KFoldRLS - RLS with Kfold cross-validation regularization parameter selection

class rlscore.learner.rls.KfoldRLS(X, Y, folds, kernel='LinearKernel', basis_vectors=None, regparams=None, measure=None, save_predictions=False, **kwargs)

Bases: rlscore.predictor.predictor.PredictorInterface

Regularized least-squares regression/classification. Wrapper code that selects regularization parameter automatically based on K-fold cross-validation.

Parameters:

X : {array-like, sparse matrix}, shape = [n_samples, n_features]

Data matrix

Y : {array-like}, shape = [n_samples] or [n_samples, n_labels]

Training set labels

folds : list of index lists, shape = [n_folds]

Each list within the folds list contains the indices of samples in one fold, indices must be from range [0,n_samples-1]

kernel : {‘LinearKernel’, ‘GaussianKernel’, ‘PolynomialKernel’, ‘PrecomputedKernel’, …}

kernel function name, imported dynamically from rlscore.kernel

basis_vectors : {array-like, sparse matrix}, shape = [n_bvectors, n_features], optional

basis vectors (typically a randomly chosen subset of the training data)

regparams : {array-like}, shape = [grid_size] (optional)

regularization parameter values to be tested, default = [2^-15,…,2^15]

measure : function(Y, P) (optional)

a performance measure from rlscore.measure used for model selection, default sqerror (squared error)

Other Parameters:  

Typical kernel parameters include:

bias : float, optional

LinearKernel: the model is w*x + bias*w0, (default=1.0)

gamma : float, optional

GaussianKernel: k(xi,xj) = e^(-gamma*<xi-xj,xi-xj>) (default=1.0) PolynomialKernel: k(xi,xj) = (gamma * <xi, xj> + coef0)**degree (default=1.0)

coef0 : float, optional

PolynomialKernel: k(xi,xj) = (gamma * <xi, xj> + coef0)**degree (default=0.)

degree : int, optional

PolynomialKernel: k(xi,xj) = (gamma * <xi, xj> + coef0)**degree (default=2)

Notes

Uses fast solve and holdout algorithms, complexity depends on the sizes of the folds. Complexity when using K-fold cross-validation is:

m = n_samples, d = n_features, l = n_labels, b = n_bvectors, r=grid_size, k = n_folds

O(m^3 + dm^2 + r*(m^3/k^2 + lm^2)): basic case

O(md^2 + r*(min(m^3/k^2 + lm^2/k, kd^3 + kld^2) + ldm)): Linear Kernel, d < m

O(mb^2 + r*(min(m^3/k^2 + lm^2/k, kb^3 + klb^2) + lbm)): Sparse approximation with basis vectors

Basic information about RLS can be found in [1]. The K-fold algorithm is based on results published in [2] and [3].

References

[1] Ryan Rifkin, Ross Lippert. Notes on Regularized Least Squares. Technical Report, MIT, 2007.

[2] Tapio Pahikkala, Jorma Boberg, and Tapio Salakoski. Fast n-Fold Cross-Validation for Regularized Least-Squares. Proceedings of the Ninth Scandinavian Conference on Artificial Intelligence, 83-90, Otamedia Oy, 2006.

[3] Tapio Pahikkala, Hanna Suominen, and Jorma Boberg. Efficient cross-validation for kernelized least-squares regression with sparse basis expansions. Machine Learning, 87(3):381–407, June 2012.

Attributes:

predictor : {LinearPredictor, KernelPredictor}

trained predictor

cv_performances : array, shape = [grid_size]

K-fold performances for each grid point

cv_predictions : list of 1D or 2D arrays, shape = [grid_size, n_folds]

predictions for each fold, shapes [fold_size] or [fold_size, n_labels]

regparam : float

regparam from grid with best performance

predict(X)

Predicts outputs for new inputs

Parameters:

X : {array-like, sparse matrix}, shape = [n_samples, n_features]

input data matrix

Returns:

P : array, shape = [n_samples, n_tasks]

predictions