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The properties of a linear discriminant function are discussed for the case of arbitrary distributions with equal covariance matrices. Using two examples, a comparison is made showing how the difference of the means relates to the covariance matrices.
Interpolating scattered data points is a problem of wide ranging interest. A number of approaches for interpolation have been proposed both from theoretical domains such as computational geometry and in applications' fields such as geostatistics. Our motivation arises from geological and mining applications. In many instances data can be costly to compute and are available only at nonuniformly scattered positions. Because of the high cost of collecting measurements, high accuracy is required in the interpolants. One of the most popular interpolation methods in this field is called ordinary kriging. It is popular because it is a best linear unbiased estimator. The price for its statistical optimality is that the estimator is computationally very expensive. This is because the value of each interpolant is given by the solution of a large dense linear system. In practice, kriging problems have been solved approximately by restricting the domain to a small local neighborhood of points that lie near the query point. Determining the proper size for this neighborhood is a solved by ad hoc methods, and it has been shown that this approach leads to undesirable discontinuities in the interpolant. Recently a more principled approach to approximating kriging has been proposed based on a technique called covariance tapering. This process achieves its efficiency by replacing the large dense kriging system with a much sparser linear system. This technique has been applied to a restriction of our problem, called simple kriging, which is not unbiased for general data sets. In this paper we generalize these results by showing how to apply covariance tapering to the more general problem of ordinary kriging. Through experimentation we demonstrate the space and time efficiency and accuracy of approximating ordinary kriging through the use of covariance tapering combined with iterative methods for solving large sparse systems. We demonstrate our approach on large data sizes arising both from synthetic sources and from real applications.
An understanding of how an initially Gaussian error volume becomes non-Gaussian over time is an important consideration for space-vehicle conjunction assessment. Traditional assumptions applied to the error volume artificially suppress the true non-Gaussian nature of the space-vehicle position uncertainties. For typical conjunction assessment objects, representation of the error volume by a state error covariance matrix in a Cartesian reference frame is a more significant limitation than is the assumption of linearized dynamics for propagating the error volume. In this study, the impact of each assumption is examined and isolated for each point in the volume. Limitations arising from representing the error volume in a Cartesian reference frame is corrected by employing a Monte Carlo approach to probability of collision (Pc), using equinoctial samples from the Cartesian position covariance at the time of closest approach (TCA) between the pair of space objects. A set of actual, higher risk (Pc >= 10 (exp -4)+) conjunction events in various low-Earth orbits using Monte Carlo methods are analyzed. The impact of non-Gaussian error volumes on Pc for these cases is minimal, even when the deviation from a Gaussian distribution is significant.
A long-sought goal in data assimilation is to build more fully the time dimension into the data assimilation process. Truly four-dimensional data assimilation requires evolving second-moment information, namely the estimation error covariance, along with the state estimate itself. The continuum evolution equations for the estimation error covariance constitute a system of partial differential equations in six space dimensions, forced by a model error covariance and by the observations themselves. The high dimensionality of this system poses a difficult computational problem. Numerous methods have been proposed for approximate, discrete solution of this system of equations. Four-dimensional variational schemes solve these equations indirectly, while schemes based on Kalman filtering solve the equations more directly, usually on a small subspace of the full six-dimensional space. In both cases, most proposed solution methods are not derived from the continuum covariance evolution equations themselves. Instead, most methods simply inherit the discretization used for evolving the state estimate, either in the form of a discrete tangent linear model or, in the case of ensemble schemes, in the form of a fully nonlinear discrete model. In this lecture we show that solutions of the continuum covariance evolution equations possess simple properties that are not readily satisfied upon discretization of the equations. we give numerical examples illustrating that "inherited" discretizations sometimes fail to produce meaningfully accurate solutions. Finally, we suggest alternative discretization methods that may yield more faithful approximate solutions of the continuum problem.
We show how to solve two problems of optimal linear estimation from a finite set of phase data. Clock noise is modeled as a stochastic process with stationary dth increments. The covariance properties of such a process are contained in the generalized autocovariance function (GACV). We set up two principles for optimal estimation: with the help of the GACV, these principles lead to a set of linear equations for the regression coefficients and some auxiliary parameters. The mean square errors of the estimators are easily calculated. The method can be used to check the results of other methods and to find good suboptimal estimators based on a small subset of the available data.
Stable linear system /shaping filter/ synthesis transforming stationary white noise into random process having given covariance function
Three flutter suppression control law design techniques are presented. Each uses multiple control surfaces and/or sensors. The first uses linear combinations of several accelerometer signals together with dynamic compensation to synthesize the modal rate of the critical mode for feedback to distributed control surfaces. The second uses traditional tools (pole/zero loci and Nyquist diagrams) to develop a good understanding of the flutter mechanism and produce a controller with minimal complexity and good robustness to plant uncertainty. The third starts with a minimum energy Linear Quadratic Gaussian controller, applies controller order reduction, and then modifies weight and noise covariance matrices to improve multi-variable robustness. The resulting designs were implemented digitally and tested subsonically on the Active Flexible Wing (AFW) wind tunnel model. Test results presented here include plant characteristics, maximum attained closed-loop dynamic pressure, and Root Mean Square control surface activity. A key result is that simultaneous symmetric and antisymmetric flutter suppression was achieved by the second control law, with a 24 percent increase in attainable dynamic pressure.
The new algorithms developed under this project included a principled procedure for classification of objects, events or circumstances according to a target variable when a very large number of potential predictor variables is available but the number of cases that can be used for training a classifier is relatively small. These "high dimensional" problems require finding a minimal set of variables -called the Markov Blanket-- sufficient for predicting the value of the target variable. An algorithm, the Markov Blanket Fan Search, was developed, implemented and tested on both simulated and real data in conjunction with a graphical model classifier, which was also implemented. Another algorithm developed and implemented in TETRAD IV for time series elaborated on work by C. Granger and N. Swanson, which in turn exploited some of our earlier work. The algorithms in question learn a linear time series model from data. Given such a time series, the simultaneous residual covariances, after factoring out time dependencies, may provide information about causal processes that occur more rapidly than the time series representation allow, so called simultaneous or contemporaneous causal processes. Working with A. Monetta, a graduate student from Italy, we produced the correct statistics for estimating the contemporaneous causal structure from time series data using the TETRAD IV suite of algorithms. Two economists, David Bessler and Kevin Hoover, have independently published applications using TETRAD style algorithms to the same purpose. These implementations and algorithmic developments were separately used in two kinds of studies of climate data: Short time series of geographically proximate climate variables predicting agricultural effects in California, and longer duration climate measurements of temperature teleconnections.
As a motivation for this lecture, we begin by stating a paradox that challenges our fundamental understanding of covariance evolution (at least it challenged my own). Attempting to resolve this 'divergence paradox' leads us to introduce the continuum fundamental solution operator for the dynamics under consideration, which will be advection dynamics in this lecture. This operator is the object that is approximated by the discrete 'tangent linear model. We then show how the fundamental solution operator can be used to describe the solution of the continuum covariance evolution equation. This description is complete enough to resolve fully the divergence paradox.
A linear state-space system will be said to be estimable if in estimating its state from its output the posterior error covariance matrix is strictly smaller than the prior covariance matrix. It will be said to be regulable if the quadratic cost of state feedback control is strictly smaller than the cost when no feedback is used. These properties, which are shown to be dual, are different from the well known observability and controllability properties of linear systems. Necessary and sufficient conditions for estimability and regulability are derived for time variant and time invariant systems, in discrete and continuous time.
This paper presents an algorithm for localization and mapping for a mobile robot using monocular vision and odometry as its means of sensing. The approach uses the Variable State Dimension filtering (VSDF) framework to combine aspects of Extended Kalman filtering and nonlinear batch optimization. This paper describes two primary improvements to the VSDF. The first is to use an interpolation scheme based on Gaussian quadrature to linearize measurements rather than relying on analytic Jacobians. The second is to replace the inverse covariance matrix in the VSDF with its Cholesky factor to improve the computational complexity. Results of applying the filter to the problem of localization and mapping with omnidirectional vision are presented.
Polarimetric SAR data can provide a great deal of information about the scattering behavior of the surface under observation. Polarimetric SAR systems often measure the scattering matrices of the areas under observation in linear polarizations (H and V). From the scattering matrix commonly used forms such as the covariance matrix and the Stokes matrix can be easily derived. Other measures derived from polarimetric SAR data include the standard deviation of texture, correlation coefficients between scattering matrix terms, and the mode and variance of phase differences between scattering matrix terms. The effects of additive system noise on these measurements is not often considered in the literature on this subject.
Polarimetric SAR data can provide a great deal of information about the scattering behavior of the surface under observation. Polarimetric SAR systems often measure the scattering matrices of the areas under observation in linear polarizations (H and V). From the scattering matrix commonly used forms such as the covariance matrix and the Stokes matrix can be easily derived. Other measures derived from polarimetric SAR data include correlation coefficients between scattering matrix terms and the mode and variance of phase differences between scattering matrix terms. The effects of additive system noise on these measurements is not often considered in the literature on this subject. In this paper, the effects of additive system noise on measurements derived from polarimetric SAR data will be examined. It will be shown how first-order noise effects can be removed and how second-order noise effects can be reduced for some measurements...
BACKGROUND: During Lunar missions, astronauts may live and operate in conditions where altered atmospheric pressure and oxygen concentrations may result in a mildly hypoxic environment. While the compensatory hemodynamic mechanisms ensuring adequate oxygen delivery to contracting muscles during exercise in hypoxic conditions are well-studied, less research has focused on potential sex differences in the responses to hypoxia exposure during exercise. As female astronauts make up half of the Artemis astronaut corps, understanding whether physiologic responses in a hypoxic environment differ between sexes may inform recommendations for exercising safely in exploration environments. METHODS: Fourteen subjects (7M/7F) from NASA’s Exploration Atmosphere Study performed two submaximal cycle exercise trials (10 min of exercise at 40% peak aerobic capacity [VO 2 pk]) ergometer under normobaric normoxic gas (21% O 2 ) and normobaric hypoxic (18% O 2 and balanced N 2 ) conditions in randomized order. Linear mixed models with Bonferroni post hoc corrections (fixed effects: condition, sex, VO 2 pk [covariate], body mass [BM, covariate]; random effects: subject, mission) were performed to evaluate the effect of condition and sex on physiologic responses to exercise (oxygen uptake [VO 2 ], carbon dioxide production [VCO 2 ], ventilation [VE], oxygen saturation [SpO 2 ], and heart rate [HR]). RESULTS: Males were comparable to females for age (36.6±4.7 vs 36.4±9.3 yrs; p>0.05) but had greater BM (87.3±10.9 vs 64.5±7.4 kg; p<0.001) and absolute VO2pk (3.5±0.6 vs 2.4±4.7 L/min; p<0.001). Additionally, males had higher VO2 (p=0.005), VCO2 (p=0.02), and VE (p=0.01) during exercise trials, independent of condition; however, when VO 2 pk was added as a covariate, the effect of sex was no longer significant. SpO 2 was reduced during hypoxic exercise compared to the normoxic condition (p<0.001), but neither sex nor environmental condition impacted HR. CONCLUSIONS: Though most physiologic responses to submaximal, short-duration exercise between normoxia and mild hypoxia were similar, females exhibited lower VO 2 , VCO 2 , and VE during both conditions, likely driven by lower aerobic capacity. Future research is needed to determine whether similar findings result from multiday hypobaric hypoxia experienced during missions.
A linear state-space system will be said to be estimable if in estimating its state from its output the posterior error covariance matrix is strictly smaller than the prior covariance matrix. It will be said to be regulable if the quadratic cost of state feedback control is strictly smaller than the cost when no feedback is used. These properties, which are are shown to be dual, are different from the well known observability and controllability properties of linear systems. Necessary and sufficient conditions for estimability and regulability are derived for time variant and time invariant systems, in discrete and continuous time.
A linear state-space system will be said to be estimable if in estimating its state from its output the posterior error covariance matrix is strictly smaller than the prior covariance matrix. It will be said to be regulable if the quadratic cost of state feedback control is strictly smaller than the cost when no feedback is used. Estimability and regulability are shown to be dual properties, equivalent to the nonreducibility of the Kalman filter and of the optimal linear quadratic regulator, respectively.
The computational procedure and associated computer program for a linear feature selection technique are presented. The technique assumes that: a finite number, m, of classes exists; each class is described by an n-dimensional multivariate normal density function of its measurement vectors; the mean vector and covariance matrix for each density function are known (or can be estimated); and the a priori probability for each class is known. The technique produces a single linear combination of the original measurements which minimizes the one-dimensional probability of misclassification defined by the transformed densities.
Minimal formula for bounding the cross correlation between a random forcing function and the state error when this correlation is unknown is used in optimal linear filter theory applications. Use of the bound results in overestimation of the estimation-error covariance.