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docs/indicators/averages/afirma/calc.md

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+# 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 $t$<br>
+- $c$ is a constant<br>
+- $\phi_i$ are the parameters of the autoregressive term<br>
+- $\theta_j$ are the parameters of the moving average term<br>
+- $\epsilon_t$ is white noise<br>
+
+### 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.