High-resolution multidimensional parametric estimation for nuclear magnetic resonance spectroscopy

<p>Nuclear Magnetic Resonance spectroscopy (NMR) is a powerful technique for rapid and efficient quantitation of compounds in chemical samples. NMR causes the nuclei in the molecules to resonate and various chemical arrangements appear as peaks in the Fourier spectrum of a free induction decay (FID). The spectral parameters elicited from the peaks serve as a fingerprint of the chemical components contained in the molecule. These fingerprints can be employed to understand the chemical structure. Signal acquired from a NMR spectrometer is ideally modelled as a superposition of multiple damped complex exponentials (cisoids) in Additive White Gaussian Noise (AWGN). The number as well as the spectral parameters of the cisoids need to be estimated for characterisation of the underlying chemicals. The estimation, however, suffers from numerous difficulties in practice. These include: unknown number of cisoids, large signal length, large dynamic range, large peak density, and numerous distortions caused by experimental artefacts. This thesis aims at the development of estimators that, in view of the above-mentioned practical features, are capable of rapid, high-resolution and apriori-information-free quantitation of NMR signals. Moreover, for the analytic evaluation of the performance of such estimators, the thesis aims to derive interpretable analytic results for the fundamental estimation theory tool for assessing the performance of an unbiased estimator: the Cramer Rao Lower Bound (CRLB). By such results, we mean those that analytically allow the determination, in terms of the CRLB, of the impact of the free model parameters on the estimator performance. For the CRLB, we report analytic expressions on the variance of unbiased parameter estimates of damping factors, frequencies and complex amplitudes of an arbitrary number of damped cisoids embedded in AWGN. In addition to the CRLB, analytic expressions for the determinant and the condition number of the associated Fisher Information Matrix (FIM) are also reported. Further results, in similar order, are reported for two special cases of the damped cisosid model: the Magnetic Resonance Relaxometry model and the amplitude-only model (employed in quantitative NMR - qNMR). Some auxiliary results for the above-mentioned models are also presented, i.e., on the multiplicity of the eigenvalues and the factorisation of the characteristic polynomial associated with their respective FIMs. These results have not been previously reported. The reported theoretical results successfully account for various physical and chemical phenomena observed in experimental NMR data, and quantify their impact on the accuracy of an unbiased estimator as a function of both model and experimental parameters, e.g., influence of prior knowledge, peak multiplicity, multiplet symmetry, solvent peak, carbon satellites, etc. For rapid, high-resolution and apriori-information-free quantitation of NMR signals, a sub-band Steiglitz-McBride algorithm is reported. The developed algorithm directly converts the time-domain FID data into a table of estimated amplitudes, phases, frequencies and damping factors, without requiring any previous knowledge or pre-processing. A 2D sub-band Steiglitz-McBride algorithm, for the quantitation of 2D NMR data in a similar manner, is also reported. The performance of the developed algorithms is validated by their application to experimental data, which manifests that they outperform the state-of-the-art in terms of speed, resolution and apriori-information-free operation.</p>

High-resolution multidimensional parametric estimation for nuclear magnetic resonance spectroscopy

<p>Nuclear Magnetic Resonance spectroscopy (NMR) is a powerful technique for rapid and efficient quantitation of compounds in chemical samples. NMR causes the nuclei in the molecules to resonate and various chemical arrangements appear as peaks in the Fourier spectrum of a free induction decay (FID). The spectral parameters elicited from the peaks serve as a fingerprint of the chemical components contained in the molecule. These fingerprints can be employed to understand the chemical structure. Signal acquired from a NMR spectrometer is ideally modelled as a superposition of multiple damped complex exponentials (cisoids) in Additive White Gaussian Noise (AWGN). The number as well as the spectral parameters of the cisoids need to be estimated for characterisation of the underlying chemicals. The estimation, however, suffers from numerous difficulties in practice. These include: unknown number of cisoids, large signal length, large dynamic range, large peak density, and numerous distortions caused by experimental artefacts. This thesis aims at the development of estimators that, in view of the above-mentioned practical features, are capable of rapid, high-resolution and apriori-information-free quantitation of NMR signals. Moreover, for the analytic evaluation of the performance of such estimators, the thesis aims to derive interpretable analytic results for the fundamental estimation theory tool for assessing the performance of an unbiased estimator: the Cramer Rao Lower Bound (CRLB). By such results, we mean those that analytically allow the determination, in terms of the CRLB, of the impact of the free model parameters on the estimator performance. For the CRLB, we report analytic expressions on the variance of unbiased parameter estimates of damping factors, frequencies and complex amplitudes of an arbitrary number of damped cisoids embedded in AWGN. In addition to the CRLB, analytic expressions for the determinant and the condition number of the associated Fisher Information Matrix (FIM) are also reported. Further results, in similar order, are reported for two special cases of the damped cisosid model: the Magnetic Resonance Relaxometry model and the amplitude-only model (employed in quantitative NMR - qNMR). Some auxiliary results for the above-mentioned models are also presented, i.e., on the multiplicity of the eigenvalues and the factorisation of the characteristic polynomial associated with their respective FIMs. These results have not been previously reported. The reported theoretical results successfully account for various physical and chemical phenomena observed in experimental NMR data, and quantify their impact on the accuracy of an unbiased estimator as a function of both model and experimental parameters, e.g., influence of prior knowledge, peak multiplicity, multiplet symmetry, solvent peak, carbon satellites, etc. For rapid, high-resolution and apriori-information-free quantitation of NMR signals, a sub-band Steiglitz-McBride algorithm is reported. The developed algorithm directly converts the time-domain FID data into a table of estimated amplitudes, phases, frequencies and damping factors, without requiring any previous knowledge or pre-processing. A 2D sub-band Steiglitz-McBride algorithm, for the quantitation of 2D NMR data in a similar manner, is also reported. The performance of the developed algorithms is validated by their application to experimental data, which manifests that they outperform the state-of-the-art in terms of speed, resolution and apriori-information-free operation.</p>

The Ecological Interpretation of Unbiased Estimator for the Taxonomic Ratio: Different Approaches for Local and Regional Flora

Taxonomic ratio in an ecological context is considered as an indicator of the level of competitive exclusion. In spite of more than a century of discussions on taxonomic ratio, the problem of finding an unbiased estimator for flora characterisation remains unsolved. The traditional form of taxonomic ratio (species/genus or species/families ratio) is biased, which depends on the area of territory for which the floral composition was established. This circumstance makes the taxonomic ratio an inadequate characteristic of the flora. To solve the problem of finding an unbiased estimator for the taxonomic ratio, we have combined two fundamental ecological generalisations. The first is that species that belong to the same genus usually live in similar habitats and have similar morphological features. The struggle for life between species from the same genus is, therefore, more intense than between species from different genera. The second is species–area relationship. We have considered the problem of finding an unbiased taxonomic relationship using the Arrhenius curves to fit species–area relationships. This combination allowed us to find a form of unbiased taxonomic relationship. The example of Cyanophyceae flora shows that this indicator is closely related to a wide range of ecological and biogeographical characteristics of vegetation. The residual of the linear equation of dependence of the logarithm of the number of species on the logarithm of the number of genera is an unbiased indicator of the taxonomic relation, which is independent of the number of genera (or number of families) and the sampling size (or area). An unbiased taxonomic relationship is a characteristic of regional flora, which depends on a wide range of its ecological and biogeographical features.

Ratio in Ratio Type Exponential Strategy for the Estimation of Population Mean

This paper is an attempt to develop an estimator for finite population mean. Motivated by Kiregyera (1984), a ratio in ratio type exponential strategy is developed for estimation of population mean in double sampling for stratification. To compare with relevant considered estimators, expressions for bias and mean squared error of the developed estimator have been derived. The developed estimator has been compared with usual unbiased estimator, Ige and Tripathi (1987), ratio estimator and ratio type exponential estimator given by Tailor et al (2014) theoretically as well as empirically.

Equalities between the BLUEs and BLUPs under the partitioned linear fixed model and the corresponding mixed model

In this article we consider the partitioned fixed linear model F : y = X1β1 + X2β2 + ε" and the corresponding mixed model M : y =X1β1+X2u+ ε, where ε is a random error vector and u is a random effect vector. In 2006, Isotalo, M¨ols, and Puntanen found conditions under which an arbitrary representation of the best linear unbiased estimator (BLUE) of an estimable parametric function of β1 in the fixed model F remains BLUE in the mixed model M . In this paper we extend the results concerning further equalities arising from models F and M.

Sensitive proportion in ranked set sampling

This paper considers the concomitant-based rank set sampling (CRSS) for estimation of the sensitive proportion. It is shown that CRSS procedure provides an unbiased estimator of the population sensitive proportion, and it is always more precise than corresponding sample sensitive proportion (Warner SL (1965)) that based on simple random sampling (SRS) without increasing sampling cost. Additionally, a new estimator based on ratio method is introduced using CRSS protocol, preserving the respondent’s confidentiality through a randomizing device. The numerical results of these estimators are obtained by using numerical integration technique. An application to real data is also given to support the methods.

Visible Light Positioning with Multiple Transmitters and Multiple Receiving Sensors

Visible light communications (VLC) based positioning systems will form an important part of the future generation wireless communication systems because they offer higher accuracy for indoor positioning as compared to radio frequency based systems. In this paper, we have used non Bayesian statistical signal processing techniques for the hybrid time-of-arrival/received-signal-strength and hybrid time-difference-of-arrival/received-signal-strength based positioning. These hybrid measurements are combined with the following fusion algorithms: weighted least squares and the best linear unbiased estimator. These two fusion algorithms are compared in terms of the mean Euclidean error as a function of various parameters such as signal-to-noise ratio, transmitter arrangement and synchronization error. Even though the performance of the weighted least squares algorithm is better, the best linear unbiased estimator is still an attractive algorithm for systems that require a lower complexity.

Determination of CRLB and Minimum Variance Unbiased Estimator of a DC Signal in AWGN Using Laplace Transform

This paper presents an alternative approach for the determination of Cramer-Rao Lower Bound (CRLB) and Minimum Variance Unbiased Estimator (MVUE) using Laplace transformation. In this work, a DC signal in Additive White Gaussian Noise (AWGN) was considered. During the investigation, a number of experiments were conducted to analyze different possible outputs under different conditions, and then the patterns of the outcomes were studied. Finally closed-form expressions for the CRLB and MVUE were deduced employing the Laplace transformation. The resulting expressions show that the proposed method has almost the same number of steps as the existing method. However, the later requires only the knowledge of algebra to arrive at the CRLB expressions contrary to the existing approach where a strong mathematical background is required and hence making it superior over the existing method, in that sense.