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Miha Kralj
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- 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.
56 lines
3.4 KiB
Markdown
56 lines
3.4 KiB
Markdown
# CWT: Continuous Wavelet Transform
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CWT computes the magnitude of the Continuous Wavelet Transform at a specified scale using the Morlet wavelet, providing a time-frequency decomposition that measures the energy content of a specific frequency band at each point in time. Unlike Fourier analysis which loses time localization, the wavelet transform maintains both time and frequency information simultaneously. The output is a non-negative magnitude series where peaks indicate strong presence of the target frequency (determined by the scale parameter) and troughs indicate absence of that frequency component.
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## Historical Context
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The wavelet transform emerged from seismology and signal processing in the 1980s, with foundational work by Jean Morlet (a geophysicist analyzing seismic reflections) and Alex Grossmann. The Morlet wavelet — a complex sinusoid modulated by a Gaussian envelope — became the standard analyzing wavelet due to its optimal time-frequency resolution (it achieves the Heisenberg uncertainty lower bound). In financial applications, CWT provides multi-resolution analysis: by varying the scale parameter, traders can identify dominant cycles at different timeframes without the windowing artifacts of short-time Fourier transforms. The scale parameter directly controls which frequency band is analyzed: larger scales capture lower frequencies (longer cycles), smaller scales capture higher frequencies (shorter cycles). The relationship between scale $s$ and approximate cycle period is $P \approx \frac{2\pi s}{\omega_0}$ where $\omega_0$ is the central frequency (default 6.0).
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## Architecture & Physics
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### Morlet Wavelet Convolution
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The CWT at scale $s$ is computed as the inner product of the signal with a scaled, translated Morlet wavelet:
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$$W(t, s) = \frac{1}{\sqrt{s}} \sum_{k=-K}^{K} x(t-k) \cdot \psi^*\!\left(\frac{k}{s}\right)$$
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The Morlet wavelet $\psi(t) = e^{-t^2/2} e^{i\omega_0 t}$ decomposes into real (cosine) and imaginary (sine) parts, both modulated by a Gaussian envelope.
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### Implementation Details
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- **Half-window:** $K = \text{round}(3s)$, ensuring the Gaussian envelope decays to $<0.01$ at the edges ($e^{-4.5} \approx 0.011$).
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- **Real and imaginary sums:** Computed separately, then combined as $|W| = \sqrt{\text{Re}^2 + \text{Im}^2}$.
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- **Normalization:** The $1/\sqrt{s}$ factor ensures energy preservation across scales.
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### Complexity
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$O(K)$ per bar where $K = 6s + 1$. For scale = 10, this is 61 multiply-adds per bar.
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## Mathematical Foundation
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**Morlet wavelet:**
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$$\psi(t) = e^{-t^2/2} \cdot e^{i\omega_0 t}$$
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**CWT at scale $s$ and time $t$:**
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$$W(t, s) = \frac{1}{\sqrt{s}} \sum_{k=-K}^{K} x_{t+k} \cdot e^{-k^2/(2s^2)} \cdot e^{-i\omega_0 k/s}$$
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**Magnitude (power at scale $s$):**
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$$|W(t,s)| = \sqrt{\left(\sum_k x_k \cdot g_k \cos\theta_k\right)^2 + \left(\sum_k x_k \cdot g_k \sin\theta_k\right)^2} \cdot \frac{1}{\sqrt{s}}$$
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where $g_k = e^{-k^2/(2s^2)}$ and $\theta_k = \omega_0 k / s$
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**Scale-to-period relationship:**
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$$P \approx \frac{2\pi s}{\omega_0}$$
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**Default parameters:** scale = 10.0, omega = 6.0 (corresponding to period $\approx 10.5$ bars).
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## Resources
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- Morlet, J. et al. (1982). "Wave propagation and sampling theory." *Geophysics*, 47(2): 203-236
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- Torrence, C. & Compo, G.P. (1998). "A Practical Guide to Wavelet Analysis." *Bulletin of the American Meteorological Society*
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- PineScript reference: [`cwt.pine`](cwt.pine)
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