Biweight midcorrelation

Here we find the biweight midcorrelation of two vectors $x$ and $y$, with $i=1,2,\; \backslash ldots,m$ items, representing each item in the vector as $x\_1,\; x\_2,\; \backslash ldots,\; x\_m$ and $y\_1,\; y\_2,\; \backslash ldots,\; y\_m$. First, we define $\backslash operatorname\{med\}(x)$ as the median of a vector $x$ and $\backslash operatorname\{mad\}(x)$ as the median absolute deviation (MAD), then define $u\_i$ and $v\_i$ as,

:$$

\begin{align}

u_i &= \frac{x_i - \operatorname{med}(x)}{9 \operatorname{mad}(x)},\\

v_i &= \frac{y_i - \operatorname{med}(y)}{9 \operatorname{mad}(y)}.

\end{align}

Now we define the weights $w\_i^\{(x)\}$ and $w\_i^\{(y)\}$ as,

:$$

\begin{align}

w_i^{(x)} &= \left(1-u_i^2\right)^2 I\left(1-|u_i|\right)\\

w_i^{(y)} &= \left(1-v_i^2\right)^2 I\left(1-|v_i|\right)

\end{align}

where $I$ is the identity function where,

:$$

I(x) = \begin{cases}1, & \text{if } x >0\\

0, & \text{otherwise}\end{cases}

Then we normalize so that the sum of the weights is 1:

:$$

\begin{align}

\tilde{x}_i &= \frac{\left(x_i - \operatorname{med}(x)\right) w_i^{(x)}}{\sqrt{\sum_{j=1}^m \left[(x_j -\operatorname{med}(x)) w_j^{(x)}\right]^2}}\\

\tilde{y}_i &= \frac{\left(y_i - \operatorname{med}(y)\right) w_i^{(y)}}{\sqrt{\sum_{j=1}^m \left[(y_j -\operatorname{med}(y)) w_j^{(y)}\right]^2}}.

\end{align}

Finally, we define biweight midcorrelation as,

:$$

\mathrm{bicor}\left(x, y\right) = \sum_{i=1}^m \tilde{x}_i \tilde{y}_i

Biweight midcorrelation has been shown to be more robust in evaluating similarity in gene expression networks,{{cite journal|last1=Song|first1=Lin|title=Comparison of co-expression measures: mutual information, correlation, and model based indices|journal=BMC Bioinformatics|date=9 December 2012|volume=13|issue=328|page=328 |doi=10.1186/1471-2105-13-328|pmid=23217028|pmc=3586947 |doi-access=free }} and is often used for weighted correlation network analysis.

Biweight midcorrelation has been implemented in the R statistical programming language as the function bicor as part of the WGCNA package{{cite web|last1=Langfelder|first1=Peter|title=WGCNA: Weighted Correlation Network Analysis (an R package)|url=https://cran.r-project.org/package=WGCNA|website=CRAN|accessdate=2018-04-06}}

Also implemented in the Raku programming language as the function bi_cor_coef as part of the Statistics module.{{cite web|last1=Khanal|first1=Suman|title=Statistics: Raku module for doing statistics | url=https://github.com/sumanstats/Statistics|website=GitHub|accessdate=2022-03-11}}

• Steve Horvath

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Category:Parametric statistics

Category:Covariance and correlation