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JTB
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Joined: August 20th, 2010, 9:28 am

Kalman Filter

May 31st, 2012, 3:35 pm

I'm running the Kalman filter in real-time, and I am not worried about getting a "100 % -optimal" estimate of the states. So my questions are, what are the most "dangerous" thing for my state estimation:1) A little bit of fat tails? If the noise(s) are not Gaussian, but e.g. Student-T; how much will this affect the state-estimation?2 Will an "almost right" signal-to-noise" ratio (ratio between variance for state equation noise and variance for the measurement equation noise) give good estimate of the states, or is the state estimation VERY sensitive to the size of the variances for the noises-processes?3) What is a robust and easy initialization of the state?I don't have time to optimize the hyper parameters (variances for measurement/state equation) using MLE, so I trying to find out what "practical shortcuts" I can do in terms of getting an "OK" estimate of the state at different time-periods. So what "shortcuts" are most important to get "right": Gaussian noise? signal-to-noise-ratio? exact estimate of the variances? JTB
 
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JTB
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Posts: 221
Joined: August 20th, 2010, 9:28 am

Kalman Filter

June 5th, 2012, 5:24 pm

It seems my questions are not good enough formulated, so I will be shorter this time.What is the most important assumption for good performance in the "Standard " Kalman filter? Normality in the noise-terms or "Right" signal-to-noise-ratio?
 
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OOglesby
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Joined: August 26th, 2011, 5:34 am

Kalman Filter

June 6th, 2012, 11:00 pm

I give you my thoughts on your questions with the caveat that I have used Kalman filters in engineering but not finance. Additionally, I rarely use linear Kalman filters, but Extended Kalman filters.Quote1) A little bit of fat tails? If the noise(s) are not Gaussian, but e.g. Student-T; how much will this affect the state-estimation?If your measurement noise is not Gaussian and uncorrelated in time, your state estimates will be slightly more noisy since the filter is trying to adjust the state estimates so that the innovation sequence (measurement - prediction) is uncorrelated Gaussian noise. If there is time correlation, then your state estimate error may be more of a random walk + noise. For the most part, Kalman filters handle non-Gaussian noise quite well from my experience.Quote2) Will an "almost right" signal-to-noise" ratio (ratio between variance for state equation noise and variance for the measurement equation noise) give good estimate of the states, or is the state estimation VERY sensitive to the size of the variances for the noises-processes?I feel that the actual noise variances are more important that signal-to-noise ratio, but I have not performed or read any studies on this matter. Good state estimates are a function of the state equations and the ratio of the state covariance to measurement variance. Below I elaborate a bit more on this distinction.Quote3) What is a robust and easy initialization of the state?Initialization is very dependent on the problem and the actual filter implementation. I have done simple initializations such as the initial state = 0, and the covariance is huge as well as complex domain specific algorithms. Figuring out a the initialization scheme is what I consider the work of Kalman filter design. It also depends on how many data samples you want to use waiting for the filter to converge to good estimates. A good initialization reduces the convergence time dramatically, but can be much more difficult to develop.QuoteSo what "shortcuts" are most important to get "right": Gaussian noise? signal-to-noise-ratio? exact estimate of the variances?In my experience, the important parts are: System model equations - if they are non-linear then use an Extended Kalman filter, Unscented Kalman filter, or solve the differential equations Measurement noise variance - The ratio of the covariance matrix to the measurement noise variance adjusts the weighting between the model prediction and the measurement update.For example, a large covariance with a small measurement variance causes the filter to believe the measurement. In contrast, a small covariance-to-noise ratio causes the measurement to be mostly ignored. State equation noise - This term expresses how uncertain you are about your model. If you are trying to model a GBM process as a constant, then this term should be very high. Whereas if your system equations are also GBM, then this should be a small value.Very rarely should this term be zero. Another way to view this quantity is that it determines the smallest value of the estimated covariance matrix.Similar to initialization, this quantity is very problem and implementation specific. There are papers that describe adaptive-bandwidth Kalman filters that estimate the state equation noise.Another issues Kalman filters face is computer arithmetic. If you are using vectors instead of scalars, be careful to ensure your covariance matrices are symmetric and positive definite at each step. Also do not use matrix inverse, but instead solve the system of equations. As numerical error builds up, it can corrupt your estimates and this is not an error source that is accounted for in the filter equations.I hope this helps some.
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