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A standard CMA has lag 17 bars for n=35
Millard's Smoothed is an n-period MA smoothed by an n/2-period MA. So
the lag is (n-1)/2 + (n/2-1)/2
For n=35 this equals (35-1)/2 + (17-1)/2 = 17+8 = 25
In his book Millard calculates an 11-week average of a 21-week average.
See his Table 7.2
He also states further on that "a 15-week smoothed average would cause
the loss of 10 points at the end of the plot"
If n=15 then the lag is (15-1)/2 + (7-1)/2 = 10
A centred SMA of n=15 would have lag of just 7
I don't agree with what you say in 1). Millard uses this CMA stuff in a
build up to his Cycle Highlighter (CH) indicator in Chapter 9. This
indicator does not attempt to extract the "dominant cycle" per se. When
I hear that phrase it reminds me of Ehler's language in "Cybernetic..."
but I won't divert onto that here.
There are lots of cycles present in most price series, once you allow
for the noise and for long-term fundamental-driven trends and they can
all be "dominant" depending on what time-frame you are looking at and
what you are trading on. For example, I might be trading a 21-day cycle
and you might be more interested in a 52-WEEK cycle, depending on our
trading styles. So, *IF those cycles are present to trade on AND are
strong enough not to get lost in noise* then the CH indicator should be
able to pick EITHER of them out, depending on how you set the parameters.
None of this is "touchy feely". *USED CORRECTLY* it works. I've traded
with it and I've had repeated success doing so. (Please, no calls for
trading records!) The success I have had though is through incorporating
it into my overall strategy. I do not rely on one indicator and I most
definitely do not automate. I agree that there needs to be some element
of automation in there, if for no other reason than for
scanning/exploring for suitable issues that show good cyclic behaviour.
To that end I have tried to automate the CH indicator as I discussed in
previous posts. So I think I answered your point 2) already. Let me know
if you need further clarification.
However, I don't subscribe to the "Mechanical Or Nothing" school of
thought. Yes, the CH indicator works in a "general way"...and that is
good enough for me. It is not "very" general though. One of my other
analyses is based on divergence. Divergence works very well indeed, when
it works at all. And therein lies a problem. My own divergence indicator
probably has a 50-60% hit rate. I could make this work on its own with
decent money management rules, but when I combine it with an
*appreciation* of the cycles that hit rate number goes up quite
significantly. So you can see that I am using mechanical signals from
one method and then applying a discretionary filter based on my
appreciation of the cycle I wish to trade on. I do not need to be "no
hands"...I like my hands!
A while back you sent a chart.png of the work you had been doing to
extract all the cycles from a waveform using Cleeton's methods.
Conceptually, there's no real difference between what you are trying to
do there and what the CH indicator does. The differences are that (a)
the CH indicator extracts cycles one-at-a-time whereas your tries to do
them all at once, and (b) there are no fancy mathematics (YET!) for
extrapolating to the right-hand edge. The extrapolation method I have is
quite crude...but please remember that cycles are quite crude too. The
amplitude and wavelengths are *never* constant. Over-engineering
something can sometimes be as dangerous as under-engineering.
Regards,
Andy
Fred wrote:
>
> Andy,
>
> In looking at your spreadsheet I understand what's there except
> for ...
>
> I'm not sure why a Millards smoothed should have more lag then a
> standard CMA ...
>
> Although I thought an even number would be required i.e. 34 or 38
> instead of 35 ...
>
> Millard would have calc'ed a 23 bar CMA and then a 11 bar of
> that ... or 25 and 13 if you prefer ... the lag would have been 11 + 5
> = 16, or 12 + 6 = 18 respectively ... where'd you get a lag of 25 for
> 35 bars ?
>
> In any case what all these seem ? to be missing imho is ...
>
> 1. I don't think the dominant cycle is enough to do the job in
> terms of extrapolation and/or prediction by itself except in a very
> general way ...
> 2. What at least semi automated ( no touchy feely allowed )
> method are you gonna use to determine the CMA length to be used and
> then
> 3. How to process that info ...
>
> The AFL I have for Trig Fit, the output of which I posted on AB takes
> care of all 3 of the above with no hands ... run time is of course a
> different issue
>
> --- In amibroker@xxxxxxxxxxxxxxx <mailto:amibroker%40yahoogroups.com>,
> Andy Davidson <AndyDavidson@xxx>
> wrote:
> >
> > Fred,
> >
> > Long post I'm afraid, but bear with me...
> >
> > Since our last conversation I've been doing some head-scratching on
> > which method of CMA is best for extracting cycles. After re-reading
> > Millard and then trying to theorise my way in ever-decreasing
> circles
> > about what *should* be the best way, I decided to try to experiment
> and
> > find out what works best *in practice*. I've attached the results
> in the
> > form of plots and a summary spreadsheet for your (or anyone else's)
> > interest. Here's the logic behind the method (AFL code posted below
> for
> > sake of completeness) :
> >
> > 1. I created two independant sine waves and a 'noise' component.
> These
> > individual components are plotted in the top pane.
> > 2. I then added these together to create a composite 'price-like'
> plot -
> > plotted in grey in the bottom pane.
> > 3. Four different kinds of Centred-MA (see below) were then plotted
> on
> > the bottom pane, with the composite as an input. The aim was to
> select a
> > periodicity for each CMA that would filter out the noise and the
> shorter
> > wavelength cycle (cycle 2), leaving the closest possible
> representation
> > of the longer cycle 1.
> > 4. The lag, wavelength and amplitude of this CMA plot were then
> > *measured* (i.e. they weren't deduced theoretically, but were
> actual
> > observed values).
> > 5. The values were compared on the spreadsheet.
> >
> > The four different CMAs were based on:
> > (a) Simple MA. The most basic centred SMA
> > (b) Millard's "Smoothed Average" from Chapter 7...i.e. an MA of
> > n-periods which has been smoothed again by an MA of n/2 periods.
> > (c) Triangular MA. This is an n/2 MA of an n/2 MA
> > (d) Custom MA. This is per your last email with the first MA being
> > n*0.75 and the second being half that.
> >
> > The Triangular MA has the same lag characteristics as a Simple MA.
> > However, in order to get the same *filtering* effect (i,e, to take
> out
> > cycle 2 completely) you have to near-enough double the periodity,
> which
> > then obviously takes the lag up. Experimenting seems to suggest
> that you
> > don't actually have to double it, which I guess is why I settled on
> a
> > multiplying factor of 1.5 for my Cycle Highlighter indicator. I
> think I
> > originally settled on 1.5 after mis-reading Millard's section on
> the
> > "Weighted MA" and have therefore been using something which was
> > nearly-correct but for the wrong reasons! Oh well, at least I have
> a
> > better idea now. However, the results of this seem to suggest that
> 1.75
> > would be a better number, so I've changed my indicator accordingly.
> I'm
> > sure there's good theory behind why this should be so, but I can't
> think
> > it through. Can you?
> >
> > So anyway, all that testing seems to show is that Millard's
> Smoothed
> > Average is the best for this purpose. My triangular MA seems to
> have
> > been suffering too much lag than necessary, for an output which
> also
> > suffers more damping. There seems to be nothing to choose between
> your
> > "Custom" CMA and the "Smoothed Average". This is obviously because
> they
> > are basically the same thing. Both are MAs smoothed by another MA
> half
> > the first's length. The fact that the "Smoothed Average" starts off
> with
> > and n-period MA and the "Custom" one starts with n-periods*0.75,
> just
> > means that the latter has to have "n" ramped up to provide the same
> > filtering/smoothing effect.
> >
> > OK, so far so good. I've decided to ditch the Tri-CMA in favour of
> the
> > "Smoothed CMA". But here's another question. Millard states
> ("Weighted
> > Average" section) that for those of us with computers(!!) it is
> > preferable to chose a centrally-weighted MA. Anyone know how to do
> that
> > without slowing things down even more? Is that the same as the
> geometric
> > mean?? My maths really is too rusty. The standard WMA function is
> no
> > good as it applies the maximum weighting to the *most recent* bar.
> We
> > would need, for example, in a 7-bar MA to have a weighting sequence
> of
> > 1-2-3-4-3-2-1
> >
> > That'll do for now. Tomorrow's job is to add a third, longer, cycle
> and
> > see how extracting the middle cycle goes.
> >
> > Cheers,
> > Andy
> >
> >
> > Fred wrote:
> > >
> > > You won't need the math texts to get though Hurst's course
> > > material ... What you will need is time and patience ...
> > >
> > > The 2 / 3 factor is in essence I thought what you were advocating
> > > i.e. the first cycle length being twice the second ...and the lag
> > > being the combo of 1 less then half of both ... Millard suggests
> such
> > > a methodology in chapter 7.
> > >
> > > The Hurst "Like" DE AFL I posted in the library was an interesting
> > > project ... It seems however that the points could be better
> picked
> > > then by using CMA's ... But that's another exercise ...
> > >
> > > --- In amibroker@xxxxxxxxxxxxxxx
> <mailto:amibroker%40yahoogroups.com> <mailto:amibroker%
> 40yahoogroups.com>,
> > > Andy Davidson <AndyDavidson@>
> > > wrote:
> > > >
> > > > Don't worry Fred, straight talk is good for us all :-)
> > > >
> > > > I'll think about that 2/3 factor tomorrow - it's late here and
> my
> > > brain
> > > > is aching.
> > > >
> > > > I ordered the Cleeton book a while back but it still hasn't
> > > arrived. I
> > > > think it'll make for a nice relaxing Xmas read! I've got the
> book
> > > by
> > > > Hurst (Profit Magic), but I froze when I got to Appendix 6 and
> so I
> > > > think I need Cleeton as you suggest! The Hurst course is on the
> > > list as
> > > > well, but first I think I'll have to get some old Maths texts
> out
> > > of the
> > > > attic and get the grey matter working again in that respect. My
> > > maths is
> > > > sadly lacking also and I feel it's really not adequate to take
> me
> > > any
> > > > further than I've got without some hard graft. Oh well, needs
> must
> > > I
> > > > suppose.
> > > >
> > > > As far as channels go, I had a look at your Hurst DE quickly
> today.
> > > I
> > > > played with Hurst-like channel trading myself a while back
> (when I
> > > was
> > > > still a naive Metastock user - yeah, I know, but it was OK for
> at
> > > least
> > > > that). I found that my skills were below that needed to tackle
> the
> > > > extrapolation problem and so it was simply a matter of using
> > > discretion
> > > > and 'eyeballing' a la Hurst.
> > > >
> > > > That was when I found Millard's book and latched on to his Cycle
> > > > Highlighter. To me it was (and still is) a simple and effective
> way
> > > of
> > > > determining the cycles if you have a bias towards discretionary
> > > trading
> > > > as I currently do. And by nature it is a normalised plot, so it
> > > seemed
> > > > logical to me to go about extrapolating on that plane before I
> > > tried to
> > > > tackle the price plot. However, I am now convinced (thanks in no
> > > small
> > > > part to yourself) that it is worth pursuing further with the
> > > ultimate
> > > > aim of automating the whole cycle-extraction process.
> > > >
> > > > So here's to the next step of the journey...hard graft and all.
> > > >
> > > >
> > > > Fred wrote:
> > > > >
> > > > > Thanks for the description ... It wasn't a sarcastic comment
> per
> > > > > se ... It is imho a benefit to be able to hear from authors of
> > > code
> > > > > what the process is that is going on as opposed to someone
> > > > > unfamiliar with the code having to dig it out ...
> > > > >
> > > > > I agree with your comments in 1 & 2 ... I had initially
> > > implemented
> > > > > Millard's CMA in the Hurst DE I posted in the library this
> way ...
> > > > >
> > > > > Lag = int(Period / 2);
> > > > > CMA = Ref(MA(MA(Data, Lag), Lag), Lag);
> > > > >
> > > > > It would seem though after reading Millard more carefully
> that a
> > > > > better implementation is something like
> > > > >
> > > > > CMAL1 = Int(Period * 2 / 3);
> > > > > if (CMAL1 < 5)
> > > > > CMAL1 = 5;
> > > > > If (CMAL1 % 2 == 0)
> > > > > CMAL1 = CMAL1 + 1;
> > > > > CMAL2 = Period - CMAL1;
> > > > > If (CMAL2 % 2 == 0)
> > > > > CMAL2 = CMAL2 + 1;
> > > > > Lag = (CMAL1 - 1) / 2 + (CMAL2 - 1) / 2;
> > > > >
> > > > > CMA = Ref(MA(MA(Data, CMAL1), CMAL2), Lag)
> > > > >
> > > > > The only potential problem I see with this approach is it
> makes
> > > the
> > > > > minimum overall CMA Length 8.
> > > > >
> > > > > For the current AFL I implemented a simple CMA ... no muss /
> > > > > fuss ... The reason is that the CMA would be sampled and
> > > potentially
> > > > > smoothed again ...
> > > > >
> > > > > I don't know whether or not you have Hurst's PM but he covers
> (
> > > very
> > > > > quickly ) the topic of pulling out the coeff's for multiple
> cycles
> > > > > simultaneously in what is to me any way some rather complex
> math
> > > in
> > > > > Appendix 6 ... But then I'm hardly a math Wiz ... If you are
> > > > > interested in this kind of thing I would strongly recommend
> > > > > Cleeton's book which while out of print is still readily
> available
> > > > > at Amazon and other places for a few bucks used. He discusses
> how
> > > > > to perform a similar operation for one cycle and for multiple
> > > cycles
> > > > > simultaneously with one of the early steps being sampling of
> the
> > > > > CMA ... He uses those points directly and as you can tell
> from my
> > > > > description I opted for this approach more or less as well
> which
> > > > > seems to produce some interesting results without requiring
> > > Gaussian
> > > > > Elimiation to solve multiple simultaneous equations.
> > > > >
> > > > > --- In amibroker@xxxxxxxxxxxxxxx
> <mailto:amibroker%40yahoogroups.com>
> > > <mailto:amibroker%40yahoogroups.com> <mailto:amibroker%
> > > 40yahoogroups.com>,
> > > > > Andy Davidson <AndyDavidson@>
> > > > > wrote:
> > > > > >
> > > > > > Hi Fred,
> > > > > >
> > > > > > It's good to be able to get back on this subject again,
> > > especially
> > > > > as it
> > > > > > looks like there's a few of us who are 'into' cycles.
> > > > > >
> > > > > > Your work-in progress looks very interesting I must say. I
> > > > > particularly
> > > > > > like the idea in step 5 to reduce the data before finding a
> > > > > > fit...brilliant in its simplicity. I also think your
> equation in
> > > > > step 6
> > > > > > will help me out...but without getting into that, here's the
> > > > > general
> > > > > > logic of my approach for comparison (and I take the
> sarcastic(?)
> > > > > comment
> > > > > > about explaining in English...I didn't do a good job of
> notating
> > > > > the
> > > > > > script properly!)
> > > > > >
> > > > > > 1. Calculate *two* CMAs using triangular-smoothed MAs. CMA1
> is
> > > n-
> > > > > periods
> > > > > > length and CMA2 is n/2-periods. Both periods are rounded up
> to
> > > the
> > > > > > nearest odd number.
> > > > > > 2. CMA1 allows wavelengths > n-periods to pass and filters
> out <
> > > > > > n-period waves. CMA2 allows through all cycle wavelengths >
> n/2-
> > > > > periods
> > > > > > and filters out those < n/2. Therefore, subtracting CMA2
> from
> > > CMA1
> > > > > will
> > > > > > give us the cycle (or combination of cycles if we're unlucky
> > > > > enough, or
> > > > > > have our value of n wrong) that lies between n/2 and n.
> > > > > >
> > > > > > Steps 1 and 2 are as per Millard's "Cycle Highlighter" (CH),
> > > > > except he
> > > > > > states that the best results are obtained with CMA1 being
> an SMA
> > > > > and
> > > > > > CMA2 being a Weighted MA. He also says CMA1 periods should
> be
> > > > > *equal* to
> > > > > > the wavelength to be isolated. This does work but, through
> > > > > > experimenting, I have found that Triangular-MAs are best for
> > > both
> > > > > as
> > > > > > they offer the superior smoothing-to-lag trade off.
> Furthermore,
> > > > > the
> > > > > > periodicity of CMA1 should be x1.5 the cycle you want
> (making
> > > CMA2
> > > > > > therefore x0.75). The logic still holds up and the results
> are
> > > > > better
> > > > > > IMO, with a more sine-like output.
> > > > > >
> > > > > > 3. Based on user-inputs (see below) I then generate an
> > > artificial
> > > > > sine
> > > > > > wave. This is *anchored to the CH at its most recent (i.e.
> > > > > confirmed)
> > > > > > peak or trough*.
> > > > > > 4. Correlation coefficients are calculated between (a) the
> sine
> > > > > wave and
> > > > > > the CH (or price - depending on user input) over
> the 'lookback'
> > > > > period
> > > > > > (see below) and (b) the sine wave and the price in the 'end
> > > zone'
> > > > > (i.e.
> > > > > > the no-data zone for the CH at the right-hand edge).
> > > > > >
> > > > > > Inputs:
> > > > > > "SINE WAVELENGTH" - this determines if the wavelength of the
> > > sine
> > > > > is (a)
> > > > > > "as per the base cycle (CH)" (i.e. there is no attempt
> to 'fit'
> > > > > the two
> > > > > > curves beyond the anchor point) or (b) a "best fit". In the
> > > second
> > > > > case,
> > > > > > the sine wavelength will depend on:
> > > > > > "BEST FIT # RECENT CYCLES" - this is the number of full,
> > > completed
> > > > > > cycles of the CH where the correlation is measured. The
> start
> > > > > point of
> > > > > > X-cycles back is shown by a blue and red tick on the
> indicator.
> > > If
> > > > > > option (b) is chosen above the average wavelength of the CH
> is
> > > > > measured
> > > > > > in the zone from the blue tick to the end of its plot. This
> > > value
> > > > > is
> > > > > > assigned to the sine plot. If option (a) above then we just
> get
> > > X-
> > > > > cycles
> > > > > > back of both plots at the same periodicity.
> > > > > >
> > > > > > All the above is as per the first indicator I posted. The
> > > > > following
> > > > > > loops are done in the auto-fit version:
> > > > > >
> > > > > > 5. A loop from "Wavelength Min" to "Wavelength Max" is
> performed
> > > > > to find
> > > > > > the highest total correlation coefficient (a weighted
> average of
> > > > > the
> > > > > > 'CH/sine' and the 'sine/end-zone price' values).
> > > > > > 6. The series of loops is repeated for "#Cycles Min"
> lookback up
> > > > > to 5
> > > > > > cycles lookback. I chose 5 as an arbitrary number...it's
> slow
> > > > > enough as
> > > > > > is and very rarely do you get a decent correlation going
> that
> > > far
> > > > > back.
> > > > > > Obviously though when you do, you take notice.
> > > > > >
> > > > > > That's as much as I can tell you right now about the logic.
> Does
> > > > > it
> > > > > > work? Well, with the usual caveats blah-blah-blah, I would
> say
> > > > > that it
> > > > > > has been a very useful tool for me for a while now *in
> > > conjunction
> > > > > with
> > > > > > other confirming and entry methods*
> > > > > >
> > > > > > Bear in mind that the purpose of the indicator is to find
> the
> > > > > *clearest*
> > > > > > cycle amongst those present, i.e. the one that conforms most
> > > > > closely to
> > > > > > a sine wave, and is therefore tradeable *on that time
> frame*. I
> > > > > will
> > > > > > manually switch between time-frames to get the various major
> > > > > cycles
> > > > > > (e.g. 1-hour, 4-hour, daily and weekly charts). Work
> on 'auto-
> > > ing'
> > > > > all
> > > > > > that would be very processor intensive and requires further
> > > > > thinking.
> > > > > >
> > > > > > The plot you sent seems to bear out a further truth about
> > > trading
> > > > > with
> > > > > > cycles, one that I've experienced with this indicator more
> than
> > > > > once:
> > > > > > i.e. short-term cycles (measured in hours and a few days)
> are
> > > less
> > > > > > tradeable than longer-term ones (measured in a few days
> upwards
> > > to
> > > > > weeks
> > > > > > & months). Certainly, in the plot you sent, most of the
> smoothed
> > > > > price
> > > > > > behaviour can be explained by the interaction of the two
> longest
> > > > > > measured cycles (dark blue and cyan).
> > > > > >
> > > > > > Anyway, I look forward to ploughing through all the good
> stuff
> > > > > you've
> > > > > > already posted and hope you can help keep this thread going.
> > > > > There's
> > > > > > lots of really cool stuff going on here.
> > > > > >
> > > > > > Cheers for now,
> > > > > > Andy
> > > > > >
> > > > > >
> > > > > > Fred Tonetti wrote:
> > > > > > >
> > > > > > > Andy,
> > > > > > >
> > > > > > >
> > > > > > >
> > > > > > > Can you describe in English what your AFL does ? ...
> > > > > > >
> > > > > > >
> > > > > > >
> > > > > > > I've been playing with a Trig Fit a la Claud Cleeton the
> steps
> > > > > for
> > > > > > > which I would describe as follows ...
> > > > > > >
> > > > > > >
> > > > > > >
> > > > > > > 1. Optional - Normalize the input i.e. Data = log10((H +
> L) /
> > > 2)
> > > > > > >
> > > > > > > 2. Calc an arbitrary length ( Parameterized but 11 at the
> > > > > moment )
> > > > > > > centered moving average ( CMA ) of the data
> > > > > > >
> > > > > > > 3. Calc a 1st order least squares fit ( LSF ) of the CMA
> over
> > > > > the
> > > > > > > period desired ( from / to range marker )
> > > > > > >
> > > > > > > 4. Subtract the LSF points from the data points resulting
> in
> > > > > detrended
> > > > > > > data.
> > > > > > >
> > > > > > > 5. Take an n-bar sampling of the detrended data. This
> array
> > > > > with
> > > > > > > "holes" or "gaps" in it needs either to be compressed or
> have
> > > > > the
> > > > > > > "gaps" filled ... I elected ( for the moment ) to calc a
> cubic
> > > > > spline
> > > > > > > to fill the gaps ( interpolation ) ...
> > > > > > >
> > > > > > > 6. Calc a LSF of the detrended data resulting in the
> coeffs
> > > for
> > > > > the
> > > > > > > Trig equation Y = A Cos wX + B * Sin wX
> > > > > > >
> > > > > > > 7. Calc the correlation of the resulting sin wave to the
> > > > > original
> > > > > > > detrended data.
> > > > > > >
> > > > > > >
> > > > > > >
> > > > > > > Repeat steps 5 & 6 varying n from 1 to ? looking for n
> where
> > > the
> > > > > > > correlation is the highest. This should yield the
> equation or
> > > > > data
> > > > > > > points that most closely correlate to the detrended data.
> > > > > > >
> > > > > > >
> > > > > > >
> > > > > > > 8. Subtract the points in the sin wave from the detrended
> data
> > > > > > > resulting in a modified detrended data.
> > > > > > >
> > > > > > >
> > > > > > >
> > > > > > > Repeat steps 5 - 8 looking for the next most significant
> > > cycle.
> > > > > This
> > > > > > > can be done repeatedly until overall correlation stops
> getting
> > > > > better
> > > > > > > and usually results in 2 - 6 cycles ...
> > > > > > >
> > > > > > >
> > > > > > >
> > > > > > > See attached ...
> > > > > > >
> > > > > > >
> > > > > > >
> > > > > > > The white line in the upper graph is detrended price ...
> > > > > > >
> > > > > > > The alternating green / red line is the trig fit, in
> sample up
> > > > > to the
> > > > > > > vertical line and out of sample projection afterwards ...
> > > > > > >
> > > > > > > The lines in the bottom section are the individual cycles
> > > found
> > > > > in the
> > > > > > > data.
> > > > > > >
> > > > > > >
> > > > > > >
> > > > > > > Sometimes the projections are almost clairvoyant ... run
> time
> > > > > however
> > > > > > > is anything but quick ...
> > > > > > >
> > > > > > >
> > > > > > >
> > > > > > >
> > > > > > >
> > > > > > >
> > > > > > >
> > > > > > >
> > > > > > >
> > > > > > >
> > > > > > >
> > > > > > >
> > > > > > > ----------------------------------------------------------
> > > > > -------
> > > > > > > I am using the free version of SPAMfighter for private
> users.
> > > > > > > It has removed 8649 spam emails to date.
> > > > > > > Paying users do not have this message in their emails.
> > > > > > > Try SPAMfighter <http://www.spamfighter.com/go.asp?t=249
> <http://www.spamfighter.com/go.asp?t=249>
> > > <http://www.spamfighter.com/go.asp?t=249
> <http://www.spamfighter.com/go.asp?t=249>>
> > > > > <http://www.spamfighter.com/go.asp?t=249
> <http://www.spamfighter.com/go.asp?t=249>
> > > <http://www.spamfighter.com/go.asp?t=249
> <http://www.spamfighter.com/go.asp?t=249>>>> for
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> > > > > > >
> > > > > >
> > > > >
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> > > >
> > >
> > >
> >
>
>
Content-Description: "AVG certification"
No virus found in this incoming message.
Checked by AVG Free Edition.
Version: 7.1.409 / Virus Database: 268.13.27/517 - Release Date: 11/3/2006
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