https://rjtk.github.io/posts/ Recent content in Posts on Quant Out of Water Hugo -- gohugo.io en-us Sun, 24 Sep 2023 00:00:00 -0700 https://rjtk.github.io/posts/constraint-programming-puzzles-ilp-and-z3/ Sun, 24 Sep 2023 00:00:00 -0700 https://rjtk.github.io/posts/constraint-programming-puzzles-ilp-and-z3/ <p>Solving puzzles with computers.</p> https://rjtk.github.io/posts/all-about-quadratic-forms/ Tue, 18 Jul 2023 00:00:00 -0700 https://rjtk.github.io/posts/all-about-quadratic-forms/ <p>Quadratic forms are functions defined through symmetric matrices and represent a ubiquitous class of functions for which there is an enormous amount of useful theoretical and computational results. Indeed, in &ldquo;linear-quadratic&rdquo; models, it is possible to provide analytic solutions to more-or-less any question you would like to ask. This post is a tour of some foundations and results relating to quadratic forms.</p> https://rjtk.github.io/posts/generalized-eigenvalue-problems-and-trace-optimization/ Tue, 18 Jul 2023 00:00:00 -0700 https://rjtk.github.io/posts/generalized-eigenvalue-problems-and-trace-optimization/ <p>Most people are familiar with the concept of matrix eigenvalues. Less well known is that this concept can be fruitfully expanded to the <em>generalized eigenvalues</em> of <em>pairs</em> of matrices. Closely related are matrix trace optimization problems, which extremizes the trace of certain matrix products. Trace optimization constites a large class of practically solvable <em>non-convex</em> optimization problems commonly useful for dimensionality reduction and which includes the unsupervised <em>weighted</em> principle component analysis and the supervised method of Fisher&rsquo;s Linear Discriminant. The purpose of this post is to explore some of these problems, their intuition, and their applications.</p> https://rjtk.github.io/posts/natures-dartboard-the-axioms-of-probability/ Sun, 23 Apr 2023 00:00:00 -0700 https://rjtk.github.io/posts/natures-dartboard-the-axioms-of-probability/ <p>The basic axioms of probability and the idea of a random variable as a function from the sample space to \(\mathbb{R}\) are quite abstract and rather confusing at first pass. However, the axioms can be well motivated from our intuition, and defining random variables simply as functions turns out to be a brilliant and intuitive abstraction. My goal in this post is to try to explain the ideas behind the axiomatization of probability theory, and hopefully make the study of measure-theoretic probability seem a bit less intimidating.</p> https://rjtk.github.io/posts/group-theory-symmetry-and-a-brain-teaser/ Sun, 19 Feb 2023 00:00:00 -0800 https://rjtk.github.io/posts/group-theory-symmetry-and-a-brain-teaser/ <p>I review some elementary discrete math concepts, use them to describe and solve a neat brain-teaser, and generalize the solution into a form of robust breadth first search.</p> https://rjtk.github.io/posts/lyapunov-stability-linear-systems-and-semidefinite-programming/ Sat, 17 Dec 2022 00:00:00 -0800 https://rjtk.github.io/posts/lyapunov-stability-linear-systems-and-semidefinite-programming/ <p>Dynamical systems are ubiquitous models occuring in science, engineering, and mathematics. Not only are they used to model real-world dynamic phenomena like the dynamics of chemical plants, population growth, and physical engineered systems, they can also be applied to model algorithms themselves. This post focuses on <em>linear</em> dynamical systems, their analysis by means of semidefinite programming, and connections with control theory through the computation of quadratic functionals of their paths.</p> https://rjtk.github.io/posts/solving-equations-with-jacobi-iteration/ Sun, 11 Dec 2022 00:00:00 -0800 https://rjtk.github.io/posts/solving-equations-with-jacobi-iteration/ <p>Jacobi iteration is a natural idea for solving certain types of nonlinear equations, and reduces to a famous algorithm for linear systems. This post discusses the algorithm, its convergence, benefits and drawbacks, along with a discussion of examples and pretty pictures 🖼️.</p> https://rjtk.github.io/posts/introduction/ Fri, 09 Dec 2022 00:00:00 -0800 https://rjtk.github.io/posts/introduction/ The title of this blog, Quant out of Water, is more-or-less the first thing that came to mind. However, as I (as of 2022) work as a quant at an hedge fund, and I wanted to write a blog that was not explicitly about finance, this title reflects that motivation. I produced the fish with money using a stable diffusion model. I hope you enjoy some of my writings. You can learn more on the &ldquo;about me&rdquo; page.