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    Error propagation through generalized high dimensional model representation for data partitioning
    (Wiley-V C H Verlag GMBH, 2004) Tunga, Mehmet Alper; Demiralp, Metin
    In many circumstances the explicit form of a multivariate function is not known; rather a finite number of data is listed from some physical experiments. In such cases a function can be constructed only by imposing some analytical structures containing a finite number of adjustable parameters to fit the function with the given values at some specified points. This means interpolation. The given data is collected or produced by some devices or means which may cause unavoidable errors. This results in an uncertainty band for each datum. The propagation of these errors through the interpolation is the focus of this work. It uses a new form of a partitioning technique called Generalized High Dimensional Model Representation (GHDMR). GHDMR is a divide-and-conquer approach starting from a constant component and proceeding upto high variate terms, univariate, bivariate and so on in the representation. The representation is truncated by keeping only constant and univariate terms for approximation. In other words just a single N variate problem is approximated by N univariate problem.
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    A factorized high dimensional model representation on the nodes of a finite hyperprismatic regular grid
    (Elsevier Science inc, 2005-05-25) Tunga, Mehmet Alper; Demiralp, Metin
    When the values of a multivariate function f(x(1),...,x(N)), having N independent variables like x(1),...,x(N) are given at the nodes of a cartesian, product set in the space of the independent variables and ail interpolation problem is defined to find out the analytical structure of this function some difficulties arise in the standard methods due to the multidimensionality of the problem. Here, the main purpose is to partition this multivariate data into low-variate data and to obtain the analytical structure of the multivariate function by using this partitioned data. High dimensional model representation (HDMR) is used for these types of problems. However, if HDMR requires all components, which means 2(N) number of components, to get a desired accuracy then factorized high dimensional model representation (FHDMR) can be used. This method uses the components of HDMR. This representation is needed when the sought multivariate function has a multiplicative nature. In this work we introduce how to utilize FHDMR for these problems and present illustrative examples.
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    Hybrid high dimensional model representation (HHDMR) on the partitioned data
    (Elsevier B.V., 2006-01-01) Tunga, Mehmet Alper; Demiralp, Metin
    A multivariate interpolation problem is generally constructed for appropriate determination of a multivariate function whose values are given at a finite number of nodes of a multivariate grid. One way to construct the solution of this problem is to partition the given multivariate data into low-variate data. High dimensional model representation (HDMR) and generalized high dimensional model representation (GHDMR) methods are used to make this partitioning. Using the components of the HDMR or the GHDMR expansions the multivariate data can be partitioned. When a cartesian product set in the space of the independent variables is given, the HDMR expansion is used. On the other band, if the nodes are the elements of a random discrete data the GHDMR expansion is used instead of HDMR. These two expansions work well for the multivariate data that have the additive nature. If the data have multiplicative nature then factorized high dimensional model representation (FHDMR) is used. But in most cases the nature of the given multivariate data and the sought multivariate function have neither additive nor multiplicative nature. They have a hybrid nature. So, a new method is developed to obtain better results and it is called hybrid high dimensional model representation (HHDMR). This new expansion includes both the HDMR (or GHDMR) and the FHDMR expansions through a hybridity parameter. In this work, the general structure of this hybrid expansion is given. It has tried to obtain the best value for the hybridity parameter. According to this value the analytical structure of the sought multivariate function can be determined via HHDMR.