bluebell

bluebell.MVBE(x, tol=0.001, maxiter=1000)

Return the mean and covariance matrix that define the minimum-volume ellipsoid that bounds the points in x using algorithm 3.2 of Moshtagh (2005).

Parameters:

• x (2-d NumPy array of shape (N, D)) –

• tol (float, optional) – Tolerance parameter for precision of bounding ellipsoid (default=1e-3).

• maxiter (int, optional) – Maximum number of iterations to perform (default=1000).

Returns:

• mu (1-d NumPy array of shape (D,)) – Mean (or center) of the ellipsoid.

• C (2-d NumPy array of shape (D,D)) – Covariance matrix that defines the ellipsoid.

Example

import matplotlib.pyplot as pl
import bluebell as bb
import bluebell.plot as bbplot

x = np.random.rand(10,2)
mu, C = bb.MVBE(x)

pl.plot(*x.T, 'o')
bbplot.cov_ellipse(mu, C, fc='none', ec='k')

(Source code, png, hires.png, pdf)

bluebell.select_by_relative_chi2(chi2, sigma_inner=1e-07, sigma_outer=inf)

Returns a logical array that is True where chi2 relative to its minimum is within the range defined by sigma_inner and sigma_outer. i.e. where sigma_inner**2 < chi2-np.min(chi2) < sigma_outer**2.

bluebell.estimate(x, chi2, sigma_inner=1e-07, sigma_outer=inf, tol=0.001, maxiter=1000)

Estimate uncertainties for a sample of points given their correspond values of chi-squared.

Parameters:

• x (2-d NumPy array of shape (N, D)) – Points at which chi-squared has been evaluated.

• chi2 (1-d NumPy array of shape (N,)) – Values of chi-squared for the points in x.

• sigma_inner (float, optional) – Only use the points that are more than this many sigma from the minimum.

• sigma_outer (float, optional) – Only use the points that are less than this many sigma from the minimum.

• tol (float, optional) – Tolerance parameter for finding minimum-volume bounding ellipsoid.

• maxiter (int, optional) – Maximum number of iterations to use when finding minimum-volume bounding ellipsoid.

Returns:

• mu (1-d NumPy array of shape (D,)) – Mean (or center) of the minimum-volume bounding ellipsoid around the rescaled points in x.

• C (2-d NumPy array of shape (D,D)) – Covariance matrix that defines the ellipsoid.

bluebell.propagate(x, y, mu, C, weights=None)

Propagate the uncertainties on x to other dependent variables y. Returns full mean and uncertainty so that the correlations between main parameters and dependent variables are given.

Parameters:

• x (2-d NumPy array) – Points at which function has been evaluated.

• y (2-d NumPy array) – Values of function at points in x.

• mu (1-d NumPy array of shape (D,)) – Mean (or center) of the ellipsoid that characterizes the chi2 values of the points in x.

• C (2-d NumPy array of shape (D,D)) – Covariance matrix of the ellipsoid.

• weights (1-d NumPy array, optional) – Relative weights for the points in the fit.

Returns:

• mu (1-d NumPy array of shape (D,)) – Mean (or center) of both the underlying parameters x and the dependent variables y.

• C (2-d NumPy array of shape (D,D)) – Covariance matrix of both the underlying parameters x and the dependent variables y.

bluebell.linearize(x, y, weights=None)

Given a set of points x and the values y of some (vector) function evaluated at those points, find the parameters of the best-fitting affine map between x and y. i.e. returns arrays Q and B such that y == Q + x.dot(B.T).

Parameters:

• x (2-d NumPy array of shape (N, D)) – Points at which function has been evaluated.

• y (2-d NumPy array of shape (N, K)) – Values of function at points in x.

• weights (1-d NumPy array, optional) – Relative weights for the points in the fit.

Returns:

• Q (1-d NumPy array of shape (K,)) – Constant offset of affine map from x to y.

• B (2-d NumPy array of shape (K, D)) – Matrix transformation of affine map from x to y.

bluebell.linear_optimum(x, y, obs, err, weights=None, rcond=None)

Given a set of points x, the values y of some (vector) function evaluated at those points, observed values of the function obs and uncertainties on those observations err, finds the best-fitting parameters an affine model of y as a function of x.

Parameters:

• x (2-d NumPy array of shape (N, D)) – Points at which function has been evaluated.

• y (2-d NumPy array of shape (N, K)) – Values of function at points in x.

• obs (1-d NumPy array of shape (D,)) – Observed values.

• err (1-d NumPy array of shape (D,)) – Uncertainties on observed values.

• weights (1-d NumPy array, optional) – Relative weights for the points in the fit.

• rcond (float) – rcond parameter passed to numpy.linalg.lstsq.

bluebell.uniform_on_unit_sphere(N, D)

Draw N points uniformly distributed on the surface of a D-dimensional unit sphere.

Example

import matplotlib.pyplot as pl
import bluebell as bb
import bluebell.plot as bbplot

x = bb.uniform_on_unit_sphere(10, 2)
pl.plot(*x.T, 'o')
bbplot.cov_ellipse(np.zeros(2), np.eye(2), fc='none', ec='k')

(Source code, png, hires.png, pdf)

bluebell.uniform_on_ellipsoid(mu, C, N)

Draw N points uniformly distributed on the surface of the ellipsoid define by mean mu and covariance C.

Example

import matplotlib.pyplot as pl
import bluebell as bb
import bluebell.plot as bbplot

mu = np.random.rand(2)
A = np.random.rand(2,2)
C = A.T.dot(A)
x = bb.uniform_on_ellipsoid(mu, C, 10)
pl.plot(*x.T, 'o')
bbplot.cov_ellipse(mu, C, fc='none', ec='k')

(Source code, png, hires.png, pdf)

bluebell.uniform_in_ellipsoid(mu, C, N)

Draw N points uniformly distributed inside the ellipsoid define by mean mu and covariance C.

Example

import matplotlib.pyplot as pl
import bluebell as bb
import bluebell.plot as bbplot

mu = np.random.rand(2)
A = np.random.rand(2,2)
C = A.T.dot(A)
x = bb.uniform_in_ellipsoid(mu, C, 100)
pl.plot(*x.T, 'o')
bbplot.cov_ellipse(mu, C, fc='none', ec='k')

(Source code, png, hires.png, pdf)

bluebell.vertices(mu, C)

Return the 2*D points at the ends of each axis of the ellipsoid defined by mean mu and covariance C.

Example

import matplotlib.pyplot as pl
import bluebell as bb
import bluebell.plot as bbplot

mu = np.random.rand(2)
A = np.random.rand(2,2)
C = A.T.dot(A)
x = bb.vertices(mu, C)
pl.plot(*x.T, 'o')
bbplot.cov_ellipse(mu, C, fc='none', ec='k')

(Source code, png, hires.png, pdf)

bluebell.is_in_ellipsoid(x, mu, C)

Returns True if the point(s) in x are inside the ellipsoid defined by mean mu and covariance C.

bluebell.is_in_simplex(x, s)

Returns True if the 1-d array x is in the simplex s, where s is a 2-d array with dimensions (D+1,D) (each row is a point).

bluebell.discard(x, iterations=1)

Drop points that don’t help to define the bounding ellipsoid.

bluebell.grid(N, lower, upper)

Create a regular grid with N points between lower and upper.