Linear Model. Formal Definition General Linear Model.
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Transcript of Linear Model. Formal Definition General Linear Model.
Formal Definition
• , - observed values of covariate variables (i.e. temperature, precipitation)
• - observed value of the response variable (i.e. tree height)
• - y intercept:
General Linear Model
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y = 0.1983x - 0.3216R² = 0.9694
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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
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• 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
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
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)
Generalized Linear Models
– Linear model without the error• is a “link” function• = )• Random component is from a known
probability distribution
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
Binomial
Number of successes of yes/no experiments Wikipedia
Poisson
Number of events in time T, k=number of occurrences Wikipedia
Gamma Distribution
Wait times, seed distribution, etc. Wikipedia
Deviance
• Where:– = Maximum log-likelihood for model– = Maximum log-likelihood for the most
complex model possible (i.e. fits observed data perfectly)
Degrees of Freedom
• number of observations• number of parameters (branches)• Degrees of freedom =