Non-parametric model

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. 

A few programs are now available that allow the efficient simultaneous data analysis from a population of subjects. This approach has the significant advantage that the number of data points per subject can be small. However, using data from many subjects, it is possible to complete the analyses and obtain both between- and within-subject variance information. These programs include NONMEM and WinNON-MIX for parametric (model dependent) analyses and NPEM when non-parametric (model independent) analyses are required. This approach nicely complements the Bayesian approach. Once the population values for the pharmacokinetic parameters are obtained, it is possible to use the Bayesian estimation approach to obtain estimates of the individual patient s pharmacokinetics and optimize their drug therapy. 

In practice, one may want to know in what cases the parametric or non-parametric model are suitable. However, one should always do a MCF plot, as it... 

For the large group of dynamic lumped models, which are used for optimization and control, also a classification between parametric and non-parametric models can be made. This classification resembles the classification in white-box and black-box models. The latter distinction is based on the difference in knowledge content, whereas the former distinction refers to the form of the set of equations. 

Parametric models are more or less white box or first principle models. They consist of a set of equations that express a set of quantities as explicit functions of several independent variables, known as parameters . Parametric models need exact information about the inner stmcture and have a limited number of parameters. For instance, for the description of the dynamics, the order of the system should be known. Therefore, for these models, process knowledge is required. Examples are state space models and (pulse) transfer functions. Non-parametric models have many parameters and need little information about the inner stmcture. For instance, for the dynamics, only the relevant time horizon shoirld be known. By their stmcture, they are predictive by nature. These models are black box and can be constructed simply from experimental data. Examples are step and pulse response functions. 

Structural considerations concern the order (parametric models) or the time-horizon (non-parametric models) of the model. Usually, the model is identified by minimizing the error between data and model. This can lead to over-modeling by including noise or eventualities. The cotmnon strategy is to use separate data sets for model identification and for testing. Statistical tests can be applied to test the parameter significance. 

In this paper we will introduce a non-parametric model based on Hammerstein-Wiener model with use of sigmoid network in nonlinear block of model which can be classified in neural network model category. Section II will discuss on structure of Hammerstein-Wiener model. In section III and TV, data acquisition and simulation procedure will be described and finally a conclusion remark will be represented. 

However, there arise acute problems when the reaction mechanism undertake a more complex, multi-step pathway [531-535]. It is worth noting that such a composite process might not be detected be mere variation in the apparent values of E but a more multifaceted kinetics is necessarily applied or the other, often non-parametric (model-free methods) ought to be considered as more convenient to such application [533]. 

Song, D., Z. Wang, V.Z. Marmarehs, and T.W. Berger 2009. Parametric and non-parametric modeling of short-term synaptic plasticity. Part 1 and 11. /. Comput. Neurosci. 26 1. 

The non-parametric model is conceptually simpler as it loosely follows the same steps as LLE. Given a new datapoint, x and a set of previously seen data points X, the low dimensional representation y is found using the following steps. Firstly, the -nearest neighbours of x in X are found. The weights, w, that best reconstruct x ... 

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. 

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. 

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 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. 

Another non - parametric approach is deconvolution by discrete Fourier transformation with built - in windowing. The samples obtained in pharmacokinetic applications are, however, usually short with non - equidistant sample time points. Therefore, a variety of parametric deconvolution methods have been proposed (refs. 20, 21, 26, 28). In these methods an input of known form depending on unknown parameters is assumed, and the model response predicted by the convolution integral (5.66) is fitted to the data. 

Two non-parametric methods for hypothesis testing with PCA and PLS are cross-validation and the jackknife estimate of variance. Both methods are described in some detail in the sections describing the PCA and PLS algorithms. Cross-validation is used to assess the predictive property of a PCA or a PLS model. The distribution function of the cross-validation test-statistic cvd-sd under the null-hypothesis is not well known. However, for PLS, the distribution of cvd-sd has been empirically determined by computer simulation technique [24] for some particular types of experimental designs. In particular, the discriminant analysis (or ANOVA-like) PLS analysis has been investigated in some detail as well as the situation with Y one-dimensional. This simulation study is referred to for detailed information. However, some tables of the critical values of cvd-sd at the 5 % level are given in Appendix C. 

Leontaritis and Billings, 1985] Leontaritis, I. J. and Billings, S. A. (1985). Input-output parametric models for non-linear systems, part I deterministic non-linear systems part II Stochastic non-linear systems. Int. J. Control, 41(2) 303-344. 

In this Chapter we describe the extension of the parametric model used for 4f" spectra to calculations of absorption and emission spectra for the 4f 15d configuration. We also illustrate how they can be applied to calculate other properties of interest, such as non-radiative relaxation rates. Finally, we discuss the relationship between parametrized calculations and other approaches, such as ab initio calculations. 

In this Chapter we have discussed the use of parametrized models to calculate the energy levels of the 4f and 4f 15d configurations of lanthanide ions in condensed-matter environments. Radiative and non-radiative transitions between these configurations have also been... 

In practice the reconstructions of the basic phenomenological effects, position of giant arcs and of multiple images, seen in images of galaxy clusters can be obtained through the superposition of one or very few such elliptical potentials. However the reconstruction of complete galaxy cluster mass maps may eventually requires the use of more complicated models and it can be necessary to perform non-parametric mass reconstructions. A number of important results have been obtained from such observations - see Mellier (1999) and references therein. 

With regards to the analysis of the quality of the various parts of the model, one may use the same methods as are used for practical identifiability analysis. Since the same methods are used, albeit with different objectives, one sometimes refers to this model quality analysis as a posteriori identifiability (and the previous analysis as a priori identifiability). Now, however, one is also interested in how the parametric uncertainty translates to an uncertainty in the various model predictions. For instance, it might be so that even though two individual parameters have a high uncertainty, they are correlated in such a manner that their effect on a specific (non-measured) model output is always the same. Such a translation may be obtained by simulations of the model using parameters within the determined confidence ellipsoids. A global alternative to this is to consider the outputs for all parameters that correspond to a cost function that is below a certain threshold, for example 2% above the found minimum. 

Probability distribution models can be used to represent frequency distributions of variability or uncertainty distributions. When the data set represents variability for a model parameter, there can be uncertainty in any non-parametric statistic associated with the empirical data. For situations in which the data are a random, representative sample from an unbiased measurement or estimation technique, the uncertainty in a statistic could arise because of random sampling error (and thus be dependent on factors such as the sample size and range of variability within the data) and random measurement or estimation errors. The observed data can be corrected to remove the effect of known random measurement error to produce an error-free data set (Zheng Frey, 2005). 

A probability distribution is a mathematical description of a function that relates probabilities with specified intervals of a continuous quantity, or values of a discrete quantity, for a random variable. Probability distribution models can be non-parametric or parametric. A non-parametric probability distribution can be described by rank ordering continuous values and estimating the empirical cumulative probability associated with each. Parametric probability distribution models can be fit to data sets by estimating their parameter values based upon the data. The adequacy of the parametric probability distribution models as descriptors of the data can be evaluated using goodness-of-fit techniques. Distributions such as normal, lognormal and others are examples of parametric probability distribution models. 

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