What is the Theory Behind JMA
?
PART 1 : PRICE GAPS
Smoothing time series data, such as daily
stock prices, in order to remove unwanted noise will inevitably produce a graph
(indicator) that moves slower than the original time series. This
"slowness" will cause the plot to lag somewhat behind the original
series. For example, a 31 day simple moving average will lag the price time
series by 15 days.
Lag is very undesirable because a trading
system using that information will have its trading delayed. Late trades can
many times be worse than no trades at all, as you might buy or sell on the
wrong side of the market's cycle. Consequently, many attempts were made to
minimize lag, each with their own failings.
Conquering lag while making no simplifying
assumptions (e.g., that data consists of superimposed cycles, daily price
changes having a Gaussian distribution, all prices are equally important, etc.)
is not a trivial task. In the end, JMA had to based on the same technology the
military uses to track moving objects in the air using nothing more than their
noisy radar. JMA sees the price time series as a noisy image of a moving target
(the underlying smooth price) and tries to estimate the location of the real
target (smooth price). The proprietary mathematics is modified to take into
consideration the special properties of a financial time series.
The result is a silky smooth curve that makes
no assumptions about the data having any cyclic components whatsoever.
Consequently JMA can turn "on a dime" if the market (moving target)
decides to turn direction or gap up/down by any amount. No price gap is too
large.
PART 2 : EVERYTHING ELSE
After several years of research, we Jurik
Research determined that the perfect noise reduction filter for financial data
has the following requirements:
 Minimum lag between signal and price,
otherwise trade triggers come late.
 Minimum overshoot, otherwise signal produces
false price levels.
 Minimum undershoot, otherwise time is lost
waiting for convergence after price gaps.
 Maximum smoothness, except at the moment
when price gaps to a new level.
When measured up to these four requirements,
all popular filters (except JMA) perform poorly. Here is a summary of the more
popular filters....
 Weighted Moving Average  not responsive to
gaps
 Exponential Moving Average  excessive
undershoot; noisy
 Adaptive Moving Averages  (not ours)
typically based on oversimplified assumptions about market activity; easily
fooled
 Regression Line  not responsive to gaps;
excessive overshoot
 FFT filters  easily distorted by
nonGaussian noise in data; window is typically too small to accurately
determine true cycles.
 FIR filters  has lag known as "group
delay". No way around it unless you want to cut some corners. See
"BandPass" filters.
 BandPass filters  no lag only at center
of frequency band; tends to oscillate and overshoot actual prices.
 Maximum Entropy filters  easily distorted
by nonGaussian noise in data; window is typically too small to accurately
determine true cycles.
 Polynomial Filters  not responsive to
gaps; excessive overshoot
In contrast, JMA integrates information
theory and adaptive nonlinear filtering in a unique way. By combining an
assessment of the information content in a time series with the power of
adaptive nonlinear transformation, the result pushes the theoretical
"envelope" on financial time series filtering almost as far as it can
go. Any more and we'd be up against Heisenburg's Uncertainty Principle
(something no one has overcome, or ever will).
As far as we know, JMA is the best. We invite
anyone to show us otherwise.
For more comparative analysis of the failings
of popular filters, download our report "The Evolution of Moving
Averages" from our Special Reports
department.
See our comparison against other
popular filters.

Why does JMA have a PHASE
parameter ?
There are two ways to decrease
noise in a time series using JMA. Increasing the LENGTH parameter will make JMA
move slower and thereby reduce noise at the expense of added lag.
Alternatively, you can change the amount of
"inertia" contained within JMA. Inertia is like physical mass, the
more you have, the more difficult it is to turn direction. So a filter with
lots of inertia will require more time to reverse direction and thereby reduce
noise at the expense of overshooting during reversals in the time
series.
All strong noise filters have lag and
overshoot, and JMA is no exception. However, the JMA's adjustable parameters
PHASE and LENGTH offer you a way to select the optimal tradeoff between lag and
overshoot. This gives you the opportunity to finetune various technical
indicators.
For example, the chart (at
right) shows a fast JMA line crossing over a slower JMA line. To make the fast
JMA line turn "on a dime" whenever the market reverses, it was set to
have no inertia. In contrast, the slow JMA was set to have large inertia,
thereby slowing down its ability to turn during market reversals. This
arrangement causes the faster line to cross over the slower line as quickly as
possible, thereby producing low lag crossover signals. Clearly, user control of
a filter's inertia offers considerable power over filters lacking this
capability.



Does JMA forecast a
timeseries ?
It does not forecast into the
future. JMA reduces noise pretty much the same way as an exponential moving
average, but many times better.

Will prior JMA values,
already plotted, change as new data arrives ?
No. For any point on a JMA plot,
only historical and current data is used in the formula. Consequently, as new
price data arrives on later time slots, those values of JMA already plotted are
not affected and NEVER change.
Also consider the case when the most recent
bar on a chart is updated in real time as each new tick arrives. Since the
closing price of the most recent bar is likely to change, JMA is automatically
reevaluated to reflect the new closing price. However, historical values of
JMA (on all prior bars) remain unaffected and do not change.
One can create impressive looking indicators
on historical data when it analyzes both past and future values surrounding
each data point being processed. However, any formula that needs to see future
values in a time series cannot be applied in real world trading. This is
because when calculating today's value of an indicator, future values don't
exist. All Jurik indicators use only current and previous timeseries data in
its calculations. This allows all Jurik indicators to work in all real time
conditions.


Can I improve other
indicators using JMA?
Yes. We typically replace most
moving average calculations in classical technical indicators with JMA. This
produces smoother and more timely results. For example, by simply inserting JMA
into the standard DMI technical indicator, we produced the DMX indicator, which
comes free with your order of JMA.


Does JMA have any special
guarantee?
If you show us a nonproprietary
algorithm for a moving average that, when coded to run in either TradeStation,
Matlab or Excel VBA, it performs "better" than our moving average in
short, medium and long time frames of a random walk, we'll refund your
purchased user license for JMA.
What we mean by "better" is that it
must be, on average, smoother with no greater average lag than ours, no greater
average overshoot and no greater average undershoot than ours. What we mean by
"short, medium and long time frames" is that the comparisons must
include three separate JMA lengths: 7 (short), 35 (medium), 175 (long). What we
mean by a random walk is a time series produced by a cumulative sum of 5000
zeromean, Cauchy distributed random numbers.
This limited guarantee is good for only the
first month of your having purchased a user license for JMA from us or one of
our worldwide distributors.

How does JMA compare to other
filters ?
The Kalman filter is similar to
JMA in that both are powerful algorithms used for estimating the behavior of a
noisy dynamical system when all you have to work with is noisy data
measurements. The Kalman filter creates smooth forecasts of the time series,
and this method is not entirely appropriate for financial time series as the
markets are prone to produce violent gyrations and price gaps, behaviors not
typical of smoothly operating dynamical systems. Consequently, Kalman filter
smoothing frequently lags behind or overshoots market price time series. In
contrast, JMA tracks market prices closely and smoothly, adapting to gaps while
avoiding unwanted overshoots. See chart below for an example.
A filter described in popular
magazines is the Kaufmann moving average. It is an exponential moving average
whose speed varies according to price action efficiency. In other words, when
price action is in a clear trend with little retracement, the Kaufmann filter
speeds up and when the action is congesting, the filter slows down. (See chart
above) Although its adaptive nature helps it overcome some of the lag typical
of exponential moving averages, it still lags significantly behind JMA. Lag is
a fundamental issue to all traders. Remember, every bar of lag may delay your
trades and deny you profit.
Another moving
average described in popular magazines is Chande's VIDYA (Variable Index
Dynamic Average). The index used most often inside VIDYA to govern its speed is
price volatility. As shortterm volatility increases, VIDYA's exponential
moving average is designed to move faster, and as volatility decreases, VIDYA
slows down.
On the surface this makes sense.
Unfortunately, this design has an obvious flaw. Although sideways congestion
should be thoroughly smoothed out regardless of its volatility, a highly
volatile period of congestion would be closely tracked (not smoothed) by VIDYA.
Consequently, VIDYA may fail to remove unwanted noise.
For example, the chart compares
JMA with VIDYA, both set to track a downward trend equally well. However,
during the ensuing congestion, VIDYA fails to smooth out the price spikes while
JMA successfully glides through the chatter.
In another comparison where both VIDYA and
Jurik's JMA were set to have the same smoothness, we see in the chart that
VIDYA lags behind. As mentioned earlier, late timing can easily steal away your
profits in any trade.


Two other popular indicators are
T3 and TEMA. They are smooth and have little lag. T3 is the better of the two.
Nonetheless, T3 can exhibit a serious overshoot problem, as seen in the chart
below. Depending on your application, you may not want an indicator showing a
price level the real market never attained, as this may inadvertently initiate
unwanted trades.
Here are two comments found
posted on relevant Internet forums:
"The T3 indicator is very good (and I've
sung its praises before, on this list). However, I've had the opportunity to
derive some alternate market measurements and I smooth them. They're pretty
badly behaved at times. When smoothing them, T3 becomes unstable and overshoots
badly, whereas JMA sails right through them."  Allan Kaminsky [allank @
xmission .com]
"My own view of JMA is consistent with what
other people have written (I've spent a good deal of time visually comparing
JMA to TEMA; I wouldn't think now of using TEMA instead of JMA)." Steven
Buss [sbuss @ pacbell .net]

An article in the Jan. 2000 issue
of TASC describes a moving average designed in the 1950's to have low lag. Its
inventor, Robert Brown, designed the "Modified Moving Average" (MMA)
to reduce lag in estimating inventories. In his formula, linear regression
estimated the curve's current momentum, which in turn is used to estimate
vertical lag. The formula then subtracts estimated lag from the moving average
to get low lag results. This technique works OK on well behaved (smoothly
transitioning) price charts, but then again, so do most other advanced filters.
The problem is that the real market is anything but well behaved.
A true measure of fitness is how well any
filter works on realworld financial data, a property that can be measured with
our well established battery of benchmark
tests. These tests reveal that MMA overshoots price charts, as illustrated
below. In comparison, the user can set a parameter in JMA to adjust the amount
of overshoot, even completely eliminating it. The choice is yours. Remember,
the last thing you want is an indicator showing a price level the real market
never attained, as this may inadvertently initiate unwanted trades. With MMA,
you have no choice and must put up with overshoot whether you like it or not.
(See chart below)
The July 2000 issue of TASC
contained an article by John Ehlers describing a "Modified Optimal
Elliptical Filter" (Abbreviated here as "MEF"). This is a superb
example of classical signal analysis. The chart below compares MEF to JMA whose
parameters (JMA length=7, phase=50) were set to make JMA be as similar to MEF
as possible.
The comparison reveals these
advantages when using JMA:
 JMA responds to extreme price swings more
quickly. Consequently, any threshold values used to trigger signals will be
executed sooner by JMA.
 JMA has almost no overshoot, permitting the
signal line to more accurately track price action right after large price
movement.
 JMA glides through small market movements.
This permits you to focus on real price action and not small market activity
that has no real consequence.
A favorite method among
engineers for smoothing time series data is to fit the data points with a
polynomial (eq, a parabolic or cubic spline). An efficient design of this type
is a class known as SavitzyGolay filters. The chart below compares JMA to a
cubicspline (3rd order) SavitzyGolay filter, whose parameter settings were
chosen top make it perform as close to JMA as possible. Note how smoothly JMA
glides through regions of trading congestion. In contrast, the SG filter is
quite jagged. Clearly JMA is, once again, the winner.


Another technique used to reduce lag in a
moving average filter is to add some momentum (slope) of the signal to the
filter. This reduces lag, but with two penalties: more noise and more overshoot
at price pivot points. To compensate for noise, one can employ a symmetrically
weighted FIR filter, which is smoother than a simple moving average, whose
weights might be: 1234321 and then adjust these weights to add some lag
reducing momentum.
The effectiveness of this approach is shown
in the figure below (red line). Although the FIR filter tracks price closely,
it still lags behind JMA as well as exhibit greater overshoot. In addition, the
FIR filter has fixed smoothness and needs to be redesigned for each different
desired smoothness. In comparison, the user only needs to change one
"smoothness" parameter of JMA to get any desired effect.
Not only does JMA produce better
price chart plots, but it can improve other classical indicators, as well. For
example, consider the classical MACD indicator, which is a comparison of two
moving averages. Their convergence (moving closer) and divergence (moving
apart) provide signals that a market trend is changing direction. It is
critical that you have as little delay as possible with these signals or your
trades will be late. In comparison, a MACD created with JMA has significantly
less lag than a MACD using exponential moving averages.
To illustrate this claim, the figure below is
a hypothetical price chart simplified to enhance the salient issues. We see
equalsized bars in a rising trend, interrupted by a sudden downward gap. The
two colored lines are exponential moving averages that make up a MACD. Note
that crossover occurs a long time after the gap, causing a trading strategy to
wait and trade late, if at all.
If you tried to speed up the timing of this
indicator by making the moving averages faster, the lines would become noisier
and more jagged. This tends to create false triggers and bad trades. On the
other hand, the chart below shows the blue JMA adjusting rapidly to the new
price level, permitting earlier crossovers and earlier designation of an
uptrend in progress. Now you can enter the market earlier and ride a larger
portion of the trend.
Unlike the exponential moving average, JMA
has an additional parameter (PHASE) that lets the user adjust the extent of
overshoot. In the chart above, the JMA yellow line was permitted to overshoot
more than the blue. This gives ideal crossovers.
One of the most difficult features to design
into a smoothing filter is an adaptive response to price gaps without
overshooting the new price level. This is especially true for filter designs
that employ the filter's own momentum as a way to reduce lag. The following
chart compares overshoot by JMA and the Hull moving average (HMA). The
parameter settings for the two filters were set so that their steady state
performance were almost identical.
Another design issue is whether or not the
filter can retain the same apparent smoothness during reversals as during
trends. The chart below shows how JMA retains near constant smoothness
throughout the entire cycle, whereas HMA oscillates at reversals. This would
pose problems for strategies that trigger trades based on whether the filter is
moving up or down.
Lastly, there is the case when price gaps up
and then retreats in a downward trend. This is especially difficult to track at
the moment of retreat. Fortunately, adaptive filters have a much easier time
indicating when a reversal occured than fixed filters, as shown in the chart
below.
Of course there are better filters than JMA,
mostly used by the military. But if you are in the business of tracking down
good trades and not enemy aircraft, JMA is the best **affordable** noise
reducing filter available for financial market data. We
guarantee it.

