[SciPy-User] scipy.interpolate.rbf: how is "smooth" defined?

josef.pktd@gmai... josef.pktd@gmai...
Mon Aug 30 11:45:27 CDT 2010


On Mon, Aug 30, 2010 at 12:22 PM, Gael Varoquaux
<gael.varoquaux@normalesup.org> wrote:
> On Mon, Aug 30, 2010 at 12:18:58PM -0400, josef.pktd@gmail.com wrote:
>> Please do, I can also use it for other things. (I never read the small
>> print in Ledoit-Wolf.)
>
> There you go (BSD licensed). This will land in scikit learn at some point
> because we are going to need it in different places, but I haven't found
> the time to polish it.
>
> Gael
>
> def ledoit_wolf(x, return_factor=False):
>    """ Estimates the shrunk Ledoit-Wolf covariance matrix.
>
>        Parameters
>        ----------
>        x: 2D ndarray, shape (n, p)
>            The data matrix, with p features and n samples.
>        return_factor: boolean, optional
>            If return_factor is True, the regularisation_factor is
>            returned.
>
>        Returns
>        -------
>        regularised_cov: 2D ndarray
>            Regularized covariance
>        regularisation_factor: float
>            Regularisation factor
>
>        Notes
>        -----
>        The regularised covariance is::
>
>            (1 - regularisation_factor)*cov
>                    + regularisation_factor*np.identity(n_features)
>    """
>    n_samples, n_features = x.shape
>    if n_features == 1:
>        if return_factor:
>            return np.atleast_2d(x.std()), 0
>        return np.atleast_2d(x.std())
>    cov = np.dot(x.T, x)/n_samples
>    i = np.identity(n_features)
>    mu = np.trace(cov)/n_features
>    delta = ((cov - mu*i)**2).sum()/n_features
>    x2 = x**2
>    beta_ = 1./(n_features*n_samples) * np.sum(
>                            np.dot(x2.T, x2)/n_samples - cov**2
>                )
>
>    beta = min(beta_, delta)
>    alpha = delta - beta
>    if not return_factor:
>        return beta/delta * mu * i + alpha/delta * cov
>    else:
>        return beta/delta * mu * i + alpha/delta * cov, beta/delta
>

Thanks,
If you use this with moment/normal equations for penalized
least-squares, do you have to multiply also `dot(x,y)/n_samples` by
`alpha/delta` ?
In the normalization of RBF, smooth would be `beta/delta * mu /
(alpha/delta)`  or maybe additionally also divided by n_samples or
something like this.

Josef
>
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