{Random sampling is the default approach, this is where across the input domain of interest, we just choose to select samples randomly (perhaps uniformly, or if we believe there’s an underlying distribution
Let the input values \(\mathbf{ x}_1, \dots, \mathbf{ x}_n\) be a random sample from \(f(\mathbf{ x})\). This method of sampling is perhaps the most obvious, and an entire body of statistical literature may be used in making inferences regarding the distribution of \(Y(t)\).
Using stratified sampling, all areas of the sample space of \(\mathbf{ x}\) are represented by input values. Let the sample space \(S\) of \(\mathbf{ x}\) be partitioned into \(I\) disjoint strata \(S_t\). Let \(\pi = P(\mathbf{ x}\in S_i)\) represent the size of \(S_i\). Obtain a random sample \(\mathbf{ x}_{ij}\), \(j = 1, \dots, n\) from \(S_i\). Then of course the \(n_i\) sum to \(n\). If \(I = 1\), we have random sampling over the entire sample space.
The same reasoning that led to stratified sampling, ensuring that all portions of \(S\) were sampled, could lead further. If we wish to ensure also that each of the input variables \(\mathbf{ x}_k\) has all portions of its distribution represented by input values, we can divide the range of each \(\mathbf{ x}_k\) into \(n\) strata of equal marginal probability \(1/n\), and sample once from each stratum. Let this sample be \(\mathbf{ x}_{kj}\), \(j = 1, \dots, n\). These form the \(\mathbf{ x}_k\) component, \(k = 1, \dots , K\), in \(\mathbf{ x}_i\), \(i = 1, \dots, n\). The components of the various \(\mathbf{ x}_k\)’s are matched at random. This method of selecting input values is an extension of quota sampling (Steinberg 1963), and can be viewed as a \(K\)-dimensional extension of Latin square sampling (Raj 1968).
Introduce your own surrogate models.
To building your own model see this notebook.
while stopping condition is not met:
optimize acquisition function
evaluate user function
update model with new observation
https://pythonhosted.org/pyDOE/
\[ f(x) = (6x-2)^2\sin(12 x-4). \]
\[ a_{US}(\mathbf{ x}) = \sigma^2(\mathbf{ x}). \]
\[ \begin{align*} a_{\text{IVR}} & = \int_{\mathbb{X}}[\sigma^2(\mathbf{ x}') - \sigma^2(\mathbf{ x}'; \mathbf{ x})]\text{d}\mathbf{ x}' \\ & \approx \frac{1}{\# \text{samples}}\sum_i^{\# \text{samples}}[\sigma^2(\mathbf{ x}_i) - \sigma^2(\mathbf{ x}_i; \mathbf{ x})]. \end{align*} \]
\[ a_{LCB} \approx \frac{1}{\# \text{samples}}\sum_i^{\# \text{samples}}\frac{k^2(\mathbf{ x}_i, \mathbf{ x})}{\sigma^2(\mathbf{ x})}. \]
GPyOpt
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