INTRODUCTION TO MAXIMUM LIKELIHOOD METHODS FOR ECOLOGY MIDIS NUMÉRIQUES APRIL 3 RD 2014 ALYSSA BUTLER

A Method to Estimate Parameters Likelihood of a parameter (λ) given the observed data x Outline The Basics Example 1.1: How to estimate a mean Example 1.2: How to estimate a function Application Example 2: An example using a Poisson Regression

Known: Data Underlying Distribution ƒ(x|λ) Probability Density Function = ƒ(x|λ) A function f(x) that defines a probability distribution for continuous distributions Probability of observing data x given the parameter λ

The Poisson Distribution: Discrete probability of counts that occur randomly during a given time interval. Scenario: 5 = mean number of birds seen during one time interval (λ) What is the probability of observing 10 birds P(X=10)? ƒ(x|λ) = e -λ λ x x !

R 2 = Σ (i=1 -> n) (y i – yhat i ) 2

Simple, universal framework (well accepted foundation) Easy model comparison (AIC methods) Tend to be more robust and versatile relative to ordinary least-squares (especially for non-linear / non-normal data) Become unbiased as sample sizes increase Confidence limits easy to calculate Can have lower variance compared to other methods

The Poisson Distribution: Presence of birds in a forest What value of λ would make a dataset most probable? Collect a small dataset 10; x = (1,3,4,5,8,6,6,6,3,4)

The Poisson Distribution: Presence of birds in a forest x = (1,3,4,5,8,6,6,6,3,4) ƒ(x 1,…x 10 |λ) = e -λ λ x1 * … * e -λ λ x1 x 1 ! x 10 ! ƒ(x 1,…x 10 |λ) = Π e -λ λ xi x i ! ƒ(x|λ) = e -λ λ x x !

Solve for a Poisson Distribution L(λ)= Π e -λ λ x x! 1.Take Log 2.Simplify 3.Take negative Remember log rules… ln(y) + ln(x) = ln(y*x) ln(y) – ln(x) = ln(y/x) ln(x y ) = y*ln(x)

EXAMPLE 1.2: SOLVING FOR REGRESSION DATA Poisson Distribution; Replace; μ = yhat (our predicted values) σ = y i (our observed values)