Geert Verdoolaege wrote: >>> How can I estimate the probability distribution of a non-stationary >>> signal? > >> ... it depends what you mean by the whole question and by the various >> parts of the question. >> >> ... in particular, what do you mean by "non-stationary" (for your >> specific application) ? For example, if the siognal is non-stationary >> because of seasonal effects, this leads to a set of possible answers >> to your question that aren't available otherwise. > > > It's no seasonal effect. In a first approximation, one might say that > the signal consists of several time intervals where the signal is > stationary, but with a different mean for each interval.
Even so, the signal might still be stationary in a technical sense if the overall properties don't change over a long period of time. Simple models for such as you describe have been used in the past. For example, time is divided into intervals with a random length, and a random mean is specified for each interval where the mean is generated from some underlying stochastic process. (Sometimes this is called a shifting mean process.) It may or may not be sensible to treat this as stationary in a particular application. The term non-stationary is rather imprecise, and the meaning can depend on how you want to view/treat things. A process which "looks" non-stationary over a moderate time-period can appear to stationary when seen over a long time-period. Recall also that a simple sine wave can be turned into a stationary process simply by viewing it (a single realisation) as a sine wave generated to have a random phase shift (shift in time) with respect to some origin. > >> ... again, if you have available a set of multiple realisations of >> the signal, this opens up another set of possibilities. > > I don't have multiple realisations. > >> ... what do you mean by "the probability distribution" ? Do you mean >> the conditional distribution at some time-point or the marginal >> distribution. If you mean the marginal distibution, this might be >> something well-defined and might be estimable given some basic >> assumptions, but it depends on what you mean by "non-stationary". > > What do you mean here by conditional distribution? Conditional on > what? The simplest case is where you condition on the value at a given time-point. such conditional distributions may be well defined even if the process is non-stationary in a drift-off type of way. > And what by the marginal distribution? Integrated over what? I had meant integrated over time, in the case that the process can be treated as stationary. > I have only one signal, isn't that just one random variable? Maybe, but if you are prepared to make assumptions that there is a limit to the memory in the process (so that values separated by a reasonable amount of time are independent), then your sample may be long enough to effective provide several realisations for the purposes of estimating a marginal distribution. . . ================================================================= Instructions for joining and leaving this list, remarks about the problem of INAPPROPRIATE MESSAGES, and archives are available at: . http://jse.stat.ncsu.edu/ . =================================================================
