API Reference

l2

L2

A class representing the L2-space of random variables defined on a given probability space.

basis `property`

basis

Get the vector space basis of the L2-space.

The basis vectors of the L2-space are the indicator functions of the atoms in the sigma algebra with nonzero probability, each scaled by the reciprocal of the square root of the atom's probability.

Parameters:

Name	Type	Description	Default
`tol`	`float`	The tolerance below which an atom is deemed to have probability 0.	`1e-8`

Returns:

Name	Type	Description
`basis`	`dict[str, RandomVariable]`	A dictionary mapping the name of each basis vector to the corresponding basis vector of the L2-space.

Examples:

>>> from sigalg.core import ProbabilityMeasure, SampleSpace, SigmaAlgebra
>>> from sigalg.l2 import L2
>>> Omega = SampleSpace().from_sequence(size=3)
>>> F = SigmaAlgebra(sample_space=Omega).from_dict({0: 0, 1: 0, 2: 1})
>>> # A probability measure assigning nonzero probability to all atoms
>>> P = ProbabilityMeasure(sample_space=Omega).from_dict({0: 0.2, 1: 0.5, 2: 0.3})
>>> H = L2(
...     sample_space=Omega,
...     sigma_algebra=F,
...     probability_measure=P,
... )
>>> V_0, V_1 = H.basis.values()
>>> V_0
Random variable 'V_0':
            V_0
sample
0       1.195229
1       1.195229
2       0.000000
>>> # A probability measure assigning zero probability to one atom
>>> Q = ProbabilityMeasure(sample_space=Omega).from_dict({0:0.2, 1:0.8, 2:0})
>>> G = L2(
...     sample_space=Omega,
...     sigma_algebra=F,
...     probability_measure=Q,
...     name="G"
... )
>>> G.basis
{'V_0': Random variable 'V_0':
        V_0
sample
0       1.0
1       1.0
2       0.0}

name `property` `writable`

name

The name of the L2-space.

Returns:

Name	Type	Description
`name`	`Hashable`	The name of the L2-space.

probability_measure `property`

probability_measure

The probability measure on which the L2-space is defined.

Returns:

Name	Type	Description
`probability_measure`	`ProbabilityMeasure`	The probability measure on which the L2-space is defined.

probability_space `property`

probability_space

The probability space on which the L2-space is defined.

Returns:

Name	Type	Description
`prob_space`	`ProbabilitySpace`	The probability space on which the L2-space is defined.

sample_space `property`

sample_space

The sample space on which the L2-space is defined.

Returns:

Name	Type	Description
`sample_space`	`SampleSpace`	The sample space on which the L2-space is defined.

sigma_algebra `property`

sigma_algebra

The sigma-algebra on which the L2-space is defined.

Returns:

Name	Type	Description
`sigma_algebra`	`SigmaAlgebra`	The sigma-algebra on which the L2-space is defined.

distance

distance(first, second)

Compute the distance between two random variables in the L2-space.

Parameters:

Name	Type	Description	Default
`first`	`RandomVariable`	The first random variable.	required
`second`	`RandomVariable`	The second random variable.	required

Raises:

Type	Description
`ValueError`	If one of the two random variables is not in the L2-space.

Returns:

Name	Type	Description
`distance`	`Real`	The distance between the two random variables in the L2-space.

fourier_coefficients

fourier_coefficients(rv)

Compute the Fourier coefficients of a random variable with respect to the basis of the L2-space.

Parameters:

Name	Type	Description	Default
`rv`	`RandomVariable`	The random variable whose Fourier coefficients are to be computed.	required

Raises:

Type	Description
`ValueError`	If `rv` is not in the L2-space.

Returns:

Name	Type	Description
`coefficients`	`dict[str, Real]`	A dictionary mapping the name of each basis vector to the corresponding Fourier coefficient of `rv` with respect to that basis vector.

Examples:

>>> from sigalg.core import ProbabilityMeasure, RandomVariable, SampleSpace, SigmaAlgebra
>>> from sigalg.l2 import L2
>>> Omega = SampleSpace().from_sequence(size=3)
>>> F = SigmaAlgebra(sample_space=Omega).from_dict({0: 0, 1: 0, 2: 1})
>>> P = ProbabilityMeasure(sample_space=Omega).from_dict({0: 0.2, 1: 0.5, 2: 0.3})
>>> H = L2(
...     sample_space=Omega,
...     sigma_algebra=F,
...     probability_measure=P,
... )
>>> # Get the Fourier coefficients of X with respect to the basis of H
>>> X = RandomVariable(domain=Omega, name="X").from_dict({0: 2, 1: 2, 2: 3})
>>> coeffs = H.fourier_coefficients(rv=X)
>>> coeffs
{'V_0': np.float64(1.6733200530681511), 'V_1': np.float64(1.6431676725154982)}
>>> # Reconstruct X from its Fourier coefficients and the basis of H
>>> sum(coeffs[basis_name] * basis_vec for basis_name, basis_vec in H.basis.items()).with_name("X_reconstructed")
Random variable 'X_reconstructed':
X_reconstructed
sample
0                   2.0
1                   2.0
2                   3.0
>>> # Define a new probability measure Q that assigns zero probability to an atom in the sigma algebra, and define a new L2-space
>>> Q = ProbabilityMeasure(sample_space=Omega).from_dict({0: 0.2, 1: 0.8, 2: 0.0})
>>> K = L2(
...     sample_space=Omega,
...     sigma_algebra=F,
...     probability_measure=Q,
... )
>>> # Compute the Fourier coefficients of X with respect to the basis of K, and note that there is only one coefficient
>>> K.fourier_coefficients(rv=X)
{'V_0': np.float64(2.0)}
>>> # Reconstruct X from its Fourier coefficients and the basis of K, and note that the reconstruction differs from the original X on a set of probability zero
>>> (2.0 * K.basis["V_0"]).with_name("X_reconstructed")
Random variable 'X_reconstructed':
X_reconstructed
sample
0                   2.0
1                   2.0
2                   0.0

inner

inner(first, second)

Compute the inner product of two random variables.

Both random variables must be in the L2-space.

Parameters:

Name	Type	Description	Default
`first`	`RandomVariable`	The first random variable.	required
`second`	`RandomVariable`	The second random variable.	required

Raises:

Type	Description
`ValueError`	If one of the random variables is not in the L2-space.

Returns:

Name	Type	Description
`inner_product`	`Real`	The inner product of the two random variables.

Examples:

>>> from sigalg.core import ProbabilityMeasure, RandomVariable, SampleSpace, SigmaAlgebra
>>> from sigalg.l2 import L2
>>> Omega = SampleSpace().from_sequence(size=3)
>>> F = SigmaAlgebra(sample_space=Omega).from_dict({0: 0, 1: 0, 2: 1})
>>> P = ProbabilityMeasure(sample_space=Omega).from_dict({0: 0.2, 1: 0.5, 2: 0.3})
>>> H = L2(
...     sample_space=Omega,
...     sigma_algebra=F,
...     probability_measure=P,
... )
>>> X = RandomVariable(domain=Omega, name="X").from_dict({0: 1, 1: 1, 2: 3})
>>> Y = RandomVariable(domain=Omega, name="Y").from_dict({0: 4, 1: 4, 2: 6})
>>> float(H.inner(X, Y))
8.2
>>> # Example of orthogonal RVs: two indicator functions of disjoint events
>>> A, B = F.to_atoms()
>>> I_A = RandomVariable.indicator_of(A)
>>> I_B = RandomVariable.indicator_of(B)
>>> float(H.inner(I_A, I_B))
0.0

integrate

integrate(rv)

Integrate a random variable with respect to the probability measure of the L2-space.

Parameters:

Name	Type	Description	Default
`rv`	`RandomVariable`	The random variable to be integrated.	required

Raises:

Type	Description
`ValueError`	If `rv` is not in the L2-space.

Returns:

Name	Type	Description
`integral`	`Real`	The integral of the random variable with respect to the probability measure of the L2-space.

Examples:

>>> from sigalg.core import ProbabilityMeasure, RandomVariable, SampleSpace, SigmaAlgebra
>>> from sigalg.l2 import L2
>>> Omega = SampleSpace().from_sequence(size=3)
>>> F = SigmaAlgebra(sample_space=Omega).from_dict({0: 0, 1: 0, 2: 1})
>>> P = ProbabilityMeasure(sample_space=Omega).from_dict({0: 0.2, 1: 0.5, 2: 0.3})
>>> H = L2(
...     sample_space=Omega,
...     sigma_algebra=F,
...     probability_measure=P,
... )
>>> X = RandomVariable(domain=Omega, name="X").from_dict({0: 1, 1: 1, 2: 3})
>>> float(round(H.integrate(X), 2))
1.6

norm

norm(X)

Compute the norm of a random variable in the L2-space.

Parameters:

Name	Type	Description	Default
`X`	`RandomVariable`	The random variable whose norm is to be computed.	required

Raises:

Type	Description
`ValueError`	The random variable must be in the L2-space.

Returns:

Name	Type	Description
`norm`	`Real`	The norm of the random variable in the L2-space.

Examples:

>>> from sigalg.core import ProbabilityMeasure, RandomVariable, SampleSpace, SigmaAlgebra
>>> from sigalg.l2 import L2
>>> Omega = SampleSpace().from_sequence(size=3)
>>> F = SigmaAlgebra(sample_space=Omega).from_dict({0: 0, 1: 0, 2: 1})
>>> P = ProbabilityMeasure(sample_space=Omega).from_dict({0: 0.2, 1: 0.5, 2: 0.3})
>>> H = L2(
...     sample_space=Omega,
...     sigma_algebra=F,
...     probability_measure=P,
... )
>>> A, _ = F.to_atoms()
>>> I_A = RandomVariable.indicator_of(A)
>>> # The squared norm of an indicator function is the probability of the corresponding event
>>> float(round(H.norm(I_A) ** 2, 1))
0.7

API Reference

l2

L2

basis property

name property writable

probability_measure property

probability_space property

sample_space property

sigma_algebra property

distance

fourier_coefficients

inner

integrate

norm

basis `property`

name `property` `writable`

probability_measure `property`

probability_space `property`

sample_space `property`

sigma_algebra `property`