# Linearity in local sensitivity analysis

As it is know local sensitivity analysis attempt to quantify the local impact of input factors on the model, through partial derivatives: a derivative of the outputs accordingly to the inputs, when the last ones are slightly perturbed. Many methods are used to estimates this derivatives: Direct method, Green Function Method, etc.

However, the input-output relationship is generally assumed to be linear (from Saltelli textbooks on the subject). Why it is assumed to be linear in a local neighborhood, even the whole system may not be linear. I guess it may have a relation with Taylor expansion expression. I am trying to understand the real reason behind this assumption. Thank you.

• There's no need to invoke Taylor: "locally linear" means the same thing as differentiable. As soon as you frame the sensitivity analysis in terms of partial derivatives you have, a fortiori, assumed local linearity.
– whuber
Dec 5, 2012 at 17:27
• Okay thanks a lot. I guess this answer to the question. However, the local sensitivity analysis is reduced only to analysis via derivates, I am not aware of other techniques. Is it correct therefore to assume this linearity, knowing that local analysis is carried on systems which may not be linear. Thus, the shape of output function can be completly unknown (even the continuity is not sure), as some hard optimization problems.
– omar
Dec 5, 2012 at 18:17
• If the system is not even continuous, you will have a hard time computing partial derivatives! You might want to check whether people aren't actually computing finite differences--which work regardless of differentiability or continuity--and are perhaps only calling them derivatives.
– whuber
Dec 5, 2012 at 18:18
• Yes, Finite-Differences are used to approximate these derivatives, but I think they are slower and less accurate.
– omar
Dec 5, 2012 at 18:39
• Your last two comments are confusing: the penultimate one assumes the system is differentiable while the ultimate one asks about non-differentiable methods, even though the penultimate complains that these are "slower and less accurate." It would be nice if in your question you would clarify what assumptions you want to make and precisely what kinds of alternatives you are looking for.
– whuber
Dec 5, 2012 at 21:05