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Extra resources for Parameter Estimation and Inverse Problems (International Geophysics)

Example text

72) i=1 is a nondifferentiable function of m at each point where m = di . The good news is that f (m) is a convex function of m. Thus any local minimum point is also a global minimum point. We can proceed by finding f (m) at those points where it is defined, and then separately consider the points at which the derivative is not defined. Every minimum point must either have f (m) undefined or f (m) = 0. 73) i=1 where the signum function, sgn(x), is −1 if its argument is negative, 1 if its argument is positive, and 0 if its argument is zero.

38). 50) will include the true values of the parameters. The reason is that these intervals, considered together as a region, include many points that lie outside of the 95% confidence ellipsoid. 25) contains information only about where and how often we made measurements, and on what the standard deviations of those measurements were. Covariance is thus exclusively a characteristic of experimental design that reflects how much influence the noise in a general data set will have on a model estimate.

For example, in linear regression to a line, it might be that all of the residuals are negative for small and large values of the independent variable x but positive for intermediate values of x. This would suggest that perhaps an additional quadratic term is required in the regression model. Parameter estimates obtained via linear regression are linear combinations of the data. 18). If the data errors are normally distributed, then the parameter estimates will also be normally distributed because a linear combination of normally distributed random variables is normally distributed [5, 22].

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