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How to calculate cosine similarity of array in BigQuery

How can we calculate cosine similarity in BigQuery ?

BigQuery(BQ) is very useful for data analysis or processing.

It is good at handling huge data. It returns summary result in short time.


BQ has one useful data format array.

When we consider array as vector, we may want cosine similarity of vectors.

So how can we get cosine similarity ?

So today I introduce about "How to calculate cosine similarity of array in BigQuery".

Author


Mid-carieer engineer (AI, system). Good at Python and SQL.

Advantage to read

You can understand "How to calculate cosine similarity of array in BigQuery". Then you don't have to concern about similarity.


What is cosine similarity

what

Cosine similarity is a measure of similarity between two vectors.

Cosine cos is one of the trigonometric functions.

It takes 1 in case of 0°, 0 in case of 90° and -1 in case of 180°.

So it means that we can get an angle if we know value of cosine.

And if an angle between 2 vectors is close to zero, it means 2 vectors are similar.

So the cosine is used as similarity.

Cosine similarity is a measure of similarity between two non-zero vectors of an inner product space.

Reference: Cosine similarity - Wikipedia


The formula of cosine similarity is below.


cosine similarity formula


I remember that I used to learn it...



Data

Before calculate it, we should prepare data.

In order to make array data, we can use the table which we create in previous topic.


You can add CREATE TABLE to SQL, and create table.

SQL

CREATE TABLE test.array_sample2
AS
SELECT
key,
ARRAY_AGG(val) as val_array
FROM test.array_sample
GROUP BY key

Result table

Row key val_array
1 a 1
2
3
2 b 2
4
5
3 c 3
2
-1



How to calculate cosine similarity of array in BigQuery

how

In order to calculate cosine similarity of array in BigQuery, we should calculate product between elements of vectors.

SQL is below.

SQL

SELECT
  t1.key AS key1,
  t2.key AS key2,
  (
  SELECT
    SUM(value1 * value2)/ SQRT(SUM(value1 * value1))/ SQRT(SUM(value2 * value2))
  FROM
    UNNEST(t1.val_array) AS value1
  WITH
  OFFSET
    pos1
  JOIN
    UNNEST(t2.val_array) AS value2
  WITH
  OFFSET
    pos2
  ON
    pos1 = pos2 ) AS cosine_similarity
FROM
  test.array_sample2 AS t1,
  test.array_sample2 AS t2
ORDER BY
  key1,
  key2,
  cosine_similarity


It decomposed array to each element bt UNNEST(array), and added order number by WITH OFFSET pos.

Then use pos as join key for multiplication of each element.


Result is below.

Result of SQL

Row key1 key2 cosine_similarity
1 a a 1
2 a b 0.9960238411
3 a c 0.2857142857
4 b a 0.9960238411
5 b b 1
6 b c 0.3585685828
7 c a 0.2857142857
8 c b 0.3585685828
9 c c 1


Vector a and b are similar. So similarity is high.

Vector c faces different direction. So similarity is low.




Conclusion

Today I explained about "How to calculate cosine similarity of array in BigQuery".

In order to calculate cosine similarity of array, we can take solution below.

Point

  • Decompose array by "UNNEST(array)"
  • Add number to array element by "WITH OFFSET pos"
  • Calculate cosine similarity by multiplication between each array element

  • Using UNNEST(array) is little difficult.



    There are some other articles about BigQuey.

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