Suppose that there are $N$ pieces of data, each of which contains $M$ parameter values such that $N >> M$. If we put this information into matrix form ($N$ rows, $M$ columns) and then compute the covariance matrix of this data, $C$, then we can consider that $C_{i,j}$ is the covariance between parameter $i$, $p_i$, and parameter $j$, $p_j$.I have always thought of this as The amount that $p_i$ would change if $p_j$ changed by 1. While this may not be exactly correct (is it correct?), there is at least a fundamental relationship between $p_i$ and $p_j$ which is reflected in $C_{i,j}$ and can consider that if $C_{i,j} = 0$, then there is no fundamental relationship and the change in one will not necessarily reflect a change in the other.Now, consider $\frac{\partial{p_i}}{\partial{p_j}}$, this is the amount of change in $p_i$ due to a small change in $p_j$ and this partial derivative will once again be zero if there is no relationship between $p_i$ and $p_j$.Is it possible to consider, in any way, $C_{i,j} \approx \frac{\partial{p_i}}{\partial{p_j}}$? Are there conditions under which this may be the case or any other relationship between these two measures?
Is there a relationship between the covariance matrix and the partial derivative?
statistics;data analysis;calculus