Comment author:iDante
28 November 2012 01:05:41AM
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1 point
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I don't think correlation coefficient is the cosine of the angle between them. In this picture you can see that the middle row has 1 or -1 and yet different angles. Instead I think it's a measure of how well they correlate with each other. Maybe I'm crazy-wrong though. Edit: For only two variables you might be right, but I don't see correlation coefficients reported between just two variables.

I often see the entire covariance matrix reported, not just the correlation coefficient. This matrix has some handy properties. Its eigenstuff contains all the information about the arrows in this picture.

Comment author:Sniffnoy
28 November 2012 01:33:11AM
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9 points
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I don't think correlation coefficient is the cosine of the angle between them. In this picture you can see that the middle row has 1 or -1 and yet different angles.

Sorry, I guess I was unclear on what I meant. I didn't mean the angles pictured in that graph. Rather I meant what was described in the footnote -- consider covariance as an inner product and consider the angles you get this way.

In other words, I'm not claiming any sort of theorem; I'm not claiming that the correlation coefficient in fact tells you some other information that isn't obviously already in there. I've defined the angles here such that "the correlation coefficient is the cosine of the angle between them" is a tautology. I'm just suggesting that this geometric viewpoint might help with the intuition.

Comment author:iDante
28 November 2012 02:41:49AM
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4 points
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Whoa. I didn't have this geometric point of view down so I didn't understand at first what you meant, but now that I've thought about it for a few minutes it makes sense.

The prevalent intuitive meaning of correlation coefficient (at least the one I learned in the watered-down statistics-for-physics-students class) is "a measure of how well two variables correlate. 1 is well, 0 is not at all, -1 is backwards correlation." Hence, the first thing I thought of was that image. Many people who need to use this coefficient won't have taken linear algebra and it'll be a complication for them to learn that it's "the inner product of two random variablesies, so 0 means lots of correlation, pi means the opposite direction, and pi/2 means no correlation." Or maybe you use degrees, idk.

## Comments (24)

Best*1 point [-]I don't think correlation coefficient is the cosine of the angle between them. In this picture you can see that the middle row has 1 or -1 and yet different angles. Instead I think it's a measure of how well they correlate with each other. Maybe I'm crazy-wrong though. Edit: For only two variables you might be right, but I don't see correlation coefficients reported between just two variables.

I often see the entire covariance matrix reported, not just the correlation coefficient. This matrix has some handy properties. Its eigenstuff contains all the information about the arrows in this picture.

*9 points [-]Sorry, I guess I was unclear on what I meant. I didn't mean the angles pictured in that graph. Rather I meant what was described in the footnote -- consider covariance as an inner product and consider the angles you get this way.

In other words, I'm not claiming any sort of theorem; I'm not claiming that the correlation coefficient in fact tells you some other information that isn't obviously already in there. I've defined the angles here such that "the correlation coefficient is the cosine of the angle between them" is a tautology. I'm just suggesting that this geometric viewpoint might help with the intuition.

*4 points [-]Whoa. I didn't have this geometric point of view down so I didn't understand at first what you meant, but now that I've thought about it for a few minutes it makes sense.

The prevalent intuitive meaning of correlation coefficient (at least the one I learned in the watered-down statistics-for-physics-students class) is "a measure of how well two variables correlate. 1 is well, 0 is not at all, -1 is backwards correlation." Hence, the first thing I thought of was that image. Many people who need to use this coefficient won't have taken linear algebra and it'll be a complication for them to learn that it's "the inner product of two random variablesies, so 0 means lots of correlation, pi means the opposite direction, and pi/2 means no correlation." Or maybe you use degrees, idk.

I like it, thanks for making this thread :D