Samplers¶
High Level Interface¶

class
psdr.
Sampler
(fun, X=None, fX=None)[source]¶ Generic sampler interface
Parameters:  fun (Function) – Function for which to preform a design of experiments
 X (arraylike (?,m)) – Existing samples from the domain
 fX (arraylike (?,nfun)) – Existing evaluations of the function at the points in X

X
¶ Samples from the function’s domain

fX
¶ Outputs from the function corresponding to samples X

class
psdr.
SequentialMaximinSampler
(fun, L=None, X=None, fX=None)[source]¶ Sequential maximin sampling with a fixed metric
Given a distance metric provided by \(\mathbf{L}\), construct a sequence of samples \(\widehat{\mathbf{x}}_i\) that are local solutions to
\[\widehat{\mathbf{x}}_j = \arg\max_{\mathbf{x} \in \mathcal{D}} \min_{i=1,\ldots,j} \\mathbf{L}(\mathbf{x}  \widehat{\mathbf{x}}_i)\_2.\]Parameters:  fun (Function) – Function for which to preform a design of experiments
 L (arraylike (?, m)) – Matrix defining the metric
 X (arraylike (?,m)) – Existing samples from the domain
 fX (arraylike (?,nfun)) – Existing evaluations of the function at the points in X
Low Level Functions¶

psdr.
seq_maximin_sample
(domain, Xhat, Ls=None, Nsamp=1000, X0=None, slack=0.9)[source]¶ A multiobjective sequential maximin sampling
Given an existing set of samples \(\lbrace \widehat{\mathbf{x}}_j\rbrace_{j=1}^M\subset \mathcal{D}\) from the domain \(\mathcal{D} \subset \mathbb{R}^m\), this algorithm finds a point \(\mathbf{x} \in \mathcal{D}\) that approximately maximizes the distance of the new point to all other points in multiple distance metrics give by matrices \(\mathbf{L}_i \in \mathbb{R}^{m\times m}\)
\[\max_{\mathbf{x} \in \mathcal{D}} \left\lbrace \min_{j=1,\ldots,M} \\mathbf{L}_i (\mathbf{x}  \widehat{\mathbf{x}}_j)\_2 \right\rbrace_{i}\]This algorithm uses
psdr.voronoi_vertex_sample()
to generate local maximizers of this problem for each metric and then tries to greedily satisfy the distance requirements for each metric.A typical use case will have Ls that are of size (1,m) This greedy sequential approach for constructing a maximin design is the CoffeeHouse Designs of Muller [Mul01]. However, the approach of Muller allows for a generic nonlinear solve for each sample point. Here though we restrict the domain to a polytope specified by linear inequalities so we can invoke
psdr.voronoi_vertex_sample()
to solve each step.Parameters:  domain (Domain) – The domain from which we will be sampling
 Xhat (arraylike (M, m)) – Previously existing samples from the domain
 Ls (list of arraylike (?, m) matrices, optional) – The weight matrix (e.g., Lipschitz matrix) corresponding to each metric; defaults to the identity matrix
 Nsamp (int, optional (default 1000)) – Number of samples to use when finding Voronoi vertices
 slack (float [0,1], optional (default 0.1)) – Rather than taking the point that maximizes the product of the distances in each metric, we choose the point x with greatest unweighted Euclidean distance from those candidates that are at least slack times the score of the best.
References
[Mul01] CoffeeHouse Designs. Werner G. Muller in Optimimum Design 2000, A. Atkinson et al. eds., 2001

psdr.
fill_distance_estimate
(domain, Xhat, L=None, Nsamp=1000, X0=None)[source]¶ Estimate the fill distance of the points Xhat in the domain
The fill distance (Def. 1.4 of [Wen04]) or dispersion [LC05] is the furthest distance between any point \(\mathbf{x} \in \mathcal{D}\) and a set of points \(\lbrace \widehat{\mathbf{x}}_j \rbrace_{j=1}^m \subset \mathcal{D}\):
\[\sup_{\mathbf{x} \in \mathcal{D}} \min_{j=1,\ldots,M} \\mathbf{L}(\mathbf{x}  \widehat{\mathbf{x}}_j)\_2.\]Similar to
psdr.seq_maximin_sample()
, this usespsdr.voronoi_vertex_sample()
to find a subset of local maximizers and returns the best of these.Parameters:  domain (Domain) – Domain on which to compute the dispersion
 Xhat (arraylike (?, m)) – Existing samples on the domain
 L (arraylike (?, m) optional) – Matrix defining the distance metric on the domain
 Nsamp (int, default 1e4) – Number of samples to use for vertex sampling
 X0 (arraylike (?, m)) – Samples from the domain to use in
psdr.voronoi_vertex_sample()
Returns: d – Fill distance lower bound
Return type: float
References
[Wen04] Scattered Data Approximation. Holger Wendland. Cambridge University Press, 2004. https://doi.org/10.1017/CBO9780511617539 [LC05] Iteratively Locating Voronoi Vertices for Dispersion Estimation Stephen R. Lindemann and Peng Cheng Proceedings of the 2005 Interational Conference on Robotics and Automation