Miha Kralj
90d5638008
- LTMA (Linear Trend Moving Average): Introduces a predictive moving average using dual cascaded EMAs for trend estimation. - MCNMA (McNicholl EMA): Implements a zero-lag TEMA using a cascaded EMA structure for enhanced responsiveness. - NLMA (Non-Lag Moving Average): Utilizes a damped cosine kernel to achieve reduced lag in moving averages. - NMA (Natural Moving Average): Adapts smoothing based on volatility profiles using a square-root kernel. - NYQMA (Nyquist Moving Average): Applies the Nyquist-Shannon theorem to prevent aliasing in cascaded moving averages. - RAIN (Rainbow Moving Average): Combines multiple SMA layers with weighted averages for multi-scale smoothing. - TRAMA (Trend Regularity Adaptive Moving Average): Adapts smoothing based on the frequency of new highs and lows in price data.
4.6 KiB
KAISER: Kaiser Window Moving Average
"James Kaiser gave signal processing a knob. Turn beta up, sidelobes go down, transition band widens. Turn it down, you get an SMA. One parameter to rule them all."
KAISER applies the Kaiser-Bessel window function as FIR filter weights, providing a single parameter (\beta) that continuously controls the trade-off between main lobe width (transition band sharpness) and sidelobe attenuation (stopband rejection). At \beta = 0 it degenerates to a rectangular window (SMA); at \beta \approx 5.65 it approximates the Blackman window; at \beta \approx 8.6 it matches the Hamming window's sidelobe profile. This makes KAISER the most flexible single-parameter window-based moving average, allowing traders to tune frequency selectivity without changing the window length.
Historical Context
James F. Kaiser and Ronald W. Schafer published the Kaiser window in 1980, building on Kaiser's earlier work at Bell Labs in the 1960s. The window was motivated by a practical problem: given a desired sidelobe attenuation level, what is the shortest FIR filter that achieves it? Kaiser showed that the modified Bessel function of the first kind, I_0, produces near-optimal windows that closely approximate the prolate spheroidal wave functions (the theoretically optimal windows derived by Slepian in 1964) while being far simpler to compute.
The Kaiser window became the default design tool in DSP textbooks (Oppenheim & Schafer, Parks & Burrus) because of its parametric flexibility. In financial applications, this flexibility maps directly to a smoothness-responsiveness knob: low \beta preserves fast price movements (less smoothing, sharper transitions), while high \beta produces smoother output with greater lag (more attenuation of high-frequency price noise).
The I_0 Bessel function is computed via power series: I_0(x) = \sum_{m=0}^{M} \left[\frac{(x/2)^m}{m!}\right]^2. Twenty-five terms provide double-precision convergence for \beta \leq 20.
Architecture & Physics
1. Bessel Function Approximation
The zeroth-order modified Bessel function I_0(x) is evaluated via its power series with 25 terms. The series converges rapidly because the terms are squared factorials, guaranteeing monotonic decrease after the peak term.
2. Weight Computation (One-Time)
For each position k \in [0, N-1], the normalized coordinate t = 2k/(N-1) - 1 maps to [-1, 1]. The Kaiser window value is:
w(k) = \frac{I_0\left(\beta \sqrt{1 - t^2}\right)}{I_0(\beta)}
Weights are normalized to sum to 1.0. The \sqrt{1-t^2} argument is clamped to non-negative to handle floating-point edge cases.
3. FIR Convolution
Standard weighted sum over the circular buffer using precomputed weights. O(N) per bar.
Mathematical Foundation
The Kaiser window function for a filter of length N:
w[k] = \frac{I_0\left(\beta\sqrt{1 - \left(\frac{2k}{N-1} - 1\right)^2}\right)}{I_0(\beta)}, \quad k = 0, 1, \ldots, N-1
where I_0(x) is the zeroth-order modified Bessel function of the first kind:
I_0(x) = \sum_{m=0}^{\infty} \left[\frac{(x/2)^m}{m!}\right]^2
Key \beta values and their equivalences:
\beta |
Equivalent Window | Sidelobe (dB) | Transition BW |
|---|---|---|---|
| 0 | Rectangular (SMA) | -13 |
0.92/N |
| 3.0 | General-purpose | -33 |
2.4/N |
| 5.65 | Blackman-like | -57 |
3.6/N |
| 8.6 | Hamming-like | -90 |
5.0/N |
Kaiser's empirical formulas (for filter design):
\beta = \begin{cases} 0.1102(A - 8.7) & A > 50 \\ 0.5842(A-21)^{0.4} + 0.07886(A-21) & 21 \leq A \leq 50 \\ 0 & A < 21 \end{cases}
where A = -20\log_{10}(\delta) is the desired stopband attenuation in dB.
Default parameters: period = 14, beta = 3.0, minPeriod = 2.
Pseudo-code (streaming):
// One-time: compute I0 and weights
bessel_i0(x):
sum = 1.0; term = 1.0; hx = x/2
for m = 1 to 25: term *= hx/m; sum += term²
return sum
i0_beta = bessel_i0(beta)
for k = 0 to period-1:
t = 2k/(N-1) - 1
arg = sqrt(max(0, 1 - t²))
w[k] = bessel_i0(beta * arg) / i0_beta
normalize(w)
// Per-bar convolution
buffer.push(price)
if count < period: return price
return Σ buffer[j] * w[j]
Resources
- Kaiser, J.F. & Schafer, R.W. (1980). "On the Use of the I0-Sinh Window for Spectrum Analysis." IEEE Trans. Acoust., Speech, Signal Process., ASSP-28(1), 105-107.
- Oppenheim, A.V. & Schafer, R.W. (2009). Discrete-Time Signal Processing, 3rd ed. Prentice Hall. Section 7.4.
- Slepian, D. (1964). "Prolate Spheroidal Wave Functions, Fourier Analysis and Uncertainty." Bell System Technical Journal, 43(6), 3009-3057.