Yes, Bo, that looks like a fair statement of the problem, in terms of least-squares minimization.
On Thu, Jun 21, 2012 at 4:46 PM, Bo Jacoby <bojac...@yahoo.dk> wrote: > 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 ---------------------------------------------------------------------- For information about J forums see http://www.jsoftware.com/forums.htm
