How to calculate the cosine similarity

Cosine similarity is a measure used to determine how similar two vectors are, often in the context of text mining and recommendation systems. Its formula is given as:

where$\Vert{A}\Vert$and$\Vert{B}\Vert$represent the magnitudes of arbitrary vectors$\Vert{A}\Vert$and$\Vert{B}\Vert.$

To further understand this formula, let's solve a simple example.

Example

Suppose we have two vectors,$A$and$B$defined as:

Let's start off by computing the dot product of these vectors.

Step 1: Compute the dot product of vectors

The dot product of any two vectors can be calculated by multiplying each vectors’ element and summing them, which can mathematically be written as:

Here, the dot product can be calculated as:

Next, we will calculate the magnitudes of the vectors$A$and$B.$

Step 2: Compute vector magnitudes

Now, we will evaluate the magnitude of each vector. The magnitude of an arbitrary vector$A$is calculated as the square root of the sum of the squares of its elements, mathematically represented as:

Here, the magnitudes of vectors$A$and$B$can be written as:

Step 3: Compute final answer

Finally, we can calculate the cosine similarity between the vectors$A$and$B,$which turns out to be:

Note: The value of the cosine similarity varies from $-1$ to $1$ where:

• $1$ implies that the vectors are in the same direction.

• $0$ means that the vectors are orthogonal (no similarity).

• $-1$ implies that the vectors are in opposite directions.

Code

The example computed above can be implemented in Python using the numpy library as shown below:

Code explanation

In the code above:

• Line 1: Importing the numpy package for performing the relevant mathematical operations.

• Lines 4–5: Initializing vectors A and B.

• Line 8: Computing the dot product between the vectors A and B.

• Lines 11–12: Computing the magnitudes of the vectors A and B.

• Lines 15–16: Calculating the cosine similarity by taking the magnitudes and dot product of the vectors A and B calculated above and printing the result.

Conclusion

In conclusion, evaluating this metric enables comparisons and clustering of data points in a multidimensional space. Its intuitive interpretation, ranging from $-1$ (opposite directions) to $1$ (same direction), makes it a valuable method for measuring the likeness or similarity between various items or documents.