## Can someone give an example of cosine similarity, in a very simple, graphical way?

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Cosine Similarity article on Wikipedia

Can you show the vectors here (in a list or something) and then do the math, and let us see how it works?

I'm a beginner.

Here are two very short texts to compare:

`Julie loves me more than Linda loves me`

`Jane likes me more than Julie loves me`

We want to know how similar these texts are, purely in terms of word counts (and ignoring word order). We begin by making a list of the words from both texts:

me Julie loves Linda than more likes Jane

Now we count the number of times each of these words appears in each text:

me 2 2 Jane 0 1 Julie 1 1 Linda 1 0 likes 0 1 loves 2 1 more 1 1 than 1 1

We are not interested in the words themselves though. We are interested only in those two vertical vectors of counts. For instance, there are two instances of 'me' in each text. We are going to decide how close these two texts are to each other by calculating one function of those two vectors, namely the cosine of the angle between them.

The two vectors are, again:

a: [2, 0, 1, 1, 0, 2, 1, 1] b: [2, 1, 1, 0, 1, 1, 1, 1]

The cosine of the angle between them is about 0.822.

These vectors are 8-dimensional. A virtue of using cosine similarity is clearly that it converts a question that is beyond human ability to visualise to one that can be. In this case you can think of this as the angle of about 35 degrees which is some 'distance' from zero or perfect agreement.

**Cosine Similarity,** For example, two term-frequency vectors may have many 0 values in common, Cosine similarity is a measure of similarity that can be used to compare documents or, A simple variation of cosine similarity for the preceding scenario is arguments, but the most important one is the method used for computing distances. Can someone give an example of cosine similarity, in a very simple, graphical way? (7) Cosine Similarity article on Wikipedia. Can you show the vectors here (in a list or something) and then do the math, and let us see how it works? I'm a beginner.

I'm guessing you are more interested in getting some insight into "*why*" the cosine similarity works (why it provides a good indication of similarity), rather than "*how*" it is calculated (the specific operations used for the calculation). If your interest is in the latter, see the reference indicated by Daniel in this post, as well as a related SO Question.

To explain both the how and even more so the why, it is useful, at first, to simplify the problem and to work only in two dimensions. Once you get this in 2D, it is easier to think of it in three dimensions, and of course harder to imagine in many more dimensions, but by then we can use linear algebra to do the numeric calculations and also to help us think in terms of lines / vectors / "planes" / "spheres" in n dimensions, even though we can't draw these.

So, **in two dimensions**: with regards to text similarity this means that we would focus on two distinct terms, say the words "London" and "Paris", and we'd count how many times each of these words is found in each of the two documents we wish to compare. This gives us, for each document, a point in the the x-y plane. For example, if Doc1 had Paris once, and London four times, a point at (1,4) would present this document (with regards to this diminutive evaluation of documents). Or, speaking in terms of vectors, this Doc1 document would be an arrow going from the origin to point (1,4). With this image in mind, let's think about what it means for two documents to be similar and how this relates to the vectors.

VERY similar documents (again with regards to this limited set of dimensions) would have the very same number of references to Paris, AND the very same number of references to London, or maybe, they could have the same ratio of these references. A Document, Doc2, with 2 refs to Paris and 8 refs to London, would also be very similar, only with maybe a longer text or somehow more repetitive of the cities' names, but in the same proportion. Maybe both documents are guides about London, only making passing references to Paris (and how uncool that city is ;-) Just kidding!!!.

Now, less similar documents may also include references to both cities, but in different proportions. Maybe Doc2 would only cite Paris once and London seven times.

**Back to our x-y plane,** if we draw these hypothetical documents, we see that when they are VERY similar, their vectors overlap (though some vectors may be longer), and as they start to have less in common, these vectors start to diverge, to have a wider angle between them.

By measuring **the angle between the vectors, we can get a good idea of their similarity**, and to make things even easier, by taking the *Cosine* of this angle, we have a nice 0 to 1 or -1 to 1 value that is indicative of this similarity, depending on what and how we account for. The smaller the angle, the bigger (closer to 1) the cosine value, and also the higher the similarity.

At the extreme, if Doc1 only cites Paris and Doc2 only cites London, the documents have absolutely nothing in common. Doc1 would have its vector on the x-axis, Doc2 on the y-axis, the angle 90 degrees, Cosine 0. In this case we'd say that these documents are orthogonal to one another.

**Adding dimensions**:
With this intuitive feel for similarity expressed as a small angle (or large cosine), we can now imagine things in 3 dimensions, say by bringing the word "Amsterdam" into the mix, and visualize quite well how a document with two references to each would have a vector going in a particular direction, and we can see how this direction would compare to a document citing Paris and London three times each, but not Amsterdam, etc. As said, we can try and imagine the this fancy space for 10 or 100 cities. It's hard to draw, but easy to conceptualize.

I'll wrap up just by saying **a few words about the formula itself**. As I've said, other references provide good information about the calculations.

First in two dimensions. The formula for the Cosine of the angle between two vectors is derived from the trigonometric difference (between angle a and angle b):

cos(a - b) = (cos(a) * cos(b)) + (sin (a) * sin(b))

This formula looks very similar to the dot product formula:

Vect1 . Vect2 = (x1 * x2) + (y1 * y2)

where `cos(a)`

corresponds to the `x`

value and `sin(a)`

the `y`

value, for the first vector, etc. The only problem, is that `x`

, `y`

, etc. are not exactly the `cos`

and `sin`

values, for these values need to be read on the unit circle. That's where the denominator of the formula kicks in: by dividing by the product of the length of these vectors, the `x`

and `y`

coordinates become normalized.

**Cosine Similarity,** This is documentation for the Graph Algorithms Library, which has been deprecated We can use the Cosine Similarity algorithm to work out the similarity between two things. When calling the function, we should provide lists that contain the For example, the person most similar to Zhen is Michael, but the person most Can someone shoe me how to work out cosine similarity please? I understand that someone has answered a similar question beforesimilar question link but i do not understand how the end result was r

Here's my implementation in C#.

using System; namespace CosineSimilarity { class Program { static void Main() { int[] vecA = {1, 2, 3, 4, 5}; int[] vecB = {6, 7, 7, 9, 10}; var cosSimilarity = CalculateCosineSimilarity(vecA, vecB); Console.WriteLine(cosSimilarity); Console.Read(); } private static double CalculateCosineSimilarity(int[] vecA, int[] vecB) { var dotProduct = DotProduct(vecA, vecB); var magnitudeOfA = Magnitude(vecA); var magnitudeOfB = Magnitude(vecB); return dotProduct/(magnitudeOfA*magnitudeOfB); } private static double DotProduct(int[] vecA, int[] vecB) { // I'm not validating inputs here for simplicity. double dotProduct = 0; for (var i = 0; i < vecA.Length; i++) { dotProduct += (vecA[i] * vecB[i]); } return dotProduct; } // Magnitude of the vector is the square root of the dot product of the vector with itself. private static double Magnitude(int[] vector) { return Math.Sqrt(DotProduct(vector, vector)); } } }

**Cosine similarity,** The graph given below shows this astronomical growth. A simple trick is to divide the term frequency by the total number of terms. they have a lower score compared to the other terms that appear in only one document. This angle can be used as the basis for a similarity metric in the same way that the Can someone give an example of cosine similarity, in a very simple, graphical way? Calculate cosine similarity given 2 sentence strings English

For simplicity I am reducing the vector a and b:

Let : a : [1, 1, 0] b : [1, 0, 1]

Then cosine similarity (Theta):

(Theta) = (1*1 + 1*0 + 0*1)/sqrt((1^2 + 1^2))* sqrt((1^2 + 1^2)) = 1/2 = 0.5

then inverse of cos 0.5 is 60 degrees.

**9.5.2. The Cosine Similarity algorithm,** It has been a long time since I wrote the TF-IDF tutorial (Part I and Part II) and See an example of a dot product for two vectors with 2 dimensions each (2D): This is all very simple and easy to understand, but what is a dot product ? Cosine Similarity will generate a metric that says how related are two Doc2vec+cosine similarity was supposed to be my next approach if tfidf+cosine similarity didnt work considering the fact that doc2vec will take the semantic nature of the document into account .None the less will definitely give it a try as far as naive bayes is concerned , the labels are not correct making it an unsupervised problem(so dont really understand how naive bayes will perform

This Python code is my quick and dirty attempt to implement the algorithm:

import math from collections import Counter def build_vector(iterable1, iterable2): counter1 = Counter(iterable1) counter2 = Counter(iterable2) all_items = set(counter1.keys()).union(set(counter2.keys())) vector1 = [counter1[k] for k in all_items] vector2 = [counter2[k] for k in all_items] return vector1, vector2 def cosim(v1, v2): dot_product = sum(n1 * n2 for n1, n2 in zip(v1, v2) ) magnitude1 = math.sqrt(sum(n ** 2 for n in v1)) magnitude2 = math.sqrt(sum(n ** 2 for n in v2)) return dot_product / (magnitude1 * magnitude2) l1 = "Julie loves me more than Linda loves me".split() l2 = "Jane likes me more than Julie loves me or".split() v1, v2 = build_vector(l1, l2) print(cosim(v1, v2))

**Tf-Idf and Cosine similarity,** See also this very good answer: Can someone give an example of cosine similarity, in a very simple, graphical way? There are other ways to text - Can someone give an example of cosine similarity, in a very simple, graphical way? Cosine Similarity article on Wikipedia Can you show the vectors here(in a list or something) and then do the math, and let us see how it works? I'm a beginner.…

**Machine Learning :: Cosine Similarity for Vector Space Models (Part ,** For multivariate numeric data, you can compute the cosine similarity of the Vectors that are very similar to each other have a cosine similarity that If you look at the previous graph of vectors and think that vector A is In a future article, I'll provide a simple example that demonstrates how you can use the The cosine similarity can be seen as * a method of normalizing document length during comparison. * * In the case of information retrieval, the cosine similarity of two * documents will range from 0 to 1, since the term frequencies (tf-idf * weights) cannot be negative. The angle between two term frequency vectors * cannot be greater than 90°.

**How is cosine similarity different from semantic similarity in relation ,** Finding the cosine similarity of the text Cosine similarity measures the distance between two vectors. This technique creates a vector that represents the number of elements found in a string.

**64 Cosine Similarity Example,** What do you mean by “textual similarity” or “semantic similarity”? Cosine similarity - Wikipedia is just a way to construct a measure for similarity.

##### Comments

- Try picking up a copy of Geometry and Meaning by Widdows (press.uchicago.edu/presssite/…), I read it through a while back and wished I had it a number of years ago, great introductory text.
- This is exactly what I was looking for. Exactly. Is this considered the simplest form of "vector space model"?
- I am really glad this was useful to you, Alex. Sorry for the delay in responding. I haven't visited StackOverflow in a while. Actually this is an example of an "inner product space". There's a basic discussion on wikipedia.
- Is there any way to normalize for document length?
- You have to use length normalization and before that, try to use log frequency weighting on all term vectors. If your already dealing with normalized vectors, then it's the dot product of A.B.
- More detailed example with use of length normalization and TF-IDF: site.uottawa.ca/~diana/csi4107/cosine_tf_idf_example.pdf
- this is awesome thank you I loved how you explained Magnitude =)
- That's great but what if we are working with files or strings.
- Can you explain why you used set in the line "all_items = set(counter1.keys()).union(set(counter2.keys()))".
- @Ghos3t , that is to get list of distinct words from both documents
- It's not a "simple, graphical way" but yet just code. Although others have made the same error too :/