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Introduction to Maximum Likelihood Methods For Ecology

Introduction to Maximum Likelihood Methods For Ecology. Midis numériques April 3 rd 2014 Alyssa Butler. Outline. 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

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Introduction to Maximum Likelihood Methods For Ecology

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  1. Introduction to Maximum Likelihood Methods For Ecology Midis numériques April 3rd 2014 Alyssa Butler

  2. Outline • 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

  3. WHAT is Maximum Likelihood? • 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

  4. Why Maximum Likelihood? • 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-λλx x !

  5. Why Maximum Likelihood? R2 = Σ(i=1-> n) (yi – yhati)2

  6. Why Maximum Likelihood? • • 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

  7. Background of Maximum Likelihood • 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)

  8. Background of Maximum Likelihood • The Poisson Distribution: Presence of birds in a forest • x = (1,3,4,5,8,6,6,6,3,4) ƒ(x1,…x10|λ) =e-λλx1 * … *e-λλx1 x1! x10! ƒ(x1,…x10|λ) = Πe-λλxi xi ! = L(λ) ƒ(x|λ) = e-λλx x !

  9. Example 1.1: solving for a mean • Solve for a Poisson Distribution L(λ)= Πe-λλx x! • Take Log • Simplify • Take negative • Remember log rules… • ln(y) + ln(x) = ln(y*x) • ln(y) – ln(x) = ln(y/x) • ln(xy) = y*ln(x)

  10. Example 1.2: Solving for Regression Data • Poisson Distribution; • Replace; • μ = yhat (our predicted values) • σ = yi(our observed values)

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