# How can I generate correlated timeseries made up of 0s and 1s?

I want to generate series of 0s and 1s that exhibit some clustering. By this I mean that 1s and 0s should occur together. So I envisage series of 0s and 1s that will exhibit similar clustering of these elements, and not just random series of 0s and 1s.

In essence, for a single time series, I would go about that by thresholding a Markov chain with a 2x2 transition matrix, with some stochastic added on to it. Now, I'm not too certain on how to do this, but since I would like to produce several of these series, I was wondering whether there's something straightforward that I have missed.

I plan to use these series to simulate data availability (0 or 1) in a data acquisition system and to do some Monte Carlo simulations of how this affects what we can do with the data.

In order for the above simulations to be realistic, I would like to fit real observations to this model, so that the simulated time series share temporal correlation with the data. I would initially do this by calculating lag autocorrelations of both series and tweaking model parameters until I get something that resembles my observations, but unsure whether this is the best way.

Thanks!

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A standard method is to begin by generating an autocorrelated Gaussian process $z_i$. (It doesn't have to be Gaussian, but such processes are easy to generate.) Take the logistic (inverse logit) of the values, producing a series of numbers $p_i = 1/\left(1 + \exp(-z_i)\right)$ in the interval $(0,1)$. Independently draw values from Bernoulli($p_i$) distributions to create a series of $0$ and $1$ values. Clustering will tend to occur with positive autocorrelation.

As a bonus, this procedure allows you to perform two stages of simulation: you can fix the underlying realization of the Gaussian process and iterate the second stage of Bernoulli draws. Or you can generate a separate realization of the Gaussian process each time.

There are probably R packages to do all this directly. The geoRGLM package performs this simulation in two dimensions (using a Matern autocorrelation function, which includes exponential and Gaussian autocorrelations as special cases); you could simulate along a straight line (or $1$ by $n$ grid) to obtain a time series.

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Why logistic over the inverse cdf? Numerical reasons? –  JMS Jun 9 '11 at 14:36
OK, this is quite neat. I've put together some python code to do this here: pastebin.com/y5SyzuVu –  Jose Jun 9 '11 at 14:49
Or you could just threshold the process $(z_i)$. The relationship between the autocorrelation of $(z_i)$ and that of the binary thresholded process is a little more direct. –  cardinal Jun 9 '11 at 17:39
@Jose In your application that's a good simple solution, because you do not have a particular model in mind, nor do you aim to use the model for estimation, prediction, or inference: all you need are simulations that exhibit some form of correlation. In general, though, the GLM approach has a nice tractable theory, good likelihood estimators, and an existing code base, all of which make it an attractive solution. –  whuber Jun 10 '11 at 13:36
@Jose Diggle & Ribeiro (Model-Based Geostatistics, 2007) say likelihood methods are "hampered by computational difficulties." They describe three solutions. (1) MCMC. (2) "Hierarchical likelihood" (Lee & Nelder). (3) GEE. D&R provide likelihood estimation, as well as simulation, in their geoR package. –  whuber Jun 14 '11 at 14:55