Mathew W. McLean

We have data from a study comparing certain white matter tracts of multiple sclerosis (MS) patients with control subjects. MS is a central nervous system disorder that leads to lesions in the white matter of the brain which disrupts the ability of cells in the brain to communicate with each other. Below is a human brain with two major white matter tracts, the corpus callosum and corticospinal tracts in red and blue, respectively (thanks to Jeff Goldsmith).

Using a Magnetic Resonance Imaging technique called Diffusion Tensor Imaging we obtain several measurements at each location in the white matter tracts. Here it makes sense to treat these measurements as being from a continuous function X(t) where t is location along the tract and use tools from functional data analysis. We wish to use the functions X(t) to predict a scalar health outcome Y. For example, Y might be score on a cognitive test or disease status (case or control). The typical approach in functional data analysis is to use the Functional Linear Model (FLM)

where β(t) is an unknown coefficient function that needs to be estimated and g(•) is a known monotonic function called the link function. Below is a plot of some of the curves X_i(t) as well as a corresponding estimated β(t).

However, this model is often not flexible enough to model the true underlying relationship between the response and predictor. We propose the new model

where F(x,t) is an unknown smooth bivariate function that we estimate using penalized regression splines. The model is much more general than the FLM, but retains the ease of interpretation. Below is an example estimated surface.

Here Y is 1 if the subject has MS and 0 if they do not and g[E(Y|X)] gives the estimated probability that subject i has MS. The red plotted curve is X(t) for one subject with MS and blue curve is for a subject without MS. We see that since the red curve traces out a higher path on the surface than the red curve, the subject corresponding to the red curve is more likely to be classified as having MS.

You may fit the FGAM and FLM yourself to several different data sets using a Shiny app I created. See the Software tab.

In this project we are interested in determining how to best deploy ambulances in a given city to minimize response times to emergency calls while keeping costs low. More specifically, we have data on every ambulance trip and every emergency call received by Toronto EMS between January 2007 and December 2008. The operations research models that address these deployment problems require accurate estimates of the number of calls that will be received during each time period. Below is a plot of the mean number of calls per hour for each day of the week.

Methods used in practice to obtain estimates are quite ad hoc. Instead, what we do is build what is called a factor model with constraints. This provides dimension reduction, accounts for seasonal and intra-day patterns in the call arrival rate process, and lets us incorporate additional covariates. The factor model is estimated using penalized regression splines so that the factors and loading in the factor model vary smoothly over time. Using the factor model we obtain smooth estimates for the arrival process for every day of 2008 which is plotted below. As is to be expected, there is a very noticeable difference between the estimates for weekdays and weekends.

Finally, an integer-value GARCH model is fit to the residual process to account for any remaining dependence. Below is a plot of the estimated number of calls for each hour of weeks 8 and 9 of 2007 as well as a plot of the estimated integer-value GARCH model fit to the residual process.

How do we check whether our model improves staffing levels and response times?

By simulating an M/M/s queueing system where the servers represent ambulances, calls for emergency medical services in hour, t, occur at rate λ_t, and each ambulance services callers at rate ν. We use our model to obtain estimates of the number of callers for each period and these determine the number of ambulances to staff, say s_t, in each period. After fitting the model to 2007 data, to initialize the queueing system we assume the number of arrivals each hour is the observed number of calls for the corresponding hour for 2008 and simulate inter-hour arrival times and service times for each caller.

RefManageR provides tools for importing and working with bibliographic references. It greatly enhances the bibentry class by providing a class BibEntry which stores BibTeX and BibLaTeX references, supports UTF-8 encoding, and can be easily searched by any field, by date ranges, and by various formats for name lists (author by last names, translator by full names, etc.). Entries can be updated, sorted, combined, printed in a number of styles, and exported. BibTeX and BibLaTeX .bib files can be read into R and converted to BibEntry objects. Interfaces to NCBI's Entrez, CrossRef and Zotero are provided for importing references and references can be created from locally stored .bib files or PDFs using Poppler. The package can also be used in RMarkdown and RHTML documents for including citations and printing a bibliography of all cited entries. Both the bibliography and the citations can include automatically generated hyperlinks. The package is available now on CRAN and a vignette is available here. The package was peer-reviewed and accepted by ROpenSci.

• Development version
• CRAN sources

The following functions were contributed to the refund package available on CRAN and GitHub:

• fgam - a wrapper for the gam function in package mgcv to fit functional generalized additive models.
• predict.fgam - wrapper for the predict.gam function in package mgcv for prediction with fgam fits.
• vis.fgam - for visualizing estimated surfaces produced by fgam.
• af - internal function for building fgam terms specified in model formulas passed to fgam.
• lf - internal function for building functional linear model terms specified in model formulas passed to fgam.

Provided assistance getting the the R package accepted to CRAN, including setting up package namespace and including compiled code.

Below is a Shiny app I created that performs estimation, visualization, inference, and prediction for the functional generalized additive model (FGAM) and the functional linear model (FLM). More information on these models can be found in the Research tab and in my papers from the Papers tab. Begin by selecting a data set and choosing model parameters and an estimation method (if you are familar with penalized splines). You may also specify options for displaying the fits, conducting a hypothesis test of FLM vs. FGAM, and performing out-of-sample prediction. When you have selected the setting you want, you may fit the models by clicking the "Fit Models" button. Source code for the app is available on GitHub.