Introduction to Numerical Integration

[Pages:36]Introduction to Numerical Integration

Biostatistics 615/815 Lecture 21

Last Series of Lectures

z Numerical Optimization

z Progressively sophisticated techniques

? Optimization in a single dimension ? Optimization along multiple dimensions ? Stochastic optimization strategies

z Last Lecture: Gibbs Sampler

Today: Numerical Integration

z Strategies for numerical integration

z Simple strategies with equally spaced abscissas

z Gaussian quadrature methods

z Introduction to Monte-Carlo Integration

The Problem

z Evaluate:

b

I = f (x)dx a

z When no analytical solution is readily available

z Many applications in statistics

? Analysis of censored data, ? Evaluation of cumulative distributions, etc.

The Challenge

z Evaluate f(x) as few times as possible z Select appropriate set of abscissas z Select appropriate set of weights

The Basic Approach

Notation

z Consider a series of abscissas

? x0, x1, x2, ..., xn, xn+1

z Let these be a constant step size h apart

? xi = x0 + i h

z Further define:

? fi = f(xi)

Two Point Trapezoidal Rule

x2 x1

f

( x)dx

h

1 2

f1

+

1 2

f

2

z Exact for polynomials up to degree 1

? For example, f(x) = 2x + 1

z Error proportional to h3 and f(2)

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