3,036 research outputs found
Geoadditive Regression Modeling of Stream Biological Condition
Indices of biotic integrity (IBI) have become an established tool to quantify the condition of small non-tidal streams and their watersheds. To investigate the effects of watershed characteristics on stream biological condition, we present a new technique for regressing IBIs on watershed-specific explanatory variables. Since IBIs are typically evaluated on anordinal scale, our method is based on the proportional odds model for ordinal outcomes. To avoid overfitting, we do not use classical maximum likelihood estimation but a component-wise functional gradient boosting approach. Because component-wise gradient boosting has an intrinsic mechanism for variable selection and model choice, determinants of biotic integrity can be identified. In addition, the method offers a relatively simple way to account for spatial correlation in ecological data. An analysis of the Maryland Biological Streams Survey shows that nonlinear effects of predictor variables on stream condition can be quantified while, in addition, accurate predictions of biological condition at unsurveyed locations are obtained
Adaptive Monotone Shrinkage for Regression
We develop an adaptive monotone shrinkage estimator for regression models
with the following characteristics: i) dense coefficients with small but
important effects; ii) a priori ordering that indicates the probable predictive
importance of the features. We capture both properties with an empirical Bayes
estimator that shrinks coefficients monotonically with respect to their
anticipated importance. This estimator can be rapidly computed using a version
of Pool-Adjacent-Violators algorithm. We show that the proposed monotone
shrinkage approach is competitive with the class of all Bayesian estimators
that share the prior information. We further observe that the estimator also
minimizes Stein's unbiased risk estimate. Along with our key result that the
estimator mimics the oracle Bayes rule under an order assumption, we also prove
that the estimator is robust. Even without the order assumption, our estimator
mimics the best performance of a large family of estimators that includes the
least squares estimator, constant- ridge estimator, James-Stein
estimator, etc. All the theoretical results are non-asymptotic. Simulation
results and data analysis from a model for text processing are provided to
support the theory.Comment: Appearing in Uncertainty in Artificial Intelligence (UAI) 201
Item response model estimation via penalized likelihood
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Estimation of variance components, heritability and the ridge penalty in high-dimensional generalized linear models
For high-dimensional linear regression models, we review and compare several estimators of variances τ2 and σ2 of the random slopes and errors, respectively. These variances relate directly to ridge regression penalty λ and heritability index h2, often used in genetics. Several estimators of these, either based on cross-validation (CV) or maximum marginal likelihood (MML), are also discussed. The comparisons include several cases of the high-dimensional covariate matrix such as multi-collinear covariates and data-derived ones. Moreover, we study robustness against model misspecifications such as sparse instead of dense effects and non-Gaussian errors. An example on weight gain data with genomic covariates confirms the good performance of MML compared to CV. Several extensions are presented. First, to the high-dimensional linear mixed effects model, with REML as an alternative to MML. Second, to the conjugate Bayesian setting, shown to be a good alternative. Third, and most prominently, to generalized linear models for which we derive a computationally efficient MML estimator by re-writing the marginal likelihood as an n-dimensional integral. For Poisson and Binomial ridge regression, we demonstrate the superior accuracy of the resulting MML estimator of λ as compared to CV. Software is provided to enable reproduction of all results.Gwenaël Leday was supported by the Medical Research Council, grant number MR/M004421
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