Mark Daniel Ward

Data Science in Statistics Curricula: Preparing Students to “Think with Data”

A growing number of students are completing undergraduate degrees in statistics and entering the workforce as data analysts. In these positions, they are expected to understand how to use databases and other data warehouses, scrape data from Internet sources, program solutions to complex problems in multiple languages, and think algorithmically as well as statistically. These data science topics have not traditionally been a major component of undergraduate programs in statistics. Consequently, a curricular shift is needed to address additional learning outcomes. The goal of this article is to motivate the importance of data science proficiency and to provide examples and resources for instructors to implement data science in their own statistics curricula. We provide case studies from seven institutions. These varied approaches to teaching data science demonstrate curricular innovations to address new needs. Also included here are examples of assignments designed for courses that foster engagement of undergraduates with data and data science. [Received November 2014. Revised July 2015.]

Asymptotic distribution of two-protected nodes in random binary search trees

Abstract We derive exact moments of the number of 2-protected nodes in binary search trees grown from random permutations. Furthermore, we show that a properly normalized version of this tree parameter converges to a Gaussian limit.

Number of survivors in the presence of a demon

This paper complements the analysis of Louchard and Prodinger [LP08] on the number of rounds in a coin-flipping selection algorithm that occurs in the presence of a demon. We precisely analyze a very different aspect of the selection algorithm, using different methods of analysis. Specifically, we obtain precise descriptions of the distribution and all moments of the number of participants ultimately selected during the execution of the algorithm. The selection algorithm is robust in at least two significant ways. The presence of a demon allows for the precise analysis even when errors may occur between the rounds of the selection process. (The analysis also handles the more traditional case, in which no demon is involved.) The selection algorithm can also use either biased or unbiased coins.

Inverse auctions: Injecting unique minima into random sets

We consider auctions in which the winning bid is the smallest bid that is unique. Only the upper-price limit is given. Neither the number of participants nor the distribution of the offers are known, so that the problem of placing a bid to win with maximum probability looks, a priori, ill-posed. Indeed, the essence of the problem is to inject a (final) minimum into a random subset (of unique offers) of a larger random set. We will see, however, that here no more than two external (and almost compelling) arguments make the problem meaningful. By appropriately modeling the relationship between the number of participants and the distribution of the bids, we can then maximize our chances of winning the auction and propose a computable algorithm for placing our bid.

On the Asymptotic Probability of Forbidden Motifs on the Fringe of Recursive Trees

Abstract We use analytic methods to study the probability of a family of motifs not occurring on the fringe of a random recursive tree. We obtain an asymptotic formula for this probability by means of singularity analysis. Two regimes are treated in particular: the case that a fixed proportion of motifs of size γ is forbidden, and the case that a fixed number of motifs of size γ is forbidden. In both cases, we observe phase transitions as the size of the random tree and the size of the motif tend to infinity. The required asymptotic expansions of the dominant singularities were first found by computer experiments and only later made rigorous.

BUILDING RANDOM TREES FROM BLOCKS

Many modern networks grow from blocks. We study the probabilistic behavior of parameters of a blocks tree, which models several kinds of networks. It grows from building blocks that are themselves rooted trees. We investigate the number of leaves, depth of nodes, total path length, and height of such trees. We use methods from the theory of Pólya urns and martingales.

Analytic methods for select sets

We analyze the asymptotic number of items chosen in a selection procedure. The procedure selects items whose rank among all previous applicants is within the best 100p percent of the number of previously selected items. We use analytic methods to obtain a succinct formula for the first-order asymptotic growth of the expected number of items chosen by the procedure.

Building Bridges: The Role of an Undergraduate Mentor

ABSTRACT I share some advice and lessons that I have learned from working with many wonderful students and colleagues, in my role as Undergraduate Chair of Statistics at Purdue University since 2008. I also reflect on developing, implementing, and sustaining a new living, learning community environment for statistics students.

Fostering Undergraduate Data Science

ABSTRACT Data Science is one of the newest interdisciplinary areas. It is transforming our lives unexpectedly fast. This transformation is also happening in our learning styles and practicing habits. We advocate an approach to data science training that uses several types of computational tools, including R, bash, awk, regular expressions, SQL, and XPath, often used in tandem. We discuss ways for undergraduate mentees to learn about data science topics, at an early point in their training. We give some intuition for researchers, professors, and practitioners about how to effectively embed real-life examples into data science learning environments. As a result, we have a unified program built on a foundation of team-oriented, data-driven projects.

On the Variety of Shapes in Digital Trees

We study the joint distribution of the number of occurrences of members of a collection of nonoverlapping motifs in digital data. We deal with finite and countably infinite collections. For infinite collections, the setting requires that we be very explicit about the specification of the underlying measure-theoretic formulation. We show that (under appropriate normalization) for such a collection, any linear combination of the number of occurrences of each of the motifs in the data has a limiting normal distribution. In many instances, this can be interpreted in terms of the number of occurrences of individual motifs: They have a multivariate normal distribution. The methods of proof include combinatorics on words, integral transforms, and poissonization.