# What techniques are used for empirical, stochastic simulation of a time series?

Suppose you have recorded a set of paths in the $y,t$ plane, with $y = f(t)$, $f$ is a stochastic function (i.e. there is a noise term), and $t$ might be time or some other monotonic increasing independent variable.

Suppose further that you know the underlying process is random, meaning the same set of initial conditions (even, "hidden variables") do not produce the same path. Do not assume that changes in $y$ are independent, or even stationary (though, stationary but dependent changes are an important special case).

What techniques are available to simulate an incomplete curve into the future?
What I mean by "incomplete" curve is: your set of historical curves are each over some range of interest, $T=[0,t_{max}]$. You are given a curve that represents data $y_1(t)$ recorded up until some intermediate value of $t' \in T$.

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Are you willing to assume that the historical data form a set of identically distributed and perhaps even independent curves? – NRH Aug 12 '11 at 20:07
Yes, good point; an important case is with each curve independent from the other, with the same underlying distribution throughout $T$.  However, it is important that changes in $y$ on a given curve are generally dependent. – Pete Aug 12 '11 at 23:38
Coudl you use Markov Chains? – Manoel Galdino Aug 16 '11 at 2:08

## 2 Answers

Would dynamic linear models be applicable? (State Space formulation, Kalman filter, etc.) The `dlm` package has some nice tools to create and simulate from models.

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Conceptually, the question fits into the framework of functional data analysis, see, for instance Applied Functional Data Analysis by Ramsay and Silverman. The usual assumption here is that we have a data set of independent, and perhaps even identically distributed, smooth curves. Fitting an fda model to your data over $[0, t_{\max}]$ you are able to predict a future $y_1(t)$ based on the fitted model, or, in principle, the continuation of $y_1(t)$ for $t > t'$ based on the conditional distribution from the fitted model of $y_1(t)$ for $t > t'$ given $y_1(t)$ for $0 \leq t \leq t'$. However, this may be easier said than done.

A simple example is when your curves can be modeled via an ordinary differential equation (with random initial values). Then you can predict $y_1(t)$ for $t > t'$ by solving the ode with the observed initial condition $y_1(t')$.

I would recommend that you take a look in the book above, and perhaps also their theoretical version Functional data analysis or their web page for inspiration. I think there are many ways to proceed depending on the relative impact of inhomogeneity in the $t$-variable and information content in the partial observation of $y_1$ on the continuation of the curve.

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(+1) Good suggestions. Even the "theoretical version" is an easy and enjoyable read. :) – cardinal Oct 1 '11 at 23:35
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