QuanTAlib/docs/indicators/averages/afirma/calc.md

The Math Behind AFIRMA

Components of AFIRMA

AFIRMA is a hybrid beast, combining two main components:

• Autoregressive Moving Average (ARMA)
• Finite Impulse Response (FIR) filter

ARMA Component

X_t = c + \epsilon_t + \sum_{i=1}^p \phi_i X_{t-i} + \sum_{j=1}^q \theta_j \epsilon_{t-j}

Where:

• X_t is the time series value at time $t$
  
• c is a constant
  
• \phi_i are the parameters of the autoregressive term
  
• \theta_j are the parameters of the moving average term
  
• \epsilon_t is white noise
  

FIR Component

y[n] = \sum_{i=0}^{N-1} b_i \cdot x[n-i]

Where:

• y[n] is the output signal
• x[n] is the input signal
• b_i are the filter coefficients
• N is the filter order

AFIRMA: Putting It All Together

AFIRMA combines these components and adds cubic spline fitting to the mix. The general form can be expressed as:

AFIRMA_t = ARMA_t + FIR_t + CS_t

Where:

• ARMA_t is the ARMA component at time t
• FIR_t is the FIR component at time t
• CS_t is the cubic spline fitting component at time t

Digital Filtering Process

• The price data is passed through the digital filter to smooth out fluctuations.
• The filter coefficients are optimized based on the specified parameters (Periods, Taps, Window).

Cubic Spline Fitting

For the most recent bars:

• A cubic spline is fitted to the data points using the least squares method.
• This ensures a smooth transition between the filtered data and the most recent price movements.

Parameter Definitions

The AFIRMA indicator allows for the adjustment of three main parameters:

• Periods: Affects the overall smoothness of the indicator.
• Taps: Influences the complexity of the digital filter.
• Window: Determines the number of recent bars to which the cubic spline fitting is applied.

Computational Process

• Apply the ARMA model to the price data.
• Pass the result through the FIR filter.
• Apply cubic spline fitting to the most recent data points.
• Combine the results to produce the final AFIRMA value.