Triangular subspace

The method is easy to apply to three axes, yielding a maximum of three end-members. Because real data sets are complex, a perfectly triangular positive subspace is rarely observed. However, by considering the changing compositions around the perimeter of the subspace, realistic estimates of the end-members can be identified. For example, the sediment in Pinto Lake, California (Fig. 4) has a broadly triangular subspace, with five apices. By comparing the compositions represented by these apices, it is possible to constrain the end-member compositions. 

In order to model the retention in a hybrid micellar mobile system, Strasters et al. [8] proposed a procedure that used the retention data of only five mobile phases four measurements at the comers of the selected two-dimensional variable space, defined by the concentrations of surfactant and modifier, and the fifth in the center (design VI in Fig. 8.4). In this method, the rectangular variable space is divided into four triangular subspaces defined by three of the five measurements two neighbor comer... 

To understand this, take the matrix group G — GL2, with H the upper triangular group. Here G acts on k1 = kei ke2, and H is the stabilizer of ev In fact G acts transitively on the set of one-dimensional subspaces and since H is the stabilizer of one of them, the coset space is the collection of those subspaces. But they form the projective line over k, which is basically different from the kind of subsets of fc" that we have considered. In the complex case, for instance, it is the Riemann sphere, and all analytic functions on it are constant whereas on subsets of n-space we always have the coordinate projection functions. 

Next, we construct a finite dimensional subspace f2h consisting of piece-wise linear functions. We define subintervals of length Az = Zj+ — Zj, j = 1,2,..., K. As parameters to describe how the function change over the subintervals, we choose the basis functions as the set of triangular functions defined as ... 

The key concept behind the algorithm is the idea of the fuzzy partition of the input space Assuming a process with Ni input variables, the space of each input variable is evenly partitioned into a number of triangular fuzzy subsets. Then, fuzzy partitioning is extended to the entire input space so that a number of fuzzy subspaces are created, where each fuzzy subspace is defined as a combination of Ni particular fuzzy sets. 

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