Input vectors similarity

The Kohonen network or self-organizing map (SOM) was developed by Teuvo Kohonen [11]. It can be used to classify a set of input vectors according to their similarity. The result of such a network is usually a two-dimensional map. Thus, the Kohonen network is a method for projecting objects from a multidimensional space into a two-dimensional space. This projection keeps the topology of the multidimensional space, i.e., points which are close to one another in the multidimensional space are neighbors in the two-dimensional space as well. An advantage of this method is that the results of such a mapping can easily be visualized. 

An input vector is fed into the network and that neuron is determined whose weights are most similar to the input data vector. 

The unit in the Kohonen map that is most similar to the input vector is declared as the winning unit and is activated (i.e. its output is set to 1). The output of a Kohonen unit is typically 0 (not activated) or 1 (activated). 

For the next input vector the similarity, p,, with the weight vector, W , of each active unit, k is calculated ... 

The similarity between x, and the winning unit is compared with a threshold value, p, in the range from zero to one. When p, < P the input pattern, x, is not considered to fall into the existing class. It is decided that a so-called novelty is detected and the input vector is copied into one of the unused dummy units. Otherwise the input pattern, x, is considered to fall into the existing class (to resonate with it). A large p will result in many novelties, thus many small clusters. A small p results in few novelties and thus in a few large clusters. 

Both cases can be dealt with both by supervised and unsupervised variants of networks. The architecture and the training of supervised networks for spectra interpretation is similar to that used for calibration. The input vector consists in a set of spectral features yt(Zj) (e.g., intensities at selected wavelengths zi). The output vector contains information on the presence and absence of certain structure elements and groups fixed by learning rules (Fig. 8.24). Various types of ANN models may be used for spectra interpretation, viz mainly such as Adaptive Bidirectional Associative Memory (BAM) and Backpropagation Networks (BPN). The correlation... 

The input is similar to that of the module 1463. No end condition flag is used since only natural splines can be fitted. On the other hand, you should specify the maximum number IM of iterations. The module returns the array S defined in the description of the module M63, and hence the function value, the derivatives and the integral at a specified X can be computed by calling the module 1464. The important additional inputs needed by the module 1465 are the standard errors given in the vector D. With all D(I) = 0, the module... 

We next describe the first steps in the derivation of the best linear unbiased predictor (BLUP) of Y x) at an untried input vector x (see, for example, Sacks et al 1989). Similar steps are used in Section 4 to estimate the effects of one, two, or more input variables. It is then apparent how to adapt results and computational methods for predicting Y(x) to the problem of estimating such effects. 

Inputs, outputs and disturbances will be denoted as u, y, and d, respectively. For multivariable processes where ui k), U2 k), , Um k) are the m inputs, the input vector u k) at time k is written as a column vector. Similarly, the p outputs are defined by a column vector ... 

FIGURE 4.13 A Kohonen network in three dimensions is a combination of neuron vectors, in which the number of components n matches the ones in the input vector v. The weights w in the Kohonen network are adapted during training. The most similar neuron is determined by the Euclidean distance the resulting neuron is the central neuron, from which the adaptation of the network weights starts. 

The molecules and infrared spectra selected for training have a profound influence on the radial distribution function derived from the CPG network and on the quality of 3D structure derivation. Training data are typically selected dynamically that is, each query spectrum selects its own set of training data by searching the most similar infrared spectra, or most similar input vector. Two similarity measures for infrared spectra are useful ... 

Figure 5.13 Topology of a SOM with 6x6 output neurons. The input vectors are connected to the output neurons (the connections to neuron j is shown in the figure). The output layer in the SOM structure displays the similarity of pattern so that the similar patterns are adjacent, and different ones are well separated.

The input test vector is not, typically, similar to what the chip will see in actual operation. Instead, the goal is to select a set of input vectors that when apphed to the chip will cause every internal node to change state at least once. The timing verification program is used for this task. The program keeps track of the nodes toggled, as each input vector is apphed, and also saves the output vector. When the finished devices are received, the same input test vectors are apphed in sequence, and the resulting output vectors are captured and... 

The learning problem is to find the weights vectors tv" such that the vector of the computed outputs of all units, o, is as close as possible, if not equal, to the vector of the desired output of all units, y, for all the available input vectors x. The system works in a manner similar to the simple perceptron... 

Here, Wo, is an H-dimensional vector of weights of connections between neurons of the hidden layer and the i-th neuron of the output layer, and Z>o,i is the bias of the t-th neuron of the output layer, whereas Wej is an i-dimensional vector of weights of coimections between neurons of the input layer and the j-th neuron of the hidden layer, buj is the bias of the j-th neuron of the hidden layer, and stands for the transpose of a vector w. Similarly, an MLP with two hidden layers, an architecture ( i, hi, h2 o) and an activation function / assigned only to hidden neurons computes a function F = (Fi,...,F such that for an input vector x, the function F,- returns... 

It is noteworthy that the laws are robust due to their design incorporating a sufficient margin of stability pUP 04 FAR 89 FAV 94 KUB 95 LAV 05 LIV 95]. In addition, if the input vector of the system is largely outside the certified maximum flight envelope, only a simple law, using the position of the controller and of the control surfaces, is enabled (this law is similar to the type of command available on a conventional aircraft). 

The natural neural network is such an incredibly complex creation that it would be futile to even attempt to manufacture an exact copy. However, it is possible to create a biologically inspired empirical model containing many densely linked nonlinear processing units (called artificial neurons). The artificial neuron carries out the conversion (in general, nonlinear) of input vector U into output value Y (approximation of the representation being the basis of empirical models) in a manner similar to that of the brain neuron (Fig. 3.5). 

The Kohonen Self-Organizing Maps can be used in a. similar manner. Suppose Xj., k = 1,. Nis the set of input (characteristic) vectors, Wy, 1 = 1,. l,j = 1,. J is that of the trained network, for each (i,j) cell of the map N is the number of objects in the training set, and 1 and j are the dimensionalities of the map. Now, we can compare each with the Wy of the particular cell to which the object was allocated. This procedure will enable us to detect the maximal (e max) minimal ( min) errors of fitting. Hence, if the error calculated in the way just mentioned above is beyond the range between e and the object probably does not belong to the training population. 

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