sim_ex_cluster {ftsa}

Simulated multiple sets of functional time series

Description

We generate 2 groups of m functional time series. For each i in {1, ..., m} in a given cluster c, c in {1,2}, the t th function, t in {1,..., T}, is given by

Y_{it}^{(c)} (x)= \mu^{(c)}(x) + \sum_{k=1}^{2}\xi_{tk}^{(c)} \rho_k^{(c)} (x) + \sum_{l=1}^{2}\zeta_{itl}^{(c)} \psi_l^{(c)} (x) + \upsilon_{it}^{(c)} (x)

Usage

data("sim_ex_cluster")

Details

The mean functions for each of these two clusters are set to be \mu^{(1)}(x) = 2(x-0.25)^{2} and \mu^{(2)}(x) = 2(x-0.4)^{2}+0.1.

While the variates \mathbf{\xi_{tk}^{(c)}}=(\xi_{1k}^{(c)}, \xi_{2k}^{(c)}, \ldots, \xi_{Tk}^{(c)})^{\top} for both clusters, are generated from autoregressive of order 1 with parameter 0.7, while the variates \zeta_{it1}^{(c)} and \zeta_{it2}^{(c)} for both clusters, are generated from independent and identically distributed N(0,0.5) and N(0,0.25), respectively.

The basis functions for the common-time trend for the first cluster, \rho_k^{(1)} (x), for k in {1,2} are sqrt(2)*sin(\pi*(0:200/200)) and sqrt(2)*cos(\pi*(0:200/200)) respectively; and the basis functions for the common-time trend for the second cluster, \rho_k^{(2)} (x), for k in {1,2} are sqrt(2)*sin(2\pi*(0:200/200)) and sqrt(2)*cos(2\pi*(0:200/200)) respectively.

The basis functions for the residual for the first cluster, \psi_l^{(1)} (x), for l in {1,2} are sqrt(2)*sin(3\pi*(0:200/200)) and sqrt(2)*cos(3\pi*(0:200/200)) respectively; and the basis functions for the residual for the second cluster, \psi_l^{(2)} (x), for l in {1,2} are sqrt(2)*sin(4\pi*(0:200/200)) and sqrt(2)*cos(4\pi*(0:200/200)) respectively.

The measurement error \upsilon_{it} for each continuum x is generated from independent and identically distributed N(0, 0.2^2)

Examples

data(sim_ex_cluster)