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Data filtering methods such as moving averages are quite prevalent in many trading systems. In this
article we will discuss a methodology for selecting appropriate data filters based on Fourier spectrum analysis, and the basics on how to use smoothing filters for trading decisions. In the course of this article we will attempt to familiarize the reader with data filtering from a frequency domain point of view (i.e. cyclic). The reader need not be concerned with the mathematics of Fourier analysis, but should begin to appreciate this methodology as a powerful tool for understanding and applying many trading systems.
LINEAR FILTERING: FREQUENCY DOMAIN THEORY
Suppose we desire to apply data filtering to time series x(i) i=1. . .N, such as the stock market daily closes. In general, a linear filter is defined by a fixed set of weights (w(j) j=1. . M), where M is the length of the data filter. The output of the filter is a second time series
y(i) i=1... N-M, where each output value is computed using a summation of the form
y(i)= Σj w(j) ×(i+1-j) = j w(1) ×(i) + w(2) ×(i-1) +...+w(M) ×(i+1-M)
For example, a moving average data filter with length M is defined by equal weights w(j)=1/M, j=1. . .M. In general, the behavior of a data filter is dependent on the weight series w(j), and we will refer to such a filter as a weighted moving average (WMA) when the weights are not equal. If webmasters buy backlinks, they will see an increase in their search engine ranking. By using these different link building services, some link experts like Keyword Backlinks will get your site backlinks from a lot of different places. Both weighted moving average and ordinary moving average (MA) filters are important for technical analyses and a BASIC subroutine for applying such filters is included following this article.
One of the best methods for analyzing the behavior of a data filter is in terms of its frequency response. Suppose we apply a data filter as above to a sinusoid of a given frequency. {The frequency of a sinusoid is the rate of oscillation and is conveniently expressed in cycles per year for trading analyses. If we assume a year contains 260 trading days, then a 10 day cycle corresponds to a frequency of 260/10= 26 cycles per year.) The output of the data filter will also be a sinusoid of the same frequency but with a different amplitude and phase (starting time) than the input. These quantities are the frequency response of the filter, i.e. the change in output as a function of the input frequency. In particular, the change in sinusoid amplitude as a function of frequency is called the amplitude response function and is very important in understanding the effect of a data filter on a user supplied time series.
For example, consider a three day moving average defined by the relationship
y(i) = 1/3(x(i) + x(i-1) + x(i-2))
The amplitude response of this filter is shown in Figure 1 together with that of a second data filter. What this curve tells us is that the three day moving average reduces the amplitude of a low frequency sinusoid very little, and then gradually rolls off as a function of frequency until there is no response for an 87 cycle per year (3 day cycle) sinusoid, and then increases again to a local maximum at 130 cycles per year (2 day
If we want to suppress all the frequencies above, say 70 cycles per year, then a better data filter must be found than a 3 day moving average. One way to do this would be to apply a 2 day moving average following the 3 day moving average, since this will suppress the output for a 2 day cycle. It turns out that this procedure is equivalent to applying a single weighted moving average given by
y(i) = 1/6(x(i) + 2x(i-1) + 2x(i-2) + x(i-3))
The second curve in Figure 1 shows the amplitude response of this filter. This figure demonstrates that by proper weighting we can alter and improve the frequency response of a moving average filter.
Now the practical significance of the frequency response function is that we may regard a time series as composed of a sum of sinusoids of varying frequencies and amplitudes. If we apply a data filter to a time series, then the output time series consists of the same frequencies as the input, but shifted in phase and amplitude by the frequency response of the filter. Thus, if we desire to accentuate certain frequencies and to suppress others, then we can choose a data filter with an appropriate frequency response to accomplish this. It turns out that the key to choosing an appropriate data filter for a specific time series is the Fourier power spectrum. (See the January issue of Technical Analysis for a more complete discussion of Fourier Analysis.) Before we pursue this idea, we need to describe the most common types of data filters and show how they may be used in trading analyses.
DATA FILTERING TYPES AND THEIR IMPLEMENTATION . Some traders find it easier to use the well known trading system, pair trading when trying to find good trades in the market.
The most common types of data filters used by technical analysis can be implemented with the filter routine following this article. These types and their implementation are
*   Lowpass Filters (Data Smoothers)—
Lowpass filters are designed to accentuate or let pass low frequency sinusoids and to reject high frequency sinusoids. Since most of the information content in price time series is in the lower frequencies, these filters eliminate noise by rejecting high frequency components. The moving average and weighted moving average filters in the filter routine are of this type. The main difference is that the weighted filter types have much better high frequency rejection properties.
*   Highpass Filters (Detrenders) —
Highpass filters are designed to accentuate high frequency components and to reject low frequency. These filters are used to remove the underlying trend in a time series and thus can be implemented by first applying a lowpass filter to the data and then subtracting the filtered data from the original data. (Actually, we must compensate for the filter time lag first, but this is already done in our filter routine.) The Lambert Commodity Channel Index for example is based on using a highpass filter. RE: Donald R. Lambert, Commodities, Oct., 1980, pp. 40-41.
*   Bandpass Filters (Oscillators)—
Bandpass filters are designed to accentuate only the frequencies in an intermediate band and to reject both high and low frequencies. This type of filter can be implemented using two passes of the filter routine. In the first pass we apply a lowpass filter designed to reject the upper frequency band. The second pass we apply a highpass filter to the filtered data. For example, we could apply a 5 day ading noise followed by a 30 day highpass filter to suppress the current trend.
*   Momentum Filters
Momentum filters are designed to measure the momentum or slope of the underlying trend in a time series. Such filters can be implemented by first applying a lowpass filter to the data and then applying a differencing operation to the result:
M(i) = (y(i) - y(i-k))/k
where k is chosen of comparable length to that of the lowpass filter. ( It is important to use a weighted moving average for this type of filter since the differencing operation tends to accentuate high frequency components in the data.)
NOTE: The BASIC subroutine following this article does not contain a momentum filter option, but could easily be added.
In order to illustrate these concepts, suppose we want to design a data filter which will emphasize cycles longer than 10 days and will reject cycles shorter than 6 days. In the frequency domain we want to pass frequencies less than 260/10 = 26 and to reject frequencies greater than 260/6 = 43 cycles per year. Now with the filter types implemented in our routine, an N day moving average will eliminate an N day cycle, and an N day weighted moving average (either triangular or Hanning weights) will eliminate an m = (N+1)/2 day cycle. Thus we can reject sinusoids with cycles shorter than 6 days using either a 6 day MA or an 11 day Hanning WMA. Figure 2 shows the amplitude response of these data filters. We see that the Hanning filter does a much better job of rejecting the undesired high frequencies, but requires about twice as long a filter to accomplish this. In some applications the moving average is better to use since it requires the shortest filter length whereas in others the weighted filters are best because of their superior frequency response.
One of the effects of data filtering is to introduce a time lag into the output compared to the input series. For a moving average or the symmetric weighted moving averages in the filter routine, the time lag is the difference between the current time point and the mid-point of the data filter. In other words, the input data and the filtered output are in phase if we reference the output series relative to the filter mid-point. For example, a 3 day MA filter may be represented in the form:
y(i) = 1/3(x(i+1) + x(i) + x(i-1))
This representation is used in the filter routine and the user has the option of lagging or shifting forward the time reference of the output series as desired. Thus, one must shift an N = 2M+1 length filter by M samples in order to reference the output series at the time of the most current data sample. We will refer to filters shifted from the midpoint as lagged filters in the following discussions.
USE OF SMOOTHING FILTERS FOR TRADING DECISIONS

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One of the simplest and best known techniques for determining market buy and sell points is to chart the daily (or weekly) market prices and a lagged, smoothed version of the market data, and then locate the points where these two curves cross. Basically, one sells when the market value crosses the smoothed curve on the downside and buys when the market value crosses on the upside. Figure 3 illustrates the theory behind this method for a time series consisting of a 60 day sinusoid, where the data filter is an 11 day MA, lagged by 5 days so that the output is referenced to the current daily price. In this example, we would be alerted to sell shortly after the market value tops, as confirmed by the downward crossing at 15 days, and would be signaled to buy on the upward crossing at around 45 days.
The above example, however, does not answer the subtle questions involved in selecting a filter type, and an appropriate filter length and lag value to use in this method. This is not an easy task since individual markets may have vastly different noise characteristics, and cycle lengths for short term corrections. Fortunately, most of the information for selecting an appropriate data filter can be obtained from analysis of a long time series of past market prices. A systematic method for selecting filter constants is given by:
Step 1: Obtain the Fourier power spectrum of the price data and determine a frequency such that only high-frequency noise components are above this frequency, and such that most of the signal power is below this frequency. Convert this frequency to a cycle length N and determine an appropriate MA or WMA filter length (N or 2N+1) which will reject this cycle.
Step 2: Apply the data filter to the long time series with several lag values and determine the amount of lag necessary for the two curves to cross only when a major market turn occurs, or when the market prices are flat. {The lag must be sufficient to prevent trading during short term market corrections on the prevailing trend.)
We will illustrate this process by designing a lagged filter for the Fidelity Equity mutual fund, using weekly closing prices. See Figure 4 for recent weekly prices. Using a power spectrum code similar to that in the January issue, we obtain the power spectrum shown in Figure 5. The spectrum is shown in db or log scales, and reveals that most of the power in the spectrum is at frequencies less than 3 cycles per year. (The spectrum is actually defined out to 26 cycles per year, but is blown up to show the low frequency components more clearly.) A second power spectrum was also obtained for this fund using earlier price data, which verified that frequencies above 3 cycles per year are primarily noise.
Consequently, a 17 week MA smoother was selected since this filter eliminates 52/17 = 3.1 cycles per year components and reduces the power in all higher frequencies. The natural time lag associated with a 17 week moving average is 8 weeks, and the 8 week lagged curve does a reasonable job of predicting buy and sell points over a 40 month time series of past data values. However, based on past experience, the author wanted slightly more lag to reduce the probability of trading whipsaws. Thus a lag value of 12 weeks was selected for determining buy and sell points, and figure 6 shows the fund price history and lagged MA curve over a 3 year time period. (The 12 week lag also has the great advantage that the smoothed curve is predicted several weeks ahead of the current weekly value. It is thus not necessary to rechart the raw and smoothed data curves for several weeks, since we can easily determine if a crossing has occurred by manual chart updates.)
Figure 6 also reveals the weak point with the lagged curve trading method, namely that the probability of getting whipsawed is great in sideways trending markets.
One is not liable to lose much by selling out on the downside crossings, but figure 6 reveals several buy/sell whipsaws when buying in with this method during the mid-interval bear market. Consequently, we must rely on other indicators or trading analyses to verify indicated buy points with this method, and what seems to be a simple, objective trading technique in reality turns out to be rather subjective and based on the user's experience and outside judgmental factors. Unfortunately, this can not be helped without greatly complicating the application of this method, and we thus leave it to the reader to apply common sense. intuition, and other technical analyses to overcome this problem.
In the next issue we will show how the use of more sophisticated prediction methods combined with
74-77): Data Filtering Methods For Technical Analysis by ANTHONY WARREN, Ph.D./Technical Analysis staff
trend channel analyses can be used to determine price break outs for trading decisions. Try it, you'll like it!

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Thursday 25th August, 2011 - A couple of tips on how to perfect your hunting skills from a couple of Wyoming hunters

It can seem that practicing for all these scenarios could take hours, or a number of different shooting stations, although that doesn't always have to be the case. You should be able to work all of the variables covered here into a single practice session, by working as many as possible into each shot. The best way to do this is with a shooting partner. Get together your bows, treestands, and a paper target and head for the closest forest.

The shooter should put his stand in a suitable tree and get his or her bow ready. The other person can then place the target at different distances and angles, using natural terrain and obstructions to simulate actual hunting conditions as much as possible. The shooter then gets one shot at each target placement. As mentioned above, be sure and vary your own body position with each shot in order to make your practice sessions as effective as possible. After five to ten shots, you can then try another angle.

Also worth mentioning are the clothing you wear and the arrow points that you use. Since the objective is to simulate an actual hunting environment as much as possible, it makes sense to practice in your hunting clothes using the same broadheads that you will be using when you hunt. Clothing such as gloves, face nets, hats, jackets or heavy shirts can all have an impact on your shooting ability. This especially holds true for late season hunting, when a bowhunter is often bundled heavily in an attempt to stay warm. Don't wait until a nice buck is standing in front of you at twenty yards only to discover that some of your clothing interferes with your accuracy.

It is very important that you practice in the same clothes and gear that you plan on hunting in.
Don't wait for an important shot opportunity to discover that your hat, gloves or facenet get in the way.

The same goes for your arrow points as well. Fixed blade broad heads often fly differently than field points when shot from the same bow. For some bows, this may mean that the bow is not tuned to it's peak level, and should be tuned accordingly.

You do not want to find out that your arrows are hitting off target on opening day of deer season. It's best that you pursue to have your equipment tuned and sighted in for a clean shot. Check with the state of Wyoming government for licensing requirements related to hunting.


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