Statistics basic terms

The term statistics basically is a summary value calculated from the observed values in... 

Multivariate statistical analysis is considered a useful tool for evaluating the significance of geochemical anomalies in relation to both any individual variable and the mutual influence of variables on each other. In basic terms, when applied to geochemistry, multivariate analysis aims to identify spatial correlations between groups of elements—lithological characteristics, enrichment phenomena, anthropogenic pollution, etc.—in a complex system and reduce a multidimensional data set to more basic components. 

The collection of data in Table 3-11 can be used to provide an illustration of the basic terms used in statistics. In the first column of figures are the observed estimates, x for a sample of radioactivity. The average or mean value of these estimates, x> is 26,644 and is a better estimate of the true amount of radioactivity in the sample than any of the observed data. The mean value approaches the true amount of radioactivity in the sample as the number of independent estimates approaches infinity. A measure of the scatter observed in the 10 estimates is shown in the second column. These values, termed the deviations, are obtained by subtracting the mean from each of the estimates. 

Application of statistics in expert systems is a topic that fills more than a single book. However, some of the investigations presented in the next chapters are based on methods of descriptive statistics. The terms and basic concepts of importance for the interpretation of these methods should be introduced first. Algorithms and detailed descriptions can be found in several textbooks [43-47]. 

Earlier in the chapter we discussed the concept of the statistical test and defined some basic terms. In this section we take a closer look at this idea and see, through an example, how this is actually done. 

Dittrich et al. [1], described the basic terms to characterize and classify artificial chemistries. Given the statistical and qualitative features of the reaction laws and element/component representation, our proposed algorithm is an abstraction of the chemical reaction process and can be described as a constructive dynamical... 

It finely recovers the basic mono-regression correlation results of Eq. (2.28) and (2.29), thus confirming the validity of the multi-linear matrix formalism reliability nevertheless, it becomes ciunbersome in analytical terms for higher regression forms, for which the actual Spectral-SAR or Spectral-Diagonal SAR successfully overcome, see Chapter 3 of the present volume. For the moment, we go further with the statistical basics analysis. 

London [11] was the first to describe dispersion forces, which were originally termed London s dispersion forces. Subsequently, London s name has been eschewed and replaced by the simpler term dispersion forces. Dispersion forces ensue from charge fluctuations that occur throughout a molecule that arise from electron/nuclei vibrations. They are random in nature and are basically a statistical effect and, because of this, a little difficult to understand. Some years ago Glasstone [12] proffered a simple description of dispersion forces that is as informative now as it was then. He proposed that,... 

This expression has a formal character and has to be complemented with a prescription for its evaluation. A priori, we can vary the values of the fields independently at each point in space and then we deal with uncountably many degrees of freedom in the system, in contrast with the usual statistical thermodynamics as seen above. Another difference with the standard statistical mechanics is that the effective Hamiltonian has to be created from the basic phenomena that we want to investigate. However, a description in terms of fields seems quite natural since the average of fields gives us the actual distributions of particles at the interface, which are precisely the quantities that we want to calculate. In a field-theoretical approach we are closer to the problem under consideration than in the standard approach and then we may expect that a simple Hamiltonian is sufficient to retain the main features of the charged interface. A priori, we have no insurance that it... 

Multiparticle collision dynamics describes the interactions in a many-body system in terms of effective collisions that occur at discrete time intervals. Although the dynamics is a simplified representation of real dynamics, it conserves mass, momentum, and energy and preserves phase space volumes. Consequently, it retains many of the basic characteristics of classical Newtonian dynamics. The statistical mechanical basis of multiparticle collision dynamics is well established. Starting with the specification of the dynamics and the collision model, one may verify its dynamical properties, derive macroscopic laws, and, perhaps most importantly, obtain expressions for the transport coefficients. These features distinguish MPC dynamics from a number of other mesoscopic schemes. In order to describe solute motion in solution, MPC dynamics may be combined with molecular dynamics to construct hybrid schemes that can be used to explore a variety of phenomena. The fact that hydrodynamic interactions are properly accounted for in hybrid MPC-MD dynamics makes it a useful tool for the investigation of polymer and colloid dynamics. Since it is a particle-based scheme it incorporates fluctuations so that the reactive and nonreactive dynamics in small systems where such effects are important can be studied. 

The main concept addressed in this new multi-part series is the idea of correlation. Correlation may be referred to as the apparent degree of relationship between variables. The term apparent is used because there is no true inference of cause-and-effect when two variables are highly correlated. One may assume that cause-and-effect exists, but this assumption cannot be validated using correlation alone as the test criteria. Correlation has often been referred to as a statistical parameter seeking to define how well a linear or other fitting function describes the relationship between variables however, two variables may be highly correlated under a specific set of test conditions, and not correlated under a different set of experimental conditions. In this case the correlation is conditional and so also is the cause-and-effect phenomenon. If two variables are always perfectly correlated under a variety of conditions, one may have a basis for cause-and-effect, and such a basic relationship permits a well-defined mathematical description. 

This chapter has considered two key aspects related to quality assurance - the use of control charts and the evaluation of measurement uncertainty. These activities, along with method validation, require some knowledge of basic statistics. The chapter therefore started with an introduction to the most important statistical terms. 

The demand planning module is used for short-term and midterm sales planning. It covers basic statistical forecasting methods, but is also capable of taking additional aspects into account. For example, these may be promotions in shortterm sales planning or the consideration of product lifecycles in midterm sales planning. 

From the thermodynamic viewpoint, the basic statistical theory is still too complex to provide useful working equations, but it does suggest forms of equations with some purely theoretical terms, and other terms including parameters to be evaluated empirically. In general, the theoretical terms arise from the electrostatic interactions which are simple and well-known while the empirical, terms relate to short-range interionic forces whose characteristics are qualitatively but not quantitatively known from independent sources. But, as we shall see, this division is not complete - there are interactions between the two categories. 

This is a selection of terms related to statistics. You will find more detailed descriptions in chapter 8 - Basic Statistics . 

These are the terms that are associated with validation. Many of these terms are explained in detail in the chapters 11- Fit for Pnrpose and Validation of Analytical Methods , and 8 - Basic Statistics . So please check there. A few examples (of the more often nsed terms) are presented on the next slide. 

Most terms on this slide are explained in the chapters Glossary or Basic Statistics and thns are not repeated here. 

Derivation of the Boltzmann distribution function is based on statistical mechanical considerations and requires use of Stirling s approximation and Lagrange s method of undetermined multipliers to arrive at the basic equation, (N,/No) = (g/go)exp[-A Ae/]. The exponential term /3 defines the temperature scale of the Boltzmann function and can be shown to equal t/ksT. In classical mechanics, this distribution is defined by giving values for the coordinates and momenta for each particle in three-coordinate space and the lin-... 

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