Chapter 7: done... ugggghhhhh

[oharac]

finished the unending chapter. I struggled to figure out how to approach 7.7, and not at all sure I got it "correct" though I got something that resembled the basic concepts.

Where I struggled:

• For a given set of data, the NLL is the sum of all the neg log likelihood NLL_i for each observation within the dataset, i.e. how likely is a given observation to occur given the parameters.
• to find the best set of parameters for the data set, just find the params that give the lowest sum(NLL_i).

So far so good I think?

Here our parameters are r, p, q, to calculate an index of abundance, basically the expected number of observed individuals I_obs for an actual population D. So the deterministic index I_det = max[0, (p + qD)/(1 + rD)].

• This I_det can be zero, if p is negative and qD is small.
• The NLL_i equation for a given observation should be I_det,i - I_obs,i * log(I_det,i) + log(I_obs,i!)
• if I_det,i = 0, then NLL_i = NA.

So here's the problem: even if we set p = -3 and r = 0.03, and let q vary around 1, different values of q*D will result in different numbers of NAs. Same if we let p and r vary, but the first case is easier.

• Then adding all the NLL_i values, excluding NAs, means we are summing with different numbers of elements.
• So more NAs means a lower sum, because you just have fewer numbers in the sum.
• So the sums aren't really comparable! mean() might help some, though since the I_det is dependent on D, mean() doesn't do a great job either...

Anyway, I got something that vaguely resembled Figure 7.9, but only vaguely. So maybe I'm making some unfounded assumption that f's it all up? Let me know what you think when you get to pseudocode 7.7...

Cheers

[oharac]

an option to avoid this problem entirely is to ignore p = -3, and just set p >= 0 (and then your testing range over p for 7.7B should stay in the positive range too). The plots look better!