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.
5.3 KiB
WINS: Winsorized Mean Moving Average
The Winsorized Mean Moving Average computes a rolling average after replacing (not discarding) the most extreme values in each tail with the boundary values at the trim point. Unlike the trimmed mean (TRIM) which removes outliers entirely, Winsorization preserves the full sample size by clamping extreme values to the nearest non-extreme observation. At winPct = 0 it degenerates to the SMA; at winPct = 50 all values equal the median pair. The default 10% Winsorization provides a robust central tendency estimator that dampens outlier impact while maintaining the statistical efficiency advantages of the full sample size.
Historical Context
Winsorization is named after Charles P. Winsor, a biostatistician at Harvard, though the technique was popularized by John Tukey (1962) who credited Winsor with the idea. The concept arises naturally from the question: "what if instead of throwing away extreme values, we replace them with the most extreme non-discarded value?" This produces an estimator that is more efficient than the trimmed mean under light contamination models while retaining comparable robustness.
The distinction between trimming and Winsorizing is subtle but consequential. Consider a 20-bar window with 10% processing: TRIM discards the 2 lowest and 2 highest values, averaging the remaining 16. WINS replaces the 2 lowest with the 3rd-lowest value and the 2 highest with the 3rd-highest, averaging all 20. Both have the same breakdown point (10%), but WINS has higher asymptotic efficiency because it uses all n observations in the average.
In financial applications, Winsorization is standard practice in factor modeling: Fama-French factor returns are typically Winsorized at 1% or 5% to prevent a handful of extreme observations from dominating cross-sectional regressions. The Winsorized mean is also used in the construction of robust risk measures like the Winsorized variance and the Winsorized covariance matrix.
Architecture and Physics
The computation has three steps per bar:
Step 1: Collection gathers the most recent period values into an array, substituting 0 for NaN via nz().
Step 2: Sort and clamp arranges values in ascending order, then replaces the lowest winCount values with the value at index winCount (the lower boundary) and the highest winCount values with the value at index period - 1 - winCount (the upper boundary):
\text{winCount} = \left\lfloor \frac{\text{period} \times \text{winPct}}{100} \right\rfloor
The clamping preserves the boundary values themselves; only values beyond them are replaced.
Step 3: Average computes the arithmetic mean of all period values (including the replaced ones). Since replaced values equal the boundary values, this is equivalent to:
\text{WINS} = \frac{\text{winCount} \cdot x_{(k+1)} + \sum_{i=k+1}^{n-k} x_{(i)} + \text{winCount} \cdot x_{(n-k)}}{n}
where k = \text{winCount} and x_{(i)} is the $i$-th order statistic.
Edge case: If winCount would reach or exceed period / 2, it is clamped to (period - 1) / 2, producing the median pair (two middle values) replicated across all positions.
Mathematical Foundation
The Winsorized mean for a sample of size n with k replacements per tail:
\bar{x}_W = \frac{1}{n}\left[k \cdot x_{(k+1)} + \sum_{i=k+1}^{n-k} x_{(i)} + k \cdot x_{(n-k)}\right]
where x_{(i)} is the $i$-th order statistic and k = \lfloor \alpha n \rfloor with \alpha = \text{winPct}/100.
Winsorized variance (used for inference on the Winsorized mean):
s_W^2 = \frac{1}{n-1} \sum_{i=1}^{n} (w_i - \bar{x}_W)^2
where w_i are the Winsorized values.
Influence function: Bounded like TRIM, but the boundary behavior differs:
\text{IF}(x; \bar{x}_W) = \begin{cases} x_{(\alpha)} - \bar{x}_W & \text{if } x \le x_{(\alpha)} \\ x - \bar{x}_W & \text{if } x_{(\alpha)} < x < x_{(1-\alpha)} \\ x_{(1-\alpha)} - \bar{x}_W & \text{if } x \ge x_{(1-\alpha)} \end{cases}
Breakdown point: \alpha (the Winsorization fraction).
Asymptotic efficiency relative to SMA under normality (higher than TRIM at same percentage):
| Win % | WINS Efficiency | TRIM Efficiency |
|---|---|---|
| 0% | 100% | 100% |
| 10% | ~97% | ~95% |
| 25% | ~90% | ~85% |
Parameter constraints: period \ge 3, winPct \in [0, 49].
WINS(source, period, winPct):
winCount = floor(period * winPct / 100)
if winCount >= period/2: winCount = (period-1)/2
// Collect and sort
vals = [source[0], source[1], ..., source[period-1]]
sort(vals, ascending)
// Replace tails with boundary values
lowerBound = vals[winCount]
upperBound = vals[period - 1 - winCount]
for i = 0 to winCount-1:
vals[i] = lowerBound
vals[period - 1 - i] = upperBound
// Average all values (full sample size)
return mean(vals)
Resources
- Tukey, J.W. "The Future of Data Analysis." Annals of Mathematical Statistics, 1962.
- Huber, P.J. & Ronchetti, E. "Robust Statistics." 2nd edition, Wiley, 2009.
- Wilcox, R.R. "Introduction to Robust Estimation and Hypothesis Testing." 4th edition, Academic Press, 2017.
- Fama, E.F. & French, K.R. "Common Risk Factors in the Returns on Stocks and Bonds." Journal of Financial Economics, 1993.
- Dixon, W.J. & Tukey, J.W. "Approximate Behavior of the Distribution of Winsorized t." Technometrics, 1968.