Zeros of weakly holomorphic modular forms of levels 2 and 3

Let $M_k^\sharp(N)$ be the space of weakly holomorphic modular forms for $\Gamma_0(N)$ that are holomorphic at all cusps except possibly at $\infty$. We study a canonical basis for $M_k^\sharp(2)$ and $M_k^\sharp(3)$ and prove that almost all modular forms in this basis have the property that the majority of their zeros in a fundamental domain lie on a lower boundary arc of the fundamental domain.Comment: Added a reference, corrected typo

Bootstrap confidence intervals for the contributions of individual variables to a Mahalanobis distance

Hotelling's T 2 and Mahalanobis distance are widely used in the statistical analysis of multivariate data. When either of these quantities is large, a natural question is: How do individual variables contribute to its size? The Garthwaite–Koch partition has been proposed as a means of assessing the contribution of each variable. This yields point estimates of each variable's contribution and here we consider bootstrap methods for forming interval estimates of these contributions. New bootstrap methods are proposed and compared with the percentile, bias-corrected percentile, non-studentized pivotal and studentized pivotal methods via a large simulation study. The new methods enable use of a broader range of pivotal quantities than with standard pivotal methods, including vector pivotal quantities. In the context considered here, this obviates the need for transformations and leads to intervals that have higher coverage, and yet are narrower, than intervals given by the standard pivotal methods. These results held both when the population distributions were multivariate normal and when they were skew with heavy tails. Both equal-tailed intervals and shortest intervals are constructed; the latter are particularly attractive when (as here) squared quantities are of interest

A Bayesian mixture model for term re-occurrence and burstiness

This paper proposes a model for term reoccurrence in a text collection based on the gaps between successive occurrences of a term. These gaps are modeled using a mixture of exponential distributions. Parameter estimation is based on a Bayesian framework that allows us to fit a flexible model. The model provides measures of a term’s re-occurrence rate and withindocument burstiness. The model works for all kinds of terms, be it rare content word, medium frequency term or frequent function word. A measure is proposed to account for the term’s importance based on its distribution pattern in the corpus

Using discovered, polyphonic patterns to filter computer-generated music

A metric for evaluating the creativity of a music-generating system is presented, the objective being to generate mazurka-style music that inherits salient patterns from an original excerpt by Frédéric Chopin. The metric acts as a filter within our overall system, causing rejection of generated passages that do not inherit salient patterns, until a generated passage survives. Over fifty iterations, the mean number of generations required until survival was 12.7, with standard deviation 13.2. In the interests of clarity and replicability, the system is described with reference to specific excerpts of music. Four concepts–Markov modelling for generation, pattern discovery, pattern quantification, and statistical testing–are presented quite distinctly, so that the reader might adopt (or ignore) each concept as they wish

Modified confidence intervals for the Mahalanobis distance

Reiser (2001) proposes a method of forming confidence interval for a Mahalanobis distance that yields intervals which have exactly the nominal coverage, but sometimes the interval is View the MathML source (0,0). We consider the case where Mahalanobis distance quantifies the difference between an individual and a population mean, and suggest a modification that avoids implausible intervals

A comparative evaluation of algorithms for discovering translational patterns in Baroque keyboard works

We consider the problem of intra-opus pattern discovery, that is, the task of discovering patterns of a specified type within a piece of music. A music analyst undertook this task for works by Domenico Scarlattti and Johann Sebastian Bach, forming a benchmark of 'target' patterns. The performance of two existing algorithms and one of our own creation, called SIACT, is evaluated by comparison with this benchmark. SIACT out-performs the existing algorithms with regard to recall and, more often than not, precision. It is demonstrated that in all but the most carefully selected excerpts of music, the two existing algorithms can be affected by what is termed the 'problem of isolated membership'. Central to the relative success of SIACT is our intention that it should address this particular problem. The paper contrasts string-based and geometric approaches to pattern discovery, with an introduction to the latter. Suggestions for future work are given

Running an international survey in a small country : challenges and opportunities

Background: National and international authorities recognize that health surveys are major sources of information on health conditions. Smaller states may prefer using health surveys to registries because they are cheaper to maintain. Nevertheless, smaller states carry out far fewer national health surveys than larger states. One reason could be that the value of surveys depends on the number of people interviewed rather than the proportion of the population. Therefore, survey costs per capita are substantially higher in smaller states. Methods: Malta is a small state with a population of under half a million. It forms part of the European Union, which has provided financial assistance and external expertise in performing international health surveys. We present the European Health Interview Survey in Malta as a case study to review the challenges for small states and the typical adaptations necessary for implementing national health surveys and meeting international health data obligations. Results: We identified the lack of health survey infrastructure, difficulties in recruiting the large samples recommended by international organizations, survey fatigue, and a lack of resources for marketing, incentivization, analysis and dissemination. Low-cost solutions have been devised to address some issues, such as marketing and incentives, which exploit specific characteristics of small states. Conclusion: In the absence of administrative data or epidemiological registers, surveys are important tools for evidence-based policy-making in small states. The experience of Malta could help other small states to minimize the resources required to run national health surveys.peer-reviewe

Contribution of individual variables to the regression sum of squares

In applications of multiple regression, one of the most common goals is to measure the relative importance of each predictor variable. If the predictors are uncorrelated, quantification of relative importance is simple and unique. However, in practice, predictor variables are typically correlated and there is no unique measure of a predictor variable’s relative importance. Using a transformation to orthogonality, new measures are constructed for evaluating the contribution of individual variables to a regression sum of squares. The transformation yields an orthogonal approximation of the columns of the predictor scores matrix and it maximizes the sum of the covariances between the cross-product of individual regressors and the response variable and the cross-product of the transformed orthogonal regressors and the response variable. The new measures are compared with three previously proposed measures through examples and the properties of the measures are examined

Elicitation of prior distributions of variable-selection problems in regression

This paper addresses the problem of quantifying expert opinion about a normal linear regression model when there is uncertainty as to which independent variables should be included in the model. Opinion is modeled as a mixture of natural conjugate prior distributions with each distribution in the mixture corresponding to a different subset of the independent variables. It is shown that for certain values of the independent variables, the predictive distribution of the dependent variable simplifies from a mixture of $t$-distributions to a single $t$-distribution. Using this result, a method of eliciting the conjugate distributions of the mixture is developed. The method is illustrated in an example

Use of expert knowledge in evaluating costs and benefits of alternative service provisions: A case study

Objectives: A treatment pathway model was developed to examine the costs and benefits of the current bowel cancer service in England and to evaluate potential alternatives in service provision. To use the pathway model, various parameters and probability distributions had to be specified. They could not all be determined from empirical evidence and, instead, expert opinion was elicited in the form of statistical quantities that gave the required information. The purpose of this study is to describe the procedures used to quantify expert opinion and note examples of good practice contained in the case study. Methods: The required information was identified and preparatory discussion with four experts refined the questions they would be asked. In individual elicitation sessions they quantified their opinions, mainly in the form of point and interval estimates for specified variables. New methods have been developed for quantifying expert opinion and these were implemented in specialized software that uses interactive graphics. This software was used to elicit opinion about quantities related to measurable covariates. Results: Assessments for thirty-four quantities were elicited and available checks supported their validity. Eight points of good practice in eliciting and using expert judgment were evident. Parameters and probability distributions needed for the pathway model were determined from the elicited assessments. Simulation results from the pathway model were used to inform policy on bowel cancer service provision. Conclusions: The study illustrates that quantifying and using expert judgment can be acceptable in real problems of practical importance. For full benefit to be gained from expert knowledge, elicitation must be conducted carefully and should be reported in detail