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Data is # of individuals through time. I fit logistic and exponential growth curves to the data. And event (population bottleneck) takes # to nearly zero at one point. The growth curves are fit in parts: one on either side of the event.

How do I calculate the "total" AIC for the two curves which describe the entire population history? I assume it is not simply adding the AIC scores you can get from each piece separately.

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    $\begingroup$ What are we talking about here? Some context might be nice... $\endgroup$
    – gung - Reinstate Monica
    Commented Jul 3, 2017 at 17:00
  • $\begingroup$ I added an image above to make it more concrete. To be specific, the "population" I'm referring to is the number of species found through time. The number of species within this group grows, but is periodically hit by an extinction. After such an event, growth resumes, though with different controlling parameters. Essentially, I want to know how to calculate the "total" AIC score for all three blue lines considered together, vs all three yellow lines. $\endgroup$
    – Larry Taylor
    Commented Jul 4, 2017 at 3:02

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The AIC for the piecewise function is indeed the sum of the AIC scores of each piece, so yes, you can just add them together.

Let $X_j$ denote the set of points which "belong" to segment $j$ of your model. The AIC can be derived in the following way. We use $k$ to denote the number of parameters in the model, with $k_j$ being the number of parameters in segment $j$.

\begin{align} AIC &= 2k-2\log(P(X|\theta))\\ &= 2 \sum_j k_j - 2 \log \prod_j P(X_j | \theta_j) \\ &= \sum_j 2k_j - 2\log P(X_j|\theta_j)\\ &= \sum_j AIC_j \end{align}

For this proof to go through, there is the caveat that each group $X_j$ must be independent from all other groups in your statistical model. That is, the population growth in each era is independent from all other eras. This is true under a piecewise model of population growth that you are using.

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  • $\begingroup$ I appreciate it! Just to be clear, does this hold true when the time series are of unequal lengths? In the curves above, the first segment is 48 observations, the second is 30, and the third is 18. $\endgroup$
    – Larry Taylor
    Commented Jul 5, 2017 at 5:42
  • $\begingroup$ Yes, it holds even if the segments are uneven in length. $\endgroup$
    – shimao
    Commented Jul 5, 2017 at 5:59

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