It’s a lot of splitting hairs about subtly different geometric criteria, coupled with vastly different highly customized solving algorithms, producing designs that are visually and practically indistinguishable from one another. Plenty of other model-free, space-filling designs enjoy wide popularity, but they’re mostly variations on a theme. They seek spread in terms of relative distance. Maximin designs are more deterministic, even if many solvers for such designs deploy stochastic search. LHSs are random, so they disperse in a probabilistic sense, targeting a certain uniformity property. Both are based on geometric criteria but offer optimal spread in different senses. Our development will focus on variations between, and combinations of, two of the most popular space-filling schemes: Latin hypercube sampling (LHS), and maximin distance designs. A spread of training examples, the thinking goes, will ultimately yield fitted models which smooth/interpolate/extrapolate best, leading to more accurate predictions at out-of-sample testing locations.
Here we seek so-called space-filling designs, ones which spread out points with the aim of encouraging a diversity of data once responses are observed. Later in Chapter 6 we’ll develop model-specific analogs and find striking similarity, and in some sense inferiority – a bizarre result considering their optimality – compared to these model-free analogs. Designs here are model-free, meaning that we don’t need to know anything about the (Gaussian process) models we intend to use with them, except in the loose sense that those are highly flexible models which impose limited structure on the underlying data they’re trained on. One of the goals here is pragmatic from an organizational perspective: to have some simple, good designs for illustrations and comparisons in later chapters. Nonparametric spatial regression, emphasizing Gaussian processes in Chapter 5, benefits from a more agnostic approach to design compared to classical, linear modeling-based, response surface methods. This segment puts the cart before the horse a little. A Numerical Linear Algebra for Fast GPs.10.3.4 Optimization, level sets, calibration and more.10.3.1 Integrated mean-squared prediction error.10.1.2 Efficient inference and prediction under replication.10.1 Replication and stochastic kriging.9.3.4 Global/local multi-resolution effect.9.2.3 Regression tree extensions, off-shoots and fix-ups.9.2 Partition models and regression trees.9.1.3 Practical Bayesian inference and UQ.9.1.2 Sharing load between mean and variance.8.2.3 First-order and total sensitivity.7.3.5 Augmented Lagrangian Bayesian optimization (ALBO).7.2.5 Illustrating conditional improvement and noise.6.2.2 A more aggregate criteria: active learning Cohn.6.2.1 Whack-a-mole: active learning MacKay.6.1.2 Minimizing predictive uncertainty.5.3 Some interpretation and perspective.5.2.4 Lengthscale: rate of decay of correlation.5.2.3 Derivative-based hyperparameter optimization.3.2.4 Confidence in the stationary point.2.1.2 Sequential design and nonstationary surrogate modeling.1.1.3 General models, inference and sequential design.I successfully used this method with sample sizes as low as 25. Also, you're not restricted to sample sizes of $6n$. This has the advantage over LHC that you can decide to add more samples later, while retaining relatively even sample coverage, and low correlations between variables. Basically, you take a sequence over the real space $^5$, and then map each dimension to your variables (so if you get something in the lower half of your $$ dimension for your first 2-level variable, then you choose level 1, etc.). If you expect that your effect size is going to be small relative to noise, then choose a larger sample size.Īnother method that might be sensible is to use a Low-discrepancy sequence, like the Sobol sequence. The number of samples you choose is up to you, but more samples will give you more reliable results, and will also help avoid correlation between variables (you should check this when you decide what your samples are, before you actually take them). However, you can do LHC if you use $6n$ (lowest common multiple of 3 and 2) levels, and then map that to your 2- and 3-level spaces. You can't technically do standard LHC sampling, or orthogonal sampling, because it requires each dimension to have the same number of levels. Depending on your experiment (and the difficulty of taking samples), you should ideally just sample everything.
The total number of sample combinations you have is $2\times 3 \times 2 \times 3 \times 3 = 108$ (or what ever).
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