Intraday Trading : EMA, MacD Enhanced and other Strange Non-linear Functions

Notes on the MACD Enhanced Signal

The so called MACD Enhanced signal comes is based on Mathematical Techniques in Financial Market Trading by Don K. Mak, World Scientific Publishing, 2006, Chapter 4. This filter is described as "Zero-lag" EMA and is attributed to J.F. Ehlers. This filter is most certainly not "zero-lag", especially with the window sizes listed here.

Although I have cited it here, I cannot recommend Don Mak's book. While Don Mak attempts to put technical analysis on a more scientific and mathematical footing, the book is needlessly obscure.

M = window size
p_t is the tick price at time t
m is the EMA time series

a = 2 / (M+1) Example: for M³⁰⁰, a = 2 / (300 +1)

m_t+1 = (a * p_t ) + (1 - a) * m_t

If there is a series of tick prices p₁, p₂, p₃... how does the EMA start? Is it

m₁ = (a * p₂ ) + (1 - a) * p₁

or, do we consider m₀ = 0 and the equation is

m₁ = (a * p₂ ) + (1 - a) * 0 ==> m₁ = (a * p₁ )

For m₁, I decided to treat m₀ = 0.

The full MACD Enhanced is

S_t = m³⁰⁰_t - m⁵⁰⁰_t

Where m³⁰⁰ and m⁵⁰⁰ are the m EMA series with 300 and 500 element "windows".

X²⁰⁰_t+1 = ( a₂₀₀ * S_t ) + (1 - a) *X²⁰⁰_t

Where a₂₀₀ ==> a = 2/(200+1)

If we have the S time series S₁, S₂, S₃... how does the X²⁰⁰_t series start. Is it

X²⁰⁰₁ = ( a₂₀₀ * S₂ ) + (1 - a) *S₁

or, do we consider X₀ = 0 and the equation is

X²⁰⁰₁ = ( a₂₀₀ * S₁ ) + (1 - a) * 0 ==> X²⁰⁰₁ = ( a₂₀₀ * S₁ )

For X²⁰⁰₁ , I decided to treat X₀ = 0.