Each dataset has 2 groups with n curves each, defined in the interval
\(t=[0, 1]\) with p equidistant points. The first n curves are
generated fron the following model
\(X_1(t)=E_1(t)+e(t)\) where \(E_1(t)=E_1(X(t))=30t^{ \frac{3}{2}}(1-t)\)
is the mean function and \(e(t)\) is a centered Gaussian process with
covariance matrix \(Cov(e(t_i),e(t_j))=0.3 \exp(-\frac{\lvert t_i-t_j \rvert}{0.3})\)
The remaining 50 functions are generated from model i_sim with
i_sim \(\in \{1, \ldots, 8\}.\)
The first three models contain changes in the mean, while the covariance
matrix does not change. Model 4 and 5 are obtained by multiplying the
covariance matrix by a constant. Model 6 is obtained from adding to
\(E_1(t)\) a centered Gaussian process \(h(t)\) whose covariance matrix
is given by \(Cov(e(t_i),e(t_j))=0.5 \exp (-\frac{\lvert t_i-t_j\rvert}{0.2})\).
Model 7 and 8 are obtained by a different mean function.
\(X_1(t)=30t^{\frac{3}{2}}(1-t)+0.5+e(t).\)
\(X_2(t)=30t^{\frac{3}{2}}(1-t)+0.75+e(t).\)
\(X_3(t)=30t^{\frac{3}{2}}(1-t)+1+e(t).\)
\(X_4(t)=30t^{\frac{3}{2}}(1-t)+2 e(t).\)
\(X_5(t)=30t^{\frac{3}{2}}(1-t)+0.25 e(t).\)
\(X_6(t)=30t^{\frac{3}{2}}(1-t)+ h(t).\)
\(X_7(t)=30t{(1-t)}^2+ h(t).\)
\(X_8(t)=30t{(1-t)}^2+ e(t).\)
sim_model_ex1(n = 50, p = 30, i_sim = 1)data matrix of size \(2n \times p\).
Number of curves to generate for each of the two groups. Set to 50 by default.
Number of grid points of the curves. Curves are generated over the interval \([0, 1]\). Set to 30 grid point by default.
Integer set to \(1, \ldots, 8\).
sm1 <- sim_model_ex1()
dim(sm1)
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