recursive least square algorithm

Daniel Parker

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18-03-2025

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The Recursive Least Squares (RLS) algorithm is a powerful tool for estimating the parameters of a linear model online. Unlike batch least squares, which requires processing all data at once, RLS updates its estimate incrementally with each new data point. This adaptability makes it ideal for applications with streaming data or systems that change over time. This article will explore the RLS algorithm in detail, explaining its mechanics, advantages, and applications.

Understanding the Core Concepts

At its heart, the RLS algorithm aims to minimize the sum of squared errors between predicted and actual values. This is done iteratively, updating the parameter estimates with each new observation. The key is its ability to efficiently update the estimate without re-calculating everything from scratch. This efficiency is achieved through matrix manipulations.

The Linear Model

We assume a linear relationship between the input vector x and the output y:

y = x<sup>T</sup>θ + ε

Where:

• y is the observed output (scalar).
• x is the input vector (column vector).
• θ is the parameter vector we want to estimate (column vector).
• ε is the error term (noise).

The goal is to find the optimal θ that best fits the data.

The Recursive Update

The magic of RLS lies in its recursive update equations. Instead of solving a large system of equations for every new data point, it updates the parameter estimate iteratively using the following formulas:

• Gain Vector Calculation: k<sub>t</sub> = P<sub>t-1</sub>x<sub>t</sub> / (λ + x<sub>t</sub><sup>T</sup>P<sub>t-1</sub>x<sub>t</sub>)

• Parameter Update: θ<sub>t</sub> = θ<sub>t-1</sub> + k<sub>t</sub>(y<sub>t</sub> - x<sub>t</sub><sup>T</sup>θ<sub>t-1</sub>)

• Covariance Matrix Update: P<sub>t</sub> = (1/λ)[P<sub>t-1</sub> - k<sub>t</sub>x<sub>t</sub><sup>T</sup>P<sub>t-1</sub>]

Where:

• t is the time index (iteration).
• k<sub>t</sub> is the Kalman gain vector.
• P<sub>t</sub> is the estimated error covariance matrix.
• λ is the forgetting factor (0 < λ ≤ 1).

The forgetting factor, λ, is crucial. It controls how much weight is given to past data. A smaller λ gives more weight to recent data, making the algorithm more responsive to changes in the system. A λ of 1 gives equal weight to all data.

Advantages of the RLS Algorithm

• Online Adaptability: RLS processes data sequentially, making it ideal for real-time applications and situations with streaming data.

• Computational Efficiency: The recursive updates are computationally efficient, especially compared to batch least squares for large datasets.

• Tracking Time-Varying Systems: The forgetting factor allows RLS to adapt to changes in the system over time.

Disadvantages of the RLS Algorithm

• Sensitivity to Noise: The algorithm can be sensitive to noisy data, especially if the forgetting factor is small.

• Computational Complexity: While more efficient than batch methods for large datasets, each update still involves matrix operations, which can be computationally demanding for high-dimensional input vectors.

• Forgetting Factor Tuning: Choosing the appropriate forgetting factor can be challenging and often requires experimentation.

Applications of the RLS Algorithm

RLS finds applications in various fields, including:

• System Identification: Estimating the parameters of dynamic systems.

• Adaptive Filtering: Adapting to changing signal characteristics.

• Control Systems: Designing controllers that adapt to changing plant dynamics.

• Signal Processing: Estimating signal parameters in noisy environments.

Implementing RLS

Numerous libraries provide implementations of the RLS algorithm. For example, in Python, you can find implementations in libraries like NumPy and SciPy. However, careful consideration must be given to numerical stability when implementing this algorithm; proper handling of matrix inversions is critical.

Conclusion

The Recursive Least Squares algorithm offers a powerful and efficient approach to parameter estimation in linear models, particularly in situations where data arrives sequentially or the system's dynamics change over time. Understanding its mechanics, advantages, and limitations is essential for successfully applying it in various engineering and scientific domains. Remember that careful selection of the forgetting factor is vital for optimal performance.