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Transformations of sparse linear systems by row-column permutations are considered and various algorithms are constructed to transform arbitrary symmetric positive definite sparse matrices, as well as matrices in band form, doubly bordered band form, and doubly bordered block diagonal form.
Blocked algorithms have much better properties of data locality and therefore can be much more efficient than ordinary algorithms when a memory hierarchy is involved. On the other hand, they are very difficult to write and to tune for particular machines. The reorganization is considered of nested loops through the use of known program transformations in order to create blocked algorithms automatically. The program transformations used are strip mining, loop interchange, and a variant of loop skewing in which invertible linear transformations (with integer coordinates) of the loop indices are allowed. Some problems are solved concerning the optimal application of these transformations. It is shown, in a very general setting, how to choose a nearly optimal set of transformed indices. It is then shown, in one particular but rather frequently occurring situation, how to choose an optimal set of block sizes.
A new cascade basis reduction method of computing the optimal least-squares set of basis functions steering a given function is presented. The method combines the Lie group-theoretic and the singular value decomposition approaches in such a way that their respective strengths complement each other. Since the Lie group-theoretic approach is used, the set of basis and steering functions computed can be expressed analytically. Because the singular value decomposition method is used, this set of basis and steering functions is optimal in the least-squares sense. Furthermore, the computational complexity in designing basis functions for transformation groups with large numbers of parameters is significantly reduced. The efficiency of the cascade basis reduction method is demonstrated by designing a set of basis functions that steers a Gabor function under the four-parameter linear transformation group.
The frequency transform method of Hayashi (1983a) and the asynoptic space-time transform method of Salby (1982b) for deriving space-time spectra are compared. The comparison reveals that the Galilean transformation used by Hayashi and the asynoptic coordinate transformation used by Salby produce identical spectra. The mapping between asynoptic data and the space-time Fourier spectrum is examined. The operation which maps a series of asynoptic measurements into the correct sequence of synoptic maps is described. It is observed that this linear transformation is unique and recovers the correct spectrum over wavenumber and frequency for a satellite observed field.
The problem is considered of classifying into one of m distinct n-variate classes an arbitrary n-channel multispectral measurement vector x. The classification procedure used is the maximum likelihood procedure. Information loss in compressing the n-channel data to k channels is taken to be the difference in the average interclass divergences (or probability of misclassification) in n-space and in k-space. Data compression is accomplished by kxn linear transformation i.e., multiplication of the spectral n-vector by a kxn matrix of rank k.
Several theorems related to the Householder transformation and separability criteria are proven. Orthogonal transformations, topology, divergence, mathematical matrices, and group theory are discussed.
Gravity anomalies and linear transforms over rotating spheres treated as random process with spectrum analysis and applications for geodesy
Linear transformation method for random data vector reduction by matrix algebra
Linear transformation for random-data vector z reduction to smaller dimension vector
Constant coefficient linear systems sensitivity functions computational procedure using nth order differential equations linear transformations
A functional description of a multispectral airborne scanner system that provides spectral bands along a single optical line of sight is reported. The airborne scanner consists of an optical telescope for scanning plane perpendicular to the longitudinal axis of the aircraft and radiation detectors for converting radiation to electrical signals. The system makes a linear transformation of input radiation to voltage recorded on analog magnetic tape.
The control of linear time-invariant systems with respect to a quadratic performance criterion was considered, subject to the constraint that the control vector be a constant linear transformation of the output vector. The optimal feedback matrix, f*, was selected to optimize the expected performance, given the covariance of the initial state. It is first shown that the expected performance criterion can be expressed as the ratio of two multinomials in the element of f. This expression provides the basis for a feasible method of determining f* in the case of single-input single-output systems. A number of iterative algorithms are then proposed for the calculation of f* for multiple input-output systems. For two of these, monotone convergence is proved, but they involve the solution of nonlinear matrix equations at each iteration. Another is proposed involving the solution of Lyapunov equations at each iteration, and the gradual increase of the magnitude of a penalty function. Experience with this algorithm will be needed to determine whether or not it does, indeed, possess desirable convergence properties, and whether it can be used to determine the globally optimal f*.
Linear transformations of variable in differential correction processes cannot always be accomplished in the equations of condition, yet may be desirable to reduce high correlations. The means by which this may be accomplished in the system of normal equations is derived and the process reduced to a very simple computational algorithm.
A theory is developed for a digital equalizer for use in reducing intersymbol interference (ISI) on high speed data communications channels. The equalizer is initialized with a single isolated transmitter pulse, provided the signal-to-noise ratio (SNR) is not unusually low, then switches to a decision directed, on-line mode of operation that allows tracking of channel variations. Conditions for optimal tap-gain settings are obtained first for a transversal equalizer structure by using a mean squared error (MSE) criterion, a first order gradient algorithm to determine the adjustable equalizer tap-gains, and a sequence of isolated initializing pulses. Since the rate of tap-gain convergence depends on the eigenvalues of a channel output correlation matrix, convergence can be improved by making a linear transformation on to obtain a new correlation matrix.
Standard NASA color composites combine the most relevant 3 bands from the 4 MSS bands available. An alternate approach is to extract the principal components of the data by a linear transformation of the 4 bands. This approach leads to a low dimensionality representation of ERTS-1 data with the least degradation, in the mean square sense, of the radiometric accuracy. The technique has been applied with success to ERTS-1 MSS data for several geographic areas in California. For all examples considered the mean square representation error is less than one percent. By combining this dimensionality reduction which our previous results on image enhancement for visual display, color composites are obtained which contain and display most of the information provided by the ERTS-1 sensors.
The B-average divergence for m-distinct classes, resulting from the linear transformation y = Bx, is proposed as a feature selection criterion, where B is a k by n matrix of rank k not greater than n. It is shown that if the B-average divergence resulting from B is large enough, then the probability of misclassification, considered as a function f the class of all k by n matrices, is essentially minimized by B. A computer program, utilizing a gradient procedure, is developed to numerically maximize the B-average divergence and results are presented for the Cl flight line. For this example, corresponding to 9-distinct classes, most of the discriminatory information is found to lie in a 3-dimensional subspace, defined by an appropriately chosen 3 by 12 matrix B.
Autocorrelated normal random variates were generated via computer and the effects of various levels of correlation on goodness of fit problems were studied. The results are useful in determining the distribution or estimating the parameters of populations that correlate observations such as wind speeds and temperature. The model used to generate the autocorrelated data is an autoregressive process of order 1. The Kolmogorov-Smirnov and chi-square statistics are used in the analysis. It was observed in the simulation that high positive correlations tend to shift the sample mean away from the population mean and negative correlations tend to shift the sample mean towards the population mean. In many cases, it was observed that positive and negative correlations tend to decrease the standard deviation. However, since this did not occur in all cases, no definite conclusion can be made regarding the standard deviation. Since the autoregressive process is a linear transformation, it is not surprising that normality was preserved. However, a possible extension of this problem could be to generate non-normal data and observe how the distribution is affected by correlation. Another extension might utilize another model such as autoregressive of order k or a moving average process of order k.
The linear feature selection problem in multi-class pattern recognition is described as that of linearly transforming statistical information from n-dimensional (real Euclidean) space into k-dimensional space, while requiring that average interclass divergence in the transformed space decrease as little as possible. Divergence is the expected interclass divergence derived from Hajek two-class divergence; it is known that there always exists a k x n matrix B such that the transformation determined by B maximizes the divergence in k-dimensional space. It is known that, if Q is any k x k invertible matrix, and B is as defined above, then QB again maximizes the divergence in k-space. It is shown that the converse of this result is false: two matrices exist, B sub 1 and B sub 2, each of which maximizes transformed divergence, which are not related in the fashion B sub 2 = QB sub 1 for any k x k matrix Q.