kurtosis - Scipy 中使用的偏度和峰度的默认表达式是什么？

Question

Scipy（迄今为止，版本 0.19.1）统计函数模块（又名scipy.stats）包含scipy.stats.skew和scipy.stats.kurtosis的函数，用于计算数据集的偏度和峰度（分别为第 3 和第 4 统计矩）。此外，scipy.stats.describe调用这些函数。

偏度和峰度的定义可能会有所不同；因此，文献中没有对它们达成共识。那么，在 Scipy 中使用哪些数学表达式来定义上述两个函数的偏度和峰度以及它们的默认设置？

Answer 1

scipy.stats.skew和scipy.stats.kurtosis都调用scipy.stats.moment的函数，该函数计算数据样本的第 k 个中心矩：

<script src="https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.0/MathJax.js?config=TeX-AMS_HTML-full"></script> <script type="text/x-mathjax-config"> MathJax.Hub.Config({"HTML-CSS": { preferredFont: "TeX", availableFonts:["STIX","TeX"], linebreaks: { automatic:true }, EqnChunk:(MathJax.Hub.Browser.isMobile ? 10 : 50) }, tex2jax: { inlineMath: [ ["$", "$"], ["\\\\(","\\\\)"] ], displayMath: [ ["$$","$$"], ["\\[", "\\]"] ], processEscapes: true, ignoreClass: "tex2jax_ignore|dno" }, TeX: { noUndefined: { attributes: { mathcolor: "red", mathbackground: "#FFEEEE", mathsize: "90%" } }, Macros: { href: "{}" } }, messageStyle: "none" });   </script>

$$m_k = \frac{1}{n} \sum_{i = 1}^n (x_i - \bar{x})^k$$

因此，具有默认设置（例如）的scipy.stats.skewbias=True计算：

<script src="https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.0/MathJax.js?config=TeX-AMS_HTML-full"></script> <script type="text/x-mathjax-config"> MathJax.Hub.Config({"HTML-CSS": { preferredFont: "TeX", availableFonts:["STIX","TeX"], linebreaks: { automatic:true }, EqnChunk:(MathJax.Hub.Browser.isMobile ? 10 : 50) }, tex2jax: { inlineMath: [ ["$", "$"], ["\\\\(","\\\\)"] ], displayMath: [ ["$$","$$"], ["\\[", "\\]"] ], processEscapes: true, ignoreClass: "tex2jax_ignore|dno" }, TeX: { noUndefined: { attributes: { mathcolor: "red", mathbackground: "#FFEEEE", mathsize: "90%" } }, Macros: { href: "{}" } }, messageStyle: "none" });   </script>

$$ S = \frac{m_3}{(m_2)^{1.5}} $$

具有默认设置的scipy.stats.kurtosis ：

<script src="https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.0/MathJax.js?config=TeX-AMS_HTML-full"></script> <script type="text/x-mathjax-config"> MathJax.Hub.Config({"HTML-CSS": { preferredFont: "TeX", availableFonts:["STIX","TeX"], linebreaks: { automatic:true }, EqnChunk:(MathJax.Hub.Browser.isMobile ? 10 : 50) }, tex2jax: { inlineMath: [ ["$", "$"], ["\\\\(","\\\\)"] ], displayMath: [ ["$$","$$"], ["\\[", "\\]"] ], processEscapes: true, ignoreClass: "tex2jax_ignore|dno" }, TeX: { noUndefined: { attributes: { mathcolor: "red", mathbackground: "#FFEEEE", mathsize: "90%" } }, Macros: { href: "{}" } }, messageStyle: "none" });   </script>

$$ K = \frac{m_4}{(m_2)^{2}} $$