GEOGG121: Methods Monte Carlo methods, revision Dr. Mathias (Mat) Disney UCL Geography Office: 113, Pearson Building Tel: 7670 0592 Email: mdisney@ucl.geog.ac.uk www.geog.ucl.ac.uk/~mdisney

Very brief intro to Monte Carlo Brute force method(s) for integration / parameter estimation / sampling Powerful BUT essentially last resort as involves random sampling of parameter space Time consuming – more samples gives better approximation Errors tend to reduce as 1/N1/2 N = 100 -> error down by 10; N = 1000000 -> error down by 1000 Fast computers can solve complex problems Applications: Numerical integration (eg radiative transfer eqn), Bayesian inference (posterior), computational physics, sensitivity analysis etc etc Numerical Recipes in C ch. 7, p304 http://apps.nrbook.com/c/index.html http://en.wikipedia.org/wiki/Monte_Carlo_method http://en.wikipedia.org/wiki/Monte_Carlo_integration

Basics: MC integration Pick N random points in a multidimensional volume V, x1, x2, …. xN MC integration approximates integral of function f over volume V as Where and +/- term is 1SD error – falls of as 1/N1/2 Choose random points in A Integral is fraction of points under curve x A From http://apps.nrbook.com/c/index.html

Basics: MC integration Why not choose a grid? Error falls as N-1 (quadrature approach) BUT we need to choose grid spacing. For random we sample until we have ‘good enough’ approximation Is there a middle ground? Pick points sort of at random BUT in such a way as to fill space more quickly (avoid local clustering)? Yes – quasi-random sampling: Space filling: i.e. “maximally avoiding of each other” FROM: http://en.wikipedia.org/wiki/Low-discrepancy_sequence Sobol method v pseudorandom: 1000 points

MC approximation of Pi? A simple example of MC methods in practice

MC approximation of Pi? A simple example of MC methods in practice In Python? import numpy as np a = np.random.rand(10,2) np.sum(a*a,1)<1 array([ True, True, False, False, True, False, True, False, True, True], dtype=bool) 4*np.mean(np.sum(a*a,1)<1) 2.3999999999999999

Markov Chain Monte Carlo (MCMC) Integration / parameter estimation / sampling From 80s: “It was rapidly realised that most Bayesian inference could be done by MCMC, whereas very little could be done without MCMC” (Geyer, 2010) Formally MCMC methods sample from probability distribution (eg a posterior) based on constructing a Markov Chain with the desired distribution as its equilibrium (tends to) distribution Markov Chain: system of random transitions where next state dpeends on only on current, not preceding chain (ie no “memory” of how we got here) Many implementations of MCMC including Metropolis-Hastings, Gibbs Sampler etc. From: http://homepages.inf.ed.ac.uk/imurray2/teaching/09mlss/slides.pdf See also: http://www.mcmchandbook.net/HandbookChapter1.pdf

MCMC: Metropolis-Hastings Initialise: pick a state x at random Pick a new candidate state x’ at random. Accept based on criteria Where A is the acceptance distribution, is the proposal distribution (conditional prob of proposing state x’, given x) Transition probability P of x -> x’ If not accepted then x’ = x (no change) OR state transits to x’ Repeat N times, save the new state x’ Repeat whole process From: http://en.wikipedia.org/wiki/Metropolis%E2%80%93Hastings_algorithm

Revision: key topics, points Model inversion – why? Forward model: model predicts system behaviour based on given set of parameter values (system state vector) f(x) BUT we usually want to observe system and INFER parameter values Inversion: f-1(x) - estimate the parameter values (system state) that give rise to observed values Forward modelling useful for understanding system, sensitivity analysis etc. Inverse model allows us to estimate system state

Revision: key topics, points Model inversion – How? Linear: pros and cons? Can be done using linear algebra (matrices) V fast but … Non-linear: pros and cons? Many approaches, all based around minimising some cost function: eg RMSE – difference between MODEL & OBS for a given parameter set Iterative – based on getting to mimimum as quickly as possible OR as robustly as possible OR with fewest function evaluations Gradient descent (L-BFGS); simplex, Powell (no gradient needed); LUT (brute force); simulated annealing; geneatic algorithms; artifical neural networks etc etc

Revision: key topics, points Model inversion – application Linear kernel-driven BRDF modelling requirement for global, near real-time satellite data product SO must be FAST MODIS BRDF product 3 param model: Isotropic (brightness) + Geometric-Optic (shadowing) + Volumetric (volume scattering) Two are (severe) approximations to radiative transfer models – only dependent on view/illum angles

Revision: key topics, points Analytical v Numerical Analytical Can write down equations for f-1(x) Can do fast Numerical No written expression for f-1(x) or perhaps even f(x) Need to approximate parts of it numerically Hard to differentiate (for inversion, gradient descent)

Don’t forget: Course feedback Short MC practical (now) Thanks! And have a great Christmas and New Year