Returns the bitwise XOR of all input values in 2's complement representation.
::

    SELECT bitwise_xor_agg(flags) AS result
    FROM (
    VALUES
        (7),   -- 0b0111
        (3),   -- 0b0011
        (1)    -- 0b0001
    ) AS t(flags);
    --(5) //0b0101

Returns a map containing the count of the number of times each input value occurs.
::

    SELECT histogram(age) AS age_histogram
    FROM (
    VALUES
        (30),
        (25),
        (30),
        (22),
        (25),
        (30)
    ) AS t(age);
    --{22->1, 25->2, 30->3}

Returns a map created from the input ``key`` / ``value`` pairs.
::

    SELECT map_agg(name, age) AS name_age_map
    FROM (
    VALUES
        ('Alice', 30),
        ('Bob', 25),
        ('Lucy', 22)
    ) AS t(name, age);
    --{Bob->25, Alice->30, Lucy->22}

    SELECT map_union(maps) AS merged_map
    FROM (
    VALUES
        (MAP(ARRAY['a', 'b'], ARRAY[1, 2])),
        (MAP(ARRAY['b', 'c'], ARRAY[3, 4])),
        (MAP(ARRAY['d'], ARRAY[5]))
    ) AS t(maps);
    --{a->1, b->2, c->4, d->5}

  Returns the union of all the input maps summing the values of matching keys in all
  the maps. All null values in the original maps are coalesced to 0.
  ::

        SELECT map_union_sum(maps) AS merged_sum_map
        FROM (
        VALUES
            (MAP(ARRAY['a', 'b'], ARRAY[1, 2])),
            (MAP(ARRAY['b', 'c'], ARRAY[3, 4])),
            (MAP(ARRAY['a', 'd'], ARRAY[5, 6]))
        ) AS t(maps);
        --{'a'->6,'b'->5,'c'->4,'d'->6}

Returns a multimap created from the input ``key`` / ``value`` pairs.
Each key can be associated with multiple values.

::

    SELECT multimap_agg(name, age) AS name_age_multimap
    FROM (
    VALUES
        ('Alice', 30),
        ('Bob', 25),
        ('Alice', 32),
        ('Lucy', 22),
        ('Bob', 28)
    ) AS t(name, age);
    --{Bob->[25, 28], Alice->[30, 32], Lucy->[22]}

Returns the approximate number of distinct input values.
This function provides an approximation of ``count(DISTINCT x)``.
Zero is returned if all input values are null.

This function should produce a standard error of 2.3%, which is the
standard deviation of the (approximately normal) error distribution over
all possible sets. It does not guarantee an upper bound on the error for
any specific input set.
::

    SELECT approx_distinct(name) AS distinct_names
    FROM (
    VALUES
        ('Alice'),
        ('Bob'),
        ('Alice'),
        ('Lucy'),
        ('Bob'),
        ('Tom')
    ) AS t(name);
    --(4)

Returns the approximate number of distinct input values.
This function provides an approximation of ``count(DISTINCT x)``.
Zero is returned if all input values are null.

This function should produce a standard error of no more than ``e``, which
is the standard deviation of the (approximately normal) error distribution
over all possible sets. It does not guarantee an upper bound on the error
for any specific input set. The current implementation of this function
requires that ``e`` be in the range of ``[0.0040625, 0.26000]``.
::

    SELECT approx_distinct(gender, 0.01) AS estimated_distinct_gender
    FROM (
    VALUES 
        ('Alice', 30, 'female'),
        ('Bob', 25, 'male'),
        ('Lucy', 22, 'female'),
        ('Tom', 40, 'male'),
        ('Amy', 35, 'female')
    ) AS t(name, age, gender);
    --(2)

Returns the approximate percentile for all input values of ``x`` at the
given ``percentage``. The value of ``percentage`` must be between zero and
one and must be constant for all input rows.
::

    SELECT approx_percentile(age, 0.5) AS median_age
    FROM (
    VALUES
        (30),
        (25),
        (22),
        (35),
        (33),
        (28)
    ) AS t(age);
    --(30)

As ``approx_percentile(x, percentage)``, but with a maximum rank error of
``accuracy``. The value of ``accuracy`` must be between zero and one
(exclusive) and must be constant for all input rows. Note that a lower
"accuracy" is really a lower error threshold, and thus more accurate. The
default accuracy is ``0.01``.
::

    SELECT approx_percentile(age, 0.5, 0.9) AS median_age
    FROM (
    VALUES
        (30),
        (25),
        (22),
        (35),
        (33),
        (28)
    ) AS t(age);
    --(30)

Returns the approximate percentile for all input values of ``x`` at each of
the specified percentages. Each element of the ``percentages`` array must be
between zero and one, and the array must be constant for all input rows.
::

    SELECT approx_percentile(age, ARRAY[0.25, 0.5, 0.75]) AS percentiles
    FROM (
    VALUES
        (22),
        (25),
        (28),
        (30),
        (33),
        (35)
    ) AS t(age);
    --[25,30,33]

As ``approx_percentile(x, percentages)``, but with a maximum rank error of
``accuracy``.
::

    SELECT approx_percentile(age, ARRAY[0.25, 0.5, 0.75], 0.9) AS percentiles
    FROM (
    VALUES
        (22),
        (25),
        (28),
        (30),
        (33),
        (35)
    ) AS t(age);
    --[25,30,33]

Returns the approximate weighed percentile for all input values of ``x``
using the per-item weight ``w`` at the percentage ``p``. The weight must be
an integer value of at least one. It is effectively a replication count for
the value ``x`` in the percentile set. The value of ``p`` must be between
zero and one and must be constant for all input rows.
::

    SELECT approx_percentile(age, weight, 0.5) AS weighted_median
    FROM (
    VALUES
        (22, 1),
        (25, 2),
        (28, 1),
        (30, 3),
        (33, 1),
        (35, 2)
    ) AS t(age, weight);
    --(30)

As ``approx_percentile(x, w, percentage)``, but with a maximum rank error of
``accuracy``.
::

    SELECT approx_percentile(age, weight, 0.5, 0.9) AS weighted_median
    FROM (
    VALUES
        (22, 1),
        (25, 2),
        (28, 1),
        (30, 3),
        (33, 1),
        (35, 2)
    ) AS t(age, weight);
    --(30)

Returns the approximate weighed percentile for all input values of ``x``
using the per-item weight ``w`` at each of the given percentages specified
in the array. The weight must be an integer value of at least one. It is
effectively a replication count for the value ``x`` in the percentile set.
Each element of the array must be between zero and one, and the array must
be constant for all input rows.
::

    SELECT approx_percentile(age, weight, ARRAY[0.25, 0.5, 0.75]) AS weighted_percentiles
    FROM (
    VALUES
        (22, 1),
        (25, 2),
        (28, 1),
        (30, 3),
        (33, 1),
        (35, 2)
    ) AS t(age, weight);
    -[25,30,33]

As ``approx_percentile(x, w, percentages)``, but with a maximum rank error of
``accuracy``.
::

    SELECT approx_percentile(age, weight, ARRAY[0.25, 0.5, 0.75],0.9) AS weighted_percentiles
    FROM (
    VALUES
        (22, 1),
        (25, 2),
        (28, 1),
        (30, 3),
        (33, 1),
        (35, 2)
    ) AS t(age, weight);
    -[25,30,33]

See :doc:`hyperloglog`.
::

    SELECT approx_set(user_id) AS hll_data
    FROM (
    VALUES
        (1001),
        (1002),
        (1003),
        (1001),
        (1004)
    ) AS t(user_id);
    --(020C0400401E4D1D4081707280E083BD444759E9)//hex format

See :doc:`hyperloglog`.
::

    WITH hll_data AS (
    SELECT region, approx_set(user_id) AS hll
    FROM (
        VALUES
        ('east', 1),
        ('east', 2),
        ('west', 2),
        ('west', 3),
        ('west', 4)
    ) AS t(region, user_id)
    GROUP BY region
    )
    SELECT cardinality(merge(hll)) AS total_unique_users
    FROM hll_data;
    --(4)

See :doc:`khyperloglog`.
::

    SELECT cardinality(khyperloglog_agg(x,y))
    FROM (
    VALUES (1,101), (2,102), (3,103), (4,101), (5,104)
    ) AS t(x,y);
    --(5)

See :doc:`qdigest`.

See :doc:`qdigest`.

See :doc:`qdigest`.

See :doc:`qdigest`.

Computes an approximate histogram with up to ``buckets`` number of buckets
for all ``value``\ s with a per-item weight of ``weight``.  The keys of the
returned map are roughly the center of the bin, and the entry is the total
weight of the bin.  The algorithm is based loosely on [BenHaimTomTov2010]_.

``buckets`` must be a ``bigint``. ``value`` and ``weight`` must be numeric.
::

    SELECT numeric_histogram(3, v, 1.0)
    FROM (
     VALUES (10),
            (15),
            (20),
            (25),
            (30)
    ) AS t(v);
    --{30.0->1.0, 22.5->2.0, 12.5->2.0}

Computes an approximate histogram with up to ``buckets`` number of buckets
for all ``value``\ s. This function is equivalent to the variant of
:func:`!numeric_histogram` that takes a ``weight``, with a per-item weight of ``1``.
In this case, the total weight in the returned map is the count of items in the bin.
::

    SELECT numeric_histogram(3, v)
    FROM (
    VALUES (10.0), (15.0), (20.0), (25.0), (30.0)
    ) AS t(v);
    --{30.0->1.0, 22.5->2.0, 12.5->2.0}

Returns correlation coefficient of input values.
::

    SELECT corr(score, study_hours)
    FROM (
    VALUES 
        (85, 2),
        (90, 3),
        (95, 4),
        (70, 1),
        (80, 2)
    ) AS t(score, study_hours);
    --(0.95751756)

Returns the population covariance of input values.
::

    SELECT covar_pop(score, study_hours)
    FROM (
    VALUES 
        (85, 2),
        (90, 3),
        (95, 4),
        (70, 1),
        (80, 2)
    ) AS t(score, study_hours);
    --(8.4)

Returns the sample covariance of input values.
::

    SELECT covar_samp(score, hours)
    FROM (
    VALUES 
        (85, 2),
        (90, 3),
        (95, 4),
        (70, 1),
        (80, 2)
    ) AS t(score, hours);
    --(10.5)

Returns the log-2 entropy of count input-values.

.. math::

    \mathrm{entropy}(c) = \sum_i \left[ {c_i \over \sum_j [c_j]} \log_2\left({\sum_j [c_j] \over c_i}\right) \right].

``c`` must be a ``bigint`` column of non-negative values.

The function ignores any ``NULL`` count. If the sum of non-``NULL`` counts is 0,
it returns 0.
::

    SELECT entropy(category)
    FROM (
    VALUES 
        (1),
        (1),
        (2),
        (1),
        (2)
    ) AS t(category);
    --(2.2359263506290326)

Returns the excess kurtosis of all input values. Unbiased estimate using
the following expression:

.. math::

    \mathrm{kurtosis}(x) = {n(n+1) \over (n-1)(n-2)(n-3)} { \sum[(x_i-\mu)^4] \over \sigma^4} -3{ (n-1)^2 \over (n-2)(n-3) }

where :math:`\mu` is the mean, and :math:`\sigma` is the standard deviation.
::

    SELECT kurtosis(salary)
    FROM (
    VALUES (1000),
            (1200),
            (1100),
            (1500),
            (900),
            (2500)
    ) AS t(salary);
    --(3.549458572481891)

Returns linear regression intercept of input values. ``y`` is the dependent
value. ``x`` is the independent value.
::

    SELECT regr_intercept(salary, age)
    FROM (
    VALUES 
        (25, 3000),
        (30, 4000),
        (35, 5000),
        (40, 6000)
    ) AS t(age, salary);
    --(-2000)

Returns linear regression slope of input values. ``y`` is the dependent
value. ``x`` is the independent value.
::

    SELECT regr_slope(salary, age)
    FROM (
    VALUES 
        (25, 3000),
        (30, 4000),
        (35, 5000),
        (40, 6000)
    ) AS t(age, salary);
    --(200)

Returns the average of the independent value in a group. ``y`` is the dependent
value. ``x`` is the independent value.
::

    SELECT regr_avgx(salary, age)
    FROM (
    VALUES 
        (25, 3000),
        (30, 4000),
        (35, 5000),
        (NULL, 6000),
        (40, NULL)
    ) AS t(age, salary);
    --(30)

Returns the average of the dependent value in a group. ``y`` is the dependent
value. ``x`` is the independent value.
::

    SELECT regr_avgy(salary, age)
    FROM (
    VALUES 
        (25, 3000),
        (30, 4000),
        (35, 5000),
        (NULL, 6000),
        (40, NULL)
    ) AS t(age, salary);
    --(4000)

Returns the number of non-null pairs of input values. ``y`` is the dependent
value. ``x`` is the independent value.
::

    SELECT regr_count(salary, age)
    FROM (
    VALUES 
        (25, 3000),
        (30, 4000),
        (35, 5000),
        (NULL, 6000),
        (40, NULL)
    ) AS t(age, salary);
    --(3)

Returns the coefficient of determination of the linear regression. ``y`` is the dependent
value. ``x`` is the independent value.
::

    SELECT regr_r2(salary, age)
    FROM (
    VALUES 
        (25, 3000),
        (30, 4000),
        (35, 5000),
        (NULL, 6000),
        (40, NULL)
    ) AS t(age, salary);
    --(1)

Returns the sum of the product of the dependent and independent values in a group. ``y`` is the dependent
value. ``x`` is the independent value.
::

    SELECT regr_sxy(salary, age)
    FROM (
    VALUES 
        (25, 3000),
        (30, 4000),
        (35, 5000),
        (NULL, 6000),
        (40, NULL)
    ) AS t(age, salary);
    --(10000)

Returns the sum of the squares of the dependent values in a group. ``y`` is the dependent
value. ``x`` is the independent value.
::

    SELECT regr_syy(salary, age)
    FROM (
    VALUES 
        (25, 3000),
        (30, 4000),
        (35, 5000),
        (NULL, 6000),
        (40, NULL)
    ) AS t(age, salary);
    --(2000000)

Returns the sum of the squares of the independent values in a group. ``y`` is the dependent
value. ``x`` is the independent value.
::

    SELECT regr_sxx(salary, age)
    FROM (
    VALUES 
        (25, 3000),
        (30, 4000),
        (35, 5000),
        (NULL, 6000),
        (40, NULL)
    ) AS t(age, salary);
    --(50)

Returns the skewness of all input values.
::

    SELECT skewness(salary)
    FROM (
    VALUES 
        (3000),
        (4000),
        (5000),
        (6000),
        (10000)
    ) AS t(salary);
    --(0.8978957037987336)

This is an alias for :func:`!stddev_samp`.
::

    SELECT stddev(salary)
    FROM (
    VALUES 
        (3000),
        (4000),
        (5000),
        (6000),
        (7000)
    ) AS t(salary);
    --(1581.1388300841897)

Returns the population standard deviation of all input values.
::

    SELECT stddev_pop(salary)
    FROM (
    VALUES 
        (3000),
        (4000),
        (5000),
        (6000),
        (7000)
    ) AS t(salary);
    --(1414.213562373095)

Returns the sample standard deviation of all input values.
::

    SELECT stddev_samp(salary)
    FROM (
    VALUES 
        (3000),
        (4000),
        (5000),
        (6000),
        (7000)
    ) AS t(salary);
    --(1581.1388300841897)

This is an alias for :func:`!var_samp`.
::

    SELECT variance(salary)
    FROM (
    VALUES 
        (3000),
        (4000),
        (5000),
        (6000),
        (7000)
    ) AS t(salary);
    --(2500000.0)

Returns the population variance of all input values.
::

    SELECT var_pop(salary)
    FROM (
    VALUES 
        (3000),
        (4000),
        (5000),
        (6000),
        (7000)
    ) AS t(salary);
    --(2000000.0)

Returns the sample variance of all input values.
::

    SELECT var_samp(salary)
    FROM (
    VALUES 
        (3000),
        (4000),
        (5000),
        (6000),
        (7000)
    ) AS t(salary);
    --(2500000.0)

WITH
    recall_precision AS (
        SELECT
            CLASSIFICATION_RECALL(10000, correct, pred) AS recalls,
            CLASSIFICATION_PRECISION(10000, correct, pred) AS precisions
        FROM
            classification_dataset
    )
SELECT
    recall,
    precision
FROM
    recall_precision
CROSS JOIN UNNEST(recalls, precisions) AS t(recall, precision)

WITH
    recall_precision AS (
        SELECT
            CLASSIFICATION_THRESHOLDS(10000, correct, pred) AS thresholds,
            CLASSIFICATION_RECALL(10000, correct, pred) AS recalls,
            CLASSIFICATION_PRECISION(10000, correct, pred) AS precisions
        FROM
            classification_dataset
    )
SELECT
    threshold,
    recall,
    precision
FROM
    recall_precision
CROSS JOIN UNNEST(thresholds, recalls, precisions) AS t(threshold, recall, precision)

WITH
    fallout_recall AS (
        SELECT
            CLASSIFICATION_FALLOUT(10000, correct, pred) AS fallouts,
            CLASSIFICATION_RECALL(10000, correct, pred) AS recalls
        FROM
            classification_dataset
    )
SELECT
    fallout
    recall,
FROM
    recall_fallout
CROSS JOIN UNNEST(fallouts, recalls) AS t(fallout, recall)

Computes the miss-rate with up to ``buckets`` number of buckets. Returns
an array of miss-rate values.

``y`` should be a boolean outcome value; ``x`` should be predictions, each
between 0 and 1; ``weight`` should be non-negative values, indicating the weight of the instance.

The
`miss-rate <https://en.wikipedia.org/wiki/Type_I_and_type_II_errors#False_positive_and_false_negative_rates>`_
is defined as a sequence whose :math:`j`-th entry is

.. math ::

    {
        \sum_{i \;|\; x_i \leq t_j \bigwedge y_i = 1} \left[ w_i \right]
        \over
        \sum_{i \;|\; x_i \leq t_j \bigwedge y_i = 1} \left[ w_i \right]
        +
        \sum_{i \;|\; x_i > t_j \bigwedge y_i = 1} \left[ w_i \right]
    },

where :math:`t_j` is the :math:`j`-th smallest threshold,
and :math:`y_i`, :math:`x_i`, and :math:`w_i` are the :math:`i`-th
entries of ``y``, ``x``, and ``weight``, respectively.

This function is equivalent to the variant of
:func:`!classification_miss_rate` that takes a ``weight``, with a per-item weight of ``1``.

Computes the fall-out with up to ``buckets`` number of buckets. Returns
an array of fall-out values.

``y`` should be a boolean outcome value; ``x`` should be predictions, each
between 0 and 1; ``weight`` should be non-negative values, indicating the weight of the instance.

The
`fall-out <https://en.wikipedia.org/wiki/Information_retrieval#Fall-out>`_
is defined as a sequence whose :math:`j`-th entry is

.. math ::

    {
        \sum_{i \;|\; x_i > t_j \bigwedge y_i = 0} \left[ w_i \right]
        \over
        \sum_{i \;|\; y_i = 0} \left[ w_i \right]
    },

where :math:`t_j` is the :math:`j`-th smallest threshold,
and :math:`y_i`, :math:`x_i`, and :math:`w_i` are the :math:`i`-th
entries of ``y``, ``x``, and ``weight``, respectively.

This function is equivalent to the variant of
:func:`!classification_fall_out` that takes a ``weight``, with a per-item weight of ``1``.

Computes the precision with up to ``buckets`` number of buckets. Returns
an array of precision values.

``y`` should be a boolean outcome value; ``x`` should be predictions, each
between 0 and 1; ``weight`` should be non-negative values, indicating the weight of the instance.

The
`precision <https://en.wikipedia.org/wiki/Positive_and_negative_predictive_values>`_
is defined as a sequence whose :math:`j`-th entry is

.. math ::

    {
        \sum_{i \;|\; x_i > t_j \bigwedge y_i = 1} \left[ w_i \right]
        \over
        \sum_{i \;|\; x_i > t_j} \left[ w_i \right]
    },

where :math:`t_j` is the :math:`j`-th smallest threshold,
and :math:`y_i`, :math:`x_i`, and :math:`w_i` are the :math:`i`-th
entries of ``y``, ``x``, and ``weight``, respectively.

This function is equivalent to the variant of
:func:`!classification_precision` that takes a ``weight``, with a per-item weight of ``1``.

Computes the recall with up to ``buckets`` number of buckets. Returns
an array of recall values.

``y`` should be a boolean outcome value; ``x`` should be predictions, each
between 0 and 1; ``weight`` should be non-negative values, indicating the weight of the instance.

The
`recall <https://en.wikipedia.org/wiki/Precision_and_recall#Recall>`_
is defined as a sequence whose :math:`j`-th entry is

.. math ::

    {
        \sum_{i \;|\; x_i > t_j \bigwedge y_i = 1} \left[ w_i \right]
        \over
        \sum_{i \;|\; y_i = 1} \left[ w_i \right]
    },

where :math:`t_j` is the :math:`j`-th smallest threshold,
and :math:`y_i`, :math:`x_i`, and :math:`w_i` are the :math:`i`-th
entries of ``y``, ``x``, and ``weight``, respectively.

This function is equivalent to the variant of
:func:`!classification_recall` that takes a ``weight``, with a per-item weight of ``1``.

Computes the thresholds with up to ``buckets`` number of buckets. Returns
an array of threshold values.

``y`` should be a boolean outcome value; ``x`` should be predictions, each
between 0 and 1.

The thresholds are defined as a sequence whose :math:`j`-th entry is the :math:`j`-th smallest threshold.

h(x) = - \int x \log_2\left(f(x)\right) dx,

Returns the approximate log-2 differential entropy from a random variable's sample outcomes. The function internally
creates a reservoir (see [Black2015]_), then calculates the
entropy from the sample results by approximating the derivative of the cumulative distribution
(see [Alizadeh2010]_).

``sample_size`` (``long``) is the maximal number of reservoir samples.

``x`` (``double``) is the samples.

For example, to find the differential entropy of ``x`` of ``data`` using 1000000 reservoir samples, use

.. code-block:: none

    SELECT
        differential_entropy(1000000, x)
    FROM
        data

.. note::

    If :math:`x` has a known lower and upper bound,
    prefer the versions taking ``(bucket_count, x, 1.0, "fixed_histogram_mle", min, max)``,
    or ``(bucket_count, x, 1.0, "fixed_histogram_jacknife", min, max)``,
    as they have better convergence.

Returns the approximate log-2 differential entropy from a random variable's sample outcomes. The function
internally creates a weighted reservoir (see [Efraimidis2006]_), then calculates the
entropy from the sample results by approximating the derivative of the cumulative distribution
(see [Alizadeh2010]_).

``sample_size`` is the maximal number of reservoir samples.

``x`` (``double``) is the samples.

``weight`` (``double``) is a non-negative double value indicating the weight of the sample.

For example, to find the differential entropy of ``x`` with weights ``weight`` of ``data``
using 1000000 reservoir samples, use

.. code-block:: none

     SELECT
         differential_entropy(1000000, x, weight)
     FROM
         data

.. note::

    If :math:`x` has a known lower and upper bound,
    prefer the versions taking ``(bucket_count, x, weight, "fixed_histogram_mle", min, max)``,
    or ``(bucket_count, x, weight, "fixed_histogram_jacknife", min, max)``,
    as they have better convergence.

Returns the approximate log-2 differential entropy from a random variable's sample outcomes. The function
internally creates a conceptual histogram of the sample values, calculates the counts, and
then approximates the entropy using maximum likelihood with or without Jacknife
correction, based on the ``method`` parameter. If Jacknife correction (see [Beirlant2001]_) is used, the
estimate is

.. math ::

    n H(x) - (n - 1) \sum_{i = 1}^n H\left(x_{(i)}\right)

where :math:`n` is the length of the sequence, and :math:`x_{(i)}` is the sequence with the :math:`i`-th element
removed.

``bucket_count`` (``long``) determines the number of histogram buckets.

``x`` (``double``) is the samples.

``method`` (``varchar``) is either ``'fixed_histogram_mle'`` (for the maximum likelihood estimate)
or ``'fixed_histogram_jacknife'`` (for the jacknife-corrected maximum likelihood estimate).

``min`` and ``max`` (both ``double``) are the minimal and maximal values, respectively;
the function will throw if there is an input outside this range.

``weight`` (``double``) is the weight of the sample, and must be non-negative.

For example, to find the differential entropy of ``x``, each between ``0.0`` and ``1.0``,
with weights 1.0 of ``data`` using 1000000 bins and jacknife estimates, use

.. code-block:: none

     SELECT
         differential_entropy(1000000, x, 1.0, 'fixed_histogram_jacknife', 0.0, 1.0)
     FROM
         data

To find the differential entropy of ``x``, each between ``-2.0`` and ``2.0``,
with weights ``weight`` of ``data`` using 1000000 buckets and maximum-likelihood estimates, use

.. code-block:: none

    SELECT
        differential_entropy(1000000, x, weight, 'fixed_histogram_mle', -2.0, 2.0)
    FROM
        data

.. note::

    If :math:`x` doesn't have known lower and upper bounds, prefer the versions taking ``(sample_size, x)``
    (unweighted case) or ``(sample_size, x, weight)`` (weighted case), as they use reservoir
    sampling which doesn't require a known range for samples.

    Otherwise, if the number of distinct weights is low,
    especially if the number of samples is low, consider using the version taking
    ``(bucket_count, x, weight, "fixed_histogram_jacknife", min, max)``, as jacknife bias correction,
    is better than maximum likelihood estimation. However, if the number of distinct weights is high,
    consider using the version taking ``(bucket_count, x, weight, "fixed_histogram_mle", min, max)``,
    as this will reduce memory and running time.

Computes the top frequent values up to ``buckets`` elements approximately.
Approximate estimation of the function enables us to pick up the frequent
values with less memory. Larger ``capacity`` improves the accuracy of
underlying algorithm with sacrificing the memory capacity. The returned
value is a map containing the top elements with corresponding estimated
frequency.

The error of the function depends on the permutation of the values and its
cardinality. We can set the capacity same as the cardinality of the
underlying data to achieve the least error.

``buckets`` and ``capacity`` must be ``bigint``. ``value`` can be numeric
or string type.

The function uses the stream summary data structure proposed in the paper
`Efficient computation of frequent and top-k elements in data streams <https://www.cse.ust.hk/~raywong/comp5331/References/EfficientComputationOfFrequentAndTop-kElementsInDataStreams.pdf>`_ by A.Metwally, D.Agrawal and A.Abbadi.

Computes a new reservoir sample given:

- ``initial_sample``: an initial sample array, or ``NULL`` if creating a new
  sample.
- ``initial_processed_count``: the number of records processed to generate
  the initial sample array. This should be 0 or ``NULL`` if
  ``initital_sample`` is ``NULL``.
- ``values_to_sample``: the column to sample from.
- ``desired_sample_size``: the size of reservoir sample.

The function outputs a single row type with two columns:

#. Processed count: The total number of rows the function sampled
   from. It includes the total from the ``initial_processed_count``,
   if provided.

#. Reservoir sample: An array with length equivalent to the minimum of
   ``desired_sample_size`` and the number of values in the
   ``values_to_sample`` argument.


.. code-block:: sql

    WITH result as (
        SELECT
            reservoir_sample(NULL, 0, col, 5) as reservoir
        FROM (
            VALUES
            1, 2, 3, 4, 5, 6, 7, 8, 9, 0
        ) as t(col)
    )
    SELECT 
        reservoir.processed_count, reservoir.sample
    FROM result;

.. code-block:: none

     processed_count |     sample
    -----------------+-----------------
                  10 | [1, 2, 8, 4, 5]

To merge older samples with new data, supply valid arguments to the
``initial_sample`` argument and ``initial_processed_count`` arguments.

.. code-block:: sql

    WITH initial_sample as (
        SELECT
            reservoir_sample(NULL, 0, col, 3) as reservoir
        FROM (
            VALUES
            0, 1, 2, 3, 4
        ) as t(col)
    ),
    new_sample as (
        SELECT
            reservoir_sample(
                (SELECT reservoir.sample FROM initial_sample), 
                (SELECT reservoir.processed_count FROM initial_sample), 
                col, 
                3
            ) as result
        FROM (
            VALUES
            5, 6, 7, 8, 9
        ) as t(col)
    )
    SELECT 
        result.processed_count, result.sample
    FROM new_sample;

.. code-block:: none

     processed_count |  sample
    -----------------+-----------
                  10 | [8, 3, 2]

To sample an entire row of a table, use a ``ROW`` type input with 
each subfield corresponding to the columns of the source table.

.. code-block:: sql

    WITH result as (
        SELECT
            reservoir_sample(NULL, 0, CAST(row(idx, val) AS row(idx int, val varchar)), 2) as reservoir
        FROM (
            VALUES
            (1, 'a'), (2, 'b'), (3, 'c'), (4, 'd'), (5, 'e')
        ) as t(idx, val)
    )
    SELECT 
        reservoir.processed_count, reservoir.sample
    FROM result;

.. code-block:: none

     processed_count |              sample
    -----------------+----------------------------------
                   5 | [{idx=1, val=a}, {idx=5, val=e}]