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RLS - Recursive least squares in C++

part of QDaq (https://gitlab.com/qdaq/qdaq) - Qt-based Data Acquisition

About

Recursive least squares (RLS) refers to algorithms that recursively find the coefficients that minimize a weighted linear least squares cost function.

It is assumed that a measured time-dependent quantity $y(t)$ follows a linear model plus noise

$$ y(t) = \theta^T \cdot \phi(t) + e(t) $$

where $\theta$ is a parameter vector, $\phi(t)$ is a vector of known functions of time and $e(t)$ is white noise.

A least squares cost function is defined as

$$ J(\theta) = \sum_t {w(t) \left[ y(t) - \theta^T \cdot \phi(t) \right]^2} $$

where $w(t)$ is a weighting function.

RLS algorithms seek to estimate $\theta$ that minimizes $J$ and recursively refine this estimation as new data become available. This is acheived with a very elegant numerical technique which is given here very briefly.

The solution of the least squares problem (for $w(t)=1$) is given by the normal equations

$$ \left[ \Phi^T \cdot \Phi \right] \theta = A\cdot \theta = \Phi^T\cdot Y \quad \rightarrow \quad \theta = A^{-1}\cdot \Phi^T\cdot Y$$

where $\Phi = [\phi(1) ; \phi(2) \dots ]^T$, $Y = [y(1) ; y(2) \dots ]^T$

When a new data point becomes available the matrix $A$ is modified as

$$ A' = A + \phi \cdot \phi^T $$

The inverse of $A'$, $P'=A'^{-1}$ can be obtained from $P=A^{-1}$ using the Wilkinson matrix formula:

$$ P' = (A + \phi \cdot \phi^T)^{-1} = P - \frac{P\cdot\phi\cdot\phi^T\cdot P}{1 + \phi^T\cdot P\cdot \phi} $$

Thus, in RLS algorithms we keep a copy of $P$ which is updated by the above formula. Note that $P$ is the covariance matrix of $\theta$.

The parameters are updated by

$$ e(t) = y(t) - \theta^T(t-1)\cdot \phi(t) $$

$$ \theta(t) = \theta(t-1) + k, e(t) $$

where $k$ is the "gain" vector, which is defined below.

Examples

  1. A system with input $u(t)$ and output $y(t)=s\cdot u(t) + y_0$, where $s$ and $y_0$ vary with time. RLS is used to obtain an estimate of $s$ and $y_0$. Example 1

  2. A timeseries $y(t)$ is fitted with a polynomial $f(t;\theta)=\theta_0 + \theta_1\cdot t$ to estimate the time-dependent parameters $(\theta_0,\theta_1)$ and thus obtain the derivative $dy/dt$

Example 2

Implemented Algorithms

1. Exponentially weighted RLS

In this algorithm the weighting function is

$$ w(t) = \lambda ^ {n-t}, \quad t\leq n$$

where $0 < \lambda \leq 1$ is the "forgetting factor". The data are exponentially weighted so that older data are "forgotten". The effective forgetting time constant $\tau$ can be found from

$$ \lambda = e^{-1/\tau} \approx 1 - 1/\tau $$

The recursion is given by

$$ e(t) = y(t) - \theta^T(t-1)\cdot \phi(t) $$ $$ u = P(t-1)\cdot \phi(t) $$ $$ k = u / \left[ \lambda + \phi^T(t)\cdot u\right] $$ $$ \theta(t) = \theta(t-1) + k, e(t) $$ $$ P(t) = \left[P(t-1) - k \cdot u^T\right]/\lambda $$ $$ J(t) = \lambda J(t-1) + e^2(t) $$

The algorithm is still valid for $\lambda=1$ but with infinite memory length ($\tau \to \infty$)

2. Exponentially weighted RLS with square-root algorithm

The square root algorithm (initially due to Potter (1963)) utilizes the fact that $A$ and $P$ are symmetric, positive definite matrices. This can be understood e.g. by the fact that $J(\theta)$ close to the minimum can be expanded to 2nd order in $\theta - \theta_0$

$$ J = J_0 + (\theta - \theta_0)^{T} P^{-1} (\theta - \theta_0) $$

thus the matrix $A=P^{-1}$ (and $P$) must be positive definite so that there is a minimum.

Such matrices can be written as $P=Q\cdot Q^T$ where $Q$ is called the square root of $P$ (thus the name of the algorithm). The covariance update can be turned into an update of $Q$:

$$ P' = Q'\cdot Q'^T = Q\left[ I - \beta u\cdot u^T \right] Q^T$$

where $u=Q^T \cdot\phi$ and $\beta = (1 + u^T\cdot u )^{-1}$.

It can be easily shown that

$$ \left[ I - \beta u\cdot u^T \right] = \left[ I - \alpha u\cdot u^T \right]^2 $$

with $\alpha = \beta / \left[ 1 + \sqrt{1 - \beta u^T\cdot u}\right]$. Finally

$$ Q' = Q \left[ I - \alpha u\cdot u^T \right]$$

The benefit of $Q$ updating is that it ensures that $P$ retains its positive-definetness.

The RLS with recursion for $Q$ becomes

$$ e(t) = y(t) - \theta^T(t-1)\cdot \phi(t) $$

$$ u = Q^T(t-1) \cdot \phi(t) $$

$$ \beta = \lambda + u^T \cdot u $$

$$ \alpha = 1 / \left[\beta + \sqrt{\beta,\lambda}\right] $$

$$ k = Q(t-1) \cdot u $$

$$ \theta(t) = \theta(t-1) + k , [e(t)/\beta] $$

$$ Q(t) = \left[Q(t-1) - \alpha , k \cdot u^T\right]/\sqrt{\lambda} $$

$$ J(t) = \lambda J(t-1) + e^2(t) $$

This algorithm is taken from Ljung & Soederstroem (1987) "Theory & Practice of Recursive Identification", p. 328

3. Block RLS

The sliding-block RLS uses the last $N$ points to estimate the parameters. In each recursion the estimate is first updated with the new data point and then downdated by removing the oldest point.

The updating sequence is given by

$$ e(t) = y(t) - \theta^T(t-1)\cdot \phi(t) $$

$$ u = P(t-1)\cdot \phi(t) $$

$$ \beta = \left[ 1 + \phi(t)^T \cdot u \right]^{-1} $$

$$ k = \beta , u $$

$$ \bar{\theta}(t) = \theta(t-1) + k, e(t) $$

$$ \bar{P}(t) = \left[P(t-1) - k \cdot u^T \right] $$

$$ e'(t) = y(t) - \theta(t)^T \cdot \phi(t) $$

$$ \bar{J}(t) = J(t-1) + e^2(t) , \beta,(1-\beta) + e'^2(t) $$

And the down-dating sequence is

$$ e(t-N) = y(t-N) - \bar{\theta}^T(t)\cdot \phi(t-N) $$

$$ u = \bar{P}(t)\cdot \phi(t-N) $$

$$ \beta = \left[ 1 - \phi(t-N)^T \cdot u \right]^{-1} $$

$$ k = \beta , u $$

$$ \theta(t) = \bar{\theta}(t) - k , e(t-N) $$

$$ P(t) = \left[\bar{P}(t) + k \cdot u^T\right] $$

$$ e'(t-N) = y(t) - \theta(t)^T\cdot \phi(t-N) $$

$$ J(t) = \bar{J}(t) - e^2(t-N) , \beta,(1-\beta) - e'^2(t-N) $$

4. Block RLS with square root update

Similar to the above, the block RLS algorithm can be formulated with update of the square root of $P$.

Polynomial fitting of timeseries

A time series $y(t)$ can be fitted to a polynomial $f(t;\theta)=\sum_n \theta_n t^n$ using RLS. This is usefull for on-line monitoring to obtain, e.g., the rate of change $dy/dt$ of a signal.

However, during long term operation of a polynomial RLS algorithm $t$ becomes very large and the numerical calculation of covariance update fails.

A solution to this problem is to operate sychronously 2 RLS loops with a phase difference of $\Delta T$. When the operation time of the 1st loop reaches $2\Delta T$ the loop is reset and the output switches to the 2nd loop. This is repeated periodically every $\Delta T$ interval and the two loops are interchanged. Thus $t$ in each loop is bounded to $0\leq t \leq \Delta T$ and numerical failure of the algorithm due to large $t$ values is prevented.

In sliding-window block RLS $\Delta T$ is equal to the block size. For exponentially weighted RLS $\Delta T$ is a multiple of the forgetting time constant, i.e. $\Delta T = n/(1-\lambda)$. Thus, when we change from loop 1 to loop 2 we essentially delete older points which were scaled by $w(t-\Delta t) = e^{-n}$ or lower. A value of $n\sim 10$ is recommended.

This algorithm is implemented in the RLS::PolyRLS C++ class which is defined in PolyRLS.h.

Getting started

The algorithms are organized in a number of templated objects which are defined in 2 header files:

  • RLS.h defines the RLS algorithms
  • PolyRLS.h defines the PolyRLS object for fitting timeseries with polynomials

In the filter folder there are 2 command line applications which can be used for testing and show most of the library features. More details can be found in the relevant README.

The project can be built with cmake using the following commands:

> mkdir .build
> cd .build
> cmake ..
> make / nmake

TODO

  • Implement the $U D U^T$ factorization of covariance matrix and the updating algorithm by Bierman (1977)
  • Document the C++ classes

References

Contributors

  • @nickspanos
    • Initial implementation of the algorithms
  • @MatinaDt
    • Implemented the use of Eigen library
    • Tested Cholesky decomposition methods