Pattern vector

Feed-back models can be constructed and trained. In a constructed model, the weight matrix is created by adding the output product of every input pattern vector with itself or with an associated input. After construction, a partial or inaccurate input pattern can be presented to the network and, after a time, the network converges to one of the original input patterns. Hopfield and BAM are two well-known constructed feed-back models. 

Figure 20 shows more definitively how the location and orientation of a hyperplane is determined by the projection directions, a and the bias, o- Given a pattern vector x, its projection on the linear discriminant is in the a direction and the distance is calculated as d(x ) / cf The problem is the determination of the weight parameters for the hyper-plane ) that separate different pattern classes. These parameters are typically learned using labeled exemplar patterns for each of the pattern classes. 

ART2 forms clusters from training patterns by first computing a measure of similarity (directional rather than distance) of each pattern vector to a cluster prototype vector, and then comparing this measure to an arbitrarily specified proximity criterion called the vigilance. If the pattern s similarity measure exceeds the vigilance, the cluster prototype or center is updated to incorporate the effect of the pattern, as shown in Fig. 25 for pattern 3. If the pattern fails the similarity test, competition resumes without the node... 

Both the sample vector and the node vector contain n entries x is the -th entry in the pattern vector for sample q, while a is the -th entry in the weights vector at node p. This comparison of pattern and node weights is made for each node in turn across the entire map. 

The problem of determining the class to which a target belongs directly relies upon the amount of information available. ATRs are systems that contain an input sub-system that accepts pattern vectors from the... 

However, this procedure failed completely with the hourly data set collected on July 7th and 8th in the same location (See Figure 6). Here the evolution in time of the air conposition pattern vector circles around in the boundary area between complaint and non-complaint situations. There are complaints registered at 12 and 13 hours, however, why is not clear fron the picture. This is another illustration of the observation that features other than physical or chemical ones may be involved in triggering complaints ty the population. 

Pattern Recognition. An alternative treatment of the data is possible and has been discussed by some of us (70). This approach involves the application of pattern recognition, a subject which has received considerable attention in the recent literature. Essentially, the technique involves the transformation of the concentrations of the five target (fingerprint) elements into points in 5-dimensional space which is represented by "pattern vector", for example ... 

Fig. 9. Designation of configuration pattern vector in polyhedral geometry a tetrahedral reference configuration pattern and the corresponding sample configurational pattern b Generation and comparison of the configurational pattern vectors for two enantiomer polyhedral geometric patterns...

Quantitative evaluation of molecular similarity. The fragment spectrum obtained in the above can be described as a kind of multidimensional pattern vector. Consequently, using this pattern representation of a spectrum it is possible to apply diverse quantitative methods for the evaluation of similarity. 

The rows of the matrix represent the observations, and the columns are the values of the descriptors. In other words, each row is a data or pattern vector, and the components of the data vector are physically measurable quantities called descriptors. It is essential that descriptors encode the same information for all samples in the data set. If variable 5 is the area of a gas chromatographic (GC) peak for phenol in sample 1, it must also be the area of the GC peak for phenol in samples 2, 3, etc. Hence, peak matching is crucial when chromatograms or spectra are translated into data vectors. 

The set of transformed, scaled, and normalized data for the ith compound is expressed as a pattern vector of the form ... 

An object is characterized by a set of measures, and it may be represented as a point in multidimensional space defined by the axes, each of which corresponds to a variate. In Figure 5, a data matrix X defines measures of two variables on two objects A and B. Object A is characterized by the pattern vector, a = Xu, and B by the pattern vector, b = X21, X22. Using a distance measure, objects or points closest together are assigned to the same cluster. Numerous distance metrics have been proposed and applied in the scientific literature. 

Calculation of the left-hand side of this equation results in a value for each object which is a function of x, the pattern vector, and which is referred to as the discriminant score. 

The basic idea underlying nearest-neighbour methods is conceptually very simple, and in practice it is mathematically simple to implement. The general method is based on applying the so-called AT-nearest neighbour classification rule, usually referred to as AT-NN. The distance between the pattern vector of... 

For objects 1 and 2 characterized by multivariate pattern vectors Xi and X2 defined by... 

For a total training set of N objects comprised of /i samples known to belong to each group i, the procedure adopted is to determine the /Tth nearest neighbour to the unclassified object defined by its pattern vector x, ignoring group membership. From this, the conditional probability of the pattern vector... 

As well as the Euclidean distance, other metrics have been proposed and employed to measure similarities of pattern vectors between objects. One method used for comparing and classifying spectroscopic data is the Hamming distance. For two pattern vectors i and X2 defined by Equation (38), the... 

Figure 10(a) illustrates a bivariate data set comprising two groups, each of two objects. These four objects are defined by their pattern vectors, including... 

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