Non-parametric

C. G. Lambert, Multipole-based Algorithms in Molecular Biophysics and Non-parametric Statistics, Ph.D. Dissertation, Duke University Department of Computer Science, 1997. 

Multiple linear regression is strictly a parametric supervised learning technique. A parametric technique is one which assumes that the variables conform to some distribution (often the Gaussian distribution) the properties of the distribution are assumed in the underlying statistical method. A non-parametric technique does not rely upon the assumption of any particular distribution. A supervised learning method is one which uses information about the dependent variable to derive the model. An unsupervised learning method does not. Thus cluster analysis, principal components analysis and factor analysis are all examples of unsupervised learning techniques. 

Ideally, to characterize the spatial distribution of pollution, one would like to know at each location x within the site the probability distribution of the unknown concentration p(x). These distributions need to be conditional to the surrounding available information in terms of density, data configuration, and data values. Most traditional estimation techniques, including ordinary kriging, do not provide such probability distributions or "likelihood of the unknown values pC c). Utilization of these likelihood functions towards assessment of the spatial distribution of pollutants is presented first then a non-parametric method for deriving these likelihood functions is proposed. 

These various covariance models are Inferred directly from the corresponding indicator data i(3 z ), i-l,...,N. The indicator kriging approach is said to be "non-parametric, in the sense that it draws solely from the data, not from any multivariate distribution hypothesis, as was the case for the multi- -normal approach. 

We also make a distinction between parametric and non-parametric techniques. In the parametric techniques such as linear discriminant analysis, UNEQ and SIMCA, statistical parameters of the distribution of the objects are used in the derivation of the decision function (almost always a multivariate normal distribution... 

Non-linear models, such as described by the Michaelis-Menten equation, can sometimes be linearized by a suitable transformation of the variables. In that case they are called intrinsically linear (Section 11.2.1) and are amenable to ordinary linear regression. This way, the use of non-linear regression can be obviated. As we have pointed out, the price for this convenience may have to be paid in the form of a serious violation of the requirement for homoscedasticity, in which case one must resort to non-parametric methods of regression (Section 12.1.5). 

GET p VALUE FROM NON PARAMETRIC COMPARISON OF AGE MEANS. proc nparlway g... 

Hollander, M., and D. A. Wolf, Non Parametric Statistical Methods. John Wiley and Sons, Inc., New York, NY (1973). 

However, as pointed out in [2], it remains to be seen to what extent the Meynet Maeder [4] yields for N in the intermediate mass star range would increase once the hot bottom burning (HBB) is taken into account. Although Meynet Maeder did not formally include the third dredge-up and HBB, they predict an important N production in low and intermediate mass stars, at low metallicities. In absence of a real quantitative assessment of the importance of the HBB it is interesting to study the importance of this new process, which produce non-parametric yields, independently of HBB. 

The kinetics of the CTMAB thermal decomposition has been studied by the non-parametric kinetics (NPK) method [6-8], The kinetic analysis has been performed separately for process I and process II in the appropriate a regions. The NPK method for the analysis of non-isothermal TG data is based on the usual assumption that the reaction rate can be expressed as a product of two independent functions,/ and h(T), where f(a) accounts for the kinetic model while the temperature-dependent function, h(T), is usually the Arrhenius equation h(T) = k = A exp(-Ea / RT). The reaction rates, da/dt, measured from several experiments at different heating rates, can be expressed as a three-dimensional surface determined by the temperature and the conversion degree. This is a model-free method since it yields the temperature dependence of the reaction rate without having to make any prior assumptions about the kinetic model. 

The non-parametric kinetics method has proved helpful in analyzing separately two partly overlapping processes. 

Graphical and statistical data analysis will be carried out at various scales (regional, States/Northern Territory, and National). Non-parametric univariate and multivariate analysis along with the production of geochemical maps will be carried out. 

The results have been statistically processed by means of Spearman s non-parametrical correlation analysis and by multiple regression analysis to assess the complex effects induced by toxic and essential elements (Evstafyeva, Slusarenko, 2003 Evstafyeva et al., 2004). 

From semi-quantitative results to non-parametric measurement, there exist several levels of qualitative information ... 

A third mode of qualitative measurement is a non-parametric one [2]. This concept is based on the direct exploitation of an analytical signal (absorbance, intensity, potential,...) without parameter calculation, leading to the simple characterization of the studied sample (Fig. 2). For example, fingerprinting or image analysis can be considered as non-parametric measurement. 

A final up-and-coming application, based on non-parametric measurement, is used more and more for process control and hazards prevention, for example for shock load prevention or toxic events. This qualitative approach uses integrated information coming from multiple physical signals ... 

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 range of values that lie between the first and third quartiles and, therefore, represent 50°/o of the data points. Used in non-parametric tests. 

Siegel, S. Non-Parametric Statistics for the Behavioral Sciences. McGraw-Hill, N.Y. 

The models used can be either fixed or adaptive and parametric or non-parametric models. These methods have different performances depending on the kind of fault to be treated i.e., additive or multiplicative faults). Analytical model-based approaches require knowledge to be expressed in terms of input-output models or first principles quantitative models based on mass and energy balance equations. These methodologies give a consistent base to perform fault detection and isolation. The cost of these advantages relies on the modeling and computational efforts and on the restriction that one places on the class of acceptable models. 

The data obtained was analyzed using SPSS-9.0 software. Any differences with a significance level p<0.05 were considered valid. To assess the validity of the differences and correlation, non-parametric tests were used as indicated vide infra). The data are displayed as mean SE. 

Statistical analysis of the obtained research data was performed by parametric and non-parametric statistics using the softwares EXCEL (Microsoft, 2003, USA) and STATISTICA 6.1 (StatSoft Inc., 1984-2004, USA). 

Friedman, J. A. (1977) Recursive partitioning decision rules for non-parametric classification. IEEE Trans. Comput. 26, 404-408. 

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