Hi Ian! No, I have not written a Kálmán Filter in J, but I do have a suggestion for your problem. Subject your vector X to a discrete Fourier transform. Y=:FT X. The DC component is the constant K. The ground frequency component account for M1 and M2 being different. The high frequency components constitute the noise G+V1=:V2=:0 (The variances V1 and V2 cannot be distinguished from the noise G anyway, so just put them down and forget about it.) The power of M2-M1 should be compared to the power of G to estimate significance. - Bo
>________________________________ > Fra: Ian Clark <earthspo...@gmail.com> >Til: Programming forum <programming@jsoftware.com> >Sendt: 6:33 torsdag den 21. juni 2012 >Emne: [Jprogramming] Kalman filter in J? > >Can anyone help? Has anyone written a Kalman filter in J? > >I'm not a specialist in either statistics or control theory, so I'm >only guessing a Kalman filter is what I need. Though I do have a >passing acquaintance with the terms: stochastic control and linear >quadratic Gaussian (LQG) control. I am aware that a "Kalman filter" >(like ANOVA) is more a topic than a black-box. > >So let me explain what I want it for. > >I have a time series X which I am assuming can be modelled like this: > >X=: K + G + (X1,X2) > >where > >K is constant >G is Gaussian noise >X1 is a random variable with mean: M1 and variance: V1 >X2 is a random variable with mean: M2 and variance: V2 > >Typically X is a sequence of sensor readings, but may also be >measurements from a series of user trials conducted on a working >prototype, which suffers a design-change at a given point T. > >Simplifying assumptions (which unfortunately I may need to relax in due >course): > >(a) X is not multivariate >(b) X1 and X2 are Gaussian >(c) V1=V2 (only the mean value changes, not the variance). > >The problem: > >1. Estimate T=: 1+#X1 -- the point at which X1 gives way to X2. > >2. Given T, estimate (M2-M1) -- the "underlying improvement", if any, >of the change to the prototype. > >3. (subcase of 2.) Given T, test the null hypothesis: M1=M2, viz that >there has been no underlying improvement. > >4. Estimate U=: #X2 -- the minimum number of samples needed after T in >order to achieve 1-3 above with 95% confidence. > >In other words, detect the signal-in-noise: M1-->M2, and do so in real-time. > >Because of 4, the need to estimate T and (M2-M1) on an ongoing basis, >I can't do a randomised block design. I gather that a Kalman filter, >or some sort of adaptive filter, will handle this problem. > >But maybe something simpler will turn out good enough? > >Supposing I can get hold of a "black box" Kalman filter, I propose to >test it out on generated data and compare its performance to some >simple-minded approach, like estimating M1 / M2 from a simple moving >average of the last U samples, or applying the F-test to 2 sets of U >samples taken either side of T. > >But since the technique aims to be published, or at least critically >scrutinised (and maybe incorporated in a software product), I'd rather >depend on a state-of-art packaged solution than reinvent the wheel: a >large and very well-turned wheel it appears to me. > >Ian Clark >---------------------------------------------------------------------- >For information about J forums see http://www.jsoftware.com/forums.htm > > > ---------------------------------------------------------------------- For information about J forums see http://www.jsoftware.com/forums.htm
