Isabella Verdinelli

Isabella Verdinelli, Larry Wasserman

Because of the widespread use of black box prediction methods such as random forests and neural nets, there is renewed interest in developing methods for quantifying variable importance as part of the broader goal of interpretable prediction. A popular approach is to define a variable importance parameter - known as LOCO (Leave Out COvariates) - based on dropping covariates from a regression model. This is essentially a nonparametric version of R-squared. This parameter is very general and can be estimated nonparametrically, but it can be hard to interpret because it is affected by correlation between covariates. We propose a method for mitigating the effect of correlation by defining a modified version of LOCO. This new parameter is difficult to estimate nonparametrically, but we show how to estimate it using semiparametric models.

Isabella Verdinelli, Larry Wasserman

We use the output of a random forest to define a family of local smoothers with spatially adaptive bandwidth matrices. The smoother inherits the flexibility of the original forest but, since it is a simple, linear smoother, it is very interpretable and it can be used for tasks that would be intractable for the original forest. This includes bias correction, confidence intervals, assessing variable importance and methods for exploring the structure of the forest. We illustrate the method on some synthetic examples and on data related to Covid-19.

Christopher Genovese, Marco Perone-Pacifico, Isabella Verdinelli et al.

We present a method for finding high density, low-dimensional structures in noisy point clouds. These structures are sets with zero Lebesgue measure with respect to the $D$-dimensional ambient space and belong to a $d<D$ dimensional space. We call them "singular features." Hunting for singular features corresponds to finding unexpected or unknown structures hidden in point clouds belonging to $\R^D$. Our method outputs well defined sets of dimensions $d<D$. Unlike spectral clustering, the method works well in the presence of noise. We show how to find singular features by first finding ridges in the estimated density, followed by a filtering step based on the eigenvalues of the Hessian of the density.

Christopher Genovese, Marco Perone-Pacifico, Isabella Verdinelli et al.

We derive nonparametric confidence intervals for the eigenvalues of the Hessian at modes of a density estimate. This provides information about the strength and shape of modes and can also be used as a significance test. We use a data-splitting approach in which potential modes are identified using the first half of the data and inference is done with the second half of the data. To get valid confidence sets for the eigenvalues, we use a bootstrap based on an elementary-symmetric-polynomial (ESP) transformation. This leads to valid bootstrap confidence sets regardless of any multiplicities in the eigenvalues. We also suggest a new method for bandwidth selection, namely, choosing the bandwidth to maximize the number of significant modes. We show by example that this method works well. Even when the true distribution is singular, and hence does not have a density, (in which case cross validation chooses a zero bandwidth), our method chooses a reasonable bandwidth.