Cram�r-Rao bounds: an evaluation tool for quantitation (original) (raw)
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Journal of Magnetic Resonance, 2000
We have derived analytical expressions of the Cramer-Rao lower bounds on spectral parameters for singlet, doublet, and triplet peaks in noise. We considered exponential damping (Lorentzian lineshape) and white Gaussian noise. The expressions, valid if a sufficiently large number of samples is used, were derived in the time domain for algebraic convenience. They enable one to judge the precision of any unbiased estimator as a function of the spectral and experimental parameters, which is useful for quantitation objectives and experimental design. The influence of constraints (chemical prior knowledge) on parameters of the peaks of doublets and triplets is demonstrated both analytically and numerically and the inherent benefits for quantitation are shown. Our expressions also enable analysis of spectra comprising many peaks. Copyright 2000 Academic Press.
Information and Statistical Efficiency When Quantizing Noisy DC Values
IEEE Transactions on Instrumentation and Measurement, 2015
This paper considers estimation of a quantized constant in noise when using uniform and non-uniform quantizers. Estimators based on simple arithmetic averages, on sample statistical moments and on the maximum-likelihood procedure are considered. It provides expressions for the statistical efficiency of the arithmetic mean by comparing its variance to the proper Cramér-Rao lower bound. It is conjectured that the arithmetic mean is optimal among all estimators with an exactly known bias. Conditions under which its statistical performance are improved by the other estimation procedures when the exact bias is not known are found and analyzed. Using simulations and analysis of experimental data, it is shown that both momentbased and maximum-likelihood-based estimators provide better results when the noise standard deviation is comparable to the quantization step and the noise model of quantization can not be applied.
Environmental Science & Technology, 2000
The effects of resolution, spectral window, and background type on the predictive ability of classical least squares regression (CLS) on spectra measured by an open-path Fourier transform (OP/FT-IR) spectrometer were tested in some detail. It is shown that the most accurate quantitative results are obtained by using equidistant backgrounds, reduced spectral windows, and low resolution. The effect of interfering compounds is shown to be particularly serious when CLS regression is used to process OP/FT-IR spectra. FIGURE 1. "Equidistant background" spectrum obtained by ratioing two single-beam spectra measured over a path of 300 m. These two spectra were not measured on the same day, but conditions were very similar. This background is representative of the type used for the EQDI 0 data. The noise in the region of strong atmospheric absorption would be greater were it not for the presence of about 1% stray light in the single-beam background spectra. Inset are baseline spectra in the atmospheric windows with the ordinate scale expanded by a factor of 10 (from -0.01 to 0.03 AU).
An esprit-based parameter estimator for spectroscopic data
2012
Abstract The pulse spin-locking sequence is a common excitation sequence for magnetic resonance and nuclear quadrupole resonance signals, with the resulting measurement data being well modeled as a train of exponentially damped sinusoidals. In this paper, we derive an ESPRIT-based estimator for such signals, together with the corresponding Cramér-Rao lower bound.
Evaluation of Approximate Methods for Calculating the Limit of Detection and Limit of Quantification
Environmental Science & Technology, 1999
In a previous paper, a computational method was presented for determining statistically rigorous limits of detection and quantification. The main purpose of this study is to evaluate similar but less computationally complex methods. These "approximate" methods use data at multiple spiking concentrations, are iterative, can be derived from either prediction intervals or statistical tolerance intervals, and require at a minimum ordinary least-squares regression for calculating the intercept and slope. Approximate detection and quantification limits calculated for various PCB congeners were similar to those calculated using the computationally exact method. Although the exact methods should be employed whenever possible (they include uncertainty in the calibration function and provide greater weight to less variable data), approximate methods can provide detection and quantification limits that are sufficiently accurate for most applications. Practical application of multiconcentration-based methods may involve the use of routinely generated quality control data in the statistical calculations.
Fundamental noise in three chromatographic detectors
Journal of Chromatography A, 1994
Counting statistics a-e used to estimate the minimum theoretical noise of three chromatographic detectors, by assuming that the standard deviation of their baselines equals the square raot of their primary chemical. events, These primary events are taken to be the observed generation of photons in the flame photometric detector, the emission of /I rays in the electron-capture detector, and the formation of ion pairs in the flame ionization detector. The theoretically estimated and the experimentally observed noise agree in every case. This suggests that baseline noise in the three particular detectors is due, predominantly if not exclusively, to random processes involving the atomic structure of matter: therefore, It cannut be further reduced.
On the Estimation of Quantizer Reconstruction Levels
IEEE Transactions on Instrumentation and Measurement, 2000
We consider the problem of estimating the optimal reconstruction levels of a fixed quantizer, such as the amplitude quantization part of an analog-to-digital converter. A probabilistic transfer function model is applied for the quantizer. Two different assumptions are made for the transfer function, and an estimator based on order statistics is applied. The estimator turns out to give better results in terms of mean square error than the commonly applied sample mean.