Hi Ian.
The following is a tool, but not yet a solution.
FT is a slow Fourier Transform. When the problem has been solved it is time to
optimize. This FT has the nice property that FT^:_1 is identical to FT.
NB.FT =: Fourier Transformation
FT =: +/&(+*((%:%~_1&^&+:&(%~(*/])&i.))&#))
NB.rd =: round complex number.
rd =: (**|)&.+.
NB.H =: Heaviside step function
H =: (]#0:),-#1:
10 H 3
0 0 0 1 1 1 1 1 1 1
NB. FT = FT^:_1
rd FT FT 10 H 3
0 0 0 1 1 1 1 1 1 1
NB. For fixed C the values of A and B to minimize +/*:X-A+B*(#X)H C are found
by solving (2{.FT X) = 2{.FT A+B*(#X)H C
- Bo
>________________________________
> Fra: Ian Clark <earthspo...@gmail.com>
>Til: Programming forum <programming@jsoftware.com>
>Sendt: 18:15 torsdag den 21. juni 2012
>Emne: Re: [Jprogramming] Kalman filter in J?
>
>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
>>>
>>>
>>>
>> ----------------------------------------------------------------------
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