mirror of
https://github.com/ClickHouse/ClickHouse.git
synced 2024-12-14 18:32:29 +00:00
119 lines
4.4 KiB
Markdown
119 lines
4.4 KiB
Markdown
---
|
|
slug: /en/sql-reference/aggregate-functions/reference/kolmogorovsmirnovtest
|
|
sidebar_position: 156
|
|
sidebar_label: kolmogorovSmirnovTest
|
|
---
|
|
|
|
# kolmogorovSmirnovTest
|
|
|
|
Applies Kolmogorov-Smirnov's test to samples from two populations.
|
|
|
|
**Syntax**
|
|
|
|
``` sql
|
|
kolmogorovSmirnovTest([alternative, computation_method])(sample_data, sample_index)
|
|
```
|
|
|
|
Values of both samples are in the `sample_data` column. If `sample_index` equals to 0 then the value in that row belongs to the sample from the first population. Otherwise it belongs to the sample from the second population.
|
|
Samples must belong to continuous, one-dimensional probability distributions.
|
|
|
|
**Arguments**
|
|
|
|
- `sample_data` — Sample data. [Integer](../../../sql-reference/data-types/int-uint.md), [Float](../../../sql-reference/data-types/float.md) or [Decimal](../../../sql-reference/data-types/decimal.md).
|
|
- `sample_index` — Sample index. [Integer](../../../sql-reference/data-types/int-uint.md).
|
|
|
|
**Parameters**
|
|
|
|
- `alternative` — alternative hypothesis. (Optional, default: `'two-sided'`.) [String](../../../sql-reference/data-types/string.md).
|
|
Let F(x) and G(x) be the CDFs of the first and second distributions respectively.
|
|
- `'two-sided'`
|
|
The null hypothesis is that samples come from the same distribution, e.g. F(x) = G(x) for all x.
|
|
And the alternative is that the distributions are not identical.
|
|
- `'greater'`
|
|
The null hypothesis is that values in the first sample are *stochastically smaller* than those in the second one,
|
|
e.g. the CDF of first distribution lies above and hence to the left of that for the second one.
|
|
Which in fact means that F(x) >= G(x) for all x. And the alternative in this case is that F(x) < G(x) for at least one x.
|
|
- `'less'`.
|
|
The null hypothesis is that values in the first sample are *stochastically greater* than those in the second one,
|
|
e.g. the CDF of first distribution lies below and hence to the right of that for the second one.
|
|
Which in fact means that F(x) <= G(x) for all x. And the alternative in this case is that F(x) > G(x) for at least one x.
|
|
- `computation_method` — the method used to compute p-value. (Optional, default: `'auto'`.) [String](../../../sql-reference/data-types/string.md).
|
|
- `'exact'` - calculation is performed using precise probability distribution of the test statistics. Compute intensive and wasteful except for small samples.
|
|
- `'asymp'` (`'asymptotic'`) - calculation is performed using an approximation. For large sample sizes, the exact and asymptotic p-values are very similar.
|
|
- `'auto'` - the `'exact'` method is used when a maximum number of samples is less than 10'000.
|
|
|
|
|
|
**Returned values**
|
|
|
|
[Tuple](../../../sql-reference/data-types/tuple.md) with two elements:
|
|
|
|
- calculated statistic. [Float64](../../../sql-reference/data-types/float.md).
|
|
- calculated p-value. [Float64](../../../sql-reference/data-types/float.md).
|
|
|
|
|
|
**Example**
|
|
|
|
Query:
|
|
|
|
``` sql
|
|
SELECT kolmogorovSmirnovTest('less', 'exact')(value, num)
|
|
FROM
|
|
(
|
|
SELECT
|
|
randNormal(0, 10) AS value,
|
|
0 AS num
|
|
FROM numbers(10000)
|
|
UNION ALL
|
|
SELECT
|
|
randNormal(0, 10) AS value,
|
|
1 AS num
|
|
FROM numbers(10000)
|
|
)
|
|
```
|
|
|
|
Result:
|
|
|
|
``` text
|
|
┌─kolmogorovSmirnovTest('less', 'exact')(value, num)─┐
|
|
│ (0.009899999999999996,0.37528595205132287) │
|
|
└────────────────────────────────────────────────────┘
|
|
```
|
|
|
|
Note:
|
|
P-value is bigger than 0.05 (for confidence level of 95%), so null hypothesis is not rejected.
|
|
|
|
|
|
Query:
|
|
|
|
``` sql
|
|
SELECT kolmogorovSmirnovTest('two-sided', 'exact')(value, num)
|
|
FROM
|
|
(
|
|
SELECT
|
|
randStudentT(10) AS value,
|
|
0 AS num
|
|
FROM numbers(100)
|
|
UNION ALL
|
|
SELECT
|
|
randNormal(0, 10) AS value,
|
|
1 AS num
|
|
FROM numbers(100)
|
|
)
|
|
```
|
|
|
|
Result:
|
|
|
|
``` text
|
|
┌─kolmogorovSmirnovTest('two-sided', 'exact')(value, num)─┐
|
|
│ (0.4100000000000002,6.61735760482795e-8) │
|
|
└─────────────────────────────────────────────────────────┘
|
|
```
|
|
|
|
Note:
|
|
P-value is less than 0.05 (for confidence level of 95%), so null hypothesis is rejected.
|
|
|
|
|
|
**See Also**
|
|
|
|
- [Kolmogorov-Smirnov'test](https://en.wikipedia.org/wiki/Kolmogorov%E2%80%93Smirnov_test)
|