Statistical characterisation

Summarizing, the statistical characterisation of the random process (mean and covariance) can be projected through the interval tk < t < tk+1, and in this process there is an input noise that will increase the error, damaging the quality of the estimate. 

Although there are several techniques for estimating the reaction rate constants based upon the deterministic model, these methods are usually rather complicated, and the results cannot be statistically characterised. That is why from time to time estimates based upon one or another stochastic model are suggested. Such a suggestion has earlier been described under the name fluctuation-dissipation theorem , and similar methods have been presented by Mulloolly (1971, 1972, 1973), Hilden (1974), and Matis and Hartley (1971). 

However, if the probe is used as linear scanning system, the acoustic beam depends on the element characteristics which are liable to change from one element to an other. Therefore, the only two alternative proposals are to characterise the aeoustie behaviour of all active sub-set of elements or to proeeed to a statistical characterization. 

To reiterate a point that we made earlier, these problems of accurately calculating the free energy and entropy do not arise for isolated molecules that have a small number of well-characterised minima which can all be enumerated. The partition function for such systems can be obtained by standard statistical mechanical methods involving a summation over the mini mum energy states, taking care to include contributions from internal vibrational motion. 

Helfand and Tagami [75,76] introduced a model which considered the probability that a chain of polymer 1 has diffused a given distance into polymer 2 when the interactions are characterised by the Flory-Huggins interaction parameter x They predicted that at equilibrium the thickness , d c, of the interface would depend upon the interaction parameter and the mean statistical segment length, b, as follows ... 

The characterisation of the toxicity associated with a specific chemical is invariably dependent on feeding the chemical to laboratory animals in statistically-based lifetime bioassay, guidelines for which have been agreed at the international level by the OECD (OECD, 1981) and the IPCS (IPCS, 1999), and which are undertaken strictly in compliance with the principles of good laboratory practice (GLP). The animals used in long-term bioassays are... 

Whereas in the second approach of the size effects it is also assumed that fracture is controlled by defects, the strength is now considered a statistically distributed parameter rather than a physical property characterised by a single value. The statistical distribution of fibre strength is usually described by the Weibull model [22,23]. In this weakest-link model the strength distribution of a series arrangement of units of length L0 is given by... 

The eigenvalues of SG are here characterised in terms of two quantum numbers, an angular momentum quantum number m and a second quantum number counting the eigenvalues in each m manifold. If the spectra for different m are uncorrelated, one expects Poisson statistics of the total spectrum in the limit n — oo. 

The assessment and quantification of the remaining reserves and resources of fossil fuels is a very complex and broad field, characterised by a lack of internationally harmonised definitions and standards, great data uncertainties and discrepancies and, consequently, the potential danger of data abuse for political purposes. Within the scope of this publication, only an overview of the range of the currently available estimates of fossil resources is provided and the focus is rather on the general discussion of potential sources of uncertainty, than on a detailed assessment of the different methodological and statistical approaches and discrepancies at country or even field level. 

A Gaussian curve is characterised by peak position, peak height and peak width. Commonly, the half width is used, i.e. the width at half peak height. We accommodate this convention. In statistics, the Gaussian distribution is usually normalised to unit integral, however, this is not useful in the present context. 

They can easily be characterised by standard deviations (ot the statistical distributions)... 

The method which satisfies these conditions is partial least squares (PLS) regression analysis, a relatively recent statistical technique (18, 19). The basis of tiie PLS method is that given k objects, characterised by i descriptor variables, which form the X-matrix, and j response variables which form the Y-matrix, it is possible to relate the two blocks (or data matrices) by means of the respective latent variables u and 1 in such a way that the two data sets are linearly dependent ... 

There are many other distributions used in statistics besides the normal distribution. Common ones are the yl and the F-distributions (see later) and the binomial distribution. The binomial distribution involves binomial events, i.e. events for which there are only two possible outcomes (yes/no, success/failure). The binomial distribution is skewed to the right, and is characterised by two parameters n, the number of individuals in the sample (or repetitions of a trial), and n, the true probability of success for each individual or trial. The mean is n n and the variance is nn(l-n). The binomial test, based on the binomial distribution, can be used to make inferences about probabilities. If we toss a true coin a iarge number of times we expect the coin to faii heads up on 50% of the tosses. Suppose we toss the coin 10 times and get 7 heads, does this mean that the coin is biased. From a binomiai tabie we can find that P(x=7)=0.117 for n=10 and n=0.5. Since 0.117>0.05 (P=0.05 is the commoniy... 

The Austrian industrial landscape 2003 is characterised by small and medium sized enterprises (SME). A special survey of the Leistungs- und Strukturstatistik (performance and structural statistics) 2003 showed that 91.3% of the enterprises employed less than 250 people. 

The summary statistics for accuracy described in the previous section are just average statistics for the whole set of samples. They are important, because they allow monitoring of changes when the calibration model is optimised i.e. a dilferent data pretreatment or optimal number of factors is used). However, they do not provide an indication of the uncertainty for individual predicted concentrations. Uncertainty is defined as aparameter, associated with the result of a measurement, which characterises the dispersion of the values that could reasonably be attributed to the measurancT [60]. Therefore, uncertainty gives an idea of the quality of the result since it provides the range of values in which the analyst believes that the true concentration of the analyte is situated. Its estimation is a requirement for analytical laboratories [61] and is especially important when analytical results have to be compared with an established legal threshold. 

In this work, we have shown that the effects of the exchange processes and the scalar couplings can be separated in the simulation. The spin interactions are described by quantum mechanics while the dynamic effects are characterised by statistical methods. The easiest way to handle the latter one is by means of the KMC simulation. 

One says that the microstate of a macromolecule is determined, if a sequence of atoms, the distances between atoms, valence angles, the potentials of interactions and so on are determined. The statistical theory of long chains developed in considerable detail in monographs (Birshtein and Ptitsyn 1966 Flory 1969) defines the equilibrium quantities that characterise a macromolecule in a whole as functions of the macromolecular microparameters. 

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