Linear Model

1. Linear Model - residual or error

2. Formal Definition • ,- observed values of predictor variables (i.e. temperature, precipitation) • - observed value of the response variable (i.e. tree height) • - y intercept:

3. General Linear Model

4. General Linear Model • Can transform the predictor values to linearize the relationship between the predictors and the response • Also changes the variance so it only should be done if the variance is not uniform and is made uniform by the transform

5. Polynomial Regression

6. Need More • Not all phenomenon follow linear response • Not all residuals are normally distributed • This leads: • GLMs: Single function, specified regression distribution • GAMs: Multiple functions • “Non-parametric” approaches: function is determined by the computer

7. GLM • Generalized Linear Model • Not to be confused with a general linear model • Allows a linear model to be related to the response variable via a “Link” function. • Also requires to be from a defined probability distribution

8. Generalized Linear Models • - a random variable with some probability distribution • Related to the response values • - error • Residuals • Linear model without the intercept • - Expected value of • Predicted value (no error)

9. Generalized Linear Models • Linear model without the error • is a “link” function • = ) • is from a known probability distribution

10. Common Functions in R • Probability Distribution (Link Function) • Binomial (link = "logit") • True/false, alive/dead • Gaussian (link = "identity") • Continuous, normal • Gamma (link = "inverse") • Seed distribution, distance from… • Poisson (link = "log") • Counts

11. Normal Distribution Wikipedia

12. Binomial Number of successes of yes/no experiments

13. Poisson Number of events in time T, k=number of occurrences

14. Gamma Distribution Wait times, seed distribution, etc.