Correlation

In statistics, correlation defines the similarity between two random variables. The most commonly used correlation is the Pearson correlation and it is defined by the following:

The preceding formula defines the Pearson correlation as the covariance between X and Y, which is divided by the standard deviation of X and Y, or it can also be defined as the expected mean of the sum of multiplied difference of random variables with respect to the mean divided by the standard deviation of X and Y. Let's understand this with an example. Let's take the mileage and horsepower of various cars and see if there is a relation between the two. This can be achieved using the pearsonr function in the SciPy package:

>>> mpg = [21.0, 21.0, 22.8, 21.4, 18.7, 18.1, 14.3, 24.4, 22.8, 19.2, 17.8, 16.4, 17.3, 15.2, 10.4, 10.4, 14.7, 32.4, 30.4,
 33.9, 21.5, 15.5, 15.2, 13.3, 19.2, 27.3, 26.0, 30.4, 15.8, 19.7, 15.0, 21.4]
>>> hp = [110, 110, 93, 110, 175, 105, 245, 62, 95, 123, 123, 180, 180, 180, 205, 215, 230, 66, 52, 65, 97, 150, 150, 245,
 175, 66, 91, 113, 264, 175, 335, 109]

>>> stats.pearsonr(mpg,hp)

(-0.77616837182658638, 1.7878352541210661e-07)

The first value of the output gives the correlation between the horsepower and the mileage and the second value gives the p-value.

So, the first value tells us that it is highly negatively correlated and the p-value tells us that there is significant correlation between them:

>>> plt.scatter(mpg, hp)
>>> plt.show()

From the plot, we can see that as the mpg increases, the horsepower decreases.

Let's look into another correlation called the Spearman correlation. The Spearman correlation applies to the rank order of the values and so it provides a monotonic relation between the two distributions. It is useful for ordinal data (data that has an order, such as movie ratings or grades in class) and is not affected by outliers.

Let's get the Spearman correlation between the miles per gallon and horsepower. This can be achieved using the spearmanr() function in the SciPy package:

>>> stats.spearmanr(mpg,hp)

(-0.89466464574996252, 5.085969430924539e-12)

We can see that the Spearman correlation is -0.89 and the p-value is significant.

Let's do an experiment in which we introduce a few outlier values in the data and see how the Pearson and Spearman correlation gets affected:

>>> mpg = [21.0, 21.0, 22.8, 21.4, 18.7, 18.1, 14.3, 24.4, 22.8, 19.2, 17.8, 16.4, 17.3, 15.2, 10.4, 10.4, 14.7, 32.4, 30.4,
 33.9, 21.5, 15.5, 15.2, 13.3, 19.2, 27.3, 26.0, 30.4, 15.8, 19.7, 15.0, 21.4, 120, 3]
>>> hp = [110, 110, 93, 110, 175, 105, 245, 62, 95, 123, 123, 180, 180, 180, 205, 215, 230, 66, 52, 65, 97, 150, 150, 245,
 175, 66, 91, 113, 264, 175, 335, 109, 30, 600]

>>> plt.scatter(mpg, hp)
>>> plt.show()

From the plot, you can clearly make out the outlier values. Lets see how the correlations get affected for both the Pearson and Spearman correlation

The following commands show you the Pearson correlation:

>>> stats.pearsonr(mpg, hp)
>>> (-0.47415304891435484, 0.0046122167947348462)

Here is the Spearman correlation:

>>> stats.spearmanr(mpg, hp)
>>> (-0.91222184337265655, 6.0551681657984803e-14)

We can clearly see that the Pearson correlation has been drastically affected due to the outliers, which are from a correlation of 0.89 to 0.47.

The Spearman correlation did not get affected much as it is based on the order rather than the actual value in the data.