Aggregate Functions

Aggregate functions operate on a set of values to compute a single result.

Except for count(), count_if(), max_by(), min_by() and approx_distinct(), all of these aggregate functions ignore null values and return null for no input rows or when all values are null. For example, sum() returns null rather than zero and avg() does not include null values in the count. The coalesce function can be used to convert null into zero.

Some aggregate functions such as array_agg() produce different results depending on the order of input values. This ordering can be specified by writing an ORDER BY Clause within the aggregate function:

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

General Aggregate Functions

any_value(x) [same as input]

This is an alias for arbitrary().

arbitrary(x) [same as input]

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

array_agg(x) array<[same as input]>

Returns an array created from the input x elements.

avg(x) double

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

avg(time interval type) time interval type

Returns the average interval length of all input values.

bool_and(boolean) boolean

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

bool_or(boolean) boolean

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

checksum(x) varbinary

Returns an order-insensitive checksum of the given values.

count(*) bigint

Returns the number of input rows.

count(x) bigint

Returns the number of non-null input values.

count_if(x) bigint

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

every(boolean) boolean

This is an alias for bool_and().

geometric_mean(bigint) double
geometric_mean(double) double
geometric_mean(real) real

Returns the geometric mean of all input values.

max_by(x, y) [same as x]

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

max_by(x, y, n) array<[same as x]>

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

min_by(x, y) [same as x]

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

min_by(x, y, n) array<[same as x]>

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

max(x) [same as input]

Returns the maximum value of all input values.

max(x, n) array<[same as x]>

Returns n largest values of all input values of x.

min(x) [same as input]

Returns the minimum value of all input values.

min(x, n) array<[same as x]>

Returns n smallest values of all input values of x.

reduce_agg(inputValue T, initialState S, inputFunction(S, T, S), combineFunction(S, S, S)) S

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 and associative operation with initialState as the identity 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.

set_agg(x) array<[same as input]>

Returns an array created from the distinct input x elements.

If the input includes NULL, NULL will be included in the returned array.

set_union(array(T)) -> array(T)

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.

Example:

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

Returns ARRAY[1, 2, 3, 4]

sum(x) [same as input]

Returns the sum of all input values.

Bitwise Aggregate Functions

bitwise_and_agg(x) bigint

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

bitwise_or_agg(x) bigint

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

bitwise_xor_agg(x) bigint

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

Map Aggregate Functions

histogram(x)

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

map_agg(key, value)

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

map_union(x(K, V)) -> map(K, V)

Returns the union of all the input maps. If a key is found in multiple input maps, that key’s value in the resulting map comes from an arbitrary input map.

map_union_sum(x(K, V)) -> map(K, V)

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.

multimap_agg(key, value)

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

Approximate Aggregate Functions

approx_distinct(x) bigint

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.

approx_distinct(x, e) bigint

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].

approx_percentile(x, percentage) [same as x]

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.

approx_percentile(x, percentage, accuracy) [same as x]

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.

approx_percentile(x, percentages) array<[same as x]>

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.

approx_percentile(x, percentages, accuracy) array<[same as x]>

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

approx_percentile(x, w, percentage) [same as x]

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.

approx_percentile(x, w, percentage, accuracy) [same as x]

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

approx_percentile(x, w, percentages) array<[same as x]>

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.

approx_percentile(x, w, percentages, accuracy) array<[same as x]>

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

approx_set(x) HyperLogLog

See HyperLogLog Functions.

merge(x) HyperLogLog

See HyperLogLog Functions.

khyperloglog_agg(x) KHyperLogLog

See KHyperLogLog Functions.

merge(qdigest(T)) -> qdigest(T)

See Quantile Digest Functions.

qdigest_agg(x) qdigest<[same as x]>

See Quantile Digest Functions.

qdigest_agg(x, w) qdigest<[same as x]>

See Quantile Digest Functions.

qdigest_agg(x, w, accuracy) qdigest<[same as x]>

See Quantile Digest Functions.

numeric_histogram(buckets, value, weight) map<double, double>

Computes an approximate histogram with up to buckets number of buckets for all values 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.

numeric_histogram(buckets, value) map<double, double>

Computes an approximate histogram with up to buckets number of buckets for all values. This function is equivalent to the variant of 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.

noisy_count_gaussian(x, noise_scale) bigint

Counts the non-null values and then adds a random Gaussian noise with 0 mean and standard deviation of noise_scale to the true count. The noisy count is post-processed to be non-negative and rounded to bigint.

When there are no input rows, this function returns NULL.

Noise is from a secure random.

SELECT noisy_count_gaussian(orderkey, 20.0) FROM tpch.tiny.lineitem WHERE false; -- NULL (1 row)
SELECT noisy_count_gaussian(orderkey, 20.0) FROM tpch.tiny.lineitem WHERE false  GROUP BY orderkey; -- (0 row)
noisy_count_gaussian(x, noise_scale, random_seed) bigint

Counts the non-null values and then adds a random Gaussian noise with 0 mean and standard deviation of noise_scale to the true count. The noisy count is post-processed to be non-negative and rounded to bigint.

When there are no input rows, this function returns NULL.

Random seed is used to seed the random generator. This method does not use a secure random.

SELECT noisy_count_gaussian(orderkey, 20.0, 321) FROM tpch.tiny.lineitem WHERE false; -- NULL (1 row)
SELECT noisy_count_gaussian(orderkey, 20.0, 321) FROM tpch.tiny.lineitem WHERE false  GROUP BY orderkey; --  (0 row)
noisy_count_if_gaussian(x, noise_scale) bigint

Counts the TRUE values and then adds a random Gaussian noise with 0 mean and standard deviation of noise_scale to the true count. The noisy count is post-processed to be non-negative and rounded to bigint.

When there are no input rows, this function returns NULL.

Noise is from a secure random.

SELECT noisy_count_if_gaussian(orderkey > 10, 20.0) FROM tpch.tiny.lineitem WHERE false; -- NULL (1 row)
SELECT noisy_count_if_gaussian(orderkey > 10, 20.0) FROM tpch.tiny.lineitem WHERE false  GROUP BY orderkey; -- (0 row)
noisy_count_if_gaussian(x, noise_scale, random_seed) bigint

Counts the TRUE values and then adds a random Gaussian noise with 0 mean and standard deviation of noise_scale to the true count. The noisy count is post-processed to be non-negative and rounded to bigint.

When there are no input rows, this function returns NULL.

Random seed is used to seed the random generator. This method does not use a secure random.

SELECT noisy_count_if_gaussian(orderkey > 10, 20.0, 321) FROM tpch.tiny.lineitem WHERE false; -- NULL (1 row)
SELECT noisy_count_if_gaussian(orderkey > 10, 20.0, 321) FROM tpch.tiny.lineitem WHERE false  GROUP BY orderkey; --  (0 row)
noisy_sum_gaussian(x, noise_scale) double

Calculates the sum over the input values and then adds random Gaussian noise with 0 mean and standard deviation of noise_scale. All values are converted to double before being added to the sum, and the return type is double.

When there are no input rows, this function returns NULL.

Noise is from a secure random.

SELECT noisy_sum_gaussian(orderkey, 20.0) FROM tpch.tiny.lineitem WHERE false; -- NULL (1 row)
SELECT noisy_sum_gaussian(orderkey, 20.0) FROM tpch.tiny.lineitem WHERE false  GROUP BY orderkey; -- (0 row)
noisy_sum_gaussian(x, noise_scale, random_seed) double

Calculates the sum over the input values and then adds random Gaussian noise with 0 mean and standard deviation of noise_scale. All values are converted to double before being added to the sum, and the return type is double.

When there are no input rows, this function returns NULL.

Random seed is used to seed the random generator. This method does not use a secure random.

SELECT noisy_sum_gaussian(orderkey, 20.0, 321) FROM tpch.tiny.lineitem WHERE false; -- NULL (1 row)
SELECT noisy_sum_gaussian(orderkey, 20.0, 321) FROM tpch.tiny.lineitem WHERE false  GROUP BY orderkey; --  (0 row)
noisy_sum_gaussian(x, noise_scale, lower, upper) double

Calculates the sum over the input values and then adds random Gaussian noise with 0 mean and standard deviation of noise_scale. All values are converted to double before being added to the sum, and the return type is double.

Each value is clipped to the range of [lower, upper] before adding to the sum.

When there are no input rows, this function returns NULL.

Noise is from a secure random.

SELECT noisy_sum_gaussian(orderkey, 20.0, 10.0, 50.0) FROM tpch.tiny.lineitem WHERE false; -- NULL (1 row)
SELECT noisy_sum_gaussian(orderkey, 20.0, 10.0, 51.0) FROM tpch.tiny.lineitem WHERE false  GROUP BY orderkey; -- (0 row)
noisy_sum_gaussian(x, noise_scale, lower, upper, random_seed) double

Calculates the sum over the input values and then adds random Gaussian noise with 0 mean and standard deviation of noise_scale. All values are converted to double before being added to the sum, and the return type is double.

Each value is clipped to the range of [lower, upper] before adding to the sum.

When there are no input rows, this function returns NULL.

Random seed is used to seed the random generator. This method does not use a secure random.

SELECT noisy_sum_gaussian(orderkey, 20.0, 10.0, 50.0, 321) FROM tpch.tiny.lineitem WHERE false; -- NULL (1 row)
SELECT noisy_sum_gaussian(orderkey, 20.0, 10.0, 50.0, 321) FROM tpch.tiny.lineitem WHERE false  GROUP BY orderkey; --  (0 row)
noisy_avg_gaussian(x, noise_scale) double

Calculates the average (arithmetic mean) of all the input values and then adds random Gaussian noise with 0 mean and standard deviation of noise_scale. All values are converted to double before being added to the avg, and the return type is double.

When there are no input rows, this function returns NULL.

Noise is from a secure random.

SELECT noisy_avg_gaussian(orderkey, 20.0) FROM tpch.tiny.lineitem WHERE false; -- NULL (1 row)
SELECT noisy_avg_gaussian(orderkey, 20.0) FROM tpch.tiny.lineitem WHERE false  GROUP BY orderkey; -- (0 row)
noisy_avg_gaussian(x, noise_scale, random_seed) double

Calculates the average (arithmetic mean) of all the input values and then adds random Gaussian noise with 0 mean and standard deviation of noise_scale. All values are converted to double before being added to the avg, and the return type is double.

When there are no input rows, this function returns NULL.

Random seed is used to seed the random generator. This method does not use a secure random.

SELECT noisy_avg_gaussian(orderkey, 20.0, 321) FROM tpch.tiny.lineitem WHERE false; -- NULL (1 row)
SELECT noisy_avg_gaussian(orderkey, 20.0, 321) FROM tpch.tiny.lineitem WHERE false  GROUP BY orderkey; --  (0 row)
noisy_avg_gaussian(x, noise_scale, lower, upper) double

Calculates the average (arithmetic mean) of all the input values and then adds random Gaussian noise with 0 mean and standard deviation of noise_scale. All values are converted to double before being added to the avg, and the return type is double.

Each value is clipped to the range of [lower, upper] before adding to the avg.

When there are no input rows, this function returns NULL.

Noise is from a secure random.

SELECT noisy_avg_gaussian(orderkey, 20.0, 10.0, 50.0) FROM tpch.tiny.lineitem WHERE false; -- NULL (1 row)
SELECT noisy_avg_gaussian(orderkey, 20.0, 10.0, 51.0) FROM tpch.tiny.lineitem WHERE false  GROUP BY orderkey; -- (0 row)
noisy_avg_gaussian(x, noise_scale, lower, upper, random_seed) double

Calculates the average (arithmetic mean) of all the input values and then adds random Gaussian noise with 0 mean and standard deviation of noise_scale. All values are converted to double before being added to the avg, and the return type is double.

Each value is clipped to the range of [lower, upper] before adding to the avg.

When there are no input rows, this function returns NULL.

Random seed is used to seed the random generator. This method does not use a secure random.

SELECT noisy_avg_gaussian(orderkey, 20.0, 10.0, 50.0, 321) FROM tpch.tiny.lineitem WHERE false; -- NULL (1 row)
SELECT noisy_avg_gaussian(orderkey, 20.0, 10.0, 50.0, 321) FROM tpch.tiny.lineitem WHERE false  GROUP BY orderkey; --  (0 row)

Statistical Aggregate Functions

corr(y, x) double

Returns correlation coefficient of input values.

covar_pop(y, x) double

Returns the population covariance of input values.

covar_samp(y, x) double

Returns the sample covariance of input values.

entropy(c) double

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

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

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.

kurtosis(x) double

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

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

where \(\mu\) is the mean, and \(\sigma\) is the standard deviation.

regr_intercept(y, x) double

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

regr_slope(y, x) double

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

regr_avgx(y, x) double

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

regr_avgy(y, x) double

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

regr_count(y, x) double

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

regr_r2(y, x) double

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

regr_sxy(y, x) double

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.

regr_syy(y, x) double

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

regr_sxx(y, x) double

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

skewness(x) double

Returns the skewness of all input values.

stddev(x) double

This is an alias for stddev_samp().

stddev_pop(x) double

Returns the population standard deviation of all input values.

stddev_samp(x) double

Returns the sample standard deviation of all input values.

variance(x) double

This is an alias for var_samp().

var_pop(x) double

Returns the population variance of all input values.

var_samp(x) double

Returns the sample variance of all input values.

Classification Metrics Aggregate Functions

The following functions each measure how some metric of a binary confusion matrix changes as a function of classification thresholds. They are meant to be used in conjunction.

For example, to find the precision-recall curve, use

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)

To get the corresponding thresholds for these values, use

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)

To find the ROC curve, use

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)
classification_miss_rate(buckets, y, x, weight) array<double>

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 is defined as a sequence whose \(j\)-th entry is

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

where \(t_j\) is the \(j\)-th smallest threshold, and \(y_i\), \(x_i\), and \(w_i\) are the \(i\)-th entries of y, x, and weight, respectively.

classification_miss_rate(buckets, y, x) array<double>

This function is equivalent to the variant of classification_miss_rate() that takes a weight, with a per-item weight of 1.

classification_fall_out(buckets, y, x, weight) array<double>

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 is defined as a sequence whose \(j\)-th entry is

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

where \(t_j\) is the \(j\)-th smallest threshold, and \(y_i\), \(x_i\), and \(w_i\) are the \(i\)-th entries of y, x, and weight, respectively.

classification_fall_out(buckets, y, x) array<double>

This function is equivalent to the variant of classification_fall_out() that takes a weight, with a per-item weight of 1.

classification_precision(buckets, y, x, weight) array<double>

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 is defined as a sequence whose \(j\)-th entry is

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

where \(t_j\) is the \(j\)-th smallest threshold, and \(y_i\), \(x_i\), and \(w_i\) are the \(i\)-th entries of y, x, and weight, respectively.

classification_precision(buckets, y, x) array<double>

This function is equivalent to the variant of classification_precision() that takes a weight, with a per-item weight of 1.

classification_recall(buckets, y, x, weight) array<double>

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 is defined as a sequence whose \(j\)-th entry is

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

where \(t_j\) is the \(j\)-th smallest threshold, and \(y_i\), \(x_i\), and \(w_i\) are the \(i\)-th entries of y, x, and weight, respectively.

classification_recall(buckets, y, x) array<double>

This function is equivalent to the variant of classification_recall() that takes a weight, with a per-item weight of 1.

classification_thresholds(buckets, y, x) array<double>

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 \(j\)-th entry is the \(j\)-th smallest threshold.

Differential Entropy Functions

The following functions approximate the binary differential entropy. That is, for a random variable \(x\), they approximate

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

where \(f(x)\) is the partial density function of \(x\).

differential_entropy(sample_size, x)

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

SELECT
    differential_entropy(1000000, x)
FROM
    data

Note

If \(x\) 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.

differential_entropy(sample_size, x, weight)

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

SELECT
    differential_entropy(1000000, x, weight)
FROM
    data

Note

If \(x\) 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.

differential_entropy(bucket_count, x, weight, method, min, max) double

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

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

where \(n\) is the length of the sequence, and \(x_{(i)}\) is the sequence with the \(i\)-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

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

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

Note

If \(x\) 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.

approx_most_frequent(buckets, value, capacity) map<[same as value], bigint>

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 by A.Metwally, D.Agrawal and A.Abbadi.

Reservoir Sample Functions

Reservoir sample functions use a fixed sample size, as opposed to TABLESAMPLE. Fixed sample sizes always result in a fixed total size while still guaranteeing that each record in dataset has an equal probability of being chosen. See [Vitter1985].

reservoir_sample(initial_sample: array(T), initial_processed_count: bigint, values_to_sample: T, desired_sample_size: int) -> row(processed_count: bigint, sample: array(T))

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:

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

  2. 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.

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;
 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.

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;
 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.

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

[Alizadeh2010] (1,2)

Alizadeh Noughabi, Hadi & Arghami, N. (2010). “A New Estimator of Entropy”.

[Beirlant2001]

Beirlant, Dudewicz, Gyorfi, and van der Meulen, “Nonparametric entropy estimation: an overview”, (2001)

[BenHaimTomTov2010]

Yael Ben-Haim and Elad Tom-Tov, “A streaming parallel decision tree algorithm”, J. Machine Learning Research 11 (2010), pp. 849–872.

[Black2015]

Black, Paul E. (26 January 2015). “Reservoir sampling”. Dictionary of Algorithms and Data Structures.

[Efraimidis2006]

Efraimidis, Pavlos S.; Spirakis, Paul G. (2006-03-16). “Weighted random sampling with a reservoir”. Information Processing Letters. 97 (5): 181–185.

[Vitter1985]

Vitter, Jeffrey S. “Random sampling with a reservoir.” ACM Transactions on Mathematical Software (TOMS) 11.1 (1985): 37-57.