skip to main content

Title: Latent variable Gaussian process models: A rank‐based analysis and an alternative approach

Gaussian process (GP) models have been extended to emulate expensive computer simulations with both qualitative/categorical and quantitative/continuous variables. Latent variable (LV) GP models, which have been recently developed to map each qualitative variable to some underlying numerical LVs, have strong physics‐based justification and have achieved promising performance. Two versions use LVs in Cartesian (LV‐Car) space and hyperspherical (LV‐sph) space, respectively. Despite their success, the effects of these different LV structures are still poorly understood. This article illuminates this issue with two contributions. First, we develop a theorem on the effect of the ranks of the qualitative factor correlation matrices of mixed‐variable GP models, from which we conclude that the LV‐sph model restricts the interactions between the input variables and thus restricts the types of response surface data with which the model can be consistent. Second, following a rank‐based perspective like in the theorem, we propose a new alternative model named LV‐mix that combines the LV‐based correlation structures from both LV‐Car and LV‐sph models to achieve better model flexibility than them. Through extensive case studies, we show that LV‐mix achieves higher average accuracy compared with the existing two.

more » « less
Author(s) / Creator(s):
 ;  ;  ;  
Publisher / Repository:
Wiley Blackwell (John Wiley & Sons)
Date Published:
Journal Name:
International Journal for Numerical Methods in Engineering
Page Range / eLocation ID:
p. 4007-4026
Medium: X
Sponsoring Org:
National Science Foundation
More Like this
  1. null (Ed.)
    Transformation-based methods have been an attractive approach in non-parametric inference for problems such as unconditional and conditional density estimation due to their unique hierarchical structure that models the data as flexible transformation of a set of common latent variables. More recently, transformation-based models have been used in variational inference (VI) to construct flexible implicit families of variational distributions. However, their use in both nonparametric inference and variational inference lacks theoretical justification. We provide theoretical justification for the use of non-linear latent variable models (NL-LVMs) in non-parametric inference by showing that the support of the transformation induced prior in the space of densities is sufficiently large in the L1 sense. We also show that, when a Gaussian process (GP) prior is placed on the transformation function, the posterior concentrates at the optimal rate up to a logarithmic factor. Adopting the flexibility demonstrated in the non-parametric setting, we use the NL-LVM to construct an implicit family of variational distributions, deemed GP-IVI. We delineate sufficient conditions under which GP-IVI achieves optimal risk bounds and approximates the true posterior in the sense of the Kullback–Leibler divergence. To the best of our knowledge, this is the first work on providing theoretical guarantees for implicit variational inference. 
    more » « less
  2. Development of safer drugs based on epigenetic modifiers, e.g., histone deacetylase inhibitors (HDACi), requires better understanding of their effects on cardiac electrophysiology. Using RNAseq data from the genotype-tissue-expression database (GTEx), we created models that link the abundance of acetylation enzymes (HDAC/SIRT/HATs), and the gene expression of ion channels (IC) via select cardiac transcription factors (TFs) in male and female adult human hearts (left ventricle, LV). Gene expression data (transcripts per million, TPM) from GTEx donors (21–70 y.o.) were filtered, normalized and transformed to Euclidian space to allow quantitative comparisons in 84 female and 158 male LVs. Sex-specific partial least-square (PLS) regression models, linking gene expression data for HDAC/SIRT/HATs to TFs and to ICs gene expression, revealed tight co-regulation of cardiac ion channels by HDAC/SIRT/HATs, with stronger clustering in the male LV. Co-regulation of genes encoding excitatory and inhibitory processes in cardiac tissue by the acetylation modifiers may help explain their predominantly net-neutral effects on cardiac electrophysiology. ATP1A1 , encoding for the Na/K pump, represented an outlier—with orthogonal regulation by the acetylation modifiers to most of the ICs. The HDAC/SIRT/HAT effects were mediated by strong (+) TF regulators of ICs, e.g., MEF2A and TBX5 , in both sexes. Furthermore, for male hearts, PLS models revealed a stronger (+/-) mediatory role on ICs for NKX25 and TGF1B/KLF4 , respectively, while RUNX1 exhibited larger (-) TF effects on ICs in females. Male-trained PLS models of HDAC/SIRT/HAT effects on ICs underestimated the effects on some ICs in females. Insights from the GTEx dataset about the co-expression and transcriptional co-regulation of acetylation-modifying enzymes, transcription factors and key cardiac ion channels in a sex-specific manner can help inform safer drug design. 
    more » « less
  3. null (Ed.)
    Abstract The data-driven approach is emerging as a promising method for the topological design of multiscale structures with greater efficiency. However, existing data-driven methods mostly focus on a single class of microstructures without considering multiple classes to accommodate spatially varying desired properties. The key challenge is the lack of an inherent ordering or “distance” measure between different classes of microstructures in meeting a range of properties. To overcome this hurdle, we extend the newly developed latent-variable Gaussian process (LVGP) models to create multi-response LVGP (MR-LVGP) models for the microstructure libraries of metamaterials, taking both qualitative microstructure concepts and quantitative microstructure design variables as mixed-variable inputs. The MR-LVGP model embeds the mixed variables into a continuous design space based on their collective effects on the responses, providing substantial insights into the interplay between different geometrical classes and material parameters of microstructures. With this model, we can easily obtain a continuous and differentiable transition between different microstructure concepts that can render gradient information for multiscale topology optimization. We demonstrate its benefits through multiscale topology optimization with aperiodic microstructures. Design examples reveal that considering multiclass microstructures can lead to improved performance due to the consistent load-transfer paths for micro- and macro-structures. 
    more » « less
  4. Abstract. Plume-SPH provides the first particle-based simulation ofvolcanic plumes. Smoothed particle hydrodynamics (SPH) has several advantagesover currently used mesh-based methods in modeling of multiphase freeboundary flows like volcanic plumes. This tool will provide more accurateeruption source terms to users of volcanic ash transport anddispersion models (VATDs), greatly improving volcanic ash forecasts. The accuracy ofthese terms is crucial for forecasts from VATDs, and the 3-D SPH modelpresented here will provide better numerical accuracy. As an initial effortto exploit the feasibility and advantages of SPH in volcanic plume modeling,we adopt a relatively simple physics model (3-D dusty-gas dynamic modelassuming well-mixed eruption material, dynamic equilibrium and thermodynamicequilibrium between erupted material and air that entrained into the plume,and minimal effect of winds) targeted at capturing the salient features of avolcanic plume. The documented open-source code is easily obtained andextended to incorporate other models of physics of interest to the largecommunity of researchers investigating multiphase free boundary flows ofvolcanic or other origins.

    The Plume-SPH code ( also incorporates several newly developed techniques inSPH needed to address numerical challenges in simulating multiphasecompressible turbulent flow. The code should thus be also of general interestto the much larger community of researchers using and developing SPH-basedtools. In particular, the SPHε turbulence model is used to capturemixing at unresolved scales. Heat exchange due to turbulence is calculated bya Reynolds analogy, and a corrected SPH is used to handle tensile instabilityand deficiency of particle distribution near the boundaries. We alsodeveloped methodology to impose velocity inlet and pressure outlet boundaryconditions, both of which are scarce in traditional implementations of SPH.

    The core solver of our model is parallelized with the message passinginterface (MPI) obtaining good weak and strong scalability using novel techniquesfor data management using space-filling curves (SFCs), object creationtime-based indexing and hash-table-based storage schemes. These techniques areof interest to researchers engaged in developing particles in cell-typemethods. The code is first verified by 1-D shock tube tests, then bycomparing velocity and concentration distribution along the central axis andon the transverse cross with experimental results of JPUE (jet or plume thatis ejected from a nozzle into a uniform environment). Profiles of severalintegrated variables are compared with those calculated by existing 3-D plumemodels for an eruption with the same mass eruption rate (MER) estimated forthe Mt. Pinatubo eruption of 15 June 1991. Our results are consistent withexisting 3-D plume models. Analysis of the plume evolution processdemonstrates that this model is able to reproduce the physics of plumedevelopment.

    more » « less
  5. Summary

    Distyly is an intriguing floral adaptation that increases pollen transfer precision and restricts inbreeding. It has been a model system in evolutionary biology since Darwin. Although theS‐locus determines the long‐ and short‐styled morphs, the genes were unknown inTurnera. We have now identified these genes.

    We used deletion mapping to identify, and then sequence,BACclones and genome scaffolds to constructS/shaplotypes. We investigated candidate gene expression, hemizygosity, and used mutants, to explore gene function.

    Thes‐haplotype possessed 21 genes collinear with a region of chromosome 7 of grape. TheS‐haplotype possessed three additional genes and two inversions.TsSPH1was expressed in filaments and anthers,TsYUC6in anthers andTsBAHDin pistils. Long‐homostyle mutants did not possessTsBAHDand a short‐homostyle mutant did not expressTsSPH1.

    Three hemizygous genes appear to determine S‐morph characteristics inT. subulata. Hemizygosity is common to all distylous species investigated, yet the genes differ. The pistil candidate gene,TsBAHD, differs from that ofPrimula, but both may inactivate brassinosteroids causing short styles.TsYUC6is involved in auxin synthesis and likely determines pollen characteristics.TsSPH1is likely involved in filament elongation. We propose an incompatibility mechanism involvingTsYUC6andTsBAHD.

    more » « less