Overview
Consider the function.
{% f(x_{1}, x_{2}) = x_1 \times x_2^2 %}
And consider two points
{% (x_{1}, x_{2}) = (1,1) %}
{% (x_{1}, x_{2}) = (2,2) %}
Then the value of {% f %} changes from {% 1 %} to {% 8 %}. How much of the change was due to the change in {% x_1 %} versus {% x_2 %}?
A common example of non linear attribution is to ask how much a particular asset contributes to the overall risk of a portfolio. Because there correlations between assets that are less than 1, there is a diversification effect. In such a case, it may be meaningless to ask how much a particular asset contributes to the risk (except in the marginal case)