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68 lines
2.0 KiB
Markdown
68 lines
2.0 KiB
Markdown
# The Math Behind AFIRMA
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## Components of AFIRMA
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AFIRMA is a hybrid beast, combining two main components:
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- Autoregressive Moving Average (ARMA)
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- Finite Impulse Response (FIR) filter
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### ARMA Component
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$ X_t = c + \epsilon_t + \sum_{i=1}^p \phi_i X_{t-i} + \sum_{j=1}^q \theta_j \epsilon_{t-j} $
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Where:
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- $X_t$ is the time series value at time $t$<br>
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- $c$ is a constant<br>
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- $\phi_i$ are the parameters of the autoregressive term<br>
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- $\theta_j$ are the parameters of the moving average term<br>
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- $\epsilon_t$ is white noise<br>
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### FIR Component
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$ y[n] = \sum_{i=0}^{N-1} b_i \cdot x[n-i] $
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Where:
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- $y[n]$ is the output signal
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- $x[n]$ is the input signal
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- $b_i$ are the filter coefficients
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- $N$ is the filter order
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### AFIRMA: Putting It All Together
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AFIRMA combines these components and adds cubic spline fitting to the mix. The general form can be expressed as:
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$ AFIRMA_t = ARMA_t + FIR_t + CS_t $
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Where:
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- $ARMA_t$ is the ARMA component at time $t$
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- $FIR_t$ is the FIR component at time $t$
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- $CS_t$ is the cubic spline fitting component at time $t$
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### Digital Filtering Process
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- The price data is passed through the digital filter to smooth out fluctuations.
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- The filter coefficients are optimized based on the specified parameters (Periods, Taps, Window).
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### Cubic Spline Fitting
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For the most recent bars:
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- A cubic spline is fitted to the data points using the least squares method.
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- This ensures a smooth transition between the filtered data and the most recent price movements.
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### Parameter Definitions
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The AFIRMA indicator allows for the adjustment of three main parameters:
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- **Periods**: Affects the overall smoothness of the indicator.
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- *Taps*: Influences the complexity of the digital filter.
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- *Window*: Determines the number of recent bars to which the cubic spline fitting is applied.
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### Computational Process
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- Apply the ARMA model to the price data.
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- Pass the result through the FIR filter.
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- Apply cubic spline fitting to the most recent data points.
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- Combine the results to produce the final AFIRMA value.
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