Quadratic Least Squares Moving Average - Smoothing + Forecast

The Quadratic Least Squares Moving Average indicator provides a high-reactivity smoothing filter and forecasting tool based on second-degree polynomial regression. By fitting a parabolic curve to price data rather than a linear one, it offers reduced lag and better alignment with complex price movements compared to standard moving averages.

Usage

The Usage section describes how the script can be used to analyze market trends and anticipate potential price targets.

The primary application of this tool is to identify the underlying trend with minimal lag. Because it uses a quadratic formula ($ax^2 + bx + c$), the indicator curves more aggressively to follow price action than a standard Linear Regression (LSMA). This makes it effective for identifying reversals earlier, though users should be aware that the increased reactivity can lead to greater overshoots or undershoots during high volatility.

Smoothing and Trend Identification

Traders can use the main QLSMA line to determine trend direction. When the price is above the QLSMA, the short-term momentum is considered bullish; when below, it is considered bearish. The script includes built-in alerts for price crossovers to assist with timing.

Forecasting

The forecast feature projects the current quadratic curve into the future by a user-defined number of bars. This can be used to visualize potential price trajectories if the current momentum persists. To evaluate the accuracy of these projections, the "Show past predicted values" setting allows users to see how previous forecasts aligned with subsequent price action.

Details

The script implements a rolling quadratic regression model. While a standard LSMA assumes a linear relationship ($ax + b$), the QLSMA models the relationship between the closing price and a linear sequence using a second-degree polynomial.

The coefficients ($a$, $b$, and $c$) are calculated using variance and covariance estimations rather than complex matrix calculus, ensuring the script remains efficient. Specifically:

• Variance of Index: Measures the spread of the time-based independent variable.
• Covariance: Measures the relationship between the price and the time index (and the squared index).
• Intercept (c): Aligns the curve with the average price level within the lookback window.

Mathematically, the forecast is calculated by extending the quadratic formula to $n + \text{forecast}$, where $n$ is the current bar index. This assumes the parabolic trend will continue, which is most effective for short-term projections.

Settings

General Settings

• Length: Determines the lookback period used to calculate the quadratic regression. Higher values result in smoother lines, while lower values increase reactivity.
• Source: The price data used for the calculation (e.g., Close, Open, HL2).
• Forecast: The number of bars to project the curve into the future.
• Show Past Predicted Values: When enabled, the forecast line remains visible on historical bars, allowing for backtesting of the projection's accuracy. When disabled, only the current forecast is projected into the future.

FAQ

How do I use the forecast for trading?

The forecast is intended for educational and visualization purposes to show the current trajectory of the curve. It should not be used as a standalone prediction tool but rather as a way to see where current momentum is mathematically heading.

Why does this indicator overshoot the price more than a regular SMA?

Because the QLSMA is a low-lag filter of a higher degree, it attempts to fit the data points as closely as possible. This mathematical "tightness" causes the line to swing wider than the price during sharp turns, a characteristic of polynomial regression.

How can I access this indicator?

You can get access on the LuxAlgo Library for charting platforms like TradingView, MetaTrader (MT4/MT5), and NinjaTrader for free.