Hi Ian.
I shined the program up a bit. Rather than transforming each row I tranform the
whole matrix at once. It is faster, but still slow. The test function used is
rounded down to zero or up to one. Of course it is hard to reconstruct ABC from
a bitvector, but I think it does a reasonable job. Please set me know how fast
is is on your test examples, and if you manage to replace the Slow Fourier
Transform (SFT) by Fast Fourier Transform (FFT).
- Bo
NB.H =: Heaviside steps
H =: i.&<:</i.
NB.U =: Unitary matrix generator
U =: %:%~_1^(%~+:&(*/~)&i.)
NB.SFT=: Slow Fourier Transformation
SFT=: ++/ . *U&{:&$
NB.sm =: symmetrize
sm =: [:,"1/}:"1&(,:|."1)
NB.RFT=: Real Fourier Transform.
RFT=: 9 o.${.SFT&sm
NB.M1 =: zero origin index of the last item before the step
M1 =: [:(i.<./)[:+/"1[:*:[:}.[:(-"1{.)[:}."1[:(%{."1)}."1
M2 =: [:{:"1[:RFT{.,:~([:{:$){.[:{.{:
M3 =: [:(}.,-/){.,:([:%/1&{"1)*1&{
M4 =: M2&M3
M5 =: 3 : 'x,~M4 y{~0,>:x=.M1 y'
M6 =: RFT&(,H&{:&$)
NB.ABC=: constant,stepsize,stepposition
ABC=: M5&M6
round=:>[:?0$~$
ABC X=.round 0.2+0.4*33<i.200
0.264787 0.273477 31
10 20$X
0 0 0 0 0 0 0 1 0 1 1 0 0 0 0 1 0 1 0 0
0 0 0 1 0 0 0 0 1 0 0 0 1 0 1 1 1 1 0 1
1 0 1 1 0 0 1 1 1 0 1 0 1 1 0 1 0 0 1 0
1 0 1 1 1 1 0 0 0 1 0 0 0 1 0 1 0 1 1 1
0 1 1 1 0 0 1 1 0 0 0 1 1 0 1 0 0 1 0 0
0 1 1 0 0 0 1 0 1 0 1 1 1 1 1 0 1 0 0 1
0 1 0 0 0 0 1 1 1 1 1 1 0 1 1 0 1 0 1 1
0 0 1 1 0 1 0 0 1 1 0 1 0 1 1 1 1 0 1 1
1 0 1 0 0 0 1 1 0 1 1 0 1 0 0 0 1 0 1 0
0 1 0 1 0 0 1 1 0 0 1 1 1 0 1 1 1 0 0 1
>________________________________
> Fra: Bo Jacoby <bojac...@yahoo.dk>
>Til: Programming forum <programming@jsoftware.com>
>Sendt: 13:47 onsdag den 27. juni 2012
>Emne: Re: [Jprogramming] Kalman filter in J?
>
>Hi Ian.
>
>The FFT is fast and it is not worth while to try to optimize it further. My
>toy implementation of the FT is embarassingly slow and should definitely be
>replaced. NB.FT=:[:FFT+ is exactly equal to its inverse, so that you no longer
>need to distinguish. That makes life happier and safer. I included this
>observation into Wikipedia, but as Original Research is banned in Wikipedia it
>was eventually removed. I do not know if FFT X works when #X is not an integer
>power of 2.
>
>- Bo
>
>
>
>>________________________________
>> Fra: Ian Clark <earthspo...@gmail.com>
>>Til: Programming forum <programming@jsoftware.com>
>>Sendt: 9:22 onsdag den 27. juni 2012
>>Emne: Re: [Jprogramming] Kalman filter in J?
>>
>>I've just read what it says about Discrete Hartley Transform in:
>>
>>http://en.wikipedia.org/wiki/Fast_fourier_transform#FFT_algorithms_specialized_for_real_and.2For_symmetric_data
>>But this is such a weaselly assertion, I've a good mind to try it out,
>>when I get round to it.
>>
>>On Wed, Jun 27, 2012 at 8:11 AM, Ian Clark <earthspo...@gmail.com> wrote:
>>>> Why would a Hartley transform be faster than the Fast Fourier Transform?
>>>
>>> That should be "might" not "would", because it's an off-the-cuff
>>> conjecture, until I've tried it. The Hartley transform works entirely
>>> with real numbers, avoiding the need for symmetrisation, which nearly
>>> doubles the size of the processed vector. It avoids the use of (o.) to
>>> ensure complex datatype doesn't creep in. Also multiplying a complex
>>> vector (by a floating scalar) takes twice as long as multiplying a
>>> same-length real vector ...at least it does in my rough timer tests.
>>>
>>> J wiki says the addon "math/fftw" works only for Windows, so I assumed
>>> FFT was not available in J for me as a Mac user. I am wrong. I see it
>>> uses a dylib (libfftw3.3.dylib) when loaded on my iMac, and it appears
>>> to work. (I have added a note to this effect to:
>>> http://www.jsoftware.com/jwiki/Addons/math/fftw .)
>>>
>>> I'm glad I've discovered that. I last played with FFT way back in 1974
>>> and was impressed then with what it could do.
>>>
>>> So my earlier thoughts about adapting Cooley-Tukey to the Hartley
>>> transform may be wasted effort since FFTW may outperform it anyway.
>>> Even with symmetrization.
>>>
>>> However, provided it runs in minutes, not hours or days, performance
>>> is low priority for me on this task. And I rather like the fact that
>>> ABC computes C first, because then I have the option of telling it the
>>> "true" C if I know it.
>>>
>>> Ian
>>>
>>> On Tue, Jun 26, 2012 at 8:13 AM, Bo Jacoby <bojac...@yahoo.dk> wrote:
>>>> Hi Ian.
>>>>
>>>>
>>>> I'm glad that your test was successful, albeit slow. The important
>>>> optimization is to replace my slow fourier transform by a fast fourier
>>>> transform. Your example has 100 items in the time series, and the real
>>>> transform RT asks the fourier transform FT to perform 98 vector
>>>> multiplications by a 198*198 complex matrix, (98*198*198)=3841992, which
>>>> is embarassingly inefficient. The FFT will speed it up by a factor
>>>> ((%2&^.)198)=26.
>>>>
>>>>
>>>> Another optimization is to incorporate the computation of A and B into the
>>>> computation of C, rather than starting from scratch after computing C.
>>>>
>>>> The choice of index origin for C is a matter of taste.
>>>>
>>>>
>>>> Why would a Hartley transform be faster than the Fast Fourier Transform?
>>>>
>>>>
>>>> It remains to estimate the inaccuracies of A and B. It is the
>>>> root-mean-square value of the noise vector X-A+B*C<#X. Note that the noise
>>>> has only (#X)-2 degrees of freedom, because A and B have been estimated,
>>>> so the root-mean-square is
>>>>
>>>>
>>>> RMS=:(+/%#-2:)&.:*:
>>>>
>>>> - Bo
>>>>
>>>>
>>>>>________________________________
>>>>> Fra: Ian Clark <earthspo...@gmail.com>
>>>>>Til: Programming forum <programming@jsoftware.com>
>>>>>Sendt: 2:41 tirsdag den 26. juni 2012
>>>>>Emne: Re: [Jprogramming] Kalman filter in J?
>>>>>
>>>>>...I should add that my CC returns 1+(the value of your 'C'). This is
>>>>>for compatibility with how I generate the step fn, using (the
>>>>>constant) C.
>>>>>
>>>>>On Tue, Jun 26, 2012 at 1:37 AM, Ian Clark <earthspo...@gmail.com> wrote:
>>>>>> @Bo -Here is your solution running in my test rig...
>>>>>>
>>>>>> A= 100 bias constant (popn mean of X before the step)
>>>>>> B= 20 scaling factor for height of step
>>>>>> C= 90 epoch of actual occurrence of the Heaviside step
>>>>>> D= 10 scaling factor for Gaussian noise component (sigma)
>>>>>> N= 100 number of epochs in time series X
>>>>>> step detected at: (wait...)
>>>>>> 90
>>>>>> ABC X
>>>>>> 99.8246 20.0353 90
>>>>>>
>>>>>> ABC takes approx 10 s to run on my iMac.
>>>>>> CC (--your 'C') takes approx 5 s.
>>>>>>
>>>>>> I'm actually quite impressed with how accurately it estimates A, B, C,
>>>>>> compared with other methods (which don't do A or B). I'm writing up my
>>>>>> experiments and will post a wiki link.
>>>>>>
>>>>>> Whilst I like your symmetrisation idea, might instead using the
>>>>>> discrete Hartley Transform (which works entirely in the real domain)
>>>>>> speed things up?
>>>>>>
>>>>>> Ian
>>>>>>
>>>>>> On Mon, Jun 25, 2012 at 12:00 PM, Bo Jacoby <bojac...@yahoo.dk> wrote:
>>>>>>> Hi Ian. Here is your solution.
>>>>>>>
>>>>>>> ABC=: 3 : 0
>>>>>>> c=.C y
>>>>>>> y=.(0,>:c){RT"1&(,H&#)y
>>>>>>> y=.y,(%/1{"1 y)*1{y
>>>>>>> y=.y,(0{y)-2{y
>>>>>>> y=.y,(#|:y){.{.{:y
>>>>>>> y=.4 2{{:"1 RT"1 y
>>>>>>> y,c
>>>>>>> )
>>>>>>> ABC 101 103 100 210 230 200
>>>>>>> 101.15 115.101 2
>>>>>>>
>>>>>>>
>>>>>>> - Bo
>>>>>>>
>>>>>>>
>>>>>>>
>>>>>>>
>>>>>>>>________________________________
>>>>>>>> Fra: Ian Clark <earthspo...@gmail.com>
>>>>>>>>Til: Programming forum <programming@jsoftware.com>
>>>>>>>>Sendt: 6:46 mandag den 25. juni 2012
>>>>>>>>Emne: Re: [Jprogramming] Kalman filter in J?
>>>>>>>>
>>>>>>>>Interesting approach, Bo.
>>>>>>>>
>>>>>>>>Turned my attention to CUSUM (an adaptive filter) which is rather easy
>>>>>>>>to compute and gives good results, but needs parameters setting "by
>>>>>>>>eye". I've yet to try a Bayesian approach, which I suspect will be the
>>>>>>>>most sensitive of all.
>>>>>>>>
>>>>>>>>But I did play with your FT, using *:@(10&o.) to plot a power
>>>>>>>>spectrum. I'm using rather noisier data than your example: my step is
>>>>>>>>only roughly (sigma) high. Nevertheless with the onset of the step I
>>>>>>>>can clearly see a high-frequency spike appear at the far end of the
>>>>>>>>transformed time-series, due to the transformed Heaviside fn. Same
>>>>>>>>problem as with the CUSUM statistic however: designing a detector
>>>>>>>>which can be adjusted for false negatives and positives, and doesn't
>>>>>>>>take too many subsequent samples to detect the step.
>>>>>>>>
>>>>>>>>Will try out your C function in my context, and try to tweak it. Your
>>>>>>>>solution detects when the step happens, solving 1 in my stated
>>>>>>>>objectives: 1-4. I fear however that I am more interested in 2-4.
>>>>>>>>
>>>>>>>>
>>>>>>>>
>>>>>>>>On Sun, Jun 24, 2012 at 6:06 PM, Bo Jacoby <bojac...@yahoo.dk> wrote:
>>>>>>>>> Hi Ian
>>>>>>>>> The Fourier Transform FT transforms a complex n-vector X into a
>>>>>>>>> complex n-vector Y.If X is real, then Y is symmetric. And if X is
>>>>>>>>> symmetric then Y is real. So I think it is a good idea to symmetrize
>>>>>>>>> X before taking the FT. (This makes a slow program even slower.
>>>>>>>>> Optimize later!)
>>>>>>>>>
>>>>>>>>>
>>>>>>>>> NB.sm=: symmetrize
>>>>>>>>> sm=:[:,}:"1&(,:|.)
>>>>>>>>> sm i.6 NB. see what I mean
>>>>>>>>>
>>>>>>>>> 0 1 2 3 4 5 4 3 2 1
>>>>>>>>> NB.RT=: Real Transform. (9 o. removes every trace of complexity)
>>>>>>>>> RT=:9 o.#{.FT&sm
>>>>>>>>> NB.H=:Heaviside steps
>>>>>>>>> H=:i.&<:</i.
>>>>>>>>>
>>>>>>>>> NB.C=:Index of the last item before the change
>>>>>>>>>
>>>>>>>>> C=:[:(i.<./)[:+/"1[:*:[:}.[:(-"1{.)[:}."1[:(%{."1)[:}."1[:RT"1],[:H#
>>>>>>>>> NB.test
>>>>>>>>> C 101 204 202 200 203
>>>>>>>>> 0
>>>>>>>>> C 101 104 202 200 203
>>>>>>>>> 1
>>>>>>>>> C 101 104 102 200 203
>>>>>>>>> 2
>>>>>>>>> C 101 104 102 100 203
>>>>>>>>> 3
>>>>>>>>>
>>>>>>>>> This is not a Kálmán filter, but it solves the problem.
>>>>>>>>> - Bo
>>>>>>>>>
>>>>>>>>>
>>>>>>>>>
>>>>>>>>>>________________________________
>>>>>>>>>> Fra: Ian Clark <earthspo...@gmail.com>
>>>>>>>>>>Til: Programming forum <programming@jsoftware.com>
>>>>>>>>>>Sendt: 3:20 fredag den 22. juni 2012
>>>>>>>>>>Emne: Re: [Jprogramming] Kalman filter in J?
>>>>>>>>>>
>>>>>>>>>>Thanks, Bo. I'll look at it when I get back, probably over the
>>>>>>>>>>weekend.
>>>>>>>>>>
>>>>>>>>>>Ian
>>>>>>>>>>
>>>>>>>>>>On Fri, Jun 22, 2012 at 12:44 AM, Bo Jacoby <bojac...@yahoo.dk> wrote:
>>>>>>>>>>> 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
>>>>>>>>>>>>>>
>>>>>>>>>>>>>>
>>>>>>>>>>>>>>
>>>>>>>>>>>>> ----------------------------------------------------------------------
>>>>>>>>>>>>> For information about J forums see
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>>>>>>>>>>>>
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>>>>>>>>>>>>
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>>>>>>>>>>
>>>>>>>>>>
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