STAT 541: Poisson Regression
Statistics 541: Poisson Regression
- Read section 7.5 and 7.6
- Any questions about the project so far?
Poisson RV review
- If time involved: e-t*mu (t*mu)y/y!
- E(Y) = mu (or t*mu)
- Var(Y) = mu (or t*mu)
- Y is approximately normal if mean is large
- So we could use weighted least squares since variance depends only on the mean which we will have to know anyway.
- Note: We don't have to estimate the variance unlike regular regression
Poisson regression model
- E(Y) = X beta
- E(Y) = t * X beta (if using time)
- E(Y) = expX beta
- E(Y) = t * expX beta (if using time)
- Called "link function"
- Y = X beta + Z where E(Z) = 0, var(Z) = X beta
- Y/E(Y) = X beta/E(Y) + Z where E(Z) = 0, var(Z) = 1
- So as long as we know E(Y) we can estimate E(Y) using regression
- IRLS to the rescue
- Works when mean is large and poisson looks like a normal distribution
Poisson likelihood (no regression case)
- goal is maximize: product e-mumuy/y!
- same as maximize: sum -mu + y log(mu) - log y!
- max at: sum (-1 + y/mu) = 0
- max at: mu-hat = sum(y)/n
Poisson likelihood (regression case)
- If mu is a function of beta, say mu = t X beta
- maximize: sum -t X beta + y log(t X beta) - log y!
- max is at: sum -t X + y /(X beta) * X = 0
- max is at: sum (Y - Y-hat)/(y-hat) * X = 0
- close to: sum (Y - Y-hat)/(sigma-hat) * X = 0
- Which is exactly weighted least squares (IRLS)
- Again, we don't have to estimate the variance
Poisson likelihood (regression with link function)
- If mu is a function of beta, say mu = expt X beta
- Usually written: ln(E(Y)) = t X beta
- maximize: sum -expt X beta + y t X beta - log y!
- max is at: sum -t X expt X beta + y t X = 0
- max is at: sum (Y - Y-hat) * t X = 0
- Since Y-hat is non-linear this isn't as easy as it appears
Last modified: Wed Mar 7 13:45:06 2001