Statistics basic ideas

Bickel PJ, Doksum KA (1977) Mathematical statistics basic ideas and selected topics. Holden Day, San Francisco... 

The experimental data and arguments by Trassatti [25] show that at the PZC, the water dipole contribution to the potential drop across the interface is relatively small, varying from about 0 V for An to about 0.2 V for In and Cd. For transition metals, values as high as 0.4 V are suggested. The basic idea of water clusters on the electrode surface dissociating as the electric field is increased has also been supported by in situ Fourier transfomr infrared (FTIR) studies [26], and this model also underlies more recent statistical mechanical studies [27]. 

Statistical mechanics methods such as Cluster Variation Method (CVM) designed for working with lattice statics are based on the assumption that atoms sit on lattice points. We extend the conventional CVM [1] and present a method of taking into account continuous displacement of atoms from their reference lattice points. The basic idea is to treat an atom which is displaced by r from its reference lattice point as a species designated by r. Then the summation over the species in the conventional CVM changes into an integral over r. An example of the 1-D case was done successfully before [2]. The similar treatments have also been done for... 

In this section, we take an approach that is characteristic of conventional perturbation theories, which involves an expansion of a desired quantity in a series with respect to a small parameter. To see how this works, we start with (2.8). The problem of expressing ln(exp (—tX)) as a power series is well known in probability theory and statistics. Here, we will not provide the detailed derivation of this expression, which relies on the expansions of the exponential and logarithmic functions in Taylor series. Instead, the reader is referred to the seminal paper of Zwanzig [3], or one of many books on probability theory - see for instance [7], The basic idea of the derivation consists of inserting... 

There are many variants on this basic idea, such as the weighted kappa described by Cohen himself (20) and the Rand statistic (21), which is perhaps the most widely used of the measures available for comparing different clusterings of the same set of objects. 

Chemical Kinetics of Solids covers a special part of solid state chemistry and physical chemistry. It has been written for graduate students and researchers who want to understand the physical chemistry of solid state processes in fair depth and to be able to apply the basic ideas to new (practical) situations. Chemical Kinetics of Solids requires the standard knowledge of kinetic textbooks and a sufficient chemical thermodynamics background. The fundamental statistical theory underlying the more or less phenomenological approach of this monograph can be found in a recent book by A. R. Allnatt and A.B. Lidiard Atomic Transport in Solids, which complements and deepens the theoretical sections. 

The basic idea of the model was the representation of a network as a statistical combination of molecular cycles with a different number of diepoxides and amines in each of them. This approach gives the possibility to describe network topology by a distribution function of cycle sizes or by its moments. Modelling also includes small... 

Since percolation is a property of macroscopic many-particle systems, it can be analyzed in terms of statistical mechanics. The basic idea of statistical mechanics is the relaxation of the perturbed system to the equilibrium state. In general the distribution function p(p,q t) of a statistical ensemble depends on the generalized coordinates q, momentum p, and time t. However, in the equilibrium state it does not depend explicitly on time [226-230] and obeys the equation... 

Kuhn was the first to point out that the dimensions of a chain with given persistence p may always be described as if it were completely flexible (see (5.1.1)) by grouping a number of monomer units together into statistical chain elements (s.c.e.) or Kuhn segments. The number a of bonds in such an s.c.e. is the larger the stlffer the chain. The basic idea is that such s.c.e. s may be considered as orlentatlonally independent they are then independent subsystems as defined in sec. 1.3.6. The real chain of N bonds is now modelled as an equivalent ideal chain of = N/a s.c.e. s and the Kuhn length becomes bt (where a > 1, b > 1). Then (r ) = vdilch equals = 6pN(, provided that a... 

Of the statistical simulations, two major types are distinguished cellular automata (CA) and Monte Carlo (MC) simulations. The basic ideas concerning CA go back to Wiener and Rosenblueth [1] and Von Neumann [2]. CA exist in many variants, which meikes the distinction between MC and CA not always clear. In general, in both techniques, the catalyst surface is represented by a matrix of m x n elements (cells, sites) with appropriate boundary conditions. Each element can represent an active site or a collection of active sites. The cells evolve in time according to a set of rules. The rules for the evolution of cells include only information about the state of the cells and their local neighborhoods. Time often proceeds in discrete time steps. After each time step, the values of the cells are updated according to the specified rules. In cellular automata, all cells are updated in each time step. In MC simulations, both cells and rules are chosen randomly, and sometimes the time step is randomly chosen as well. Of course, all choices have to be made with the correct probabilities. 

In conclusion, let us summarize the main principles of the equilibrium statistical mechanics based on the generalized statistical entropy. The basic idea is that in the thermodynamic equilibrium, there exists a universal function called thermodynamic potential that completely describes the properties and states of the thermodynamic system. The fundamental thermodynamic potential, its arguments (variables of state), and its first partial derivatives with respect to the variables of state determine the complete set of physical quantities characterizing the properties of the thermodynamic system. The physical system can be prepared in many ways given by the different sets of the variables of state and their appropriate thermodynamic potentials. The first thermodynamic potential is obtained from the fundamental thermodynamic potential by the Legendre transform. The second thermodynamic potential is obtained by the substitution of one variable of state with the fundamental thermodynamic potential. Then the complete set of physical quantities and the appropriate thermodynamic potential determine the physical properties of the given system and their dependences. In the equilibrium thermodynamics, the thermodynamic potential of the physical system is given a priori, and it is a multivariate function of several variables of state. However, in the equilibrium... 

The basic idea is that these are experiments on molecular spins in a fluid at room temperature. At this temperature, the states tend to be completely mixed (a statistical mixture of all states) but adding an external field introduces some correlations that can be written as small perturbations... 

The distribution of spot sample compositions of a certain size, taken from a randomly mixed batch of A and B, can be calculated theoretically. The methods of calculation are standard statistical techniques, and several papers have shown how various aspects of these basic ideas can be applied to solids mixing. Most of the calculations and discussion center around three distributions binomial, normal, and Poisson. 

An approach based on the sequential use of Monte Carlo simulation and Quantum Mechanics is suggested for the treatment of solvent effects with special attention to solvatochromic shifts. The basic idea is to treat the solute, the solvent and its interaction by quantum mechanics. This is a totally discrete model that avoids the use of a dielectric continuum. Statistical analysis is used to obtain uncorrelated structures. The radial distribution function is used to determine the solvation shells. Quantum mechanical calculations are then performed in supermolecular structures and the spectral shifts are obtained using ensemble average. Attention is also given to the case of specific hydrogen bond between the solute and solvent. 

Recently, Manning and Mohanty [83] have developed a statistical mechanical theory of end effects for a finite size polyion of radius a. The basic idea exploited by these authors relies upon the system being at equilibrium. Under these conditions the chemical potential /x is independent of the distance s measured along the backbone from one end of the polyion [84,83] ... 

Another internal validation technique is the posterior predictive check (PPC), which has been used in the Bayesian literature for years, but only recently reported in the PopPK literature by Yano, Beal, and Sheiner (2001). The basic idea is an extension of the predictive check method just described but include hyperparameters on the model parameters. Data are then simulated, some statistic of the data that is not based on the model is calculated, e.g., half-life or AUC by noncompartmental method, and then compared to the observed statistic obtained with real data. The underlying premise is that the simulated data should be similar to the observed data and that any discrepancies between the observed and simulated data are due to chance. With each simulation the statistic of interest is calculated and after all the simulations are complete, a p-value is determined by... 

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