By G. Larry Bretthorst
This booklet is essentially a examine record at the program of chance thought to the parameter estimation challenge. the folk who may be attracted to this fabric are physicists, chemists, economists, and engineers who've to house information each day; accordingly, we have now integrated loads of introductory and instructional fabric. anyone with the similar of the math heritage required for the graduate-level learn of physics might be capable of stick with the fabric contained during this booklet, even though no longer with no attempt. during this paintings we observe chance thought to the matter of estimating parameters in really common types. particularly while the version involves a unmarried desk bound sinusoid we exhibit that the direct program of chance conception will yield frequency estimates an order of value greater than a discrete Fourier rework in signal-to-noise of 1. Latter, we generalize the matter and exhibit that likelihood thought can separate shut frequencies lengthy after the peaks in a discrete Fourier rework have merged.
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Additional resources for Lecture Notes In Statistics Bayesian Spectrum Analysis And Parameter Estimation
But before we can turn to the model selection problem, the results of this chapter must be generalized to more complex models and it is to this task that we now turn. Chapter 3 THE GENERAL MODEL EQUATION PLUS NOISE The results of the previous chapter already represent progress on the spectral analysis problem because we were able to remove consideration of the amplitude, phase and noise level, and nd what probability theory has to say about the frequency alone. In addition, it has given us an indication about how to proceed to more general problems.
The noise is white. 1. 2. 3. 4. 5. If any of these six conditions is not met, the discrete Fourier transform may give misleading or simply incorrect results in light of the more realistic models. Not because the discrete Fourier transform is wrong, but because it is answering what we should regard as the wrong question. The discrete Fourier transform will always interpret the data in terms of a single harmonic frequency model! In Chapter 6 we illustrate the eects of violating one or more of these assumptions and demonstrate that when they are violated the estimated parameters are always less certain than when these conditions are met.
4. THE SIGNAL-TO-NOISE RATIO 47 The estimate depends on the number m of expansion functions used in the model. 18) the larger models t the data better and (d mN h ) decreases. But this should not decrease our estimate of unless that factor decreases by more than we would expect from tting the noise. The factor N=(N m 2) takes this into account. In eect probability theory tells us that m + 2 degrees of freedom should go to estimating the model parameters and the variance, and the remaining degrees of freedom should go to the noise: everything not explicitly accounted for in the model is noise.