The problem with indicators and oscillators is that they suffer lag. For instance, a 19-bar moving average has a 9-bar lag i.e almost half the filter length. Exponential moving averages also suffer lag which varies over the sampling frequency range. The are many reasons why one should opt for a minimum lag indicator when filtering input data to a neural network. Generally speaking, you would not want your neural network to learn the lag present in the input data, or atleast you want to minimise this as much as possible. Here I describe a way of deriving an almost zero lag indicator in the Z-domain, as outlined in [Ehlers]

The time domain difference equation for an EMA is

equation1.png

where alpha is a number between 0 and 1. Taking the z-transform of both sides and re-arranging we get

equation2.png

I want to create a high-pass filter using the z-domain expression for the low-pass EMA filter. Hence I subtract the expression from 1. The reasoning behind this is that a filter with transfer response 1 represents all frequencies. Hence the residual of subtracting a low-pass filter from 1 should be a high-pass filter. But care must be taken. The transfer response of the EMA does not cover all frequencies right through to infinity because it is 0 at the nyquist frequency (i.e. when z = -1). Doing a direct subtraction would lead to a gain error in the frequency response. To eleminate this problem we take two sequential inputs and average them together, rather than using one input sample at a time. Thus the transfer response as a result of averaging becomes

equation3.png

We can now subtract this equation from 1 without fear of getting a gain error. Thus we have

equation5.png

equation6.png

equation7.png

Inorder to improve attenuation of the derived high-pass filter, we increase its order. But increasing the order of a filter is equivalent to introducing lag. However, the second-order Gaussain filter suffers the least amount of lag amongst all second order filters. Squaring the above equation would yield a second-order Gaussain filter. Thus we have

equation8.png

If we apply this filter to price data, we would essentially see cyclic components at the output. We are interested in trending components, or the low-pass components because we are designing a low-pass indicator filter. Hence we subtract the above equation from 1 again. After much simplification we get

equation9.png

By inspection we see that the number of delay operators in the numerator and denominator are equal. This implies the filter has zero lag. Hence we have derived an indicator with zero lag. The simulations below show different characteristics of the filter. I have also shown an equivalent MA filter for comparison purposes, since the lag relationship between an EMA and a MA is governed by the following equation

equation4.png

IRcomp3.png

The impulse respose of our new filter is finite, unlike an its equivalent EMA filter. This shows our new filter demonstrates short memory over finite duration, which might be a desired feature if too many past values are not required to be remembered.

MAGcomp3.png

The frequency response of our new filter is comparable to that of its EMA counterpart. Notice that the attenuation at nyquist frequency is an asymptote. That is the benefit of taking two sequential input samples in deriving the new filter.

And now the most important graph - showing the filter lags:

LAGcomp3.png

We have successfully designed a filter with virtually zero lag.

Filter design summary

  1. Get z-domain EMA filter equation for general alpha.
  2. Convert z-domain EMA equation to take two sequential inputs at a time so as to eliminate gain error.
  3. Subtract expression from 1 to get high-pass filter.
  4. Convert expression to 2nd order Gaussain inorder to improve filter attenuation.
  5. Finally subtract expression from 1 to get your zero lag low-pass filter.

In my next post I shall compare the performace of our new filter using real price time-series.

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