Metric statistics

The major difficulty with the methods detailed above is the reliance on conventional metric statistics. Vector distances in an n-dimensional space including... 

In both studies, nonmetric clustering outperformed the metric tests, although both principal components analysis and correspondence analysis yielded some additional insight into large-scaled patterns, which was not provided by the nonmetric clustering results. However, nonmetric clustering provided information without the use of inappropriate assumptions, data transformations, or other dataset manipulations that usually accompany the use of multivariate metric statistics. The success of these studies and techniques led to the examination of community dynamics in a series of two multispecies toxicity tests. 

These differences may be tested by statistical methods, which are not described here. The reader is referred to Chapter 14 and to standard textbooks of parametric and nonpara-metric statistics. ... 

In 2003, I wrote a book, Applied Statistical Designs for the Researcher (Marcel Dekker, Inc.), in which I covered experimental designs commonly encountered in the pharmaceutical, applied microbiological, and healthcare-product-formulation industries. It included two sample evaluations, analysis of variance, factorial, nested, chi-square, exploratory data analysis, nonpara-metric statistics, and a chapter on linear regression. Many researchers need more than simple linear regression methods to meet their research needs. It is for those researchers that this regression analysis book is written. 

Production and consumption statistics for sucrose are shown ia Table 1. World production of sucrose dufing 1993—1994 was - 110 million metric tons, of which - 64% was derived from sugarcane. The largest producer is the European Union (EU), followed closely by India and Bra2il. In 1993—1994, the United States ranked fourth in production. World raw sugar prices from 1990—1995 ranged from 20.20—32. l0/kg (10). 

According to statistics collected by the U.S. Geological Survey (3), U.S. production of cmde talc in 1995 was 1,050,000 metric tons. Montana, Texas, Vermont, and New York were the principal producing U.S. states. Worldwide production was estimated to be 5,845,000 t. China, having 2,400,000 t, was the largest producer in the world after China and the United States, Finland, India, Brazil, France, Italy, and Canada are the next principal producers. World production of talc in 1994 is Hsted in Table 1. 

Mono- and dichlorotoluenes ate used chiefly as chemical iatermediates ia the manufacture of pesticides, dyestuffs, pharmaceuticals, and peroxides, and as solvents. Total annual production was limited prior to 1960 but has expanded greatly siace that time. Chlorinated toluenes ate produced ia the United States, Germany, Japan, and Italy. Siace the number of manufacturers is small and much of the production is utilised captively, statistics covering production quantities ate not available. Worldwide annual production of o- and -chlorotoluene is estimated at several tens of thousands of metric tons. Yearly productions of polychlorotoluene ate ia the range of 100—1000 tons. 

Given a space G, let g (x) be the closest model in G to the real function, fix). As it is shown in Appendbc 1, if /e G and the L°° error measure [Eq. (4)] is used, the real function is also the best function in G, g = f, independently of the statistics of the noise and as long as the noise is symmetrically bounded. In contrast, for the measure [Eq. (3)], the real function is not the best model in G if the noise is not zero-mean. This is a very important observation considering the fact that in many applications (e.g., process control), the data are corrupted by non-zero-mean (load) disturbances, in which cases, the error measure will fail to retrieve the real function even with infinite data. On the other hand, as it is also explained in Appendix 1, if f G (which is the most probable case), closeness of the real and best functions, fix) and g (x), respectively, is guaranteed only in the metric that is used in the definition of lig). That is, if lig) is given by Eq. (3), g ix) can be close to fix) only in the L -sense and similarly for the L definition of lig). As is clear,... 

That is, the real function fix) is the solution to the minimization of Eq. (3) only in the absence of noise id = 0) or when the noise has zero mean (d = 0). This is, in fact, true for all L" norms with 2 [Pg.201]

Metrics for this might include number of excursions from statistical process control, but one very useful metric for controllability is process capability, or more accurately, process capability indices. Process capability compares the output of an in-control process to the specification limits by using capability indices. The comparison is made by forming the ratio of the spread between the process specifications (the specification width ) to the spread of the process values. In a six-sigma environment, this is measured by six standard deviation units for the process (the process width ). A process under control is one where almost all the measurements fall inside the specification limits. The general formula for process capability index is ... 

In the case of health effects, other methods than stated or revealed preference methods are often used to estimate the impact of externalities and valuating the human health damages. Both productivity losses and costs for hospital admissions or other hospital-related activities are used to monetize health effects. Of special importance for the valuation of health effects are the metrics Value of a Statistical Life / Value of Prevented Fatality (VSL, VOSL or VPF) and Value of a Life Year Lost (VOLY). 

WHO (2011) Health statistics and health information systems - metrics disability-adjusted life year (DALY)... 

The t value associated with each model descriptor, defined as the descriptor coefficient divided by its standard error, is a useful statistical metric. Descriptors with large f values are important in the predictive model and, as such, can be examined in order to gain some understanding of the nature of the property or activity of interest. It should be stated, however, that the converse is not necessarily true, and thus no conclusions can be drawn with respect to descriptors with small f values. 

For circumstances where wide variability is observed, or a statistical evaluation of f2 metric is desired, a bootstrap approach to calculate a confidence interval can be performed (8). 

Data belonging to distribution profiles may be compared either vertically along the release/response ordinate or horizontally along the time abscissa. The semi-invariants (moments) provide a complete set of metrics, representing both aspects in logical sequence AUC accounts (vertically) for the difference of the extent, the mean compares (horizontally) the rates, and higher-order moments and higher-order statistics (variance, etc.) characterize the shape aspect from coarse to finer. 

Summation of absolute differences (I) results in an ME in which all differences have the same statistical weight. Summation of squared differences (II) is the more common practice and gives an MSE in which large deviations have higher weight than small ones. In order to make the metric independent of the number N of observations, the error sum must be related to N or an equivalent sum of the observations ... 

The definition of these metrics appears somewhat arbitrary and is hard to understand in the framework of statistical reasoning. In particular, the meaning of maximum and minimum terms in the definition of the Chinchilli and the rho metrics cannot be easily verified. The fact that an arbitrarily defined index performs better for an arbitrarily selected set of experimental data cannot be accepted as a general proof of validation. 

The focus is on multivariate statistical methods typically needed in chemo-metrics. In addition to classical statistical methods, also robust alternatives are introduced which are important for dealing with noisy data or with data including outliers. Practical examples are used to demonstrate how the methods can be applied and results can be interpreted however, in general the methodical part is separated from application examples. 

Frank, I. E., Friedman, J. Technometrics 35, 1993, 109. A statistical view of some chemo-metrics regression tools. 

Only the more important and more frequently used variables are listed here. Chemo-metrics literature sometimes uses different symbols as statistics or economics. In some cases, the same symbol has to be used for different variables however, the particular meaning should be always clear from the definition. 

Because of the nature of the scientific method. Metrics is an indispensable tool of scientific research. It can provide rigorous indices of the internal consistency and the predictive power of "accepted knowledge" about study systems. Thus, it can aid with theory testing. It can also provide rigorous indices of the strength of correlations between the attributes of the study system and the external factors that might influence it. Thus, it can assist with statistical hypothesis formulation and testing. 

Summarizing, sequence comparison allows us to compute the quantitative distance between any two linguistic assertions. In this research, the assertions are those derived from the procedures described above. Once distances (similarity data) are computed that satisfy the metric axioms, the matrix of inter-assertion distances can be analyzed using a variety of multivariate statistical techniques. 

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