Consider the step functions H=:(]#0:),-#1: 10 H 0 1 1 1 1 1 1 1 1 1 1 10 H 3 0 0 0 1 1 1 1 1 1 1 10 H 10 0 0 0 0 0 0 0 0 0 0 The problem is to find constants A,B,C to minimize the expression
+/*:X-A+B*(#X)H C Right? -Bo >________________________________ > Fra: Ian Clark <earthspo...@gmail.com> >Til: Programming forum <programming@jsoftware.com> >Sendt: 14:34 torsdag den 21. juni 2012 >Emne: Re: [Jprogramming] Kalman filter in J? > >Thanks, Bo and Ric. Yes, I'd tried searching jwiki on "Kalman" and >Pieter's page was the only instance. > >I neglected to mention that in the most general case I can't really be >confident that X1 and X2 have the same distribution function, let >alone the same variance. But looking at it again, I see that under the >restrictions I've placed the problem simplifies immensely to fitting a >step-function H to: X=: K+G+H. If I can just do that, I'll be happy >for now. > >Repeated application of FFT should allow me to subtract the noise >spectrum F(G), or at least see a significant change in the overall >spectrum emerge after point T, and that might handle the more general >cases as well. > >Anyway it's simple-minded enough for me, and worth a try. FFT is, >after all, "fast" :-) > >But won't an even faster transform do the trick, such as (+/X)? On the >above model, X performs a drunkard's walk around a value M1 until some >point T, after which it walks around M2. Solution: simply estimate M1 >and M2 on an ongoing basis. > >I get the feeling I ought to be searching on terms like "edge >detection", "step detection" and CUSUM. > >Anyway, there's enough here to try. > >On Thu, Jun 21, 2012 at 10:28 AM, Ric Sherlock <tikk...@gmail.com> wrote: >> Hi Ian, >> >> A quick search of the J wiki finds this: >> http://www.jsoftware.com/jwiki/Stories/PietdeJong >> Sounds like he might have what you're after? >> >> Cheers, >> Ric >> >> On Thu, Jun 21, 2012 at 4:33 PM, Ian Clark <earthspo...@gmail.com> wrote: >>> 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 > > > ---------------------------------------------------------------------- For information about J forums see http://www.jsoftware.com/forums.htm
