QuanTAlib/lib/trends/afirma/Afirma.md

AFIRMA: Autoregressive Finite Impulse Response Moving Average

"When ARMA met FIR at a signal processing conference and they had a baby with cubic spline DNA. The result filters noise like a surgeon and tracks price like a stalker."

AFIRMA is a hybrid smoothing filter that combines three signal processing techniques: autoregressive (AR) modeling, finite impulse response (FIR) filtering with windowed sinc coefficients, and cubic spline fitting for the leading edge. The result is a filter that achieves superior noise reduction while maintaining signal fidelity and minimizing lag.

Historical Context

AFIRMA emerged from the intersection of econometric time series analysis (ARMA models from Box-Jenkins methodology, circa 1970) and digital signal processing (FIR filters with window functions). The combination addresses a fundamental problem: traditional moving averages either lag badly (SMA, EMA) or introduce ringing artifacts (sharp cutoff filters). AFIRMA uses the mathematically optimal sinc function—the ideal low-pass filter impulse response—tempered by window functions that trade off main lobe width against sidelobe suppression.

Architecture & Physics

AFIRMA operates through a convolution of the input signal with pre-computed windowed sinc coefficients.

The Sinc Function

The sinc function is the impulse response of an ideal low-pass filter:

\text{sinc}(x) = \begin{cases} 1 & \text{if } x = 0 \\ \frac{\sin(x)}{x} & \text{otherwise} \end{cases}

In practice, the sinc function extends infinitely—inconvenient for real-time processing. AFIRMA truncates it to a finite number of taps and applies a window function to minimize the resulting spectral leakage.

Window Functions

Window functions control the trade-off between frequency resolution (main lobe width) and spectral leakage (sidelobe suppression).

Window	Main Lobe	Sidelobe	Use Case
Rectangular	Narrowest	Worst (-13 dB)	Maximum frequency resolution, high leakage
Hanning	Moderate	Good (-31 dB)	General purpose smoothing
Hamming	Moderate	Better (-42 dB)	Reduced leakage with decent resolution
Blackman	Wide	Excellent (-58 dB)	Low leakage, good for noisy data
Blackman-Harris	Widest	Best (-92 dB)	Minimum leakage, maximum smoothing

The default Blackman-Harris window provides the best sidelobe suppression, making AFIRMA robust to impulsive noise in price data.

Cubic Spline Component

The ARMA polynomial coefficients are precomputed during initialization to support least-squares cubic fitting at the leading edge. This reduces end-point distortion common in FIR filters, where the filter "sees" incomplete data at the boundaries.

Mathematical Foundation

1. Windowed Sinc Coefficients

For tap k of N total taps:

w_k = W(k) \cdot \text{sinc}\left(\frac{\pi (k - c)}{P}\right)

Where:

• c = \frac{N-1}{2} is the center tap
• P is the period parameter
• W(k) is the window function value at tap k

2. Window Functions

Hanning:

W(k) = 0.5 - 0.5 \cos\left(\frac{2\pi k}{N-1}\right)

Hamming:

W(k) = 0.54 - 0.46 \cos\left(\frac{2\pi k}{N-1}\right)

Blackman:

W(k) = 0.42 - 0.5 \cos\left(\frac{2\pi k}{N-1}\right) + 0.08 \cos\left(\frac{4\pi k}{N-1}\right)

Blackman-Harris:

W(k) = 0.35875 - 0.48829 \cos\left(\frac{2\pi k}{N-1}\right) + 0.14128 \cos\left(\frac{4\pi k}{N-1}\right) - 0.01168 \cos\left(\frac{6\pi k}{N-1}\right)

3. Convolution

\text{AFIRMA}_t = \frac{\sum_{k=0}^{N-1} w_k \cdot P_{t-k}}{\sum_{k=0}^{N-1} w_k}

Parameters

Parameter	Default	Range	Description
Period	-	≥ 1	Controls the cutoff frequency. Higher values = more smoothing.
Taps	6	≥ 1 (odd preferred)	Filter length. More taps = sharper frequency response.
Window	BlackmanHarris	Enum	Window function for sidelobe control.

Parameter Selection Guide

• Period: Start with half your expected cycle length. For intraday on 1-minute bars with 20-minute cycles, use Period=10.
• Taps: Use odd numbers (5, 7, 9...) for symmetric response. More taps = more lag but sharper cutoff. 6-12 is typical.
• Window: Blackman-Harris for noisy data, Hamming for faster response, Rectangular only for experimentation.

Usage

Streaming (Real-time)

var afirma = new Afirma(period: 10, taps: 7, window: Afirma.WindowType.BlackmanHarris);

foreach (var bar in marketData)
{
    var smoothed = afirma.Update(new TValue(bar.Time, bar.Close));
    Console.WriteLine($"{bar.Time}: {smoothed.Value:F4}");
}

Batch Processing

var series = new TSeries(timestamps, prices);
var smoothed = Afirma.Batch(series, period: 10, taps: 7);

Span API (Zero-Allocation)

ReadOnlySpan<double> prices = GetPrices();
Span<double> output = stackalloc double[prices.Length];

Afirma.Batch(prices, output, period: 10, taps: 7);

Event-Driven (Chaining)

var source = new TSeries();
var afirma = new Afirma(source, period: 10, taps: 7);

// AFIRMA automatically updates when source changes
source.Add(new TValue(DateTime.UtcNow, 100.0));
Console.WriteLine(afirma.Last.Value);

Performance Profile

Metric	Score	Notes
Throughput	~50 ns/bar	O(n) per update where n = taps
Allocations	0	Zero-allocation in hot paths
Complexity	O(taps)	Linear in filter length
Accuracy	9	Excellent noise reduction
Timeliness	7	Lower lag than equivalent SMA
Overshoot	2	Minimal with proper window selection
Smoothness	9	Very smooth output

Validation

Library	Status	Notes
Internal	✅	Batch, Streaming, and Span modes match
Mathematical	✅	Variance reduction verified

AFIRMA is a QuanTAlib-specific implementation. No direct external library comparison is available, but internal consistency across all API modes has been verified.

Window Type Comparison

For the same Period and Taps, different windows produce different smoothing characteristics:

Window	Smoothness	Responsiveness	Best For
Rectangular	Low	Highest	Testing/comparison only
Hanning	Medium	High	General use
Hamming	Medium-High	Medium-High	Balanced applications
Blackman	High	Medium	Noisy data
BlackmanHarris	Highest	Lower	Very noisy data, maximum smoothing

Common Pitfalls

1. Too Many Taps: More taps mean more lag. Don't use 50 taps "just because." Start with 5-9.

2. Period vs. Taps Confusion: Period controls smoothness (like EMA period). Taps control filter sharpness. They're independent parameters.

3. Rectangular Window: Almost never the right choice for financial data. The severe sidelobe leakage introduces ringing.

4. Cold Values: AFIRMA needs taps bars of history to be fully warmed up. The IsHot property indicates when the filter is primed.

See Also

• ALMA - Gaussian-weighted moving average with offset
• CONV - General convolution filter
• SSF - Ehlers Super Smooth Filter (2-pole IIR)