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.
4.5 KiB
ILRS: Integral of Linear Regression Slope
"John Ehlers took the slope of a regression line, integrated it, and got a smoother trend follower. Differentiate to find direction, integrate to find position. Calculus: still useful after 300 years."
ILRS computes the linear regression slope over a rolling window, then accumulates it via discrete integration (running sum) to reconstruct a smoothed price-level signal. By differentiating (slope extraction) and reintegrating, ILRS acts as a low-pass filter that preserves trend direction while suppressing high-frequency noise more aggressively than LSMA. The integration step introduces a natural momentum quality: the output continues rising even as slope magnitude diminishes, making ILRS particularly effective for trend-following systems that need early exit signals based on slope deceleration.
Historical Context
John Ehlers introduced ILRS in Rocket Science for Traders (Wiley, 2001) as part of his signal-processing approach to technical analysis. Ehlers recognized that most moving averages are essentially low-pass filters applied directly to price, but the differentiate-then-integrate approach offers a different noise profile. The linear regression slope acts as a first-derivative estimator, and the running sum reconstructs the original signal minus the high-frequency components that the regression window cannot track.
The concept has deep roots in control theory and signal processing. The "differentiate and integrate" technique is standard in PID controllers and phase-locked loops, where it provides better noise rejection than direct filtering when the signal's derivative is smoother than the signal itself. In financial time series, this condition holds when price changes are more persistent than price levels, a reasonable assumption during trending regimes.
ILRS differs from LSMA (which evaluates the regression line at the endpoint) in a crucial way: LSMA's output is bounded by the regression window, while ILRS accumulates indefinitely. This makes ILRS a non-stationary filter whose output drifts with the integrated slope, requiring periodic resynchronization to avoid floating-point drift over very long series.
Architecture & Physics
1. Rolling Linear Regression Slope
The slope is computed via the standard least-squares formula over the circular buffer:
\text{slope} = \frac{N \sum x_i y_i - \sum x_i \sum y_i}{N \sum x_i^2 - \left(\sum x_i\right)^2}
The x-index sums (\sum x, \sum x^2) are computed analytically (Faulhaber's formulas), reducing the per-bar cost to a single O(N) pass for the y-dependent sums.
2. Discrete Integration
The integral is a simple running sum:
\text{ILRS}_t = \text{ILRS}_{t-1} + \text{slope}_t
This is O(1) per bar after the slope is computed.
3. Initialization
The integral is initialized to the first price value, ensuring the output starts at a reasonable level rather than zero.
4. Drift Management
Because ILRS accumulates slope indefinitely, floating-point precision degrades over millions of bars. A periodic resynchronization (e.g., every 1000 bars, re-anchor to slope-implied price) prevents meaningful drift.
Mathematical Foundation
Given a window of N prices y_0, y_1, \ldots, y_{N-1} (oldest to newest), the regression slope is:
b = \frac{N \sum_{i=0}^{N-1} i \cdot y_i - \left(\sum_{i=0}^{N-1} i\right)\left(\sum_{i=0}^{N-1} y_i\right)}{N \sum_{i=0}^{N-1} i^2 - \left(\sum_{i=0}^{N-1} i\right)^2}
With analytical x-sums:
\sum i = \frac{N(N-1)}{2}, \quad \sum i^2 = \frac{N(N-1)(2N-1)}{6}
The ILRS output:
\text{ILRS}_t = \text{ILRS}_{t-1} + b_t, \quad \text{ILRS}_0 = y_0
Default parameters: period = 14, minPeriod = 2.
Pseudo-code (streaming):
buffer ← circular_buffer(period)
buffer.push(price)
n ← min(bar_count, period)
if n < 2:
integral ← price
return integral
// Analytical x-sums
sumX = n*(n-1)/2
sumX2 = n*(n-1)*(2n-1)/6
// Data-dependent y-sums (O(n) pass)
sumY = 0; sumXY = 0
for i = 0 to n-1:
sumY += buffer[i]
sumXY += i * buffer[i]
denomX = n * sumX2 - sumX * sumX
slope = (n * sumXY - sumX * sumY) / denomX
integral += slope
return integral
Resources
- Ehlers, J.F. (2001). Rocket Science for Traders: Digital Signal Processing Applications. John Wiley & Sons.
- Ehlers, J.F. (2004). Cybernetic Analysis for Stocks and Futures. John Wiley & Sons.
- Kendall, M.G. & Stuart, A. (1979). The Advanced Theory of Statistics, Vol. 2. Griffin. Chapter 29: Regression.