Stochastic differential equations with time uncertainty....

[asimov42]

Hi all,

I'm wondering if anyone is able to point me in a direction regarding an aspect of stochastic differential equations. I have a situation in which I need to propagate a stochastic DE through time using measurement updates - however, the exact time at which each measurement arrives is uncertain (with some distribution, assumed known). Would anyone have a pointer on where to look for information on this?

Thanks.

 

[Stephen Tashi]

. I have a situation in which I need to propagate a stochastic DE through time using measurement updates

What does it mean to "propagate" the equation?

Do you have a known stochastic differential equation or are you trying to find the specific equation by fitting a family of equations to the data?

 

[asimov42]

Ah, very good question, @Stephen Tashi - sorry I should have specified.

In the simplest example case: I have a time series representing a series of, say, accelerometer measurements in 1-D, and I want to determine position as a function of time, given a known, exact initial velocity. In theory, the accelerometer data could be continuous (in practice, it's discrete, but let's leave that for later).

Now, not only are the acceleration values noisy (with some known standard deviation), but the exact time at which each sample is taken is also stochastic. One can assume that data never arrives out-of-sequence, also.

The main question then, is how to determine the mode of the resulting distribution (best estimate of position as a function of time) as well as the standard deviation of the position at any time.

 

[Stephen Tashi]

In control systems, that general type of problem is often handled with a "Kalman filter". Unfortunately, on this forum, questions about Kalman filters often go unanswered, so we may be lacking in Kalman filter experts.

The term "stochastic differential equation" is most often used when the equation models Brownian motion. By contrast, the usual scenario for a Kalman filter is that the basic model is a deterministic differential equation and the stochastic aspects are due to errors in measurements that have discrete distributions.

but the exact time at which each sample is taken is also stochastic.

Does that mean that the "time stamp" telling when an acceleration measurement is taken may be in error? Or do you merely mean that you can't predict in advance when an acceleration measurement will be made?

 

[asimov42]

Right - very familiar with the Kalman filter framework, and there has been work on stochastic measurement arrival times.

I'm referring to the situation in which the time stamp is in error by some amount. So I have a process model (in the KF context) driven by the acceleration updates, but when a time stamped acceleration value arrives, it represents the acceleration at some point in the past $\delta t$, where $\delta t$ is a random variable.

I've seen some work as I noted above on the case with uncertain measurement times (observation updates), but never where the driving process has time uncertainty.

 

[Stephen Tashi]

To make progress, I think you must state a specific probability model for the process.

but never where the driving process has time uncertainty.

In the Kalman filter model, don't we have random variables that represent the things "driving" the process in the sense of random effects? There can be a random acceleration $a(t)$ , which is a variable in the model, but not known precisely from the data. There can be another variable $b(t_1)$ that is the known measured acceleration with time stamp $t_1$. We use $b(t_1)$ to estimate $a(t_1)$. The error in estimating $a(t_1)$ from $b(t_1)$ is due both to the fact that the measurement might not be taken exactly at time $t_1$ and also that $b(t_1)$ may differ from $a(t_1)$ even when the times coincide.

It seems to me if you can model the error between the estimate of $a(t_1)$ from $b(t_1)$ then it doesn't matter that one contributing cause to that error is making a measurement at the wrong time. It's only necessary to model the "bottom line" of the distribution of the error, not the details of how it arises.

 

[mpresic]

Sounds like Kalman filtering. I like Tashi's post but to some degree I disagree with it. I sort of remember a Kalman-Bucy filter. The continuous aspects of the problem are propagated by the Ricatti equation, and the discrete aspects are propagated using the usual Kalman filter equations for observation.

Be sure to look into this if you are interested.

In looking at your later post I now see my previous comment may have limited value. You are saying you have stochastic measurement update times. I am coming up empty for ideas. Usually measurement times are assumed to be under our control.

 

[mpresic]

I have not examined this material for a long time. It seems your time stamps are stochastic, but if you are not doing this analysis in real time, the time stamps are known. If I remember correctly, the ricatti equation can propagate the continuous process (perhaps your acceleration process) state between times TS1, and TS2, assuming these TS's are known. then the measurement in made (via the kalman filter). Between measurements the Ricatti equation is used between the known time stamps.