Data Analysis in the Shot Noise Limit Part II: Methods for Data Regression

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Analytical Chemistry

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The equations for multiple parameter estimation are found by applyhg the method of maximum Ilkellhood to Bayes’ theorem probability dlstrlbution for data obtained In the shot noise limit. Bayes’ theorem limits the estimates by the a prlorl probability restrlctlng the Poisson distribution parameters to be positlve. Bayes’ theorem is applied prior to formulation of the total probability. Subsequent applcation of the maxlmum Ilkellhood method results in nonlinear equations describing the coefficients for the independent processes. The main innovation of these equatlons over the usual mlnlmum covariance ones is that there is an additional term which can take on real values such that the solution is obtained within the bounds prescribed by the Bayes probability. It is reasoned that by the presence of this term, the simultaneous equations for the maxlmum Ilkellhood estlmates will be Independent of regressors which would yield negative parameters by their inclusion in the basis set. Three methods for coefficient estimatlon based on these nonlinear equatlons using conventional matrix methods are tested on synthetic data. It is found that the row-welghted least-squares equations do not perform as well as does a direct interpolatlon solution of the governing equations.

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