Department of Mathematics and Statistics

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An adaptive method for statistical detection with applications to drug discovery
(2003) Zhu, Mu; Chipman, Hugh A.; Su, Wanhua
Researchers have tried to tackle various statistical detection problems using state-of-the-art classification techniques but are often disappointed at the results. The reason is two-fold. First of all, as classification problems, these statistical detection problems are heavily unbalanced: the class of interest is rare in the training data; an overwhelming majority of the training data belong to what can be called a background class. A primary example is drug discovery, where most of the chemical compounds in the data set are inactive whereas the goal is to detect a small number of active compounds. Secondly, the goal of statistical detection is fundamentally different from that of classification, making misclassification rate the wrong criterion to focus on. In this article, we develop an adaptive method for statistical detection and demonstrate that it can be an effective tool for drug discovery.
Point sets and dynamical systems in the autocorrelation topology
(2004) Strungaru, Nicolae; Moody, R. V.
This paper is about the topologies arising from statistical coincidence on locally finite point sets in locally compact Abelian groups G . The first part defines a uniform topology (autocorrelation topology) and proves that, in effect, the set of all locally finite subsets of G is complete in this topology. Notions of statistical relative denseness, statistical uniform discreteness, and statistical Delone sets are introduced. The second part looks at the consequences of mixing the original and autocorrelation topologies, which together produce a new Abelian group, the autocorrelation group. In particular the relation between its compactness (which leads then to a G -dynamical system) and pure point diffractivity is considered. Finally for generic regular model sets it is shown that the autocorrelation group can be identified with the associated compact group of the cut and project scheme that defines it. For such a set the autocorrelation group, as a G -dynamical system, is a factor of the dynamical local hull.
Almost periodic measures and long-range order in Meyer sets
(2005) Strungaru, Nicolae
The main result of this paper is that the diffraction pattern of any Meyer set with a well-defined autocorrelation has a relatively dense set of Bragg peaks. In the second part of the paper we provide a necessary and sufficient condition for a positive pure point measure to have a continuous Fourier transform. In particular, one can get a necessary and sufficient condition for a point set to have no Bragg peaks in its diffraction.
LAGO: a computationally efficient approach for statistical detection
(2006) Su, Wanhua; Zhu, Mu; Chipman, Hugh A.
We study a general class of statistical detection problems where the underlying objective is to detect items belonging to a rare class from a very large database. We propose a computationally efficient method to achieve this goal. Our method consists of two steps. In the first step we estimate the density function of the rare class alone with an adaptive bandwidth kernel density estimator. The adaptive choice of the bandwidth is inspired by the ancient Chinese board game known today as Go. In the second step we adjust this density locally depending on the density of the background class nearby. We show that the amount of adjustment needed in the second step is approximately equal to the adaptive bandwidth from the first step, which gives us additional computational savings. We name the resulting method LAGO, for “locally adjusted Go-kernel density estimator.” We then apply LAGO to a real drug discovery dataset and compare its performance with a number of existing and popular methods.
Academic procrastination: the pattern and correlates of behavioural postponement
(2006) Powell, Russell A.; Howell, Andrew J.; Watson, David; Buro, Karen
Using a series of computer-based assignments, we examined whether students’ submission patterns revealed a hyperbolic pattern of temporal discounting, such that few assignments are submitted far ahead of the deadline and submission of assignments accelerates at an increasing rate as the deadline becomes imminent. We further examined whether variables related to self-regulation – namely, self-reported procrastination, implementation intentions, say-do correspondence, and perceived academic control – correlated with behavioural postponement. Results revealed strong behavioural evidence of temporal discounting, especially among those who identified themselves as procrastinators. Among the self-regulation measures, only say-do correspondence consistently correlated with procrastination.
Circular symmetry of pinwheel diffraction
(2006) Strungaru, Nicolae; Moody, R. V.; Postnikoff, D.
A method is given for explicitly determining the autocorrelation of the pinwheel tiling by use of the substitution system generating the tiling. Using this a new proof of the circular symmetry of the diffraction of the pinwheel tiling is given.
Unitary groups of nondegenerate Hermitian forms and the homotopy groups of pseudospheres
(2008) Farenick, D.R.; Ramsey, Christopher
The Witt Extension Theorem states that the unitary group of a finite-dimensional vector space V equipped with a nondegenerate hermitian form acts transitively on the pseudosphere induced by the form. We provide a new, constructive proof of this result for finite-dimensional vector spaces V over R , C , or H . This constructive proof is then used to prove a similar result for the unitary group of a finitely generated free right module over an abelian AW ∗-algebra. The topology of these unitary groups is examined and as an application we determine the homotopy groups π 1 and π 2 of the induced real, complex, and quaternionic pseudospheres.
The isomorphism problem for some universal operator algebras
(2011) Davidson, Kenneth R.; Ramsey, Christopher; Shalit, Orr Moshe
This paper addresses the isomorphism problem for the universal (nonself-adjoint) operator algebras generated by a row contraction subject to homogeneous polynomial relations. We find that two such algebras are isometrically isomorphic if and only if the defining polynomial relations are the same up to a unitary change of variables, and that this happens if and only if the associated subproduct systems are isomorphic. The proof makes use of the complex analytic structure of the character space, together with some recent results on subproduct systems. Restricting attention to commutative operator algebras defined by a radical ideal of relations yields strong resemblances with classical algebraic geometry. These commutative operator algebras turn out to be algebras of analytic functions on algebraic varieties. We prove a projective Nullstellensatz connecting closed ideals and their zero sets. Under some technical assumptions, we find that two such algebras are isomorphic as algebras if and only if they are similar, and we obtain a clear geometrical picture of when this happens. This result is obtained with tools from algebraic geometry, reproducing kernel Hilbert spaces, and some new complex-geometric rigidity results of independent interest. The C*-envelopes of these algebras are also determined. The Banach-algebraic and the algebraic classification results are shown to hold for the wot-closures of these algebras as well.
Pseudo-likelihood inference underestimates model uncertainty: evidence from bayesian nearest neighbours
(2011) Su, Wanhua; Chipman, Hugh A.; Zhu, Mu
When using the K-nearest neighbours (KNN) method, one often ignores the uncertainty in the choice of K. To account for such uncertainty, Bayesian KNN (BKNN) has been proposed and studied (Holmes and Adams 2002 Cucala et al. 2009). We present some evidence to show that the pseudo-likelihood approach for BKNN, even after being corrected by Cucala et al. (2009), still significantly underestimates model uncertainty.
Positive definite measures with discrete Fourier transform and pure point diffraction
(2011) Strungaru, Nicolae
In this paper we characterize the positive definite measures with discrete Fourier transform. As an application we provide a characterization of pure point diffraction in locally compact Abelian groups.
Attitudes and decisions about sexual offenders: a comparison of laypersons and professionals
(2012) Jung, Sandy; Jamieson, Lisa; Buro, Karen; DeCesare, John; Jung, Sandy
The current study examines the inherent biases about sexual offending held by 123 laypersons and 120 professionals (i.e. probation officers and therapists). In order to determine the extent of these biases, a series of brief newspaper articles were constructed to depict cases of sexual offenders. Each article comprised several combinations of key variables, including offender type, level of admission, and the presence of alcohol. Participants read a series of three fabricated articles and then completed a questionnaire regarding attitudes about the various offenders. The results indicate important differences between the lay and professional samples. Laypersons deemed sex offenders more favourably in terms of character, accountability, and risk for sexual recidivism. However, both groups showed some similar perceptions about sexual offending. Specifically, both groups evaluated child molesters more negatively than exhibitionists and in some cases, rapists. These findings highlight the need for continuing education for professionals in order to attenuate the effects of prejudicial attitudes.
Statistical inference on recall, precision and average precision under random selection
(2012) Su, Wanhua; Zhang, P.
The objective of a rare target detection problem is to identify the rare targets as early as possible. Recall, precision and average precision are three popular performance measures for evaluating different detection methods. However, there is little literature on the statistical properties of these three measures. We develop a framework for conducting statistical inference on recall, precision and average precision through establishing their asymptotic properties. Simulations are used to illustrate the idea. The proposed methods can also be applied in other areas where ranking systems need to be evaluated, such as information retrieval.
On the Bragg diffraction spectra of a Meyer set
(2013) Strungaru, Nicolae
Meyer sets have a relatively dense set of Bragg peaks, and for this reason they may be considered as basic mathematical examples of (aperiodic) crystals. In this paper we investigate the pure point part of the diffraction of Meyer sets in more detail. The results are of two kinds. First, we show that, given a Meyer set and any positive intensity a less than the maximum intensity of its Bragg peaks, the set of Bragg peaks whose intensity exceeds a is itself a Meyer set (in the Fourier space). Second, we show that if a Meyer set is modified by addition and removal of points in such a way that its density is not altered too much (the allowable amount being given explicitly as a proportion of the original density), then the newly obtained set still has a relatively dense set of Bragg peaks.
Automorphisms and dilation theory of triangular UHF algebras
(2013) Ramsey, Christopher
We study the triangular subalgebras of UHF algebras which provide new examples of algebras with the Dirichlet property and the Ando property. This in turn allows us to describe the semicrossed product by an isometric automorphism. We also study the isometric automorphism group of these algebras and prove that it decomposes into the semidirect product of an abelian group by a torsion free group. Various other structure results are proven as well.
Almost periodic measures and Bragg diffraction
(2013) Strungaru, Nicolae
In this paper we prove that the cone $\mathcal {PDS}(G)$ of positive, positive definite, discrete and strong almost periodic measures over a σ-compact, locally compact Abelian group G has an interesting property: given any positive and positive definite measure μ smaller than some measure in $\mathcal {PDS}(G)$, the strong almost periodic part μS of μ is also in $\mathcal {PDS}(G)$. We then use this result to prove that given a positive-weighted Dirac comb ω with finite local complexity and pure point diffraction, any positive Dirac comb less than ω has either a trivial Bragg spectrum or a relatively dense set of Bragg peaks.
On weighted Dirac combs supported inside model sets
(2014) Strungaru, Nicolae
In this paper we prove that given a weakly almost periodic measure μ supported inside some model set $\Lambda (W)$ with closed window W, then the strongly almost periodic component ${{\mu }_{S}}$ and the null weakly almost periodic component μ0 are both supported inside $\Lambda (W)$. As a consequence we prove that given any translation bounded measure ω, supported inside some model set, then each of the pure point diffraction spectrum ${{\hat{\gamma }}_{{\rm pp}}}$ and the continuous diffraction spectrum ${{\hat{\gamma }}_{c}}$ is either trivial or has a relatively dense support.
Evidence for vicarious hope and vicarious gratitude
(2015) Howell, Andrew J.; Bailie, Thomas; Buro, Karen
Theorists posit that well-being reflects an optimal balance of self- and other-interest. An index of other-interest may be the degree to which hope and gratitude concern others (termed vicarious hope and vicarious gratitude) in addition to concerning the self. We examined the frequency of vicarious responses generated by participants (N = 350) invited to list ten things for which they were hopeful or grateful. Results showed that, on average, about 13 % of participants’ responses were other-oriented, that such responses were more likely to occur in the hope than in the gratitude condition, and that they were more likely to occur in conditions where task instructions primed inclusion of others. The generation of vicarious responses correlated with the trait of empathic concern. Implications of these findings for future work on vicarious hope and vicarious gratitude are discussed.
Measuring and predicting student well-being: further evidence in support of the flourishing scale and the scale of positive and negative experiences
(2015) Howell, Andrew J.; Buro, Karen
An increased focus on well-being in university settings has spurred the development of brief scales of both functioning well and feeling good. The objectives of the current study were to generate descriptive findings concerning psychometric properties (e.g., factor structure; reliability) of the recently devised Flourishing Scale (FS) and Scale of Positive and Negative Experiences (SPANE; Diener et al., Soc Indic Res 97:143–156, 2010) with an English-speaking university student sample, and to test associations between the scales and potential predictors of eudaimonic and hedonic aspects of well-being. The FS and SPANE scales were completed by 478 undergraduate students, along with scales measuring 10 human values and both time and material affluence. Descriptive statistics (e.g., means, standard deviations, reliability coefficients) for the FS and SPANE scales were highly similar to those reported by Diener et al. (Soc Indic Res 97:143–156, 2010) and confirmatory factor analysis supported the hypothesized three-factor model (i.e., flourishing, positive feelings, and negative feelings). Self-transcendence and conservation value types were significant predictors of FS scores, whereas only the conservation value type predicted affect balance scores from the SPANE. Time and material affluence were significant predictors of both FS and affect balance scores. Results are discussed in relation to the distinction between eudaimonic and hedonic aspects of well-being.
Threshold-free measures for assessing the performance of medical screening tests
(2015) Yuan, Yan; Su, Wanhua; Zhu, Mu
Background: The area under the receiver operating characteristic curve (AUC) is frequently used as a performance measure for medical tests. It is a threshold-free measure that is independent of the disease prevalence rate. We evaluate the utility of the AUC against an alternate measure called the average positive predictive value (AP), in the setting of many medical screening programs where the disease has a low prevalence rate. Methods: We define the two measures using a common notation system and show that both measures can be expressed as a weighted average of the density function of the diseased subjects. The weights for the AP include prevalence in some form, but those for the AUC do not. These measures are compared using two screening test examples under rare and common disease prevalence rates. Results: The AP measures the predictive power of a test, which varies when the prevalence rate changes, unlike the AUC, which is prevalence independent. The relationship between the AP and the prevalence rate depends on the underlying screening/diagnostic test. Therefore, the AP provides relevant information to clinical researchers and regulators about how a test is likely to perform in a screening population. Conclusion: The AP is an attractive alternative to the AUC for the evaluation and comparison of medical screening tests. It could improve the effectiveness of screening programs during the planning stage.
Operator algebras for analytic varieties
(2015) Davidson, Kenneth R.; Ramsey, Christopher; Shalit, Orr Moshe
We study the isomorphism problem for the multiplier algebras of irreducible complete Pick kernels. These are precisely the restrictions MV of the multiplier algebra M of Drury-Arveson space to a holomorphic subvariety V of the unit ball Bd. We find that MV is completely isometrically isomorphic to MW if and only if W is the image of V under a biholomorphic automorphism of the ball. In this case, the isomorphism is unitarily implemented. This is then strengthened to show that, when d<∞, every isometric isomorphism is completely isometric. The problem of characterizing when two such algebras are (algebraically) isomorphic is also studied. When V and W are each a finite union of irreducible varieties and a discrete variety in Bd with d<∞, then an isomorphism between MV and MW determines a biholomorphism (with multiplier coordinates) between the varieties; and the isomorphism is composition with this function. These maps are automatically weak-∗ continuous. We present a number of examples showing that the converse fails in several ways. We discuss several special cases in which the converse does hold---particularly, smooth curves and Blaschke sequences. We also discuss the norm closed algebras associated to a variety, and point out some of the differences.

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