lmlib.statespace.cost.RLSAlssm#

class lmlib.statespace.cost.RLSAlssm(cost_model, **kwargs)#

Bases: lmlib.statespace.cost.RLSAlssmBase

Filter and Data container for Recursive Least Sqaure Alssm Filters

RLSAlssm computes and stores intermediate values such as covariances, as required to efficiently solve recursive least squares problems between a model-based cost function CompositeCost or CostSegment and given observations. The intermediate variables are observation dependant and therefore the memory consumption of RLSAlssm scales linearly with the observation vector length.

Main intermediate variables are the covariance W, weighted mean xi, signal energy kappa, weighted number of samples nu, see Equation (4.6) in [Wildhaber2019] PDF

Parameters

cost_model (CostSegment, CompositeCost) – Cost Model
**kwargs – Forwarded to RLSAlssmBase

Examples

Methods

`__init__`(cost_model, **kwargs)
`eval_errors`(xs[, ks])	Evaluation of the squared error for multiple state vectors xs.
`filter`(y[, v])	Computes the intermediate parameters for subsequent squared error computations and minimizations.
`filter_minimize_v`(y[, v, H, h])	Combination of `RLSAlssm.filter()` and `RLSAlssm.minimize_v()`.
`filter_minimize_x`(y[, v, H, h])	Combination of `RLSAlssm.filter()` and `RLSAlssm.minimize_x()`.
`minimize_v`([H, h, return_constrains])	Returns the vector v of the squared error minimization with linear constraints
`minimize_x`([H, h])	Returns the state vector x of the squared error minimization with linear constraints
`set_backend`(backend)	Setting the backend computations option

Attributes

`W`	Filter Parameter \(W\)
`betas`	Segment scalars weights the cost function per segment
`cost_model`	Cost Model
`kappa`	Filter Parameter \(\kappa\)
`nu`	Filter Parameter \(\nu\)
`xi`	Filter Parameter \(\xi\)

property W#

Filter Parameter \(W\)

Type: ndarray

property betas#

Segment scalars weights the cost function per segment

Type: ndarray

property cost_model#

Cost Model

Type: CostSegment, CompositeCost

eval_errors(xs, ks=None)#

Evaluation of the squared error for multiple state vectors xs.

The return value is the squared error

\[J(x) = x^{\mathsf{T}}W_kx -2*x^{\mathsf{T}}\xi_k + \kappa_k\]

for each state vector \(x\) from the list xs.

Parameters

xs (array_like of shape=(K, N)) – List of state vectors \(x\)
ks (None, array_like of int of shape=(XS,)) – List of indices where to evaluate the error

Returns

J – Squared Error for each state vector

Return type

np.ndarray of shape=(XS,)

K : number of samples
XS : number of state vectors in a list
N : ALSSM system order, corresponding to the number of state variables

filter(y, v=None)#

Computes the intermediate parameters for subsequent squared error computations and minimizations.

Computes the intermediate parameters using efficient forward- and backward recursions. The results are stored internally, ready to solve the least squares problem using e.g., minimize_x() or minimize_v(). The parameter allocation allocate() is called internally, so a manual pre-allcation is not necessary.

Parameters

y (array_like) –
Input signal
RLSAlssm or RLSAlssmSteadyState

Single-channel signal is of shape =(K,) for
Multi-channel signal is of shape =(K,L)

RLSAlssmSet or RLSAlssmSetSteadyState
Single-channel set signals is of shape =(K,S) for
Multi-channel set signals is of shape =(K,L,S)

Multi-channel-sets signal is of shape =(K,L,S)
v (array_like, shape=(K,), optional) – Sample weights. Weights the parameters for a time step k and is the same for all multi-channels. By default the sample weights are initialized to 1.

K : number of samples
L : output order / number of signal channels
S : number of signal sets

filter_minimize_v(y, v=None, H=None, h=None)#

Combination of RLSAlssm.filter() and RLSAlssm.minimize_v().

This method has the same output as calling the methods

rls.filter(y)
xs = rls.minimize_v()