constraints.quadratic¶
Module: constraints.quadratic¶
Inheritance diagram for selectinf.constraints.quadratic:
Classes¶
constraints¶
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class
selectinf.constraints.quadratic.constraints(quad_part, lin_part=None, offset=None, covariance=None, mean=None, rank=None)[source]¶ Bases:
selectinf.constraints.base.constraintsObject solving the problems of slices for quadratic and linear inequalities
\[\forall i, y^T Q_i y + a_i^t < b_i\]-
__init__(quad_part, lin_part=None, offset=None, covariance=None, mean=None, rank=None)[source]¶ Create a new object for quadratic constraints
- Parameters
quad_part : np.float(l, p, p)
3-dimensions array. The lines are some quadratic forms
lin_part : np.float(l, p)
The lines are the vector of the linear forms in the inquality Default to np.zeros(l, p)
offset : np.float(l)
The offsets of all inequalities Defaults to np.zeros(l)
covariance : np.float((p,p))
Covariance matrix of Gaussian distribution to be truncated. Defaults to np.identity(self.dim).
mean : np.float(p)
Mean vector of Gaussian distribution to be truncated. Defaults to np.zeros(self.dim).
rank : int
If not None, this should specify the rank of the covariance matrix. Defaults to self.dim.
WARNING : The shapes of the three parameters must fit
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bounds(nu, y)[source]¶ Return the intervals of the slice in a direction nu, which respects the inequality
- Parameters
nu : np.float(p)
The direction of the slice
y : np.float(p)
A point on the affine slice
- Returns
intervals : array of couples
Array of (a, b), which means that the set is the union of [a, b]
Examples
The constraints are : (x - 1)^2 + y ^ 2 < 4 10 * x^2 - 10 * y^2 < -1 (x + 1)^2 + y ^ 2 < 4
>>> q1, lin1, off1 = np.identity(2), np.array([2., 0.]), 3. >>> q2, lin2, off2 = np.array([[10., 0], [0, -10.]]), np.zeros(2), -1. >>> q3, lin3, off3 = np.identity(2), np.array([-2., 0.]), 3. >>> cons = constraints(np.array([q1, q2, q3]), np.array([lin1, lin2, lin3]), np.array([off1, off2, off3]))
>>> y = np.array([[0., -1.]]).T
>>> nu1 = np.array([[0., 1.]]).T >>> I1 = np.array(cons.bounds(nu1, y).intersection()) >>> I1_expected = np.array([[- np.sqrt(3) + 1, 1. -1./np.sqrt(10)], [1. + 1./np.sqrt(10), 1. + np.sqrt(3)]]) >>> #np.all(np.fabs(I1 - I1_expected) < 1.e-10)
>>> nu2 = 2*nu1 ## Try with a nu not unitary >>> I2 = np.array(cons.bounds(nu2, y).intersection()) >>> I2_expected = I1_expected /2 >>> #np.all(np.fabs(I2 - I2_expected) < 1.e-10)
>>> nu3 = np.array([[1., 0]]).T >>> I3 = np.array(cons.bounds(nu3, y).intersection()) >>> I3_expected = np.array([[1. - np.sqrt(3), -1. + np.sqrt(3)]]) >>> #np.all(np.fabs(I3 - I3_expected) < 1.e-10)
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distr_norm(X_s, y, sigma=1.0)[source]¶ Return the value of the norm of X_s.T*y and an instance of truncated : the distribution of X_s.T*y This is implementing the forward stepwise paper in the general case
- Parameters
X_s : np.float(p, k):
X_s is a full ranked matrix
y : np.float(p):
y is the data, and satisfies the constraints
sigma: float
sigma is the variance of the normal distribution under wich y is chosen
- Returns
distr : truncated_chi
distr is an object used to study the distribution of np.linalg.norm(np.dot(X_s.T, y)), when y is a gaussian vector, chosen under the constraints and on the slice given by nu
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constraints_vecnorm¶
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class
selectinf.constraints.quadratic.constraints_vecnorm(alpha, beta, dim)[source]¶ Bases:
selectinf.constraints.quadratic.constraints-
__init__(alpha, beta, dim)[source]¶ Create a new object for quadratic constraints
- Parameters
quad_part : np.float(l, p, p)
3-dimensions array. The lines are some quadratic forms
lin_part : np.float(l, p)
The lines are the vector of the linear forms in the inquality Default to np.zeros(l, p)
offset : np.float(l)
The offsets of all inequalities Defaults to np.zeros(l)
covariance : np.float((p,p))
Covariance matrix of Gaussian distribution to be truncated. Defaults to np.identity(self.dim).
mean : np.float(p)
Mean vector of Gaussian distribution to be truncated. Defaults to np.zeros(self.dim).
rank : int
If not None, this should specify the rank of the covariance matrix. Defaults to self.dim.
WARNING : The shapes of the three parameters must fit
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bounds(nu, y)[source]¶ Return the intervals of the slice in a direction nu, which respects the inequality
- Parameters
nu : np.float(p)
The direction of the slice
y : np.float(p)
A point on the affine slice
- Returns
intervals : array of couples
Array of (a, b), which means that the set is the union of [a, b]
Examples
The constraints are : (x - 1)^2 + y ^ 2 < 4 10 * x^2 - 10 * y^2 < -1 (x + 1)^2 + y ^ 2 < 4
>>> q1, lin1, off1 = np.identity(2), np.array([2., 0.]), 3. >>> q2, lin2, off2 = np.array([[10., 0], [0, -10.]]), np.zeros(2), -1. >>> q3, lin3, off3 = np.identity(2), np.array([-2., 0.]), 3. >>> cons = constraints(np.array([q1, q2, q3]), np.array([lin1, lin2, lin3]), np.array([off1, off2, off3]))
>>> y = np.array([[0., -1.]]).T
>>> nu1 = np.array([[0., 1.]]).T >>> I1 = np.array(cons.bounds(nu1, y).intersection()) >>> I1_expected = np.array([[- np.sqrt(3) + 1, 1. -1./np.sqrt(10)], [1. + 1./np.sqrt(10), 1. + np.sqrt(3)]]) >>> #np.all(np.fabs(I1 - I1_expected) < 1.e-10)
>>> nu2 = 2*nu1 ## Try with a nu not unitary >>> I2 = np.array(cons.bounds(nu2, y).intersection()) >>> I2_expected = I1_expected /2 >>> #np.all(np.fabs(I2 - I2_expected) < 1.e-10)
>>> nu3 = np.array([[1., 0]]).T >>> I3 = np.array(cons.bounds(nu3, y).intersection()) >>> I3_expected = np.array([[1. - np.sqrt(3), -1. + np.sqrt(3)]]) >>> #np.all(np.fabs(I3 - I3_expected) < 1.e-10)
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bounds_unknownSigma(P, y)¶
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distr_norm(X_s, y, sigma=1.0)¶ Return the value of the norm of X_s.T*y and an instance of truncated : the distribution of X_s.T*y This is implementing the forward stepwise paper in the general case
- Parameters
X_s : np.float(p, k):
X_s is a full ranked matrix
y : np.float(p):
y is the data, and satisfies the constraints
sigma: float
sigma is the variance of the normal distribution under wich y is chosen
- Returns
distr : truncated_chi
distr is an object used to study the distribution of np.linalg.norm(np.dot(X_s.T, y)), when y is a gaussian vector, chosen under the constraints and on the slice given by nu
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distr_norm_unknownSigma(P, y, n, p)¶
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gen_p_value(y, f, sigma=1.0, n_samples=1000)¶ - Parameters
f : z -> x
observed value
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p_value(X_s, y, sigma=1.0)¶
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p_value_unknownSigma(P, y, n, p)¶
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sample(n_sample, initial_point)¶
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sample_dum(n_sample)¶
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