Gramian Matrix

The Gramian

API Reference

rolch.init_forget_vector

init_forget_vector(forget: float, size: int) -> np.ndarray

Initialise an exponentially discounted vector of weights.

Recursively initialise a vector of exponentially discounted weights of size N.

The weight for \(n\)-th observation is defined as \((1 - \text{forget})^{(N - n)}\)

Note that this functions assumes that the first observation is the oldest observation and the last observation is the newest observation. This is in line with the standard pandas way of sorting pd.DataFrames with Datetime-indices.

Numba

This function uses numba just-in-time-compilation.

Parameters:

Name	Type	Description	Default
forget	float	Forget factor.	required
size	int	Length of the output vector.	required

Returns:

Type	Description
ndarray	np.ndarray: Vector of exponentially discounted weights.

rolch.init_gram

init_gram(X: np.ndarray, w: np.ndarray, forget: float = 0) -> np.ndarray

Initialise the Gramian Matrix.

The Gramian Matrix is defined as $$ G = X^T \Gamma WX $$ where \(X\) is the design matrix, \(W\) is a diagonal, user-defined weight matrix, \(\Gamma\) is a diagonal matrix of exponentially discounting weights.

Numba

This function uses numba just-in-time-compilation.

Parameters:

Name	Type	Description	Default
X	ndarray	Design matrix \(X\)	required
w	ndarray	Weights vector	required
forget	float	Forget factor. Defaults to 0.	0

Returns:

Type	Description
ndarray	np.ndarray: Gramian Matrix.

rolch.init_y_gram

init_y_gram(X: np.ndarray, y: np.ndarray, w: np.ndarray, forget: float = 0) -> np.ndarray

Initialise the y-Gramian Matrix.

The Gramian Matrix is defined as $$ H = X^T \Gamma WY $$ where \(X\) is the design matrix, \(Y\) is the response variable, \(W\) is a diagonal, user-defined weight matrix, \(\Gamma\) is a diagonal matrix of exponentially discounting weights.

Numba

This function uses numba just-in-time-compilation.

Parameters:

Name	Type	Description	Default
X	ndarray	Design matrix \(X\)	required
y	ndarray	Response variable \(Y\)	required
w	ndarray	Weights vector	required
forget	float	Forget factor. Defaults to 0.	0

Returns:

Type	Description
ndarray	np.ndarray: y-Gramian Matrix.

rolch.init_inverted_gram

init_inverted_gram(X: np.ndarray, w: np.ndarray, forget: float = 0) -> np.ndarray

Initialise the inverted Gramian Matrix.

The inverted Gramian Matrix is defined as $$ G = (X^T \Gamma WX)^{-1} $$ where \(X\) is the design matrix, \(W\) is a diagonal, user-defined weight matrix, \(\Gamma\) is a diagonal matrix of exponentially discounting weights.

Numba

This function uses numba just-in-time-compilation.

Parameters:

Name	Type	Description	Default
X	ndarray	Design matrix \(X\)	required
w	ndarray	Weights vector	required
forget	float	Forget factor. Defaults to 0.	0

Returns:

Type	Description
ndarray	np.ndarray: Gramian Matrix.

Raises:

Type	Description
ValueError	If the matrix is not invertible (if rank(X.T @ X) < X.shape[0]).

rolch.update_gram

update_gram(gram: np.ndarray, X: np.ndarray, forget: float = 0, w: float = 1) -> np.ndarray

Update the Gramian Matrix.

Numba

This function uses numba just-in-time-compilation.

Parameters:

Name	Type	Description	Default
gram	ndarray	Gramian Matrix	required
X	ndarray	New observations for \(X\)	required
forget	float	Forget factor. Defaults to 0.	0
w	float	Weights for the new observations. Defaults to 1.	1

Returns:

Type	Description
ndarray	np.ndarray: Updated Gramian Matrix.

rolch.update_y_gram

update_y_gram(gram: np.ndarray, X: np.ndarray, y: np.ndarray, forget: float = 0, w: float = 1) -> np.ndarray

Update the Y-Gramian Matrix.

Numba

This function uses numba just-in-time-compilation.

Parameters:

Name	Type	Description	Default
gram	ndarray	Gramian Matrix	required
X	ndarray	New Observations for \(X\)	required
y	ndarray	New Observations for \(Y\)	required
forget	float	Forget Factor. Defaults to 0.	0
w	float	Weights for the new observations. Defaults to 1.	1

Returns:

Type	Description
ndarray	np.ndarray: Updated Gramian Matrix.

rolch.update_inverted_gram

update_inverted_gram(gram: np.ndarray, X: np.ndarray, forget: float = 0, w: float = 1) -> np.ndarray

Update the inverted Gramian Matrix.

Numba

This function uses numba just-in-time-compilation.

Parameters:

Name	Type	Description	Default
gram	ndarray	Inverted Gramian Matrix.	required
X	ndarray	New observations for \(X\).	required
forget	float	Forget Factor. Defaults to 0.	0
w	float	Weights for new observations. Defaults to 1.	1

Returns:

Type	Description
ndarray	np.ndarray: Updated inverted Gramian matrix.