查看“︁费雪变换”︁的源代码

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[[File:fisher transformation.svg|300px|right|thumb|费雪变换]]
'''费雪变换'''（{{lang-en|Fisher transformation}}）是[[统计学]]中用于[[相关系数]]假设检验的一种方法。对样本相关系数进行费雪变换后，可以用来检验关于总体相关系数ρ的假设。<ref>{{cite journal| last=Fisher | first= R.A. | year=1915 | title= Frequency distribution of the values of the correlation coefficient in samples of an indefinitely large population | url=https://archive.org/details/sim_biometrika_1915-05_10_4/page/507 | journal=Biometrika | volume=10 | pages=507–521 | jstor=2331838| issue=4| publisher=Biometrika Trust | doi=10.2307/2331838}}</ref><ref>{{cite journal | authorlink=Ronald Fisher | last=Fisher | first=R.A. | year=1921 | title=On the `probable error' of a coefficient of correlation deduced from a small sample | journal=Metron | volume=1 | pages=3–32 | url=http://digital.library.adelaide.edu.au/dspace/bitstream/2440/15169/1/14.pdf | access-date=2015-09-03 | archive-date=2021-02-12 | archive-url=https://web.archive.org/web/20210212075523/https://digital.library.adelaide.edu.au/dspace/bitstream/2440/15169/1/14.pdf | dead-url=no }}</ref>

== 定义 ==
已知{{Mvar|N}}组双变量样本{{Math|1=(''X''<sub>''i''</sub>,&nbsp;''Y''<sub>''i''</sub>), ''i''&nbsp;=&nbsp;1,&nbsp;...,&nbsp;''N''}}，样本相关系数{{Mvar|r}}为
:<math>r = \frac{\sum ^N _{i=1}(X_i - \bar{X})(Y_i - \bar{Y})}{\sqrt{\sum ^N _{i=1}(X_i - \bar{X})^2} \sqrt{\sum ^N _{i=1}(Y_i - \bar{Y})^2}}</math>

于是，{{Mvar|r}}的费雪变换可定义为 

:<math>z := {1 \over 2}\ln\left({1+r \over 1-r}\right) = \operatorname{arctanh}(r).</math>


当{{Math|(''X'',&nbsp;''Y'')}}为[[多元正态分布|二元正态分布]]且{{Math|(''X''<sub>''i''</sub>,&nbsp;''Y''<sub>''i''</sub>)}}对相互独立时，{{Mvar|z}}近似为[[正态分布]]。其均值为
:<math>{1 \over 2}\ln\left({{1+\rho} \over {1-\rho}}\right),</math>

[[标准差]]为
:<math>{1 \over \sqrt{N-3}},</math>

其中{{Mvar|N}}是样本大小，{{Mvar|ρ}}是变量{{Mvar|X}}与{{Mvar|Y}}的总体相关系数。

费雪变换及其逆变换
:<math>r = {{\exp(2z)-1} \over {\exp(2z)+1}} = \operatorname{tanh}(z),</math>

可以用于构造{{Mvar|ρ}}的[[置信区间]]。

== 参考文献 ==
{{reflist}}

{{统计学}}

[[Category:统计学]]
[[Category:变换]]