The Theoretically Perfect Moving Average and Oscillator

Readers of this blog will probably have noticed that I am partial to using concepts from digital signal processing in the development of my trading system. Recently I "rediscovered" this PDF, written by Tim Tillson, on Google Docs, which has a great opening introduction:

" 'Digital filtering includes the process of smoothing, predicting, differentiating,

integrating, separation of signals, and removal of noise from a signal. Thus many people
who do such things are actually using digital filters without realizing that they are; being
unacquainted with the theory, they neither understand what they have done nor the
possibilities of what they might have done.'

This quote from R. W. Hamming applies to the vast majority of indicators in technical
analysis."

The purpose of this blog post is to outline my recent work in applying some of the principles discussed in the linked PDF.

Long time readers of this blog may remember that back in April this year I abandoned work I was doing on the AFIRMA trend line because I was dissatisfied with the results. The code for this required projecting prices forward using my leading signal code, and I now find that I can reuse the bulk of this code to create a theoretically perfect moving average and oscillator. I have projected prices forward by 10 bars and then taken the average of the forwards and backwards exponential moving averages, as per the linked PDF, to create a series of averages that theoretically are in phase with the underlying price, and then taken the mean of all these averages to produce my "perfect" moving average. Similarly, I have done the same to create a "perfect" oscillator and the results are shown in the images below.
Sideways Market

Trending up Market

Trending down Market

The upper panes show "price" and its perfect moving average, the middle panes show the perfect oscillator, its one bar leading signal, and exponential standard deviation bands set at 1 standard deviation above and below an exponential moving average of the oscillator. The lower panes show the %B indicator of the oscillator within the bands, offset so that the exponential standard deviation bands are at 1 and -1 levels.

Even cursory examination of many charts such as these shows the efficacy of the signals generated by the crossovers of price with its moving average, the oscillator with its leading signal and the %B indicator crossing the 1 and -1 levels, when applied to idealised data. My next post will discuss the application to real price series.

Post to TradingBlox Forum

I have recently posted a reply to a thread on the TradingBlox forum here which readers of this blog might be interested in. The Octave code below is that used to generate the graphic in the linked forum reply.

clear all

a_returns = [2,-1,2,-1,2,-1,2,-1,2,-1];
b_returns = [3,-1.5,3,-1.5,3,-1.5,3,-1.5,3,-1.5];
c_returns = shift(b_returns,1);

a_equity = cumsum([1,a_returns]);
b_equity = cumsum([1,b_returns]);
c_equity = cumsum([1,c_returns]);

ab_equity = a_equity .+ b_equity;
ac_equity = a_equity .+ c_equity;

ab_equity_correl = corrcoef(a_equity,b_equity)
ac_equity_correl = corrcoef(a_equity,c_equity)

ab_returns_correl = corrcoef(a_returns,b_returns)
ac_returns_correl = corrcoef(a_returns,c_returns)

ab_centre_returns_correl = corrcoef(center(a_returns),center(b_returns))
ac_centre_returns_correl = corrcoef(center(a_returns),center(c_returns))

plot(a_equity,"k",b_equity,"b",c_equity,"c",ab_equity,"r",ac_equity,"g")
legend("a equity","b equity","c equity","ab equity","ac equity")