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Kurtosis
https://www.itl.nist.gov/div898/handbook/eda/section3/eda35b.htm
https://en.wikipedia.org/wiki/Kurtosis
The kurtosis is a measure of the "tailedness" of the probability distribution. https://en.wikipedia.org/wiki/Kurtosis
- The kurtosis of any univariate normal distribution is 3. A univariate normal distribution is usually called just normal distribution.
- Platykurtic: Kurtosis less than 3 (Negative Kurtosis if we talk about the adjusted version of Pearson's kurtosis, the excess kurtosis).
- A negative value means that the distribution has a light tail compared to the normal distribution (which means that there is little data in the tail).
- An example of a platykurtic distribution is the uniform distribution, which does not produce outliers.
- Leptokurtic: Kurtosis greater than 3 (Positive excess kurtosis).
- A positive Kurtosis tells that the distribution has a heavy tail (outlier), which means that there is a lot of data in the tail.
- An example of a leptokurtic distribution is the Laplace distribution, which has tails that asymptotically approach zero more slowly than a Gaussian and therefore produce more outliers than the normal distribution.
- This heaviness or lightness in the tails usually means that your data looks flatter (or less flat) compared to the normal distribution.
- It is also common practice to use the adjusted version of Pearson's kurtosis, the excess kurtosis, which is the kurtosis minus 3, to provide the comparison to the standard normal distribution. Some authors use "kurtosis" by itself to refer to the excess kurtosis. https://en.wikipedia.org/wiki/Kurtosis
- It must be noted that the Kurtosis is related to the tails of the distribution, not its peak; hence, the sometimes-seen characterization of kurtosis as "peakedness" is incorrect. https://en.wikipedia.org/wiki/Kurtosis
In Python: https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.kurtosis.html
scipy.stats.kurtosis(a, axis=0, fisher=True, bias=True, nan_policy='propagate')
- Compute the kurtosis (Fisher or Pearson) of a dataset.
import numpy as np
from scipy.stats import kurtosis
data = norm.rvs(size=1000, random_state=3)
data2 = np.random.randn(1000)
kurtosis(data2)
from scipy.stats import kurtosis
import matplotlib.pyplot as plt
import scipy.stats as stats
x = np.linspace(-5, 5, 100)
ax = plt.subplot()
distnames = ['laplace', 'norm', 'uniform']
for distname in distnames:
if distname == 'uniform':
dist = getattr(stats, distname)(loc=-2, scale=4)
else:
dist = getattr(stats, distname)
data = dist.rvs(size=1000)
kur = kurtosis(data, fisher=True)
y = dist.pdf(x)
ax.plot(x, y, label="{}, {}".format(distname, round(kur, 3)))
ax.legend()
