Least Absolute Deviations

lad.lad.lad(X, y, yerr=None, l1_regularizer=0.0, cov=False, maxiter=50, rtol=0.0001, eps=0.0001, session=None)

Linear least absolute deviations with L1 norm regularization using Majorization-Minimization. See [1] for a similar mathematical derivation.

Parameters:

X : (n, m)-matrix

Design matrix.

y : (n, 1) matrix

Vector of observations.

yerr : (n, 1) matrix

Vector of standard deviations on the observations.

l1_regularizer : float

Factor to control the importance of the L1 regularization.

cov : boolean

Whether or not to return the covariance matrix of the best fitted coefficients. Standard errors on the coefficients can be computed as the square root of the diagonal of the covariance matrix.

maxiter : int

Maximum number of iterations of the majorization-minimization algorithm. If maxiter equals zero, then this function returns the Weighted Least-Squares coefficients.

rtol : float

Relative tolerance on the coefficients used as an early stopping criterion. If |x_{k+1} - x_{k}|/max(1, |x_{k}|) < rtol, where |x| is the L1-norm of x, the algorithm stops.

eps : float

Increase this value if tensorflow raises an exception saying that the Cholesky decomposition was not successful.

session : tf.Session object

A tensorflow.Session object.

Returns:

x : (m, 1) matrix

Vector of coefficients that minimizes the least absolute deviations with L1 regularization.

cov : (m, m) matrix

Covariance matrix of x.

References

[1] Phillips, R. F. Least absolute deviations estimation via the EM algorithm. Statistics and Computing, 12, 281-285, 2002.

lad.lad.lad_polyfit(x, y, order=1, **kwargs)

Least absolute deviations polynomial fitting.

Fit a polynomial p(x) = p[0]  + ... + p[order] * x**order of degree order to points (x, y). Returns a vector of coefficients p that minimises the absolute error.

Parameters:

x : (n, 1)-matrix

x-coordinate of the observations.

y : (n, 1) matrix

Vector of observations.

order : int

Degree of the fitting polynomial.

**kwargs : dict

See the docstrings of lad.

Returns:

p : (m, 1) matrix

Vector of coefficients that minimizes the least absolute deviations with L1 regularization.