Source-linked AI summary
The relation between Pearson's correlation coefficient r and Salton's cosine measure
Leo Egghe, Loet Leydesdorff
TL;DR
There is no one-to-one correspondence between a zero Pearson-correlation cutoff and a cosine similarity value. The paper models their relation as a sheaf of increasingly straight lines and reports that the theoretical findings are confirmed on two datasets.
Problem
A zero cutoff for Pearson’s r has no one-to-one corresponding value for cosine similarity.
Source: §1 · p. 3Method
The paper models the relation between Pearson’s r and Salton’s cosine as a cloud described by a sheaf of increasingly straight lines.
Source: §8 · p. 21Results
The theoretical findings are confirmed on two datasets.
Source: §8 · p. 21Takeaways & Limitations
The relation between Pearson’s r and Salton’s cosine is not a pure function.
Source: §8 · p. 21The formalization considers vectors whose coordinates are all positive.
Source: §3 · p. 5
Abstract
from arXiv · showhide
The relation between Pearson's correlation coefficient and Salton's cosine measure is revealed based on the different possible values of the division of the L1-norm and the L2-norm of a vector. These different values yield a sheaf of increasingly straight lines which form together a cloud of points, being the investigated relation. The theoretical results are tested against the author co-citation relations among 24 informetricians for whom two matrices can be constructed, based on co-citations: the asymmetric occurrence matrix and the symmetric co-citation matrix. Both examples completely confirm the theoretical results. The results enable us to specify an algorithm which provides a threshold value for the cosine above which none of the corresponding Pearson correlations would be negative. Using this threshold value can be expected to optimize the visualization of the vector space.
1. Introduction
Pearson’s r and Salton’s cosine differ geometrically and may produce different similarity results, even when those differences are negligible in some cases. Because there is no direct cosine cutoff corresponding to r = 0, choosing a visualization threshold remains difficult.
1. Introduction
- Ahlgren, Jarneving & Rousseau’s critique challenged Pearson’s use in Author Cocitation Analysis, while White defended it partly by arguing that practical differences between measures can be neglected.Bensman argued that Pearson’s statistical grounding and normalization give it advantages over cosine.Source: §1 · p. 2
- Pearson measures linear dependency, so nonlinear relationships may yield a different statistical correlation from the one suggested by Pearson coefficients.The authors favor cosine for similarity analysis and visualization but note that multivariate statistics may still begin with a Pearson correlation matrix.Source: §1 · pp. 2–3
- Pearson’s r ranges from –1 to +1, while cosine ranges from zero to one in a single quadrant.Geometrically, Pearson centers the vector space at the vectors’ arithmetic means, whereas cosine uses an origin where every vector has value zero.Source: §1 · pp. 2–3
- Pearson’s r and Salton’s cosine can differ, although their differences may be negligible in many practical cases and cannot be estimated in advance.The cosine is insensitive to added zeros, whereas Pearson’s correlation is sensitive to them.Source: §1 · pp. 2–3
- Negative correlations can still correspond to positive cosine values, so the cosine has no natural cutoff and threshold choice remains somewhat arbitrary.Changing the threshold can produce different visualizations, while using the lower-triangle mean often fails to align visible clusters with factor-analytically informed clustering.Source: §1 · p. 4
2. Data
The analysis uses two co-citation datasets for 24 informetricians: an asymmetric occurrence matrix and a symmetric co-citation matrix. Together, they allow comparisons of similarity matrices derived from both data forms.
2. Data
- The symmetric co-citation matrix is based on a 24-author set drawn from 913 references in JASIST articles published from 1996 to 2000.The set contains 12 information-retrieval authors and 12 bibliometric-scientometric authors; adding diagonal values to the earlier co-citation table yielded the symmetric data.Source: §2 · pp. 4–5
- The two datasets comprise asymmetric occurrence data and symmetric co-citation data for author co-citation analysis of 24 informetricians.In the asymmetric matrix, each paper coordinate is 1 when an author is cited and 0 otherwise; 279 papers contain co-citations to at least two listed authors.Source: §2 · p. 4
- The two matrix forms allow comparisons of similarity matrices based on symmetric co-citation data and asymmetric occurrence data.The paper turns to analysis of these data after establishing its mathematical model for the relation between Pearson’s r and Salton’s cosine.Source: §2 · p. 5
3. Formalization of the problem
The formalization draws on earlier work relating similarity criteria and considers vectors with positive coordinates. It also notes the proposed conceptual advantages of Jaccard normalization for symmetric co-citation data.
3. Formalization of the problem
- For symmetric co-occurrence data, Small’s Jaccard normalization is presented as having conceptual advantages over cosine.The passage attributes the proposal to Small (1973) and names Jaccard (1901) and Tanimoto (1957).Source: §3 · p. 5
- Egghe showed that several similarity criteria can be functionally related using the same data.This result was based on Leydesdorff’s figure and data.Source: §3 · p. 5
- The formalization introduces two vectors whose coordinates are all positive and considers their Jaccard index.The Jaccard index is described as measuring the vectors’ similarity.Source: §3 · p. 5
X Y Cos
The paper addresses the relation between Pearson’s correlation coefficient and Salton’s cosine, with the simultaneous appearance of vector norms complicating the relationship. It establishes a linear relation for each relevant norm configuration, yielding a model consistent with experimental findings.
X Y Cos
- Pearson’s r is distinctive among the discussed measures because its definition involves both L1- and L2-norms.The paper contrasts these norms with the other similarity measures defined earlier.Source: X Y Cos · p. 7
- Pearson’s r remains important for measuring how well a regression line fits an experimental two-dimensional cloud of points.The paper points to examples in library and information science.Source: X Y Cos · p. 7
- Each relevant vector norm configuration yields a linear relation between Pearson’s r and cosine, and the resulting simple relation agrees with experimental findings.The paper identifies the simultaneous occurrence of vector L1- and L2-norms in the Pearson coefficient as the complication behind the relation.Source: X Y Cos · pp. 6–7
4. The mathematical model for the relation between r and Cos
The relation between Pearson’s r and Salton’s cosine is linear for fixed parameters a and b, but varies with those parameters. Their variation produces a sheaf of increasingly straight lines and a cloud of points rather than a single-valued function.
4. The mathematical model for the relation between r and Cos
- For fixed a and b, equation (13) gives a linear relation between r and cosine, with its form dependent on those parameters.The vector length n is constant in this relation.Source: §4 · p. 9
- As a and b vary, equations (17) and (18) have opposite signs, producing a sheaf of increasingly straight lines.The lines together form an increasing cloud of points, so the relation is not a function.Source: §4 · p. 10
- For fixed a and b, the slope of (13) goes to 1 as vector length n becomes large; the passage says this is readily seen.(18) decreases with the length of the vector.Source: §4 · p. 10
5. One example and two applications
The proposed model is tested on two matrices derived from co-citation data for 24 informetricians. In both cases, it explains the observed correlation–cosine cloud and its limiting ranges.
5. One example and two applications
- Both matrices use the same co-citation dataset but yield very different vectors: one is a 279 × 24 binary occurrence matrix and the other a 24 × 24 symmetric matrix.The dataset represents 24 informetricians.Source: §5 · p. 10
- In both matrices, the model explains the observed correlation–cosine cloud, with its limiting ranges shown alongside the data points.The comparison is shown for the asymmetric occurrence matrix in Figure 2 and the symmetric co-citation matrix in Figure 3.Source: Figure 2 · p. 13, §5.1 · p. 13, Figure 3 · p. 15
5.1 The case of the binary asymmetric occurrence matrix
- For the binary occurrence matrix, the model explains the observed cloud; its r-range narrows as cosine rises, while negative r-values occur at low cosine.Figure 2 shows the experimental points together with the model ranges.Source: Figure 2 · p. 13, §5.1 · p. 13
5.2 The case of the symmetric co-citation matrix
- For the symmetric co-citation matrix, the model also explains the cloud, and its r-range narrows more strongly than in Figure 2 as cosine increases.Figure 3 presents the points and model ranges for this matrix.Source: Figure 3 · p. 15, §5.2 · p. 16
6. The effects of the predicted threshold values on the visualization
The predicted cosine cutoffs alter which author relationships appear in the network: the upper cutoff removes edges corresponding to negative correlations but can also hide positive relationships.
6. The effects of the predicted threshold values on the visualization
- At cosine > 0.068, the two author groups are separated but remain connected by the positive Croft–Tijssen correlation, above the cutoff.Cronin is also linked to Moed (r = −0.02), Nederhof (r = −0.03), and Glänzel (r = −0.05), despite those correlations being negative.Source: §6 · pp. 17–18
- At cosine > 0.222, the groups disconnect, omitting the Croft–Tijssen correlation (r = 0.31) and Cronin’s links to Van Raan and Callon, both with r < 0.1.The passage also states that all correlations with r > 0.1 are made visible.Source: §6 · p. 18
- The upper cosine cutoff prevents edges corresponding to negative correlations, but is conservative.The paper describes this as a desirable property for most representations.Source: §6 · pp. 18–19
- In the application, cosine enhances edges between Research Policy and Research Evaluation and Scientometrics; these relations were depressed, and cosine improves visualizations while its model-predicted value provides a useful threshold.Source: §6 · p. 19
7. The relation between r and similarity measures other than Cos
The section extends the Pearson–cosine relationship to other similarity measures and describes how the Pearson–Jaccard relation behaves for the binary asymmetric occurrence matrix.
7. The relation between r and similarity measures other than Cos
- Combining the derived relations yields relations between Pearson r and other similarity measures, including Jaccard and Dice.The discussion refers to earlier results for these measures and assumes equality of the relevant L-norms.Source: §7 · p. 20
- Under the L-norm equality assumption, equations (13) and (20) are used to obtain the Pearson–Jaccard relation.The displayed expression is introduced as the result of applying those equations.Source: §7 · p. 20
- For the binary asymmetric occurrence matrix, the r–J point cloud increases faster than in Figure 2 because J is convexly increasing below the first bisector.The passage states that the relation is depicted in Figure 8.Source: §7 · pp. 20–21, Figure 8 · p. 21
8. Conclusion
The paper models the Pearson–cosine relation as an increasing cloud of points described by a sheaf of increasingly straight lines. Its theoretical findings are confirmed in co-citation data, and an algorithm gives a sample-specific cosine threshold above which no corresponding Pearson correlation is negative.
8. Conclusion
- The relation is not a pure function: it forms an increasing cloud described by increasingly straight lines whose slopes decrease higher in the sheaf.Source: §8 · p. 21
- The model explains the negative-r portion of the cloud and reports a decreasing r-range, or cloud thickness.Source: §8 · p. 21
- Theoretical findings are confirmed in co-citation vectors from 24 informetricians, using asymmetric occurrence and symmetric co-citation matrices.Source: §8 · p. 21
- An algorithm calculates a sample-specific cosine threshold above which no corresponding Pearson correlation is negative.For the asymmetric matrix example, the stated cutoff is 61.97/279; Equation 18 automates calculation for any dataset.Source: §8 · p. 22