Statistical analysis physical observations

The classical microscopic description of molecular processes leads to a mathematical model in terms of Hamiltonian differential equations. In principle, the discretization of such systems permits a simulation of the dynamics. However, as will be worked out below in Section 2, both forward and backward numerical analysis restrict such simulations to only short time spans and to comparatively small discretization steps. Fortunately, most questions of chemical relevance just require the computation of averages of physical observables, of stable conformations or of conformational changes. The computation of averages is usually performed on a statistical physics basis. In the subsequent Section 3 we advocate a new computational approach on the basis of the mathematical theory of dynamical systems we directly solve a... 

In view of this difficulty, verifiable statements about measured physical quantities cannot be made with unlimited exactness, but always imply an uncertainty. Nevertheless, from a large number of individual measurements one can arrive by statistical analysis at values that have a high probability of being correct and this probability can also be calculated. Properly speaking, then, any distance from the nucleus is possible for the electron however, some distances are more probable than others and there is also a most probable distance. The discovery of Heisenberg thus forces us into devising a theory that not only makes pertinent predictions but at the same time qualifies these predictions by stating a probability that the expectation is observed. 

Consideration of the thermohaline structure of the Black Sea provides new results on the statistical and physical analysis of the historical data of ship-borne observations of the vertical profiles of the temperature and salinity of the waters. The general features of the vertical thermohaline structure of the Black Sea waters, the seasonal and interannual variabilities of the horizontal structure of the temperature and salinity in all the main water layers are described. The relations of the large-scale features of the hydrology of the Black Sea waters to external forcing (heat and moisture fluxes across the water surface, river mouths and straits, fluxes of the momentum and relative vorticity of wind) are shown. The generalization of the results of the studies of the T,S-structure of the Black Sea waters and of its seasonal and interannual variability allows the following conclusions to be made. 

General uncertainty of measurement is said to be dispersed between the measured values and parameter values that are interrelated. In other words, in the course of measurement of physical quantities, such as personnel, equipment and measurement of environmental impact, the degree of uncertainty for measurement result. And evaluation of uncertainty in measurement is by means of a quantitative determine the extent of such uncertainty, the purpose of which is to determine the distribution of measurement range and measurement result reliability level. Observation column evaluation results of the statistical analysis of standard uncertainty is called type A standard uncertainty. Are different from those used for observing statistical analysis to evaluate the uncertainty of standard uncertainty is known as type B standard uncertainty . 

Unfortunately, some of the observations and interpretations presented in Chapter 7 ("Microscopical Interpretation of Clinkers") do not appear to be founded in systematic experimental design or statistical analysis. Statistical measures to determine the degrees of correlation and association of the observations, and their relationship to the various physical and chemical causal factors of the production process, are essential and urgently needed for several very important reasons ... 

In Voortman, two different possibilities of modeling the distribution of water levels are examined (a) by direct statistic analysis of the observed water levels and (b) by combining the probability distribution of wind speed with a relevant physical model. Realizing that physical models are often imperfect, in option (b) an estimate of the model uncertainty is necessary. 

The data features are being used for a couple of different reasons. One reason is that a complex process frace with thousands of points can be summarized by only a few data features that are representative of the observed behavior. This helps to reduce the dimensionahty of the data, which will lead to high quality predictive noodels and allows for instantaneous multivariate statistical analysis. Another reason is that certain process behaviors may be evaluated through specific data features, wha-eas, the raw data may not catch the behaviors. For example, there may be some physical variables that are manifested in a combination of process signals. For example, the melt viscosity is a function of the material type, melt pressure, ram velocity, and melt tenperature. Without defining representative data features, it is highly unlikely that the effect of this variable would be explicit the system or operator would not be easily able to quantify the external variable. 

In an attempt to address these questions, a modern method of statistical physics was recently applied by Varotsos et al. (2007) to C02 observations made at Mauna Loa, Hawaii. The necessity to employ a modern method of C02 data analysis stems from the fact that most atmospheric quantities obey non-linear laws, which usually generate non-stationarities. These non-stationarities often conceal existing correlations between the examined time series and therefore, instead of applying the conventional Fourier spectral analysis to atmospheric time series, new analytical techniques capable of eliminating non-stationarities in the data should be utilized (Hu et al., 2001 Chen et al., 2002 Grytsai et al., 2005). 

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