I am trying to figure out how to decompose Mahalanobis Distance into its marginal contributors on a granular level. I've already split it up into marginal contribution from each variable: $$ MD_i = r_i\frac{\partial MD}{\partial r_i} = \frac{r_i\sum_j(r_j\Sigma^{-1}_{i,j})}{MD} $$ But would like to split it up even further into individual volatility and correlation components. Below is my attempt at that: \begin{align} MD_i &= \frac{r_i\sum_j(r_j\Sigma^{-1}_{i,j})}{MD} \\ &= \frac{r_i^2\Sigma^{-1}_{i,i}}{MD} + \frac{r_i\sum_{j\neq i}r_j\Sigma^{-1}_{i,j}}{MD} \\ &= \underbrace{\frac{r_i^2\Sigma^{-1}_{i,i}}{MD}}_{\text{Volatility Component}_i} + \underbrace{\frac{r_ir_j\Sigma^{-1}_{i,j}}{MD} + \frac{r_ir_k\Sigma^{-1}_{i,k}}{MD} + \dots + \frac{r_ir_l\Sigma^{-1}_{i,l}}{MD}}_{\text{Correlation Component}_i} \end{align} My logic is that if I split up all terms into those that:
- Contain data just from variable $i$ ($r_i^2$)
- Contain data from both variables $i$ and $j\neq i$ ($r_ir_j$)
Then I will have split up this marginal contribution into terms that pertain to volatility (1) and correlation (2). However, I've read a bit about how the inverse covariance matrix (precision matrix) and learned that each element contains data regarding volatilities and correlation, which makes me think that my attempt isn't valid.
Can anybody shed some light on whether my logic is indeed off, and perhaps how to go about accomplishing my goal?
