Change point detection

I have a specific question about the formulation of offline multiple change point detection given in Burg and Williams.

Where the change points are denoted $$\{\tau_i\}$$, and the slice of a time series from $$a$$ to $$b$$ is $$\textbf{y}_{a:b}$$. $$\ell$$ is a loss function, and $$P$$ is a penalty on the number of change points.

My question is: why is it written as $$\ell(\textbf{y}_{\tau_{i-1}:\tau_{i}-1})$$ and not $$\ell(\textbf{y}_{\tau_{i-1}:\tau_{i}})$$? In other words, why is there a second $$-1$$ term in the loss function?

• This notation is fully explained two paragraphs before the appearance of the formula. Plugging in some numbers for the change points (take $n=2$) will make it obvious.
– whuber
Sep 28, 2021 at 15:06

So $$\textbf{y}_{\tau_{i-1}:\tau_{i}}$$ are the elements of the time series going from changepoint number $$i-1$$ going up to changepoint number $$i$$.
Then $$\textbf{y}_{\tau_{i-1}:\tau_{i}-1}$$ (notice the additional -1 is not a subscript of the $$\tau$$ but rather a subscript of the $$\textbf{y}$$). So this time series starts at changepoint number $$i-1$$ and goes up to one element before changepoint number $$i$$ ($$\tau_i-1$$).