Nonlinear Estimation - USGS
Nonlinear Estimation
A Simple Explanation of Nonlinear Estimation
Nonlinear estimation uses linear estimation iteratively applied to linear approximations of the model until coefficients converge.
Review of Linear Estimation
Linear Model
[pic]
y is an N by 1 vector, X is N by K matrix, where K is the number of explanatory variables, θ is a K by 1 vector of unknown coefficients, and e is an N by 1 vector of errors.
Coefficient Estimates
[pic]
Covariance Matrix
[pic]
where [pic] is the estimated mean squared model error.
Nonlinear Model Estimation
[pic]
where [pic] is an N by 1 vector function of the K by vector of coefficients.
First-order Taylor Series Approximation
[pic]
where [pic] is the N by 1 vector function evaluated at the initial coefficient estimate [pic], and [pic] is the N by K matrix of derivatives of G with respect to θ, evaluated at the initial coefficient estimate [pic].
Redefine the following variables,
[pic]
The next iteration estimate of θ is least squares applied to the transformed data:
[pic]
Replace [pic] by [pic], and iterate until convergence at [pic].
Coefficient Asymptotic Covariance Matrix
[pic]
where [pic] is the estimated mean squared error (i.e., the variance of [pic]), and [pic] is evaluated at [pic].
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