Cosine Similarity Calculator

The cosine similarity calculator will teach you all there is to know about the cosine similarity measure, which is widely used in machine learning and other fields of data science.

Read on to discover:

• What the cosine similarity is;
• What the formula for the cosine similarity is;
• Whether the cosine similarity can be negative; and
• How to calculate the cosine similarity in Python.

Here's how to use this cosine similarity calculator:

1. Enter your vectors $\vec{a}$ and $\vec{b}$ into the calculator, one element at a time.

   • More fields will appear as you need them.

   • Empty fields are treated as zeroes.

   • The vectors will automatically be extended to matching lengths.

2. The cosine similarity $\rm S_C$ (and derivative values, like the angle between the vectors, $\theta$, and the cosine distance, $\rm D_C$) are displayed below the vector inputs.

3. The calculations for finding the cosine similarity are shown below the results so that you may understand your specific result.

The cosine similarity measure indicates how similar two vectors are using the cosine of the angle between them. It gives no information on the comparative magnitudes of the vectors.

Cosine similarity is widely used in data analysis and data science, particularly in the field of natural language processing.

It helps to know what the cosine similarity is conceptually, but how do we calculate it? Let's explore the formula.

The cosine similarity between two $N$-dimensional vectors $\vec{a}$ and $\vec{b}$, which is denoted as ${\rm S_C}(\vec{a}, \vec{b})$, is defined as the cosine of the angle between the two vectors, $\theta$:

However, we don't always know the angle $\theta$ — then what? Well, a more complex yet more helpful formula can be derived from the dot product. Let's investigate!

The dot product of the two vectors is denoted as $\vec{a}\cdot\vec{b}$, and is defined as:

where $\Vert\vec{a}\Vert$ is the magnitude of the vector $\vec{a}$ (and similar for $\Vert\vec{b}\Vert$). We can rearrange this equation to become:

And so we have a handy formula for the cosine similarity that doesn't rely on the angle directly:

And what's more, we can rewrite the formula with sums:

Lovely! A formula for the cosine distance that relies only on the known elements of the vectors.

The cosine similarity, $\rm S_C$, falls within the range $[-1, 1]$, which of course, are the limits of the cosine function.

• When the two vectors are in the same direction, $\theta = 0^\circ$ and so $\rm S_C = 1$.

• When the two vectors are orthogonal, $\theta = 90^\circ$ and $\rm S_C = 0$.

• When the two vectors are in opposite directions, $\theta = 180^\circ$ and so the cosine similarity is -1.

Note that we say "similar" and not "identical" — $\rm S_C$ only measures the angle and is not influenced by the comparative magnitudes. $\rm S_C = 1$ only means the two vectors' angles are the same, not that the two vectors are equal. See if you can prove this mathematically!

The cosine similarity is not defined when either vector is a zero-vector — a vector with all elements as zeroes and thus zero magnitude.

To calculate the cosine similarity between two vectors, follow these steps:

1. If you know the angle between the vectors, the cosine similarity is the cosine of that angle.

2. If you don't know the angle, calculate the dot product of the two vectors.

3. Calculate both vectors' magnitudes.

4. Divide the dot product by the product of the magnitudes.

5. The result is the cosine similarity.

Let's look at an example of two 2D vectors and their cosine similarity. Let's use:

• $\vec{a} = [1, 5]$ and
• $\vec{b} = [-1, 3]$.

Already, we can visualize that the two vectors point in the same general direction, i.e., up. We can guess that $\theta < 90^\circ$ and therefore that ${\rm S_C} > 0$, but let's calculate it properly using the formula we learned above.

1. The dot product is:

   $\vec{a}\cdot\vec{b} = 1\cdot (-1) + 5 \cdot 3 = 14$

2. The vectors' magnitudes are:

   $\Vert\vec{a}\Vert = \sqrt{1^2+5^2} = 5.099$

   and

   $\Vert\vec{b}\Vert = \sqrt{(-1)^2+3^2} = 3.162$

3. The cosine similarity is, therefore:

   ${\rm S_C} = (\vec{a}\cdot\vec{b}) / (\Vert\vec{a}\Vert\ \Vert\vec{b}\Vert)$
   $\textcolor{transparent}{\rm S_C} = 14 / (5.099 \cdot 3.162)$
   $\textcolor{transparent}{\rm S_C} = 0.868$

Our guesses were right!

As it's arguably the best language for data science, you might need to calculate the cosine similarity in Python. If you're implementing it yourself, you can use NumPy's dot function for the dot product and the norm function from the numpy.linalg submodule for the vector magnitude. Here's how it might be done:

from numpy import dot
from numpy.linalg import norm

def calc_cosine_similarity(a, b):
    return dot(a,b)/(norm(a)*norm(b))

Then you can call the function as:

a = [1, 1, 1]
b = [3, 4, 5]
calc_cosine_similarity(a, b)
# delivers 0.9797958971132713

The cosine distance is used to measure the dissimilarity between two vectors. It's simply the complement of the cosine similarity, i.e.,

However, the cosine distance is not a true distance metric, because it does not have the triangle inequality property, i.e., the inequality:

does not hold for all possible values of $\vec{a}$, $\vec{b}$, and $\vec{c}$.

Yes, cosine similarity can be negative because the cosine of some angles can be negative. A negative cosine similarity means that the two vectors are more dissimilar than similar and that the angle between them is greater than 90°.

A cosine similarity of -1 means that the two vectors point in opposite directions. This does not mean that their magnitudes are equal, but simply that their angle is 180°.