Recursive Least Squares Filter¶
A sequential linear-model estimator that maintains least-squares matrix state and uses each new prediction error to update its coefficients.
Core Idea¶
A recursive least squares (RLS) filter updates a linear model's coefficients after each input–response observation while maintaining matrix information from earlier observations. The new prediction error is multiplied by a gain shaped by that matrix and input, and both coefficients and matrix state are corrected. Under declared initialization and ideal arithmetic, conventional updates follow a cumulative weighted squared-error objective without re-solving the entire past batch.[rls][online]
Scope of Application¶
An adaptive FIR filter can estimate unknown taps from input windows and observed output; an online valve model can update gain and offset in its above-dead-band line-fit segment, not across its entire piecewise response. In both, a sequential linear residual, maintained least-squares state and gain-weighted correction fill the same roles. Forgetting factors below one can help track changing parameters, but \(\lambda=1\) is also a valid RLS setting.[ident][valve][^online]
Clarity¶
RLS is not every recursive parameter update. Batch least squares lacks the sequential matrix update, LMS uses a gradient-style correction, and generic Bayesian filtering updates a posterior under a different model. “Exact” describes the specified algebraic objective under assumptions, not guaranteed finite-precision results or universally superior convergence.[rls][online]
Manages Complexity¶
To recognize RLS, identify the linear prediction and new residual, the persistent inverse-correlation or equivalent factorized state, and the data-dependent gain correcting both coefficients and state. Only then examine tuning choices such as forgetting and numerical realization. This separates identity from tracking and speed claims.[rls][online]
Abstract Reasoning¶
For real regressors, a common objective is \(J_t(w)=\sum_{i=1}^{t}\lambda^{t-i}(d_i-u_i^Tw)^2\) plus any declared initialization term, \(0<\lambda\leq1\). A conventional gain uses the prior matrix \(P_{t-1}\) and new regressor: \(k_t=P_{t-1}u_t/(\lambda+u_t^TP_{t-1}u_t)\). The coefficient update is \(w_t=w_{t-1}+k_t(d_t-u_t^Tw_{t-1})\), with a coupled matrix update. The gain therefore reflects accumulated input geometry, not one fixed learning rate.[rls][online]
Knowledge Transfer¶
The regressor can be a FIR sample window or \([u(t),1]\) for the active line-fit segment of a valve response. These applications share the least-squares update relation but not a physical interpretation. Similar-looking Kalman gains or LMS error steps require their own objective and assumptions before they are treated as equivalent.[ident][valve][^online]
[^rls]: MathWorks, “dsp.RLSFilter”, Algorithms and initialization, directly checked. [^online]: MathWorks, “Recursive Algorithms for Online Parameter Estimation”, forgetting-factor and comparison equations, directly checked. [^ident]: MathWorks, “System Identification Using RLS Adaptive Filtering”, FIR worked example, directly checked. [^valve]: MathWorks, “Line Fitting with Online Recursive Least Squares Estimation”, valve model, directly checked.
Neighborhood in Abstraction Space¶
Recursive Least Squares Filter sits in a sparse region of the domain-specific corpus (78th percentile for distinctiveness): few abstractions share its structure, so a faithful description tends to retrieve it precisely.
Family — Statistical Learning & Model Failure Modes (41 abstractions)
Nearest neighbors
- Structural Risk Minimization — 0.83
- Particle Filter — 0.83
- Lifting Scheme — 0.83
- Leabra — 0.82
- FWL theorem — 0.82
Computed from structural-signature embeddings · 2026-10-08