Dr Michelle Carey is a Lecturer/Assistant Professor in Statistics at University College Dublin. She is a graduate of the University of Limerick (UL) with a BSc in Financial Mathematics and received her PhD in Statistics from UL in 2012.
In 2011, she joined the Department of Accounting and Finance, Kemmy Business School, UL as a Lecturer in Finance. In 2013, she started a postdoctoral fellowship in biostatistics at the Department of Biostatistics and Computational Biology in the University of Rochester School of Medicine and Dentistry, Rochester, New York, USA. During her postdoc, she researched the evolution of biological systems in terms of differential equations under the direction of Prof Hulin Wu, a prominent expert in biostatistics and data science. In 2015, she started a postdoctoral fellowship in statistics at the Department of Mathematics and Statistics, McGill University, Montreal, Canada. During her postdoc, she researched statistical methods for the analysis of functional data under the direction of the internationally acclaimed founder of functional data analysis, Prof James O. Ramsay, and a leading authority in the field of multivariate analysis, nonparametric statistics and extreme-value theory, Prof Christian Genest.
In 2017, she joined the School of Mathematics and Statistics at the University College Dublin, Ireland. Michelle is a member of the UCD Energy Institute, the VistaMilk SFI Research Centre, the SFI Centre for Research Training in Foundations of Data Science and the SFI Centre for Research Training in Genomics Data Science.
Michelle's research is on dynamic data analysis: Statistical Modeling with Differential Equations, which combines techniques in statistics and applied mathematics and has applications in climatology, medicine, finance and agriculture.
Dynamic Data Analysis is a section of Functional Data Analysis.
An excellent text on Dynamic Data Analysis is:
Differential Equation Models
Differential equation models are mechanistic models built from the theoretical behaviour of complex systems. A single differential equation model can describe a wide variety of behaviours including oscillations, steady states and exponential growth and decay, with few but readily interpretable parameters. Consequently, differential equation models are routinely used to describe chemical reactions, predator-prey interactions, heat transfer, economic growth, epidemiological outbreaks, climate and weather prediction, gene regulatory pathways, etc.
Functional Data Analysis
Functional data analysis (FDA) is a branch of statistics that analyzes data that describes a process that evolves smoothly over one-or multi-dimensional domains. Functional data analysis concerns the analysis of information on curves, surfaces or objects.
Some example of functional data:
- Providing information about a curve denoting how a process varies over one dimension (i.e. changes over time, space, or frequency). For example, Figure (1) shows accelerometer readings of the brain tissue before and after a blow to the cranium indicated by the circles. The fitted curve approximating the average trajectory (solid line), the approximated 95% point-wise confidence interval for the curve (dashed line) and the approximated 95% point-wise prediction interval for the curve (grey region).
- Providing information about multiple curves denoting how multiple processes vary over one dimension (i.e. change over time, space, or frequency). For example, Figure 2 shows gene expression patterns for two gene response modules (GRMs) in the spleen and two GRMs in the lung. The total number of genes in each GRM is marked at the top of each plot. Black curves represent the expression of each gene in the cluster and the grey curves represent the mean expression for each GRM.
- Providing information about surfaces denoting how a process changes over two dimensions (i.e latitude and longitude). For example, Figure 3 depicts 467 daily rainfall measurements recorded in Switzerland on the 8th of May 1986. The size and colour of point markers represent the value of the rainfall at each location, highlighting a strong spatial anisotropy, with higher rainfall values alternating with lower rainfall values along elongated regions oriented in the northeast-southwest direction.
What is functional data?
The data describe a process or multiple processes that change smoothly, and continuously over the domain (i.e. space, time, etc). The key assumption is that the dependence between the measurements is a smooth process.
- We usually do not want to make parametric assumptions about our processes.
- There can be multiple measurements of the same process.
- The processes can be measured at unequal intervals.
FDA allows us to acquire:
- Representations of the distribution of the functions that is their mean, variation, and covariation.
- Relationships of functional data to covariates, responses, or other functions.
- Relationships between derivatives of functions.
Some excellent texts on FDA: