Behavior of time series is tricky to judge from simple plots. The process generating the series can have multiple underlying mechanics and simple plots make it hard to discern these. Sometime back I read about time maps, which try to solve a part of this problem (I encourage you to read that post before this one). Time maps target visualizing time differences between discrete events. This is helpful, for example, to see if there are multiple modes in the repetition of some events. If on a single day you eat 5 times from 9 AM to 11 AM and call it a day, your eating time plot will have repetitions for those small intervals and for the daily one. A time map captures these two modes easily.
Consider the following time series. The x axis shows date and the y axis shows my lastfm scrobbles per day.
Nothing much to it. If I plot it as a time map, I get this (scale is log; sorry for the not that useful tick labels in this whole post, will fix it).
Each dot here is a listen. The \( x \) value being the time difference between it and the previous listen, \( y \) being the difference with next listen. Note that the plot is log scaled (thus the repeated pattern in lower values) which helps us understand the diffs at multiple scales.
Another important point is the almost symmetry along \( x = y \) line. When you use the pre-event and post-event time diffs of each event as \( x \) and \( y \) value for plotting, one points \( x \) will be previous one’s (according to event ordering) \( y \) value. Now this has consequences on whether a time map is useful for you. What follows is the same plot with marginal histograms along the axis. No doubt the \( x \) and \( y \) marginals are similar. This is not because of the data but because of the way both axis values are derived, resulting in a non-exact symmetry.
Think about a point in top left corner. This refers to an event which was preceded by another event shortly but is followed by the next event after a long time gap. The opposite happens with point in bottom right. Because adjacent events share \( x \) and \( y \), the mass of points has similar distributions (notice that this is not the case with very few points). Unless you are displaying another data dimension using color / size of the circles (like in the original blog, where we see points colored according to the time of day), the two dimensions here just add to the visual clutter.
Lets tweak the lastfm data a little bit. Now, the scrobbles are filtered to show only the first listen of each song. To give this some meaning, a lot of these filtered scrobbles in a short time span would mean that I explored more as compared to repeating the same old songs.
Notice how easy it is to find the bumps in the marginal plot. A plot of \( x \) marginal follows.
As a side note this plot makes me wonder about the origin of the bumps. The initial rise up to \( x = 3, 4 \) (around 20, 50 minutes) is mostly due to radios listens (which give you fresh songs frequently) or binging on some new album/artist. The one around 7 (around 18 hours) might be a session change. A new session, with fresh items probably. Need to dig in the actual songs to understand this.
The original blog made a time map for tweets of @BarackObama. I did a re-crawl. Here is the time plot of tweets per day.
Next is the full time map for the series.
As argued earlier, unless we are showing extra information, its much easier to see the marginal to get the frequency behavior of the series. See the plot below.
- Making sense of a histogram in log scale (the kind I used, with uniform bins over log scaled data; You can have non-uniform, log scaled bins too. I haven’t tried that) is tricky. The bins and density don’t exactly go as you would think. Additionally you would see repeated pattern (exposing the discrete values) in the beginning and smoothing in the end. More rigorous analysis should be done to derive something other than qualitative meanings from these.
- Add to it the number-of-bins problem. Plots above use Freedman-Diaconis rule to get the number of bins. Changing this number can result in different views as shown below.
Time maps are neat exploratory tools. To me, they have more qualitative value than quantitative. Most of the visualizations with more than a few dozen points make more sense qualitatively and are better without unnecessary details, that’s why we prefer heatmaps instead of regular scatter in certain cases. A marginal time map follows the same idea.