Skewness and Kurtosis
Skewness and Kurtosis是表征分布形状的两个经典指标，简要介绍其含义及Intuition。
Skewness偏度
衡量一个分布的不对称性程度和方向。
分两种情况
Positive Skewness
When the tail on the right side of the distribution is longer or fatter. The mean and median will be greater than the mode.
Negative Skewness
When the tail of the left side of the distribution is longer or fatter than the tail on the right side. The mean and median will be less than the mode.
具体数值的intuition, rule of thumb
- skewness在$(-0.5, 0.5)$, the data are fairly symmetrical
- skewness在$[-1, -0.5]$ or $[0.5, 1]$, the data are moderately skewed
- skewness在$(-00 ,-1)$ or $[1, +00]$ the data are highly skewed
Kurtosis峰度
最直观的，Kurtosis体现在峰值上，即
表征分布在均值处峰值的高低，直观上反应了峰部的尖度。可以据此判断数据分布相对于正态分布而言是更陡峭(peakedness)还是平缓(flattening)。
但其实更本质的是尾部的分布
Kurtosis is all about the tails of the distribution — not the peakedness or flatness. It is used to describe the extreme values in one versus the other tail. It is actually the measure of outliers present in the distribution.
正态分布的Kurtosis值是3，其它分布的Kurtosis高低是相对正态分布的，有的实现里面会将Kurtosis减去3，其正负能反应相对于正态分布的形态。
High kurtosis，峰值更尖，尾部占比也就更多
High kurtosis in a data set is an indicator that data has heavy tails or outliers. Sharper than a normal distribution, with values concentrated around the mean and longer tails.This means high probability for extreme values.
Low kurtosis，峰值更平，尾部占比也就更少
Low kurtosis in a data set is an indicator that data has light tails or lack of outliers. Flatter than a normal distribution with a wider peak. The probability for extreme values is less than for a normal distribution, and the values are wider spread around the mean.
如下图几种常见分布的Kurtosis值
- D: Laplace distribution, red curve (two straight lines in the log-scale plot), excess kurtosis = 3
- S: hyperbolic secant distribution, orange curve, excess kurtosis = 2
- L: Logistic distribution, green curve, excess kurtosis = 1.2
- N: Normal distribution, black curve, excess kurtosis = 0
- C: Raised cosine distribution, cyan curve, excess kurtosis = −0.593762…
- W: Wigner semicircle distribution, blue curve, excess kurtosis = −1
- U: Uniform distribution, magenta curve, excess kurtosis = −1.2
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