Encountered invalid value when I use pearsonr

Question

Maybe I made a mistake. If so, I am sorry to ask this.

I want to calculate Pearson's correlation coefficent by using scipy's pearsonr function.

from scipy.stats.stats import pearsonr

X = [4, 4, 4, 4, 4, 4]
Y = [4, 5, 5, 4, 4, 4]

pearsonr(X, Y)

I get an error below

RuntimeWarning: invalid value encountered in double_scalars ###

The reason why I get an error is E[X] = 4 (Excepted Value of X is 4)

I look at the code of pearsonr function in scpy.stats.stats.py. Some part of the pearsonr function is as follows.

mx = x.mean() # which is 4
my = y.mean() # not necessary
xm, ym = x-mx, y-my # xm = [0 0 0 0 0 0]
r_num = n*(np.add.reduce(xm*ym)) #r_num = 0, because xm*ym 1x6 Zero Vector.
r_den = n*np.sqrt(ss(xm)*ss(ym)) #r_den = 0
r = (r_num / r_den) # Invalid value encountered in double_scalars

At the end, pearsonr returns (nan, 1.0)

Should pearsonr return (0, 1.0)?

I think if a vector has same value for every row/column, covariance should be zero. Thus Pearson's Correleation Coefficient should also be zero by the definition of PCC.

Pearson's correlation coefficient between two variables is defined as the covariance of the two variables divided by the product of their standard deviations.

Is it bug or where do I make a mistake?

Answer 1

Pearson's correlation coefficient between two variables is defined as the covariance of the two variables divided by the product of their standard deviations.

So it's the covariance over

• the standard deviation of [4, 5, 5, 4, 4, 4] times
• the standard deviation of [4, 4, 4, 4, 4, 4].

The standard deviation of [4, 4, 4, 4, 4, 4] is zero.

So it's the covariance over

• the standard deviation of [4, 5, 5, 4, 4, 4] times
• zero.

So it's the covariance over

• zero.

Anything divided by zero is nan. The value of the covariance is irrelevant.