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.
3.9 KiB
FOSC: Forecast Oscillator
The Forecast Oscillator measures the percentage deviation of the current price from its linear regression forecast value, producing a zero-centered oscillator that quantifies how far price has moved beyond what a least-squares trend projection would predict. Positive values indicate price is above the regression forecast (bullish divergence from trend), negative values indicate price is below (bearish divergence), and zero crossings mark the points where price meets its statistically expected value. The implementation achieves O(1) per-bar complexity through incremental running sums for the linear regression calculation.
Historical Context
The Forecast Oscillator was introduced by Tushar Chande in his exploration of regression-based indicators during the 1990s, published in Technical Analysis of Stocks & Commodities magazine and later in his book on technical analysis. The indicator builds on the Time Series Forecast (TSF) concept but reframes it as an oscillator by expressing the relationship as a percentage deviation rather than an absolute price level. This normalization makes FOSC comparable across instruments of different price scales. Chande positioned FOSC as a complementary tool to his other regression-based indicators (R-Squared, Linear Regression Slope), where R-Squared measures trend quality, slope measures trend direction, and FOSC measures the current price's position relative to trend extrapolation. The percentage formulation also makes FOSC functionally similar to a detrended price series, connecting it to the broader family of detrending oscillators used in cycle analysis.
Architecture & Physics
Linear Regression via Running Sums
FOSC requires computing a linear regression forecast at each bar. The standard OLS formula needs \Sigma x, \Sigma x^2, \Sigma y, and \Sigma xy. Since the x values are sequential integers, \Sigma x and \Sigma x^2 are closed-form functions of n. Only \Sigma y and \Sigma xy require incremental maintenance via circular buffer.
The key optimization for \Sigma xy is identical to CTI: when the window slides, subtracting \Sigma y (before removal) shifts all position indices down by one, and adding (n-1) \times y_{\text{new}} places the new value at the highest position. This avoids recomputing n products per bar.
Forecast Point
The linear regression yields slope m and intercept b. The forecast value is evaluated at the endpoint of the window: \hat{y} = m \cdot (n-1) + b. This represents the trend-projected value for the current bar.
Percentage Deviation
The oscillator output is \frac{x_t - \hat{y}}{x_t} \times 100, which normalizes the deviation by the current price. Division by zero is guarded when source equals zero.
Mathematical Foundation
Given source values y_t over a window of n observations with indices x_i = 0, 1, \ldots, n-1:
Closed-form time sums:
\Sigma_x = \frac{n(n-1)}{2}, \quad \Sigma_{x^2} = \frac{n(n-1)(2n-1)}{6}
Running sums (O(1) incremental):
\Sigma_y = \sum y_i, \quad \Sigma_{xy} = \sum i \cdot y_i
OLS linear regression:
m = \frac{n \cdot \Sigma_{xy} - \Sigma_x \cdot \Sigma_y}{n \cdot \Sigma_{x^2} - \Sigma_x^2}, \quad b = \frac{\Sigma_y - m \cdot \Sigma_x}{n}
Forecast at endpoint:
\hat{y}_t = m \cdot (n - 1) + b
Forecast Oscillator:
FOSC_t = \frac{y_t - \hat{y}_t}{y_t} \times 100
Streaming update pseudo-code:
// When buffer full, oldest y_old exits:
Σy -= y_old; Σxy -= Σy_before
Σy += y_new; Σxy += (n-1) × y_new
m = (n×Σxy - Σx×Σy) / (n×Σx² - Σx²)
b = (Σy - m×Σx) / n
forecast = m×(n-1) + b
FOSC = (y_new ≠ 0) ? (y_new - forecast) / y_new × 100 : 0
Default parameters: period = 14.
Resources
- Chande, T.S. (1997). Beyond Technical Analysis. Wiley
- Chande, T.S. & Kroll, S. (1994). The New Technical Trader. Wiley
- PineScript reference:
fosc.pine