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.
4.9 KiB
PARZEN: Parzen (de la Vallée-Poussin) Window Moving Average
"Emanuel Parzen convolved two triangular windows and got a piecewise cubic with zero sidelobe discontinuity. When your window function is its own proof of smoothness, the spectral leakage has nowhere to hide."
PARZEN applies the Parzen (de la Vallée-Poussin) window function as FIR filter weights, producing a moving average with exceptional sidelobe suppression (-24 dB/octave rolloff) and a smooth bell-shaped kernel. The Parzen window is the self-convolution of two triangular (Bartlett) windows at half-length, which guarantees continuous first and second derivatives at all points. This makes it one of the few windows whose frequency response has no discontinuities in its first three derivatives, yielding the fastest sidelobe decay rate among common windows without requiring the computational cost of Bessel functions (Kaiser) or specialized polynomials (Henderson).
Historical Context
Emanuel Parzen (1929-2016) introduced the window in a 1961 paper on spectral estimation in Technometrics, though the underlying function was studied earlier by de la Vallée-Poussin in the context of Fourier series summability. Parzen's contribution was to recognize the window's optimality properties for spectral density estimation: among all non-negative windows with continuous derivatives up to order 2, the Parzen window minimizes the integrated squared bias of the spectral estimate.
The Parzen window's construction as a convolution of two Bartlett windows gives it a natural interpretation: it is equivalent to computing the SMA of an SMA of half the period, twice. This "double triangular smoothing" produces the piecewise cubic shape without explicit polynomial computation. In the spectral domain, the convolution translates to multiplication: the Parzen frequency response is the square of the Bartlett frequency response, which explains the doubled sidelobe rolloff rate (-24 dB/octave vs. -12 dB/octave for Bartlett).
Compared to competing windows, Parzen trades main-lobe width for sidelobe suppression. Its main lobe is wider than Hann or Hamming (meaning more lag in the time domain), but its sidelobes decay faster than any other polynomial-based window. For financial applications where smooth trend extraction matters more than sharp frequency cutoff, this trade-off favors Parzen.
Architecture & Physics
1. Piecewise Cubic Weight Function
The Parzen window is defined in two regions based on the normalized coordinate |u| = |k - (N-1)/2| / ((N-1)/2):
- Inner region (
|u| \leq 0.5): Cubic spline with positive curvature tapering from the peak. - Outer region (
0.5 < |u| \leq 1.0): Cubic taper to zero at the window edge.
The two pieces join with continuous first and second derivatives at |u| = 0.5, ensuring no spectral artifacts from weight discontinuities.
2. Weight Normalization
Weights are normalized to sum to 1.0. Because all Parzen weights are non-negative, the filter output is always a convex combination of input prices (no overshoot possible from negative weights).
3. FIR Convolution
Standard weighted convolution over the circular buffer. O(N) per bar. The symmetric structure allows paired-element optimization for SIMD.
Mathematical Foundation
For a window of length N, with normalized coordinate u = (k - (N-1)/2) / ((N-1)/2), k = 0, \ldots, N-1:
w(k) = \begin{cases} 1 - 6u^2 + 6|u|^3 & |u| \leq 0.5 \\ 2(1 - |u|)^3 & 0.5 < |u| \leq 1.0 \\ 0 & |u| > 1.0 \end{cases}
Frequency response properties:
| Property | Value |
|---|---|
Main lobe width (-3 dB) |
\approx 2.0/N |
| First sidelobe | -53 dB |
| Sidelobe rolloff | -24 dB/octave |
| All weights non-negative | Yes |
Equivalence to double convolution:
w_{\text{Parzen}}[n] = w_{\text{Bartlett}}[n] * w_{\text{Bartlett}}[n]
where * denotes discrete convolution and the Bartlett windows are of length N/2.
Normalized output:
\text{PARZEN}_t = \frac{\sum_{k=0}^{N-1} w[k] \cdot x_{t-k}}{\sum_{k=0}^{N-1} w[k]}
Default parameters: period = 14, minPeriod = 2.
Pseudo-code (streaming):
// One-time weight computation
half_N = (period - 1) / 2
for k = 0 to period-1:
u = (k - half_N) / half_N
abs_u = |u|
if abs_u <= 0.5:
w[k] = 1 - 6*abs_u² + 6*abs_u³
else if abs_u <= 1.0:
w[k] = 2*(1 - abs_u)³
else:
w[k] = 0
normalize(w)
// Per-bar convolution
buffer.push(price)
if count < period: return price
return Σ buffer[j] * w[j]
Resources
- Parzen, E. (1961). "Mathematical Considerations in the Estimation of Spectra." Technometrics, 3(2), 167-190.
- Harris, F.J. (1978). "On the Use of Windows for Harmonic Analysis with the Discrete Fourier Transform." Proceedings of the IEEE, 66(1), 51-83.
- Nuttall, A.H. (1981). "Some Windows with Very Good Sidelobe Behavior." IEEE Trans. Acoust., Speech, Signal Process., 29(1), 84-91.