# Predicting long-memory processes

I'm working with a two-state process with $x_t$ in $\{1, -1\}$ for $t = 1, 2, \ldots$

The autocorrelation function is indicative of a process with long-memory, i.e. it displays a power law decay with an exponent < 1. You can simulate a similar series in R with:

> library(fArma)
> x<-fgnSim(10000,H=0.8)
> x<-sign(x)
> acf(x)


My question: is there a canonical way to optimally predict the next value in the series given just the autocorrelation function? One way to predict is simply to use

$\hat{x}(t) = x(t-1)$

which has a classification rate of $(1 + \rho_1) / 2$, where $\rho$ is the lag-1 autocorrelation, but I feel like it must be possible to do better by taking into account the long-memory structure.

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I think part of the problem lies in the fact that the process you've laid out is not fully defined by the characteristics you've listed. For a sample of size $n$, you've given $n\choose 2$ linear constraints for $2^n$ parameters. Many processes could satisfy the constraints and yet lead to different achievable classification rates. Your $R$ code does uniquely define a process, but it seemed you intended that as a concrete example instead of as the main object of interest. –  cardinal Feb 12 '11 at 4:43
@cardinal, the problem should have to have known solution, which is probably found in W.Palma Long Memory time series: Theory and Methods. The point is that autocorrelation function may be used to obtain by Yule Walker system of equations the parameters of $AR(\infty)$ representation of the process, the point is when such representation exists (invertability) and what truncation is acceptable by the means of say MSE. For $R$ code in my PhD I used fracdiff package. –  Dmitrij Celov Feb 14 '11 at 12:21
@Dmitrij, @Chris, the OP specifically states he is interested in binary-valued processes (I've got a pretty good guess at what he's likely interested in), for which an AR formulation via Yule-Walker would strike me as ad-hoc at the least. Perhaps you could throw a logistic around it to estimate a conditional probability, but it's still important to recognize the assumptions one is making in that case. Also, for long-memory processes, the choice of truncation can be important and induce nontrivial artifacts. –  cardinal Feb 14 '11 at 16:16
@cardinal, @Chris. oh, I as usually missed the part of the task ^__^ In the case of binary-valued process it seems to be a very well known (studied) problem of traffic measurement that comes from communication networks or so called ON/OFF process that exhibits long range dependence (long memory) property. As for the particular example, I'm a bit confused, since in "one way to predict" Chris actually takes the previous value, not using the ACF only (or I'm even more confused by the term "classification rate"). –  Dmitrij Celov Feb 15 '11 at 8:44
I imagine it would be possible to take the code for an autoregressive fractionally integrated model and change the likelihood function to incorporate a probit effects. Then you could get the probability of $1$ or $-1$. –  John Aug 16 '12 at 20:53