Recall Poisson Suppose we have some predictor We refer to the above function as a Link Function since it links to .
Suppose is the number of cigarette butts at the location.
From Poisson, . We denote distance as .
We have Log Likelihood For Poisson we have the log likelihood function
= \sum_{i=1}^n \left( x_i \log \lambda_i - \lambda_i - \log(x_i!) \right)$$ In this case, we wish to optimize for $\beta_0$ and $\beta_1$ where $\text{log}(\lambda_i) = \beta_0 + \beta_1 k_i$ As an example, we suspect there might be a relationship between $\lambda_i$ and $d_i$. Fix $\beta_0 = 3.55$. Then $\text{log}(\lambda_i)=3.55 + \beta_1d_i$. Now $$\text{log}(\lambda_i)=3.55 + \beta_1d_i \iff \lambda_i= e^{3.55 + \beta_1d_i}$$ Thus $$\ell(\beta_1) = \sum_{i=1}^n \left[ x_i(3.55+\beta_1 d_i) - e^{\,3.55+\beta_1 d_i} - \log(x_i!) \right].$$ We take the first derivative and find the maximum. We can use `glm(...)` in `R` to to fit Poisson Regression. ```r cig.glm <- glm (x ~ d, family = poisson) ``` - In summary, we have some observations $x_i$. - For each observation $x_i$ we suspect it has some parameter $\lambda_i$ for Poisson. - In this case, we suspect that some *predictor* $d _i$ is associated with $\lambda_i$ for each $x_i$. - We aggregate our observations $x_i$ and construct a log likelihood for our Poisson distribution. - In our example, we fixed $\beta_0$. However, we would typically optimize and find $\beta_0$ and $\beta_1$. - When fixing one value, we find the maximum of our log likelihood function. - This optimization and easily be accomplished using `glm(...)`