Why there are no breakpoints in time series

r time-series

Regarding breakpoint (aka changepoint) detection, here I borrow from the headline of a blog post from Dr. Andrew Gelman (https://statmodeling.stat.columbia.edu/2016/03/18/i-definitely-wouldnt-frame-it-as-to-determine-if-the-time-series-has-a-change-point-or-not-the-time-series-whatever-it-is-has-a-change-point-at-every-time-the-question/):

I definitely wouldn’t frame it as “To determine if the time series has a change-point or not.” The time series, whatever it is, has a change point at every time. The question might be, “Is a change point necessary to model these data?” That’s a question I could get behind.

So, given that time series segmentation is model-based, when and how many breakpoints occur are more or less model-dependent (e.g., assumptions-dependent); that is also why there are numerous alternative breakpoint detection models available in R--the same argument applies essentially to all data analyses. With that said, here are some quick results from two Bayesian changepoint packages: bcp and Rbeast (as a disclaimer, I developed the latter). Unlike those freqentisit-based models seeking the single set of best possible locations of breakpoints, Bayesian approaches try to estimate probabilities of breakpoint occurrence for any given point of time.

# Your sample time series; this is a pure data vector without the time info.
z = c( 0.03,0.03,0.01,0.02,0.03,0.02,0.01,0.03,0.02,0.01,0.03,0.03,0.03,0.02,0.01,0.03,
 0.03,0.03,0.01,0.02,0.03,0.03,0.02,0.01,0.03,0.02,0.03,0.00,0.03,0.03,0.03,0.00,
 0.03,0.03,0.03,0.01,0.02,0.03,0.03,0.03,0.02,0.01,0.03,0.03,0.03,0.01,0.02,0.03,
 0.03,0.00,0.03,0.03,0.02,0.01,0.03,0.03,0.00,0.03,0.03,0.01,0.02,0.03,0.02,0.01,
 0.03,0.03,0.00,0.03,0.03,0.00,0.03,0.02)

library(bcp)
out = bcp(z)
plot(out)

No breakpoints found, as shown below, but somehow in the posterior probability curve, there is a tiny bit of probability somewhere to find changepoints. Overall, statistical evidence suggesting a breakpoint is very weak.

enter image description here

A try with Rbeast, which aims to both detect breakpoints and decompose time series (i.e., separate seasonality from trend), but your data contains no periodic/seasonal component; that is why season='none' is used in the beast function.

library(Rbeast)
out = beast(z, season='none')
plot(out)

Similarly, the overall evidence of suggesting the presence of breakpoints is low, but Rbeast finds a few locations that are more likely to be breakpoints than others, as indicated by the tiny peaks in the Pr(tcp) curve. Not surprisingly, the magnitudes of these peak probabilities are very small. On average, beast finds 1 breakpoint and if it has to pinpoint its location, the most probable location is the last peak--the one marked by the vertical dash line.

enter image description here

Again, how to segment a time series depends on how you define breakpoints. I am pretty sure if you try a different approach, the result will vary. If you intend to find any locations that have a literal change (i.e., not constant). You can try something like this