Reports

Reports and a few talks

Most of these papers are based upon work supported by the National Science Foundation under Grants: IIS-1837931, DMS-1521145, DMS-1407397, DMS-0906056, DMS-0604939, DMS-0306612, and DMS-0072445. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.

Slides from talks are beside some of the articles. More talks are here.

There is work on empirical likelihood, Monte Carlo & quasi-Monte Carlo, computer experiments, transposable data, hypothesis testing and bioinformatics. You may have to scroll down for some of those.

Papers by year of initial posting

Each article goes under the year when it was first written, usually as a technical report. Revisions don't usually make an article move up the list.

2019

• C. R. Hoyt and A. B. Owen
Mean dimension of ridge functions PDF
Ridge functions $$f(\boldsymbol{x})=g(\Theta^{\mathsf{T}}\boldsymbol{x})$$ depend on d dimensional inputs only through an $$r\ll d$$ projection. If g is Lipschitz, then f has bounded mean dimension as $$d\to\infty$$ making it potentially very amenable to quasi-Monte Carlo integration. For discontinuous g, the mean dimension depends on sparsity of $$\Theta$$ and could grow like $$\sqrt{d}$$. Pre-integration of f then improves its smoothness and can reduce mean dimension to O(1) if the pre-integrated variable's importance does not vanish as $$d\to\infty$$. This article is mostly about integration over $$\mathbb{R}^d$$ under a spherical Gaussian measure.
• A. Gelman, B. Haig, C. Hennig, A. B. Owen, R. Cousins, S. Young, C. Robert, C. Yanovsky, E. J. Wagenmakers, R. Kennet and D. Lakeland
Many perspectives on Deborah Mayo's "Statistical Inference as Severe Testing: How to Get Beyond the Statistics Wars" arXiv
• Owen, A. B. and Zhou, Y.
The square root rule for adaptive importance sampling PDF | arXiv
Suppose you have uncorrelated unbiased estimates $$\hat\mu_k, 1\leqslant k\leqslant K$$ with variances that decay like $$k^{-y}$$ for $$0\leqslant y\leqslant 1$$. Not knowing $$y$$, you weight them proportionally to $$k^x$$ for $$0\leqslant x\leqslant 1$$. If you choose $$x=1/2$$, your estimate will remain unbiased and will have variance at most 9/8 times that of the unknown best unbiased combination. Weighting by an estimated variance would be prone to bias in rare event settings because the sample values are prone to being correlated with the variance estimates.
This theorem comes from the appendix of an unpublished technical report from 1999.

2018

• Owen, A. B.
Unreasonable effectiveness of Monte Carlo PDF
This is a comment on a paper by Briol, Oates, Girolami, Osborne and Sejdinovic entitled "Probabilistic Integration: A Role in Statistical Computation?"
My take is that the Bayesian approach holds tremendous promise but is better directed at more complicated problems than estimating an integral. For instance most Bayesian computation is now done by frequentist MCMC methods. Here is their rejoinder. The paper, all discussions, and rejoinder will appear in Statistical Science.
• Owen, A. B. and Varian, H.
Optimizing the tie-breaker regression discontinuity design PDF
We consider settings where one could use either a Randomized Controlled Trial (RCT) or a Regression Discontinuity Design (RDD) to analyze the causal impact a treatment that offers customers some sort of bonus or incentive. The RDD is expected to have superior immediate payoff on the customers under study. The RCT is well known to have greater statistical efficiency. We study a hybrid tie-breaker design in which a fraction $$\Delta\in(0,1)$$ of the customers get a randomized treatment. We find statistical efficiency increases monotonically with $$\Delta$$ in a two regression line model. The cost of experimentation decreases monotonically in $$\Delta$$ and we find an expression for optimizing the tradeoff.
This is work I did for Google, and it was not part of my Stanford duties.
• Dobriban, E. and Owen, A. B.
Deterministic parallel analysis PDF
It is really hard to pick the number k of factors in a factor analysis. Parallel analysis is one of the most popular ways. People say it works well. In Buja and Eyoboglu's version we take a data matrix X and apply uniform random permutations to each column some number of times, computing the factor analysis each time. If the real first eigenvalue is larger than say 95% of the simulated ones then we take $$k\geqslant 1$$ and look at the second eigenvalues. And so on. This paper is about how to do that without actually running all those permutations which are expensive and introduce unneeded randomness into the outcome. Instead we can use random matrix theory to determine where the upper edge of the spectrum would be if there were no correlations among the data.
• A. Ben Abdellah, P. L'Ecuyer, A. B. Owen, F. Puchhammer
Density estimation by randomized quasi-Monte Carlo arXiv
We have a deterministic function $$x = g(\boldsymbol{u})$$ on the unit cube $$(0,1)^s$$, or some other domain that can be mapped to from the unit cube. We want to estimate the probability density function of $$x=g(\boldsymbol{u})$$ with respect to $$\boldsymbol{u}\sim U(0,1)^s$$. We do this by kernel density estimation and by histograms applied to randomized quasi-Monte Carlo input points in the unit cube. Using RQMC reduces the variance leaving the bias unaffected. That makes it possible to use smaller bandwidths in the KDE. The gain can be mitigated by the reduced smoothness of the implied integrand when h is small. There can be large improvements in low dimensions.
• Rosenman, E. and Baiocchi, M. and Owen, A. B.
Propensity score methods for merging observational and experimental datasets PDF
Consider a large observational dataset (ODB) and a much smaller randomized controlled trial (RCT) about the same treatement intervention. The RCT may support better causal conclusions, while the ODB has much more data and may have better external validity (a more representative sample). We consider merging the two, using the propensity score that the RCT data would have had, had they been in the ODB. We develop two approaches, comparing them by simulations and by the bias and variance they attain in a delta method approximation. Even a small RCT can greatly improve the estimation of treatment effects over what the ODB alone can do.
• Owen, A. B and Glynn, P. (editors)
Monte Carlo and Quasi-Monte Carlo Methods MCQMC, Stanford CA, August 2016
Springer web site for this book
The proceedings volume of MCQMC 2016 has been published by Springer. It has 26 contributions on Monte Carlo and quasi-Monte Carlo. It includes 3 tutorials on QMC, written by Fred Hickernell, Pierre L'Ecuyer and Frances Kuo.

2017

• Owen, A. B., Chertkov, M. and Maximov, Y.
Importance sampling the union of rare events with an application to power systems analysis arXiv
We consider importance sampling to estimate the probability $$\mu$$ of a union of $$J$$ rare events $$H_j$$ defined by a random variable $$\boldsymbol{x}$$. The sampler we study has been used in spatial statistics, genomics and combinatorics going back at least to Frigessi and Vercellis (1985). It works by sampling one event at random, then sampling $$\boldsymbol{x}$$ conditionally on that event happening and it constructs an unbiased estimate of $$\mu$$ by multiplying an inverse moment of the number of occuring events by the union bound. We prove some variance bounds for this sampler. For a sample size of $$n$$, it has a variance no larger than $$\mu(\bar\mu-\mu)/n$$ where $$\bar\mu$$ is the union bound. It also has a coefficient of variation no larger than $$\sqrt{(J+J^{-1}-2)/(4n)}$$ regardless of the overlap pattern among the events. Our motivating problem comes from power system reliability, where the phase differences between connected nodes have a joint Gaussian distribution and the $$J$$ rare events arise from unacceptably large phase differences. In the grid reliability problems even some events defined by $$5772$$ constraints in $$326$$ dimensions, with probability below $$10^{-22}$$, are estimated with a coefficient of variation of about $$0.0024$$ with only $$n=10{,}000$$ sample values.
• Owen, A. B.
Effective dimension of some weighted pre-Sobolev spaces with dominating mixed partial derivatives arXiv| PDF of SINUM article
Old title: Effective dimension of weighted Sobolev spaces: non-periodic case
This paper considers two notions of effective dimension for quadrature in weighted pre-Sobolev spaces with dominating mixed partial derivatives. We begin by finding a ball in those spaces just barely large enough to contain a function with unit variance. If no function in that ball has more than $$\varepsilon$$ of its variance from ANOVA components involving interactions of order $$s$$ or more, then the space has effective dimension at most $$s$$ in the superposition sense. A similar truncation sense notion replaces the cardinality of the ANOVA component by the largest index it contains. Some Poincar\'e type inequalities are used to bound variance components by multiples of these space's squared norm and those in turn provide bounds on effective dimension. Very low effective dimension in the superposition sense holds for some spaces defined by product weights in which quadrature is strongly tractable. The superposition dimension is $$O( \log(1/\varepsilon)/\log(\log(1/\varepsilon)))$$ just like the superposition dimension used in the multidimensional decomposition method. Surprisingly, even spaces where all subset weights are equal, regardless of their cardinality or included indices, have low superposition dimension in this sense. This paper does not require periodicity of the integrands.
• Owen, A. B.
On $$L^2$$ norms of derivatives of orthogonal polynomials with respect to Sobolev inner products PDF
For $$\lambda\ge0$$, let $$\langle f,g\rangle_\lambda := \int_{-1}^1 f(x)g(x) d x +\lambda\int_{-1}^1 f'(x)g'(x) d x$$ define an inner product for differentiable functions on $$[-1,1]$$. For $$n\ge0$$, let $$S_n=S_n^\lambda$$ be the orthogonal polynomials of degree $$n$$ obtained by applying the Gram-Schmidt algorithm in this inner product to monomials, normalized so that $$S_n(1)=1$$. Then the derivatives $$S_n'$$ are given as explicit Legendre series, it is proved that $$\arg\min_{n\ge1}\int_{-1}^1 S_n'(x)^2 d x/\int_{-1}^1S_n(x)^2 d x=1$$, and an expression is given for $$\int_{-1}^1 S'_n(x)S'_{n'}(x) d x$$.
• Owen, A. B.
A randomized Halton algorithm in R PDF
Randomized quasi-Monte Carlo (RQMC) sampling can bring orders of magnitude reduction in variance compared to plain Monte Carlo (MC) sampling. The extent of the efficiency gain varies from problem to problem and can be hard to predict. This article presents an R function rhalton that produces scrambled versions of Halton sequences. On some problems it brings efficiency gains of several thousand fold. On other problems, the efficiency gain is minor. The code is designed to make it easy to determine whether a given integrand will benefit from RQMC sampling. An RQMC sample of n points in $$[0,1]^d$$ can be extended later to a larger n and/or d.
• Gao, K. and Owen, A. B.
Estimation and inference for very large linear mixed effects models PDF | PDF (supplement) | Slides | Code
Linear mixed models with large imbalanced crossed random effects structures pose severe computational problems for maximum likelihood estimation and for Bayesian analysis. The costs can grow as fast as $$N^{3/2}$$ when there are N observations. Such problems arise in any setting where the underlying factors satisfy a many to many relationship (instead of a nested one) and in electronic commerce applications, the N can be quite large. Methods that do not account for the correlation structure can greatly underestimate uncertainty. We propose a method of moments approach that takes account of the correlation structure and that can be computed at O(N) cost. The method of moments is very amenable to parallel computation and it does not require parametric distributional assumptions, tuning parameters or convergence diagnostics. For the regression coefficients, we give conditions for consistency and asymptotic normality as well as a consistent variance estimate. For the variance components, we give conditions for consistency and we use consistent estimates of a mildly conservative variance estimate. All of these computations can be done in O(N) work. We illustrate the algorithm with some data from Stitch Fix where the crossed random effects correspond to clients and items. This one appeared in 2016, but has been very substantially revised.
Katelyn Gao's thesis

2016

• Owen, A. B. and Launay, T.
Multibrand geographic experiments PDF
It is about geographic experiments. The idea is to run B experiments on G regions at once and then use a Stein shrinkage method from Xie, Kou and Brown (2012), or Bayesian statistics via Stan, to pool the results. Pooling means you can work with smaller experiments than if you did them one at a time. B and G are both even. In the design, each experiment is toggled up in G/2 regions and down in G/2 regions. Each region is up for B/2 experiments and down for B/2 experiments. The Bayesian approach seemed to work a bit better. The design is found using a Markov chain studied by Diaconis and Gangolli. For G>B the design is a two level factorial for each experiment and is also a supersaturated design for the regions. It would be nice if the GxB matrix had no duplicate rows or columns and no rows or columns that are exact opposites. The paper provides some sufficient conditions for the existence of such designs (with constructions).
This is work I did for Google, and it was not part of my Stanford duties.
• Owen, A. B.
(new title) Refiltering hypothesis tests to control sign error revised PDF (Aug 2019) | arXiv| WHOA-PSI 2019 slides
A common, though not recommended statistical practice is to report confidence intervals if and only if they exclude a null value of 0. The resulting filtered confidence intervals generally do not have their nominal confidence level. More worryingly, in low power settings they will frequently lie on the wrong side of zero. We assume that the confidence intervals were constructed using an asymptotically Gaussian parameter estimate accompanied by a weakly consistent estimate of its variance. In these cases, we can subject the given confidence interval(s) to a second filtering step such that the probability of a sign error is controled. This refiltering step retains only those confidence intervals that are sufficiently well separated from the origin.
• Gao, K. and Owen, A. B.
Estimation and inference for very large linear mixed effects models PDF | PDF (supplement)
Very large crossed data sets are increasingly common especially in e-commerce. It is often appropriate to model them with crossed random effects. Their size provides challenges for statistical analysis. For such large data sets, the computational costs of estimation and inference should grow at most linearly with the sample size. Commonly used maximum likelihood and Bayesian approaches do not scale at that rate. We propose a method of moments based algorithm that scales linearly and can be easily parallelized at the expense of some loss of statistical efficiency. We apply the algorithm to some data from Stitch Fix where the crossed random effects correspond to clients and items. The random effects analysis is able to account for the increased variance due to intra-client and intra-item correlations in the data. Ignoring the correlation structure can lead to standard error underestimates of over 10-fold for that data. This algorithm is proven to give estimates that are asymptotically consistent and normally distributed. We use a martingale CLT decomposition of U-statistics to establish normality for the variance components.
Katelyn Gao's thesis
• Wang, J., Sabatti, C. and Owen, A. B.
Adaptive filtering multiple testing procedures for partial conjunction hypotheses arXiv | PDF
The partial conjunction null hypothesis $$H_0^{r/n}$$ allows up to n-r+1 related basic null hypotheses to hold. Rejecting it allows us to conclude that at least r of the basic nulls are false, providing a measure of reproducibility. Motivated by genomic problems we consider a setting with a large number M of partial conjunction null hypotheses to test, based on an $$n\times M$$ matrix of p-values. When r>1 the hypothesis $$H_0^{r/n}$$ is composite. Validity versus the case with r-1 alternative hypotheses holding can lead to very conservative tests. We develop a filtering approach for $$H_0^{r/n}$$ based on the M p-values for $$H_0^{(r-1)/n}$$. This filtering approach has greater power than straightforward PC testing. We prove that it can be used to control the familywise error rate, the per family error rate, and the false discovery rate among M PC tests. In simulations we find that our filtering approach properly controls the FDR while achieving good power.
• Owen, A. B. and Prieur, C.
On Shapley value for measuring importance of dependent inputs Revised PDF | arXiv | HAL
This paper makes the case for using Shapley value to quantify the importance of random input variables to a function. Alternatives based on the ANOVA decomposition can run into conceptual and computational problems when the input variables are dependent. Our main goal here is to show that Shapley value removes the conceptual problems. We do this with some simple examples where Shapley value leads to intuitively reasonable nearly closed form values.
Errata: In Theorem 4.1 the condition that $$\Sigma$$ has full rank should instead be that $$\Sigma$$ is positive definite. The matrix in the paragraph following Theorem 4.1 should be $$\bigl(\begin{smallmatrix} 1 & \rho\\\rho & 1\end{smallmatrix}\bigr)$$ not $$\bigl(\begin{smallmatrix} 0 & \rho\\\rho & 0\end{smallmatrix}\bigr)$$.
• Basu, K. and Owen, A. B.
Quasi-Monte Carlo for an integrand with a singularity along a diagonal in the square PDF
A singularity along an arbitrary manifold works differently for QMC than one at an isolated point or along the boundary of the unit cube. We look at a singularity along the diagonal of $$[0,1]^2$$ as a basic example. There are several ways to handle this, but the best one appears to be to transform the integrand into one or more integrands with QMC-friendly singularities (along the boundary of the unit cube, in this case). For a festschrift marking Ian Sloan's 80th birthday.
• He, H. Y., Basu, K., Zhao, Q. and Owen, A. B.
Permutation p‐value approximation via generalized Stolarsky invariance Annals of Statistics| arXiv 1603.02757| latest PDF | Supplement | pipeGS R package on CRAN| data
It is really expensive to get a tiny p value via permutations. For linear test statistics, the permutation p value is the fraction of permuted data vectors in a given spherical cap. Stolarsky's invariance principal from quasi-Monte Carlo sampling gives the root mean squared discrepancy between the observed and expected proportions of points in spherical caps. We use and extend this principal to develop approximate p values with small relative errors. Their accuracy is competitive with saddlepoint approximations, but they come with an error estimate.
Hera He's thesis
• Gao, K. and Owen, A. B.
Efficient moment calculations for variance components in large unbalanced crossed random effects models PDF | supplement
Large unbalanced crossed random effects models provide extreme challenges. Both maximum likelihood and MCMC scale superlinearly in problem size. We look at method of moments estimates that scale linearly, while retaining competitive accuracy with the MLE on problems small enough to allow computation of the MLE.
Katelyn Gao's thesis

2015

• Basu, K. and Owen, A. B.
Transformations and Hardy-Krause variation PDF | arXiv | SINUM
A transformation $$\tau$$ may be used to map the unit cube onto the simplex, disk, sphere and related spaces. For use with QMC integration of f we want $$f\circ\tau$$ to be of bounded variation in the sense of Hardy and Krause. We give such conditions using a multivariable Faa di Bruno formula of Constantine and Savits. A similar condition makes $$f\circ\tau$$ smooth enough to benefit from scrambled net quadrature. We also apply Faa di Bruno to find conditions for importance sampling and Rosenblatt-Hlawka-Muck sequential inversion to be suitable for QMC. Finally we give an importance sampled mapping from the unit cube to the simplex that results in RQMC quadratures with RMSE $$O(n^{-3/2+\epsilon})$$ for a class of functions generalizing polynomials.
Kinjal Basu's thesis
• Owen, A. B.
Statistically efficient thinning of a Markov chain sampler JCGS| revised PDF | arXiv | R code
Thinning or subsampling an MCMC is well known to decrease efficiency if we simply discard computed values. Suppose that it costs one unit of time to advance the Markov chain, and $$\theta$$ units to compute the quantity of interest. If we thin to every k'th value, $$k\geqslant2$$ then we don't have to compute the quantity of interest at every iteration and so we can run the chain longer. If the autocorrelations in the process are non-increasing and non-negative then there is alway a $$\theta$$ large enough to make thinning to every k'th value more efficient than not thinning. Very commonly autocorrelations resemble those of an AR(1) process, $$\rho_\ell = \rho^{|\ell|}$$. This paper gives an algorithm for computing the optimal subsampling factor k for the AR(1) case. When $$\rho$$ is large the optimal thinning value k can be large. When $$\theta$$ is large the efficiency gain can be important. Taking k=1 (no thinning) is optimal for $$\rho>0$$ if and only if $$\theta \leqslant (1-\rho)^2/(2\rho)$$. If $$-1<\rho<0$$ then k=1 is optimal for any $$\theta\geqslant0$$.
• Wang, J, Zhao, Q., Hastie, T. and Owen, A. B.
Confounder adjustment in multiple hypothesis testing arXiv
We look at the problem of large scale hypothesis testing when there are latent variables that induce correlations among the tests, and more seriously, correlate with the primary variable (e.g. treatment) under study. LEAPP handles this by assuming sparsity of effects and RUV-4 handles it by assuming some negative controls: tests known a priori to be null. We provide theoretical gaurantees for versions of these algorithms and generalize them to handle multiple primary variables. When the confounding variables are strong enough, then the methods can perform as well asymptotically as an oracle that observes the latent variables.
Jingshu Wang's thesis | Yunting Sun's thesis
• Wang, J and Owen, A. B.
Admissibility in partial conjunction testing arXiv | PDF (to appear in JASA)
Admissibility of meta-analysis has been well understood since Allan Birnbaum's work in the 1950s. Any valid combined p-value obeying a monotonicity constraint is optimal at some alternative and hence admissible. In an exponential family context, the admissible tests reduce to those with a convex acceptance region. The partial conjunction null hypothesis is that at most r - 1 of n independent component hypotheses are non-null with r = 1 corresponding to a usual meta-analysis. Benjamini and Heller (2008) provide a valid test for this null by ignoring the r - 1 smallest p-values and applying a valid meta-analysis p-value to the remaining n - r + 1 p-values. We provide sufficient conditions for the admissibility of their test among monotone tests. A generalization of their test also provides admissible monotone tests and we show that admissible monotone tests are necessarily of that generalized form. If one does not require monotonicity then their test is no longer admissible, but the dominating tests are too unreasonable to be used in practice.
• Lee, M. and Owen, A. B.
Single nugget Kriging PDF
We propose a method with better predictions at extreme values than the standard method of Kriging. We construct our predictor in two ways: by penalizing the mean squared error through conditional bias and by penalizing the conditional likelihood at the target function value. Our prediction exhibits robustness to model mismatch in the covariance parameters, a desirable feature for computer simulations with a restricted number of data points. Applications on several functions show that our predictor is robust to the non-Gaussianity of the function.
• Minyong Lee's thesis
• Dobriban, E, Fortney, K. Kim, S. K. and Owen, A. B.
Optimal multiple testing under a Gaussian prior on effect sizes Biometrika| arXiv | PDF
We develop a new method for frequentist multiple testing with Bayesian prior information. Our procedure finds a new set of optimal p-value weights called the Bayes weights. Prior information is relevant to many multiple testing problems. Existing methods assume fixed, known effect sizes available from previous studies. However, the case of uncertain information is more usual. For a Gaussian prior on effect sizes, we show that finding the optimal weights is a non-convex problem. Despite the non-convexity, we give an efficient algorithm that solves this problem nearly exactly. We show that our method can discover new loci in genome-wide association studies. On several data sets it compares favorably to other methods. Open source code is available.
• Lawrence, M., Huntley, M. A., Stawiski, E., Owen, A. B., Wu, T. D., Goldstein, L., Cao, Y., Degenhardt, J., Young, J., Guillory, J., Heldens, S., Jackson, A., Seshagiri, S., Gentleman, R.
Genomic variant calling: Flexible tools and a diagnostic data set bioR$$\chi$$iv | Supplement
This is about identifying low frequency genetic variants in tumors. The supplement includes a use of the Cauchy link in binomial regression. A quasi-binomial logistic regression gave overdispersions like $$10^9$$; Cauchy is much better.
• Fortney, K., Dobriban, E., Garagnani, P., Pirazzini, C., Monti, D., Mari, D., Atzmon, G., Barzilai, N., Franceschi, C., Owen, A. B., Kim, S. K.
Genome-wide scan informed by age-related disease identifies loci for exceptional human longevity
We find some new SNPs associated with extreme longevity using frequentist multiple testing based on Bayesian priors from the companion Biometrika paper. PLoS | genetics
• Basu, K. and Owen, A. B.
Scrambled geometric net integration over general product spaces Foundations of Computational Mathematics | arXiv | PDF
We develop scrambled net quadrature over Cartesian products of triangles, disks, spherical triangles and generalizations. For smooth functions on an s-fold product of d-dimensional sets the variance is $$O(n^{-1-2/d}\log(n)^{s-1})$$.
Kinjal Basu's thesis
• He, Z. and Owen, A. B.
Discussion on `Sequential Quasi-Monte-Carlo Sampling', by Gerber & Chopin PDF
Text from copyright form: This is the pre-peer reviewed version of the discussion named above that was published in its final form at JRSS-B.
(Technically this is a comment that was not peer reviewed, but the journal did copy edit what we sent.)

2014

• He, H. Y. and Owen, A. B.
Optimal mixture weights in multiple importance sampling PDF
We show that the mixture weights in multiple importance sampling can be optimized via convex optimization.
• He, Z. and Owen, A. B.
Extensible grids: uniform sampling on a space-filling curve PDF
We study QMC in $$[0,1]^d$$ by QMC and randomized QMC on $$[0,1]$$ followed by applying Hilbert's space-filling curve. We find that the result has the same high-dimensional performance as sampling on a grid. As bad as that is in high dimensions, it is best possible for some hard problems, and the proposed method does not require sample sizes of the form $$n=m^d$$. An RQMC version attains MSE $$O(n^{-1-2/d})$$ for integration of Lipshitz continuous functions.
• Larson, J. L. and Owen, A. B.
Moment based gene set tests BMC bioinformatics| npGSEA Bioconductor package| old PDF
Permutation tests are a popular way to test whether a set of genes is associated with a treatment, or reversing the hoped-for causal arrow, a phenotype. But they are expensive. We develop moment based alternatives that are much faster. Both beta and Gaussian approximations are available for linear statistics (similar to the JG score). We also develop a scaled chisquare approximation to a sum of squared regression coefficients. Such a test was best overall among 261 gene set tests investigated by Ackermann and Strimmer (2009). We illustrate the method on three public data sets related to Parkinson's disease, and find some enriched sets not noticed in the original publications.
• Owen, A. B.
A constraint on extensible quadrature rules Numerische Mathematik preprint
Suppose that the best possible rate for a quadrature problem is $$O(n^{-\alpha})$$ for $$\alpha>1$$, for a simple average of function values. Suppose further that a rate optimal sequence uses sample sizes $$n_k$$. This paper extends an idea from Sobol' (1998) to give a lower bound on $$\rho = n_{k+1}/n_k$$. The bound is between 1 and 2, so it always rules out arithmetic sequences and never rules out sample size doubling. This version (Jan 2015) fixes an error in the proof of Theorem 1. (Neither statement nor conclusion had to change.)
• Basu, K. and Owen, A. B.
Low discrepancy constructions in the triangle SIAM Journal on Numerical Analysis | PDF
We give two explicit constructions of point sets in the triangle with vanishing discrepancy. One adapts the van der Corput sequence to the triangle. It has discrepancy at most 12/$$\sqrt{N}$$. The other scales a regular grid then rotates it through an angle with a badly approximable tangent attaining discrepancy $$O(\log(N)/N)$$. For smooth functions, randomizing the van der Corput sequence gives RMSE $$O(1/N)$$.
Kinjal Basu's thesis
• Owen, A. B. and Roediger, P. A.
The sign of the logistic regression coefficient American Statistician (teacher's corner)| arXiv | PDF
This paper settles a conjecture that Paul Roediger made (with D. M. Ray and B. T. Neyer) in a comment on a paper by Jeff Wu and Y. Tang. In logistic regression on a scalar $$x$$, the MLE of the slope coefficient $$\beta$$ satisfies sign($$\hat\beta$$) = sign($$\bar x_1-\bar x_0$$), where $$\bar x_y$$ is the sample mean of x for Y=y. That this should usually be the case is intuitively obvious. That one cannot wiggle out of it by tweaking the variances, skewnesses and/or outliers in the two x groups is less obvious. One might imagine it follows from the means being sufficient statistics, but they are only conditionally sufficient. Besides it holds also for Probit models and others whose inverse link is the CDF of a log-concave density, and where there is no tiny sufficient statistic conditional or otherwise. There is a generalization to vector valued predictors.

2013

• Owen, A. B.
Sobol' indices and Shapley value PDF
Sobol' indices are used to measure the importance of input variables in black box functions. Shapley value is used by economists to apportion the value of a team's efforts among its individual members. This paper compares the methods. Neither kind of Sobol' index yields the Shapley value for variance explained. Compared to Shapley value, Sobol's lower index ignores interactions while Sobol's upper index overcounts them.
• Billman, D. and He, H. and Owen, A. B.
Grouping tasks and data display items via the non-negative matrix factorization PDF
Our data are a list of 119 tasks that a pilot must perform in flying a modern air liner, 210 input and output variables available to the pilot, and a matrix indicating whether a given IO variable is required for a given task. We use biclustering of this data to aid in designing an interface.
• Owen, A. B. and Dick, J. and Chen. S.
Higher order Sobol' indices original PDF | Published version: Information and Inference 2014(3)59-81
We generalize Sobol' indices from an $$L^2$$ to $$L^p$$ (integer $$p\ge2$$) setting in order to emphasize those variables that most affect extreme values of the function. Our generalizations have integral representations that allow Monte Carlo or quasi-Monte Carlo estimation.
• Chen, A., Owen, A. B. and Shi, M.
Data enriched linear regression arXiv | revised Nov 2014
We have a small high quality data set of (X,Y) values following a Gaussian linear regression, and a potentially much larger data set following a possibly different Gaussian linear regression. We apply a new form of Stein shrinkage to connect the two. It becomes inadmissible to just use the small data set when there are p$$\ge$$5 predictors and the error df $$\ge$$10. The new form of shrinkage outperforms ordinary Stein shrinkage in simulations and we provide an explanation by comparing them both to an oracle.

2012

• Owen, A. B.
Self-concordance for empirical likelihood PDF
Empirical likelihood computations for the mean are typically made through a quadratic extension of logarithm to the interval $$(-\infty,1/n]$$. The quadratic extension is not self-condordant, but a quartic extension is self-concordant. Self concordant functions have Hessians that do not change too rapidly and convex optimization has strong gaurantees under self-concordance.
• Hickernell, F. J., Jiang, L., Yuewei, L. and Owen, A. B.
Guaranteed Conservative Fixed Width Confidence Intervals Via Monte Carlo Sampling PDF
We construct two stage fixed with confidence intervals for the mean. The first stage estimates variance. The second stage estimates the mean. The confidence level holds so long as the random variables have kurtosis below a known/computable bound. We use new Berry-Esseen inequalities of Nefedova and Shevtsova.
• Owen, A. B.
Quasi-regression for heritability PDF
Quasi-regression is employed to estimate missing heritability. The method assumes complete linkage equilibrium, i.e., all SNPs are independent. It then estimates the proportion of heritability by direct moment calcultions. With n subjects and d genes the mean squared error is $$O( 1/n + d^2/n^3 )$$.

• The three papers below grew out of MCQMC 2012 and the 2012 MASCOT NUM meeting.

• Owen, A. B.
Better estimation of small Sobol' sensitivity indices Revised Sept 2012
A new method for estimating Sobol' indices is proposed. © ACM, 2012. This is the author's version of the work. It is posted here by permission of ACM for your personal use. Not for redistribution. The definitive version was published in ACM TOMACS, {VOL\#, ISS\#, (2012)} http://doi.acm.org/10.1145/nnnnnn.n nnnnn" The new method makes use of 3 independent input vectors rather than the usual 2. It attains much greater accuracy on problems where the target Sobol' index is small, even outperforming some oracles which adjust using the true but unknown mean of the function. When the target Sobol' index is quite large, the oracles do better than the new method. The new method attains a better rate of convergence than three others do in an asymptotic limit of effects growing small. Six other asymptotes are considered and different methods prevail in those other limits but usually the effect is in the lead constant only.
Here is a further improvement.
• Owen, A. B.
Variance components and generalized Sobol' indices PDF| slides
This paper introduces generalized Sobol' indices, compares strategies for their estimation, and makes a systematic search for efficient estimators. Of particular interest are contrasts, sums of squares and indices of bilinear form which allow a reduced number of function evaluations compared to alternatives. The bilinear framework includes some efficient estimators from Saltelli (2002) and Mauntz (2002) as well as some new estimators for specific variance components and mean dimensions. This paper also provides a bias corrected version of the estimator of Janon et al. (2012), and extends the bias correction to generalized Sobol' indices. Some numerical comparisons are given.
• Owen, A. B.
Effective dimension for weighted function spaces PDF revised July 2014
This paper was written in 2012. The 2014 revision corrects it as follows: the results actually require periodicity that that 2012 version did not assume.
This paper introduces notions of effective dimension for weighted Sobolev spaces. The space has low effective dimension in the truncation sense if the smallest ball of functions containing a function of variance 1 contains no functions depending materially on high index variables. It has low effective dimension in the superposition sense if that ball has no functions depending materially on higher order interactions. For product weights it is possible to explicitly compute the effective dimension of a space.

2011

• Ma, L, Wong, W. H. and Owen, A. B.
A sparse transmission disequilibrium test for haplotypes based on Bradley-Terry graphs PDF
• Owen, A. B. and Eckles, D.
Bootstrapping data arrays of arbitrary order PDF| slides
McCullagh (2000) showed that there is no bootstrap for the crossed random effects model. But resampling each factor (rows and columns) independently works well and is slightly conservative. Here we replace resampling by reweighting and then extend the result to data arrays of arbitrary order. Reweighting is better suited to large data warehouses than resampling is.

• Gleich, D. G. and Owen, A. B.
Moment based estimation of stochastic Kronecker graph parameters PDF
It is hard to estimate the parameters of Kronecker random graphs by maximum likelihood. Here is a method of moments strategy based on simple feature counts. We find that moments are more robust and give better fits than maximum likelihood.

• Sun, Y. and Zhang, N. and Owen, A. B.
Multiple hypothesis testing, adjusting for latent variables PDF| R package on CRAN| R package tar file
We introduce LEAPP (latent effect adjustment after primary projection), a method for taking account of unmeasured latent variables when doing multiple hypothesis testing. Simulations show good performance compared to alternatives, EIGENSTRAT and SVA (surrogate variable analysis). When applied to the 16 tissue AGEMAP data, LEAPP gives lists of age-related genes with much more reproducibility (over tissues) than the other methods. We prove some results on the LEAPP estimates.
Yunting Sun's thesis

2010

• Owen, A. B.
Moment based estimation of stochastic Kronecker graph parameters Deprecated. See above article with David Gleich.
It is hard to estimate the parameters of Kronecker random graphs by maximum likelihood. Here is a method of moments strategy based on simple feature counts. For large data sets the error is dominated by model lack of fit and so the extra efficiency of likelihood over moments is less important than the speed advantage of moments.

• Chen, S. and Matsumoto, M. and Nishimura, T. and Owen, A. B.
New inputs and methods for Markov chain quasi-Monte Carlo PDF
We present some new ways of generating small CUD sequences. We introduce some embedded antithetic and round trip variance reductions into MCMC and prove that they preserve the CUD property. In some simulations of GARCH and stochastic volatility models, the new methods greatly outperform standard IID sampling. The original publication is available at www.springerlink.com (or will be so, once it has been completed).
Su Chen's thesis

• Dyer, J. and Owen, A.B.
Visualizing bivariate long tailed data revised PDF| slides from NIPS
Suppose that we observe two or more categorical variables with a long tail, such as movies and customers in ratings data. This paper looks at a way to visualize the joint distribution of such data. We use a copula plot based on the observed ranks of the data. We prove under a generative model that the observed ranks are asymptotically close to some underlying true ranks under a Zipf-Mandelbrot-Poisson assumption. Some ratings data show a strong head to tail affinity: busy raters are over represented at rarely rated items and conversely. We present two simple generative models that produce such an effect. One is a saturation model and the other is bipartite preferential attachment. We prove bounds on the marginal distributions for these models.
Justin Dyer's thesis

• Dyer, J. and Owen, A.B.
Correct ordering in the Zipf-Poisson ensemble PDF Revised Jan 2012 PDF
Counted rank data arise commonly, such as most popular baby names, English words and web sites. This paper analyzes the reliability of such ordering. We use a model where $$X_i$$ is Poisson with mean following a Zipf law. We get estimates for the number n of leading items $$i=1\dots n$$ correctly ordered by their observed counts $$X_i$$. If grows at the rate $$(AN/\log(N))^{1/(\alpha+2)}$$ where $$\alpha$$ is the Zipf parameter and $$A = \alpha^2(\alpha+2)/4$$. For a Zipf-Poisson model of the British National Corpus of 100,000,000 words, we estimate that the 72 most frequent words are in their correct order.
Justin Dyer's thesis

• She, Y. and Owen, A.B.
Outlier detection using nonconvex penalized regressions PDF (orig) | PDF (revised)
We put in a dummy variable for all n observations in a regression but regularize their coefficients via a thresholding rule. The result is robust regression that empirically is very good at identifying outliers. A key step is a clean model for which the outliers become the signal and in which BIC is applicable.
Yiyuan She's thesis

2009

• L'Ecuyer, P. and Owen, A.B. (eds)
Monte Carlo and Quasi-Monte Carlo Methods 2008
Proceedings of MCQMC 2008, July 6-11 2008, Montreal Canada.
Springer-Verlag. ISBN 978-3-642-04106-8 List of articles

• S.C. Emerson and Owen, A.B.
Calibration of the empirical likelihood method for a vector mean Electronic Journal of Statistics
This paper presents an approach for getting outside of the 'convex hull problem' in empirical likelihood.
Sarah Emerson's thesis

• Owen, A.B.
Recycling physical random variables EJS
This paper shows how to get n(n-1)/2 pairwise independent random vectors out of just n fully independent ones. Similar constructions are widely used. What is new here is a statistical analysis of the consequences for Monte Carlo sampling: the resulting means are degenerate U-statistics with non-normal limits. Quite surprisingly, their asymptotic distributions come out symmetric, based on recent results on the spectrum of circulant matrices.

• Southworth, L.K., Owen, A.B. and Kim, S.K.
Aging mice show a decreasing correlation of gene expression within genetic modules. PLoS Genetics PDF
As mice age, the correlations among sets of related genes grow weaker.

• Chen, S., Dick, J. and Owen, A.B.
Consistency of Markov chain quasi-Monte Carlo on continuous state spaces PDF
Tribble has made over 1000-fold efficiency improvements by inserting QMC sampling into MCMC problems. The earlier consistency results for use of QMC in MCMC only worked for discrete state spaces. This paper extends them to continuous problems like the ones in Seth Tribble's thesis
Su Chen's thesis

• Xu, Y., Dyer, J.S. and Owen, A.B.
Empirical stationary correlations for semi-supervised learning on graphs PDF| Talk
Many methods for semi-supervised learning on graphs turn out to be forms of kriging. They use a correlation structure derived from the graph, but without taking account of correlations among the observed response values. We incorporate the empirical correlations into the covariance. In two example data sets we find greatly improved prediction.
Ya Xu's thesis

2008

• Owen, A.B.
Monte Carlo and Quasi-Monte Carlo for Statistics PDF | Proceedings of MCQMC 2008 (Montreal)
This is a survey which sketches some topics in statistics that use Monte Carlo and quasi-Monte Carlo methods. The emphasis is on problems with some open research issues.

• Owen, A.B.
Karl Pearson's meta-analysis revisited
Annals of Statistics | Slides | Supplementary figures (web) | Supplementary figures (pdf)
A test of Karl Pearson, thought to be inadmissible for over 50 years, is shown to be admissible. Furthermore it has good power against alternatives in which all or most of the non-zero parameters share the same sign. An earlier paper below, used a big Monte Carlo simulation, where this one uses an FFT to get exact power. This one also compares to standard tests not ordinarily thought of as meta-analysis.

• Southworth, L.K., Kim. S.K. and Owen, A.B.
Properties of balanced permutations
Journal of Computational Biology, April 2009, 16(4): 625-638
Balanced permutations are a fascinating idea for microarray analyses. But we find that they can give very misleading p values.

2007

• Perry, P.O. and Owen, A.B.
A rotation test to verify latent structure JMLR Feb 2010
We test the presence of a latent variable in correlated noise by employing projection pursuit non-normality measures.
Patrick Perry's thesis

• Zahn, Poosala, Owen, Ingram, Lustig, Carter, Weeratna, Taub, Gorospe, Mazan-Mamczarz, Lakatta, Boheler, Zu, Mattson, Falco, Ko, Schlessinger, Firman, Kummerfeld, Wood, Longo, Zonderman, Kim, Becker
"AGEMAP: a gene expression database for aging in mice" PLOS Genetics
This is a preliminary online version of the article. There may be changes.
We look at patterns of aging in mice for 16 different tissues, as measured by gene expression.

• Owen, A.B. and Perry, P.O.
Bi-cross-validation of the SVD and the non-negative matrix factorization final PDF from AOAS | JSM 2009 slides | talk at Cambridge University | older slides
We look at how to pick the rank k when approximating a matrix by a truncated SVD. We hold out a rectangular submatrix, fit an SVD to a complementary submatrix, truncate it and predict. The method extends to the non-negative matrix factorization among other models.
Patrick Perry's thesis

• Owen, A.B. "Pearson's test in a large scale multiple meta-analysis" PDF
The AGEMAP study generated an 8932 x 16 matrix of p values. We apply meta-analysis to each row. A method originally proposed by Pearson (1934) and thought for over 50 years to be inadmissible performs best in a simulation. We also show that Pearson's test really is admissible.

• Owen, A.B. "The pigeonhole bootstrap" Annals of Applied Statistics | PDF | Slides
McCullagh (2000) showed that large crossed random effects data sets, such as are now studied for recommender engines and information retrieval are impossible to bootstrap. This means that even for balanced homoscedastic random effects models with no missing data, no bootstrap correctly estimates the variance of a sample mean (let alone a more complicated procedure). But one of the methods he studied, that of independently resampling rows and columns, comes pretty close. This article shows the expected bootstrap variance in that method tracks the desired variance, even for severely unbalanced and heteroscedastic data sets.

2006

• Owen, A.B. "A robust hybrid of lasso and ridge regression" PDF | Slides
A penalty that behaves like lasso for small coefficients and like ridge for large coefficients is developed. This penalty is a reversed Huber function. The penalty is convex. Like the Huber function it requires scaling. The scaling parameter can be incorporated into a criterion that is jointly convex in it and the regression coefficient vector.

• Owen, A.B. "Infinitely imbalanced logistic regression" JMLR
Many binary classfication problems are very unbalanced with one category much more common than the other. This paper shows what happens to logistic regression in the limit where one category's sample size tends to infinity while the other remains finite. For example one could use logistic regression to separate a Gaussian measure from a finite data set. Under mild conditions, the limiting coefficient vector (apart from the intercept) is finite. It can be expressed in terms of an exponential tilt and solved for by a convex optimization.
Journal of Machine Learning Research (v 8, pp 761-773, 2007)

• Zahn, Sonu, Vogel, Crane, Mazan-Mamczarz, Rabkin, Davis, Becker, Owen, Kim
"Transcriptional profile of aging in human muscle reveals a common aging signature" PLOS Genetics
This paper relates human aging to various genes and gene groups. It includes a new version of Gene Set Enrichment Analysis (GSEA) geared to handle regressions and covariates. The electron transport group of genes are found to be age related in human kidney, muscle and brain, and in other species.

2005

• Tribble, S.D and Owen, A.B.
"Construction of weakly CUD sequences for MCMC sampling Electronic Journal of Statistics
An earlier paper below showed that MCMC can be driven by completely uniformly distributed (CUD) or weakly CUD (WCUD) point sequences. This paper shows that a construction of Liao's that permutes QMC vectors leads to WCUD point sequences. A theorem of Niederreiter (1977) implies that certain lattice constructions satisfy a triangular array version of CUD. We find QMC methods for MCMC reduce variance by factors ranging from 10 to several hundred in a 42 dimensional Gibbs sampling probit example. A proposal of Liao's for incorporating acceptance rejection into QMC-MCMC
Seth Tribble's thesis

• Owen, A.B.
"Local antithetic sampling with scrambled nets" Annals of Statistics 2008 | @ arXiv | @ Project Euclid
A local antithetic reflection strategy can improve the variance rate of scrambled nets from $$n^{-3/2+\epsilon}$$ to $$n^{-3/2-2/d+\epsilon}$$ in dimension d. The benefit is similar to that which Haber gets when moving from stratified sampling to stratified antithetic sampling. The method also looks like a merger of scrambled nets and monomial cubature.

• Owen, A.B.
"On the Warnock-Halton quasi-standard error" Monte Carlo Methods and Applications
v12 n 1 pp 47--54 DOI: 10.1515/156939606776886652
Warnock and Halton have proposed a method of treating multiple QMC estimates as replicates. This paper reproduces an example where the method seems to work, but cautions that the method can fail arbitrarily badly.

• Lin, Z & Owen, A.B. & Altman, R. Science

2004

• Owen, A.B. and Tribble, S.D
"A quasi-Monte Carlo Metropolis algorithm" PNAS
This paper proves that QMC methods can be applied to Metropolis-Hastings style MCMC. The key idea is to use QMC points that are "completely uniformly distributed". This is like using the entire period of a small random number generator.
Seth Tribble's thesis

• Rodwell, Sonu, Zahn, Lund, Wilhelmy, Wang, Xiao, Mindrinos, Crane, Segal, Myers, Brooks, Davis, Higgins, Owen and Kim
"A transcriptional profile of aging in the human kidney" Public Library of Science
We found genes that change expression with age, in the human kidney. These genes do not tend to be the ones that serve to distinguish kidney from other tissue types, consistent with a model that aging is similar in different tissue types. The genes do not overlap with aging related genes in other species that we looked at.

• Owen, A.B.
"Randomized QMC and point singularities" PDF
Randomized QMC is shown to have a superior rate of convergence to ordinary MC on some functions with square integrable singularities at unknown locations. Surprisingly that means RQMC will generally beat importance sampling asymptotically. Of course one might combine them.

• Lin, Z & Owen, A.B. & Altman, R.
"Genetic research and human subject privacy" Science, Vol 305, Issue 5681, 183, 9 July 2004

• Owen, A. B.
"Variance of the number of false discoveries". PDF | R functions (beta) | R function documentation (beta)
Given d hypothesis tests at level $$\alpha$$ we expect $$d \alpha$$ false positives. When the tests are dependent, the variance of the number of false positives can be $$O(d^2)$$ higher than the independence value of $$d \alpha(1-\alpha)$$. This paper shows how to estimate such a variance taking account of dependency in the tests. The R functions are beta and very subject to change. Feedback is welcomed.

• Owen, A. B. "Halton sequences avoid the origin". SIAM Review v 48 n 3 pp 487--503
Gives rates of convergence for QMC on unbounded integrands, using growth conditions on f and a singularity avoidance pattern for x's. Halton sequences and randomized QMC avoid the origin suitably.

• Owen, A. B. "Multidimensional variation for quasi-Monte Carlo". PDF | BiBTeX
Survey, and some new results, on multidimensional variation (Vitali and Hardy-Krause) for Quasi-Monte Carlo.

2003

• Nomogram for predicting the likelihood of delayed graft function in adult cadeveric renal transplant patients Journal of the American Society of Nephrology

• Liu, R. Owen, A.B.
"Estimating Mean Dimensionality of ANOVA decompositions"
JASA 2006
Relationships between Sobol's sensitivity indices and moments of the dimension distribution are established. The mean dimension is computed for some functions arising in finance and extreme value theory. The minimum of d independent uniform random variables is seen to have strong low dimensional components.

• Troyanskaya, O.G., Dolinski, K., Owen, A.B., Altman, R.B., Botstein, D.
"A Bayesian framework for combining heterogeneous data sources for gene function prediction (in S. cerevisiae)" PNAS| Online Supplement

• Owen, A.B. "Quasi-Monte Carlo Sampling" PDF | BiBTeX
A Chapter on QMC for a SIGGRAPH 2003 course. It motivates QMC as a deterministic law of large numbers. The algorithms are presented as extensions of stratification methods, like those already well known in graphics (jittering, n rooks, multi-jittered sampling).

2002

• Owen, A.B., Stuart, J., Mach, K. Villeneuve, A,M., Kim, S.
"A gene recommender algorithm to identify co-expressed genes in C elegans" Paper | Software
This paper imitates algorithms from movie and book recommenders to find new genes related to a group of old genes. Given a query of genes with common function, we identify experiments in which the query genes are strongly co-expressed. Then we rank all the organisms genes according to the extent to which they agree with the query group, in the selected experiments. RNA interference knockouts confirmed two new Retinoblastoma related genes in C elegans.

• Owen, A.B. "Variance and discrepancy with alternative scramblings" PostScript | PDF
There are many computationally efficient proposals for scrambling digital nets. Generally they preserve mean squared discrepancy. This paper shows that one alternative can be detrimental to the sampling variance, adversely affecting the rate of convergence. Another scrambling improves the rate of convergence, at least for d=1.

• Owen, A.B. "Necessity of low effective dimension" PostScript | PDF
This paper explores the extent to which low superposition dimension is necessary for QMC to beat MC.

• Jiang, T. and Owen, A.B. "Quasi-regression for visualization and interpretation of black box functions" PostScript | PDF
Quasi-regression is applied to the output of a support vector machine and to a neural network. The method allows one to peer into a black box and identify important variables and interactions. The most vexing issue is how to reconcile a decomposition derived for independent variables with a function fit to highly dependent data.

• Hickernell, F. and Lemieux, C. and Owen, A.B. "Control variates for quasi-Monte Carlo" PostScript | PDF
It is easy and natural to combine quasi-Monte Carlo with control variates. But the proper control variate coefficients can change, as can the choice of what constitutes a good control variate. In MC a good control variate correlates with the integrand. In QMC it is better to correlate with some derivative or high frequency component of the target integrand.

2001

• Jiang, T. and Owen, A.B. "Quasi-regression with shrinkage" PostScript | PDF | Software | Slides
Quasi-regression is a method of Monte Carlo approximation useful for global sensitivity analysis. This paper presents a new version, incorporating shrinkage parameters of the type used in wavelet approximation. As an example application, a black box function from machine learning is analyzed. That function is nearly a superposition of functions of one and two variables and the first variable acting alone accounts for more than half of the variance.

• Owen, A.B. "The dimension distribution and quadrature test functions" PostScript | PDF
A "dimension distribution" is introduced through which various measures of effective dimension of a function can be defined. The idea is explored on some widely used quadrature test functions. Some isotropic functions are shown to be of low effective dimension, explaining the success of QMC methods on them.