During Persuasive 2012 (great conference by the way) I was experimenting with some ways of looking at time series data. I have been very appealed by a bunch of visualizing options coming from people working on Chaos and nonlinear systems. One very simple approach is to plot a time series by plotting the state of the system at t on the x-axis and t-1 on the y-axis. More advance methods exits (see, e.g. http://en.wikipedia.org/wiki/Phase_space), but I thought this simple one was appealing for a lot of social science time series data.
However, after running the below simulations I am kind of unsure. The plots show time lagged presentations of the following functions (from left to right in each panel:
f(x) = 0.5
f(x) = 1.5x / 20 (the 20 added for rescaling)
f(x) = sin(x)
All “time series” are computed in the range [-10, 10]. The first three panels (rows) however show different “sampling” frequencies from the respective functions that generate the time series (1, .1, .01). A striking observation here for me is the fact that with high sampling frequency a sinoid and a straight line are very hard to distinguish using the time-lag plots. Their use thus depends on the sampling frequency however this latter is hard to determine when the functional form is unknown.
Now, the first 3 panels (rows) represent simulations without any noise, so they still seem relatively useful in distinghuising the different functional forms. The lower three panels are again for different sampling frequencies, but this time with noise ( ~Unif(-1,1), added to the process afterwards, no random walk.)
The fourth row basically shows that with low sampling frequency and some noise on top of the actual signal, the time-lag diagram will not at all allow one to distinguish between the different functional form of the time-series.
The simulations were informative for me since recently I have been looking at some data coming from an experiment reported in Science a while ago (“A wandering mind is an unhappy mind” by Killingsworth and Gilbert).
Now, I was doing these same time-lag plots of people’s happiness. Below is a plot showing the plots for a number of randomly selected participants. Which functional form do you think we are looking at?