ocd_CI

This repository contains implementation of the ocd_CI algorithm described in the submitted manuscript. In this algorithm, we output a confidence interval for the changepoint location and estimate the set of indices of coordinates in which the mean changes, all under online changepoint monitoring. A real data example is also included in the repository.

Main function

ocd_CI.R is the main function of this repository. The function requires the 'MASS' library installed. You can do it by running library('MASS'). The inputs of the main function are as follows:

Auguments

• p: the dimension of each observation;
• s: the sparsity of the post-change vector;
• z: the changepoint location;
• n_max: maximum monitoring time;
• vartheta: the $\ell_2$ norm of the vector of change;
• signal_shape: the shape of the post-change vector, can be one of four: random, uniform, inverse square root or harmonic;
• spatial: the covariance matrix structure, can be one of two: identity $\Sigma= I_p$ or Toeplitz $\Sigma = (\rho^{|j-k|})_{j,k}$;
• rho: the Toeplitz parameter $\rho$;
• gamma: patience level of the online changepoint monitoring;
• beta: a lower bound of the $\ell_2$ norm of the vector of change;
• ocd_thresholds: declaration thresholds for the ocd base procedure, can be one of two: theoretical choices or Monte Carlo thresholds;
• l_choice: extra sample size post declaration, can be zero, or given based on theory, or otherwise specified;
• l: user-specified extra sample size;
• d1_choice: the value of parameter d1, can be set to default, or two variant values, or otherwise specified;
• d1: user-specified d1 value;
• alpha: the desired confidence level is 1-alpha;
• kaul_comp: whether to compare with a confidence interval procedure based on Kaul et al., (2021); default option is False;
• print_type: options to print results during the running, can be one of four: CI only, support only, both, or muted.

Details
The function outputs a list of three (when kaul_comp=FALSE) elements. The first element of the output list is an ocd_CI confidence interval for the changepoint location; the second is an estimate of the set of indices of coordinates in which the mean changes; the third is an augmented support estimate (with addition of the anchor coordinate). When kaul_comp=TRUE, it also outputs a confidence interval constructed via a procedure based on Kaul et al. (2021).

Examples
ret <- ocd_CI(p=100, s=10, vartheta=1, signal_shape='random', spatial='identity', beta=1, ocd_thresholds='MC', l_choice='zero', d1_choice='default', alpha=0.05, kaul_comp = TRUE, print_type ='both') ret$CI ret$support

Other functions

Other functions are collected in ingredients.R and auxiliary.R. As the names suggest, the former contains many functions that are building blocks of the main ocd_CI function, while the latter contains other auxiliary functions.

Tables and figures

The codes for generating all the tables and figures in the submitted manuscript are provided in tables.R. In order, the readers are able to reproduce Table 1, Table 2, Table S1 (in supplement), Figure 1, and Figure 2 of the paper by simply running the corresponding blocks of codes. Readers are warned that the running time can be very long, as the current implementation does not use parallel computing.

Real dataset

US_weekly_deaths.csv is a dataset of weekly deaths in the United States between January 2017 and June 2020 (available at: https://www.cdc.gov/nchs/nvss/vsrr/covid19/excess_deaths.htm). There are 12506 rows (excl. headers) and 12 columns in this dataset. The columns are:

• country: This is always United States;
• region: All 50 states plus Washington, D.C., as well as the entire United States;
• region code: The abbreviation of the state;
• start date: The start date of a reporting week;
• end date: The end date of a reporting week;
• days: Days in a week (always 7);
• year: The year this reporting week belongs to;
• week: The week number in the year of this reporting week;
• population: The population of the region;
• total_deaths: The total number of deaths in this reporting week;
• covid_deaths: The number of covid deaths in this reporting week;
• expected_deaths: The number of expected deaths in this reporting week;

Among the three columns regarding deaths numbers, we only make use of the total_deaths for our analyses.

Running ocd_CI on this dataset

US_excess_deaths.R provides the complete codes for running the ocd_CI procedure on the above US excess deaths dataset. It also generates Figure 3 of the submitted manuscript. This figure is also available in the repository as US_excess_deaths_ocd_CI.pdf.

Reference

Main paper
Inference in high-dimensional online changepoint detection. Preprint.

Other reference
Kaul, A., Fotopoulos, S. B., Jandhyala, V. K., and Safikhani, A. (2021) Inference on the change point under a high dimensional sparse mean shift. Electron. J. Statist., 15, 71--134.