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Photonic Reservoir Computing
Published in Paul R. Prucnal, Bhavin J. Shastri, Malvin Carl Teich, Neuromorphic Photonics, 2017
Paul R. Prucnal, Bhavin J. Shastri, Malvin Carl Teich
An example of a classification problem that is not linearly separable is the XOR operation. Nonlinear combinations of the input features can be used to increase the feature space dimensionality in a way that enables linear separability. In the case of XOR, the product of original inputs is taken, and the new feature space becomes (x1, x2, x3) where x3 = x1x2. The linear separator is then () yXOR(x1,x2,x3)=f(−x3) where f is the same Heaviside step function. This simple example is pictured in Fig. 13.2. In Fig. 13.2(a), there is no line that separates red from yellow points. By mapping the original inputs into a three dimensional space in Fig. 13.2, the z = 0 plane can now separate the classes. In this case, z corresponds exactly to x3 in Eq. (13.3).
Nonparametric Decision Theoretic Classification
Published in Sing-Tze Bow, Pattern Recognition and Image Preprocessing, 2002
Some properties relating to the classification, such as linear separability of patterns, are discussed next. Pattern classes are said to be linearly separable if they are classifiable by any linear function, as shown in Figure 3.7a, whereas the classes in Figure 3.7b and c are not classifiable by any linear function. Such types of problems will be discussed in later chapters.
Fuzzy Pattern Recognition
Published in Bogdan M. Wilamowski, J. David Irwin, Intelligent Systems, 2018
In virtue of the classification rule, the membership of Y(x) is highly asymmetric. The slope at one side of the classification line is reflective of the distribution of patterns belonging to class ω1. The geometry of the classifier is still associated with a linear boundary. What fuzzy sets offer is a fuzzy set of membership associated with this boundary. Linear separability is an idealization of the classification problem. In reality there could be some patterns located in the boundary region which does not satisfy the linearity assumption. So even though the linear classifier comes as a viable alternative as a first attempt, further refinement is required. The concept of the nearest neighbor (NN) classifier could form a sound enhancement of the fuzzy linear classifier. The popularity of the NN classifiers stems from the fact that in their development we rely on lazy learning, so no optimization effort is required at all. Any new pattern is assigned to the same class as its closest neighbor. The underlying classification rule reads as follows: given x, determine xi0 in the training set such that i0 = argi min║x – xi║ assuming that the class membership of xi0 is ω2, x is classified as ω2 as well. The NN neighbor classifier could involve a single closest neighbor (in which case the classification rule is referred to as 1-NN classifier), 3 neighbors giving rise to 3-NNs, 5 neighbors resulting in 5-NNs, or k-NNs where “k” is an odd number (k-NN classifier). The majority vote implies the class membership of x. The extension of the k-NN classification rule can be realized in many ways. The intuitive one is to compute a degree of membership of x to a certain class by looking at the closest “k” neighbors, determining the membership degrees ui(x), i = 1, 2, …, L, and choosing the highest one as reflective of the allocation of x to given class. Here Card (Γ) = L (Figure 22.4).
Brazilian Forest Dataset: A new dataset to model local biodiversity
Published in Journal of Experimental & Theoretical Artificial Intelligence, 2022
Ricardo A. Rios, Tatiane N. Rios, Gabriel R. Palma, Rodrigo F. De Mello
In order to hold all those requirements, some kernel function must produce a (semi) positive definite matrix for all examples , thus making possible the linear separability even for nonlinear classification problems, once the original input is mapped into some high-dimensional Hilbert feature space . As matter of fact, it is not even necessary to know the map itself, but only its resultant kernel matrix which contains the inner products of in form , provided the space transformation (Scholkopf & Smola, 2001).
Neural activities classification of left and right finger gestures during motor execution and motor imagery
Published in Brain-Computer Interfaces, 2021
Chao Chen, Peiji Chen, Abdelkader Nasreddine Belkacem, Lin Lu, Rui Xu, Wenjun Tan, Penghai Li, Qiang Gao, Duk Shin, Changming Wang, Dong Ming
The algorithm implementation of SVM is as follows: known test samples … are the test data and corresponding categories (take 1 or – 1), in which . Firstly, the data that can’t be linearly distinguished in low space are transformed into high-dimensional space according to some non-linear transformation method. Then, the data can be classified according to linear separability. The corresponding classification discriminant is shown in formula (1).
Pedestrian detection based on a hybrid Gaussian model and support vector machine
Published in Enterprise Information Systems, 2022
Feng Du, Wan-Liang Wang, Zhi Zhang
Support Vector Machine (SVM) is implemented to minimise the risk and find the best solution. In video detection feature classification, it is aimed at the linear inseparability problem in low-dimensional space. The kernel function is mapped to the high-dimensional space to achieve linear separability, and then linear segmentation is used to achieve feature classification (Darmanjian and Principe 2008).