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)= μ^{(c)}(x) + ∑_{k=1}^{2}ξ_{tk}^{(c)} ρ_k^{(c)} (x) + ∑_{l=1}^{2}ζ_{itl}^{(c)} ψ_l^{(c)} (x) + υ_{it}^{(c)} (x)

Usage

Details

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

While the variates \mathbf{ξ_{tk}^{(c)}}=(ξ_{1k}^{(c)}, ξ_{2k}^{(c)}, …, ξ_{Tk}^{(c)})^{\top} for both clusters, are generated from autoregressive of order 1 with parameter 0.7, while the variates ζ_{it1}^{(c)} and ζ_{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, ρ_k^{(1)} (x), for k in {1,2} are sqrt(2)*sin(π*(0:200/200)) and sqrt(2)*cos(π*(0:200/200)) respectively; and the basis functions for the common-time trend for the second cluster, ρ_k^{(2)} (x), for k in {1,2} are sqrt(2)*sin(2π*(0:200/200)) and sqrt(2)*cos(2π*(0:200/200)) respectively.

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

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

Examples