Choice between an adjusted cosine similarity vs regular cosine similarity
<p>I'm working on an item-based CF which uses an adjusted cosine similarity. I have recently added a regular cosine similarity and I got totally different results. Now my question is which fits better considering my data? </p>

<p>This is a possible scenario of users, items and ratings</p>

<pre><code>         User 1 | User 2 | User 3 | User 4 | User 5
Item 1 |   5    |    1   |   1    |   5    |   5
Item 2 |   5    |    1   |   2    |   4    |   5
Item 3 |   1    |    5   |   4    |   2    |   3
</code></pre>

<p>Considering this data, you'd conclude that item 1 and item 2 are relatively 'similar'. Here are the results of the different similarity coefficients:</p>

<p>Similarity between Item 1 and Item 2<br>
<code>Adjusted cosine similarity = 0.865</code><br>
<code>Regular cosine similarity = 0.987</code><br>
<em>I rounded them off for this example</em></p>

<p>You can see this is basically the same, but when you try to calculate a similarity between Item 2 and 3 (Which aren't similar at all) it results in totally different results:</p>

<p>Similarity between Item 2 and Item 3<br>
<code>Adjusted cosine similarity = -0.955</code><br>
<code>Regular cosine similarity = 0.656</code><br>
<em>I rounded them off for this example</em></p>

<p>Which of these would be 'better'? I assume using an adjusted cosine similarity works better since it take the average rating of the user into account, but why would a regular cosine similarity result in a positive number for such 'different' items? Should I just refrain from using the regular cosine similarity in general or only for certain scenarios?</p>

<p>Any help would be appreciated!</p>