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Kalman Filtering with missing observations

  1. Aug 10, 2018 at 7:12 AM #1
    Hi All,

    Apologies if this is not the right place for this question.

    I have been digging quite a bit around Kalman Filtering recently and I understand how the base case works. However, I am trying to solve quite a complicated problem using the Kalman filter, but am not sure whether the Kalman Filter can be applied in this context.

    I am doing work on UK interest rates and for the data inputs for the analysis I am using Government Spot Rates (government bonds) = C (below), SWAP rates = A (below) and a War Bond Perpetuity = B (below). Now, I have the history of Government bonds (for example) from 01/01/1980 until 31/12/2017. I have the history of the perpetuity from 01/01/1932 until 31/12/2017. I have the history of SWAPs from 01/01/2002 until 31/12/2017. What I am trying to do is to use a Kalman Filter to backfill the SWAP history, so say from 01/01/2002 back to 01/01/1980 using inputs and relationships from the available data that I have described above.

    The problem can be described in more simple terms imagining the following:

    We have three cars – A, B and C. A is driving on a road in the middle, B is driving on a road above A and C is driving on a road below A (as graph below):

    B--------------------------------------------------------------------

    A----------------------------------------------------X

    C--------------------------------------------------------------------

    A simple model to calculate the position of A would be to take the difference between B and C and the difference between A and C so that A = (B-C) – (A-C)

    We also know that there is correlation between the movements of the cars (i.e. B is correlated to A, A is correlated to C and C is correlated to B).

    I am going to use the observed values I have as my measurement inputs (so A, B and C) which feed into the Kalman Filter algorithm to get my estimated state based on my measurements (so by saying measurements I mean the observed data I have) and my simple equation above (theoretical state) – also introducing some noise around these.

    The question I have is what happens when I run out of observations for A and I also loose the relationship between A and C (indicated by the X in the above chart) as the Kalman Filter relies on receiving observations. I have thought of setting the measurement of A to 0 and giving it infinitely big variance, so it gets ignored. So the big questions is - is there a way to use the Kalman Filter to “predicted” where my A will go, given I have lost measurement updates for A at point X (if I was using a GPS for the measurement the GPS has stopped working) but I still know where my B and C are, and I also know the distance between B and C but I have lost the distance between A and C (so my model starts relying on less information)? I was thinking something along the lines of Markov Chain Monte Carlo applying the known correlation structure from before but not sure how this can feed into a Kalman Filter process.

    Any thoughts will be much appreciated.

    Thanks in advance,

    Todor
     
  2. jcsd
  3. Aug 10, 2018 at 11:51 PM #2

    marcusl

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    Caveat, I'm not an expert in Kalman filtering so I hope my comments are helpful and not misleading...
    Kalman filters assume some model for the "plant" or process being simulated, so that state variables can be chosen intelligently. A classic case is tracking the position of an airplane using ground-based radar observations. Variables are position (x, y, z), velocity (vx, vy, vz) and, sometimes, acceleration (ax, ay, az). Training these parameters using past observations permits prediction of future positions. What are the mechanisms underpinning bond prices, and how are they correlated? Your success at setting up a KF for this problem will depend keenly on the validity of your model.

    As for missed observations, the variance of the prediction grows as predictions are made increasingly far from observations. At some point the predictions may become worthless. In radar tracking, tracks are carried along for some time in the absence of new data, but are eventually "killed off."
     
  4. Aug 12, 2018 at 10:44 PM #3

    scottdave

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    Your description is how I remembered it. The RADAR analogy is a good one for people to picture.
     
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