projnormal.formulas.projected_normal_Bc

Formulas for the distribution of \(y=x/\sqrt{x^T B x + c}\).

empirical_moments(mean_x, covariance_x, const, n_samples, B=None, B_chol=None)

Compute the mean, covariance and second moment of the variable \(y = x/\sqrt{x^T B x + c}\) where \(x \sim \mathcal{N}(\mu_x, \Sigma_x)\), \(B\) is a symmetric positive definite matrix, and \(c\) is a positive constant added to the denominator, by sampling from the distribution.

Parameters:

mean_x (torch.Tensor) – Mean of x. Shape is (n_dim,).
covariance_x (torch.Tensor) – Covariance of x. Shape is (n_dim, n_dim).
n_samples (int) – Number of samples to draw.
const (torch.Tensor) – Constant added to the denominator. Shape is ().
B (torch.Tensor, optional) – Matrix B used in the denominator of the projection. If not provided, the identity matrix is used. Shape is (n_dim, n_dim).
B_chol (torch.Tensor, optional) – Cholesky decomposition matrix L, such that B = LL’. Can be provided to avoid recomputing it. Shape is (n_dim, n_dim).

Returns:

Dictionary with the keys mean, covariance, and second_moment, containing the empirical moments of the projected normal distribution.

Return type:

dict

log_pdf(mean_x, covariance_x, y, const, B=None, B_chol=None)

Compute the log-pdf at points y for the distribution of the variable \(y = x/\sqrt{x^T B x + c}\), where \(x \sim \mathcal{N}(\mu_x, \Sigma_x)\), \(B\) is a symmetric positive definite matrix and \(c\) is a positive constant. (\(y\) has a projected normal distribution.).

Parameters:

mean_x (torch.Tensor) – Mean of x. Shape is (n_dim,).
covariance_x (torch.Tensor) – Covariance of x. Shape is (n_dim, n_dim).
y (torch.Tensor) – Points where to evaluate the PDF. Shape is (n_points, n_dim).
const (torch.Tensor) – Constant added to the denominator. Must be positive. Shape is ().
B (torch.Tensor, optional) – Matrix B used in the denominator of the projection. If not provided, the identity matrix is used. Shape is (n_dim, n_dim).
B_chol (torch.Tensor, optional) – Cholesky decomposition of B. Can be provided to avoid recomputing it. Shape is (n_dim, n_dim).

Returns:

Log-PDF evaluated at each y. Shape is (n_points,).

Return type:

torch.Tensor

mean(mean_x, covariance_x, const, B=None, B_chol=None)

Compute the mean of \(y = x/\sqrt{x^T B x + c}\), where \(x \sim \mathcal{N}(\mu_x, \Sigma_x)\), \(B\) is a symmetric positive definite matrix and \(c\) is a positive constant. Uses a Taylor approximation. (\(y\) is distributed on the ellipse defined by \(B\).).

Parameters:

mean_x (torch.Tensor) – Mean of x. Shape is (n_dim,).
covariance_x (torch.Tensor) – Covariance of x. Shape is (n_dim, n_dim).
const (torch.Tensor) – Constant added to the denominator. Shape is ().
B (torch.Tensor, optional) – Symmetric positive definite matrix defining the ellipse. Shape is (n_dim, n_dim).
B_chol (torch.Tensor, optional) – Cholesky decomposition of B. Can be provided to avoid recomputing it. Shape is (n_dim, n_dim).

Returns:

Expected value for the projected normal on ellipse. Shape is (n_dim,).

Return type:

torch.Tensor

pdf(mean_x, covariance_x, y, const, B=None, B_chol=None)

Compute the pdf at points y for the distribution of the variable \(y = x/\sqrt{x^T B x + c}\), where \(x \sim \mathcal{N}(\mu_x, \Sigma_x)\), \(B\) is a symmetric positive definite matrix and \(c\) is a positive constant.

Parameters:

mean_x (torch.Tensor) – Mean of x. Shape is (n_dim,).
covariance_x (torch.Tensor) – Covariance of x. Shape is (n_dim, n_dim).
y (torch.Tensor) – Points where to evaluate the PDF. Shape is (n_points, n_dim).
const (torch.Tensor) – Constant added to the denominator. Must be positive. Shape is ().
B (torch.Tensor, optional) – Matrix B used in the denominator of the projection. If not provided, the identity matrix is used. Shape is (n_dim, n_dim).
B_chol (torch.Tensor, optional) – Cholesky decomposition of B. Can be provided to avoid recomputing it. Shape is (n_dim, n_dim).

Returns:

PDF evaluated at each y. Shape is (n_points,).

Return type:

torch.Tensor

sample(mean_x, covariance_x, n_samples, const, B=None, B_chol=None)

Sample the variable \(y = x/\sqrt{x^T B x + c}\) where \(x \sim \mathcal{N}(\mu_x, \Sigma_x)\), \(B\) is a symmetric positive definite matrix and \(c\) is a positive constant added to the denominator.

Parameters:

mean_x (torch.Tensor) – Mean of x. Shape is (n_dim,).
covariance_x (torch.Tensor) – Covariance of x. Shape is (n_dim, n_dim).
n_samples (int) – Number of samples to draw.
const (torch.Tensor) – Constant added to the denominator. Shape is ().
B (torch.Tensor, optional) – Matrix B used in the denominator of the projection. If not provided, the identity matrix is used. Shape is (n_dim, n_dim).
B_chol (torch.Tensor, optional) – Cholesky decomposition matrix L, such that B = LL’. Can be provided to avoid recomputing it. Shape is (n_dim, n_dim).

Returns:

Samples from the projected normal. Shape is (n_samples, n_dim).

Return type:

torch.Tensor

second_moment(mean_x, covariance_x, const, B=None, B_chol=None)

Compute the second moment matrix of \(y = x/\sqrt{x^T B x + c}\), where \(x \sim \mathcal{N}(\mu_x, \Sigma_x)\), \(B\) is a symmetric positive definite matrix and \(c\) is a positive constant. Uses a Taylor approximation.

Parameters:

mean_x (torch.Tensor) – Mean of x. Shape is (n_dim,).
covariance_x (torch.Tensor) – Covariance of x. Shape is (n_dim, n_dim).
const (torch.Tensor) – Constant added to the denominator. Shape is ().
B (torch.Tensor, optional) – Symmetric positive definite matrix defining the ellipse. Shape is (n_dim, n_dim).
B_chol (torch.Tensor, optional) – Cholesky decomposition of B. Can be provided to avoid recomputing it. Shape is (n_dim, n_dim).

Returns:

Second moment matrix of \(y\). Shape is (n_dim, n_dim).

Return type:

torch.Tensor

Modules

`moments`	Approximation to the moments of the general projected normal distribution projected onto ellipse given by matrix B.
`probability`	Probability density function (PDF) for the general projected normal distribution.
`sampling`	Sampling functions for the general projected normal distribution.