# Analysis of Ordinal Value Time Series

I have to analyze a number of ordinal value time series with different lengths. Just to be clear:

y_1 = +1, 0, 0, 0, -1, +1, 0, +1, +1, 0, 0, 0, -1, +1, 0, +1
y_2 = 0, 0, 0, +1, 0, 0, 0, -1, +1, 0, +1, -1, -1, 0
y_3 = 0, +1, -1, 0, 0, +1, -1
...


Response variables can take values $\{+1, 0, -1\}$. Times have been discretized, so that if a time series has $n$ observations, the vector of times will be $t = 1,\dots,n$.

I have a bit of experience with classic time series models like AR, MA, ARIMA, but I never dealt with ordinal response variables in time series of different length. I think that this kind of analysis could be done also by means of pattern recognition techniques, but I have zero experience with that kind of stuff.

However, any reference will be very appreciated. Thanks.

• Not offering an answer, just a concept. You have just 3 states, so think about classical stochastic processes, not time series, and focus on the transition probabilities. In your y_1 data, the transition counts to (-1,0,+1) are -1:(0,0,2); 0:(2,4,2); +1:(0,4,1); so, for example, -1 is always followed by +1. For starters, a simple chi-square test could be used to test whether there is dependence. Jun 3 '15 at 2:49
• Do you have a more specific question that you're trying to answer? Jun 3 '15 at 9:37
• I'll add to my earlier comment that the chi-square test I mention wont work very well if the series are all as short as the ones you show; or if you combine the matrices of transition counts for series you think should have the same dependence. (Frankly, I'd venture to say that no analysis technique as enough power to write home about for series as short as you show.) Jun 3 '15 at 23:58
• If you are satisfied with analyzing these time series individually, there is a old model called DAR (discrete autoregressive) in the literature, proposed by Jacobs, and Lewis (1978, JRSS). Their model is to study binary discrete valued time series. You shall be able to find many extensions of their paper. For a more general framework, you can read Cui and Lund (2009, Biometrika). Their approach can deal with very general discrete dependent process. Jun 6 '15 at 19:13