Symbolic Maximum Likelihood Estimation with Mathematica

Mathematica is a symbolic programming language that empowers the user to undertake complicated algebraic tasks. One such task is the derivation of maximum likelihood estimators, demonstrably an important topic in statistics at both the research and expository level. In this paper, a Mathematica package is provided that contains a function entitled SuperLog. This function utilises pattern-matching code that enhances Mathematica's ability to simplify expressions involving the natural logarithm of a product of algebraic terms. This enhancement to Mathematica's functionality can be of particular benefit for maximum likelihood estimation

An Artificial Immune System Approach to Automated Program Verification: Towards a Theory of Undecidability in Biological Computing

We propose an immune system inspired Artificial Immune System (AIS) algorithm for the purposes of automated program verification. It is proposed to use this AIS algorithm for a specific automated program verification task: that of predicting shape of program invariants. It is shown that the algorithm correctly predicts program invariant shape for a variety of benchmarked programs. Program invariants encapsulate the computability of a particular program, e.g. whether it performs a particular function correctly and whether it terminates or not. This work also lays the foundation for applying concepts of theoretical incomputability and undecidability to biological systems like the immune system that perform robust computation to eliminate pathogens

Counterexample-Guided Polynomial Loop Invariant Generation by Lagrange Interpolation

We apply multivariate Lagrange interpolation to synthesize polynomial quantitative loop invariants for probabilistic programs. We reduce the computation of an quantitative loop invariant to solving constraints over program variables and unknown coefficients. Lagrange interpolation allows us to find constraints with less unknown coefficients. Counterexample-guided refinement furthermore generates linear constraints that pinpoint the desired quantitative invariants. We evaluate our technique by several case studies with polynomial quantitative loop invariants in the experiments

Conjugate-Gradient Preconditioning Methods for Shift-Variant PET Image Reconstruction

Gradient-based iterative methods often converge slowly for tomographic image reconstruction and image restoration problems, but can be accelerated by suitable preconditioners. Diagonal preconditioners offer some improvement in convergence rate, but do not incorporate the structure of the Hessian matrices in imaging problems. Circulant preconditioners can provide remarkable acceleration for inverse problems that are approximately shift-invariant, i.e., for those with approximately block-Toeplitz or block-circulant Hessians. However, in applications with nonuniform noise variance, such as arises from Poisson statistics in emission tomography and in quantum-limited optical imaging, the Hessian of the weighted least-squares objective function is quite shift-variant, and circulant preconditioners perform poorly. Additional shift-variance is caused by edge-preserving regularization methods based on nonquadratic penalty functions. This paper describes new preconditioners that approximate more accurately the Hessian matrices of shift-variant imaging problems. Compared to diagonal or circulant preconditioning, the new preconditioners lead to significantly faster convergence rates for the unconstrained conjugate-gradient (CG) iteration. We also propose a new efficient method for the line-search step required by CG methods. Applications to positron emission tomography (PET) illustrate the method.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/85979/1/Fessler85.pd

Preconditioning complex symmetric linear systems

A new polynomial preconditioner for symmetric complex linear systems based on Hermitian and skew-Hermitian splitting (HSS) for complex symmetric linear systems is herein presented. It applies to Conjugate Orthogonal Conjugate Gradient (COCG) or Conjugate Orthogonal Conjugate Residual (COCR) iterative solvers and does not require any estimation of the spectrum of the coefficient matrix. An upper bound of the condition number of the preconditioned linear system is provided. Moreover, to reduce the computational cost, an inexact variant based on incomplete Cholesky decomposition or orthogonal polynomials is proposed. Numerical results show that the present preconditioner and its inexact variant are efficient and robust solvers for this class of linear systems. A stability analysis of the method completes the description of the preconditioner.Comment: 26 pages, 4 figures, 4 table

Improving Indoor BlueTooth Localization By Using Bayesian Reasoning To Explore System Parameters

With the advent of smaller, more mobile electronic devices, a wide variety of services can now be augmented with the additional context that is provided by positional information. Systems commonly used for outdoor localization, such as GPS, cannot necessarily be used for indoor localization because often, separating a localizing device from system infrastructure with walls and other obstacles lowers accuracy. Instead, indoor localization systems can be deployed to replace the contextual information required for some situated services, that would otherwise be lost when a device moves indoors. For example, the trilateration algorithm that GPS uses to combine distance estimates from satellites can be repeated using Bluetooth (BT) devices spread throughout an environment. The signal strength of a set of beacons can be read by a localizing device, and those signal strengths can be equated to the distance between the localizing device and the beacon. These distances can then be combined using trilateration. A major source of error in such a system is that BT signal strength does not map directly to only one distance. Because microwave frequency propagation is susceptible to multipath effects and antenna direction, two devices at a fixed location can read a variety of signal strengths, which may not map to the ideal line-of-sight calibrated value. Therefore, any given signal strength reading cannot be interpreted as a single distance without introducing the potential for substantial error. One solution is to probabilistically model the relationship between distance and signal strength by modelling BT localization using a Bayesian network. In a Bayesian network, the distance versus signal strength relationship is stored as the conditional probability of a signal strength reading given a specific distance. Using a Bayesian inference algorithm, one can then reason backwards from a signal strength to a probability distribution representing the estimated position of the localizing BT device. In this thesis, I explore some of the effects of modelling BT localization with a Bayesian network. I first extend the probabilistic calibration to include the influence of the relative orientation of device antennae on the attenuation of BT signal strength between them. I then experiment with the effects of the position of a receiver within a discrete spatial bin, and of the proximity of the transmitters to the edges of the discrete space, because both have the potential to reduce the accuracy of localization using discrete variables. I found that neither affected the localization results in a significant, avoidable fashion. I then studied the effects of the scope of calibration, in terms of the number of distance values used, and of the number of beacons used in localization. I found that additional distance values and a smaller minimum distance used in calibration could result in increased BT localization accuracy, whereas many BT localization systems perform little calibration at distances smaller than 2 m. I also found that accuracy increased when the number of beacons was greater than four, and that accuracy did not significantly decrease when the number of beacons was three or fewer; whereas most trilateration systems use only three or four beacons. I conclude in general that a combination of probabilistic trilateration calibration and Bayesian network inference are viable techniques, and could allow for improvements to localization accuracy in a number of areas