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Register a new userMarkov basis for design of experiments with three-level factors
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Satoshi Aoki
Publication date
2008
fields
Mathematical Statistics
and research's language is
English
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We consider Markov basis arising from fractional factorial designs with three-level factors. Once we have a Markov basis, $p$ values for various conditional tests are estimated by the Markov chain Monte Carlo procedure. For designed experiments with a single count observation for each run, we formulate a generalized linear model and consider a sample space with the same sufficient statistics to the observed data. Each model is characterized by a covariate matrix, which is constructed from the main and the interaction effects we intend to measure. We investigate fractional factorial designs with $3^{p-q}$ runs noting correspondences to the models for $3^{p-q}$ contingency tables.
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Reproducible research in Machine Learning has seen a salutary abundance of progress lately: workflows, transparency, and statistical analysis of validation and test performance. We build on these efforts and take them further. We offer a principled experimental design methodology, based on linear mixed models, to study and separate the effects of multiple factors of variation in machine learning experiments. This approach allows to account for the effects of architecture, optimizer, hyper-parameters, intentional randomization, as well as unintended lack of determinism across reruns. We illustrate that methodology by analyzing Matching Networks, Prototypical Networks and TADAM on the miniImagenet dataset.
This paper presents a new Gaussian process (GP) surrogate modeling for predicting the outcome of a physical experiment where some experimental inputs are controlled by other manipulating factors. Particularly, we are interested in the case where the control precision is not very high, so the input factor values vary significantly even under the same setting of the corresponding manipulating factors. The case is observed in our main application to carbon nanotube growth experiments, where one experimental input among many is manipulated by another manipulating factors, and the relation between the input and the manipulating factors significantly varies in the dates and times of operations. Due to this variation, the standard GP surrogate that directly relates the manipulating factors to the experimental outcome does not provide a great predictive power on the outcome. At the same time, the GP model relating the main factors to the outcome directly is not appropriate for the prediction purpose because the main factors cannot be accurately set as planned for a future experiment. Motivated by the carbon nanotube example, we propose a two-tiered GP model, where the bottom tier relates the manipulating factors to the corresponding main factors with potential biases and variation independent of the manipulating factors, and the top tier relates the main factors to the experimental outcome. Our two-tier model explicitly models the propagation of the control uncertainty to the experimental outcome through the two GP modeling tiers. We present the inference and hyper-parameter estimation of the proposed model. The proposed approach is illustrated with the motivating example of a closed-loop autonomous research system for carbon nanotube growth experiments, and the test results are reported with the comparison to a benchmark method, i.e. a standard GP model.
How to measure the incremental Return On Ad Spend (iROAS) is a fundamental problem for the online advertising industry. A standard modern tool is to run randomized geo experiments, where experimental units are non-overlapping ad-targetable geographical areas (Vaver & Koehler 2011). However, how to design a reliable and cost-effective geo experiment can be complicated, for example: 1) the number of geos is often small, 2) the response metric (e.g. revenue) across geos can be very heavy-tailed due to geo heterogeneity, and furthermore 3) the response metric can vary dramatically over time. To address these issues, we propose a robust nonparametric method for the design, called Trimmed Match Design (TMD), which extends the idea of Trimmed Match (Chen & Au 2019) and furthermore integrates the techniques of optimal subset pairing and sample splitting in a novel and systematic manner. Some simulation and real case studies are presented. We also point out a few open problems for future research.
In two-way contingency tables we sometimes find that frequencies along the diagonal cells are relatively larger(or smaller) compared to off-diagonal cells, particularly in square tables with the common categories for the rows and the columns. In this case the quasi-independence model with an additional parameter for each of the diagonal cells is usually fitted to the data. A simpler model than the quasi-independence model is to assume a common additional parameter for all the diagonal cells. We consider testing the goodness of fit of the common diagonal effect by Markov chain Monte Carlo (MCMC) method. We derive an explicit form of a Markov basis for performing the conditional test of the common diagonal effect. Once a Markov basis is given, MCMC procedure can be easily implemented by techniques of algebraic statistics. We illustrate the procedure with some real data sets.
We study approaches to robust model-based design of experiments in the context of maximum-likelihood estimation. These approaches provide robustification of model-based methodologies for the design of optimal experiments by accounting for the effect of the parametric uncertainty. We study the problem of robust optimal design of experiments in the framework of nonlinear least-squares parameter estimation using linearized confidence regions. We investigate several well-known robustification frameworks in this respect and propose a novel methodology based on multi-stage robust optimization. The proposed methodology aims at problems, where the experiments are designed sequentially with a possibility of re-estimation in-between the experiments. The multi-stage formalism aids in identifying experiments that are better conducted in the early phase of experimentation, where parameter knowledge is poor. We demonstrate the findings and effectiveness of the proposed methodology using four case studies of varying complexity.
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