Toby Kenney

Associate Professor
Department of Mathematics and Statistics,
Dalhousie University
Halifax, Nova Scotia,
B3H 3J5

Office: 102 Chase Building

Toby Kenney


MATH 3090: Advanced Calculus - Fall 2006

MATH/CSCI 2112: Discrete Structures I - Winter 2007

MATH 2051: Problems in Geometry - Fall 2007

MATH/CSCI 2113: Discrete Structures II - Winter 2008

MATH 1115: Mathematics for Commerce - Winter 2011

MATH 1115: Mathematics for Liberal Arts - Winter 2012

MATH 3030X/Y: Abstract Algebra - Fall 2012 and Winter 2013

MATH/STAT 3360: Probability - Fall 2014, Fall 2013, Fall 2012, Fall 2011

MATH/STAT 2600: Theory of interest - Fall 2014, Fall 2013, Fall 2010

MATH/STAT 3460: Intermediate Statistical Theory - Winter 2014

ACSC/STAT 3703: Actuarial Models I - Winter 2015 Winter 2023

ACSC/STAT 3720: Life Contingencies I - Winter 2015, Winter 2016, Winter 2017, Winter 2018

ACSC/STAT 4703: Actuarial Models II - Fall 2015, Fall 2016, Fall 2017, Fall 2018, Winter 2020, Fall 2021, Fall 2022, Fall 2023

ACSC/STAT 4720: Life Contingencies II - Fall 2015, Fall 2016, Fall 2017, Fall 2018 , Fall 2021

ACSC/STAT 3740: Predictive Analytics - Winter 2023

Actuarial Science Program

I am responsible for maintaining and developing the Actuarial Science program at Dalhousie. Here is a website giving some details of the current program.

Society of Actuaries Exam Reimbursement Policy

The Society of Actuaries has a scheme where the university can apply for reimbursement of the exam fees of up to three students per year sitting SoA exams. Here is a summary of this program, how to apply and the policy for how the university will choose which students to apply for.


Research Interests

I have interests in a wide variety of topics in areas of pure mathematics, data science and actuarial science.

In data science, I am interested in the application of abstract mathematical modelling to statistical methodology. I am interested in general statistical methodology, and in development of statistical techniques to analyse microbiome data. Some particular topics of interest include understanding the temporal dynamics of the microbiome; and handling the unique measurement error structure in microbiome data. In other areas of data science, I am working on a project on automatic diagnosis of emergency department data. This is a big project involving dealing with many difficulties from a statistical standpoint including: the large quantities of free text data; the amount of missing data, with the potential for the missing pattern to change between training and test data; and the structure of the data and diagnoses. In statistical methodology, I have several projects relating to various areas such as ranking problems; measurement error and variable selection. I have recently been working on unfolding and deconvolution problems which aim to estimate the distribution of a quantity from a sample with measurement error.

I am also interested in general machine-learning methods. These methods use mathematical structures to create flexible functions that can approximate any true distribution of the data. While the flexibility allows these functions to approximate the complicated relationships that can arise in real data, it also results in hard-to-interpret predictions. Furthermore, avoiding overfitting can be a challenge. By studying the mathematical structures underlying the methods, I hope to find improvements that address these issues.

I am beginning to develop a research program in Actuarial Science. Topics of interest include: estimating utility functions to find optimal insurance choices for individuals; and assessing the value of information for calculating insurance costs.

In pure mathematics, my research interests mostly fall between Category theory, universal algebra, logic and combinatorics. Particular areas of interest include partition and congruence lattices; and Coxeter groups. I have also recently worked on topological convexity spaces.


C. Liu, T. Kenney, R. Beiko , H. Gu. The Community Coevolution Model with Application to the Study of Evolutionary Relationships between Genes based on Phylogenetic Profiles. Systematic Biology, (2022) syac052 (16 pages)
L. Liu, H. Gu, J. Van Limbergen and T. Kenney. SuRF: a New Method for Sparse Variable Selection, with Application in Microbiome Data Analysis. Statistics in Medicine 40 (2021), 897-919.
T. Kenney, T. Huang and H. Gu. Poisson PCA: Poisson Measurement Error corrected PCA, with Application to Microbiome Data. Biometrics 77 (2021) 1369-1384
T. Kenney, J. Gao and H. Gu. Application of OU processes to modelling temporal dynamics of the human microbiome, and calculating optimal sampling schemesBMC Bioinformatics 21(2020), 450 (32 pages)
W. Chen*, T. Kenney*, J. P. Bielawski and H. Gu. Testing Adequacy for DNA Substitution Models. BMC Bioinformatics 20(2019) 349 (16 pages). *‐co-first authors.
K. A. Dunn*, T. Kenney*, H. Gu and J. P. Bielawski. Improved inference of site- specific selection pressures under a generalized parametric model of codon evolution. BMC Evolutionary Biology 19(2019) 19:22 (19 pages). *‐co-first authors.
Y. Cai, H. Gu and T. Kenney Learning Microbial Community Structures with Supervised and Unsupervised Non-negative Matrix Factorization. Microbiome 5 (2017), (27 pages)
T. Kenney Partial Sup Lattices Theory and Applications of Categories 30 (2015), 305-331
M. Abeysunderra, T. Kenney, C. Field and H. Gu Combining Distance Matrices on Identical Taxon Sets for Multi-Gene Analysis with Singular Value Decomposition. PLoS ONE 9 (2014), e94279. doi:10.1371/journal.pone.0094279 (14 pages)
T. Kenney Coxeter Groups, Coxeter Monoids and the Bruhat Order. Journal of Algebraic Combinatorics 39 (2014), 719-731
T. Kenney and H. Gu Hessian Calculation for Phylogenetic Likelihood based on the Pruning Algorithm and its Applications Statistical Applications in Genetics and Molecular Biology, 11 (2012), issue 4, article 14
T. Kenney and R. Paré Categories as Monoids in Span, Rel and Sup, Cahiers de e Topologie et Géométrie Différentielle Catégoriques, 52 (2011), 209-240
T. Kenney The Path Relation for Directed Planar Graphs, and its Relation to the Free Diad. Discrete Mathematics 311 (2011), 441-456
T. Kenney Injective Power Objects and the Axiom of Choice Journal of Pure and Applied Algebra 215 (2011), 131-144
T. Kenney Graphical algebras - a new approach to congruence lattices Algebra Universalis 64 (2010), 313-338
H. Gu, T. Kenney and M. Zhu Partial Generalized Additive Models: an Information-Theoretic Approach to Selecting Variables and Dealing with Concurvity. Journal of computational and graphical statistics 19 (2010), 531-551
T. Kenney The General Theory of Diads Appl. Cat. Struct. 18 (2010), 523-572
T. Kenney and R. J. Wood Tensor Products of Sup Lattices and generalized sup-arrows. Theory and Applications of Categories 24 (2010), 266-287
T. Kenney Diads and Their Application to Topoi, Appl. Cat. Struct. 17 (2009), 567-590
T. Kenney Copower Objects and their applications to Finiteness in Topoi, Theory and Applications of Categories 16 (2006), 923-956
T. Kenney Generating Families in a Topos, Theory and Applications of Categories 16 (2006), 896-922

Papers submitted or under revision

T. Kenney, H. He and H. Gu. Prior Distributions for Ranking Problems.ArXiv
T. Kenney. Consistency of Ranking Estimators.ArXiv
L. Xu, X. Xu, D. Kong, L. Wang, H. Gu And T. Kenney Stochastic Generalized Lotka-Volterra Model with An Application to Learning Microbial Community Structures ArXiv
Y. Cai, H. Gu And T. Kenney Deconvolution density estimation with penalised MLE. ArXiv
X. Zhang, H. Gu And T. KenneySimultaneous Feature and Structure Selection of Dense Neural Network.
W. Zhang, T. Kenney And L. Ho. Evolutionary shift detection with ensemble variable selection. ArXiv
T. Kenney Stone Duality for Topological Convexity Spaces. ArXiv
Y. Cai, H. Gu And T. Kenney Rank Selection for Non-negative Matrix Factorization.

Papers in preparation

S. Ling, T. Kenney, C. Field, H. Gu. Model Combination for Block Missing Data.
M. Wang, T. Kenney. The Influence of Utility Functions on Life Insurance Choices.
T. Kenney. Euclidean Abstract Convexity Spaces.
R. Doig, T. Kenney, H. GuNegative Binomial PCA for Overdispersed Count Data
L. Liu, H. Gu, and T. Kenney. The influence of long tailed distributions on LASSO-based variable selection methods.

Graduate Students

Current Students

Lihui LiuPhD. (Co-supervised with H. Gu)Variable selection methods with application to microbiome data.
Xinyue ZhangPhD. (Co-supervised with H. Gu) Use of Convolutional Neural Networks in Estimating House Prices.
Wensha ZhangPhD. (Co-supervised with L. Ho)Variable selection with dependant data.
Shanglun LiPhD. (Co-supervised with H. Gu)State-space models for microbiome data.
Fatemah Tofighi KhelejanPhD. (Co-supervised with H. Gu) Model adequacy tests for phylogenetic models.
Shuangming YangPhD. (Co-supervised with H. Gu) Modelling Microbiome Temporal Dynamics Using Nonlinear Stochastic Differential Equations
Yurunyun WangPhD. (Co-supervised with H. Gu) Predicting Osteoarthritis from X-ray Images with Machine Learning

Completed Students

Yun CaiPhD. (Co-supervised with H. Gu) Measurement Error Deconvolution Methods and Rank Selection for Non-Negative Matrix Factorization with Applications in Microbiome Data.
Shen LingPhD. (Co-supervised with H. Gu and C. Field) A New Method for Multi-Class Classification with Multiple Data Sources, with Application to Abdom- inal Pain Diagnosis.
Wanru JiaMSc. (Co-supervised with H. Gu) Edge Detection Operators for X-ray Images based on Hessian Matrices.
Junqiu GaoMSc. (Co-supervised with H. Gu) Ornstein-Uhlenbeck Process and Optimal Sampling for Analysis of Microbiome Data.
Mingzhu WangMSc. The Influence of Utility Functions on Insurance Choices
Tianshu HuangMSc. (Co-supervised with H. Gu) Semi-Parametric Principal Component Analysis for Poisson Count Data with Application to Microbiome Data Analysis.
Hao HeMSc. (Co-supervised with H. Gu) Robust Ranking and Selection with Heavy-tailed Priors and its Application to Market Basket Analysis.
Li LiMSc. (Co-supervised with H. Gu)Recombination Detection Based on Likelihood and Clustering for DNA and Amino Acid Sequences.
Yun CaiMSc. (Co-supervised with H. Gu) Non-negative matrix factorisation for classification of metagenomic data.
Wei DaiMSc. (Co-supervised with H. Gu)A new Test to Build Confidence Regions using Balanced Minimum Evolution.

Guides for Graduate Students

These guides give advice on what are new topics for a number of graduate students, and are based on common mistakes I have seen when supervising students.

  • Guide to Programming
  • Guide to Using Cluster
  • Guide to Writing
  • Software

  • COLD (latest version: 1.2.2)
  • COLD (Codon Optimal Likelihood Discoverer) is a program for estimating phylogenetic parameters using maximum likelihood for codon models.

  • Simple Plot (latest version: 1.0.0)
  • Simple Plot is a program for adaptively projecting multidimensional data into two dimensions.

  • Adequate Bootstrap (latest version: 1.0.0)
  • Adequate Bootstrap is a program for estimating confidence intervals for parameter values that take into account the uncertainty due to model misspecification.

  • R packages
  • Here are a number of R packages written by me or my students. These will probably be submitted to CRAN when the relevant papers are published.