Dealing with effects of missing and outlier samples on the mentioned methods should be investigated in another study. In this study, it is supposed that there are no missing or outlier samples in datasets for training, testing and incremental learning of the classifier. HD-SVM by improving mechanism of selecting samples covers weakness of TIL for keeping information. It has shown, using HD-SVM reduce exceptionally the training time of the classifier compared with NIL (1/10), while increases the accuracy of the classifier (1.1 %), compared with TIL. In this study HD-SVM algorithm is implemented and comparison of HD-SVM, TIL and NIL is done for process FDD.
![hyperplan equation hyperplan equation](https://cdn-images-1.medium.com/max/1600/1*RCOQPerqRMduIuEz3mjlow.png)
By considering these samples, losing of information by discarding samples significantly reduces.
#Hyperplan equation plus#
In HD-SVM incremental learning algorithm, plus samples violate KTT conditions, samples which satisfy the KTT conditions are added into incremental learning. Antonio Espuña, in Computer Aided Chemical Engineering, 2016 4 Conclusions Warner, F.W.: Foundations of Differentiable Manifolds and Lie Groups.Mohammadhamed Ardakani. Spivak, M.: A Comprehensive Introduction to Differential Geometry. Klingenberg, W.: A Course in Differential Geometry. Harris, J.: Algebraic Geometry: A First Course. Goldschmidt, H.: Integrability criteria for systems of non-linear partial differential equations. Prentice-Hall, Englewood Cliffs (1976)įisher, R.J., Laquer, H.T.: Generalized immersions and the rank of the second fundamental form. Academic Press, New York (1969, 1970, 1972, 1977, 1974)ĭo Carmo, M.P.: Differential Geometry of Curves and Surfaces. Thesis, Idaho State University (1995)ĭieudonné, J.: Treatise on Analysis, vols. Birkhäuser, Boston (1985)īrunette, J.: The Clairaut equation: a study in the geometry of partial differential equations.
![hyperplan equation hyperplan equation](https://i.stack.imgur.com/R783m.png)
Kluwer Academic, Dordrecht (1990)Īrnold, V.I., Gusein-Zade, S.M., Varchenko, A.N.: Singularities of Differentiable Maps, vol. I. Springer, New York (1991)Īrnold, V.I.: Singularities of Caustics and Wave Fronts. Encyclopaedia of Mathematical Sciences, vol. Springer, New York (1988)Īlekseevskij, D.V., Vinogradov, A.M., Lychagin, V.V.: Geometry I: Basic Ideas and Concepts of Differential Geometry. With one mild assumption, namely that the associated family of linear hyperplanes is immersed, it is proven that a family of affine hyperplanes always has an envelope, and that envelope is essentially unique.Ībraham, R., Marsden, J.E., Ratiu, T.: Manifolds, Tensor Analysis, and Applications, 2nd edn. The beauty of this approach becomes apparent in the “Envelope Theorem”.
#Hyperplan equation full#
These have been described elsewhere but, briefly, every classical immersion defines a generalized immersion in a canonical way so that generalized immersions can be understood as ordinary immersions “with singularities.” Next, the concept of an envelope is given a modern definition, namely, an envelope is a generalized immersion solving the family that has a universal mapping property relative to all other full rank solutions. In the process, the classical results are generalized and unified.Ī key step in this process is the use of “generalized immersions”. This paper brings a modern perspective to the classical problem of envelopes of families of affine hyperplanes.
![hyperplan equation hyperplan equation](https://miro.medium.com/max/1280/1*D9U4SpEFIpFFHMsXG_qvJw.png)
The subject of envelopes has been part of differential geometry from the beginning.