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An algorithm is presented for performing accurate, high-speed, floating-point function generation for univariate functions defined at arbitrary breakpoints. Rapid identification of the breakpoint interval, which includes the input argument, is shown to be the key operation in the algorithm. A hardware implementation which makes extensive use of read/write memories is used to illustrate the algorithm.
Several computer subroutines have been developed that interpolate three types of nonanalytic functions: univariate, bivariate, and map. The routines use data in floating-point form. However, because they are written for use on a 16-bit Intel 8086 system with an 8087 mathematical coprocessor, they execute as fast as routines using data in scaled integer form. Although all of the routines are written in assembly language, they have been implemented in a modular fashion so as to facilitate their use with high-level languages.
The use of spline functions in the development of classification algorithms is discussed. A method is formulated for producing spline approximations to univariate density functions when each density function is described by a histogram of measurements. The resulting approximations are then incorporated into a Bayesian classification procedure for which the probability of misclassification can be readily computed. Some preliminary numerical results are presented to illustrate the method.
Discussed is an algorithm for smoothing surfaces with spline functions containing tension parameters. The bivariate spline functions used are tensor products of univariate rational-spline functions. A distinct tension parameter corresponds to each rectangular strip defined by a pair of consecutive spline knots along either axis. Equations are derived for writing the bivariate rational spline in terms of functions and derivatives at the knots. Estimates of these values are obtained via weighted least squares subject to continuity constraints at the knots. The algorithm is illustrated on a set of terrain elevation data.
Two algorithms for interpolating surfaces with spline functions containing tension parameters are discussed. Both algorithms are based on the tensor products of univariate rational spline functions. The simpler algorithm uses a single tension parameter for the entire surface. This algorithm is generalized to use separate tension parameters for each rectangular subregion. The new algorithm allows for local control of tension on the interpolating surface. Both algorithms are illustrated and the results are compared with the results of bicubic spline and bilinear interpolation of terrain elevation data.
Interpolation techniques for three types (univariate, bivariate, and map) of nonanalytic functions are described. These interpolation techniques are then implemented in scaled fraction arithmetic on a representative 16 bit microprocessor. A FORTRAN program is described that facilitates the scaling, documentation, and organization of data for use by these routines. Listings of all these programs are included in an appendix.
Sturm's Theorem is a well-known result in real algebraic geometry that provides a function that computes the number of roots of a univariate polynomial in a semiopen interval. This paper presents a formalization of this theorem in the PVS theorem prover, as well as a decision procedure that checks whether a polynomial is always positive, nonnegative, nonzero, negative, or nonpositive on any input interval. The soundness and completeness of the decision procedure is proven in PVS. The procedure and its correctness properties enable the implementation of a PVS strategy for automatically proving existential and universal univariate polynomial inequalities. Since the decision procedure is formally verified in PVS, the soundness of the strategy depends solely on the internal logic of PVS rather than on an external oracle. The procedure itself uses a combination of Sturm's Theorem, an interval bisection procedure, and the fact that a polynomial with exactly one root in a bounded interval is always nonnegative on that interval if and only if it is nonnegative at both endpoints.
As humans extend their reach beyond the earth, bioregenerative life support systems must replace the resupply and physical/chemical systems now used. The Controlled Ecological Life Support System (CELSS) will utilize plants to recycle the carbon dioxide (CO2) and excrement produced by humans and return oxygen (O2), purified water and food. CELSS design requires knowledge of gas flux levels for net photosynthesis (PS(sub n)), dark respiration (R(sub d)) and evapotranspiration (ET). Full season gas flux data regarding these processes for wheat (Triticum aestivum), soybean (Glycine max) and rice (Oryza sativa) from published sources were used to develop empirical models. Univariate models relating crop age (days after planting) and gas flux were fit by simple regression. Models are either high order (5th to 8th) or more complex polynomials whose curves describe crop development characteristics. The models provide good estimates of gas flux maxima, but are of limited utility. To broaden the applicability, data were transformed to dimensionless or correlation formats and, again, fit by regression. Polynomials, similar to those in the initial effort, were selected as the most appropriate models. These models indicate that, within a cultivar, gas flux patterns appear remarkably similar prior to maximum flux, but exhibit considerable variation beyond this point. This suggests that more broadly applicable models of plant gas flux are feasible, but univariate models defining gas flux as a function of crop age are too simplistic. Multivariate models using CO2 and crop age were fit for PS(sub n), and R(sub d) by multiple regression. In each case, the selected model is a subset of a full third order model with all possible interactions. These models are improvements over the univariate models because they incorporate more than the single factor, crop age, as the primary variable governing gas flux. They are still limited, however, by their reliance on the other environmental conditions under which the original data were collected. Three-dimensional plots representing the response surface of each model are included. Suitability of using empirical models to generate engineering design estimates is discussed. Recommendations for the use of more complex multivariate models to increase versatility are included.
In a normal healthy population, we desired to determine the most age-dependent conventional and advanced ECG parameters. We hypothesized that changes in several ECG parameters might correlate with age and together reliably characterize the functional age of the heart. Methods: An initial study population of 313 apparently healthy subjects was ultimately reduced to 148 subjects (74 men, 84 women, in the range from 10 to 75 years of age) after exclusion criteria. In all subjects, ECG recordings (resting 5-minute 12-lead high frequency ECG) were evaluated via custom software programs to calculate up to 85 different conventional and advanced ECG parameters including beat-to-beat QT and RR variability, waveform complexity, and signal-averaged, high-frequency and spatial/spatiotemporal ECG parameters. The prediction of functional age was evaluated by multiple linear regression analysis using the best 5 univariate predictors. Results: Ignoring what were ultimately small differences between males and females, the functional age was found to be predicted (R2= 0.69, P < 0.001) from a linear combination of 5 independent variables: QRS elevation in the frontal plane (p<0.001), a new repolarization parameter QTcorr (p<0.001), mean high frequency QRS amplitude (p=0.009), the variability parameter % VLF of RRV (p=0.021) and the P-wave width (p=0.10). Here, QTcorr represents the correlation between the calculated QT and the measured QT signal. Conclusions: In apparently healthy subjects with normal conventional ECGs, functional cardiac age can be estimated by multiple linear regression analysis of mostly advanced ECG results. Because some parameters in the regression formula, such as QTcorr, high frequency QRS amplitude and P-wave width also change with disease in the same direction as with increased age, increased functional age of the heart may reflect subtle age-related pathologies in cardiac electrical function that are usually hidden on conventional ECG.
Experimental aerodynamic researchers require estimated precision and bias uncertainties of measured physical quantities, typically at 95 percent confidence levels. Uncertainties of final computed aerodynamic parameters are obtained by propagation of individual measurement uncertainties through the defining functional expressions. In this paper, rigorous mathematical techniques are extended to determine precision and bias uncertainties of any instrument-sensor system. Through this analysis, instrument uncertainties determined through calibration are now expressed as functions of the corresponding measurement for linear and nonlinear univariate and multivariate processes. Treatment of correlated measurement precision error is developed. During laboratory calibration, calibration standard uncertainties are assumed to be an order of magnitude less than those of the instrument being calibrated. Often calibration standards do not satisfy this assumption. This paper applies rigorous statistical methods for inclusion of calibration standard uncertainty and covariance due to the order of their application. The effects of mathematical modeling error on calibration bias uncertainty are quantified. The effects of experimental design on uncertainty are analyzed. The importance of replication is emphasized, techniques for estimation of both bias and precision uncertainties using replication are developed. Statistical tests for stationarity of calibration parameters over time are obtained.
General ideas of functional statistical inference analysis of one sample and two samples, univariate and bivariate are outlined. ONESAM program is applied to analyze the univariate probability distributions of multi-spectral image data.
Fatigue crack growth analysis software has been available to damage tolerance analysts for many years in either commercial products or via proprietary in-house codes. The NASGRO software has been publicly available since the mid-80s (known as NASA/FLAGRO up to 1999) and since 2000 has been sustained and further developed by a collaborative effort between Southwest Research Institute® (SwRI®), the NASA Johnson Space Center (JSC), and the members of the NASGRO Industrial Consortium. Since the stress intensity factor (K) is the foundation of fracture mechanics and damage tolerance analysis of aircraft structures, a significant focus of development efforts in the past fifteen years has been geared towards enhancing legacy K solutions and developing new and efficient numerical K solutions that can handle the complicated stress gradients computed by today’s analysts using detailed finite element models of fatigue critical locations. This paper provides an overview of K solutions that have been recently implemented or improved for the analysis of geometries such as two unequal through cracks at a hole and two unequal corner cracks at a hole, as well as state-of-the-art weight function models capable of computing K in the presence of univariant and/or bivariant stress gradients and complicated residual stress distributions. Some historical background is provided to review how common K solutions have evolved over the years, including selective examples from the literature and from new research. Challenges and progress in rectifying discrepancies between older legacy solutions and newer models are reviewed as well as approaches and challenges for verification and validation of K solutions. Finally, a summary of current challenges and future research and development needs is presented. A key theme throughout the presentation of this paper will be how members of the aerospace industry have collaborated with software developers to develop a practical analysis tool that is used world-wide to support new design as well as the ongoing sustainment and airworthiness of commercial and military aircraft.
For univariate random sequences, the power spectral density acts like a probability density function of the frequencies present in the sequence. This dissertation extends that concept to bivariate random sequences. For this purpose, a function called the joint spectral density is defined that represents a joint probability weighing of the frequency content of pairs of random sequences. Given a pair of random sequences, the joint spectral density is not uniquely determined in the absence of any constraints. Two approaches to constraining the sequences are suggested: (1) assume the sequences are the margins of some stationary random field, (2) assume the sequences conform to a particular model that is linked to the joint spectral density. For both approaches, the properties of the resulting sequences are investigated in some detail, and simulation is used to corroborate theoretical results. It is concluded that under either of these two constraints, the joint spectral density can be computed from the non-stationary cross-correlation.
Previously established results demonstrate that LANDSAT data are autocorrelated and can be described by a univariate linear stochastic process known as auto-regressive-integrated-moving-average model of degree 1, 0, 1 or ARIMA (1, 0, 1). This model has two coefficients of interest for interpretation phi(1) and theta(1). In a comparison of LANDSAT thematic mapper simulator (TMS) data and LANDSAT MSS data several results were established: (1) The form of the relatedness as described by this model is not dependent upon system look angle or pixel size. (2) The phi(1) coefficient increases with decreasing pixel size and increasing topographic complexity. (3) Changes in topography have a greater influence upon phi(1) than changes in land cover class. (4) The theta(1) seems to vary with the amount of atmospheric haze. These patterns of variation in phi(1) and theta(1) are potentially exploitable by the remote sensing community to yield stochastically independent sets of observations, characterize topography, and reduce the number of bytes needed to store remotely sensed data.
The objective of this investigation is to develop a state-of-the-art optimum interpolation (O/I) objective analysis procedure for use in numerical weather prediction studies. A three-dimensional multivariate O/I analysis scheme has been developed. Some characteristics of the GLAS O/I compared with those of the NMC and ECMWF systems are summarized. Some recent enhancements of the GLAS scheme include a univariate analysis of water vapor mixing ratio, a geographically dependent model prediction error correlation function and a multivariate oceanic surface analysis.
The development of a state of the art optimum interpolation (O/I) objective analysis procedure for use in numerical weather prediction studies was investigated. A three dimensional multivariate O/I analysis scheme was developed. Some characteristics of the GLAS O/I compared with those of the NMC and ECMWF systems are summarized. Some recent enhancements of the GLAS scheme include a univariate analysis of water vapor mixing ratio, a geographically dependent model prediction error correlation function and a multivariate oceanic surface analysis.
One of the most difficult aspects of ocean state estimation is the prescription of the model forecast error covariances. The paucity of ocean observations limits our ability to estimate the covariance structures from model-observation differences. In most practical applications, simple covariances are usually prescribed. Rarely are cross-covariances between different model variables used. Here a comparison is made between a univariate Optimal Interpolation (UOI) scheme and a multivariate OI algorithm (MvOI) in the assimilation of ocean temperature. In the UOI case only temperature is updated using a Gaussian covariance function and in the MvOI salinity, zonal and meridional velocities as well as temperature, are updated using an empirically estimated multivariate covariance matrix. Earlier studies have shown that a univariate OI has a detrimental effect on the salinity and velocity fields of the model. Apparently, in a sequential framework it is important to analyze temperature and salinity together. For the MvOI an estimation of the model error statistics is made by Monte-Carlo techniques from an ensemble of model integrations. An important advantage of using an ensemble of ocean states is that it provides a natural way to estimate cross-covariances between the fields of different physical variables constituting the model state vector, at the same time incorporating the model's dynamical and thermodynamical constraints as well as the effects of physical boundaries. Only temperature observations from the Tropical Atmosphere-Ocean array have been assimilated in this study. In order to investigate the efficacy of the multivariate scheme two data assimilation experiments are validated with a large independent set of recently published subsurface observations of salinity, zonal velocity and temperature. For reference, a third control run with no data assimilation is used to check how the data assimilation affects systematic model errors. While the performance of the UOI and MvOI is similar with respect to the temperature field, the salinity and velocity fields are greatly improved when multivariate correction is used, as evident from the analyses of the rms differences of these fields and independent observations. The MvOI assimilation is found to improve upon the control run in generating the water masses with properties close to the observed, while the UOI failed to maintain the temperature and salinity structure.
FORTRAN 2 computer program calculates maximum estimates of a monotonic non-decreasing response function. The program uses the method of reversals algorithm which applies to the analysis of univariate or multivariate sensitivity experiments.