Statistical estimation

Statistical estimation theory provides a powerful framework for the development of retrieval methods. Rodgers (2000) gives an excellent overview of this subject. Algorithms which yield optimum solutions in the statistical sense have found extensive applications in the remote temperature sounding of the terrestrial atmosphere where much a priori information is available. 

As an example of the statistical approach we consider a formulation that follows the works of Foster (1961), Strand Westwater (1968), and Rodgers (1970). Consider a representative ensemble of atmospheric temperature profiles defined by a mean profile (T) and a covariance matrix 

The /th element of the vector T is the temperature at the tth atmospheric level, and the angular brackets denote an ensemble mean. Expansion of B [v, T(z)] about the mean temperature profile, and substitution into Eq. (8.2.1), permits us to write an expression in discrete form analogous to Eq. (8.2.3), 

The matrix L is chosen such that the mean square deviation of the solution from the temperature profiles of the ensemble is a minimum, i.e.. 

This approach works best when the solution does not deviate greatly from the mean properties of the statistical ensemble. By exploiting correlations between large- and small-scale vertical structure in the profiles, the method can yield solutions with higher vertical resolution than that intrinsic to the radiance measurements alone. Note that Eq. (8.2.35) is formally equivalent to the constrained solution defined by Eqs. (8.2.8) and (8.2.9) with y = 1. However, the matrix S is given a different conceptual interpretation. 

Engineers are often faced with the problem of using a set of data to calculate quantities that they hope will describe the behavior of the process from which the data were taken. Because the measured process variable may be subject to random fluctuations as well as to random errors of measurement, the engineers calculated estimate is subject to error, but how much Here is where the method of statistical estimation can help. 

Statistical estimation uses sample data to obtain the best possible estimate of population parameters. The p value of the Binomial distribution, the p value in Poison s distribution, or the p and a values in the normal distribution are called parameters. Accordingly, to stress it once again, the part of mathematical statistics dealing with parameter distribution estimate of the probabilities of population, based on sample statistics, is called estimation theory. In addition, estimation furnishes a quantitative measure of the probable error involved in the estimate. As a result, the engineer not only has made the best use of this data, but he has a numerical estimate of the accuracy of these results. 

Estimates of two kinds can be made, point estimate and interval estimate. 

Point estimate uses the sample data to calculate a single best value, which estimates a population parameter. The point estimate is one number, a point on a numeric axis, calculated from the sample and serving as approximation of the unknown population distribution parameter value from which the sample was taken. Such a point estimate alone gives no idea of the error involved in the estimation. If parameter estimates are expressed in ranges then they are called interval estimates. 

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So basic is the notion of a statistical estimate of a physical parameter that statisticians use Greek letters for the parameters and Latin letters for the estimates. For many purposes, one uses the variance, which for the sample is s and for the entire populations is cr. The variance s of a finite sample is an unbiased estimate of cr, whereas the standard deviation 5- is not an unbiased estimate of cr. 

WR Pearson. Empirical statistical estimates for sequence similarity searches. J Mol Biol 276 71-84, 1998. 

The model of simple competitive antagonism predicts that the slope of the Schild regression should be unity. However, experimental data is a sample from the complete population of infinite DR values for infinite concentrations of the antagonist. Therefore, random sample variation may produce a slope that is not unity. Under these circumstances, a statistical estimation of the 95% confidence limits of the slope (available in most... 

A first evaluation of the data can be done by running nonparametric statistical estimation techniques like, for example, the Nadaraya-Watson kernel regression estimate [2]. These techniques have the advantage of being relatively cost-free in terms of assumptions, but they do not provide any possibility of interpreting the outcome and are not at all reliable when extrapolating. The fact that these techniques do not require a lot of assumptions makes them... 

Patino-Leal, H., and P.M. Reilly, "Statistical Estimation of Parameters in Vapor-Liquid Equilibrium", AIChEJ., 28(4), 580-587 (1982). 

The National Center for Health Statistics estimates that 15.8 million Americans have signs and symptoms of OA. The true extent of the disease is much larger nearly everyone has radiographic evidence of OA by the eighth decade of life, but individuals without symptoms often go undiagnosed. Approximately 6% of United States adults have daily symptomatic knee OA, and 3% report daily symptoms affecting the hip.2 After age 60, 10% to 15% of persons report such symptoms. 

HG Boxenbaum, S Riegelman, RM Elashoff. Statistical estimation in pharmacokinetics. J Pharmacokin Biopharm 2 123-148, 1974. 

G Test statistic (estimate) of Grubb s outlier test (4.36)... 

Number of subsamples (in sampling) Test statistic (estimate) of the David test... 

A2 Test statistic (estimate) for von Neumann s trend test (4.34)... 

Fisher RA (1925) Theory of statistical estimation. Proc Cambridge Phil Soc 22 700... 

On the basis of spot position and intensity, by assuming different critical interdistance Ax o values, experimental points can be obtained and fitted by a straight line (Eq. 4.13) whose slope represents a statistical estimation of m, the estimated number of single components. The values estimated for each ID strip were added to obtain the total number of proteins (Pietrogrande et al., 2002). 

Table 12. The statistical estimation of the morbidity in different regions of biosphere, Ukraine, using the deviations from normalized mean values of morbidity (cr-values) (after Evstafyeva et al., 1999).

The statistical estimation of heavy metal concentrations in the Spruce Forest ecosystems of the Boreal climatic zone is the subject of wide variation, with coefficient of variation from 36 to 330%. However, we can note the clear trend in biogeochemical peculiarities and relevant exposure to heavy metal uptakes by dominant plant species. 

This procedure is widely used when extracting statistical estimates from DNS data. 

Although not denoted explicitly, we have seen in Section 6.8 that this estimate will depend on the grid spacing M and the number of particles Nv. In addition to the mean composition, the output data from the PDF code will usually be various composition statistics estimated at grid-cell centers. We will thus need accurate and efficient statistical estimators for determining particle fields given the ensemble of Nv notional particles. 

As already noted in Section 1.6.1, many statistical estimators rely on symmetry of the data distribution. For example, the standard deviation can be severely increased if the data distribution is much skewed. It is thus often highly recommended to first transform the data to approach a better symmetry. Unfortunately, this has to be done for each variable separately, because it is not sure if one and the same transformation will be useful for symmetrizing different variables. For right-skewed data, the log transformation is often useful (that means taking the logarithm of the data values). More flexible is the power transformation which uses a power p to transform values x into xp. The value of p has to be optimized for each variable any real number is reasonable for p, except p 0 where a log-transformation has to be taken. A slightly modified version of the power transformation is the Box Cox transformation, defined as... 

Honda M (1988) Statistical estimation of the production of cosmic-ray induced nuclides in meteorites. Meteoritics 23 3-12... 

One of the best examples of a precise statistical estimation of counting, analyzing, and drawing conclusions was the exact forecast of the eclipse of the sun In 585 B.C. made by Thales of Millet ( 1 ). 

It is not the purpose of this paper at this moment to investigate further for more detailed reasons for discrepancies in confidence bands or estimated amount intervals. That will be investigated fully at a later time. I do wish to point out that the assumptions one makes about the information he has and the statistical approaches he makes profoundly affect the resultant error calculations. Far from being a staid and dormant subject matter, statistical estimations of error are currently very actively being studied in order for scientific workers and citizens alike to be informed about the error in their work. 

A class of statistical methods frequently used to analyze kinetic and thermodynamic data. Most of these methods require a preliminary estimate of the constants followed by cycles of iterative calculations that converge on a final value(s). Cleland has presented a protocol for the statistical estimation of kinetic data. A nonlinear analysis has also been applied to progress curves. ... 

Hence, the temperature coefficient of k1 having been measured, for an absolute calculation of k only kt) and bo must be known, and not the heat of adsorption, X. At the moment we are concerned with b0. A simple statistical estimate can be based on the assumption that in the absence of adsorption energy the adsorption space is filled at a proportion given by the ratio of the molecular adsorption volume (liquid volume Fm) to the molecular gas volume... 

A particular situation where bias may be important is in statistical meta-analysis, where statistical estimates are combined across studies. When estimates from individual studies may be averaged arithmetically, it is better to average unbiased estimates (Rao 1973, Section 3a). In case of biases that are consistent across studies, an arithmetic average would have a bias of the same sign, regardless of the number of studies included in the analysis. The average of biased estimates could fail to be consistent (in the statistical sense). 

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