= Linearized Variance =

'''Linearization''' is a method for approximating variance of a non-linear estimator.

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== Description ==

If an estimator is of the form ''X = f(Y,,1,,``, Y,,2,,``, ... Y,,n,,)'' where the ''Y,,i,,'' are themselves estimators, and if ''f'' is a linear function, then the [[Analysis/Variance|variance]] of ''X'' is straightforward to calculate using the basic properties of variance under linear transformations. If ''f'' is non-linear but smooth, then the function can be approximated with [[Calculus/TaylorSeries|Taylor linearization]]. This allows for an approximation of the variance of ''X''.

Given a smooth ''f'', the first order Taylor approximation (by convention, calculated around the mean) is:

{{attachment:taylor1.svg}}

Taking the variance of both sides, noting that ''Var[a + b] = Var[a] + Var[b]'':

{{attachment:taylor2.svg}}

The first term on the RHS is a constant, so has zero variance. The equation reduces to:

{{attachment:taylor3.svg}}

More properly, the estimator should be described as being of the form ''X = F(Y) = <f,,1,,(Y)``, f,,2,,(Y)>'', and the linearization is in terms of a [[Calculus/JacobianMatricesAndDeterminants|Jacobian matrix]] ''J''. Therefore, ''X ≈ F(μ) + J(μ)(Y - μ)''. Note that linear transformations operate on a [[Analysis/Covariance|covariance matrix]] as ''Var[AX] = A Var[X] A^T^''.

{{attachment:taylor4.svg}}



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CategoryRicottone