Functional Principal Component Analysis

Introduction Functional Principal Component Analysis (FPCA) is a generalization of PCA where entire functions act as samples ($latex X in L^2(mathcal{T})&bg=ffffff$ over time a interval $latex mathcal{T}&bg=ffffff$) instead of scalar values ($latex X in mathbb{R}^p&bg=ffffff$). The FPCA can be used to find the dominant modes of a set of functions. One of the central ideas … Continue reading Functional Principal Component Analysis

Resource Constrained Scheduling

[redirect url='' sec='0'] Introduction Scheduling constrained resources over time is a tough problem. In fact, the problem is NP-hard. One part of the problem is to find a feasible solution where all constraints are satisfied simultaneously. Another part is to also find a solution which also satisfies some measure of optimality. In practice, it is … Continue reading Resource Constrained Scheduling

Solving Ordinary Linear Differential Equations with Random Initial Conditions

Introduction Ordinary linear differential equations can be solved as trajectories given some initial conditions. But what if your initial conditions are given as distributions of probability? It turns out that the problem is relatively simple to solve. Transformation of Random Variables If we have a random system described as $latex dot{X}(t) = f(X(t),t) qquad X(t_0) … Continue reading Solving Ordinary Linear Differential Equations with Random Initial Conditions

Bayesian Regression Using MCMC

[redirect url='' sec='0'] Introduction Bayesian Regression has traditionally been very difficult to work with since analytical solutions are only possible for simple problems. Hence, the frequentist method called "least-squares regression" has been dominant for a long time. Bayesian approaches, however, has the advantage of working with probability distributions such that the estimated parameters of a … Continue reading Bayesian Regression Using MCMC