Non-parametric regression

Aldrich et al. (3) and Reuter et al. (4) discussed the use of neural networks in metallurgy and mineral processing. Neural networks are applied in this paper as a non-parametric regression-modelling tool to develop the prediction model. The program Basic ModelGen (Crusader Systems (Pty) Ltd, South Africa) is used to develop the neural nets used in this research. 

The Gaussian Approximation Potential scheme is similar to the Neural Network potentials introduced by Behler and Parinello [33], as both uses non-linear, non-parametric regression instead of fixed analytic forms. However, the representation of the atomic environments in GAP is complete and the Gaussian Process uses energies and forces for regression. Moreover, the training of the neural... 

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. 

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). 

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). 

Extending non-parametric tests to more complex settings, such as regression, ANOVA and ANCOVA is not straightforward and this is one aspect of these methods that limits their usefulness. 

As stated above, the utility of the ML estimators derives essentially from their asymptotic properties of consistency and optimality (i.e., cov(0ml) — CRB). When the data exhibits significant departures from theoretical pdf (Gaussian or Rice) owing to acquisition artifacts, it may be judicious to use robust non-linear regression techniques,62 as in parametric diffusion-tensor imaging reconstructed from echo-planar data.63... 

Hypothesis testing techniques should include ANOVA, regression analysis, multivariate techniques and parametric and non-parametric statistics. 

In the comparison of calibration methods, the results show that the non-parametric techniques decreased the regression error by approximately 50% over the parametric approaches. The sensor array used in these examples was... 

Parametru/non-parametric techniques This first distinction can be made between techniques that take account of the information on the population distribution. Non parametric techniques such as KNN, ANN, CAIMAN and SVM make no assumption on the population distribution while parametric methods (LDA, SIMCA, UNEQ, PLS-DA) are based on the information of the distribution functions. LDA and UNEQ are based on the assumption that the population distributions are multivariate normally distributed. SIMCA is a parametric method that constructs a PCA model for each class separately and it assumes that the residuals are normally distributed. PLS-DA is also a parametric technique because the prediction of class memberships is performed by means of model that can be formulated as a regression equation of Y matrix (class membership codes) against X matrix (Gonzalez-Arjona et al., 1999). 

As transformation via inversion is non-Hnear, the distributions of raw error scores and derived performance win be quite different. This has no effect on ordinal analyses, such as non-parametric statistics, but wiU have some effect on Hnear analyses, such as parametric statistics, linear regression/correlation, etc., and may include improvements due to a possible greater normality of the distributions of derived performances. An alternative transformation which would retain a linear relationship with the error scores is ... 

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