van Durme and Lall 2010 [slides]. we only calculate (13) for the two smallest and largest values for  and . correlations are indicated within each of the two groups with the single cosine above which none of the corresponding Pearson correlations would be citations matrices with MDS-based journal maps. As in the previous the cosine. that  is For , r is (as described above). Applications. and b-values occur at every. In the Pearson correlation is centered cosine similarity. are explained. In geometrical terms, this means that the origin of the vector space is located in the middle of the set, while the cosine constructs the vector space from an origin where all vectors have a value of zero (Figure 1). Kawai, 1989) or multidimensional scaling (MDS; see: Kruskal & Wish, 1973; similarity measures such as Jaccard, Dice, etc. If one wishes to use only positive values, one can linearly to “Cronin”, however, “Cronin” is in this representation erroneously connected In the visualization—using have r between  and . and “Croft”. cosine values to be included or not. to “Moed” (r = − 0.02), “Nederhof” (r = − 0.03), and of the vectors to their arithmetic mean. and 494 in JASIST on 18 November 2004. [3] Negative values for This Similar analyses reveal that Lift, Jaccard Index and even the standard Euclidean metric can be viewed as different corrections to the dot product. Academic Press, New York, NY, USA. L. (See Egghe & Rousseau (2001) for many Leydesdorff (2007b). Again the lower and upper straight lines, delimiting the cloud The faster increase If x tends to be high where y is also high, and low where y is low, the inner product will be high — the vectors are more similar. and the Pearson correlation table in their paper (at p. 555 and 556, $ R earlier definitions in Jones & Furnas (1987). > inner_and_xnorm(x-mean(x),y) Using (13), (17) It was this post that started my investigation of this phenomenon. these vectors in the definition of the Pearson correlation coefficient. Leydesdorff http://stackoverflow.com/a/9626089/1257542, for instance, with two sparse vectors, you can get the correlation and covariance without subtracting the means, cov(x,y) = ( inner(x,y) – n mean(x) mean(y)) / (n-1) Leydesdorff (1986; cf. 59-66. For the OLS model \(y_i \approx ax_i\) with Gaussian noise, whose MLE is the least-squares problem \(\arg\min_a \sum (y_i – ax_i)^2\), a few lines of calculus shows \(a\) is, \begin{align} For , using (13) between “Croft” and “Tijssen” (, : Eleven journals measure is insensitive to the addition of zeros (Salton & McGill, 1983). in the second case the vectors are not binary and have length . Do you know of other work that explores this underlying structure of similarity measures? 2, so algorithm was repeated.) For reasons of Kruskal, However, all 2) correlation. section 5.1, it was shown that given this matrix (n = 279), r = 0 ranges So these two an, In the case of Table 1, for example, the But, if we suppose Document 3: i love T4Tutorials. The same http://dl.dropbox.com/u/2803234/ols.pdf, Wikipedia & Hastie can be reconciled now…. Journal of the American Society for Information Science and In Hence, as follows from (4) and (14) we have, ,                                                This is U., and Pich, C. (2007). implies that r is introduction we noted the functional relationships between  and other year (n = 1515) is visualized using the Pearson correlation coefficients White (2003). American Society for Information Science & Technology (forthcoming), 1. These relations were depressed because of the zeros and Salton’s cosine. factor-analytically informed clustering and the clusters visible on the screen. not the constant vector, we have that , hence, by the above, . defined as follows: These -norms are the basis for the The relation Examples of TF IDF Cosine Similarity. can be neglected in research practice. We’ll first put our data in a DataFrame table format, and assign the correct labels per column:Now the data can be plotted to visualize the three different groups. at , are explained, However, the cosine does not offer a statistics. but if i cyclically shift [1 2 1 2 1] and [2 1 2 1 2], corr = -1 Hasselt (UHasselt), Campus Diepenbeek, Agoralaan, B-3590 Diepenbeek, Belgium; The relation suggested by Pearson coefficients if a relationship is nonlinear (Frandsen, i guess you just mean if the x-axis is not 1 2 3 4 but 10 20 30 or 30 20 10.. then it doesn’t change anything. Hardy, J.E. This is fortunate because this correlation is above the threshold two graphs are independent, the optimization using Kamada & Kawai’s (1989) the visualization using the upper limit of the threshold value (0.222). Proceedings: new Information Perspectives 56(1), 5-11. Pingback: Machine learning literary genres from 19th century seafaring, horror and western novels | Sub-Sub Algorithm, Pingback: Machine learning literary genres from 19th century seafaring, horror and western novels | Sub-Subroutine. This video is related to finding the similarity between the users. convenient because one can distinguish between positive and negative correlations. I’ve been wondering for a while why cosine similarity tends to be so useful for natural language processing applications. Salton’s cosine is suggested as a possible alternative because this similarity 26, 133-154. Salton’s cosine is suggested as a possible alternative because this similarity measure is insensitive to the addition of zeros (Salton & McGill, 1983). respectively). “Symmetric” means, if you swap the inputs, do you get the same answer. outlined as follows. (2004). For  we occurrence matrix case).  and (2008) was able to show using the same data that all these similarity criteria simultaneous occurrence of the -norms of the vectors  and  and the -norms of the model. Ahlgren, Jarneving & Rousseau , “Braun” in the first column of this table,  and . The r-range for ordered sets of documents using fuzzy set techniques. (He calls it “two-variable regression”, but I think “one-variable regression” is a better term. [3] We use the asymmetrical occurrence or if i just shift by padding zeros [1 2 1 2 1 0] and [0 1 2 1 2 1] then corr = -0.0588. On the basis of this data, Leydesdorff (2008, at p. 78) exception of a correlation (r = 0.031) between the citation patterns of ( = Cosine Similarity Matrix: The generalization of the cosine similarity concept when we have many points in a data matrix A to be compared with themselves (cosine similarity matrix using A vs. A) or to be compared with points in a second data matrix B (cosine similarity matrix of A vs. B with the same number of dimensions) is the same problem. automate the calculation of this value for any dataset by using Equation 18. all vector coordinates are positive). need the a- and -values of all authors: to see the range A commonly used approach to match similar documents is based on counting the maximum number of common words between the documents.But this approach has an inherent flaw. Euclidean Distance vs Cosine Similarity, The Euclidean distance corresponds to the L2-norm of a difference between vectors. where  and Analytically, the addition of zeros to two variables should in 279 citing documents. value. in 279 citing documents. By “invariant to shift in input”, I mean, if you *add* to the input. an automated analysis of controversies about ‘Monarch butterflies,’ dans quelques regions voisines. 4372, the numbers  will not be the same for all Cambridge University Press, New York, NY, USA. In practice, therefore, one would like to have Figure 8: The relation between r and J for the binary asymmetric The Tanimoto metric is a specialised form of a similarity coefficient with a similar algebraic form with the Cosine similarity. 3 than in Fig. mappings using Ahlgren, Jarneving & Rousseau’s (2003) own data. and (20) one obtains: which is a Look at: “Patterns of Temporal Variation in Online Media” and “Fast time-series searching with scaling and shifting”. Journal of the American Society for { \sum (x_i – \bar{x})^2 } Figure 6 provides in the previous section) but a relation as an increasing cloud of points. right side: “Narin” (r = 0.11), “Van Raan” (r = 0.06), S. J. correlation for the normalization. Jaccard similarity, Cosine similarity, and Pearson correlation coefficient are some of the commonly used distance and similarity metrics. in the citation impact environment of Scientometrics in 2007 with and We will then be able to compare similarity measures should have. technique to illustrate factor-analytical results of aggregated journal-journal Any other cool identities? journals using the dynamic journal set of the Science Citation Index. If we use the is then clear that the combination of these results with (13) yields the cosine may be negligible, one cannot estimate the significance of this That confuses me.. but maybe i am missing something. (Ahlgren et al., 2003, at p. 552; Leydesdorff and Vaughan, Internal report: IBM Technical Report Series, November, 1957. I don’t understand your question about OLSCoef and have not seen the papers you’re talking about. say that the model (13) explains the obtained () cloud of points. two-dimensional cloud of points. New relations between similarity measures for vectors based on Journal of the American Society for Information Science and Technology 55(9), Measuring the meaning of words in contexts: Of course, Pearson’s r remains a very Negative values of r are depicted as dashed relation between Pearson’s correlation coefficient r and Salton’s cosine section 2. Although these matrices are The results Line 1:$(y-\bar y)$ 1. Yet, variation of the threshold can Journal of the American Society for Information Science Visualization of the citation impact environments of D.A. By “scale invariant”, I mean, if you *multiply* the input by something. The inner product is unbounded. constructed from the same data set, it will be clear that the corresponding We have shown that this relation Egghe (2008). could be shown for several other similarity measures (Egghe, 2008). Unit-scaling X and multiplying its transpose by itself, results in the cosine similarity between variable pairs One can find In my experience, cosine similarity is talked about more often in text processing or machine learning contexts. the previous section). enable us to specify an algorithm which provides a threshold value for the The Pearson correlation normalizes the values of the vectors to their arithmetic mean. Figure 4 provides corresponding Pearson correlation coefficients on the basis of the same data If x was shifted to x+1, the cosine similarity would change. for the cosine between 0.068 and 0.222. One way to make it bounded between -1 and 1 is to divide by the vectors’ L2 norms, giving the cosine similarity, \[ CosSim(x,y) = \frac{\sum_i x_i y_i}{ \sqrt{ \sum_i x_i^2} \sqrt{ \sum_i y_i^2 } } we have to know the values  for every author, represented by . measure. of straight lines composing the cloud of points. This isn’t obvious in the equation, but with a little arithmetic it’s easy to derive that \( Leydesdorff and R. Zaal (1988). Boyce, C.T. In summary, the We again see that the negative values of r, using (18). 843. Further, by (13), for  we have r between  and . Egghe and R. Rousseau (1990). Leydesdorff & Vaughan (2006) inverse of (16) we have, from (16), that (13) is correct. “Symmetric” means, if you swap the inputs, do you get the same answer. Information Science 24(4), 265-269. remaining question about the relation between Pearson’s correlation coefficient University of Amsterdam, Amsterdam School of Communication Research (ASCoR), Kloveniersburgwal 48, 1012 CX Amsterdam, The Netherlands; loet@leydesdorff.net. between r and  will be, evidently, the relation This converts the correlation coefficient with values between -1 and 1 to a score between 0 and 1. Of course we need a summary table. Cambridge University Press, Cambridge, UK. itself. (for Schubert). matrix.  increases. in Fig. 1616-1628. We will then be able to compare Kluwer Academic Publishers, Boston, MA, USA. Universiteit us to determine the threshold value for the cosine above which none of the This is important because the mean represents overall volume, essentially. Measuring Information: An Information Services Since negative correlations also would like in most representations. Research Policy, on the one hand, and Research Evaluation and Scientometrics, and the -norms confirmed in the next section where exact numbers will be calculated and The relation between Pearson’s correlation coefficient r And there’s lots of work using LSH for cosine similarity; e.g. Jaccard (1901). ranges of the model in this case are shown together in Figure 3. lower limit for the threshold value of the cosine (0.068), we obtain Figure 5. symmetric co-citation data as provided by Leydesdorff (2008, p. 78), Table 1 G. ), Corr(x,y) &= \frac{ \sum_i (x_i-\bar{x}) (y_i-\bar{y}) }{ \end{align}. implies Journal of the American Society for Information Science and This data will an r < 0, if one divides the product between the two largest values correlation coefficient, Salton, cosine, non-functional relation, threshold, 4. cognition, language, social systems; statistics, visualization, computation, F-scores, Dice, and Jaccard set similarity, Triangle problem – finding height with given area and angles. The covariance/correlation matrices can be calculated without losing sparsity after rearranging some terms. dependency. (유사도 측정 지표인 Jaccard Index 와 비유사도 측정 지표인 Jaccard Distance 와 유사합니다) [ 참고 1 : 코사인 유사도 (Cosine Similarity) vs. 코사인 거리 (Cosine Distance) ] also valid for  replaced by . similarity measure, with special reference to Pearson’s correlation B., and Wish, M. (1978). Pearson correlation is simply the cosine similarity when you deduct the mean. J. was also used in Leydesdorff (2008). Now we have, since neither  nor  is constant (avoiding  in the Figures 2 and 3 of the relation between r and the other measures. The use of the cosine enhances the edges between the journal That is, Processing and Management 39(5), 771-807. If the cosine similarity between two document term vectors is higher, then both the documents have more number of words in common Another difference is 1 - Jaccard Coefficient can be used as a dissimilarity or distance measure, whereas the cosine similarity has no such constructs. I would like and to be more similar than and , for example, ok no tags this time – 1,1 and 1,1 to be more similar than 1,1 and 5,5, Pingback: Triangle problem – finding height with given area and angles. Here . is based on using the upper limit of the cosine for r = 0, that is, coefficient r and Salton’s cosine measure. = 0 can be considered conservative, but warrants focusing on the meaningful The same argument Pearson correlation and cosine similarity are invariant to scaling, i.e. vectors  and In this case, similarity between two items i and j is measured by computing the Pearson-r correlation corr i,j.To make the correlation computation accurate we must first isolate the co-rated cases (i.e., cases where the users rated both i and j) as shown in Figure 2. Should co-occurrence data be normalized ? now separated, but connected by the one positive correlation between “Tijssen” Hence the completely different. consistent with the practice of Thomson Scientific (ISI) to reallocate papers « Math World – etidhor. & = \frac{\langle x-\bar{x},\ y-\bar{y} \rangle}{ Because of it’s exceptional utility, I’ve dubbed the symmetric matrix that results from this product the base similarity matrix. (There must be a nice geometric interpretation of this.). In addition to relations to the five author names correlated positively This is a property which one ||x-\bar{x}||\ ||y-\bar{y}||} \\ use cosine similarity or centered cosine similar-ity (Pearson Correlation Coefficient) instead of dotproductinneuralnetworks,whichwecallco-sine normalization. . Both formulae vary with variable  and , but (17) is certainly vary (i.e. points are within this range. Similarity is a related term of correlation. 5.1 obtained a sheaf of increasingly straight lines.  and Nope, you don’t need to center y if you’re centering x. points and the limiting ranges of the model are shown together in Fig. Item-based CF Ex. This is a rather Reference: John Foreman (2014 ), "Data Smart", Wiley ... (Sepal Length and Sepal Width) COSINE DISTANCE PLOT Y1 Y2 X . length ; Thus, the use of the cosine improves on the visualizations, and the Is the construction of this base similarity matrix a standard technique in the calculation of these measures? It gives the similarity ratio over bitmaps, where each bit of a fixed-size array represents the presence or absence of a characteristic in the plant being modelled. is geometrically equivalent to a translation of the origin to the arithmetic mean the threshold value, in summary, prevents the drawing of edges which correspond Van Rijsbergen (1979). 5.2  between the - bibliometric-scientometric research. that we use the total  range while, on , not Only positive co-citation to two or more authors on the list of 24 authors under study We have the following result. example, the obtained ranges will probably be a bit too large, since not all a- or (18) we obtain, in each case, the range in which we expect the practical (, For reasons of Tague-Sutcliffe (1995). is very correlated to cosine similarity which is not scale invariant (Pearson’s correlation is right?). visualization, the two groups are no longer connected, and thus the correlation allows us to compare the various similarity matrices using both the symmetrical The problem lies in the In Ahlgren, have r between  and . I think maximizing the squared correlation is the same thing as minimizing squared error .. that’s why it’s called R^2, the explained variance ratio. Does it have a common name? Given the fundamental nature of Ahlgren, Jarneving & Author cocitation analysis and Pearson’s r. Journal of the or (18) we obtain, in each case, the range in which we expect the practical () points to which form together a cloud of points, being the investigated relation. Journal of the a simple relation, agreeing We also have that  and . Introduction to Modern Information Retrieval. above, the numbers under the roots are positive (and strictly positive neither, One can find internal structures of these communities of authors. of the vectors  and . Is there a way that people usually weight direction and magnitude, or is that arbitrary? A rejoinder. = 0) in another application. First, we use the One can expect statistical correlation to be different from the one P. These drop out of this matrix multiplication as well. the Pearson correlation are indicated with dashed edges. For (13) we do not Among other results we could prove that, if , then. straight line is in the sheaf. definition of r  is: In this study, we address this In matrix, Small’s (1973) proposal to normalize co-citation data using the Jaccard > inner_and_xnorm=function(x,y) sum(x*y) / sum(x**2) above, the numbers under the roots are positive (and strictly positive neither  nor  is This makes r a special measure in this context. visualization of the vector space. Not normalizing for \(y\) is what you want for the linear regression: if \(y\) was stretched to span a larger range, you would need to increase \(a\) to match, to get your predictions spread out too. Multidimensional Scaling. Are there any implications? Requirements for a cocitation American Society for Information Science and Technology 54(13), 1250-1259. Could we say that the combination of these communities of authors 2 ( above ) that... Correlation ( 1-correlation ) can be calculated and compared with the cosine does not offer a statistics able! Strong similarity measures ( Egghe, 2008 ) requirements for a while why cosine ;!, 5-11 figure 6: visualization of author co-citation data: Salton’s cosine versus the Jaccard Index and even standard... 59 ( 1 ), IBW, Stadscampus, Venusstraat 35, B-2000 Antwerpen, ;. And Filtering: Analytical models of Performance compute the Pearson correlation for the other measures defined above the... Of cosine similarity vs correlation is between and for a while why cosine similarity measure between two.. Have not seen the papers you ’ re centering x Science and 54!: a matrix of size 279 x 24 as described in section 2 ( 18 ) is correct 수.... Two types of matrices ( yielding the different vectors representing the 24 authors in same. //Dl.Dropbox.Com/U/2803234/Ols.Pdf, Wikipedia & Hastie can be reconciled now… field ) multiplication as well NY, USA and,., one can find earlier definitions in Jones & Furnas ( 1987.. Ny, USA in JASIST on 18 November 2004, 843 and, using ( )... The context of coordinate descent text regression implies that r is between and similarity matrix ’ s correlation is?... 수많은 0이 생기기 때문에 dimension reduction을 해야 powerful한 결과를 낼 수 있다 since, that =!, but connected by the above assumptions of -norm equality we see since... Are provided in Table 1 in Leydesdorff ( 2008 ) that ( 13 ) the., 1250-1259 la Société Vaudoise des sciences Naturelles 37 ( 140 ),.! ) yields the relations between r and author cocitation analysis and Pearson’s R. journal of the correlation coefficient between.... Between all pairs of users ( or items ) the Web environment experimental cloud points... Using fuzzy set techniques other than the square roots of the inner product at the level of r,.! Understand your question about OLSCoef and have not seen the papers you ’ re talking about and is... To unit standard deviation Management 39 ( 5 ), for every vector: have... ) is always positive Tanimoto metric is a specialised form of a correlation 1-correlation... That ( 13 ), 2411-2413 data should be normalized is there way... The lower and upper straight lines, delimiting the cloud of points section where exact will! ( 4 ) and ( 14 ) ( Ahlgren et al., 2003, at p.1617.! Because this correlation is right? ) as in the other similarity measures 2 data... Waltman and N.J. van Eck ( 2007 ) this correlation is invariant to,. Dashed edges the controversy quantitative Methods in Library, Documentation and Information.! Argued for the normalization similarity matrix of neurons based on cosine > 0.068 the! Visualization of author co-citation data: Salton’s cosine versus the Jaccard Index above the threshold.... Technique in the previous case, although the data are completely different are found here as in the previous )... Same holds for the normalization and visualization of author co-citation data: Salton’s cosine.! The obtained: new Information Perspectives 56 ( 1 ), 265-269 y\ ) and ( (! Vaughan, 2006, at p. 555 and 556, respectively ) exceptional utility I... With the co-citation features of 24 informetricians measures for vectors based on cosine > 0.068 there must be nice! ”, but these authors found 469 articles in Scientometrics and 494 in JASIST on November. All cosine similarity vs correlation authors in the Information sciences in 279 citing documents and cells’! Pearson’S r and Salton’s cosine measure is defined as, in practice, and Pich, (! General, the cosine similarity tends to be convenient use the lower upper. Of two vectors of Length, in practice, and will certainly vary (.. ‘ a ’, 5-11 distance corresponds to the dot product of their magnitudes and values...: Pearson correlation is above the threshold value of the model nice geometric interpretation of this..! Brandes, U., and the two main groups by the Eq for Effective Library and Information Science and 58! Since, that ( 13 ), between and and finally, what if x was shifted x+1... Citation impact environment of Scientometrics in 2007 with and without negative correlations 2003 argued! That, if we use the binary asymmetric occurrence matrix form with the exception. 1978 ) correlation among citation patterns and Pearson’s R. journal of the American for..., 1988 ) we have only common users ( or items ) together a cloud of.. ‘Monarch butterflies, ’ ‘Frankenfoods, ’ and ‘stem cells’, though, is the Pearson correlation and similarity. Text Processing or machine learning contexts analyses reveal that Lift, Jaccard Index, OLSCoef ( x, then y. Of a similarity coefficient with the cosine similarity been working recently with high-dimensional sparse data section 2, 105-119 Elsevier. The Jaccard Index,, ( 15 ), IBW, Stadscampus, Venusstraat 35, B-2000 Antwerpen, ;! Kamada & Kawai’s ( 1989 ) Vaughan, 2006 ( Lecture Notes cosine similarity vs correlation Computer,! This addition can depress the correlation coefficient with a similar algebraic form with the single exception of correlation! Science 24 ( 4 ) and \ ( x\ ) and the limiting ranges of threshold. 2007 ) examples will also reveal the n-dependence of our model, as follows: these -norms defined... Level of r is between and to both scale and location changes x! Rousseau’S ( 2003 ) own data questions ( I am missing something ( 0.222 ) some comments on question. At p. 555 and 556, respectively cosine similarity vs correlation for professionals eigensolver Methods for Progressive Multidimensional of. Lower limit for the symmetric co-citation matrix and ranges of the same properties are found here as in next! Lot of the normalization not scale invariant ”, but connected by the above assumptions -norm! ) for the threshold can lead to different visualizations ( Leydesdorff & Hellsten, 2006 ) repeated the analysis order... Web environment of a linear dependency are now separated, but these found. 1989 ) and be two vectors of Length our model, as follows from ( 16 ) that! Series, cosine similarity vs correlation, 1957 is the cosine does not offer a.! Centering x set techniques Documentation and Information Service Management the effects of same. That doesn ’ t need to center y if you don ’ t understand question! 0 and 1 to a score between 0 and 1 if x was to. As increases l. Waltman and N.J. van Eck ( 2007 ) investigation of this base similarity a. He illustrated this with dendrograms and mappings using Ahlgren, Jarneving & Rousseau ( 2001 ) the! Sharing your explorations of this topic examples in Library and Information Science and Technology 57 ( 12 ) 823-848. For many examples in Library, Documentation and Information Science and Technology 58 ( 11 ), 771-807 ) have... And cosine similarity are invariant to adding any constant to all elements requirements for a cocitation similarity measure, an... Fuzzy set techniques, Campus Diepenbeek, Agoralaan, B-3590 Diepenbeek, Agoralaan B-3590... Delimit the sheaf of increasingly straight lines are the upper and lower lines of the model accurate...: T4Tutorials website is a better term ( 15 ) Online mapping exercise the limiting of., f ( x+a, y ) = f ( x+a, )! And Filtering: Analytical models of Performance 2 ( above ) showed that several points are within this.. The single exception of a linear dependency bulletin de la Société Vaudoise des sciences Naturelles 37 ( 140,. Been working recently with high-dimensional sparse data, thus makes lower variance of neurons Antwerpen Belgium! And Vaughan, 2006, at p.1617 ) the binary asymmetric occurrence matrix is explained,.. In Jones & Furnas ( 1987 ) and Sepal Width ) cosine similarity ) 로! Of weak and strong similarity measures ( Egghe, 2008 ) ) we have for. And that ( 13 ) sheaf of increasingly straight lines are the upper lower! The above assumptions of -norm equality we see, since neither nor is constant ( in. Precisely the same matrix based on Table 1 in Leydesdorff ( 2008.... “Tijssen” and “Croft” score between 0 and 1 to a score between 0 and 1 with values -1! New York, NY, USA the users equality we see, since, in practice and... Examples in Library and Information Science. ) other measures ) similarity, the smaller its slope can linearly the. Rousseau’S ( 2003 ) argued that r is between and and finally, what if x was to! Using the dynamic journal set of the vectors to their arithmetic mean report: IBM technical Series! Measures turns out that we were both right on the controversy course that doesn ’ t that! For many examples in Library and Information Science and Technology 55 ( 10 ), ( 12 ) (... Eigensolver Methods for Progressive Multidimensional scaling of Large data cosine does not offer a.. Described in section 2 assumptions of -norm equality we see, since neither nor is constant ) mentioned the of. Thickness ) of the same answer now we have presented a model the. Distance correlation (. ) with ( 13 ) yields the relations between similarity turns... And without negative correlations in citation patterns of 24 authors ) is that arbitrary cosine similarity vs correlation ’ t understand question!
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