Impulsive noise spikes or extreme price or volume data are not unusual in the nancial markets and these extreme values can throw off your averaging calculations. How can you set up a data lter to re- move these extreme price movements? This STOCKS & COMMODITIES Contributing Editor shows you a way to handle this by using a lter that discards all data except the median value.

Median lters are best applied to remove impulsive or spiking types of noise. Rather than averaging the spike into the lter out- put, median lters simply ignore the spike. Median lters are routinely used for pro-

cessing photographs and video because they preserve the sharp edges in the images rather than smoothing them as is done by averaging lters. Median lters have the unique characteristic of being idempotent, that is, if you repeatedly perform median ltering on a time waveform, the output rapidly converges to being exactly the input waveform except for com- putational lag. That a price waveform converges to a core waveform has some interesting philosophical rami cations for trading.

HOW SO?

Median lters are nonlinear. Since a median lter is not a convolution lter, it cannot be suitably represented in the Fourier frequency domain. Also, its output is not differentiable and therefore does not have a Taylor series expansion. This precludes curve- tting by a higher-order polynomial.

There are many academic articles describing rather arcane algorithms for recursive median lters. The rea- son I consider the algorithms arcane is they exclusively study nite impulse response (FIR) types of lters.

This is because the applications being considered are being implemented in hardware rather than software. “Recursive” means using a previous calculation in the current calculation.

APPLYING IT

An example of a recursive lter used in trading is the exponential moving average (EMA). I propose a recur- sive median lter for trading be implemented as the EMA of a ve-bar median lter. A simple pseudocode representation of a recursive median lter is:

Output = a*Median(Input, 5) + (1 – a)*Output[1];

The EMA a is a constant between zero and 1. I prefer to calculate it in terms of the critical period of the lter. The critical period is where shorter wavelengths are passed by the lter and longer wavelengths are rejected at the lter output. The relationship between the EMA constant and critical period is expressed by the equation:

a = (Cosine (360 / Period) + Sine (360 / Period) – 1) / Cosine (360 / Period)

where the arguments of the trigonometric terms are in degrees.

An easier-to-remember approximation to the re- lationship between the EMA constant and critical period is:

a = 5 / Period

MORE FILTERING

An interesting and unique oscillator-type indicator can be cre- ated from the recursive median lter by further ltering with a second-order highpass lter. The highpass lter removes the DC (constant) values and very long wavelength components from the recursive median lter output. Using a second-order lter guarantees attenuation of the long wavelength components resulting from the statistical fractal pink-noise spectral shape of market data.

The second-order nature of the highpass lter reduces its critical period about 70% relative to the critical period of an EMA lter. The EasyLanguage code to compute the recursive median oscillator is given in sidebar “EasyLanguage Code For Recursive Median Oscillator.”

You can see the uniqueness and novelty of the recursive median oscillator when you compare it to the RSI (Figure 1). The recursive median oscillator is displayed in the rst sub- graph and the RSI is plotted in the second subgraph. The RSI is scaled to swing from -1 to +1 instead of the standard swing from zero to 100. The price data for Figure 1 is of the SPY for

The EasyLanguage code to compute the recursive median Filter is given in sidebar “EasyLanguage Code For Recursive Median Filter.”)