sim_ex_cluster: 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")

Arguments

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

Run this code

# NOT RUN {
data(sim_ex_cluster)
# }

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