Probability Commons^™

Lectures On Mathematical Computing With Python, Jay Gopalakrishnan

PDXOpen: Open Educational Resources

This open resource is a collection of class activities for use in undergraduate courses aimed at teaching mathematical computing, and computational thinking in general, using the python programming language. It was developed for a second-year course (MTH 271) revamped for a new undergraduate program in data science at Portland State University. The activities are designed to guide students' use of python modules effectively for scientific computation, data analysis, and visualization.

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Some New Results On Stochastic Comparisons Of Coherent Systems Using Signatures, Ebrahim Amini-Seresht, Baha-Eldin Khaledi, Subhash C. Kochar

Mathematics and Statistics Faculty Publications and Presentations

We consider coherent systems with independent and identically distributed components. While it is clear that the system’s life will be stochastically larger when the components are replaced with stochastically better components, we show that, in general, similar results may not hold for hazard rate, reverse hazard rate, and likelihood ratio orderings. We find sufficient conditions on the signature vector for these results to hold. These results are combined with other well-known results in the literature to get more general results for comparing two systems of the same size with different signature vectors and possibly with different independent and identically distributed …

Counting And Coloring Sudoku Graphs, Kyle Oddson

Mathematics and Statistics Dissertations, Theses, and Final Project Papers

A sudoku puzzle is most commonly a 9 × 9 grid of 3 × 3 boxes wherein the puzzle player writes the numbers 1 - 9 with no repetition in any row, column, or box. We generalize the notion of the n² × n² sudoku grid for all n ∈ ℤ^≥2 and codify the empty sudoku board as a graph. In the main section of this paper we prove that sudoku boards and sudoku graphs exist for all such n; we prove the equivalence of [3]'s construction using unions and products of graphs to the definition of …

Reconstructability Analysis With Fourier Transforms, Martin Zwick

Complex Systems Faculty Publications and Presentations

Fourier methods used in two‐ and three‐dimensional image reconstruction can be used also in reconstructability analysis (RA). These methods maximize a variance‐type measure instead of information‐theoretic uncertainty, but the two measures are roughly collinear and the Fourier approach yields results close to that of standard RA. The Fourier method, however, does not require iterative calculations for models with loops. Moreover, the error in Fourier RA models can be assessed without actually generating the full probability distributions of the models; calculations scale with the size of the data rather than the state space. State‐based modeling using the Fourier approach is also …

Estimation Of Cumulative Incidence Functions In Competing Risks Studies Under An Order Restriction, Hammou El Barmi, Subhash C. Kochar, Hari Mukerjee, Francisco J. Samaniego

Mathematics and Statistics Faculty Publications and Presentations

In the competing risks problem an important role is played by the cumulative incidence function (CIF), whose value at time t is the probability of failure by time t for a particular type of failure in the presence of other risks. Its estimation and asymptotic distribution theory have been studied by many. In some cases there are reasons to believe that the CIFs due to two types of failure are order restricted. Several procedures have appeared in the literature for testing for such orders. In this paper we initiate the study of estimation of two CIFs subject to a type …

Multi-Level Decomposition Of Probabilistic Relations, Stanislaw Grygiel, Martin Zwick, Marek Perkowski

Complex Systems Faculty Publications and Presentations

Two methods of decomposition of probabilistic relations are presented in this paper. They consist of splitting relations (blocks) into pairs of smaller blocks related to each other by new variables generated in such a way so as to minimize a cost function which depends on the size and structure of the result. The decomposition is repeated iteratively until a stopping criterion is met. Topology and contents of the resulting structure develop dynamically in the decomposition process and reflect relationships hidden in the data.

Control Uniqueness In Reconstructability Analysis, Martin Zwick

Complex Systems Faculty Publications and Presentations

When the reconstructability analysis of a directed system yields a structure in which a generated variable appears in more than one subsystem, information from all of the subsystems can be used in modeling the relationship between generating and generated variables. The conceptualization and procedure proposed here is discussed in relation to Klir's concept of control uniqueness.

Set-Theoretic Reconstructability Of Elementary Cellular Automata, Martin Zwick, Hui Shu

Complex Systems Faculty Publications and Presentations

Set-theoretic reconstructability analysis is used to characterize the structures of the mappings of elementary cellular automata. The minimum complexity structure for each ECA mapping, indexed by parameter σ, is more effective than the λ parameter of Langton as a predictor of chaotic dynamics.

Probability Driven Heuristic Nets, Lynn Robert Carter

Dissertations and Theses

Let a probability driven switch be defined as a switch of three input paths and three output paths. The status of the input paths defines a probability for each output path (as to whether it will generate a signal or not.) One output path is linked to one input path, so the results of the switch at time t can affect the switch at time t+1. A switch so constructed can be defined (by the probabilities) to take on the function of the standard logic gates (AND, OR, …) A net constructed of these switches can be “taught” by …