Nonparametric index

The main difficulty with the RE concept is the hypothetical nature of the reference structure, its choice being somewhat arbitrary. There are many ways of defining RE (M. Dewar, C. F. Wilcox, and others). Here we shall briefly examine the topological resonance energy (TRE). TRE is a nonparametric index which is directly related to the topology of a molecule and is of great practical value in predicting the aromatic stability of an arbitrary jt-network. 

NA MacMillan, DC Creelman. Response bias Characteristics of detection-theory, threshold-theory, and nonparametric indexes. Psychol Bull 107 401-413, 1990. 

Our goal is to estimate the function P(r) from the set of discrete observations Y(tj). We use a nonparametric approach, whereby we seek to estimate the function without supposing a particular functional form or parameterization. We require that our estimated function be relatively smooth, yet consistent with the measured data. These competing properties are satisfied by selecting the function that minimizes, for an appropriate value of the regularization parameter X, the performance index ... 

Vertzoni et al. (30) recently clarified the applicability of the similarity factor, the difference factor, and the Rescigno index in the comparison of cumulative data sets. Although all these indices should be used with caution (because inclusion of too many data points in the plateau region will lead to the outcome that the profiles are more similar and because the cutoff time per percentage dissolved is empirically chosen and not based on theory), all can be useful for comparing two cumulative data sets. When the measurement error is low, i.e., the data have low variability, mean profiles can be used and any one of these indices could be used. Selection depends on the nature of the difference one wishes to estimate and the existence of a reference data set. When data are more variable, index evaluation must be done on a confidence interval basis and selection of the appropriate index, depends on the number of the replications per data set in addition to the type of difference one wishes to estimate. When a large number of replications per data set are available (e.g., 12), construction of nonparametric or bootstrap confidence intervals of the similarity factor appears to be the most reliable of the three methods, provided that the plateau level is 100. With a restricted number of replications per data set (e.g., three), any of the three indices can be used, provided either non-parametric or bootstrap confidence intervals are determined (30). 

The nonparametric bootstrap is useful when distributions cannot be assumed as true or when the sampled statistic is based on few observations. In this setting, an observed data set, for example, Zj,. .., Z , where X could be vector-valued (i.e., concentrations at fixed sampling times) can be summarized in the usual way by a mean, median, and variance. An approximate sampling distribution can be obtained drawing a sample of the same size as the original sample from the original data with replacement, for example, Z/,. .., Z , where i is the index of the bootstrap sample... 

Nonparametric approaches typically extract a set of features from each video frame. The features are then matched to a stored template. The template can be either 2D or 3D. When using nonparametric techniques, the typical procedures consist of motion detection and human tracking in the scene, which enables the construction of a sequence. Then, a periodicity index is computed and the periodicity sequence is segmented into individual cycles for recognition. 

In this text we have only considered parametric observation models. In other words, the density is indexed by a finite dimensional parameter. Bayesian inference is based on the posterior distribution of those parameters. Nonparametric Bayesian models use distributions with infinitely many parameters, as the ivobability model is defined on a function space, not a finite dimensional parameter space. The random probability model on the function space is often generated by a Dirichlet process. Interested readers are referred to Dey et al. (1998). 

Another statistic often calculated is an overall index of acceptability of the form (T- Q/(r + C), where T and C are defined as above. This statistic has an expected value of 0 when there is no difference in preference of the test and control treatments and a range from - 1 (when the test treatment is never chosen) to + 1 where the test treatment is completely preferred over the control. While this statistic allows the expression of the relative attraction and deterrence of a range of compounds, its statistical properties are not well understood. The statistic is not normally distributed, and the occurrence of negative values precludes the use of common transformations (e.g., log, arcsine, square root) to remove some deviations from normality. Thus, if this statistic is used, it should be analyzed only by nonparametric, distribution-free statistical tests based on ranks. Further discussion of procedures and criteria for selecting statistical tests can be found in many standard texts and manuals for a number of statistic analysis packages for use with computers (e.g.. Steel Torrie 1980 SAS Institute 1989 Sokal Rohlf 1995). 

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