We define a bivariate copula that captures the scale-invariant extent of dependence of a single random variable $Y$ on a set of potential explanatory random variables $X_1, \dots, X_d$. The copula itself contains the information whether $Y$ is completely dependent on $X_1, \dots, X_d$, and whether $Y$ and $X_1, \dots, X_d$ are independent. Evaluating this copula uniformly along the diagonal, i.e. calculating Spearman's footrule, leads to the so-called 'simple measure of conditional dependence' recently introduced by Azadkia and Chatterjee [1]. On the other hand, evaluating this copula uniformly over the unit square, i.e. calculating Spearman's rho, leads to a distribution-free coefficient of determination. Applying the techniques introduced in [1], we construct an estimate for this copula and show that this copula estimator is strongly consistent. Since, for $d=1$, the copula under consideration coincides with the well-known Markov product of copulas, as by-product, we also obtain a strongly consistent copula estimator for the Markov product. A simulation study illustrates the small sample performance of the proposed estimator.
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Conditionally specified models are often used to describe complex multivariate data. Such models assume implicit structures on the extremes. So far, no methodology exists for calculating extremal characteristics of conditional models since the copula and marginals are not expressed in closed forms. We consider bivariate conditional models that specify the distribution of $X$ and the distribution of $Y$ conditional on $X$. We provide tools to quantify implicit assumptions on the extremes of this class of models. In particular, these tools allow us to approximate the distribution of the tail of $Y$ and the coefficient of asymptotic independence $\eta$ in closed forms. We apply these methods to a widely used conditional model for wave height and wave period. Moreover, we introduce a new condition on the parameter space for the conditional extremes model of Heffernan and Tawn (2004), and prove that the conditional extremes model does not capture $\eta$, when $\eta<1$.
Parameter reconstructions are indispensable in metrology. Here, on wants to explain $K$ experimental measurements by fitting to them a parameterized model of the measurement process. The model parameters are regularly determined by least-square methods, i.e., by minimizing the sum of the squared residuals between the $K$ model predictions and the $K$ experimental observations, $\chi^2$. The model functions often involve computationally demanding numerical simulations. Bayesian optimization methods are specifically suited for minimizing expensive model functions. However, in contrast to least-square methods such as the Levenberg-Marquardt algorithm, they only take the value of $\chi^2$ into account, and neglect the $K$ individual model outputs. We introduce a Bayesian target-vector optimization scheme that considers all $K$ contributions of the model function and that is specifically suited for parameter reconstruction problems which are often based on hundreds of observations. Its performance is compared to established methods for an optical metrology reconstruction problem and two synthetic least-squares problems. The proposed method outperforms established optimization methods. It also enables to determine accurate uncertainty estimates with very few observations of the actual model function by using Markov chain Monte Carlo sampling on a trained surrogate model.
The Dynamic Mode Decomposition has proved to be a very efficient technique to study dynamic data. This is entirely a data-driven approach that extracts all necessary information from data snapshots which are commonly supposed to be sampled from measurement. The application of this approach becomes problematic if the available data is incomplete because some dimensions of smaller scale either missing or unmeasured. Such setting occurs very often in modeling complex dynamical systems such as power grids, in particular with reduced-order modeling. To take into account the effect of unresolved variables the optimal prediction approach based on the Mori-Zwanzig formalism can be applied to obtain the most expected prediction under existing uncertainties. This effectively leads to the development of a time-predictive model accounting for the impact of missing data. In the present paper we provide a detailed derivation of the considered method from the Liouville equation and finalize it with the optimization problem that defines the optimal transition operator corresponding to the observed data. In contrast to the existing approach, we consider a first-order approximation of the Mori-Zwanzig decomposition, state the corresponding optimization problem and solve it with the gradient-based optimization method. The gradient of the obtained objective function is computed precisely through the automatic differentiation technique. The numerical experiments illustrate that the considered approach gives practically the same dynamics as the exact Mori-Zwanzig decomposition, but is less computationally intensive.
This work studies an experimental design problem where {the values of a predictor variable, denoted by $x$}, are to be determined with the goal of estimating a function $m(x)$, which is observed with noise. A linear model is fitted to $m(x)$ but it is not assumed that the model is correctly specified. It follows that the quantity of interest is the best linear approximation of $m(x)$, which is denoted by $\ell(x)$. It is shown that in this framework the ordinary least squares estimator typically leads to an inconsistent estimation of $\ell(x)$, and rather weighted least squares should be considered. An asymptotic minimax criterion is formulated for this estimator, and a design that minimizes the criterion is constructed. An important feature of this problem is that the $x$'s should be random, rather than fixed. Otherwise, the minimax risk is infinite. It is shown that the optimal random minimax design is different from its deterministic counterpart, which was studied previously, and a simulation study indicates that it generally performs better when $m(x)$ is a quadratic or a cubic function. Another finding is that when the variance of the noise goes to infinity, the random and deterministic minimax designs coincide. The results are illustrated for polynomial regression models and the general case is also discussed.
In this paper, we study the sample complexity of {\em noisy Bayesian quadrature} (BQ), in which we seek to approximate an integral based on noisy black-box queries to the underlying function. We consider functions in a {\em Reproducing Kernel Hilbert Space} (RKHS) with the Mat\'ern-$\nu$ kernel, focusing on combinations of the parameter $\nu$ and dimension $d$ such that the RKHS is equivalent to a Sobolev class. In this setting, we provide near-matching upper and lower bounds on the best possible average error. Specifically, we find that when the black-box queries are subject to Gaussian noise having variance $\sigma^2$, any algorithm making at most $T$ queries (even with adaptive sampling) must incur a mean absolute error of $\Omega(T^{-\frac{\nu}{d}-1} + \sigma T^{-\frac{1}{2}})$, and there exists a non-adaptive algorithm attaining an error of at most $O(T^{-\frac{\nu}{d}-1} + \sigma T^{-\frac{1}{2}})$. Hence, the bounds are order-optimal, and establish that there is no adaptivity gap in terms of scaling laws.
In recent decades, one of the scientists' main concerns has been to improve the accuracy of satellite attitude, regardless of the expense. The obvious result is that a large number of control strategies have been used to address this problem. In this study, an adaptive neuro-fuzzy integrated (ANFIS) satellite attitude estimation and control system was developed. The controller is trained with the data provided by an optimal controller. A pulse modulator is used to generate the right ON/OFF commands of the thruster actuator. To evaluate the performance of the AN-FIS controller in closed-loop simulation, an ANFIS observer is used to estimate the attitude and angular velocities of the satellite using magnetometer, sun sensor and data gyro data. In addition, a new ANFIS system will be proposed and evaluated that can jointly control and estimate the system. The performance of the ANFIS controller is compared to the optimal PID controller in a Monte Carlo simulation with different initial conditions, disturbance and noise. The results show that the ANFIS controller can surpass the optimal PID controller in several aspects, including time and smoothness. In addition, the ANFIS estimator is examined and the results demonstrate the high ability of this designated observers. Both the control and estimation phases are simulated by a single ANFIS subsystem, taking into account the high capacity of ANFIS, and the results of using the ANFIS model are demonstrated.
The Analyst's Traveling Salesman Problem asks for conditions under which a (finite or infinite) subset of $\mathbb{R}^N$ is contained on a curve of finite length. We show that for finite sets, the algorithm constructed by Schul (2007)and Badger-Naples-Vellis (2019) that solves the Analyst's Traveling Salesman Problem has polynomial time complexity and we determine the sharp exponent.
We consider off-policy evaluation (OPE) in Partially Observable Markov Decision Processes (POMDPs), where the evaluation policy depends only on observable variables and the behavior policy depends on unobservable latent variables. Existing works either assume no unmeasured confounders, or focus on settings where both the observation and the state spaces are tabular. In this work, we first propose novel identification methods for OPE in POMDPs with latent confounders, by introducing bridge functions that link the target policy's value and the observed data distribution. We next propose minimax estimation methods for learning these bridge functions, and construct three estimators based on these estimated bridge functions, corresponding to a value function-based estimator, a marginalized importance sampling estimator, and a doubly-robust estimator. Our proposal permits general function approximation and is thus applicable to settings with continuous or large observation/state spaces. The nonasymptotic and asymptotic properties of the proposed estimators are investigated in detail.
There has been significant attention given to developing data-driven methods for tailoring patient care based on individual patient characteristics. Dynamic treatment regimes formalize this through a sequence of decision rules that map patient information to a suggested treatment. The data for estimating and evaluating treatment regimes are ideally gathered through the use of Sequential Multiple Assignment Randomized Trials (SMARTs) though longitudinal observational studies are commonly used due to the potentially prohibitive costs of conducting a SMART. These studies are typically sized for simple comparisons of fixed treatment sequences or, in the case of observational studies, a priori sample size calculations are often not performed. We develop sample size procedures for the estimation of dynamic treatment regimes from observational studies. Our approach uses pilot data to ensure a study will have sufficient power for comparing the value of the optimal regime, i.e. the expected outcome if all patients in the population were treated by following the optimal regime, with a known comparison mean. Our approach also ensures the value of the estimated optimal treatment regime is within an a priori set range of the value of the true optimal regime with a high probability. We examine the performance of the proposed procedure with a simulation study and use it to size a study for reducing depressive symptoms using data from electronic health records.
Ensemble methods based on subsampling, such as random forests, are popular in applications due to their high predictive accuracy. Existing literature views a random forest prediction as an infinite-order incomplete U-statistic to quantify its uncertainty. However, these methods focus on a small subsampling size of each tree, which is theoretically valid but practically limited. This paper develops an unbiased variance estimator based on incomplete U-statistics, which allows the tree size to be comparable with the overall sample size, making statistical inference possible in a broader range of real applications. Simulation results demonstrate that our estimators enjoy lower bias and more accurate confidence interval coverage without additional computational costs. We also propose a local smoothing procedure to reduce the variation of our estimator, which shows improved numerical performance when the number of trees is relatively small. Further, we investigate the ratio consistency of our proposed variance estimator under specific scenarios. In particular, we develop a new "double U-statistic" formulation to analyze the Hoeffding decomposition of the estimator's variance.
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