meanResponse {surveillance}R Documentation

Calculate mean response needed in algo.hhh

Description

Calculates the mean response for the model specified in designRes according to equations (1.2) and (1.1) in Held et al., 2005 for univariate time series and equations (3.3) and (3.2) (with extensions) for multivariate time series. See details.

Usage

meanResponse(theta, designRes)

Arguments

theta vector of parameters
theta = (α_1,...,α_m, λ, phi, β, gamma_1, ..., gamma_m, psi),
where λ=(λ_1,...,λ_m), phi=(phi_1,...,phi_m), β=(β_1,...,β_m), gamma_1=(gamma_11,...,gamma_(1,2S_1)), gamma_m=(gamma_m1,...,gamma_(m,2S_m)), psi=(psi_1,...,psi_m).
If the model specifies less parameters, those components are omitted.
designRes Result of a call to make.design

Details

Calculates the mean response for a Poisson or a negative binomial model with mean

μ_t = λ y_t-lag + nu_t

where

log(nu_t) = α + β t + sum_(j=1)^S (gamma_(2j-1) * sin(omega_j * t) + gamma_2j * cos(omega_j * t) )

and omega_j = 2 * π * j / period are Fourier frequencies with known period, e.g. period=52 for weekly data, for a univariate time series.

Per default, the number of cases at time point t-1, i.e. lag=1, enter as autoregressive covariates into the model. Other lags can also be considered.

The seasonal terms in the predictor can also be expressed as gamma_s sin(omega_s * t) + delta_s cos(omega_s * t) = A_s sin(omega_s * t + ε_s) with amplitude A_s=sqrt{gamma_s^2 +delta_s^2} and phase difference tan(ε_s) = delta_s / gamma_s. The amplitude and phase shift can be obtained from a fitted model by specifying amplitudeShift=TRUE in the coef method.

For multivariate time series the mean structure is

μ_it = λ_i * y_i,t-lag + phi_i * sum_(j ~ i) w_ji * y_j,t-lag + n_it * nu_it

where

log(nu_it) = α_i + β_i * t + sum_(j=1)^S_i (gamma_(i,2j-1) * sin(omega_j * t) + gamma_(i,2j) * cos(omega_j * t) )

and n_it are standardized population counts. The weights w_ji are specified in the columns of the neighbourhood matrix disProgObj$neighbourhood.

Alternatively, the mean can be specified as

μ_it = λ_i *π_i * y_i,t-1 + sum_(j ~ i) λ_j *(1-π_j)/|k ~ j| * y_j,t-1 + n_it * nu_it

if proportion="single" ("multiple") in designRes$control. Note that this model specification is still experimental.

Value

Returns a list with elements

mean matrix of dimension n x m with the calculated mean response for each time point and unit, where n is the number of time points and m is the number of units.
epidemic matrix with the epidemic part λ_i * y_i,t-1 + phi_i * sum_(j ~ i) y_j,t-1
endemic matrix with the endemic part of the mean n_it*nu_it
epi.own matrix with λ_i * y_i,t-1
epi.neighbours matrix with phi_i * sum_(j ~ i) y_j,t-1

Author(s)

M. Paul, L. Held

Source

Held, L., Höhle, M., Hofmann, M. (2005) A statistical framework for the analysis of multivariate infectious disease surveillance counts. Statistical Modelling, 5, p. 187–199.


[Package surveillance version 1.1-2 Index]