Appendix - Integration - Overview

Example

• 1D - Time Series
• 2D - Spatial Field
• 3D - Spatial Field
• 2D+T - Spatiotemporal Field

Exact Integration¶

These are the cases where we can find a closed-form expression of our integral. This usually stems from very simple cases, i.e., linear and Gaussian.

• Conditions - Linear & Gaussian
• Symbolic - Calculus Course
• Series Approximation
• Conjugate

Approximate Integration¶

This first section looks as many of the classical methods for approximating integrals like Newton-Cotes, Quadrature, Bayesian Quadrature, or Monte-Carlo methods. We will outline the methods

• Newton-Cotes
  • $\int f(x)dx=\sum_n f(x_n)$
  • Locally Linear Interpolation between nodes
  • Nodes - Equidistant Node
  • Interpolating - 6-degree Polynomial
  • e.g. Trapezoid - Linear, Simpsons - Quadratic
• Quadrature
  • $\int f(x)w(x)dx=\sum_n f(x_n)w(x_n)$
  • Nodes - User Defined
  • Interpolant - Roots of Orthogonal Polynomial
  • Polynomials - e.g., Hermite, Legendre, Chebychev, Laguerre
  • e.g., Gaussian
• Bayesian Quadrature
• Monte Carlo

Uncertainty Propagation¶

This is an extension to numerical integration whereby we wish to integrate a quantity defined by a distribution.

Applications

• Integration
• Dynamical Models Complexity
• function
• prob distribution
• dimensionality
• integral method Methods
• Exact - Linear, Gaussian
• Taylor - Linearized Function
• Unscented - Linearized Distribution
• Quadrature - Assumed Density
• Bayesian Quadrature - Kernels , ≈ 10
  • GP Parameterization —> For Free, .e.g., Observations,
  • Otherwise —> Function Approximation, e.g., expensive or black-box simulators
• Monte Carlo - Stochastic
• Markov Chain Monte Carlo

Applications¶

• Convolution + Filtering
• Uncertainty Propagation
• Inference
• Sensitivity Analysis
• Bayesian Filtering-Smoothing