Exercise 2.2.

Time to practice : maple distribution at Sutton

Exercise 2.2.

A species distribution problem

The model

$$y \thicksim f(E, \theta)$$

• f is a probability density function
• $y$ is the abundance (in stems per quadrat) of sugar maple
• $E$ is elevation
• $\theta$ is a set of parameters

Things to think about :

• What are the characteristics of the data ?
• What is the form of the function ?
• What is the probability density function ?

Solution 2.2.

Linear regression

### Likelihood function
ll_fn <- function(a,b,sig,E,obs) {
    # Function for the mean
    mu <- a + b*E
    # PDF 
    lik <- dnorm(x=obs, mean = mu, sd = sig)
    loglik <- log(lik)
    # Return loglikelihood
    sum(loglik)
}

Solution 2.2.

Linear regression

### Try different values 
ll_fn(a=25,b=-0.075,sig=10,E=sutton$y,obs=sutton$acsa)

## [1] -3275.689

ll_fn(a=25,b=-0.05,sig=10,E=sutton$y,obs=sutton$acsa)

## [1] -2046.312

ll_fn(a=30,b=-0.075,sig=10,E=sutton$y,obs=sutton$acsa)

## [1] -2878.139

Solution 2.2.

Linear regression

### Run a grid search 
a_vec <- seq(15,45,length.out = 100)
b_vec <- seq(-0.1,-0.01,length.out = 100)
res <- matrix(nr=100,nc=100)
for(i in 1:100) {
    for(j in 1:100) {
        res[i,j] <- ll_fn(a=a_vec[i],b=b_vec[j],sig=10,E=sutton$y,obs=sutton$acsa)
    }
}

Solution 2.2.

Linear regression

Solution 2.2.

Logistic regression

### Likelihood function
ll_fn <- function(a,b,sig,E,obs) {
    # Function for the mean
    mu <- a + b*E
    # logit transformation 
    p = exp(mu)/(1+exp(mu))
    # no PDF for logistic regression
    # use the output of the model directly 
    lik <- numeric(length(obs))
    lik[obs==1] = log(p[obs==1])
    lik[obs==0] = log(1-p[obs==0])
    # Return loglikelihood
    sum(log(lik))
}

Solution 2.2.

Poisson regression

### Likelihood function
ll_fn <- function(a,b,sig,E,obs) {
    # Function for the mean
    mu <- a + b*E
    # PDF 
    lik <- dpois(x=obs, lambda = mu)
    loglik <- log(lik)
    # Return loglikelihood
    sum(loglik)
}