Data vector

As we have mentioned, the particular characterization task considered in this work is to determine attenuation in composite materials. At our hand we have a data acquisition system that can provide us with data from both PE and TT testing. The approach is to treat the attenuation problem as a multivariable regression problem where our target values, y , are the measured attenuation values (at different locations n) and where our input data are the (preprocessed) PE data vectors, u . The problem is to find a function iy = /(ii ), such that i), za jy, based on measured data, the so called training data. 

The back wall echoes were sampled at 100 MHz and the length of these were 70 samples, yielding a size of the input data vectors, ) = 35. An example of such an echo is shown in Figure 2 together with its log spectral amplitude. 

In Figure 3 we see how the logarithm of the spectral amplitude effects the estimation results. For each component in input data vector, u, we have defined the feature relevance, Fn d), as... 

It would also be of interest to investigate if the attenuation estimates can be further improved by extending our input data vectors. Since attenuation (and porosity) is spatially correlated, we should expect improvements when including data from A-.scans in a neighbourhood around the point of interest. This is also a topic for future work. 

To enable the application of electronic data analysis methods, the chemical structures have to be coded as vectors see Chapter 8). Thus, a chemical data set consists of data vectors, where each vector, i.e., each data object, represents one chemical structure. 

For example, the objects may be chemical compounds. The individual components of a data vector are called features and may, for example, be molecular descriptors (see Chapter 8) specifying the chemical structure of an object. For statistical data analysis, these objects and features are represented by a matrix X which has a row for each object and a column for each feature. In addition, each object win have one or more properties that are to be investigated, e.g., a biological activity of the structure or a class membership. This property or properties are merged into a matrix Y Thus, the data matrix X contains the independent variables whereas the matrix Ycontains the dependent ones. Figure 9-3 shows a typical multivariate data matrix. 

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

This is done by calculating the Euclidean distance between the input data vector Xc and the weight vectors Wj of all neurons ... 

In clustering, data vectors are grouped together into clusters on the basis of intrinsic similarities between these vectors. In contrast to classification, no classes are defined beforehand. A commonly used method is the application of Kohonen networks (cf. Section 9.5.3). 

Prediction implies the generation of unknown properties. On the basis of example data, a model is established which is able to relate an object to its property. This model can then be used for predicting values for new data vectors. 

The ordered set of measurements made on each sample is called a data vector. The group of data vectors, identically ordered, for all of the samples is called the data matrix. If the data matrix is arranged such that successive rows of the matrix correspond to the different samples, then the columns correspond to the variables as in Figure 1. Each variable, or aspect of the sample that is measured, defines an axis in space the samples thus possess a data stmcture when plotted as points in that / -dimensional vector space, where n is the number of variables. 

Many of the tools are aimed at classification and prediction problems, such as the handwriting example, where a training set of data vectors for which the property is known is used to develop a classification rule. Then the rule can be appHed to a test set of data vectors for which the property is... 

Normalization is a preprocessing method often appHed to spectral data. It makes the lengths of all of the data vectors the same. Thus the sum of the squares of the elements of the data vectors is constant for all samples in the set. If is this sum for the unnormalized sample /, then to normalize the data vectors to the constant m, each element of the data vector would be multiphed by vnj.yj. A common example of this method is normalizing the area under a set of curves to unit area. AppHcation of this method effectively removes the variance in a data set because of arbitrary differences in magnitudes of a set of measurements when such variation is not meaningful and would obscure the significant variance. 

Numeric-to-symbohc transformations are used in pattern-recognition problems where the network is used to classify input data vectors into specific labeled classes. Pattern recognition problems include data interpretation, feature identification, and diagnosis. 

Data base A repository for equipment reliability information categorized to facilitate data retrieval or tabular lists of multiple data vectors, with little text except that needed to explain the data presentation format. 

Data elements The basic items that form a data set or data vector for example, component name, size, failure mode, mean, 5% confidence level, are each a data element. 

Data vector Only those data elements and numerical values mat are used to specify failure characteristics, for example mean, distribution, failure modes. 

Assignments I, J XO H A, P FNZ indices data vector scaling factor for ordinate intermediate results function to transform %-probability values to NPS-scale x is the independent variable in the polynomial. (See Section 5.1.1.)... 

Multiplication of this 4x8 transformation matrix with the 8x1 column vector of the signal results in 4 wavelet transform coefficients or N/2 coefficients for a data vector of length N. For c, = C2 = Cj = C4 = 1, these wavelet transform coefficients are equivalent to the moving average of the signal over 4 data points. Consequently,... 

Even after reading the SAS documentation Combining SAS Data Sets and understanding how the program data vector works, it can still be a bit confusing as to when it may be safe to redefine a pre-existing data set variable in place. The good news is that there is a safe and simple way to avoid the unplanned retention of variables Do... 

The projection of the X data vectors onto the first eigenvector produces the first latent variable or pseudomeasurement set, Zx. Of all possible directions, this eigenvector explains the greatest amount of variation in X. The second eigenvector explains the largest amount of variability after removal of the first effect, and so forth. The pseudomeasurements are called the scores, Z, and are computed as the inner products of the true measurements with the matrix of loadings, a ... 

Procedurally, an input data vector is presented to the ART network. 

In general, signal functions will be obtained in analogous form and consist of a large number of arranged measured points which form a data vector. There are three types of signal functions, which contain ... 

Consider a g-dimensional process sample data vector at a time k,... 

This equation has in general no solution because the data vector y should represent a linear combination of the n column vectors aua2,...,a of the matrix A, for... 

For a straight line, the linear relationship between two data vectors xm and ym can be written in three ways... 

For a least-square plane, the linear relationship between three data vectors xm, ym and zm can also be written in different ways, such as... 

We assume that a random variable vector Y of (here upper-case is used to indicate not a matrix but an ordered set of m random variables) distributed as a multivariate normal distribution has been measured through an adequate analytical protocol (e.g., CaO concentration, the 87Sr/86Sr ratio,...). The outcome of this measurement is the data vector jm. Here ym is the mean of a large number of measurements with expected... 

The value of the expectation can also be used to scale errors when the covariance matrix Sy of the data vector is unknown. The usual procedure is to assume that Sy can be approximated by... 

For a start, the pattern of an atmospheric composition and situation, i.e., a data vector catprising all available physical and/or chatiical data pertaining to that situation, is positioned in a multidimensional feature space that is spanned by all physical (i.e., meteorological) and chanical (i.e., compositional) named features. 

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