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#
- 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 isTRUE
, otherwiseFALSE
.
- bool_or(boolean) boolean #
Returns
TRUE
if any input value isTRUE
, otherwiseFALSE
.
- 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 tocount(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 ofy
over all input values.
- max_by(x, y, n) array<[same as x]> #
Returns
n
values ofx
associated with then
largest of all input values ofy
in descending order ofy
.
- min_by(x, y) [same as x] #
Returns the value of
x
associated with the minimum value ofy
over all input values.
- min_by(x, y, n) array<[same as x]> #
Returns
n
values ofx
associated with then
smallest of all input values ofy
in ascending order ofy
.
- 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 ofx
.
- 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 ofx
.
- 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, initiallyinitialState
, 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 ifinitialState
is NULL. The behavior is undefined ifinputFunction
orcombineFunction
return a NULL.Take care when designing
initialState
,inputFunction
andcombineFunction
. 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 thatcombineFunction
is commutative and associative operation withinitialState
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. IfNULL
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 thate
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 givenpercentage
. The value ofpercentage
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 ofaccuracy
. The value ofaccuracy
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 is0.01
.
- 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 thepercentages
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 ofaccuracy
.
- approx_percentile(x, w, percentage) [same as x] #
Returns the approximate weighed percentile for all input values of
x
using the per-item weightw
at the percentagep
. The weight must be an integer value of at least one. It is effectively a replication count for the valuex
in the percentile set. The value ofp
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 ofaccuracy
.
- approx_percentile(x, w, percentages) array<[same as x]> #
Returns the approximate weighed percentile for all input values of
x
using the per-item weightw
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 valuex
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 ofaccuracy
.
- approx_set(x) HyperLogLog
- merge(x) HyperLogLog
- khyperloglog_agg(x) KHyperLogLog
- merge(qdigest(T)) -> qdigest(T)
- qdigest_agg(x) qdigest<[same as x]>
- qdigest_agg(x, w) qdigest<[same as x]>
- qdigest_agg(x, w, accuracy) qdigest<[same as x]>
- numeric_histogram(buckets, value, weight) map<double, double> #
Computes an approximate histogram with up to
buckets
number of buckets for allvalue
s with a per-item weight ofweight
. 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 abigint
.value
andweight
must be numeric.
- numeric_histogram(buckets, value) map<double, double> #
Computes an approximate histogram with up to
buckets
number of buckets for allvalue
s. This function is equivalent to the variant ofnumeric_histogram()
that takes aweight
, with a per-item weight of1
. 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 abigint
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.
- 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
, andweight
, respectively.
- classification_miss_rate(buckets, y, x) array<double> #
This function is equivalent to the variant of
classification_miss_rate()
that takes aweight
, with a per-item weight of1
.
- 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
, andweight
, respectively.
- classification_fall_out(buckets, y, x) array<double> #
This function is equivalent to the variant of
classification_fall_out()
that takes aweight
, with a per-item weight of1
.
- 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
, andweight
, respectively.
- classification_precision(buckets, y, x) array<double> #
This function is equivalent to the variant of
classification_precision()
that takes aweight
, with a per-item weight of1
.
- 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
, andweight
, respectively.
- classification_recall(buckets, y, x) array<double> #
This function is equivalent to the variant of
classification_recall()
that takes aweight
, with a per-item weight of1
.
- 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
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
ofdata
using 1000000 reservoir samples, useSELECT 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 weightsweight
ofdata
using 1000000 reservoir samples, useSELECT 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
andmax
(bothdouble
) 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 between0.0
and1.0
, with weights 1.0 ofdata
using 1000000 bins and jacknife estimates, useSELECT 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
and2.0
, with weightsweight
ofdata
using 1000000 buckets and maximum-likelihood estimates, useSELECT 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. Largercapacity
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
andcapacity
must bebigint
.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.
Beirlant, Dudewicz, Gyorfi, and van der Meulen, “Nonparametric entropy estimation: an overview”, (2001)
Yael Ben-Haim and Elad Tom-Tov, “A streaming parallel decision tree algorithm”, J. Machine Learning Research 11 (2010), pp. 849–872.
Black, Paul E. (26 January 2015). “Reservoir sampling”. Dictionary of Algorithms and Data Structures.
Efraimidis, Pavlos S.; Spirakis, Paul G. (2006-03-16). “Weighted random sampling with a reservoir”. Information Processing Letters. 97 (5): 181–185.