Central limit theorem analysis

Given a regular fraction of a 2f experiment and independent response variables, the estimators described above have constant variance even if the individual response variables do not. Also, the estimators are approximately normally distributed by the Central Limit Theorem. However, if the response variables have unequal variances, this unfortunately causes the estimators to be correlated and, therefore, dependent. The use of the data analysis to assess whether the levels of some factors affect the response variability is, itself, a problem of great interest due to its role in robust product design see Chapter 2 for a discussion of available methods and analysis. In the present chapter, we consider situations in which the estimators are independent. 

CALS—See Computer automated laboratory system CAS On-Line, 22,95 Central limit theorem, 115 Certificate of analysis, 21 Clustering methods, 91,93-97 Clusters technology, 42... 

In the absence of instmment-induced correlations, stochastic errors in the frequency-domain are normally distributed. The appearance of a normal distribution of frequency-domain stochastic errors can be regarded to be a consequence of the Central Limit Theorem applied to the methodology used to measure the complex impedance. ° This result validates an essential assumption routinely used during regression analysis of impedance (and other frequency-domain) data. 

The central limit theorem also delivers another positive for the analyst. Most of the simple data analysis assumes a normal distribution of data. Much of the time for real sets of data this is not so, but by taking averages of results the distribution of the means tends to a normal distribution, even if the original population is not normally distributed. Hence taking averages of data also helps us with data analysis by removing concerns we might have had about whether our data conform to a normal distribution. 

This result is known as the central limit theorem and serves to emphasize the importance and applicability of the normal distribution function in statistical data analysis since non-normal data can be normalized and can be subject to basic statistical analysis. ... 

Also using chemical space as a framework, Agrafiotis [118] presented a very fast method for diversity analysis on the basis of simple assumptions, statistical sampling of outcomes, and principles of probability theory. This method presumes that the optimal coverage of a chemical space is that of uniform coverage. The central limit theorem of probability theory... 

In this case it is very difficult to obtain at r information about the individual scatterers from the statistical analysis of the scattered intensity. However, this behavior may be different if the number of independent scatterers is small so that the central limit theorem does not apply. This happens, for instance, in the case of a dilute sample or for a small illuminated surface area. In this situation of a small number of independent scatterers there are still two main cases first, when the number of scatterers is deterministic (i.e. N is fixed) and second, when the number of scatterers fluctuate between each realization. For fixed N and if the scatterers are independent (uncorrelated amplitude and phase of the individual scattered fields), and if all contributions to the total... 

Many important properties, such as critical dimensionality and critical exponents describing the divergence of correlation length and other quantities can thus be obtained from renormalization group analysis. It is worth noting that for X well below X only the quadratic terms of the potential contribute to the asymptotic properties of P, which reduces therefore to a multigaussian distribution in accordance with the central limit theorem [13]. There exists,however,a (frequently very narrow) vicinity of X ... 

This is the central limit theorem, a proof of which can be found in Sienfeld and Lapidus [5]. This is a very important theorem of statistics, particularly in regression analysis where experimental data are being analyzed. The experimental error is a composite of many separate errors whose probability distributions are not necessarily normal distributions. However, as the number of components contributing to the error increases, the central limit theorem justifies the assumption of normality of the error. 

There are many different ways to treat mathematically uncertainly, but the most common approach used is the probability analysis. It consists in assuming that each uncertain parameter is treated as a random variable characterised by standard probability distribution. This means that structural problems must be solved by knowing the multi-dimensional Joint Probability Density Function of all involved parameters. Nevertheless, this approach may offer serious analytical and numerical difficulties. It must also be noticed that it presents some conceptual limitations the complete uncertainty parameters stochastic characterization presents a fundamental limitation related to the difficulty/impossibility of a complete statistical analysis. The approach cannot be considered economical or practical in many real situations, characterized by the absence of sufficient statistical data. In such cases, a commonly used simplification is assuming that all variables have independent normal or lognormal probability distributions, as an application of the limit central theorem which anyway does not overcome the previous problem. On the other hand the approach is quite usual in real situations where it is only possible to estimate the mean and variance of each uncertainty parameter it being not possible to have more information about their real probabilistic distribution. The case is treated assuming that all uncertainty parameters, collected in the vector d, are characterised by a nominal mean value iJ-dj and a correlation =. In this specific... 

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