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| '''Slope One''' is a family of algorithms used for [[collaborative filtering]], introduced in a 2005 paper by Daniel Lemire and Anna Maclachlan.<ref name="lemire2005">Daniel Lemire, Anna Maclachlan, [http://arxiv.org/abs/cs/0702144 Slope One Predictors for Online Rating-Based Collaborative Filtering], In SIAM Data Mining (SDM'05), Newport Beach, California, April 21–23, 2005.</ref> Arguably, it is the simplest form of non-trivial item-based collaborative filtering based on ratings. Their simplicity makes it especially easy to implement them efficiently while their accuracy is often on par with more complicated and computationally expensive algorithms.<ref name="lemire2005" /><ref>Fidel Cacheda, Victor Carneiro, Diego Fernandez, and Vreixo Formoso. 2011. [http://portal.acm.org/citation.cfm?id=1921593 Comparison of collaborative filtering algorithms: Limitations of current techniques and proposals for scalable, high-performance recommender systems]. ACM Trans. Web 5, 1, Article 2</ref> They have also been used as building blocks to improve other algorithms.<ref>Pu Wang, HongWu Ye, [http://doi.ieeecomputersociety.org/10.1109/IIS.2009.71 A Personalized Recommendation Algorithm Combining Slope One Scheme and User Based Collaborative Filtering], IIS '09, 2009.</ref><ref>DeJia Zhang, [http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=5209738&isnumber=5209636 An Item-based Collaborative Filtering Recommendation Algorithm Using Slope One Scheme Smoothing], ISECS '09, 2009.</ref><ref>Min Gaoa, Zhongfu Wub, and Feng Jiang, Userrank for item-based collaborative filtering recommendation, Information Processing Letters Volume 111, Issue 9, 1 April 2011, pp. 440-446.</ref><ref>Mi, Zhenzhen and Xu, Congfu, A Recommendation Algorithm Combining Clustering Method and Slope One Scheme, Lecture Notes in Computer Science 6840, 2012, pp. 160-167.</ref><ref>Sun, Z., Luo, N., Kuang, W., One real-time personalized recommendation systems based on Slope One algorithm, FSKD 2011, 3, art. no. 6019830, 2012 pp. 1826-1830.</ref><ref>Gao, M., Wu, Z., Personalized context-aware collaborative filtering based on neural network and slope one, LNCS 5738, 2009, pp. 109-116</ref> They are part of major open-source libraries such as [[Apache Mahout]] and [[Easyrec]].
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| ==Item-based collaborative filtering of rated resources and overfitting==
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| When ratings of items are available, such as is the case when people are given the option of ratings resources (between 1 and 5, for example), collaborative filtering aims to predict the ratings of one individual based on his past ratings and on a (large) database of ratings contributed by other users.
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| '''Example''': Can we predict the rating an individual would give to the new Celine Dion album given that he gave the Beatles 5 out of 5?
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| In this context, item-based collaborative filtering <ref>Slobodan Vucetic, Zoran Obradovic: Collaborative Filtering Using a Regression-Based Approach. Knowl. Inf. Syst. 7(1): 1-22 (2005)</ref><ref>Badrul M. Sarwar, George Karypis, Joseph A. Konstan, John Riedl: Item-based collaborative filtering recommendation algorithms. WWW 2001: 285-295</ref> predicts the ratings on one item based on the ratings on another item, typically using [[linear regression]] (<math>f(x)=ax+b</math>). Hence, if there are 1,000 items, there could be up to 1,000,000 linear regressions to be learned, and so, up to 2,000,000 regressors. This approach may suffer from severe [[overfitting]]<ref name="lemire2005" /> unless we select only the pairs of items for which several users have rated both items.
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| A better alternative may be to learn a simpler predictor such as <math>f(x)=x+b</math>: experiments show that this simpler predictor (called Slope One) sometimes outperforms<ref name="lemire2005"/> linear regression while having half the number of regressors. This simplified approach also reduces storage requirements and latency.
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| Item-based collaborative is just one form of [[collaborative filtering]]. Other alternatives include user-based collaborative filtering where relationships between users are of interest, instead. However, item-based collaborative filtering is especially scalable with respect to the number of users.
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| ==Item-based collaborative filtering of purchase statistics==
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| We are not always given ratings: when the users provide only binary data (the item was purchased or not), then Slope One and other
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| rating-based algorithm do not apply{{citation_needed|date=August 2012}}.
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| Examples of binary item-based collaborative filtering include Amazon's [http://doi.ieeecomputersociety.org/10.1109/MIC.2003.1167344 item-to-item] patented algorithm<ref>Greg Linden, Brent Smith, Jeremy York, "Amazon.com Recommendations: Item-to-Item Collaborative Filtering," IEEE Internet Computing, vol. 07, no. 1, pp. 76-80, Jan/Feb, 2003</ref> which computes the cosine between binary vectors representing the purchases in a user-item matrix.
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| Being arguably simpler than even Slope One, the Item-to-Item algorithm offers an interesting point of reference. Let us consider an example.
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| {| class="wikitable" style="text-align:center"
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| |+Sample purchase statistics
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| ! Customer !! Item 1 !! Item 2 !! Item 3
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| | John || Bought it || Didn't buy it || Bought it
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| | Mark || Didn't buy it || Bought it || Bought it
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| | Lucy || Didn't buy it || Bought it || Didn't buy it
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| |}
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| In this case, the cosine between items 1 and 2 is:
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| <math>\frac{(1,0,0)\cdot (0,1,1) }{ \Vert (1,0,0)\Vert \Vert (0,1,1)\Vert }= 0</math>,
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| The cosine between items 1 and 3 is:
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| <math>\frac{(1,0,0)\cdot (1,1,0) }{ \Vert (1,0,0)\Vert \Vert (1,1,0)\Vert }= \frac{1}{\sqrt{2}}</math>,
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| Whereas the cosine between items 2 and 3 is:
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| <math>\frac{(0,1,1)\cdot (1,1,0)}{ \Vert (0,1,1)\Vert \Vert (1,1,0)\Vert }= \frac{1}{2}</math>.
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| Hence, a user visiting item 1 would receive item 3 as a recommendation, a user visiting item 2 would receive item 3 as a recommendation, and finally, a user visiting item 3 would receive item 1 (and then item 2) as a recommendation. The model uses a single parameter per pair of item (the cosine) to make the recommendation. Hence, if there are ''n'' items, up to ''n(n-1)/2'' cosines need to be computed and stored.
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| ==Slope one collaborative filtering for rated resources==
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| To drastically reduce [[overfitting]], improve performance and ease implementation, the '''Slope One''' family of easily implemented Item-based Rating-Based [[collaborative filtering]] algorithms was proposed. Essentially, instead of using linear regression from one item's ratings to another item's ratings (<math>f(x)=ax+b</math>), it uses a simpler form of regression with a single free parameter (<math>f(x)=x+b</math>). The free parameter is then simply the average difference between the two items' ratings. It was shown to be much more accurate than linear regression in some instances,<ref name="lemire2005" /> and it takes half the storage or less.
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| [[Image:Simplicity diagram.png]] | |
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| '''Example''':
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| # User A gave a 1 to Item I and an 1.5 to Item J.
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| # User B gave a 2 to Item I.
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| # How do you think User B rated Item J?
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| # The Slope One answer is to say 2.5 (1.5-1+2=2.5).
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| For a more realistic example, consider the following table.
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| {| class="wikitable" style="text-align:center"
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| |+Sample rating database
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| ! Customer !! Item A !! Item B !! Item C
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| | John || 5 || 3 || 2
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| | Mark || 3 || 4 || Didn't rate it
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| | Lucy || Didn't rate it || 2 || 5
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| |}
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| In this case, the average difference in ratings between item B and A is (2+(-1))/2=0.5. Hence, on average, item A is rated above item B by 0.5. Similarly, the average difference between item C and A is 3. Hence, if we attempt to predict the rating of Lucy for item A using her rating for item B, we get 2+0.5 = 2.5. Similarly, if we try to predict her rating for item A using her rating of item C, we get 5+3=8.
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| If a user rated several items, the predictions are simply combined using a weighted average where a good choice for the weight is the number of users having rated both items. In the above example, we would predict the following rating for Lucy on item A:
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| <math>\frac{2 \times 2.5 + 1 \times 8 }{2+1} = \frac{13 }{3} = 4.33</math>
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| Hence, given ''n'' items, to implement Slope One, all that is needed is to compute and store the average differences and the number of common ratings for each of the ''n''<sup>2</sup> pairs of items.
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| ==Algorithmic complexity of Slope One==
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| Suppose there are ''n'' items, ''m'' users, and ''N'' ratings. Computing the average rating differences for each pair of items requires up to ''n(n-1)/2'' units of storage, and up to ''m n''<sup>2</sup> time steps. This computational bound may be pessimistic: if we assume that users have rated up to ''y'' items, then it is possible to compute the differences in no more than ''n''<sup>2</sup>+''my''<sup>2</sup>. If a user has entered ''x'' ratings, predicting a single rating requires ''x'' time steps, and predicting all of his missing ratings requires up to (''n-x'')''x'' time steps. Updating the database when a user has already entered ''x'' ratings, and enters a new one, requires ''x'' time steps.
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| It is possible to reduce storage requirements by partitioning the data (see [[Partition (database)]]) or by using sparse storage: pairs of items having no (or few) corating users can be omitted.
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| == Footnotes ==
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| {{reflist}}
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| [[Category:Collaboration]]
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