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Register a new userConstrained Minimum Energy Designs
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Added by
Chaofan Huang
Publication date
2021
fields
Mathematical Statistics
and research's language is
English
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Space-filling designs are important in computer experiments, which are critical for building a cheap surrogate model that adequately approximates an expensive computer code. Many design construction techniques in the existing literature are only applicable for rectangular bounded space, but in real world applications, the input space can often be non-rectangular because of constraints on the input variables. One solution to generate designs in a constrained space is to first generate uniformly distributed samples in the feasible region, and then use them as the candidate set to construct the designs. Sequentially Constrained Monte Carlo (SCMC) is the state-of-the-art technique for candidate generation, but it still requires large number of constraint evaluations, which is problematic especially when the constraints are expensive to evaluate. Thus, to reduce constraint evaluations and improve efficiency, we propose the Constrained Minimum Energy Design (CoMinED) that utilizes recent advances in deterministic sampling methods. Extensive simulation results on 15 benchmark problems with dimensions ranging from 2 to 13 are provided for demonstrating the improved performance of CoMinED over the existing methods.
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This study extends power formulas proposed by Schochet (2008) assuming that the cluster-level score variable follows quadratic functional form. Results reveal that we need not be concerned with treatment by linear term interaction, and polynomial degree up to second order for symmetric truncation intervals. In comparison, every slight change in the functional form alters sample size requirements for asymmetric truncation intervals. Finally, an empirical framework beyond quadratic functional form is provided when the asymptotic variance of the treatment effect is untraceable. In this case, the CRD design effect is either computed from moments of the sample or approximate population moments via simulation. Formulas for quadratic functional form and the extended empirical framework are implemented in the cosa R package and companion Shiny web application.
We study the constrained minimum energy problem with an external field relative to the $alpha$-Riesz kernel $|x-y|^{alpha-n}$ of order $alphain(0,n)$ for a generalized condenser $mathbf A=(A_i)_{iin I}$ in $mathbb R^n$, $ngeqslant 3$, whose oppositely charged plates intersect each other over a set of zero capacity. Conditions sufficient for the existence of minimizers are found, and their uniqueness and vague compactness are studied. Conditions obtained are shown to be sharp. We also analyze continuity of the minimizers in the vague and strong topologies when the condenser and the constraint both vary, describe the weighted equilibrium vector potentials, and single out their characteristic properties. Our arguments are based particularly on the simultaneous use of the vague topology and a suitable semimetric structure on a set of vector measures associated with $mathbf A$, and the establishment of completeness theorems for proper semimetric spaces. The results remain valid for the logarithmic kernel on $mathbb R^2$ and $mathbf A$ with compact $A_i$, $iin I$. The study is illustrated by several examples.
Tie-breaker experimental designs are hybrids of Randomized Control Trials (RCTs) and Regression Discontinuity Designs (RDDs) in which subjects with moderate scores are placed in an RCT while subjects with extreme scores are deterministically assigned to the treatment or control group. The design maintains the benefits of randomization for causal estimation while avoiding the possibility of excluding the most deserving recipients from the treatment group. The causal effect estimator for a tie-breaker design can be estimated by fitting local linear regressions for both the treatment and control group, as is typically done for RDDs. We study the statistical efficiency of such local linear regression-based causal estimators as a function of $Delta$, the radius of the interval in which treatment randomization occurs. In particular, we determine the efficiency of the estimator as a function of $Delta$ for a fixed, arbitrary bandwidth under the assumption of a uniform assignment variable. To generalize beyond uniform assignment variables and asymptotic regimes, we also demonstrate on the Angrist and Lavy (1999) classroom size dataset that prior to conducting an experiment, an experimental designer can estimate the efficiency for various experimental radii choices by using Monte Carlo as long as they have access to the distribution of the assignment variable. For both uniform and triangular kernels, we show that increasing the radius of randomized experiment interval will increase the efficiency until the radius is the size of the local-linear regression bandwidth, after which no additional efficiency benefits are conferred.
We study minimum energy problems relative to the $alpha$-Riesz kernel $|x-y|^{alpha-n}$, $alphain(0,2]$, over signed Radon measures $mu$ on $mathbb R^n$, $ngeqslant3$, associated with a generalized condenser $(A_1,A_2)$, where $A_1$ is a relatively closed subset of a domain $D$ and $A_2=mathbb R^nsetminus D$. We show that, though $A_2capmathrm{Cl}_{mathbb R^n}A_1$ may have nonzero capacity, this minimum energy problem is uniquely solvable (even in the presence of an external field) if we restrict ourselves to $mu$ with $mu^+leqslantxi$, where a constraint $xi$ is properly chosen. We establish the sharpness of the sufficient conditions on the solvability thus obtained, provide descriptions of the weighted $alpha$-Riesz potentials of the solutions, single out their characteristic properties, and analyze their supports. The approach developed is mainly based on the establishment of an intimate relationship between the constrained minimum $alpha$-Riesz energy problem over signed measures associated with $(A_1,A_2)$ and the constrained minimum $alpha$-Green energy problem over positive measures carried by $A_1$. The results are illustrated by examples.
Blocking is often used to reduce known variability in designed experiments by collecting together homogeneous experimental units. A common modelling assumption for such experiments is that responses from units within a block are dependent. Accounting for such dependencies in both the design of the experiment and the modelling of the resulting data when the response is not normally distributed can be challenging, particularly in terms of the computation required to find an optimal design. The application of copulas and marginal modelling provides a computationally efficient approach for estimating population-average treatment effects. Motivated by an experiment from materials testing, we develop and demonstrate designs with blocks of size two using copula models. Such designs are also important in applications ranging from microarray experiments to experiments on human eyes or limbs with naturally occurring blocks of size two. We present methodology for design selection, make comparisons to existing approaches in the literature and assess the robustness of the designs to modelling assumptions.
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