[R-sig-dyn-mod] differential equations with unknown inputs?

[Spencer Graves]

Hi, Thomas:


       Thanks for the reply.  I'm concerned that if I don't know the 
inputs, I have to estimate them somehow, because otherwise I get only 
the standard homogeneous solution of the differential equation system.  
I don't see how it can support the rich behavior of real physical 
systems subject to substantive but unknown inputs.


       With sufficiently short time between observations, a differential 
equation system can can be turned into difference equations and solved 
using a Kalman filtering approach.  With continuous observations with 
normal errors, the best package I know for that in R is dlm.  It has a 
good vignette, a companion book that appeared less than 2 years ago, and 
seems to be actively maintained.  When the time between observations is 
not sufficiently short, one could still use dlm overall with either a 
theoretical solution or deSolve for the behavior between observations.  
I have not yet tried this, because I uncovered data quality problems 
that need to be fixed before I can proceed.


       At least that is what I'm thinking now.


       Best Wishes,
       Spencer


On 3/26/2011 12:14 PM, Thomas Petzoldt wrote:
> Hello Spencer,
>
> your question is a little bit vague, because I don't know what kind of 
> inputs you have and how is your model constructed. Nevertheless, I 
> would probably start with deSolve and then explore other methods if 
> required.
>
> I'm not sure if the developers of dlm and sde are already on this 
> list, so let's send them an invitation.
>
> Thomas
>
> On 3/24/2011 10:40 PM, Spencer Graves wrote:
>> Hello:
>>
>>
>> How would you approach solving a linear differential equation system
>> with constant coefficients and unknown inputs sampled at irregular time
>> intervals? I'm trying to model the motion of a bridge driven by traffic
>> and heating with a 6-dimensional linear state space model with constant
>> coefficients but with no knowledge of the traffic and large gaps in my
>> records on the temperature.
>>
>>
>> I perceive 3 primary options: dlm, deSolve, and sde.
>>
>>
>> deSolve: I'm concerned that the difference between observation and
>> transition noise could be large in my current application.
>>
>>
>> dlm: This seems like the best option, because unknown inputs can be
>> handled as transition noise. The primary difficulty I see is in
>> translating the differential equation into a difference equation,
>> including estimating the noise variance as proportional to
>> integral(exp(A(t-tau))d.tau).
>>
>>
>> sde: It might make most sense to model this as a stochastic differential
>> equation. However, I have the impression that sde will not handle a
>> multivariate state vector.
>>
>>
>> Thanks in advance for any thoughts you may have on this.
>>
>>
>> Best Wishes,
>> Spencer Graves
>>
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>
>