Multiscale Inference for Nonparametric Time Trend(s)

Description

This package performs a multiscale analysis of a single nonparametric time trends (Khismatullina and Vogt (2020)) or multiple nonparametric time trends (Khismatullina and Vogt (2022), Khismatullina and Vogt (2023)).

In case of a single nonparametric regression, the multiscale method to test qualitative hypotheses about the nonparametric time trend m in the model Y_t = m(t/T) + \epsilon_t with time series errors \epsilon_t is provided. The method was first proposed in Khismatullina and Vogt (2020). It allows to test for shape properties (areas of monotonic decrease or increase) of the trend m.

This method require an estimator of the long-run error variance \sigma^2 = \sum_{l=-\infty}^{\infty} Cov(\epsilon_0, \epsilon_l). Hence, the package also provides the difference-based estimator for the case that the errors belong to the class of AR(\infty) processes. The estimator was also proposed in Khismatullina and Vogt (2020).

In case of multiple nonparametric regressions, we provide the multiscale method to test qualitative hypotheses about the nonparametric time trends in the context of epidemic modelling. Specifically, we assume that the we observe a sample of the count data \{\mathcal{X}_i = \{ X_{it}: 1 \le 1 \le T \}\}, where X_{it} are quasi-Poisson distributed with time-varying intensity parameter \lambda_i(t/T). The multiscale method allows to test whether intenisty parameters are different or not, and if they are, it detects with a prespicified significance level the regions where these differences most probably occur. The method was introduced in Khismatullina and Vogt (2023) and can be used for comparing the rates of infection of COVID-19 across countries.

References

Khismatullina M, Vogt M (2020). “Multiscale inference and long-run variance estimation in nonparametric regression with time series errors.” Journal of the Royal Statistical Society: Series B (Statistical Methodology).

Khismatullina M, Vogt M (2023). “Nonparametric comparison of epidemic time trends: The case of COVID-19.” Journal of Econometrics, 232(1), 87-108. ISSN 0304-4076, doi:10.1016/j.jeconom.2021.04.010.

Computes estimator of the AR(p) coefficients by the procedure from Khismatullina and Vogt (2020).

Description

Computes estimator of the AR(p) coefficients by the procedure from Khismatullina and Vogt (2020).

Usage

ar_coef(data, l1, l2, correct, p)

Arguments

Value

. Vector of length p of estimated AR coefficients.

Computes the set of minimal intervals as described in Duembgen (2002)

Description

Given a set of intervals, this function computes the corresponding subset of minimal intervals which are defined as follows. For a given set of intervals \mathcal{K}, all intervals \mathcal{I}_k \in \mathcal{K} such that \mathcal{K} does not contain a proper subset of \mathcal{I}_k are called minimal.

This function is needed for illustrative purposes. The set of all the intervals where our test rejects the null hypothesis may be quite large, hence, we would like to focus our attention on the smaller subset, for which we are still able to make simultaneous confidence intervals. This subset is the subset of minimal intervals, and it helps us to to precisely locate the intervals of further interest.

More details can be found in Duembgen (2002) and Khismatullina and Vogt (2019, 2020)

Usage

compute_minimal_intervals(dataset)

Arguments

Value

Subset of minimal intervals

Examples

startpoint   <- c(0, 0.5, 1)
endpoint     <- c(2, 2, 2)
dataset      <- data.frame(startpoint, endpoint)
minimal_ints <- compute_minimal_intervals(dataset)

Computes quantiles of the gaussian multiscale statistics.

Description

Quantiles from the gaussian version of the test statistics which are used to approximate the critical values for the multiscale test.

Usage

compute_quantiles(
  t_len,
  n_ts = 1,
  grid = NULL,
  ijset = NULL,
  sigma = 1,
  deriv_order = 0,
  sim_runs = 1000,
  probs = seq(0.5, 0.995, by = 0.005),
  correction = TRUE,
  epidem = FALSE
)

Arguments

Value

Matrix with 2 rows where the first row contains the vector of probabilities (probs) and the second contains corresponding quantiles of the gaussian statistics distribution.

Examples

compute_quantiles(100)

Computes quantiles of the gaussian multiscale statistics.

Description

Quantiles from the gaussian version of the test statistics which are used to approximate the critical values for the multiscale test.

Usage

compute_quantiles_2(
  t_len,
  n_ts = 1,
  grid = NULL,
  ijset = NULL,
  sigma = 1,
  deriv_order = 0,
  sim_runs = 1000,
  probs = seq(0.5, 0.995, by = 0.005),
  correction = TRUE,
  epidem = FALSE,
  numCores = NULL
)

Arguments

Value

Matrix with 2 rows where the first row contains the vector of probabilities (probs) and the second contains corresponding quantiles of the gaussian statistics distribution.

Examples

compute_quantiles_2(100, numCores = 2)

Calculates the value of the test statistics both for single time series analysis and multiple time series analysis.

Description

Calculates the value of the test statistics both for single time series analysis and multiple time series analysis.

Usage

compute_statistics(
  data,
  sigma = 1,
  sigma_vec = 1,
  n_ts = 1,
  grid = NULL,
  ijset = NULL,
  deriv_order = 0,
  epidem = FALSE
)

Arguments

Value

In case of n_ts = 1, the function returns a list with the following elements:

In case of n_ts > 1, the function returns a list with the following elements:

Computes the location-bandwidth grid for the multiscale test.

Description

Computes the location-bandwidth grid for the multiscale test.

Usage

construct_grid(t, u_grid = NULL, h_grid = NULL, deletions = NULL)

Arguments

Value

A list with the following elements:

Examples

construct_grid(100)
construct_grid(100, u_grid = seq(from = 0.05, to = 1, by = 0.05),
               h_grid = c(0.1, 0.2, 0.3, 0.4))

Computes the location-bandwidth weekly grid for the multiscale test.

Description

Computes the location-bandwidth weekly grid for the multiscale test.

Usage

construct_weekly_grid(t, min_len = 7, nmbr_of_wks = 4)

Arguments

Value

A list with the following elements:

Examples

construct_weekly_grid(100)
construct_weekly_grid(100, min_len = 7, nmbr_of_wks = 2)

Computes vector of correction terms for second-stage estimator of AR parameters as described in Khismatullina, Vogt (2019).

Description

Computes vector of correction terms for second-stage estimator of AR parameters as described in Khismatullina, Vogt (2019).

Usage

corrections(coefs, var_eta, len)

Arguments

Value

. Vector of the corrections terms of length len.

Number of daily new cases of infections of COVID-19 per country.

Description

Data on the geographic distribution of COVID-19 cases worldwide (© ECDC [2005-2019])

Usage

data("covid")

Format

A matrix with 99 rows and 41 columns. Each column corresponds to one coutnry, with the name of the country (denoted by three letter) being the name of the column.

Details

Each entry in the dataset denotes the number of new cases of infection per day and per country. In order to make the data comparable across countries, we take the day of the 100th confirmed case in each country as the starting date t = 1. This way of “normalizing” the data is common practice (Cohen and Kupferschmidt (2020)).

Source

https://www.ecdc.europa.eu/en

Computes autocovariances at lags 0 to p for the ell-th differences of data.

Description

Computes autocovariances at lags 0 to p for the ell-th differences of data.

Usage

emp_acf(data, ell, p)

Arguments

Value

Vector of length (p + 1) that consists of empirical autocoavariances for the corresponding lag - 1.

Computes estimator of the long-run variance of the error terms.

Description

A difference based estimator for the coefficients and long-run variance in case of a nonparametric regression model are AR(p).

Specifically, we assume that we observe Y(t) that satisfy the following equation:

Y(t) = m(t/T) + \epsilon_t.

Here, m(\cdot) is an unknown function, and the errors \epsilon_t are AR(p) with p known. Specifically, we ler \{\epsilon_t\} be a process of the form

\epsilon_t = \sum_{j=1}^p a_j \epsilon_{t-j} + \eta_t,

where a_1,a_2,\ldots, a_p are unknown coefficients and \eta_t are i.i.d.\ with E[\eta_t] = 0 and E[\eta_t^2] = \nu^2.

This function produces an estimator \widehat{\sigma}^2 of the long-run variance

\sigma^2 = \sum_{l=-\infty}^{\infty} cov(\epsilon_0,\epsilon_{l})

of the error terms, as well as estimators \widehat{a}_1, \ldots, \widehat{a}_p of the coefficients a_1,a_2,\ldots, a_p and an estimator \widehat{\nu}^2 of the innovation variance \nu^2.

The exact estimation procedure as well as description of the tuning parameters needed for this estimation can be found in Khismatullina and Vogt (2020).

Usage

estimate_lrv(data, q, r_bar, p)

Arguments

Value

A list with the following elements:

References

Khismatullina M., Vogt M. Multiscale inference and long-run variance estimation in non-parametric regression with time series errors //Journal of the Royal Statistical Society: Series B (Statistical Methodology). - 2020.

Carries out the multiscale test given that the values the estimatates of long-run variance have already been computed.

Description

Carries out the multiscale test given that the values the estimatates of long-run variance have already been computed.

Usage

multiscale_test(
  data,
  sigma = 1,
  sigma_vec = 1,
  n_ts = 1,
  grid = NULL,
  ijset = NULL,
  alpha = 0.05,
  sim_runs = 1000,
  deriv_order = 0,
  correction = TRUE,
  epidem = FALSE
)

Arguments

Value

In case of n_ts = 1, the function returns a list with the following elements:

In case of n_ts > 1, the function returns a list with the following elements:

Plots SiZer map from the test results of the multiscale testing procedure.

Description

Plots SiZer map from the test results of the multiscale testing procedure.

Usage

plot_sizer_map(
  u_grid,
  h_grid,
  test_results,
  plot_title = NA,
  greyscale = FALSE,
  ...
)

Arguments

Value

No return value, called for plotting a SiZer map.

Calculates different information criterions for a single time series or multiple time series with AR(p) errors based on the long-run variance estimator(s) for a range of tuning parameters and different orders p.

Description

This function fits AR(1), ... AR(9) models for all given time series and calculates different information criterions (FPE, AIC, AICC, SIC, HQ) for each of these fits. The result is the best fit in terms of minimizing the infromation criteria.

Usage

select_order(data, q = NULL, r = 5:15)

Arguments

Value

A list with a number of elements:

Hadley Centre Central England Temperature (HadCET) dataset, Monthly Mean Central England Temperature (Degrees C)

Description

The CET dataset is the longest instrumental record of temperature in the world. It contains the mean monthly surface air temperatures (in degrees Celsius) from the year 1659 to the present. These monthly temperatures are representative of a roughly triangular area of the United Kingdom enclosed by Lancashire, London and Bristol. Manley (1953, 1974) compiled most of the monthly series, covering 1659 to 1973. These data were updated to 1991 by Parker et al (1992). It is now kept up to date by the Climate Data Monitoring section of the Hadley Centre, Met Office.

Usage

data("temperature")

Format

A numeric vector of length 359.

Details

Since 1974 the data have been adjusted to allow for urban warming: currently a correction of -0.2 C is applied to mean temperatures. CET datasets are freely available for use under Open Government License.

Source

https://www.metoffice.gov.uk/hadobs/hadcet/

Computes variance of AR(p) innovation terms eta.

Description

Computes variance of AR(p) innovation terms eta.

Usage

variance_eta(data, coefs, p)

Arguments

Value

Variance of the innovation term