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I'm wondering if there is any relationship among these 3 measures. I can't seem to make a connection among them by referring to the definitions (possibly because I am new to these definitions and am having a bit of a rough time grasping them).

I know the range of the cosine similarity can be from 0 - 1, and that the pearson correlation can range from -1 to 1, and I'm not sure on the range of the z-score.

I don't know, however, how a certain value of cosine similarity could tell you anything about the pearson correlation or the z-score, and vice versa?

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    $\begingroup$ z score of what? z scores of some things might be related to Pearson correlation, Z scores of other things may not. For example, if you internally standardize your original variables then the Pearson correlation between x and y is the expected product of their z-scores. Or you might be talking about z-scores of Pearson correlations (Pearson correlations minus their expectation under some condition all divided by the standard error of the Pearson correlation), which would certainly be related to the Pearson correlation. $\endgroup$
    – Glen_b
    Commented Sep 19, 2016 at 5:17
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    $\begingroup$ Direct relation: stats.stackexchange.com/a/22520/3277 $\endgroup$
    – ttnphns
    Commented Dec 4, 2017 at 16:52

2 Answers 2

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The cosine similarity between two vectors $a$ and $b$ is just the angle between them $$\cos\theta = \frac{a\cdot b}{\lVert{a}\rVert \, \lVert{b}\rVert}$$ In many applications that use cosine similarity, the vectors are non-negative (e.g. a term frequency vector for a document), and in this case the cosine similarity will also be non-negative.

For a vector $x$ the "$z$-score" vector would typically be defined as $$z=\frac{x-\bar{x}}{s_x}$$ where $\bar{x}=\frac{1}{n}\sum_ix_i$ and $s_x^2=\overline{(x-\bar{x})^2}$ are the mean and variance of $x$. So $z$ has mean 0 and standard deviation 1, i.e. $z_x$ is the standardized version of $x$.

For two vectors $x$ and $y$, their correlation coefficient would be $$\rho_{x,y}=\overline{(z_xz_y)}$$

Now if the vector $a$ has zero mean, then its variance will be $s_a^2=\frac{1}{n}\lVert{a}\rVert^2$, so its unit vector and z-score will be related by $$\hat{a}=\frac{a}{\lVert{a}\rVert}=\frac{z_a}{\sqrt n}$$

So if the vectors $a$ and $b$ are centered (i.e. have zero means), then their cosine similarity will be the same as their correlation coefficient.

TL;DR Cosine similarity is a dot product of unit vectors. Pearson correlation is cosine similarity between centered vectors. The "Z-score transform" of a vector is the centered vector scaled to a norm of $\sqrt n$.

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    $\begingroup$ +1. latexnazi comment: \| often looks better than ||, and \lVert ... \rVert is the best way to write it. $\endgroup$
    – amoeba
    Commented Dec 4, 2017 at 15:14
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To convert a z-score to a cosine, use the cumulative distribution function for a Gaussian distribution. Find the value of the Gaussian cdf corresponding to the z-score value. Subtract 0.5 from that value, multiply by 2, and assume that value is the sine of an angle. Use the arcsine function to find that angle. Then take the cosine of that angle. Voila!

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