Re: How can you use a Stochastic Differential Equation
From: Simo Särkkä (ssarkka_at_lce.hut.fi)
Date: 02/13/05
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Date: Mon, 14 Feb 2005 00:28:43 +0200
On Thu, 9 Feb 2005, james wrote:
> I have a question. If you have a stochastic Differential Equation, and
> you can determine a analitical solution for it. What does This
> actually mean or how can I use it.
>
> If, for example with a stochastic linear ODE could I determine the
> probablity ofthe finial value being in a certain set in state space.Eg
> using indicator functions.
>
> Can I take expectations of the finial answer and interpret things
One way of using SDE's is to model the uncertainties in dynamics as
"noise" in the system. For example, movement of a car could be described
as being "almost constant velocity", which could be modeled such that the
second derivative the position trajectory (acceleration) is a zero-mean 2D
white noise process:
d^2 x/dt^2 = w1(t)
d^2 y/dt^2 = w2(t)
Now given the initial conditions you can compute the distribution of the
position and velocity of the car at any time instance in the future by
solving the SDE. However, without any additional information
(measurements) this distribution will be such that the mean is a straight
line and the variances increases in time.
In case of Gaussian initial conditions and linear stochastic differential
equation the solution is a Gaussian process and you can often compute the
mean and covariance of the process analytically. Or if not analytically,
at least there exists very efficient numerical methods for computing them.
> My mathematical background isn't too strong but I have heard of the
> kalman filter. How could this fit into this framework.
Kalman and Kalman-Bucy filters come into play when you have some sensors
for measuring the SDE (i.e., the movement of the car). For example, GPS
could give you measurements of the position and velocity of the car, and
intertial sensors could measure the accelerations and angular velocity.
Given the SDE and noisy linear measurements from it, there is optimal way
of inferring the state of the system and this optimal way is given by
Kalman and Kalman-Bucy filters. Kalman filter considers discrete-time
(sampled) measurements and Kalman-Bucy filter assumes that sensor
measurement is a random function (not sampled).
-- - Simo
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