What's the formula of normalized correlation?

I read a paper which used normalized correlation to evaluate the distance between two vectors. But I searched on the Internet and found little about normalized correlations, but I still got some clues. In one paper the formula for normalized correlation is given as follows:

$$dc(y_1,y_2) = \frac{y_1^T y_2}{|y_1||y_2|}$$

I was confused that it's just the cosine similarity formula! So what on earth is the formula of normalized correlation?

PS: I found some explanation in this article The Corrected Normalized Correlation Coefficient: A Novel Way Of Matching Score Calculation for LDA-Based Face Verification

$$\delta_{cos}(y,\bar{y_{j}}) = \frac {y^{T} \bar{y_{j}}} { \|y\|\|\bar{y_{j}}\|}$$

In statistics, the expression above is often referred to as the normalized correlation coefficient and is used for measuring the extent to which two samples, in our case the vectors $$y$$ and $$\bar{y_{j}}$$ are linearly related. When the absolute value of the normalized correlation coefficient equals one, then there exists a linear relation between the two samples, while on the other hand, when the value of the normalized correlation coefficient equals zero, then the two samples have no linear relation. Generally, the higher the absolute value of the coefficient, the stronger the linear relation between the two samples. Based on this fact the absolute value of the normalized correlation coefficient is commonly employed for computing the matching score between the input vector $$y$$ in the client template $$\bar{y_{j}}$$

• I've never heard of "normalized correlation", and cosine is a better word. Correlation itself is the cosine after centering of variables (see). Commented May 2, 2013 at 10:15
• You don't need pictures to post nice formulas here -- MathJAX will render $\LaTeX$ input.
– user88
Commented May 2, 2013 at 11:46

I haven't come across this usage, but it seems easy to decode.

Matters may differ in your field, but within mainstream statistics, and all statistics-using disciplines I know about, correlation is understood as being by definition scaled to fall within [-1, 1]. When calculated similarly to your formula correlation is a cosine.

So the term "normalized" is just emphasizing that fact; it is not flagging a special case.

The unnormalized correlation would just be called the covariance.

So, you can't find this term being used because it is very unusual.

• Thanks for your help. I added some explanation in my post which I get it from a paper. Commented May 2, 2013 at 13:42
• I'd insert "not" in one sentence you quote "In statistics, the expression above is often referred to as the normalized correlation coefficient", i.e. "not often". But the explanation you cite seems fine, so far as it goes. This is just a standard correlation. If the name is qualified ever, it is as the Pearson product-moment correlation. I'd say that "product-moment" was a quite common label decades ago, but has faded in use, while "Pearson" remains common. But this is all, and only, about terminology. Commented May 2, 2013 at 13:52

I think they might refer (with a somehow misleading notation) to the cosine similarity.

• Thanks, I got some hints in a paper and I added it in my post, from the paper the absolute value of cosine similarity maybe the normalized correlation Commented May 2, 2013 at 13:44
• Just to clarify and summarise it for other people who might read this: "cosine similarity" = "normalized correlation coefficient". The difference is that Machine Learning and Information Retrieval people would use the former term, while statisticians might use the latter. Commented May 2, 2013 at 14:00