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This work contributes to the limited literature on estimating the diffusivity or drift coefficient of nonlinear SPDEs driven by additive noise. Assuming that the solution is measured locally in space and over a finite time interval, we show that the augmented maximum likelihood estimator introduced in Altmeyer, Reiss (2020) retains its asymptotic properties when used for semilinear SPDEs that satisfy some abstract, and verifiable, conditions. The proofs of asymptotic results are based on splitting the solution in linear and nonlinear parts and fine regularity properties in $L^p$-spaces. The obtained general results are applied to particular classes of equations, including stochastic reaction-diffusion equations. The stochastic Burgers equation, as an example with first order nonlinearity, is an interesting borderline case of the general results, and is treated by a Wiener chaos expansion. We conclude with numerical examples that validate the theoretical results.

Differential Granger causality, that is understanding how Granger causal relations differ between two related time series, is of interest in many scientific applications. Modeling each time series by a vector autoregressive (VAR) model, we propose a new method to directly learn the difference between the corresponding transition matrices in high dimensions. Key to the new method is an estimating equation constructed based on the Yule-Walker equation that links the difference in transition matrices to the difference in the corresponding precision matrices. In contrast to separately estimating each transition matrix and then calculating the difference, the proposed direct estimation method only requires sparsity of the difference of the two VAR models, and hence allows hub nodes in each high-dimensional time series. The direct estimator is shown to be consistent in estimation and support recovery under mild assumptions. These results also lead to novel consistency results with potentially faster convergence rates for estimating differences between precision matrices of i.i.d observations under weaker assumptions than existing results. We evaluate the finite sample performance of the proposed method using simulation studies and an application to electroencephalogram (EEG) data.

Choosing models from a hypothesis space is a frequent task in approximation theory and inverse problems. Cross-validation is a classical tool in the learner's repertoire to compare the goodness of fit for different reconstruction models. Much work has been dedicated to computing this quantity in a fast manner but tackling its theoretical properties occurs to be difficult. So far, most optimality results are stated in an asymptotic fashion. In this paper we propose a concentration inequality on the difference of cross-validation score and the risk functional with respect to the squared error. This gives a pre-asymptotic bound which holds with high probability. For the assumptions we rely on bounds on the uniform error of the model which allow for a broadly applicable framework. We support our claims by applying this machinery to Shepard's model, where we are able to determine precise constants of the concentration inequality. Numerical experiments in combination with fast algorithms indicate the applicability of our results.

Motivated by the high-frequency data streams continuously generated, real-time learning is becoming increasingly important. These data streams should be processed sequentially with the property that the stream may change over time. In this streaming setting, we propose techniques for minimizing a convex objective through unbiased estimates of its gradients, commonly referred to as stochastic approximation problems. Our methods rely on stochastic approximation algorithms due to their computationally advantage as they only use the previous iterate as a parameter estimate. The reasoning includes iterate averaging that guarantees optimal statistical efficiency under classical conditions. Our non-asymptotic analysis shows accelerated convergence by selecting the learning rate according to the expected data streams. We show that the average estimate converges optimally and robustly to any data stream rate. In addition, noise reduction can be achieved by processing the data in a specific pattern, which is advantageous for large-scale machine learning. These theoretical results are illustrated for various data streams, showing the effectiveness of the proposed algorithms.

In randomized experiments, adjusting for observed features when estimating treatment effects has been proposed as a way to improve asymptotic efficiency. However, only linear regression has been proven to form an estimate of the average treatment effect that is asymptotically no less efficient than the treated-minus-control difference in means regardless of the true data generating process. Randomized treatment assignment provides this "do-no-harm" property, with neither truth of a linear model nor a generative model for the outcomes being required. We present a general calibration method which confers the same no-harm property onto estimators leveraging a broad class of nonlinear models. This recovers the usual regression-adjusted estimator when ordinary least squares is used, and further provides non-inferior treatment effect estimators using methods such as logistic and Poisson regression. The resulting estimators are non-inferior to both the difference in means estimator and to treatment effect estimators that have not undergone calibration. We show that our estimator is asymptotically equivalent to an inverse probability weighted estimator using a logit link with predicted potential outcomes as covariates. In a simulation study, we demonstrate that common nonlinear estimators without our calibration procedure may perform markedly worse than both the calibrated estimator and the unadjusted difference in means.

Discrete choice experiments are frequently used to quantify consumer preferences by having respondents choose between different alternatives. Choice experiments involving mixtures of ingredients have been largely overlooked in the literature, even though many products and services can be described as mixtures of ingredients. As a consequence, little research has been done on the optimal design of choice experiments involving mixtures. The only existing research has focused on D-optimal designs, which means that an estimation-based approach was adopted. However, in experiments with mixtures, it is crucial to obtain models that yield precise predictions for any combination of ingredient proportions. This is because the goal of mixture experiments generally is to find the mixture that optimizes the respondents' utility. As a result, the I-optimality criterion is more suitable for designing choice experiments with mixtures than the D-optimality criterion because the I-optimality criterion focuses on getting precise predictions with the estimated statistical model. In this paper, we study Bayesian I-optimal designs, compare them with their Bayesian D-optimal counterparts, and show that the former designs perform substantially better than the latter in terms of the variance of the predicted utility.

We study the problem of the non-parametric estimation for the density of the stationary distribution of the multivariate stochastic differential equation with jumps (Xt) , when the dimension d is bigger than 3. From the continuous observation of the sampling path on [0, T ] we show that, under anisotropic Holder smoothness constraints, kernel based estimators can achieve fast convergence rates. In particular , they are as fast as the ones found by Dalalyan and Reiss [9] for the estimation of the invariant density in the case without jumps under isotropic Holder smoothness constraints. Moreover, they are faster than the ones found by Strauch [29] for the invariant density estimation of continuous stochastic differential equations, under anisotropic Holder smoothness constraints. Furthermore, we obtain a minimax lower bound on the L2-risk for pointwise estimation, with the same rate up to a log(T) term. It implies that, on a class of diffusions whose invariant density belongs to the anisotropic Holder class we are considering, it is impossible to find an estimator with a rate of estimation faster than the one we propose.

Dynamic treatment regimes (DTRs) consist of a sequence of decision rules, one per stage of intervention, that finds effective treatments for individual patients according to patient information history. DTRs can be estimated from models which include the interaction between treatment and a small number of covariates which are often chosen a priori. However, with increasingly large and complex data being collected, it is difficult to know which prognostic factors might be relevant in the treatment rule. Therefore, a more data-driven approach of selecting these covariates might improve the estimated decision rules and simplify models to make them easier to interpret. We propose a variable selection method for DTR estimation using penalized dynamic weighted least squares. Our method has the strong heredity property, that is, an interaction term can be included in the model only if the corresponding main terms have also been selected. Through simulations, we show our method has both the double robustness property and the oracle property, and the newly proposed methods compare favorably with other variable selection approaches.

The stable under iterated tessellation (STIT) process is a stochastic process that produces a recursive partition of space with cut directions drawn independently from a distribution over the sphere. The case of random axis-aligned cuts is known as the Mondrian process. Random forests and Laplace kernel approximations built from the Mondrian process have led to efficient online learning methods and Bayesian optimization. In this work, we utilize tools from stochastic geometry to resolve some fundamental questions concerning STIT processes in machine learning. First, we show that a STIT process with cut directions drawn from a discrete distribution can be efficiently simulated by lifting to a higher dimensional axis-aligned Mondrian process. Second, we characterize all possible kernels that stationary STIT processes and their mixtures can approximate. We also give a uniform convergence rate for the approximation error of the STIT kernels to the targeted kernels, generalizing the work of [3] for the Mondrian case. Third, we obtain consistency results for STIT forests in density estimation and regression. Finally, we give a formula for the density estimator arising from an infinite STIT random forest. This allows for precise comparisons between the Mondrian forest, the Mondrian kernel and the Laplace kernel in density estimation. Our paper calls for further developments at the novel intersection of stochastic geometry and machine learning.

From only positive (P) and unlabeled (U) data, a binary classifier could be trained with PU learning, in which the state of the art is unbiased PU learning. However, if its model is very flexible, empirical risks on training data will go negative, and we will suffer from serious overfitting. In this paper, we propose a non-negative risk estimator for PU learning: when getting minimized, it is more robust against overfitting, and thus we are able to use very flexible models (such as deep neural networks) given limited P data. Moreover, we analyze the bias, consistency, and mean-squared-error reduction of the proposed risk estimator, and bound the estimation error of the resulting empirical risk minimizer. Experiments demonstrate that our risk estimator fixes the overfitting problem of its unbiased counterparts.