projnormal.formulas.projected_normal_Bc.moments

Approximation to the moments of the general projected normal distribution projected onto ellipse given by matrix B.

Functions

`mean`(mean_x, covariance_x, const[, B, B_chol])	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.
`second_moment`(mean_x, covariance_x, const[, ...])	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.

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

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