Modelling the COVID‐19 infection trajectory: A piecewise linear quantile trend model*

A new time-varying coefficient regression approach for analyzing infectious disease data

Since the beginning of the global pandemic of Coronavirus (SARS-COV-2), there has been many studies devoted to predicting the COVID-19 related deaths/hospitalizations. The aim of our work is to (1) explore the lagged dependence between the time series of case counts and the time series of death counts; and (2) utilize such a relationship for prediction. The proposed approach can also be applied to other infectious diseases or wherever dynamics in lagged dependence are of primary interest. Different from the previous studies, we focus on time-varying coefficient models to account for the evolution of the coronavirus. Using two different types of time-varying coefficient models, local polynomial regression models and piecewise linear regression models, we analyze the province-level data in Canada as well as country-level data using cumulative counts. We use out-of-sample prediction to evaluate the model performance. Based on our data analyses, both time-varying coefficient modeling strategies work well. Local polynomial regression models generally work better than piecewise linear regression models, especially when the pattern of the relationship between the two time series of counts gets more complicated (e.g., more segments are needed to portray the pattern). Our proposed methods can be easily and quickly implemented via existing R packages.

A New Class of Weighted CUSUM Statistics

A change point is a location or time at which observations or data obey two different models: before and after. In real problems, we may know some prior information about the location of the change point, say at the right or left tail of the sequence. How does one incorporate the prior information into the current cumulative sum (CUSUM) statistics? We propose a new class of weighted CUSUM statistics with three different types of quadratic weights accounting for different prior positions of the change points. One interpretation of the weights is the mean duration in a random walk. Under the normal model with known variance, the exact distributions of these statistics are explicitly expressed in terms of eigenvalues. Theoretical results about the explicit difference of the distributions are valuable. The expansions of asymptotic distributions are compared with the expansion of the limit distributions of the Cramér-von Mises statistic and the Anderson and Darling statistic. We provide some extensions from independent normal responses to more interesting models, such as graphical models, the mixture of normals, Poisson, and weakly dependent models. Simulations suggest that the proposed test statistics have better power than the graph-based statistics. We illustrate their application to a detection problem with video data.