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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.
95 lines
4.6 KiB
Markdown
95 lines
4.6 KiB
Markdown
# KAISER: Kaiser Window Moving Average
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> "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."
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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.
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## Historical Context
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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.
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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).
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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 $\beta \leq 20$.
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## Architecture & Physics
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### 1. Bessel Function Approximation
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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.
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### 2. Weight Computation (One-Time)
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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:
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$$
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w(k) = \frac{I_0\left(\beta \sqrt{1 - t^2}\right)}{I_0(\beta)}
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$$
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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.
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### 3. FIR Convolution
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Standard weighted sum over the circular buffer using precomputed weights. O(N) per bar.
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## Mathematical Foundation
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The Kaiser window function for a filter of length $N$:
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$$
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w[k] = \frac{I_0\left(\beta\sqrt{1 - \left(\frac{2k}{N-1} - 1\right)^2}\right)}{I_0(\beta)}, \quad k = 0, 1, \ldots, N-1
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$$
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where $I_0(x)$ is the zeroth-order modified Bessel function of the first kind:
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$$
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I_0(x) = \sum_{m=0}^{\infty} \left[\frac{(x/2)^m}{m!}\right]^2
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$$
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**Key $\beta$ values and their equivalences:**
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| $\beta$ | Equivalent Window | Sidelobe (dB) | Transition BW |
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| :---: | :--- | :---: | :---: |
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| 0 | Rectangular (SMA) | $-13$ | $0.92/N$ |
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| 3.0 | General-purpose | $-33$ | $2.4/N$ |
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| 5.65 | Blackman-like | $-57$ | $3.6/N$ |
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| 8.6 | Hamming-like | $-90$ | $5.0/N$ |
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**Kaiser's empirical formulas** (for filter design):
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$$
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\beta = \begin{cases} 0.1102(A - 8.7) & A > 50 \\ 0.5842(A-21)^{0.4} + 0.07886(A-21) & 21 \leq A \leq 50 \\ 0 & A < 21 \end{cases}
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$$
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where $A = -20\log_{10}(\delta)$ is the desired stopband attenuation in dB.
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**Default parameters:** `period = 14`, `beta = 3.0`, `minPeriod = 2`.
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**Pseudo-code (streaming):**
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```
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// One-time: compute I0 and weights
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bessel_i0(x):
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sum = 1.0; term = 1.0; hx = x/2
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for m = 1 to 25: term *= hx/m; sum += term²
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return sum
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i0_beta = bessel_i0(beta)
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for k = 0 to period-1:
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t = 2k/(N-1) - 1
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arg = sqrt(max(0, 1 - t²))
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w[k] = bessel_i0(beta * arg) / i0_beta
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normalize(w)
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// Per-bar convolution
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buffer.push(price)
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if count < period: return price
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return Σ buffer[j] * w[j]
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```
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## Resources
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- 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.
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- Oppenheim, A.V. & Schafer, R.W. (2009). *Discrete-Time Signal Processing*, 3rd ed. Prentice Hall. Section 7.4.
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- Slepian, D. (1964). "Prolate Spheroidal Wave Functions, Fourier Analysis and Uncertainty." *Bell System Technical Journal*, 43(6), 3009-3057.
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